ahmed t equations of state and pvt analysis applications for

Equations of State and
PVT Analysis
Applications for Improved Reservoir Modeling
Equations of State and
PVT Analysis
Applications for Improved Reservoir Modeling
Second Edition
Tarek Ahmed, PhD, PE
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
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noted herein).
Notices
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changes in research methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information,
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A catalog record for this book is available from the Library of Congress
ISBN: 978-0-12-801570-4
For information on all Gulf Professional publications
visit our website at http://www.elsevier.com/
This book is dedicated to my wife Wendy
About the Author
Tarek Ahmed, PhD, PE, served as a professor of petroleum engineering at Montana Tech of
the University of Montana from 1980 to 2002. He has worked for Anadarko Petroleum, Baker
Hughes, and Talisman Energy. In 2012, Dr. Ahmed founded the consulting company Tarek
Ahmed & Associates, Ltd. Dr. Ahmed has authored or coauthored several textbooks, including
Reservoir Engineering Handbook and Hydrocarbon Phase Behvior, among others.
xiii
Preface
The primary focus of this book is to present the basic fundamentals of equations of state and
PVT laboratory analysis, and their practical applications in solving reservoir engineering problems. The book is arranged so it can be used as a textbook for senior and graduate students, or as
a reference book for practicing petroleum engineers.
Chapter 1 reviews the principles of hydrocarbon phase behavior and illustrates the use of phase
diagrams in characterizing reservoirs and hydrocarbon systems. Chapter 2 presents numerous
mathematical expressions and graphical relationships that are suitable for characterizing the
undefined hydrocarbon-plus fractions. Chapter 3 provides a comprehensive and updated review
of natural gas properties and the associated, well-established correlations that can be used to
describe the volumetric behavior of gas reservoirs. Chapter 4 discusses the PVT properties
of crude oil systems and illustrates the use of laboratory data to generate the properties of crude
oil that can be used to perform reservoir engineering studies. Chapter 5 reviews developments
and advances in the field of empirical cubic equations of state, and demonstrates their practical
applications in solving phase equilibria problems.
xv
Acknowledgments
It is my hope that the information presented in this textbook will improve the understanding of
the subject of equations of state and phase behavior. Much of the material on which this book is
based was drawn from the publications of the Society of Petroleum Engineers, the American
Petroleum Institute, and the Gas Processors Suppliers Associations. Tribute is paid to the educators and authors who have made numerous and significant contributions to the field of phase
equilibria. I would like to especially acknowledge the significant contributions that have been
made to the field of phase behavior and equations of state to Donald Katz, M. B. Standing, Curtis
Whitson, and Bill McCain.
Special thanks to the editorial and production staff of Elsevier, in particular, Anusha
Sambamoorthy, Kattie Washington, and Katie Hammon.
I hope my children Carsen and Asia will follow my footsteps and study petroleum engineering.
xvii
Chapter
1
Fundamentals of Hydrocarbon
Phase Behavior
A PHASE IS DEFINED AS ANY homogeneous part of a system that is physically
distinct and separated from other parts of the system by definite boundaries.
For example, ice, liquid water, and water vapor constitute three separate
phases of the pure substance H2O, because each is homogeneous and physically distinct from the others; moreover, each is clearly defined by the
boundaries existing between them. Whether a substance exists in a solid,
liquid, or gas phase is determined by the temperature and pressure acting
on the substance. It is known that ice (solid phase) can be changed to water
(liquid phase) by increasing its temperature and, by further increasing the
temperature, water changes to steam (vapor phase). This change in phases
is termed phase behavior.
Hydrocarbon systems found in petroleum reservoirs are known to display
multiphase behavior over wide ranges of pressures and temperatures. The
most important phases that occur in petroleum reservoirs are a liquid phase,
such as crude oils or condensates, and a gas phase, such as natural gases.
The conditions under which these phases exist are a matter of considerable
practical importance. The experimental or the mathematical determinations
of these conditions are conveniently expressed in different types of diagrams, commonly called phase diagrams.
The objective of this chapter is to review the basic principles of hydrocarbon
phase behavior and illustrate the use of phase diagrams in describing
and characterizing the volumetric behavior of single-component, twocomponent, and multicomponent systems.
SINGLE-COMPONENT SYSTEMS
The simplest type of hydrocarbon system to consider is that containing one
component. The word component refers to the number of molecular or
atomic species present in the substance. A single-component system is
Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00001-5
Copyright # 2016 Elsevier Inc. All rights reserved.
1
2 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
composed entirely of one kind of atom or molecule. We often use the word
pure to describe a single-component system.
The qualitative understanding of the relationship between temperature T,
pressure p, and volume V of pure components can provide an excellent basis
for understanding the phase behavior of complex petroleum mixtures. This
relationship is conveniently introduced in terms of experimental measurements conducted on a pure component as the component is subjected
to changes in pressure and volume at a constant temperature. The effects
of making these changes on the behavior of pure components are
discussed next.
Suppose a fixed quantity of a pure component is placed in a cylinder fitted
with a frictionless piston at a fixed temperature T1. Furthermore, consider
the initial pressure exerted on the system to be low enough that the entire
system is in the vapor state. This initial condition is represented by point
E on the pressure-volume phase diagram (p-V diagram) as shown in
Fig. 1.1. Consider the following sequential experimental steps taking place
on the pure component:
1. The pressure is increased isothermally by forcing the piston into the
cylinder. Consequently, the gas volume decreases until it reaches point F
on the diagram, where the liquid begins to condense. The corresponding
pressure is known as the dew point pressure, pd, and defined as the
pressure at which the first droplet of liquid is formed.
2. The piston is moved further into the cylinder as more liquid condenses.
This condensation process is characterized by a constant pressure and
represented by the horizontal line FG. At point G, traces of gas remain
Van der Waals’ critical point condition
H
c
C
pc
c
c
c
c
T1 < T2 < T3 < TC < T4
Critical
point
Liquid + Vapor
G
F
B
Vc
c
V
n FIGURE 1.1 Typical pressure-volume diagram for pure component.
A
T4
TC
T3
T
E 2
T1
Single-Component Systems 3
and the corresponding pressure is called the bubble point pressure, pb,
and defined as the pressure at which the first sign of gas formation is
detected. A characteristic of a single-component system is that, at a
given temperature, the dew point pressure and the bubble point pressure
are equal.
3. As the piston is forced slightly into the cylinder, a sharp increase in
the pressure (point H) is noted without an appreciable decrease in
the liquid volume. That behavior evidently reflects the low
compressibility of the liquid phase.
By repeating these steps at progressively increasing temperatures, a family
of curves of equal temperatures (isotherms) is constructed as shown in
Fig. 1.1. The dashed curve connecting the dew points, called the dew point
curve (line FC), represents the states of the “saturated gas.” The dashed
curve connecting the bubble points, called the bubble point curve (line
GC), similarly represents the “saturated liquid.” These two curves meet a
point C, which is known as the critical point. The corresponding pressure
and volume are called the critical pressure, pc, and critical volume, Vc,
respectively. Note that, as the temperature increases, the length of the
straight-line portion of the isotherm decreases until it eventually vanishes
and the isotherm merely has a horizontal tangent and inflection point at
the critical point. This isotherm temperature is called the critical temperature, Tc, of that single component. This observation can be expressed mathematically by the following relationship:
@p
@V
¼ 0, at the critical point
(1.1)
2 @ p
¼ 0, at the critical point
@V 2 Tc
(1.2)
Tc
Referring to Fig. 1.1, the area enclosed by the area AFCGB is called the twophase region or the phase envelope. Within this defined region, vapor and
liquid can coexist in equilibrium. Outside the phase envelope, only one
phase can exist.
The critical point (point C) describes the critical state of the pure component
and represents the limiting state for the existence of two phases, that is, liquid and gas. In other words, for a single-component system, the critical point
is defined as the highest value of pressure and temperature at which two
phases can coexist. A more generalized definition of the critical point, which
is applicable to a single- or multicomponent system, is this: The critical
point is the point at which all intensive properties of the gas and liquid
phases are equal.
4 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
An intensive property is one that has the same value for any part of a homogeneous system as it does for the whole system; that is, a property independent of the quantity of the system. Pressure, temperature, density,
composition, and viscosity are examples of intensive properties.
Many characteristic properties of pure substances have been measured and
compiled over the years. These properties provide vital information for calculating the thermodynamic properties of pure components as well as their
mixtures. The most important of these properties include
n
n
n
n
n
n
n
n
Critical pressure, pc.
Critical temperature, Tc.
Critical volume, Vc.
Critical compressibility factor, Zc.
Boiling point temperature, Tb.
Acentric factor, ω.
Molecular weight, M.
Specific gravity, γ.
Those physical properties needed for hydrocarbon phase behavior calculations are presented in Table 1.1 for a number of hydrocarbon and nonhydrocarbon components.
Another means of presenting the results of this experiment is shown in
Fig. 1.2, in which the pressure and temperature of the system are the independent parameters. Fig. 1.2 shows a typical pressure-temperature diagram
(p-T diagram) of a single-component system with solids lines that clearly
represent three different phase boundaries: vapor-liquid, vapor-solid, and
liquid-solid phase separation boundaries. As shown in the illustration, line
AC terminates at the critical point (point C) and can be thought of as the
dividing line between the areas where liquid and vapor exist. The curve
is commonly called the vapor pressure curve or the boiling point curve.
The corresponding pressure at any point on the curve is called the vapor
pressure, pv, with a corresponding temperature termed the boiling point
temperature.
The vapor pressure curve represents the conditions of pressure and temperature at which two phases, vapor and liquid, can coexist in equilibrium.
Systems represented by a point located below the vapor pressure curve
are composed only of the vapor phase. Similarly, points above the curve
represent systems that exist in the liquid phase. These remarks can
be conveniently summarized by the following expressions:
Table 1.1 Physical Properties for Pure Components
See Note No. !
Number
Compound
Formula
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Methane
Ethane
Propane
Isobutane
n-Butane
Isopentane
n-Pentane
Neopentane
n-Hexane
2-Methylpentane
3-Methylpentane
Neohexane
2.3-Dimethylbutane
n-Heptane
2-Methylhexane
3-Methylhexane
3-Ethylpentane
2.2-Dimethylpetane
2.4-Dimethylpentane
3.3-Dimethylpentane
Triptane
n-Octane
Dilsobutyl
Isooctane
n-Nonane
n-Decane
Cyclopentane
Methylcyclopentane
Cyclohexane
Methyl cyclohexane
Ethene (ethylene)
Propene (propylene)
CH4
C2H6
C3H8
C4H10
C4H10
C5H12
C5H12
C5H12
C6H14
C6H14
C6H14
C6H14
C6H14
C7H16
C7H16
C7H16
C7H16
C7H16
C7H16
C7H16
C7H16
C8H18
C8H18
C8H18
C9H20
C8H22
C5H10
C6H12
C6H12
C7H14
C6H4
C6H6
A
B
C
D
Critical Constants
Molar Mass
(Molecular
Weight)
Boiling
Point (°F)
14.696
psia
Vapor
Pressure
(psia), 100°F
Freezing
Point (°F)
14.696
psia
Refractive
Index, πD Pressure
60°F
(psia)
16.043
30.070
44.097
58.123
58.123
72.150
72.150
72.150
86.177
86.177
86.177
86.177
86.177
100.204
100.204
100.204
100.204
100.204
100.204
100.204
100.204
114.231
114.231
114.231
128.258
14.285
70.134
84.161
84.161
98.188
28.054
42.081
258.73
127.49
43.75
10.78
31.08
82.12
96.92
49.10
155.72
140.47
145.89
121.52
136.36
209.16
194.09
197.33
200.25
174.54
176.89
186.91
177.58
258.21
228.39
210.63
303.47
345.48
120.65
161.25
177.29
213.68
154.73
53.84
(5000)*
(800)*
188.64
72.581
51.706
20.445
15.574
36.69
4.9597
6.769
6.103
9.859
7.406
1.620
2.272
2.131
2.013
3.494
3.293
2.774
3.375
0.53694
1.102
1.709
0.17953
0.06088
9.915
4.503
3.266
1.609
(1400)*
227.7
296.44*
297.04*
305.73*
255.28
217.05
255.82
201.51
2.17
139.58
244.62
–
147.72
199.38
131.05
180.89
–
181.48
190.86
182.63
210.01
12.81
70.18
132.11
161.27
64.28
21.36
136.91
224.40
43.77
195.87
–272.47*
301.45*
1.00042*
1.20971*
1.29480*
1.3245*
1.33588*
1.35631
1.35992
1.342*
1.37708
1.37387
1.37888
1.37126
1.37730
1.38989
1.38714
1.39091
1.39566
1.38446
1.38379
1.38564
1.39168
1.39956
1.39461
1.38624
1.40748
1.41385
1.40896
1.41210
1.42862
1.42538
(1.228)*
1.3130*
666.4
706.5
616.0
527.9
550.6
490.4
488.6
464.0
436.9
436.6
453.1
446.8
453.5
396.8
396.5
408.1
419.3
402.2
396.9
427.2
428.4
360.7
360.6
372.4
331.8
305.2
653.8
548.9
590.8
503.5
731.0
668.6
Temperature Volume
(°F)
(ft3/lb-m)
116.67
89.92
206.06
274.46
305.62
369.10
385.8
321.13
453.6
435.83
448.4
420.13
440.29
512.7
495.00
503.80
513.39
477.23
475.95
505.87
496.44
564.22
530.44
519.46
610.68
652.0
461.2
499.35
536.6
570.27
48.54
197.17
0.0988
0.0783
0.0727
0.0714
0.0703
0.0679
0.0675
0.0673
0.0688
0.0682
0.0682
0.0667
0.0665
0.0691
0.0673
0.0646
0.0665
0.0665
0.0668
0.0662
0.0636
0.0690
0.0676
0.0656
0.0684
0.0679
0.0594
0.0607
0.0586
0.0600
0.0746
0.0689
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
(Continued)
Table 1.1 Physical Properties for Pure Components Continued
See Note No. !
A
B
Boiling
Point (°F)
14.696
psia
Vapor
Pressure
(psia), 100°F
Freezing
Point (°F)
14.696
psia
20.79
38.69
33.58
19.59
85.93
51.53
24.06
93.31
120.49*
176.18
231.13
277.16
291.97
282.41
281.07
293.25
306.34
148.44
172.90
312.68
109.257*
76.497
14.11
27.99
317.8
422.955*
297.332*
320.451
29.13
212.000*
452.09
121.27
62.10
45.95
49.87
63.02
19.12
36.53
59.46
16.48
–
3.225
1.033
0.3716
0.2643
0.3265
0.3424
0.2582
0.1884
4.629
2.312
–
–
394.59
85.46
211.9
–
–
–
–
157.3
0.9501
–
906.71
301.63*
218.06
157.96
220.65
265.39
213.16
164.02
230.73
114.5
41.95
139.00
138.966
13.59
54.18
55.83
23.10
140.814
143.79
173.4
337.00*
69.83*
121.88*
103.86*
107.88*
–
435.26*
361.820*
346.00*
149.73*
32.00
–
173.52*
Number
Compound
Formula
Molar Mass
(Molecular
Weight)
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
1-Butene (butylene)
cis-2-Butene
trans-2-Butene
Isobutene
1-Pentene
1.2-Butadiene
1.3-Butadiene
Isoprene
Acetylene
Benzene
Toluene
Ethylbenzene
o-Xylene
m-Xylene
p-Xylene
Styrene
Isopropylbenzene
Methyl alcohol
Ethyl alcohol
Carbon monoxide
Carbon dioxide
Hydrogen sulfide
Sulfur dioxide
Ammonia
Air
Hydrogen sulfide
Oxygen
Nitrogen
Chlorine
Water
Helium
Hydrogen chloride
C4H8
C4H8
C4H8
C4H8
C5H10
C4H6
C4H6
C5H8
C2H2
C6H6
C7H8
C8H10
C8H10
C8H10
C8H10
C8H8
C9H12
CH4O
C2H6O
CO
CO2
H2S
SO2
NH3
N2+O2
H2
O2
N2
Cl2
H2O
He
HCl
56.108
56.108
56.108
56.108
70.134
54.092
54.092
68.119
26.038
78.114
92.141
106.167
106.167
106.167
106.167
104.152
120.194
32.042
46.069
28.010
44.010
34.08
64.06
17.0305
2.0159
2.0159
31.9988
28.0134
70.906
18.0153
4.0026
36.461
C
D
Critical Constants
Refractive
Index, πD Pressure
60°F
(psia)
1.3494*
1.3665*
1.3563*
1.3512*
1.37426
–
1.3875*
1.42498
–
1.50396
1.49942
1.49826
1.50767
1.49951
1.49810
1.54937
1.49372
1.33034
1.36346
1.00036*
1.00048*
1.00060*
1.00062*
1.00036*
1.00028*
1.00013*
1.00027*
1.00028*
1.3878*
1.33335
1.00003*
1.00042*
583.5
612.1
587.4
580.2
511.8
(653)*
627.5
(558)*
890.4
710.4
595.5
523.0
541.6
512.9
509.2
587.8
465.4
1174
890.1
507.5
1071
1300
1143
1646
546.9
188.1
731.4
493.1
1157
3198.8
32.99
1205
Temperature Volume
(°F)
(ft3/lb-m)
295.48
324.37
311.86
292.55
376.93
(340)*
305
(412)*
95.34
552.22
605.57
651.29
674.92
651.02
649.54
(703)*
676.3
463.08
465.39
220.43
87.91
212.45
315.8
270.2
221.31
399.9
181.43
232.51
290.75
705.16
450.31
124.77
0.0685
0.0668
0.0679
0.0682
0.0676
(0.065)*
0.0654
(0.065)*
0.0695
0.0531
0.0550
0.0565
0.0557
0.0567
0.0570
0.0534
0.0572
0.0590
0.0581
0.0532
0.0344
0.0461
0.0305
0.0681
0.0517
0.5165
0.0367
0.0510
0.0280
0.0497s
0.2300
0.0356
Number
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
Table 1.1 Physical Properties for Pure Components Continued
E
F
G
H
Density of Liquid 14.696 psia, 60°F
Number
Relative
Density
(Specific
Gravity)
60°F/60°F
1b-m/gal.
gal./lb-m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
(0.3)*
0.35619*
0.50699*
0.56287*
0.58401*
0.62470
0.63112
0.59666*
0.66383
0.65785
0.66901
0.65385
0.66631
0.68820
0.68310
0.69165
0.70276
0.67829
0.67733
0.69772
0.69457
0.70696
(2.5)*
2.9696*
4.2268*
4.6927*
4.4690*
5.2082
5.2617
4.9744*
5.5344
5.4846
5.5776
5.4512
5.551
5.7376
5.6951
5.7664
5.8590
5.6550
5.6470
5.8170
5.7907
5.8940
(6.4172)*
10.126*
10.433*
12.386*
11.937*
13.853
13.712
14.504*
15.571
15.713
15.451
15.809
15.513
17.464
17.595
17.377
17.103
17.720
17.745
17.226
17.304
19.381
–
–
0.00162*
0.00119*
0.00106*
0.00090
0.00086
0.00106*
0.00075
0.00076
0.00076
0.00076
0.00076
0.00068
0.00070
0.00070
0.00069
0.00070
0.00073
0.00067
0.00068
0.00064
23
24
25
26
0.69793
0.69624
0.72187
0.73421
5.8187
5.8046
6.0183
6.1212
19.632
19.679
21.311
23.245
0.00067
0.00065
0.00061
0.00057
I
J
Ideal Gas 14.696 psia, 60°F
Specific Heat
60°F 14.696
psia Btu/
(1 bm, °F)
Compressibility
Factor of Real
Gas, Z 14.696
psia, 60°F
Relative
Density
(Specific
Gravity)
Air 5 1
ft3 gas/
lb-m
0.0104
0.0979
0.1522
0.1852
0.1995
0.2280
0.2514
0.1963
0.2994
0.2780
0.2732
0.2326
0.2469
0.3494
0.3298
0.3232
0.3105
0.2871
0.3026
0.2674
0.2503
0.3977
0.9980
0.9919
0.9825
0.9711
0.9667
–
–
0.9582
–
–
–
–
–
–
–
–
–
–
–
–
–
–
0.5539
1.0382
1.5226
2.0068
2.0068
2.4912
2.4912
2.4912
2.9755
2.9755
2.9755
2.9755
2.9755
3.4598
3.4598
3.4598
3.4598
3.4598
3.4598
3.4598
3.4598
3.9441
0.3564
0.3035
0.4445
0.4898
–
–
–
–
3.9441
3.9441
4.4284
4.9127
Temperature
Coefficient
Acentric
of Density,
Factor,
1/°F
ω
ft3 gas/
gal.
liquid
Cp. I
Deal
Gas
Cp.
Liquid
Number
23.654
(59.135)*
12.620
37.476*
8.6059
36.375*
6.5291
30.639*
5.25596 27.393
5.2596
27.674
5.2596
26.163*
4.4035
24.371
4.4035
24.152
4.4035
24.561
4.4035
24.462
3.7872
21.729
3.7872
21.568
3.7872
21.838
3.7872
22.189
3.7872
21.416
3.7872
21.386
3.7872
22.030
3.7872
21.930
3.220
19.580
3.3220
19.330
3.3220
19.283
0.52669
0.40782
0.38852
0.38669
0.38448
0.38825
0.39038
0.38628
0.38526
0.37902
0.37762
0.38447
0.38041
0.37882
0.38646
0.38594
0.39414
0.38306
0.37724
0.38331
0.37571
0.38222
–
0.97225
0.61996
0.57066
0.53331
0.54363
0.55021
0.53327
0.52732
0.51876
0.51308
0.52802
0.52199
0.51019
0.51410
0.51678
0.52440
0.50138
0.49920
0.52406
0.51130
0.48951
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
2.9588
2.9588
2.9588
2.6671
0.38246
0.38222
0.38246
0.38179
0.52244
0.48951
0.52244
0.52103
23
24
25
26
17.807
17.807
17.807
16.326
(Continued)
Table 1.1 Physical Properties for Pure Components Continued
E
F
G
H
Density of Liquid 14.696 psia, 60°F
Number
Relative
Density
(Specific
Gravity)
60°F/60°F
1b-m/gal.
gal./lb-m
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
0.75050
0.75349
0.78347
0.77400
–
0.52095*
0.60107*
0.62717*
0.60996*
0.60040*
0.64571
0.65799*
0.62723*
0.68615
(0.41796)
0.88448
0.87190
0.87168
0.88467
6.2570
6.2819
6.5319
6.4529
–
4.3432*
5.0112*
5.2288*
5.0853*
5.0056*
5.3834
5.4857*
5.2293*
5.7205
(3.4842)
7.3740
7.2691
7.2673
7.3756
11.209
13.397
12.885
15.216
–
9.6889*
11.197*
10.731*
11.033*
11.209*
13.028
9.8605
10.344*
11.908
(7.473)
10.593
12.676
14.609
14.394
Temperature
Acentric
Coefficient
Factor,
of Density,
ω
1/°F
0.00073
0.00069
0.00065
0.00062
–
0.00173*
0.00112*
0.00105*
0.00106*
0.00117*
0.00089
0.00101*
0.00110*
0.00082*
–
0.00067
0.00059
0.00056
0.00052
0.1950
0.2302
0.2096
0.2358
0.0865
0.1356
0.1941
0.2029
0.2128
0.1999
0.2333
0.2840
0.2007
0.1568
0.1949
0.2093
0.2633
0.3027
0.3942
I
J
Ideal Gas 14.696 psia, 60°F
Specific Heat
60°F 14.696
psia Btu/
(1 bm, °F)
Compressibility
Factor of Real
Gas, Z 14.696
psia, 60°F
Relative
Density
(Specific
Gravity)
Air 5 1
ft gas/
lb-m
ft3 gas/
gal.
liquid
–
–
–
–
0.9936
0.9844
0.9699
0.9665
0.9667
0.9700
–
(0.969)
(0.965)
–
0.9930
–
–
–
–
2.4215
2.9059
2.9059
3.3902
0.9686
1.4529
1.9373
1.9373
1.9373
1.9373
2.4215
1.8677
1.8677
2.3520
0.8990
2.6971
3.1814
3.6657
3.6657
5.4110
4.5090
4.5090
3.8649
13.527
9.0179
6.7636
6.7636
6.7636
6.7636
5.4110
7.0156
7.0156
5.5710
14.574
4.8581
4.1184
3.5744
3.5744
33.856
28.325
29.452
24.940
–
39.167*
33.894*
35.366*
34.395*
33.856*
29.129
38.485*
36.687*
31.869
–
35.824
29.937
25.976
26.363
3
Cp. I
Deal
Gas
Cp.
Liquid
Number
0.27199
0.30100
0.28817
0.31700
0.35697
0.35714
0.35446
0.33754
0.35574
0.37690
0.36351
0.34347
0.34120
0.35072
0.39754
0.24296
0.26370
0.27792
0.28964
0.42182
0.44126
0.43584
0.44012
–
0.57116
0.54533
0.54215
0.54215
0.54839
0.51782
0.54029
0.53447
0.51933
–
0.40989
0.40095
0.41139
0.41620
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
0.86875
0.86578
0.91108
0.86634
0.79626
0.79399
0.78939*
0.81802*
0.80144*
1.3974*
0.61832*
0.87476*
0.071070*
1.1421*
0.80940*
1.4244*
1.00000
0.12510*
0.85129*
7.2429
7.2181
7.5958
7.228
6.6385
6.6196
6.5812*
6.8199*
6.6817*
11.650*
5.1550*
7.2930*
0.59252*
9.5221*
6.7481*
11.875*
8.33712
1.0430*
7.0973*
14.658
14.708
13.712
16.641
4.8267
6.9595
4.2561*
6.4532*
5.1005*
5.4987
3.3037*
3.9713*
3.4022*
3.3605*
4.1513*
5.9710*
2.1609
3.8376*
5.1373*
0.00053
0.00056
0.00053
0.00055
0.00066
0.00058
–
0.00583*
0.00157*
–
–
–
–
–
–
–
0.00009
–
0.00300*
0.3257
0.3216
(0.2412)
0.3260
0.5649
0.6438
0.0484
0.2667
0.0948
0.2548
0.2557
–
0.2202
0.0216
0.0372
0.0878
0.3443
0.
0.1259
–
–
–
–
–
–
0.9959
0.9943
0.9846
0.9802
0.9877
1.0000
1.0006
0.9992
0.9997
(0.9875)
–
1.0006
0.9923
3.6657
3.6657
3.5961
4.1500
1.1063
1.5906
0.9671
1.5196
1.1767
2.2118
0.5880
1.0000
0.06960
1.1048
0.9672
2.4482
0.62202
0.1382
1.2589
3.5744
3.5744
3.6435
3.1573
11.843
8.2372
13.548
8.6229
11.135
5.9238
22.283
13.103
188.25
11.859
13.546
5.3519
21.065
94.814
10.408
25.889
25.800
27.675
22.804
78.622
54.527
89.16*
58.807*
74.401*
69.012*
114.87*
95.557*
111.54*
112.93*
91.413*
63.554*
175.62
98.891*
73.806*
0.27427
0.27471
0.27110
0.29170
0.32316
0.33222
0.24847
0.19911
0.23827
0.14804
0.49677
0.23988
3.4038
0.21892
0.24828
0.11377
0.44457
1.2404
0.19086
0.40545 46
0.40255 47
0.41220 48
0.42053 49
0.59187 50
0.56610 51
–
52
–
53
0.50418 54
0.32460 55
1.1209 56
–
57
–
58
–
59
–
60
–
61
0.99974 62
–
63
–
64
Note: Numbers in this table do not have accuracies greater than 1 part in 1000; in some cases extra digits have been added to calculated values to achieve consistency or to permit recalculation
of experimental values.
Source: Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering Data Book, 10th edition, 1987.
10 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
D
pc
r
po
Va
id
+
Liquid
qu
Solid
Li
Solid + L
iq
uid
C
J
Pressure
I
or
+
lid
A
Vapor
p
Va
So
B
T1
T2
Tc
Temperature
n FIGURE 1.2 Typical pressure-temperature diagram for a single-component system.
If p < pv ! the system is entirely in the vapor phase;
If p > pv ! the system is entirely in the liquid phase;
If p ¼ pv ! the vapor and liquid coexist in equilibrium;
where p is the pressure exerted on the pure substance. It should be pointed
out that these expressions are valid only if the system temperature is below
the critical temperature Tc of the substance.
The lower end of the vapor pressure line is limited by the triple point A. This
point represents the pressure and temperature at which solid, liquid, and
vapor coexist under equilibrium conditions. The line AB is called the sublimation pressure curve of the solid phase, and it divides the area where solid
exists from the area where vapor exists. Points above AB represent solid systems, and those below AB represent vapor systems. The line AD is called the
melting curve or fusion curve and represents the change of melting-point
temperature with pressure. The fusion (melting) curve divides the solid
phase area from the liquid phase area, with a corresponding temperature
at any point on the curve termed the fusion or melting-point temperature.
Note that the solid-liquid curve (fusion curve) has a steep slope, which indicates that the triple point for most fluids is close to their normal meltingpoint temperatures. For pure hydrocarbons, the melting point generally
increases with pressure so the slope of the line AD is positive. Water is
the exception in that its melting point decreases with pressure, so in this
case, the slope of the line AD is negative.
Single-Component Systems 11
Each pure hydrocarbon has a p-T diagram similar to that shown in Fig. 1.2.
In addition, each pure component is characterized by its own vapor pressures, sublimation pressures, and critical values, which are different for each
substance, but the general characteristics are similar. If such a diagram is
available for a given substance, it is obvious that it could be used to predict
the behavior of the substance as the temperature and pressure are changed.
For example, in Fig. 1.2, a pure component system is initially at a pressure
and temperature represented by the point I, which indicates that the system
exists in the solid phase state. As the system is heated at a constant pressure
until point J is reached, no phase changes occur under this isobaric temperature increase and the phase remains in the solid state until the temperature
reaches T1. At this temperature, which is identified as the melting point at
this constant pressure, liquid begins to form and the temperature remains
constant until all the solid has disappeared. As the temperature is further
increased, the system remains in the liquid state until the temperature T2
is reached. At T2 (which is the boiling point at this pressure), vapor forms
and again the temperature remains constant until all the liquid has vaporized. The temperature of this vapor system now can be increased until
the point J is reached. It should be emphasized that, in the process just
described, only the phase changes were considered. For example, in going
from just above T1 to just below T2, it was stated that only liquid was present
and no phase change occurred. Obviously, the intensive properties of the
liquid are changed as the temperature is increased. For example, the increase
in temperature causes an increase in volume with a resulting decrease in the
density. Similarly, other physical properties of the liquid are altered, but the
properties of the system are those of a liquid and no other phases appear during this part of the isobaric temperature increase.
A method that is particularly convenient for plotting the vapor pressure as a
function of temperature for pure substances is shown in Fig. 1.3. The chart is
known as a Cox chart. Note that the vapor pressure scale is logarithmic,
while the temperature scale is entirely arbitrary.
Review of Pure Component Volumetric Properties
n
Density “ρ,” expressed in lbm/ft3, is defined as the ratio of mass “m” to
the volume, ie,
ρ¼
n
m
V
Number of moles “n” is defined as the ratio of mass “m” to the molecular
weight “M,” or
n¼
m
M
12 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
n FIGURE 1.3 Vapor pressure chart for hydrocarbon components. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering Data Book,
10th edition, 1987.)
n
Specific volume “v,” expressed in ft3/lbm, is defined as the ratio of
volume to mass; ie, reciprocal of density:
v¼
n
V 1
¼
m ρ
Molar volume “Vm” is defined as the ratio of volume “V” number of
moles “n,” ie,
V M
Vm ¼ ¼
n ρ
n
Molar density “ρm” is defined by the relationship
ρm ¼
1
ρ
¼
Vm M
Single-Component Systems 13
EXAMPLE 1.1
A pure propane is held in a laboratory cell at 80°F and 200 psia. Determine the
“existence state” (ie, as a gas or liquid) of the substance.
Solution
From a Cox chart, the vapor pressure of propane is read as pv ¼ 150 psi, and
because the laboratory cell pressure is 200 psi (ie, p > pv), this means that the
laboratory cell contains a liquefied propane.
The vapor pressure chart as presented in Fig. 1.3 allows a quick estimation
of the vapor pressure pv of a pure substance at a specific temperature. For
computer applications, however, an equation is more convenient. Lee and Kesler
(1975) proposed the following generalized vapor pressure equation:
pv ¼ pc exp ðA + ωBÞ
(1.3)
With the coefficients A and B as defined below:
A ¼ 5:92714 6:09648
1:2886 ln ðTr Þ + 0:16934ðTr Þ6
Tr
(1.4)
B ¼ 15:2518 15:6875
13:4721 ln ðTr Þ + 0:4357ðTr Þ6
Tr
(1.5)
The term Tr, called the reduced temperature, is defined as the ratio of the
absolute system temperature to the critical temperature of the component, or
Tr ¼
T
Tc
where
Tr ¼ reduced temperature
T ¼ substance temperature, °R
Tc ¼ critical temperature of the substance, °R
pc ¼ critical pressure of the substance, psia
ω ¼ acentric factor of the substance
The acentric factor “ω” is a concept that was introduced by Pitzer (1955),
and has proven to be very useful in the characterization of substances. It has
become a standard for the proper characterization of any single pure component,
along with other common properties, such as molecular weight, critical
temperature, critical pressure, and critical volume. The acentric factor ω is a
unique correlating parameter that is used as a measure of the centricity or the
deviation of the component molecular shape from that of a spherical. The shape
of the Argon molecule is considered completely spherical and is assigned an
acentric factor of zero. This important property is defined by the following
expression:
pv
ω ¼ log
1
(1.6)
pc T¼0:7Tc
14 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
where
pc ¼ critical pressure of the substance, psia
pv ¼ vapor pressure of the substance at a temperature equal to 70% of the
substance critical temperature (ie, T ¼ 0.7Tc), psia
The acentric factor frequently is used as a third parameter in corresponding
states and equation-of-state correlations. Values of the acentric factor for pure
substances are tabulated in Table 1.1.
EXAMPLE 1.2
Calculate the vapor pressure of propane at 80°F by using the Lee and Kesler
correlation.
Solution
Obtain the critical properties and the acentric factor of propane from
Table 1.1:
Tc ¼ 666:01°R
pc ¼ 616:3psia
ω ¼ 0:1522
Calculate the reduced temperature:
Tr ¼
T
540
¼
¼ 0:81108
Tc 666:01
Solve for the parameters A and B by applying Eqs. (1.4), (1.5), respectively,
to give:
A ¼ 5:92714 6:09648
1:2886 ln ðTr Þ + 0:16934ðTr Þ6
Tr
A ¼ 5:92714 6:09648
1:2886 ln ð0:81108Þ + 0:16934ð0:81108Þ6 ¼ 1:273590
0:81108
B ¼ 15:2518 15:6875
13:4721 ln ðTr Þ + 0:4357ðTr Þ6
Tr
B ¼ 15:2518 15:6875
13:4721 ln ð0:81108Þ + 0:4357ð0:81108Þ6 ¼ 1:147045
0:81108
Solve for pv by applying Eq. (1.3):
pv ¼ pc exp ðA + ωBÞ
pv ¼ ð616:3Þ exp ½1:27359 + 0:1572ð1:147045Þ ¼ 145psia
The densities of the saturated phases of a pure component (ie, densities of
the coexisting liquid and vapor) may be plotted as a function of temperature,
as shown in Fig. 1.4. Note that, for increasing temperature, the density of
the saturated liquid is decreasing, while the density of the saturated vapor
Single-Component Systems 15
Density
A
x
x
Densit
y curv
e of s
Avera
d liquid
ge de
(rv + r
L) / 2 =
Critical density
rc
aturate
nsity
ravg =
a + bT
C
rc = a + bTc
B
sity
x
Den
e
curv
por
d va
rate
tu
of sa
Temperature
TC
n FIGURE 1.4 Typical density-temperature diagram.
increases. At the critical point C, the densities of vapor and liquid converge
and become identical with a value equal to the critical density of the pure
substance “ρc.” It should be pointed out that at the critical point C, all other
properties of the phases become identical, such as viscosity and density.
Fig. 1.4 illustrates a useful observation that states that the arithmetic average of the
densities of the liquid and vapor phases is a linear function of the temperature
and can be represented by a straight-line relationship that passes through the
critical point. Mathematically, this straight-line relationship is expressed as
ρv + ρL
¼ ρavg ¼ a + bT
2
(1.7)
where
ρv ¼ density of the saturated vapor, lb/ft3
ρL ¼ density of the saturated liquid, lb/ft3
ρavg ¼ arithmetic average density, lb/ft3
T ¼ temperature, °R
a, b ¼ intercept and slope of the straight line
Because, at the critical point, ρv and ρL are identical, Eq. (1.7) can be expressed
in terms of the critical density as follows:
ρc ¼ a + bTc
(1.8)
where
ρc ¼ critical density of the substance, lb/ft3.
Combining Eq. (1.7) with (1.8) and solving for the critical density gives
a + bT
ρ
ρc ¼
a + bTc avg
16 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
This density-temperature diagram is useful in calculating the critical volume
from density data. The experimental determination of the critical volume
sometimes is difficult, as it requires the precise measurement of a volume at a
high temperature and pressure. However, it is apparent that the straight line
obtained by plotting the average density versus temperature intersects the
critical temperature at the critical density. The molal critical volume is obtained
by dividing the molecular weight by the critical density:
Vc ¼
where
M
ρc
Vc ¼ critical volume of pure component, ft3/lbm-mol
M ¼ molecular weight, lbm/lbm-mol
ρc ¼ critical density, lbm/ft3
Fig. 1.5 shows the saturated densities for a number of fluids of interest to the
petroleum engineer. Note that, for each pure substance, the upper curve is
(Specific gravity) Density grams in c.c.
Hydrocarbon fluid densities
Temperature (°F)
Identification
A 8 mol% CH4 – 92 mol% C3H8 (370 – 739 lbs.)
B 50.25 wt% C2H6 – 49.75 wt% n-C7H16
C 19.2 wt% CH4 – 80.8 wt% C6H14 (2412 – 2506 lbs.)
D 7.15 wt% CH4 – 92.85 wt% n-C5H12 (8540 – 1043 lbs.)
E National Standard Petroleum Tables
F 7.15 wt% CH4 – 92.85 wt% n % C5H12 (at 3000 lbs.)
G 9.78 wt% C2H6 – 90.22 wt% n-C7H16
H 75.45 mol% C3H5 – 24.55% n-C4H10
I 65.77 mol% C2H6 – 34.23% n-C4H10
n FIGURE 1.5 Fluid densities. (Courtesy of the Gas Processors Suppliers Association. GPSA Engineering Book, 10th edition, 1987.)
Single-Component Systems 17
termed the saturated liquid density curve, while the lower curve is labeled the
saturated vapor density curve with both curves meet and terminate at the critical
point “C.”
EXAMPLE 1.3
Calculate the saturated liquid and gas densities of n-butane at 200°F.
Solution
From Fig. 1.5, read both values of liquid and vapor densities at 200°F
to give
Liquid densityρL ¼ 0:475g=cm3
Vapor density ρv ¼ 0:035g=cm3
The density-temperature diagram also can be used to determine the state
of a single-component system, ie, the pure component exits in a liquefied
state, vaporous state, or two-phase state. Suppose that the overall (total)
density of the system, ρt, is known at a given temperature. If this overall
density is less than or equal to the pure component vapor density ρv, it is
obvious that the system is composed entirely of vapor. Similarly, if the
overall density ρt is greater than or equal to the pure component liquid
density ρL, the system is composed entirely of liquid. If, however, the
overall density is between ρL and ρv, it indicates that both liquid and vapor
are present. If both phases are present, the volume and weight of each
phase present can be calculated by applying volume and mass balances as
shown below:
mL + mv ¼ mt
VL + Vv ¼ Vt
where
mL, mv, and mt ¼ the mass of the liquid, vapor, and total system, respectively
VL, Vv, and Vt ¼ the volume of the liquid, vapor, and total system,
respectively
Combining the above two equations and introducing the density into the
resulting equation gives
mt mv mv
+
¼ Vt
ρL
ρv
(1.9)
18 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
EXAMPLE 1.4
Ten pounds of a hydrocarbon are placed in a 1 ft3 vessel at 60°F. The densities
of the coexisting liquid and vapor are known to be 25 and 0.05 lb/ft3, respectively,
at this temperature. Calculate the weights and volumes of the liquid and vapor
phases.
Solution
Step 1. Calculate the density of the overall system:
ρt ¼
mt 10
¼
¼ 10lb=ft3
Vt 1:0
Step 2. As the overall density of the system is between the density of the liquid
and the density of the gas, the system must be made up of both
liquid and vapor.
Step 3. Calculate the weight of the vapor from Eq. (1.9):
10 mv mv
+
¼1
25
0:05
Solving the above equation for mv and applying the mass balance to solve for mL
gives
mv ¼ 0:030lb
mL ¼ 10 mv ¼ 10 0:03 ¼ 9:97lb
Step 4. Calculate the volume of the vapor and liquid phases:
Vv ¼
mv 0:03
¼
¼ 0:6ft3
ρv 0:05
VL ¼ Vt Vv ¼ 1 0:6 ¼ 0:4ft3
EXAMPLE 1.5
A utility company stored 58 million pounds of propane, ie,
mt ¼ 58,000,000 lb, in a washed-out underground salt cavern of volume
480,000 bbl (Vt ¼ 480,000 bbl) at a temperature of 110°F. Estimate
the weight and volume of the existing propane phases in the underground
cavern.
Solution
Step 1. Calculate the volume of the cavern in ft3:
Vt ¼ ð480, 000Þð5:615Þ ¼ 2, 695,200ft3
Step 2. Calculate the total density of the system:
ρt ¼
mt 58,000, 000
¼
¼ 21:52lb=ft3
Vt 2, 695,200
Single-Component Systems 19
Step 3. Determine the saturated densities of propane from the density chart of
Fig. 1.5 at 110°F:
ρL ¼ 0:468g=cm3 ¼ ð0:468Þð62:4Þ ¼ 29:20lb=ft3
ρv ¼ 0:030g=cm3 ¼ ð0:03Þð62:4Þ ¼ 1:87lb=ft3
Step 4. Test for the existing phases. Because
ρv < ρt < ρL
1:87 < 21:52 < 29:20
both liquid and vapor are present.
Step 5. Solve for the weight of the vapor phase by applying Eq. (1.9)
58, 000,000 mv mv
+
¼ 2, 695,200
29:20
1:87
mv ¼ 1,416, 345lb
Vv ¼
mv 1,416, 345
¼
¼ 757,404ft3
ρ
1:87
Step 6. Solve for the volume and weight of propane:
VL ¼ Vt Vv ¼ 2,695, 200 757, 404 ¼ 1,937, 796ft3
The above suggests that the propane in the liquid state occupies 72% of total
cavern volume. In terms of the propane mass:
mL ¼ mt mv ¼ 58,000, 000 1,416, 345 ¼ 56,583, 655lb ð98%of totalweightÞ:
Pure Component Saturated Liquid Density
Rackett (1970) proposed a simple generalized equation for predicting
the saturated liquid density, ρL, of pure compounds. Rackett expressed
the relation in the following form:
ρL ¼
Mpc
RTc Zca
with the exponent “a” given as
a ¼ 1 + ð1 Tr Þ2=7
where
M ¼ molecular weight of the pure substance
pc ¼ critical pressure of the substance, psia
Tc ¼ critical temperature of the substance, °R
Zc ¼ critical gas compressibility factor
(1.10)
20 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
R ¼ gas constant, 10.73 ft3 psia/lb-m, °R
Tr ¼ T=Tc , reduced temperature
T ¼ temperature, °R
Spencer and Danner (1973) modified Rackett’s correlation by replacing
the critical compressibility factor Zc in Eq. (1.9) with a unique parameter
called Rackett’s compressibility factor “ZRA,” which is constant for each
compound. Spencer and Danner introduced the parameter ZRA into
Eq. (1.10) to give the following modification of the Rackett equation:
ρL ¼
Mpc
RTc ðZRA Þa
(1.11)
with the exponent “a” as defined previously by
a ¼ 1 + ð1 Tr Þ2=7
The values of ZRA are given in Table 1.2 for selected components.
The Rackett parameter ZRA can also be estimated from a correlation proposed by Yamada and Gunn (1973) as
ZRA ¼ 0:29056 0:08775ω
(1.12)
where ω is the acentric factor of the compound.
The Rackett’s compressibility factor can be approximated as a function of
the number of carbon atoms “n” for components heavier than heptane; eg,
n ¼ 8,9,…, by the following simple relationship:
ZRA ¼ 0:37748n0:16941
Table 1.2 Values of ZRA for Selected Pure Components
Carbon dioxide
Nitrogen
Hydrogen sulfide
Methane
Ethane
Propane
i-Butane
n-Butane
i-Pentane
0.2722
0.2900
0.2855
0.2892
0.2808
0.2766
0.2754
0.2730
0.2717
n-Pentane
n-Hexane
n-Heptanes
i-Octane
n-Octane
n-Nonane
n-Decane
n-Undecane
0.2684
0.2635
0.2604
0.2684
0.2571
0.2543
0.2507
0.2499
Single-Component Systems 21
EXAMPLE 1.6
Calculate the saturated liquid density of propane at 160°F by using
1. The Rackett correlation.
2. The modified Rackett equation.
Solution
1. The Rackett Approach:
Find the critical properties of propane from Table 1.1, to give
Tc ¼ 666:06°R
pc ¼ 616:0psia
M ¼ 44:097
vc ¼ 0:0727ft3 =lb
Calculate Zc by applying the real gas equation of state (EOS):
Z¼
pV
pV
pvM
¼
¼
nRT ðm=MÞRT RT
where
V ¼ substance volume, ft3
v ¼ substance specific volume; V/m, ft3/lb
Using the critical properties of propane to determine Zc, to give
Zc ¼
pc vc M ð616:0Þð0:0727Þð44:097Þ
¼
¼ 0:2763
RTc
ð10:73Þð666:06Þ
Calculate the reduced temperature Tr and determine the saturated liquid density
by applying the Rackett equation, Eq. (1.10):
Tr ¼
T 160 + 460
¼
¼ 0:93085
Tc
666:06
a ¼ 1 + ð1 Tr Þ2=7 ¼ 1 + ð1 0:93085Þ2=7 ¼ 1:4661
ρL ¼
ρL ¼
Mpc
RTc Zca
ð44:097Þð616:0Þ
ð10:73Þð666:06Þð0:2763Þ1:4661
¼ 25:05lb=ft3
2. The Modified Rackett Approach:
Using the Rackett compressibility factor of ZRA ¼ 0.2766 from Table 1.2 to
calculate the liquid density, gives
ρL ¼
ð44:097Þð616:0Þ
ð10:73Þð666:06Þð0:2766Þ1:4661
¼ 25:01lb=ft3
22 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
TWO-COMPONENT SYSTEMS
A unique feature of the single-component system is that, at a fixed temperature, two phases (vapor and liquid) can exist in equilibrium at only the
vapor pressure. For a binary system, two phases can exist in equilibrium
at various pressures at the same temperature. The following discussion concerning the description of the phase behavior of a two-component system
involves many concepts that apply to the more complex multicomponent
mixtures of oils and gases.
An important characteristic of binary systems is the variation of their thermodynamic and physical properties with the composition. Therefore, it is
necessary to specify the composition of the mixture in terms of mole or
weight fractions. It is customary to designate one of the components as
the more volatile component and the other the less volatile component,
depending on their relative vapor pressure at a given temperature.
Suppose that the examples previously described for a pure component are
repeated, but this time we introduce into the cylinder a binary mixture of
a known overall composition “zi.” Consider that the initial pressure p1
exerted on the system, at a fixed temperature of T1, is low enough that
the entire system exists in the vapor state. This initial condition of pressure
and temperature acting on the mixture is represented by point 1 on the p/V
diagram of Fig. 1.6. As the pressure is increased isothermally, it reaches
pb
Constant temperature
Liquid
Pressure
4
Bubble point
Bubble point pressure
3
pd
Liqu
id +
Dew point
Vap
or
Dew point pressure
2
Va
por
1
Volume
n FIGURE 1.6 Pressure-volume isotherm for a two-component system.
Two-Component Systems 23
point 2, at which point an infinitesimal amount of liquid is condensed. The
pressure at this point is called the dew point pressure, pd, of the mixture. It
should be noted that, at the dew point pressure, the composition of the vapor
phase is equal to the overall composition of the binary mixture. As the total
volume is decreased by forcing the piston inside the cylinder, a noticeable
increase in the pressure is observed as more and more liquid is condensed.
This condensation process is continued until the pressure reaches point 3, at
which point traces of gas remain. At point 3, the corresponding pressure is
called the bubble point pressure, pb. Because, at the bubble point, the gas
phase is only of infinitesimal volume, the composition of the liquid phase
therefore is identical with that of the whole system. As the piston is forced
further into the cylinder, the pressure rises steeply to point 4 with a corresponding decreasing volume.
Repeating the above process at progressively increasing temperatures, a
complete set of isotherms is obtained on the p/V diagram of Fig. 1.7 for a
binary system consisting of n-pentane and n-heptane. The bubble point
curve, as represented by line AC, represents the locus of the points of pressure and volume at which the first bubble of gas is formed. The dew point
curve (line BC) describes the locus of the points of pressure and volume at
which the first droplet of liquid is formed. The two curves meet at the critical
point (point C). The critical pressure, temperature, and volume are given by
pc, Tc, and Vc, respectively. Any point within the phase envelope (line ACB)
represents a system consisting of two phases. Outside the phase envelope,
only one phase can exist.
Liquid
C
45
0°
425°
Bubble point
Pressure
400
300
Critical point:
45
4°
F
Vapor
400°
Liquid + Vapor
Dew point curve
350°
200
300°
B
A
0.1
0.2
0.3
Volume (ft3/Ib)
n FIGURE 1.7 Pressure-volume diagram for a binary system.
0.4
24 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
If the bubble point pressure and dew point pressure for the various isotherms
on a p-V diagram are plotted as a function of temperature, a p-T diagram
similar to that shown in Fig. 1.8 is obtained. Fig. 1.8 indicates that the
pressure-temperature relationships no longer can be represented by a simple
vapor pressure curve, as in the case of a single-component system, but take
on the form illustrated in the figure by the phase envelope ACB. The dashed
lines within the phase envelope are called quality lines; they describe the
pressure and temperature conditions of equal volumes of liquid. Obviously,
the bubble point curve and the dew point curve represent 100% and 0%
liquid, respectively.
Fig. 1.9 demonstrates the effect of changing the composition of the binary
system on the shape and location of the phase envelope. Two of the lines
shown in the figure represent the vapor pressure curves for two pure components, designed as 1 for the lighter component and 2 for the other component. Four phase boundary curves (phase envelopes); labeled A
through D, represent various mixtures of the two components with increasing the concentration component 2. These curves pass continuously from the
vapor pressure curve of the one pure component to that of the other as the
composition is varied. The points labeled A through D represent the critical
points of the mixtures. It should be noted by examining Fig. 1.9 that, when
one of the constituents becomes predominant, the binary mixture tends to
exhibit a relatively narrow phase envelope and displays critical properties
close to the predominant component. The size of the phase envelope
enlarges noticeably as the composition of the mixture becomes evenly
distributed between the two components.
Cricondenbar
Liquid
%
100
Pressure
A
ble
Bub
t cur
poin
ve
C
Vapor + Liquid
75%
Cricondentherm
Pc
50%
B
25%
ur ve
0%
oint c
Dew p
Vapor
Temperature
n FIGURE 1.8 Typical pressure-temperature diagram for a two-component system.
Tc
Two-Component Systems 25
B
A
C
D
Critical point
Pressure
Critical point
Temperature
n FIGURE 1.9 Phase diagram of binary mixture.
Pressure-Composition Diagram “p-x”
for Binary Systems
The pressure-composition diagram, commonly called the p-x diagram, is
another means of describing the phase behavior of a binary system, as its
overall composition changes at a constant temperature. It is constructed
by plotting the dew point and bubble point pressures as a function of composition. The bubble point and dew point lines of a binary system are drawn
through the points that represent these pressures as the composition of the
system is changed at a constant temperature. Fig. 1.10 represents a typical
pressure-composition diagram for a two-component system. Component 1
is described as the more volatile fraction and component 2 as the less volatile
fraction. Point A in the figure represents the vapor pressure (dew point, bubble point) of the more volatile component, while point B represent that of the
less volatile component. Assuming a composition of 75% by weight of component 1 (ie, the more volatile component) and 25% of component 2, this
mixture is characterized by a dew point pressure represented as point C
and a bubble point pressure of point D. Different combinations of the
two components produce different values for the bubble point and dew point
26 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
A
Constant temperature
D
Bub
ble
poin
t cu
Pressure
Liquid
Liquid + Vapor
C
Vapor
r ve
X
Dew
point
Y
cur ve
B
wv
0
25
50
wt% of less volatile component
wL
75
100
n FIGURE 1.10 Typical pressure-composition diagram for a two-component system. Composition expressed
in terms of wt% of the less volatile component.
pressures. The curve ADYB represents the bubble point pressure curve for
the binary system as a function of composition, while the line ACXB
describes the changes in the dew point pressure as the composition of the
system changes at a constant temperature. The area below the dew point line
represents vapor, the area above the bubble point line represents liquid, and
the area between these two curves represents the two-phase region, where
liquid and vapor coexist.
It should be pointed out that the composition may be expressed equally well
in terms of weight percent of the more volatile component, in which case the
bubble point and dew point lines have the opposite placement; that is, dew
point curve above the bubble point curve.
As shown in Fig. 5.10, the p-x diagram can also be displayed in terms of
mole or weight percent of the more volatile component. The points X and
Y at the extremities of the horizontal line XY represent the composition of
the coexisting of the vapor phase (point X) and the liquid phase (point Y)
that exist in equilibrium at the same pressure. In other words, at the pressure
represented by the horizontal line XY, the compositions of the vapor and liquid that coexist in the two-phase region are given by wv and wL, and they
represent the weight percentages of the less volatile component in the vapor
and liquid, respectively.
In the p-x diagram shown in Fig. 1.11, the composition is expressed in terms
of the mole fraction of the more volatile component. Assume that a binary
system with an overall composition of z exists in the vapor phase state as
represented by point A. If the pressure on the system is increased, no phase
change occurs until the dew point, B, is reached at pressure P1. At this dew
point pressure, an infinitesimal amount of liquid forms whose composition
is given by x1. The composition of the vapor still is equal to the original
Two-Component Systems 27
C
p3
Liquid
Pressure
t
oin
le p
b
Bub
ve
cur
p2
Liquid + Vapor
ap
id
iqu
p1
XB
or
+V
oint
p
Dew
L
e
curv
XA
Vapor
0%
x1
x2
z
y2
Mole fraction of more volatile component
n FIGURE 1.11 Pressure-composition diagram illustrating an isothermal compression through the two-phase region.
composition z. As the pressure is increased, more liquid forms and the compositions of the coexisting liquid and vapor are given by projecting the ends
of the straight, horizontal line through the two-phase region of the composition axis. For example, at p2, both liquid and vapor are present and the
compositions are given by x2 and y2. At pressure p3, the bubble point, C,
is reached. The composition of the liquid is equal to the original composition
z with an infinitesimal amount of vapor still present at the bubble point with
a composition given by y3.
As indicated already, the extremities of a horizontal line through the twophase region represent the compositions of coexisting phases. Burcik
(1957) points out that the composition and the amount of each phase present
in a two-phase system are of practical interest and use in reservoir engineering calculations. At the dew point, for example, only an infinitesimal
amount of liquid is present with a finite mole fractions of the two components. Introducing the following nomenclatures:
n ¼ total number of moles in the binary system
nL ¼ number of moles of liquid
nv ¼ number of moles of vapor
z ¼ mole fraction of the more volatile component in the system
x ¼ mole fraction of the more volatile component in the liquid phase
y ¼ mole fraction of the more volatile component in the vapor phase
y3
100%
28 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
An equation for the relative amounts of liquid and vapor in a two-phase system may be derived on one of the two components (the more volatile or less
volatile). Selecting and developing molar material balance on the more volatile component, let
n ¼ nL + nv
n z ¼ moles of the more volatile component in the system
nL x ¼ moles of the more volatile component in the liquid
nv y ¼ moles of the more volatile component in the vapor
A material balance on the more volatile component gives
nz ¼ nL x + nv y
(1.13)
and
nL ¼ n nv
Combining these two expressions gives
nz ¼ ðn nv Þx + nv y
Rearranging the above expression, gives
nv z x
¼
n yx
(1.14)
Similarly, if nv is eliminated in Eq. (1.13) instead of nL, it gives
nL z y
¼
n xy
(1.15)
The geometrical interpretation of Eqs. (1.14), (1.15) is shown in
Fig. 1.12, which indicates that these equations can be written in terms of
the two segments of the horizontal line AC. Because z x ¼ the length of
segment AB, and y x ¼ the total length of horizontal line AC, Eq. (1.14)
becomes
nv z x AB
¼
¼
n y x AC
(1.16)
Similarly, Eq. (1.15) becomes
nL BC
¼
n AC
(1.17)
Eq. (1.16) suggests that the ratio of the number of moles of vapor to the total
number of moles in the system is equivalent to the length of the line segment
Two-Component Systems 29
e
urv
nt c
oi
le p
b
Bub
Liquid
Liquid + Vapor
B
Pressure
A
C
Liquid + Vapor
Vapor
Dew point curve
x
0%
z
y
Mole fraction of more volatile component
n FIGURE 1.12 Geometrical interpretation of equations for the amount of liquid and vapor in the two-phase region.
AB that connects the overall composition to the liquid composition divided
by the total length psia. This rule is known as the inverse lever rule. Similarly, the ratio of the number of moles of liquid to the total number of moles
in the system is proportional to the distance from the overall composition to
the vapor composition BC divided by the total length AC. It should be
pointed out that the straight line that connects the liquid composition with
the vapor composition, that is, line AC, is called the tie line. Note that results
would have been the same if the mole fraction of the less volatile component
had been plotted on the phase diagram instead of the mole fraction of the
more volatile component.
EXAMPLE 1.7
A system is composed of 3 mol of isobutene and 1 mol of n-heptanes. The
system is separated at a fixed temperature and pressure and the liquid and vapor
phases recovered. The mole fraction of isobutene in the recovered liquid and
vapor are 0.370 and 0.965, respectively. Calculate the number of moles of liquid
nl and vapor nv recovered.
Solution
Step 1. Given x ¼ 0.370, y ¼ 0.965, and n ¼ 4, calculate the overall mole fraction
of isobutane in the system:
3
z ¼ ¼ 0:750
4
100%
30 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
Step 2. Solve for the number of moles of the vapor phase by applying Eq. (1.14):
zx
0:750 0:370
nv ¼ n
¼4
¼ 2:56mol of vapor
yx
0:965 0:375
Step 3. Determine the quantity of liquid:
nL ¼ n nv ¼ 4 2:56 ¼ 1:44mol of liquid
The quantity of nL also could be obtained by substitution in Eq. (1.15):
zy
0:750 0:965
nL ¼ n
¼4
¼ 1:44
xy
0:375 0:965
If the composition is expressed in weight fraction instead of mole fraction,
similar expressions to those expressed by Eqs. (1.14), (1.15) can be derived in
terms of weights of liquid and vapor. Let
mt ¼ total mass (weight) of the system
mL ¼ total mass (weight) of the liquid
mv ¼ total mass (weight) of the vapor
wo ¼ weight fraction of the more volatile component in the original system
wL ¼ weight fraction of the more volatile component in the liquid
wv ¼ weight fraction of the more volatile component in the vapor
A material balance on the more volatile component leads to the following
equations:
mv wo wL
¼
mt wv wL
mL wo wv
¼
mt wL wv
THREE-COMPONENT SYSTEMS
The phase behavior of mixtures containing three components (ternary systems) is conveniently represented in a triangular diagram, such as that
shown in Fig. 1.13. Such diagrams are based on the property of equilateral
triangles that the sum of the perpendicular distances from any point to each
side of the diagram is a constant and equal to the length on any of the sides.
Thus, the composition xi of the ternary system as represented by point A in
the interior of the triangle of Fig. 1.13 is
Component 1 x1 ¼
L1
LT
Component 2 x2 ¼
L2
LT
Component 3 x3 ¼
L3
LT
Three-Component Systems 31
L3
L2
t3
en
on
mp
co
0%
0%
co
mp
on
en
t2
100%
component 1
L1
100%
component 3
0% component 1
100%
component 2
n FIGURE 1.13 Properties of the ternary diagram.
where
LT ¼ L1 + L2 + L3
Typical features of a ternary phase diagram for a system that exists in the
two-phase region at fixed pressure and temperature are shown in
Fig. 1.14. Any mixture with an overall composition that lies inside the binodal curve (phase envelope) will split into liquid and vapor phases. The line
that connects the composition of liquid and vapor phases that are in equilibrium is called the tie line. Any other mixture with an overall composition
that lies on that tie line will split into the same liquid and vapor compositions. Only the amounts of liquid and gas change as the overall mixture composition changes from the liquid side (bubble point curve) on the binodal
curve to the vapor side (dew point curve). If the mole fractions of component
i in the liquid, vapor, and overall mixture are xi, yi, and zi, the fraction of the
total number of moles in the liquid phase nl is given by
nl ¼
y i zi
yi xi
This expression is another lever rule, similar to that described for binary diagrams. The liquid and vapor portions of the binodal curve (phase envelope)
meet at the plait point (critical point), where the liquid and vapor phases are
identical.
32 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
co
V
po apo
sit r
io
n
m
“y”
Two-phase
region
Ti
e
lin
e
1
Liquid composition “x”
Cri
t
“pl ical p
ait
o
po int
int
”
Phase envelop
“binodal curve”
Single phase region
3
2
n FIGURE 1.14 Ternary phase diagram at a constant temperature and pressure for a system that forms a liquid and a vapor.
MULTICOMPONENT SYSTEMS
The phase behavior of multicomponent hydrocarbon systems in the twophase region, that is, the liquid-vapor region, is very similar to that of binary
systems. However, as the system becomes more complex with a greater
number of different components, the pressure and temperature ranges in
which two phases lie increase significantly.
The conditions under which these phases exist are a matter of considerable
practical importance. The experimental or the mathematical determinations
of these conditions are conveniently expressed in different types of diagrams, commonly called phase diagrams. One such diagram is called the
pressure-temperature diagram.
PRESSURE-TEMPERATURE DIAGRAM
Fig. 1.15 shows a typical pressure-temperature diagram (p-T diagram) of a
multicomponent system with a specific overall composition. Although a different hydrocarbon system would have a different phase diagram, the general configuration is similar.
These multicomponent p-T diagrams are essentially used to classify reservoirs, specify the naturally occurring hydrocarbon systems, and describe the
phase behavior of the reservoir fluid.
Pressure-Temperature Diagram 33
4000
pcb
D
Liquid phase
2
Gas phase
d
i
qu
3500
0%
li
10
3000 (Pc)mixture
3
“C” Critical point
rv
e
e
rv
cu
int id
po liqu
w
De 0%
int
cu
2500
lin
Bu
y
bb
lit
le
2000
ua
po
Q
es
Pressure (psia)
1
1500
B
1000
Two-phase region
E
500
0
–100
0
100
200
300
400
500
Temperature (°F)
n FIGURE 1.15 Typical p-T diagram for a multicomponent system.
To fully understand the significance of the p-T diagrams, it is necessary to
identify and define the following key points on the p-T diagram:
n
n
n
n
n
Cricondentherm (Tct): The cricondentherm is the maximum
temperature above which liquid cannot be formed regardless of pressure
(point E). The corresponding pressure is termed the cricondentherm
pressure, pct.
Cricondenbar (pcb): The cricondenbar is the maximum pressure
above which no gas can be formed regardless of temperature (point D).
The corresponding temperature is called the cricondenbar
temperature, Tcb.
Critical point: The critical point for a multicomponent mixture is
referred to as the state of pressure and temperature at which all intensive
properties of the gas and liquid phases are equal (point C). At the
critical point, the corresponding pressure and temperature are called the
critical pressure, pc, and critical temperature, Tc, of the mixture.
Phase envelope (two-phase region): The region enclosed by the bubble
point curve and the dew point curve (line BCA), where gas and
liquid coexist in equilibrium, is identified as the phase envelope of the
hydrocarbon system.
Quality lines: The lines within the phase diagram are called quality lines.
They describe the pressure and temperature conditions for equal
liquid percentage by volumes (% of liquid). Note that the quality lines
converge at the critical point (point C).
A
600
700 Tct
(Tc)mixture
800
34 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
n
n
Bubble point curve: The bubble point curve (line BC) is defined as the
line separating the liquid phase region from the two-phase region.
Dew point curve: The dew point curve (line AC) is defined as the line
separating the vapor phase region from the two-phase region.
CLASSIFICATION OF RESERVOIRS AND RESERVOIR
FLUIDS
Petroleum reservoirs are broadly classified as oil or gas reservoirs. These
broad classifications are further subdivided depending on
1. The composition of the reservoir hydrocarbon mixture.
2. Initial reservoir pressure and temperature.
3. Pressure and temperature of the surface production.
4. Location of the reservoir temperature with respect to the critical
temperature and the cricondentherm.
In general, reservoirs are conveniently classified on the basis of the reservoir
temperature “T” as compared with the critical temperature of the hydrocarbon mixture “(Tc)mix.” It should be noted that perhaps the most important
components are the Methane and the Plus-Fraction, eg, C7+. In general,
these two components impact the size of the phase envelope as well as
the critical temperature of the mixture. Accordingly, reservoirs can be classified into basically three types:
1. Oil reservoirs: If the reservoir temperature, T, is less than the critical
temperature, Tc, of the reservoir fluid, the reservoir is classified as an oil
reservoir.
2. Gas reservoirs: If the reservoir temperature is greater than the critical
temperature of the hydrocarbon fluid, the reservoir is considered a gas
reservoir.
3. Near-critical reservoirs: If the reservoir temperature is very close to
the critical temperature of the hydrocarbon fluid, the reservoir is
classified as near-critical reservoir.
Fig. 1.16 shows a flow diagram that summarizes the three reservoir classifications. The impact of fluid composition on the shape and size of the phase
envelope as well as the reservoir type identification are illustrated in Fig. 1.17.
Classification of Oil Reservoirs
Depending on initial reservoir pressure, pi, oil reservoirs can be subclassified into the following categories:
1. Undersaturated oil reservoir: If the initial reservoir pressure, pi (as
represented by point 1 in Fig. 1.15), is greater than the bubble point pressure,
pb, of the reservoir fluid, the reservoir is an undersaturated oil reservoir.
Classification of Reservoirs and Reservoir Fluids 35
Classification of reservoirs
is defined by T & (Tc)hydrocarbon
% of C1&C7+ in the system
will impact (Tc)mixture
(Tc)mixture = f(composition)
Gas reservoirs
T > (Tc)mixture
Oil reservoirs
T < (Tc)mixture
Near critical reservoirs
T ≈ (Tc)mixture
Special case:
Critical mixture separates
the gas-cap from the oil rim
Critical mixture
T = (Tc)mixture
n FIGURE 1.16 Reservoir classifications.
2. Saturated oil reservoir: When the initial reservoir pressure is equal
to the bubble point pressure of the reservoir fluid, as shown in Fig. 1.15
by point 2, the reservoir is a saturated oil reservoir.
3. Gas-cap reservoir: When two phases (gas and oil) initially exit in
equilibrium in a reservoir; the reservoir is defined as a gas-cap reservoir.
Because the composition of the gas and oil zones are entirely
different from each other, each phase will be represented separately by
individual phase diagrams with the oil zone exits at its bubble point
with the gas cap its dew point. Based on the gas-cap composition, the gas
cap may be either retrograde gas or dry/wet gas, as illustrated in Fig. 1.18.
It should be pointed out that the main condition for a reservoir to be initially
considered in equilibrium in that the reservoir pressure “pi,” bubble point
pressure “pb,” and dew point pressure “pd” are all equal at gas-oil contact
(GOC) (ie, pi ¼ pb ¼ pd at GOC); this is an essential requirement to signify
that the reservoir system is in equilibrium as shown in Fig. 1.20
Classification of Crude Oil Systems
Crude oil is a mixture of small and large hydrocarbon. Depending on the
mixture of hydrocarbon molecules, crude oil varies in color, composition,
and consistency. Different oil-producing areas yield significantly different
Composition defines the
size of phase envelop
Oil reservoir T < Tc
Comp.
Mix 2
Pc, psia
Tc, °R
4000
CO2
0.000
1071
547.9
3500
N2
0.000
493
227.6
C1
0.350
667.8
343.37
C2
0.100
707.8
550.09
C3
0.060
616.3
666.01
i-C4
0.020
529.1
734.98
n-C4
0.020
550.7
765.65
i-C5
0.001
490.4
829.1
n-C5
0.002
488.6
845.7
C6
0.030
436.9
913.7
C7+
0.417
320.3
1139.4
T
Low % of methane
High % of C7+
C
Pressure (psia)
3000
2500
Two-phase region
2000
1500
1000
500
0
–100
TC
0
100
200
300
400
Temperature (°F)
500
600
700
800
Gas reservoir T > Tc
Mix 1
Pc, psia
Tc, °R
CO2
0.009
1071
547.9
N2
0.003
493
227.6
C1
0.535
667.8
343.37
C2
0.115
707.8
550.09
C3
0.088
616.3
666.01
i-C4
0.023
529.1
734.98
n-C4
0.023
550.7
765.65
i-C5
0.015
490.4
829.1
n-C5
0.015
488.6
845.7
C6
0.015
436.9
913.7
C7+
0.159
320.3
1139.4
High % of methane
Low % of C7+
Liquid
Pressure
Comp.
C
75 50
25
0
TC
Temperature
n FIGURE 1.17 Impact of composition on the size of phase envelope.
Retrograde gas cap
Dry or wet gas cap
Pressure
Pressure
C
Gas-cap
phase envelope
Oil zone
phase envelope
pb = pd = pi
C
Gas-cap
phase envelope
nt c
urv
e
C
pb = pd = pi
Bubble point
curve
Oil zone
phase envelope
ble
poi
Dew point
curve
Bubble point
curve
Bub
Dew point
curve
T
Temperature
n FIGURE 1.18 Classifications of the gas-cap reservoir.
Temperature
Dew point
curve
T
C
Classification of Reservoirs and Reservoir Fluids 37
varieties of crude oil that ranges from light to heavy crude oil systems.
These variations in the type of crude oil are related to the ability of oil
to flow (viscosity), color, and density. The term “sweet” is used to describe
crude oil that is low in sulfur compounds such as hydrogen sulfide, and
the term “sour” is used to describe crude oil containing high sulfur
compounds.
Crude oils cover a wide range in physical properties and chemical compositions, and it is often important to be able to group them into broad categories of related oils. In general, crude oils are commonly classified into
the following types
n
n
n
n
Ordinary black oil
Low-shrinkage crude oil
High-shrinkage (volatile) crude oil
Near-critical crude oil.
The above classification essentially is based on the physical and PVT properties exhibited by the crude oil, including
n
n
n
n
n
Physical properties, such as API gravity of the stock-tank liquid
Composition
Initial producing gas-oil ratio (GOR)
Appearance, such as the color of the stock-tank liquid
Pressure-temperature phase diagram.
Three of the above properties generally are available that include initial
GOR, API gravity, and color of the separated liquid.
The initial producing GOR perhaps is the most important indicator of
fluid type. Color has not been a reliable means of differentiating clearly
between gas-condensates and volatile oils, but in general, dark colors indicate
the presence of heavy hydrocarbons. No sharp dividing lines separate these
categories of hydrocarbon systems, only laboratory studies could provide
the proper classification. In general, reservoir temperature and composition
of the hydrocarbon system greatly influence the behavior of the system.
The fluid physical characteristics of the four crude oil systems are summarized below.
1. Ordinary black oil:
A typical p-T phase diagram for ordinary black oil is shown in Fig. 1.19.
Note that quality lines that are approximately equally spaced
characterize this black oil phase diagram. Following the pressure
reduction path, as indicated by the vertical line EF in Fig. 1.19, the
liquid-shrinkage curve, shown in Fig. 1.20, is prepared by plotting the
38 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
Ordinary black oil
Quality lines are evenly spaced
Liquid phase
4000
1
Reservoir pressure
decline path
E
3500
Gas phase
00%
1
80%
rv
e
C
%
cu
50
po
int
2500
%
25
F
Bu
bb
le
2000
rve
cu
int d
po iqui
w
L
De 0%
Pressure (psia)
3000
1500
A
1000
Two-phase region
E
Separator
500
0%
0
–100
B
0
100
200
300
400
Temperature (°F)
500
600
700 Tct 800
(Tc)mixture
n FIGURE 1.19 A typical p-T diagram for an ordinary black oil.
100%
Liquid volume
E
F
Residual oil
0%
Pressure
n FIGURE 1.20 Liquid-shrinkage curve for black oil.
Bubble point pressure pb
Classification of Reservoirs and Reservoir Fluids 39
liquid volume percent as a function of pressure. The liquid-shrinkage
curve approximates a straight line except at very low pressures. When
produced, ordinary black oils usually yield GORs between 200 and 700
scf/STB and oil gravities of 15–40 API. The stock-tank oil usually is
brown to dark green in color.
2. Low-shrinkage oil:
A typical p-T phase diagram for low-shrinkage oil is shown in Fig. 1.21.
The diagram is characterized by quality lines that are closely spaced near
the dew point curve. The liquid-shrinkage curve, given in Fig. 1.22,
shows the shrinkage characteristics of this category of crude oils. The
other associated properties of this type of crude oil are
n
Oil formation volume factor less than 1.2 bbl/STB.
n
GOR less than 200 scf/STB.
n
Oil gravity less than 35° API.
n
Black or deeply colored.
n
Substantial liquid recovery at separator conditions as indicated by
point G on the 85% quality line of Fig. 1.22.
3. Volatile crude oil:
The phase diagram for a volatile (high-shrinkage) crude oil is given
in Fig. 1.23. Note that the quality lines are close together near the bubble
point, and at lower pressures they are more widely spaced. This type
of crude oil is commonly characterized by a high liquid shrinkage
Quality lines are distant from the bubble point curve
x
Reservoir pressure
XE
decline path
0%
10
Liquid
Gas
Two-phase
region
C
Pressure
%
95
%
85
%
A
70
xF
%
50
Separator
0%
B
Temperature
n FIGURE 1.21 A typical phase diagram for a low-shrinkage oil.
40 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
100%
E
Liquid volume
F
Residual oil
0%
Pressure
Bubble point pressure pb
n FIGURE 1.22 Oil-shrinkage curve for low-shrinkage oil.
n FIGURE 1.23 A typical p-T diagram for a volatile crude oil.
immediately below the bubble point, shown in Fig. 1.24. The other
characteristic properties of this oil include
n
Oil formation volume factor greater than 1.5 bbl/STB.
n
GORs between 2000 and 3000 scf/STB.
n
Oil gravities between 45° and 55° API.
n
Lower liquid recovery of separator conditions, as indicated by point
G in Fig. 1.23.
n
Greenish to orange in color.
Classification of Reservoirs and Reservoir Fluids 41
Liquid volume
100%
E
F
Residual oil
0%
Pressure
Bubble point pressure pb
n FIGURE 1.24 A typical liquid-shrinkage curve for a volatile crude oil.
Solution gas released from a volatile oil contains significant quantities
of stock-tank liquid (condensate) when the solution gas is produced at
the surface. Solution gas from black oils usually is considered “dry,”
yielding insignificant stock-tank liquid when produced to surface
conditions. For engineering calculations, the liquid content of released
solution gas perhaps is the most important distinction between
volatile oils and black oils. Another characteristic of volatile oil
reservoirs is that the API gravity of the stock-tank liquid increases in
the later life of the reservoirs.
4. Near-critical crude oil:
If the reservoir temperature, T, is near the critical temperature, Tc, of the
hydrocarbon system, as shown in Fig. 1.25, the hydrocarbon mixture is
identified as a near-critical crude oil. Because all the quality lines
converge at the critical point, an isothermal pressure drop (as shown by
the vertical line EF in Fig. 1.25) may shrink the crude oil from 100% of
the hydrocarbon pore volume at the bubble point to 55% or less at a
pressure 10–50 psi below the bubble point. The shrinkage characteristic
behavior of the near-critical crude oil is shown in Fig. 1.26. This high
shrinkage creates high gas saturation in the pore space and because
of the gas-oil relative permeability characteristics of most reservoir
rocks; free gas achieves high mobility almost immediately below the
bubble point pressure.
The near-critical crude oil is characterized by a high GOR, in excess
of 3000 scf/STB, with an oil formation volume factor of 2.0 bbl/STB or
42 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
4000
Near critical oil
0%
liq
10
E
cu
Two-phase region
int
2500
%
80
po
%
50
Bu
bb
le
2000
C
rve
cu
int d
po iqui
l
w
De 0%
rve
3000
Pressure (psia)
Reservoir pressure
decline path
d
ui
3500
1500
%
25
A
F
1000
500
Separator
0%
B
0
–100
–
0
Temperature (ºF)
n FIGURE 1.25 A schematic phase diagram for the near-critical crude oil.
Liquid volume
100%
E
≥ 50%
F
Residual oil
0%
Pressure
n FIGURE 1.26 A typical liquid-shrinkage curve for the near-critical crude oil.
Bubble point pressure pb
Classification of Reservoirs and Reservoir Fluids 43
How near are the quality lines to the
bubblepoint curve will determine the type of crude oil
100%
E
ge oil
-shrinka
(A) Low
Liquid volume
ary
din
) Or
k oil
balc
(B
il
age o
hrink
igh-s
(C) H
ear
(D) N
al oil
critic
0%
Pressure
Bubble point pressure pb
n FIGURE 1.27 Liquid shrinkage for crude oil systems.
higher. The compositions of near-critical oils usually are characterized
by 12.5–20 mol% heptanes-plus, 35% or more of ethane through
hexanes, and the remainder methane. It should be pointed out that nearcritical oil systems essentially are considered the borderline to very rich
gas-condensates on the phase diagram.
Fig. 1.27 compares the characteristic shape of the liquid-shrinkage curve for
each crude oil type.
Gas Reservoirs
In general, if the reservoir temperature is above the critical temperature of
the hydrocarbon system, the reservoir is classified as a natural gas reservoir.
Natural gases can be categorized on the basis of their phase diagram and the
prevailing reservoir condition into four categories:
1. Retrograde gas reservoirs.
2. Near-critical gas-condensate reservoirs.
3. Wet gas reservoirs.
4. Dry gas reservoirs.
In some cases, when condensate (stock-tank liquid) is recovered from a surface process facility, the reservoir is mistakenly classified as a retrograde
gas reservoir. Strictly speaking, the definition of a retrograde gas reservoir
depends only on reservoir temperature.
44 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
Retrograde Gas Reservoirs
If the reservoir temperature, T, lies between the critical temperature, Tc, and
cricondentherm, Tct, of the reservoir fluid, the reservoir is classified as a retrograde gas-condensate reservoir. This category of gas reservoir has a
unique type of hydrocarbon accumulation, in that the special thermodynamic behavior of the reservoir fluid is the controlling factor in the development and the depletion process of the reservoir. When the pressure is
decreased on these mixtures, instead of expanding (if a gas) or vaporizing
(if a liquid) as might be expected, they vaporize instead of condensing.
Consider that the initial condition of a retrograde gas reservoir is represented
by point 1 on the pressure-temperature phase diagram of Fig. 1.28. Because
the reservoir pressure is above the upper dew point pressure, the hydrocarbon system exists as a single phase (ie, vapor phase) in the reservoir. As the
reservoir pressure declines isothermally during production from the initial
pressure (point 1) to the upper dew point pressure (point 2), the attraction
between the molecules of the light and heavy components move farther
apart. As this occurs, attraction between the heavy component molecules
becomes more effective; therefore, liquid begins to condense. This retrograde condensation process continues with decreasing pressure until the
liquid dropout (LDO) reaches its maximum at point 3. Further reduction
in pressure permits the heavy molecules to commence the normal vaporization process. This is the process whereby fewer gas molecules
strike the liquid surface and more molecules leave than enter the liquid
phase. The vaporization process continues until the reservoir pressure
n FIGURE 1.28 A typical phase diagram of a retrograde system.
Classification of Reservoirs and Reservoir Fluids 45
n FIGURE 1.29 A typical liquid dropout curve.
reaches the lower dew point pressure. This means that all the liquid that
formed must vaporize because the system essentially is all vapor at the lower
dew point.
Fig. 1.29 shows a typical liquid-shrinkage volume curve for a relatively rich
condensate system. The curve is commonly called the liquid dropout curve.
The maximum LDO is 26.5%, which occurs when the reservoir pressure
drops from a dew point pressure of 5900–2800 psi. In most gas-condensate
reservoirs, the condensed liquid volume seldom exceeds more than 15–19%
of the pore volume. This liquid saturation is not large enough to allow any
liquid flow. It should be recognized, however, that around the well bore,
where the pressure drop is high, enough LDO might accumulate to give
two-phase flow of gas and retrograde liquid. The associated physical characteristics of this category are
n
n
n
GORs between 8000 and 70,000 scf/STB. Generally, the GOR for a
condensate system increases with time due to the LDO and the loss of
heavy components in the liquid.
Condensate gravity above 50° API.
Stock-tank liquid is usually water-white or slightly colored.
It should be pointed out that the gas that comes out of the solution from
a volatile oil and remains in the reservoir typically is classified a
retrograde gas and exhibits the retrograde condensate with pressure
declines.
46 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
There is a fairly sharp dividing line between oils and condensates from
a compositional standpoint. Reservoir fluids that contain heptanes and are in
concentration of more than 12.5 mol% almost always are in the liquid phase
in the reservoir. Oils have been observed with heptanes and heavier concentrations as low as 10% and condensates as high as 15.5%. These cases are rare,
however, and usually have very high tank liquid gravities.
Near-Critical Gas-Condensate Reservoirs
If the reservoir temperature is near the critical temperature, as shown in
Fig. 1.30, the hydrocarbon mixture is classified as a near-critical gascondensate. The volumetric behavior of this category of natural gas is
described through the isothermal pressure declines, as shown by the vertical
line 1–3 in Fig. 1.30 and the corresponding LDO curve of Fig. 1.31. Because
all the quality lines converge at the critical point as shown in Fig. 1.31, a
rapid liquid buildup immediately occurs below the dew point as the pressure
is reduced to point 2.
This behavior can be justified by the fact that several quality lines are
crossed very rapidly by the isothermal reduction in pressure. At the point
where the liquid ceases to build up and begins to shrink again, the reservoir
goes from the retrograde region to a normal vaporization region.
Liquid phase
Gas phase
Reservoir pressure
e
urv
nt c
decline path
poi
D
Pressure
1
Critical point “C”
e
v
ur
tc
0%
ew
20%
50%
0%
2
n
oi
le
p
ve
Quality lines
b
ub
ur
tc
B
n
oi
w
3
De
Temperature
n FIGURE 1.30 A typical phase diagram for a near-critical gas-condensate reservoir.
p
Classification of Reservoirs and Reservoir Fluids 47
Liquid volume
100%
≈50%
3
2
ropout
d
Liquid
curve
1
0%
Pressure
n FIGURE 1.31 Liquid-shrinkage (dropout) curve for a near-critical gas-condensate system.
Wet Gas Reservoirs
A typical phase diagram of a wet gas is shown in Fig. 1.32, where the
reservoir temperature is above the cricondentherm of the hydrocarbon mixture. Because the reservoir temperature exceeds the cricondentherm of the
hydrocarbon system, the reservoir fluid always remains in the vapor phase
region as the reservoir is depleted isothermally, along the vertical line AB.
Liquid phase
Gas phase
Reservoir pressure
decline path at
constant temperature
A
Pressure
Critical point “C”
Two-phase region
Liquid
75
50
25
5
0
Separator
Temperature
n FIGURE 1.32 Phase diagram for a wet gas.
Gas
B
48 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
However, as the produced gas flows to the surface, the pressure and temperature of the gas decline. If the gas enters the two-phase region, a liquid phase
condenses out of the gas and is produced from the surface separators. This is
caused by a sufficient decrease in the kinetic energy of heavy molecules
with temperature drop and their subsequent change to liquid through the
attractive forces between molecules. Wet gas reservoirs are characterized
by the following properties
n
n
n
n
GORs between 60,000 and 100,000 scf/STB.
Stock-tank oil gravity above 60° API.
Liquid is water-white in color.
Separator conditions (ie, separator pressure and temperature) lie within
the two-phase region.
It should be noted that when applying the material balance equation for a
wet gas.
Reservoir, as expressed in terms of cumulative gas production “GP” as a
function of p/Z, the cumulative surface condensate volume “Np STB” or condensate production rate “Qo STB/day” must be converted into an equivalent
gas volume and added to the cumulative separator gas production “GP” or to
the separator gas production rate qgas.
The equivalent gas volume “Veq” as expressed in scf/STB, is an important
property that can be used to convert a liquid volume into an equivalent
gas volume. The working basic equations to develop an expression for
Veq are based on the definition of the number of moles “n” and the gas EOS
Number of moles n:
The number of moles (liquid or gas) is defined as the ratio of mass “m” to
the molecular weight “M” with the fluid mass “m” given by fluid volume
times its density. Applying the definition to the liquid phase at stock-tank
condition, gives
no ¼
mo Vo ρo ð5:615Þð62:4γ o Þ
¼
¼
Mo
Mo
Mo
Or:
no ¼
mo Vo ρo ð5:615Þð62:4γ o Þ 350:376γ o
¼
¼
¼
Mo
Mo
Mo
Mo
where
Vo ¼ liquid volume, STB
Mo ¼ molecular weight of liquid
γ o ¼ liquid specific gravity, 60°/60°
Classification of Reservoirs and Reservoir Fluids 49
Gas EOS:
The gas EOS is expressed by
PV ¼ ZnRT
Solving the EOS for volume “V” at stock-tank conditions, ie, at 14.7 psi and
60°F, with gas constant “R” of the value 10.73, gives
Vsc ¼
Zsc RTn ð1Þð10:73Þð60 + 460Þn
¼
psc
14:7
The above indicates that 1 mol occupies an equivalent volume of 379.4 scf
at standard conditions. This suggests that no moles of liquid can be converted into an equivalent gas volume “Veq” by the expression:
350:376γ o
γ
133,000 o
Veq ¼ 379:4no ¼ 379:4
Mo
Mo
where
Veq ¼ equivalent gas volume in scf/STB
The liquid specific gravity γ o is related to the API gravity by the expression:
γo ¼
141:5
API + 131:5
For condensate and light hydrocarbon systems; the molecular weight
“Mo” can be approximated in terms of the API gravity by the following
relationship:
Mo ¼
6084
API 5:9
Combining the above two expressions and solving for the ratio γ o/Mo, gives
γo
¼ 0:001892 + 7:35 105 API 4:52 108 ðAPIÞ2
Mo
The ratio γ o/Mo in the equivalent gas volume equation can be replaced by the
above expression to give
Veq ¼ 252 + 9:776API 0:006ðAPIÞ2
The following example illustrates the practical use of the equivalent gas
volume in converting liquid volume into equivalent gas volume.
50 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
EXAMPLE 1.8
The following data is available on a wet-shale gas reservoir:
pi ¼ 3200psia T ¼ 200°F Φ ¼ 8% Swi ¼ 30%
Condensate daily flow rate Qo ¼ 400 STB/day
API gravity of the condensate ¼ 50°
Daily HP separator gas rate qgsep ¼ 4.20 MMscf/day
Daily IP separator gas rate qgsep ¼ 1.20 MMscf/day
Average separator gas specific gravity ¼ 0.65
Daily stock-tank gas rate qgst ¼ 0.15 MMscf/day
Based on the configuration of the surface facilities as shown below, calculate
the daily well stream gas flow rate in MMscf/day
4.2 MMscf/day
1.2 MMscf/day
0.15 MMscf/day
Qo= 400 STB/day
Gas
Gas
Solution
Step 1. Calculate the equivalent gas volume “Veq”:
Veq ¼ 252 + 9:776API 0:006ðAPIÞ2
Veq ¼ 252 + ð9:776Þð50Þ 0:006ð50Þ2 ¼ 725:8scf=STB
Step 2. Calculate total well rate “Qg”:
Qg ¼ 4:2 + 1:2 + 0:15 + ð400Þð725:8=1:0e6Þ ¼ 5:840MMscf=day
The equivalent gas volume equation can be further simplified to combine with
the above.
Dry Gas Reservoirs
The hydrocarbon mixture exists as a gas both in the reservoir and the surface
facilities. The only liquid associated with the gas from a dry gas reservoir is
water. Fig. 1.33 is a phase diagram of a dry gas reservoir. Usually, a system
Classification of Reservoirs and Reservoir Fluids 51
Gas phase
Liquid phase
Reservoir pressure decline path
at constant temperature
A
Pressure
Liquid
Critical point “C”
Separator
75 50
25
0
Gas
Temperature
n FIGURE 1.33 Phase diagram for a dry gas.
that has a GOR greater than 100,000 scf/STB is considered to be a dry gas.
The kinetic energy of the mixture is so high and attraction between molecules so small that none of them coalesce to a liquid at stock-tank conditions
of temperature and pressure.
It should be pointed out that the listed classifications of hydrocarbon fluids
might be also characterized by the initial composition of the system. It
should be pointed out that the heavy components (eg, C7+) in the hydrocarbon mixtures have the strongest effect on fluid characteristics. Numerous
laboratory compositional analyses on retrograde and crude oil samples suggests that the phase boundary separating the retrograde gas phase and other
fluid systems is based on the overall mole % of C7+ in the hydrocarbon system. In general, when
Mole % C7+ < 12.5: the hydrocarbon system is classified as gas phase
Mole % C7+ 12.5–20: the hydrocarbon system is classified as near
critical
Mole % C7+ > 20: the hydrocarbon system is classified as oil phase
The ternary diagram shown in Fig. 1.34 with equilateral triangles can be
conveniently used to roughly define the compositional phase boundaries
that separate different types of hydrocarbon systems.
Fluid samples obtained from a new field discovery may be instrumental in
defining the existence of a two-phase, that is, gas-cap system with an overlying gas cap or underlying oil rim. As the compositions of the gas and oil
zones are completely different from each other, both systems may be represented separately by individual phase diagrams, which bear little relation to
B
52 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
C1+N2
A
A: Dry gas
B: Wet gas
C: Retrograde gas
D: Near critical
E: Volatile oil
F: Ordinary oil
G: Low shrinkage oil
D
C
B
E
F
G
C7+
C2-C6+CO2
n FIGURE 1.34 Compositions of various reservoir fluid systems.
each other or to the composite. The oil zone will be at its bubble point and
produced as a saturated oil reservoir but modified by the presence of the gas
cap. Depending on the composition and phase diagram of the gas, the gascap gas may be a retrograde gas cap, as shown in Fig. 1.35, or dry or wet, as
shown in Fig. 1.36. Therefore, a discovery well drilled through a saturated
reservoir fluid usually requires further field delineation to substantiate the
presence of a second equilibrium phase above (ie, gas cap) or below (ie,
oil rim) the tested well. This may entail running a repeat formation tester
(RFT) tool to determine fluid-pressure gradient as a function of depth; or
a new well drilled updip or downdip to that of the discovery well may be
required.
When several samples are collected at various depths, they exhibit PVT
properties as a function of the depth expressed graphically to locate the
GOC. The variations of PVT properties can be expressed graphically in
terms of the compositional changes of C1 and C7+ with depth and in terms
of well-bore and stock-tank densities with depth, as shown in Table 1.3 and
Figs. 1.37–1.39.
Customized plot
4000
Pb
3500
Phase envelope of the
oil rim
At GOC Pb = Pd
3000
Pd
Pressure (psia)
2500
2000
1500
1000
500
0
−100
0
100
200
300
400
500
600
700
Temperature (°F)
n FIGURE 1.35 Retrograde gas-cap reservoir.
Phase envelope of the
gas-cap gas
Pressure
Critical point
C
Gas-oil contact
Phase envelope of the
oil rim
P
C
Critical point
T
Temperature
n FIGURE 1.36 Dry gas-cap reservoir.
800
Table 1.3 Locating the GOC
Depth (ft)
Pressure
(psia)
Temperature (°F)
Phase
Saturation
Pressure (psia)
Molecular
Weight
Density
(lb/ft3)
GOR (scf/STB)
STO Density
(lb/ft3)
Viscosity
(cP)
5200.0
5248.3
5296.6
5344.8
5393.1
5441.4
5489.7
5537.9
5586.2
5634.5
5682.8
5731.0
5779.3
5827.6
5875.9
5924.1
5972.4
6020.7
6034.4
6034.4
6069.0
6117.2
6165.5
6213.8
6262.1
6286.0
6310.3
6358.6
6406.9
6455.2
6503.4
6551.7
6600.0
3586.8
3591.6
3596.5
3601.4
3606.3
3611.1
3616.1
3621.0
3625.9
3630.9
3635.9
3640.9
3645.9
3650.9
3656.0
3661.1
3666.2
3671.4
3672.9
3672.9
3681.2
3693.2
3705.4
3717.9
3730.6
3737.0
3743.5
3756.6
3769.9
3783.4
3796.9
3810.7
3824.5
163.71
164.43
165.16
165.88
166.61
167.33
168.05
168.78
169.50
170.23
170.95
171.68
172.40
173.12
173.85
174.57
175.30
176.02
176.23
176.23
176.74
177.47
178.19
178.92
179.64
180.00
180.37
181.09
181.81
182.54
183.26
183.99
184.71
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Gas
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
Oil
3070.0
3089.9
3110.9
3133.1
3156.6
3181.5
3208.0
3236.1
3266.1
3298.1
3332.4
3369.0
3408.3
3450.6
3496.0
3545.1
3598.1
3655.7
3672.8
3673.0
3640.0
3597.5
3557.9
3520.2
3483.8
3466.2
3448.6
3414.2
3380.5
3347.4
3314.8
3282.7
3251.1
23.0
23.1
23.1
23.2
23.3
23.3
23.4
23.5
23.6
23.7
23.8
23.8
23.9
24.1
24.2
24.3
24.4
24.6
24.6
60.1
62.7
66.3
70.0
73.7
77.5
79.4
81.3
85.1
89.0
92.8
96.7
100.5
104.4
14.4365
14.4714
14.5079
14.5462
14.5865
14.6288
14.6735
14.7207
14.7709
14.8242
14.8812
14.9423
15.0082
15.0796
15.1575
15.2430
15.3379
15.4443
15.4767
34.4996
35.1887
36.0601
36.8474
37.5643
38.2196
38.5237
38.8198
39.3702
39.8754
40.3393
40.7656
41.1576
41.5185
43485.8
42419.5
41365.4
40322.1
39288.6
38263.5
37245.5
36233.1
35224.5
34218.1
33211.7
32203.1
31189.6
30168.0
29134.8
28085.5
27014.6
25914.9
25599.2
1793.4
1656.5
1495.0
1359.2
1243.2
1142.9
1098.1
1055.5
978.8
911.1
851.1
797.6
749.7
706.7
48.2664
48.2697
48.2733
48.2771
48.2814
48.2859
48.2910
48.2964
48.3025
48.3091
48.3164
48.3245
48.3334
48.3434
48.3545
48.3670
48.3810
48.3968
48.4016
49.2422
49.3460
49.4877
49.6253
49.7583
49.8863
49.9478
50.0089
50.1257
50.2367
50.3419
50.4413
50.5352
50.6236
0.0278
0.0279
0.0280
0.0281
0.0281
0.0282
0.0283
0.0284
0.0285
0.0286
0.0287
0.0288
0.0290
0.0291
0.0293
0.0295
0.0296
0.0299
0.0299
0.1404
0.1558
0.1800
0.2076
0.2391
0.2750
0.2946
0.3156
0.3612
0.4119
0.4678
0.5290
0.5954
0.6670
Determination of GOC from pressure gradients
7500.
De
w
po
in
7600.
oir
Reser v
tp
re
s
su
re
g
pressu
7700.
; psi / ft
7800.
ient
Depth/ft
re grad
g
7900.
Sat PRes P
pd = pb = pi
8000.
o
psi / ft
o
8100.
3300
3400
3500
3600
t
in
po e
le ur
bb ss
u
B pre
3700
3800
Pressure (psia)
n FIGURE 1.37 Determination of gas-oil contact from pressure gradients.
Saturation pressure & %C7+
Saturation pressure & %C1
pd
pd
C1
C7+
GOC
pb
Less gas in solution, ie,
less GOR
pb
Depth
n FIGURE 1.38 Compositional changes of C1 and C7+ with depth.
Less gas in solution,
ie, less GOR
Depth
3900
56 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
Stock-tank
density
Depth
GOC
Bottom-hole
density
Density
n FIGURE 1.39 Variation of well-bore and stock-tank density with depth.
Defining the fluid contacts, that is, GOC and water-oil contact (WOC), are
extremely important when determining the hydrocarbon initially in place
and planning field development. The uncertainty in the location of the fluid
contacts can have a significant impact on the reserves estimate. Contacts can
be determined by
1. Electrical logs, such as resistively tools.
2. Pressure measurements, such as a repeat formation tester (RFT) or a
modular formation dynamic tester (MDT).
3. Possibly by interpreting seismic data.
Normally, unless a well penetrates a fluid contact directly, there remains
doubt as to its locations. The RFT is a proprietary name used by
Schlumberger for an open-hole logging tool used to establish vertical
pressure distribution in the reservoir (ie, it provides a pressure-depth
profile in the reservoir) and to obtain fluid samples. At the appraisal
stage of a new field, the RFT survey provides the best-quality pressure
data and routinely is run to establish fluid contacts. The surveys are
usually straightforward to interpret compared to the drill stem tests
(DSTs), because no complex buildup analysis is required to determine
the reservoir pressure nor are any extensive depth corrections to be
applied, as the gauge depth is practically coincidental with that of the
RFT probe.
Classification of Reservoirs and Reservoir Fluids 57
The RFT tool is fitted with a pressure transducer and positioned across the
target zone. The device is placed against the side of the borehole by a packer.
A probe that consists of two pretest chambers, each fitted with a piston, is
pushed against the formation and through the mud cake. A pressure drawdown is created at the probe by withdrawing the pistons in the pretest chambers. These two chambers operate in series: The first piston is withdrawn
slowly (taking about 4 s to withdraw 10 cm3 of fluid); the second piston
is withdrawn at a faster rate, 5 s for 10 cm3. Before the tool is set against
the formation, the pressure transducer records the mud pressure at the
target depth.
As the first pretest chamber is filled (slowly) with fluid, the first main pressure drop Δp1 is observed, followed by Δp2 as the second pretest chamber is
filled. The tighter the formation, the larger Δp1 and Δp2 are. The recording,
therefore, gives a qualitative indication of the permeability of the reservoir.
Because the flow rate and pressure drop are known, the actual formation
permeability can be calculated, and this is done by the CSU (the computer
in the logging unit). Caution should be attached to the permeability values,
because a very small part of the reservoir is being tested and the analysis
assumes a particular flow regime, which may not be representative in some
cases. The reliability of the values should be regarded as indicating the order
of magnitude of the permeability.
Once both drawdowns have occurred, fluid withdrawal stops and the formation pressure is allowed to recover. This recovery is recorded as the pressure
buildup, which should stabilize at the true formation fluid pressure at the
depth. Note that the drawdown pressure drops, Δp1 and Δp2, are relative
to the final pressure buildup and not to the initial pressure, which was the
mud pressure. The rate of pressure buildup can be used to estimate permeability; the test therefore gives three permeability estimates at the same
sample point.
A tight formation leads to very large pressure drawdowns and a slow
buildup. If the pressure has not built up within 4–5 min, the pressure test
usually is abandoned for fear of the tool becoming stuck. The tool is
retracted and the process repeated at a new depth. Once the tool is unset,
the pressure reading should return to the same mud pressure as prior to setting. This is used as a quality control check on gauge drift. There is no limit
as to the number of depths at which pressure samples may be taken.
If a fluid sample is required, this can be done by diverting the fluid flow
during sampling to sample chambers in the tool.
58 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
It should be pointed out that, when plotting the pressure against depth, the
unit of pressures must be kept consistent; that is, either in absolute pressure
(psia) or gauge pressure (psig). The depth must be the true vertical depth,
preferably below the subsurface datum depth.
The basic principle of pressure versus depth plotting is illustrated in
Fig. 1.40. This illustration shows two wells:
n
n
Well 1 penetrates the gas cap with a recorded gas pressure of pg and a
measured gas density of ρg.
Well 2 penetrates the oil zone with a recorded oil pressure of po and oil
density of ρo.
The gas, oil, and water gradients can be calculated from:
dpg ρg
¼
¼ γg
dh 144
dpo ρo
¼
¼ γo
dh 144
dpw ρw
¼
¼ γw
dh 144
Well-1
Pressure
(dpg/dh)gas
Well-2
Known gas
column
GOC
Depth
GOC
Known oil
column
(dpo/dh)oil
WOC
WOC
Oil-down-to
“ODT”
Water gradient
0.45 psi/ft
n FIGURE 1.40 Pressure gradient vs. depth.
Fault
Gas-Oil Contact 59
where
dp/dh ¼ fluid gradient, psi/ft
γ g ¼ gas gradient, psi/ft
γ o ¼ oil gradient, psi/ft
γ w ¼ water gradient, psi/ft
ρg ¼ gas density, lb/ft3
ρo ¼ oil density, lb/ft3
ρw ¼ water density, lb/ft3
GAS-OIL CONTACT
As shown in Fig. 1.40, the intersection of the gas and oil gradient lines
locates the position of the GOC. Note that the oil can be seen only down
to a level termed the oil-down-to (ODT) level. The level is assigned to
the oil-water contact unless a well is drilled further downdip to locate
the WOC.
Fig. 1.41 shows another case where a well penetrated an oil column. Oil can
be seen down to the ODT level. However, a gas cap may exist. If the pressure
of the oil is recorded as po and the bubble point pressure is measured as pb at
this specific depth, the GOC can be determined from
A discovery well penetrates an oil column
what is the possibility of a gas cap updip ?
Gas zone !
GOC ?
o
b
D
o
o
po
Known oil
column
; psi / ft
o
n FIGURE 1.41 A well in the oil rim.
WOC ?
Oil-down-to
“ODT”
60 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
ΔD ¼
po pb
po pb
¼
ðdp=dhÞoil
γo
(1.18)
If the calculated value of ΔD locates the GOC within the reservoir, then
there is a possibility that a gas cap exists, but this is not certain.
Eq. (1.18) assumes that the PVT properties, including the bubble point pressure, do not vary with depth. The only way to be certain that a gas cap exists
is to drill a crestal well.
A similar scenario that can be encountered is illustrated in Fig. 1.42. Here, a
discovery well penetrates a gas column with gas as seen down to the gasdown-to (GDT) level, which allows the possibility of a downdip oil rim.
If the field operator feels that an oil rim exists, based on similarity with other
fields in the area, the distance to GOC can be estimated from
ΔD ¼
pb pg
pb pg
¼
ðdp=dhÞgas
γg
A discovery well penetrates a gas column
what is the possibility of an oil rim?
pg
b
Gas-down-to
“GDT”
g
D
g
GOC ?
g
; psi / ft
g
n FIGURE 1.42 A discovery well in gas column.
Oil zone !
Gas-Oil Contact 61
Well-1
Oil gradient ª 0.33 psi/ft
Known-gas-column
Well-2
Possible GOC
Gas-down-to
Possible oil zone
ΔD
Possible OWC
Possible GWC
b
a
n FIGURE 1.43 Two discovery wells.
Fig. 1.43 shows a case where two wells are drilled: The first well is in the gas
column and the second penetrates the water zone. Of course, only two possibilities could exist:
1. The first possibility is shown in Fig. 1.43, which suggests a known gas
column with an underlying water zone. Also, the gas-water contact
(GWC) has not been established. A “possible” GWC can be calculated
from the following relationship:
D1 ¼
pg pw ΔDðdp=dhÞwater
pg ½pw ΔDγ water ¼
γ water γ gas
ðdp=dhÞwater ðdp=dhÞgas
γ ¼ fluid gradient, psi/ft, that is, ρ/144
ρ ¼ fluid density, lb/ft3
D1 ¼ vertical distance between gas well and GWC, ft
ΔD ¼ vertical distance between the two wells
dp/dh ¼ γ ¼ fluid gradient, psi/ft
62 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
2. The second possibility is shown in Fig. 1.43, where a possible oil
zone exists between the two wells. Assuming a range of oil gradients,
such as 0.28–0.38 psi/ft, the maximum possible oil thickness can be
estimated graphically as shown in Fig. 1.45B, by drawing a line
originating at the GDT level with a slope in the range of 0.28–0.38 psi/ft
to intersect with the water gradient line.
The illustration in Fig. 1.44 shows again two discovery wells; one well
penetrated the gas cap and the other well penetrated the water zone, with
a vertical distance between the two wells of ΔD. If the bubble point pressure is known or assumed, the maximum oil column thickness as well as
the depth to different contacts can be estimated, from the following
expressions:
Possible depth to GOC “D1”
ΔD1 ¼
pb pg
@pg =@D g
Possible depth to WOC “D2”
ΔD3 ¼
ðpw pb Þ ½ð@po =@DÞðΔD ΔD1 Þ
½ð@pw =@DÞ ð@po =@DÞ
o
; psi / ft
g
; psi / ft
w
; psi / ft
o
Two discovery wells
Maximum Possible oil thickness
g
w
Gas
DD1
g
g
g
DD
DD3
w
b
o
w
n FIGURE 1.44 Two discovery wells — maximum possible oil thickness.
b
o
Gas-Oil Contact 63
Possible maximum oil zone thickness “D2”
ΔDðdpw =dhÞ ðpw pb Þ pb pg ðdpw =dhÞ= dpg =dh
ΔD2 ¼
ðdpw =dhÞ ðdpo =dhÞ
ΔDγ w ðpw pb Þ pb pg γ w =γ g
¼
γw γo
Or simply
ΔD2 ¼ ΔD ΔD1 ΔD3
where
ΔD2 ¼ maximum possible oil column thickness, ft
ΔD ¼ vertical distance between the two wells, ft
dp/dh ¼ fluid-pressure gradient, psi/ft
EXAMPLE 1.9
Fig. 1.45 shows two wells, one penetrated the gas zone at 4950 ft true vertical depth
(TVD) and the other in the water zone at 5150 ft. Pressures and PVT data follow:
pg ¼ 1745psi
ρg ¼ 14:4lb=ft3
pw ¼ 1808psi
ρw ¼ 57:6lb=ft3
A nearby field has a bubble point pressure of 1750 psi and an oil density of
50.4 lb/ft3 (ρo ¼ 50.4 lb/ft3). Calculate
1. Possible depth to GOC.
2. Maximum possible oil column thickness.
3. Possible depth to WOC.
Solution
Step 1. Calculate the pressure gradient of each phase:
ρg 14:4
dp
¼
¼
¼ 0:1psi=ft
Gas phase
dh gas 144 144
dp
ρ
50:4
Oil phase
¼ o ¼
¼ 0:35psi=ft
dh oil 144 144
dp
ρ
57:6
¼ w ¼
¼ 0:4psi=ft
Water phase
dh water 144 144
Step 2. Calculate the distance to the possible GOC:
ΔD1 ¼
pb pg
1750 1745
¼
¼ 50ft
0:1
@pg =@D g
GOC ¼ 4950 + 50 ¼ 5000ft
64 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
o
; psi / ft
g
; psi / ft
w
; psi / ft
o
g
w
Gas
zone
DD1
DD2
Possible oil rim
DD3
n FIGURE 1.45 Illustration for Example 1.9.
Step 3. Possible depth to WOC:
ΔD3 ¼
ðpw pb Þ ð@po =@DÞðΔD ΔD1 ð1808 1750Þ 0:35ð200 50Þ
¼
½ð@pw =@DÞ ð@po =@DÞ
0:4 0:35
¼ 110ft
WOC ¼ 5150 110 ¼ 5040ft
Step 4. Calculate possible maximum oil rim thickness:
ΔDγ w ðpw pb Þ pb pg γ w =γ g
ΔD2 ¼
γw γo
ΔD2 ¼ maximum oil thickness
¼
200ð0:4Þ ð1808 1750Þ ð1750 1745Þð0:4=0:1Þ
¼ 40ft
ð0:4Þ ð0:35Þ
or simply as
ΔD2 ¼ ΔD ΔD2 ΔD3 ¼ 200 50 110 ¼ 40ft
Phase Rule 65
UNDERSATURATED GOC
For gas-cap reservoirs with significant composition and temperature variations with depth, the compositional variations with depth could result in
changes in the type of hydrocarbon systems ranging from retrograde gas
at the top of the hydrocarbon column to hydrocarbon liquid with variable
volatility. In some cases, the transition from the dew point gas to the bubble
point oil occurs without an existing gas cap. A possible explanation for this
occurrence, as illustrated schematically in Fig. 1.46, is attributed to the possibility that the temperature gradient coincided with the hydrocarbon critical
temperature in the transition between gas and liquid system to form a critical
mixture. This critical mixture separated the gas and liquid without forming a
distinct GOC; labeled as undersaturated GOC
PHASE RULE
It is appropriate at this stage to introduce and define the concept of the phase
rule. Gibbs (1948 [1876]) derived a simple relationship between the number
of phases, P, in equilibrium, the number of components, C, and the number
Pressure & temperature
Gas cap
Critical region
Depth
Oil rim
rvoir
Rese
n
bo s
ar int
c
o
o
dr al p
Hy itic
cr
re
eratu
temp
n FIGURE 1.46 Undersaturated gas-oil contact.
66 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
of independent variables, F, that must be specified to describe the state of the
system completely.
Gibbs proposed the following fundamental statement of the phase rule:
F¼CP+2
(1.19)
where
F ¼ number of variables required to determine the state of the system at
equilibrium or number of degrees of freedom (such as pressure,
temperature, density)
C ¼ number of independent components
P ¼ number of phases
A phase was defined as a homogeneous system of uniform physical and
chemical compositions. The degrees of freedom, F, for a system include
the intensive properties such as temperature, pressure, density, and
composition (concentration) of phases. These independent variables must
be specified to define the system completely. In a single component
(C ¼ 1), two-phase system (P ¼ 2), there is only one degree of freedom
(F ¼ 1 2 + 2 ¼ 1); therefore, only pressure or temperature needs to be
specified to determine the thermodynamic state of the system.
The phase rule as described by Eq. (1.19) is useful in several ways. It indicates the maximum number of equilibrium phases that can coexist and the
number of components present. It should be pointed out that the phase rule
does not determine the nature, exact composition, or total quantity of the
phases. Furthermore, it applies only to a system in stable equilibrium and
does not determine the rate at which this equilibrium is attained.
The importance and the practical application of the phase rule are illustrated
through the following examples.
EXAMPLE 1.9
For a single-component system, determine the number of degrees of freedom
required for the system to exist in the single-phase region.
Solution
Applying Eq. (1.19) gives F ¼ 1 1 + 2 ¼ 2. Two degrees of freedom must be
specified for the system to exist in the single-phase region. These must be the
pressure p and the temperature T.
Problems 67
EXAMPLE 1.10
What degrees of freedom are allowed for a two-component system in two
phases?
Solution
Because C ¼ 2 and P ¼ 2, applying Eq. (1.19) yields F ¼ 2 2 + 2 ¼ 2. The
two degrees of freedom could be the system pressure p and the system
temperature T, and the concentration (mole fraction), or some other
combination of T, p, and composition.
EXAMPLE 1.11
For a three-component system, determine the number of degrees of
freedom that must be specified for the system to exist in the one-phase
region.
Solution
Using the phase rule expression gives F ¼ 3 1 + 2 ¼ 4, four independent
variables must be specified to fix the system. The variables could be
the pressure, temperature, and mole fractions of two of the three
components.
From the foregoing discussion it can be observed that hydrocarbon
mixtures may exist in either the gas or liquid state, depending on the reservoir and operating conditions to which they are subjected. The qualitative
concepts presented may be of aid in developing quantitative analyses.
Empirical equations of state are commonly used as a quantitative tool in
describing and classifying the hydrocarbon system. These equations of state
require
n
n
Detailed compositional analysis of the hydrocarbon system.
Complete description of the physical and critical properties of the
mixture of the individual components.
PROBLEMS
1. The following is a list of the compositional analysis of different
hydrocarbon systems, with the compositions expressed in terms of
mol%. Classify each of these systems.
68 CHAPTER 1 Fundamentals of Hydrocarbon Phase Behavior
Component
System 1
System 2
System 3
System 4
C1
C2
C3
C4
C5
C6
C7+
68.00
9.68
5.34
3.48
1.78
1.73
9.99
25.07
11.67
9.36
6.00
3.98
3.26
40.66
60.00
8.15
4.85
3.12
1.41
2.47
20.00
12.15
3.10
2.51
2.61
2.78
4.85
72.00
2. A pure component has the following vapor pressure:
T, °F
pv , psi
104
46:09
140
135:04
176
345:19
212
773:75
a. Plot these data to obtain a nearly straight line.
b. Determine the boiling point at 200 psi.
c. Determine the vapor pressure at 250°F.
3. The critical temperature of a pure component is 260°F. The densities of
the liquid and vapor phase at different temperatures are
T, °F
ρL , lb=ft3
ρv , lb=ft3
86
40:280
0:886
122
38:160
1:691
158
35:790
2:402
212
30:890
5:054
Determine the critical density of the substance.
4. Using the Lee and Kesler vapor correlation, calculate the vapor pressure
of i-butane at 100°F. Compare the calculated vapor pressure with that
obtained from the Cox charts.
5. Calculate the saturated liquid density of n-butane at 200°F using
a. The Rackett correlation.
b. The modified Rackett correlation.
Compare the calculated two values with the experimental value
determined from Fig. 1.5.
6. What is the maximum number of phases that can be in equilibrium at
constant temperature and pressure in one-, two-, and threecomponent systems?
7. For a seven-component system, determine the number of degrees of
freedom that must be specified for the system to exist in the
following regions:
a. One-phase region.
b. Two-phase region.
References 69
8. Fig. 1.9 shows the phase diagrams of eight mixtures of methane and
ethane along with the vapor pressure curves of the two components.
Determine
a. Vapor pressure of methane at 160°F.
b. Vapor pressure of ethane at 60°F.
c. Critical pressure and temperature of mixture 7.
d. Cricondenbar and cricondentherm of mixture 7.
e. Upper and lower dew point pressures of mixture 6 at 20°F.
f. The bubble point and dew point pressures of mixture 8 at 60°F.
9. Using Fig. 1.9, prepare and identify the different phase regions of the
pressure-composition diagram for the following temperatures:
a. 120°F.
b. 20°F.
10. Using Fig. 1.9, prepare the temperature-composition diagram
(commonly called the T/X diagram) at the following pressures:
a. 300 psia.
b. 700 psia.
c. 800 psia.
11. Derive equations for the amounts of liquid and vapor present in the
two-phase region when the composition is expressed in terms of
a. Weight fraction of the more volatile component.
b. Weight fraction of the less volatile component.
12. A 20 ft3 capacity tank is evacuated and thermostated at 60°F. Five
pounds of liquid propane is injected. What is the pressure in the tank
and the proportions of liquid and vapor present? Repeat calculations for
100 lb of propane injected.
REFERENCES
Burcik, E., 1957. Properties of Petroleum Reservoir Fluids. International Human
Resources Development Corporation (IHRDC), Boston, MA.
Gibbs, J.W., 1948. The Collected Works of J. Willard Gibbs (Connecticut Academy of
Arts and Sciences, Trans.). vol. 1. Yale University Press, New Haven, CT (original text
published 1876).
Lee, B.I., Kesler, M.G., 1975. A generalized thermodynamics correlation based on threeparameter corresponding states. AIChE J. 21 (3), 510–527.
Pitzer, K.S., 1955. The volumetric and thermodynamics properties of fluids. J. Am. Chem.
Soc. 77 (13), 3427–3433.
Rackett, H.G., 1970. Equation of state for saturated liquids. J. Chem. Eng. Data 15 (4),
514–517.
Spencer, F.F., Danner, R.P., 1973. Prediction of bubblepoint density of mixtures. J. Chem.
Eng. Data 18 (2), 230–234.
Yamada, T., Gunn, R., 1973. Saturated liquid molar volumes: the Rackett equation.
J. Chem. Eng. Data 18 (2), 234–236.
Chapter
2
Characterizing Hydrocarbon-Plus Fractions
Reservoir fluids contain a variety of substances of different chemical structure
that include hydrocarbon and nonhydrocarbon components. Because of
the enormous number components that constitute the hydrocarbon system,
a complete chemical composition as well as identification of various
components that constitute the hydrocarbon fluid are largely speculative.
The constituents of a natural reservoir fluid form hydrocarbon spectrum from
the lightest one; eg, nitrogen or methane, through intermediate to very large
molecular weight fractions. The relative proportion of these different parts of
the fluid can vary over a wide range which results in petroleum fluids showing
very different features: eg, dry gas, retrograde, volatile oil, and others.
CRUDE OIL ASSAY
Each crude oil has unique molecular and chemical characteristics. These
variances translate into crucial differences in crude oil quality. A crude
oil assay is essentially the evaluation of a crude oil by physical and chemical
testing. It provides data that gives information on the suitability of the crude
for its end use and also helps predict the commercial value of the crude. In
addition, Information obtained from the petroleum assay is used for detailed
equations of state tuning and validation. Crude oil bulk properties are useful
for initial screening and tentative identification of genetically related oils.
These bulk properties include several laboratory measurements that can
be used to quantify the quality of the crude oil. A brief discussion of these
properties is given below:
n
The American Petroleum Institute (API) gravity: The API gravity is a
standard industry property that is designed to quantify the quality of
the crude oil. This oil property is defined in a manner, as given below,
that suggests a higher value indicates a light oil and a lower value
indicates a heavy oil.
API ¼
141:5
131:5
γ
where γ is the liquid specific gravity.
Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00002-7
Copyright # 2016 Elsevier Inc. All rights reserved.
71
72 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
n
n
n
n
n
It should be noted that the API gravity of water is 10 because the water
specific gravity is 1, which suggests that the API gravity for very heavy
crudes could be less than 10 and varies between 10 and 30 for heavy
crudes, to between 30 and 40 for medium crudes, and to above 40 for
light crudes.
The cloud point: The cloud point is defined as the lowest temperature
at which wax crystals begin to form by a gradual cooling under standard
conditions. At this temperature the oil becomes cloudy and the first
particles of wax crystals are observed. Low cloud point products are
desirable under low-temperature conditions. Cloud points are measured
for oils that contain paraffins in the form of wax and therefore for light
fractions (naphtha or gasoline) no cloud point data are reported. Wax
crystals can cause severe flow assurance problems including plugging
flow lines and reduction in well flow potential.
The Reid vapor pressure (RVP): The RVP is the measure of vapor
pressure of lightweight hydrocarbons present above the crude oil
surface, the higher the value of RVP, the higher the percentage of light
hydrocarbons in the crude oil samples.
The pour point: The pour point of a petroleum fraction is defined as
the lowest temperature at which the oil can be stored and still be capable
of flowing through a pipeline without stirring.
The sulfur content: The sulfur content in natural gas is usually found
in the form of hydrogen sulfide (H2S). Some natural gas contains H2S
as high as 30% by volume. The amount of sulfur in a crude is expressed
as a percentage of sulfur by weight, and varies from 0.1% to 6%.
Crude oils with less than 1 wt.% sulfur are called low-sulfur or sweet
crude oil, and those with more than 1 wt.% sulfur are called high-sulfur
or sour crude oil.
The flash point: The flash point of a liquid hydrocarbon or any of its
derivatives is defined as the lowest temperature at which sufficient vapor is
produced above the liquid to form a mixture with air that can produce a
spontaneous ignition if a spark is present. Flash point is an important
parameter for safety considerations, especially during storage and
transportation of volatile petroleum products (ie, LPG, light naphtha,
gasoline). The surrounding temperature around a storage tank should always
be less than the flash point of the fuel to avoid the possibility of ignition.
The refractive index (RI): The RI represents the ratio of the velocity of
light in a vacuum to that in the oil. The property is considered as the
degree to which light bends (refraction) when passing through a
medium. Values of RI can be measured very accurately and are used to
correlate density and other properties of hydrocarbons with high
reliability. Information obtained from RI measurements can be used to
Crude Oil Assay 73
n
n
n
n
n
n
n
determine the behavior of asphaltenes; specifically, their tendency to
precipitate from crude oil.
The freeze point: The freeze point is defined as the temperature at which
the hydrocarbon liquid solidifies at atmospheric pressure.
The smoke point: The smoke points is a property that is designed to
describe the tendency of a fuel to burn with a smoky flame. A higher
amount of aromatics in a fuel causes a smoky characteristic for the flame
and energy loss due to thermal radiation. This property refers to the
height of a smokeless flame of fuel as expressed in millimeters beyond
which smoking takes place; ie, a high smoke point indicates a fuel with
low amount of aromatics.
The aniline point: This property represents the minimum temperature
for complete miscibility of equal volumes of aniline and petroleum
oil. The aniline point is important in characterization of petroleum
fractions and analysis of molecular type.
The Conradson carbon residue (CCR): This property measures the
tendency and ability of the crude oil to form coke. It is determined
by destructive distillation of a sample to the remaining coke residue.
The CCR property is expressed as the weight of the remaining coke
residue as a percentage of the original sample.
The acid number: The acid number is a property that determines the
organic acidity of a refinery stream.
The gross heat of combustion (high heating value): This property is a
measure of the amount of heat produced by the complete burning of a
unit quantity of fuel.
The net heat of combustion (lower heating value): This is obtained by
subtracting the latent heat of vaporization of the water vapor formed by
the combustion from the gross heat of combustion or higher heating value.
There are different ways to classify the components of the reservoir fluids.
Normally, the constituents of a hydrocarbon system are classified into the
following two categories:
n
n
Well-defined petroleum fractions
Undefined petroleum fractions
Well-Defined Components
Pure fractions are considered well-defined components because their physical properties were measured and compiled over the years. These properties
include specific gravity, normal boiling point, molecular weight, critical
properties, and acentric factor. Well-defined fractions are grouped in the
following three sets:
74 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
n
n
Nonhydrocarbon fractions that include CO2, N2, H2S, etc.
Methane C1 through normal pentane n-C5.
Hydrocarbon components that include hexanes, heptanes, and other
heavier components, (ie, C6, C7, C8, C9, etc.) where the number of
isomers rises exponentially. For example, Hexanes “C6” refers to
the hexane petroleum hydrocarbon distillation fraction that contains
a high proportion of n-hexane in addition to 2-methylpentane,
3-methylpentane, and smaller amounts of other components. Similarly,
the Heptanes refer to the heptane petroleum hydrocarbon fraction which
contains isomers of the empirical formula C7H16 with the most prevalent
components being n-heptane, dimethylcyclopentanes, 3-ethylpentane,
methylcyclohexane, and 3-methylhexane. However, the identification
and characterizing of each one of these components for us in EOS
applications can be a difficult task. Therefore, these components are
expressed as a group that consists of a number of components that have
boiling points within a certain range. These groups are commonly
denoted as single carbon number (SCN) and called pseudofractions
because they represent a group of components; eg, C6 group, C7 group,
and so forth. The tabulated laboratory data shown in Table 2.1 illustrates
both the concept of SCN and pseudofractions.
Katz and Firoozabadi (1978) presented a generalized set of physical properties for the hydrocarbon groups C6 through C45 that are expressed as a
SCN, such as the C6-group, C7-group, C8-group, and so on. These generalized properties, as shown in Table 2.2, were generated by analyzing the
physical properties of 26 condensates and crude oil systems. The tabulated
properties of these groups include the average boiling point, specific gravity,
and molecular weight as well as critical properties.
Ahmed (1985) conveniently correlated Katz and Firoozabadi’s tabulated
physical properties with SCN as represented by the number of carbon atoms
“n” by using a regression model. The generalized equation has the following
form:
θ ¼ a1 + a2 n + a3 n2 + a4 n3 +
a5
n
where
θ ¼ any physical property, such as Tc, pc, or Vc
n ¼ effective number of carbon atoms of SCN group, eg, 6, 7, …, 45
a1–a5 ¼ coefficients of the equation as tabulated in Table 2.3
Crude Oil Assay 75
Table 2.1 The Concept of Single Carbon Number
Wellstream
Single carbon
number group
“SCN”
C6 group
C7 group
C8 group
C9 group
C10 group
*
Component
Mole %
GPM*
Hydrogen sulfide
Nitrogen
Carbon dioxide
Methane
Ethane
Propane
Iso-butane
N-Butane
2,2-Dimethylpropane
Iso-pentane
N-Pentane
2,2-Dimethylbutane
Cyclopentane
2,3-Dimethylbutane
2-Methylpentane
3-Methylpentane
Other hexanes
n-Hexane
Methylcyclopentane
Benzene
Cyclohexane
2-Methylhexane
3-Methylhexane
2,2,4-Trimethylpentane
Other heptanes
n-Heptane
Methylcyclohexane
Toluene
Other C-8s
n-Octane
Ethylbenzene
m- and p-Xylene
o-Xylene
Other C-9s
n-Nonane
Other C-10s
n-Decane
Undecanes plus
Total
0.002
0.084
1.163
65.410
12.107
5.329
1.280
2.039
0.012
0.899
0.902
0.031
0.006
0.060
0.370
0.221
0.000
0.560
0.098
0.066
0.144
0.235
0.211
0.000
0.213
0.471
0.329
0.286
0.815
0.388
0.088
0.354
0.113
0.547
0.299
0.742
0.247
3.880
100.000
0.000
0.000
0.000
0.000
3.220
1.459
0.416
0.639
0.004
0.327
0.325
0.013
0.002
0.025
0.153
0.090
0.000
0.229
0.035
0.018
0.049
0.109
0.096
0.000
0.092
0.216
0.132
0.095
0.380
0.198
0.034
0.136
0.043
0.285
0.167
0.425
0.151
3.131
12.692
Gallon-per-thousand cubic feet
76 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Table 2.2 Generalized Physical Properties
Group SCN
Tb (°R)
γ
Kw
M
Tc (°R)
Pc (psia)
ω
Vc (ft3/lb)
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
C29
C30
C31
C30
C33
C34
C35
C36
C37
C38
C39
C40
C41
C42
C43
C44
607
658
702
748
791
829
867
901
936
971
1002
1032
1055
1077
1101
1124
1146
1167
1187
1207
1226
1244
1262
1277
1294
1310
1326
1341
1355
1368
1382
1394
1407
1419
1432
1442
1453
1464
1477
0.69
0.727
0.749
0.768
0.782
0.793
0.804
0.815
0.826
0.836
0.843
0.851
0.856
0.861
0.866
0.871
0.876
0.881
0.885
0.888
0.892
0.896
0.899
0.902
0.905
0.909
0.912
0.915
0.917
0.92
0.922
0.925
0.927
0.929
0.931
0.933
0.934
0.936
0.938
12.27
11.96
11.87
11.82
11.83
11.85
11.86
11.85
11.84
11.84
11.87
11.87
11.89
11.91
11.92
11.94
11.95
11.95
11.96
11.99
12.00
12.00
12.02
12.03
12.04
12.04
12.05
12.05
12.07
12.07
12.08
12.08
12.09
12.10
12.11
12.11
12.13
12.13
12.14
84
96
107
121
134
147
161
175
190
206
222
237
251
263
275
291
300
312
324
337
349
360
372
382
394
404
415
426
437
445
456
464
475
484
495
502
512
521
531
914
976
1027
1077
1120
1158
1195
1228
1261
1294
1321
1349
1369
1388
1408
1428
1447
1466
1482
1498
1515
1531
1545
1559
1571
1584
1596
1608
1618
1630
1640
1650
1661
1671
1681
1690
1697
1706
1716
476
457
428
397
367
341
318
301
284
268
253
240
230
221
212
203
195
188
182
175
168
163
157
152
149
145
141
138
135
131
128
126
122
119
116
114
112
109
107
0.271
0.310
0.349
0.392
0.437
0.479
0.523
0.561
0.601
0.644
0.684
0.723
0.754
0.784
0.816
0.849
0.879
0.909
0.936
0.965
0.992
1.019
1.044
1.065
1.084
1.104
1.122
1.141
1.157
1.175
1.192
1.207
1.226
1.242
1.258
1.272
1.287
1.300
1.316
5.6
6.2
6.9
7.7
8.6
9.4
10.2
10.9
11.7
12.5
13.3
14
14.6
15.2
15.9
16.5
17.1
17.7
18.3
18.9
19.5
20.1
20.7
21.3
21.7
22.2
22.7
23.1
23.5
24
24.5
24.9
25.4
25.8
26.3
26.7
27.1
27.5
27.9
Source: Permission to publish from the Society of Petroleum Engineers of the AIME. # SPE-AIME.
Group 1. TBP 77
Table 2.3 Coefficients of the Equation
θ
M
Tc (°R)
pc (psia)
Tb (°R)
ω
γ
Vc (ft3/
lbm-mol)
a1
a2
a3
a4
a5
131.11375
24.96156000
0.34079022
2.49411840E-03
468.32575000
926.602244514
39.729362915
0.722461850
0.005519083
1366.431748654
311.2361908
14.6869301
0.3287671
0.0027346
1690.9001135
427.2959078
50.08577848
0.88693418
6.75667E-03
551.2778516
0.31428163
7.80028E-02
1.39205E-03
1.02147E-05
0.991028867
0.86714949
3.4143E-03
2.8396E-05
2.4943E-08
1.1627984
0.232837085
0.974111699
0.009226997
3.63611E-05
0.111351508
Undefined Components
The undefined petroleum fractions are those heavy components lumped
together and identified as the plus fraction; for example, the C7+ fraction.
Nearly all naturally occurring hydrocarbon systems contain a quantity of
heavy fractions that are not well defined and not mixtures of discretely identified components.
A proper description of the physical properties of the plus fractions and other
undefined petroleum fractions in hydrocarbon mixtures is essential in performing reliable phase-behavior calculations and compositional modeling
studies. The proper description of these physical properties require the
use of thermodynamic property prediction models, such as an equation of
state, which in turn requires that one must be able to provide the acentric
factor, critical temperature, and critical pressure for both the defined and
undefined (heavy) fractions in the mixture. The problem of how to adequately characterize these undefined plus fractions in terms of their critical
properties and acentric factors has been long recognized in the petroleum
industry. Typically, undefined components are identified by a true boiling
point (TBP) distillation analysis and characterized by an average normal
boiling point, specific gravity, and molecular weight. In general, the data
available that can be used to characterize the plus fractions are divided into
the following three distinct groups:
n
n
n
TBP
Stimulated distillation by gas chromatography (GC)
The plus fraction characterization methods
A brief discussion of the three groups is given next.
GROUP 1. TBP
Boiling point of a pure petroleum fraction is the temperature at which it
starts boiling and transfers from a liquid state to a vapor state at a pressure
of 1 atm. In other words, it is the temperature at which the vapor pressure is
78 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
equal to atmospheric pressure for the petroleum fraction. There are various
standard distillation methods through which the boiling point curve can be
obtained. In the TBP distillation test, the oil is distilled and the temperature
of the condensing vapor as well as the volume of liquid formed are recorded.
The TBP test data are used to construct a distillation curve of liquid volume
percent distilled versus condensing temperature.
In this distillation process, the Heptanes-plus fraction “C7+” is subjected to a
standardized analytical distillation, first at atmospheric pressure, then in a
vacuum at a pressure of 40 mmHg. Usually the temperature is taken when
the first droplet distills over. Ten fractions (cuts) are then distilled off, the first
one at 50°C and each successive one with a boiling range of 25°C. For each
distillation cut of the C7+, the volume, specific gravity, and molecular weight
are determined. Cuts obtained in this manner are identified by the boiling
point ranges in which they were collected. The condensing temperature of
the vapor at any point in the test will be close to the boiling of the material
condensing at that point. A key result from this distillation test is the boiling
point curve that is defined as the boiling point of the petroleum fraction versus
the percentage volume distilled. The initial boiling point (IBP) is defined
as the temperature at which the first drop of liquid leaves the condenser tube
of the distillation apparatus with the final boiling point (FBP) defined as the
highest temperature recorded in the test. The difference between FBP and
IBP is called boiling point range. It should be pointed out that oil fractions
tend to crack at a temperature of approximately 650°F and one atmosphere.
As a result, when approaching this cracking temperature, the pressure is gradually reduced to as low as 40 mmHG to avoid cracking the sample and
distorting TBP measurements. A typical TBP curve is shown in Fig. 2.1.
It should be noted that each distilled cut includes a large number of components with close boiling points. Based on the data from the TBP distillation,
the C7+ fraction is divided into pseudocomponents or hypothetical components suitable for equations of state modeling application. These fractions
are commonly collected within the temperature range of two consecutive
distillation volumes, as shown in Fig. 2.1. The distilled cut is referred to
as pseudocomponent. Each pseudocomponent, eg, pseudocomponent 1, is
represented by a volumetric portion of the TBP curve with an average
normal boiling point “Tbi.” The average boiling point of each cut is taken
at the mid-volume percent of the cut; that is, between Vi and Vi1. This is
given by the integration
ð Vi
Tb ðV ÞdV
ðTb Þi + ðTb Þi1
Tb1 ¼ Vi1
2
Vi Vi1
Group 1. TBP 79
Pseudocomponent Tbi from TBP curve
Temperature
Vi
Tb1 =
∫V T dv (T ) + (T )
i−1
Vi − Vi−1
≈
b i
b i−1
2
(Tb)i
(Tb)i–1
Pseudocomponent 1
Vi–1
Vi
Volume distilled
n FIGURE 2.1 Typical distillation curve.
where
Vi Vi1 ¼ volumetric cut of the TBP curve associated with
pseudocomponent i
Tbt(V) ¼ TBP temperature at liquid volume percent distilled “V”
The pseudocomponents from the TBP test are identified primarily by the
following three laboratory measured physical properties:
n
n
n
The boiling point “Tb.”
The molecular weight “M,” is measured by the freezing point depression
experiment. In this test, the molecular weight of the distillation cut is
measured by comparing its freezing point with the freezing point of the
pure solvent with a known molecular weight.
The fraction specific gravity “γ i” of each distillation cut, is measured by
using a pycnometer or electronic densitometer.
The mass (mi) of each distillation cut is measured directly during the
TBP analysis. The cut is quantified in terms of number of moles ni, which is
given as the ratio of the measured mass (mi) to the molecular weight (Mi); ie,
ni ¼
mi
Mi
The density (ρi) and specific gravity (γ i) is then calculated from the
measured mass and volume of the distillation cut as
ρi ¼
mi
Vi
80 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Or in terms of the specific gravity as
γi ¼
ρi
62:4
The average molecular weight and density of the C7+ from the TBP experiment with N distillation cuts are given by
XN
MC7 + ¼ Xi¼1
N
mi
n
i¼1 i
and
XN
ρC7 + ¼ Xi¼1
N
mi
V
i¼1 i
The calculated molecular weight and density from the distillation test should
be compared with those of the measured C7+ values, any discrepancies
are attributed and reported as lost materials resulting from the distillation
test. Fig. 2.2 shows a typical graphical presentation of the molecular weight,
specific gravity, and the TBP as a function of the weight percent of liquid
vaporized.
The molecular weight, M, specific gravity, γ, and boiling point temperature,
Tb, are considered the key properties that reflect the chemical makeup
Tb,
M
Sp.gr
1
1600
0.9
1400
0.8
1200
Tb
800
M
600
0.6
0.5
0.4
0.3
400
0.2
200
0
0.1
0
20
40
60
80
Mass %
n FIGURE 2.2 TBP, specific gravity, and molecular weight of a stock-tank oil sample.
0
100
Specific gravity
0.7
1000
Group 1. TBP 81
of petroleum fractions. Watson et al. (1935) introduced a widely used
characterization factor, commonly known as the universal oil products or
Watson characterization factor, based on the normal boiling point and
specific gravity. This characterization parameter is given by the following
expression:
1=3
Kw ¼
Tb
γ
(2.1)
where
Kw ¼ Watson characterization factor
Tb ¼ normal boiling point temperature, °R
γ ¼ specific gravity
The average heptanes-plus characterization factor “KC7 + ” can be estimated
from the TBP distillation data by applying the following relationship:
XN h 1=3 i
Tbi =γ i mi
i¼1
K C7 + ¼
XN
m
i¼1 i
where mi is the mass of each individual cut.
The Watson characterization factor “Kw” provides a qualitative measure of
the paraffinicity of a crude oil. In general, Kw varies roughly from 8.5 to 13.5
as follows:
n
n
n
For paraffinic hydrocarbon systems, Kw ranges from 12.5 to 13.5.
For naphthenic hydrocarbon systems, Kw ranges from 11.0 to 12.5.
For aromatic hydrocarbon systems, Kw ranges from 8.5 to 11.0.
The significance of the Watson characterization property is based on the
observation that over reasonable ranges of boiling point, Kw is almost
constant for C7+ distillation cuts. Whitson (1980) suggests that the Watson
factor can be correlated with the molecular weight M and specific gravity γ
as given by the following expression:
0:15178 M
Kw 4:5579 0:84573
γ
(2.2)
Whitson pointed out that Eq. (2.2) becomes less reliable when the molecular
weight “M” of the distillation cut exceeds 250. However; the limitation on
the molecular weight can be removed by expressing the above relationship
in the following modified form:
0:149405 M
Kw 3:33475 0:935227 + 3:099934
γ
(2.2A)
82 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
If the molecular weight and specific gravity of the C7+ are available data, the
Watson characterization factor is estimated from the above expression as
0:149405 M
ðKw ÞC7 + 3:33475 0:935227
+ 3:099934
γ
C7 +
Whitson and Brule (2000) observed that a hydrocarbon mixture with multiple hydrocarbon samples, the ðKw ÞC7 + as correlated with MC7 + and γ C7 +
and expressed by Eq. (2.2) often exhibits a constant value for all PVT samples and distillation cuts. The authors suggest that a plot of molecular weight
versus specific gravity of the plus fractions is useful for checking the consistency of C7+ molecular weight and specific gravity measurements.
Austad (1983) and Whitson and Brule (2000) illustrated this observation
by plotting M versus γ for each C7+ fractions from data on two North Sea
fields. Whitson and Brule’s observations are shown schematically in
Figs. 2.3 and 2.4. The schematic plot of MC7 + versus γ C7 + of the volatile
oil shown in Fig. 2.3 has an average ðKw ÞC7 + ¼ 11:90 0:01 for a range
of molecular weights from 220 to 255. Data for the gas condensate, as shown
schematically in Fig. 2.4, indicate an average ðKw ÞC7 + ¼ 11:99 0:01 for a
range of molecular weights from 135 to 150. Whitson and Brule concluded
n FIGURE 2.3 Volatile oil specific gravity versus molecular weight of C7+.
Group 1. TBP 83
n FIGURE 2.4 Condensate-gas specific gravity versus molecular weight of C7+.
that the high degree of correlation for these two fields suggests accurate
molecular weight measurements by the laboratory. In general, the spread
in ðKw ÞC7 + values exceeds 0.01 when measurements are performed by a
commercial laboratory.
Whitson and Brule stated that, when the characterization factor for a field
can be determined, Eq. (2.8) is useful for checking the consistency of C7+
molecular weight and specific gravity measurements. Significant deviation
in ðKw ÞC7 + , such as 0.03 could indicate possible error in the measured data.
Because molecular weight is more prone to error than determination of
specific gravity, an anomalous ðKw ÞC7 + usually indicates an erroneous
molecular weight measurement.
The implication of assuming a constant Kw, eg, ðKw ÞC7 + , is that it can
be used to estimate “Mi” or “γ i” of a distillation cut if they are not
measured. Assuming a constant Kw for each fraction, Eq. (2.2A) is rearranged to solve for the molecular weight “Mi” and specific gravity “γ i”,
to give
Mi ¼ 315:5104753 106 ðKw 3:099934Þ6:6932358 γ 6:2596977
i
or
γi ¼
3:624849522 Mi0:159752123
ðKw 3:099934Þ1:0692586
(2.3)
84 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
It should be noted that the molecular weight “M,” specific gravity “γ,” and
the Watson characterization factor “Kw” of pseudocomponents provide the
key data needed for further characterization of these fractions in terms of
their critical properties and acentric factors. The following simplified two
sets of correlations for estimating Tc, pc, and ω of pseudocomponents were
generated by matching data from several literature published data. The first
set was correlated as a function of the ratio “M/γ” while the second set was
correlated as a function of “Tb/γ,” to give
Set 1. Properties as a function of the ratio “M/γ”:
0:351579776
M
M
Tc ¼ 231:9733906
0:352767639
233:3891996 (2.3A)
γ
γ
0:885326626
M
M
0:106467189
+ 49:62573013
pc ¼ 31829
γ
γ
M
M
ω ¼ 0:279354619 ln
+ 0:00141973
1:243207091
γ
γ
vc ¼ 0:054703719
(2.3B)
(2.3C)
0:984070268
M
M
+ 87:7041106
0:001007476 (2.3D)
γ
γ
0:664115311
M
M
379:4428973
1:40066452
Tb ¼ 33:72211
ðM=γ Þ
γ
γ
(2.3E)
Set 2. Properties as a function of the ratio “Tb/γ”:
Tc ¼ 1:379242274
0:967105064
Tb
Tb
+ 0:010009512 (2.3F)
+ 0:012011487
γ
γ
Tb
Tb
0:00867297
+ 0:099138742
pc ¼ 2898:631565exp 0:002020245
γ
γ
(2.3G)
1:250922701
Tb
Tb
ω ¼ 0:000115057
0:92098982
(2.3H)
+ 0:00068975
γ
γ
3
2
Tb
vc ¼ 8749:94 106 Tγb 9:75748 106 Tγb + 0:014653
γ
595:7922076
Tb
γ
0:001617123Tb
132:1481762
M ¼ 52:11038728 exp
γ
4:60231684 (2.3I)
(2.3J)
Group 1. TBP 85
If the Tb is not known, the molecular weight can be estimated from the
following expression:
M ¼ 0:048923 exp ð9:88378γ Þ 33:085468γ +
39:598437
γ
(2.3K)
where
Tb ¼ boiling point in °R
Tc ¼ critical temperature in °R
vc ¼ critical volume, ft3/lbm-mol
M ¼ molecular weight
Notice, to express critical volume in ft3/lbm, divide vc by the molecular
weight “M.”
Vc ¼
vc
M
The main issue that should be addressed after obtaining the distillation
curve is how to break down or cut the entire boiling range into a number
of cut-point ranges, which are used to define pseudocomponent. The determination of the number of cuts is essentially arbitrary; however, the number
of pseudocomponents has to be sufficient to reproduce the volatility properties in fractions with wide boiling point ranges. The following provide a
brief summary of industry experience and guidelines:
1. Pedersen et al. (1982) suggest selecting a small number of
pseudocomponents with approximately the same weight fraction.
2. As a rule of thumb, use between 10 and 15 fractions if the TBP
curve is relatively “steep” and use 5–8 fractions if the curve is
relatively “flat.”
3. To better reproduce the shape of the TBP curve and to reflect the
relative distribution of the light, middle, and heavy ends in
pseudocomponent, Miquel et al. (1992) pointed out that the
pseudocomponent breakdown can be made from either equal volumetric
fractions or regular temperature intervals in the TBP curve. To
reproduce the shape of the TBP curve better and reflect the relative
distribution of light, middle, and heavy components in the fraction, a
breakdown based on equal TBP temperature intervals is recommended.
On the other hand, when equal volumetric fractions are taken,
information concerning the light initial component and heavy end
component could be lost.
4. Most commercial process simulators (eg, the HYSYS model)
generate pseudocomponents based on boiling point ranges
86 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
representing the oil fractions. Typical boiling point ranges
for pseudocomponents in a process simulator are shown below:
Boiling Point Range
IBP to 800°F
800–1200°F
1200–1650°F
Total no. of
pseudocomponents
Suggested Number of
Pseudocomponents
30
10
8
48
It should be pointed out that a complete TBP curve is rarely available for an
entire range of percent distilled. In many cases, the data is obtained up to the
50% or 70% distilled point for any petroleum fractions or crude oils. This
incomplete distillation curve is attributed to the fact that the petroleum fractions or crude oils contain more heavy hydrocarbons toward the end of the
distillation curve; therefore it is more difficult is to achieve accurate boiling
points for that fraction or crude oil. However, it is important for accurate
fluid characterizations and in the process of equations of state tuning that
the distillation curve should be extended to 90–95% distilled point. As conceptually shown in Fig. 2.5, Riazi and Daubert (1987) developed a
(1/b)
Temperature
Tbi = To 1+
a
1
ln
1–Vi
b
Tbi
To
≈50–70%
Volume distilled “Vi”
n FIGURE 2.5 Mathematical representation of TBP curve.
Group 2. Stimulated Distillation by GC 87
mathematical representation of the TBP curve that can be used to complete
the TBP curves up to the 95% point.
The mathematical representation is given by the following equation:
(
Tbi ¼ To 1 +
a
1
ln
b
1 Vi
ð1=bÞ )
where,
Tbi ¼ TBP temperature and any distilled volume Vi
Vi ¼ distilled volume fraction
To ¼ IBP (ie, at Vi ¼ 0)
a, b ¼ correlating coefficients
The coefficients a and b are the two parameters that can be determined by
regression or alternatively by linearizing the above equation as follows:
ln
Tbi To
1
a
1
1
+
¼
ln
ln ln
To
b
b
b
1 Vi
The above relationship indicates that a plot of ln ½Tbi To =To versus
ln ½1=1 Vi would produce a straight line with a slope of (1/b) and intercept
of [(1/b)ln(a/b)] and used to calculate the parameters a and b. The mathematical expression can then be used to predict and complete the TBP curve
up to 95% distilled volume.
GROUP 2. STIMULATED DISTILLATION BY GC
Normal TBP distillation involves a long procedure and is costly. GC has
been used widely as a fast and reproducible method for simulated distillation
(SD) to analyze the carbon number distribution of various hydrocarbon
components in the crude oil. GC is a common type of chromatography used
for separating and analyzing compounds that can be vaporized without
decomposition. Like all other chromatographic techniques, GC consists
of a mobile and a stationary phase. The mobile phase is carrier gas comprised of an inert gas (ie, helium, argon, or nitrogen). The stationary phase
consists of a packed capillary column with the packing acting as a stationary
phase that is capable of absorbing and retaining hydrocarbon components
for a specific time before releasing them sequentially based on their boiling
points. This causes each compound to liberate at a different time, known as
the retention time of the compound. GC is then similar to fractional distillation, as both processes separate the components of a mixture primarily
based on boiling point.
88 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
The specific process of obtaining the SD curve from a GC experiment is
based on injecting a small oil sample into a stream of carrier gas, which displaces that sample into a column packed with materials that absorbs the
heavier components of the oil. The column is placed in a temperature programmable oven; as the temperature of the column is raised, the heavier
components are released sequentially based on their boiling point and consequently removed by the carrier gas into a detector that senses changes in
the thermal conductivity of the carrier gas. For a given program of heating,
the retention time in the column is proportional to the normal boiling point
of the released components. These are correlated with the retention times of
a calibration curve obtained by conducting the separation under the same
conditions of a known mixture of hydrocarbons, usually n-alkanes, covering
the boiling range expected in the sample. The resulting calibration curve
relates carbon number and boiling point to retention time and allows for
the determination of boiling points for the components in the hydrocarbon
sample. As shown in Fig. 2.6 and Table 2.4, all components detected by the
GC between two neighboring n-alkanes are commonly grouped together and
n FIGURE 2.6 GC data calibration sample of n-alkanes.
Group 2. Stimulated Distillation by GC 89
Table 2.4 Boiling Point Data
n-Alkane
Retention Time (min)
Boiling Point (°C)
C5
C6
C7
C8
C9
C10
C11
C12
C14
C15
C16
C17
C18
C20
C24
C28
C32
C36
C40
0.09764
0.11994
0.16272
0.23050
0.31882
0.41783
0.51717
0.61712
0.79714
0.87985
0.95957
1.03281
1.10226
1.23156
1.45693
1.64895
1.81618
1.96389
2.10297
21.9
54.7
84.1
111.9
136.3
159.7
181.9
201.9
220.8
239.7
256.9
273.0
288.0
301.9
376.9
416.9
451.9
481.9
508.0
reported as a pseudocomponent with a SCN equal to the higher n-alkane; eg,
between n-alkanes of C8 and C9 is designated as pseudocomponent with
SCN of C9.
As shown in Fig. 2.7, the distillation curve resulting from the plot of GC data
in terms of boiling point as a function of the retention time or SCN of
n-alkane is called a SD curve and represents boiling points of compounds
in a petroleum mixture at atmospheric pressure. It should be pointed out that
SD curves are very close to actual boiling points shown by TBP curves.
The main advantage of this SD process is that it requires smaller samples
and also is less expensive than TBP distillation. The GC is used for gas
and oil samples generating distillation curves as well as the composition
of hydrocarbon mixtures in terms of their weight fractions “wi”. Conversion
from weight fraction “wi” to mole fraction “xi” requires molecular weights
“Mi” of all components and can be achieved from
wi =Mi
xi ¼ X
ðwi =Mi Þ
i
90 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Stimulated distillation curve
600
Boiling point (°C)
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
Retention time (min)
n FIGURE 2.7 Stimulated distillation curve.
Similarly, if the volume fraction vi and density ρi are available, they can be
converted into mole fraction from the following relationship:
vi ρ =Mi
xi ¼ X i
ðv ρ =Mi Þ
i i i
However, it should be pointed out that a major drawback of GC analysis is
that no physical fractions are collected during this chromatographic analysis
test and, therefore, their molecular weights “Mi” and specific gravities are
not measured. As discussed later in this chapter, to describe these fractions
in terms of their critical properties and acentric factors, the molecular weight
and specific gravity must be known and used as input for most of the widely
used property correlations. Ahmed (2014) pointed out that without the availability of molecular weight and specific gravity of pseudocomponents, the
GC stimulated distillation data could still be used to characterize these fractions. The recommended proposed methodology is based on reproducing the
GC stimulated boiling point data by a corresponding set of pseudocomponents with equivalent average boiling point temperatures to that of the GC
stimulated boiling point data. These pseudocomponents are identified by a
set of effective carbon numbers (ECN) that can be used to estimate their
physical and critical properties. This approach relies on matching the boiling
point temperature “Tb” by using Microsoft Solver and regressing on the ECN
as given by the following relationship:
Tb ¼ 427:2959078 + 50:08577848ðECNÞ
0:88693418ðECNÞ2 + 0:00675667ðECNÞ3 551:2778516
ðECNÞ
Group 2. Stimulated Distillation by GC 91
where Tb is expressed in °R. The physical, critical properties, and acentric
factor of a pseudocomponent can then be calculated from the following set
of expressions:
M ¼ 131:11375 + 24:96156ðECNÞ
0:34079022ðECNÞ2 + 0:002494118ðECNÞ3 +
468:32575
ðECNÞ
γ ¼ 0:86714949 + 0:0034143408ðECNÞ 2:839627
1:1627984
105 ðECNÞ2 + 2:49433308 108 ðECNÞ3 ðECNÞ
Tc ¼ 926:6022445 + 39:729363ðECNÞ
0:7224619ðECNÞ2 + 0:005519083ðECNÞ3 1366:4317487
ðECNÞ
pc ¼ 311:236191 14:68693ðECNÞ + 0:3287671ðECNÞ2
1690:900114
0:0027346ðECNÞ3 +
ðECNÞ
Tb ¼ 427:295908 + 50:0857785ðECNÞ
0:8869342ðECNÞ2 + 0:00675667ðECNÞ3 551:277852
ðECNÞ
ω ¼ 0:3142816 + 0:0780028ðECNÞ 0:00139205ðECNÞ2 + 1:02147
0:99102887
105 ðECNÞ3 +
ðECNÞ
vc ¼ 0:2328371 + 0:9741117ðECNÞ 0:009227ðECNÞ2 + 36:3611
0:1113515
106 ðECNÞ3 +
ðECNÞ
where,
ECN ¼ effective carbon number
Tc ¼ critical temperature of the fraction with a carbon number “n,” °R
Tb ¼ boiling point temperature of the fraction with a carbon number
“n,” °R
pc ¼ critical pressure of the fraction with a carbon number “n,” psia
vc ¼ critical volume of the fraction with a carbon number “n,” ft3/lbm-mol
M ¼ molecular weight
ω ¼ acentric factor
γ ¼ specific gravity
It should be pointed out that the number of carbon atoms “n” that is equivalent to that of n-alkane is calculated from
n ¼ INT½ðECNÞ + 1
92 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where,
ECN ¼ effective carbon number
n ¼ equivalent carbon number of n-alkane
INT ¼ integer
As a validation of Ahmed’s approach, assume the tabulated data listed in
Table 2.24 are closely approximate the TBP. The objective is to estimate
the ECN for each of the listed boiling point and compare with n-alkane
carbon number and to calculate the physical and critical properties of each
carbon number.
Step 1. Using regression (Microsoft Solver), determine the ECN from the
expression
Tb ¼ 427:2959078 + 50:08577848ðECNÞ
0:88693418ðECNÞ2 + 0:00675667ðECNÞ3 551:2778516
ðECNÞ
n ¼ INT½ðECNÞ + 1
To give:
n
Tb (°C)
Tb (°R)
ECN
5
6
7
8
9
10
11
12
14
15
16
17
18
20
24
28
32
36
40
21.9
54.7
84.1
111.9
136.3
159.7
181.9
201.9
220.8
239.7
256.9
273
288
310.9
376.9
416.9
451.9
481.9
508
530.82
589.86
642.78
692.82
736.74
778.86
818.82
854.82
888.84
922.86
953.82
982.8
1009.8
1051.02
1169.82
1241.82
1304.82
1358.82
1405.8
4.8
5.8
6.7
7.7
8.7
9.7
10.7
11.7
12.6
13.6
14.6
15.6
16.5
18.1
23.2
27.1
31.1
35
38.9
Group 3. The Plus Fraction Characterization Methods 93
Step 2. Calculate the physical and critical properties of each carbon number
using the proposed working relationships and the values of ECN
given above, to give
n
M
Tc (°R)
pc (psi)
Tb (°R)
W
γ
Vc (ft3/lbm-mol)
5.000
6.000
7.000
8.000
9.000
10.000
11.000
12.000
14.000
15.000
16.000
17.000
18.000
20.000
24.000
28.000
32.000
36.000
40.000
79.151
84.980
94.678
106.586
119.791
133.750
148.123
162.688
191.849
206.271
220.515
234.547
248.342
275.171
325.661
372.107
415.049
455.213
493.416
834.591
912.423
975.996
1030.221
1077.845
1120.526
1159.332
1194.988
1258.753
1287.520
1314.526
1339.947
1363.928
1408.036
1483.330
1544.968
1596.289
1640.091
1678.898
546.140
516.176
465.156
424.744
391.568
363.599
339.539
318.518
283.332
268.402
254.890
242.602
231.383
211.673
180.771
158.115
141.146
127.974
117.046
546.140
605.461
658.000
705.768
749.899
791.089
829.797
866.342
933.821
965.074
994.834
1023.198
1050.252
1100.727
1188.915
1262.975
1325.995
1380.843
1430.278
0.240
0.271
0.309
0.350
0.393
0.436
0.479
0.522
0.604
0.643
0.681
0.718
0.753
0.820
0.938
1.038
1.122
1.194
1.257
0.651
0.693
0.724
0.747
0.766
0.782
0.796
0.807
0.826
0.835
0.842
0.849
0.855
0.866
0.885
0.900
0.912
0.922
0.931
4.434
5.306
6.162
7.002
7.826
8.633
9.424
10.200
11.704
12.433
13.147
13.846
14.530
15.855
18.338
20.610
22.685
24.577
26.298
GROUP 3. THE PLUS FRACTION CHARACTERIZATION
METHODS
It is well known that C7+ characterization methodology has a significant
impact on EOS predictions of reservoir hydrocarbon fluid behavior.
The basis for most characterization methods is the TBP data that includes
mass, mole, and volume fractions of each distillation cuts with a measured
molecular weight, specific gravity, and boiling point. Each distillation cut is
usually treated as pseudocomponent that can be characterized by a critical
pressure, critical temperature, and acentric factor. If the distillation cuts are
collected within a range of neighboring normal alkanes boiling points, they
are referred to as SCN groups. Correlations for estimating their critical properties are commonly used based on the molecular weight and specific
gravity of the SCN group. Based on the type of existing laboratory measured
data on the plus fraction, two approaches commonly are used to generate
properties of the plus fractions:
94 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
n
n
Generalized correlations,
PNA determination using analytical correlations, and
PNA determination using graphical correlations.
These two techniques of characterizing the undefined petroleum fractions
are detailed next.
Generalized Correlations
To use any of the thermodynamic property prediction models, such as an
equation of state, to predict the phase and volumetric behavior of complex hydrocarbon mixtures, one must be able to provide the acentric factor, ω, critical temperature, Tc, and critical pressure, pc, for both the
defined and undefined (heavy) fractions in the mixture. The problem
of how to adequately characterize these undefined plus fractions in terms
of their critical properties and acentric factors has been long recognized
in the petroleum industry. Whitson (1984) presented excellent documentation on the influence of various heptanes-plus (C7+) characterization
schemes on predicting the volumetric behavior of hydrocarbon mixtures
by equations of state.
Numerous correlations are available for estimating the physical properties
of petroleum fractions. Most of these correlations use the specific gravity,
γ, and the boiling point, Tb, correlation parameters. However, coefficients of
these correlations were determined from matching pure component physical
and critical properties data, which might cause abnormality when applied to
characterize the undefined fractions. Several of these correlations are presented next and referred to by the names of the authors, including:
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Riazi and Daubert
Cavett
Kesler-Lee
Winn and Sim-Daubert
Watansiri-Owens-Starling
Edmister
Critical compressibility factor correlations
Rowe
Standing
Willman-Teja
Hall-Yarborough
Magoulas-Tassios
Twu
Silva and Rodriguez
Group 3. The Plus Fraction Characterization Methods 95
Riazi and Daubert Generalized Correlations
Riazi and Daubert (1980) developed a simple two-parameter equation for
predicting the physical properties of pure compounds and undefined hydrocarbon mixtures. The proposed generalized empirical equation is based on
the use of the normal boiling point and specific gravity as correlating parameters. The basic equation is
θ ¼ aTbb γ c
(2.4)
where
θ ¼ any physical property (Tc, pc, Vc or M)
Tb ¼ normal boiling point, °R
γ ¼ specific gravity
M ¼ molecular weight
Tc ¼ critical temperature, °R
pc ¼ critical pressure, psia
Vc ¼ critical volume, ft3/lbm
a, b, c ¼ correlation constants are given in Table 2.5 for each property
The expected average errors for estimating each property are included in the
table. Note that the prediction accuracy is reasonable over the boiling point
range of 100–850°F. Riazi and Daubert (1987) proposed improved correlations for predicting the physical properties of petroleum fractions by taking
into consideration the following factors: accuracy, simplicity, generality,
and availability of input parameters; ability to extrapolate; and finally, comparability with similar correlations developed in recent years.
The authors proposed the following modification of Eq. (2.4), which maintains the simplicity of the previous correlation while significantly improving
its accuracy:
θ ¼ aθb1 θc2 exp ½dθ1 + eθ2 + f θ1 θ2 Table 2.5 Correlation Constants for Eq. (2.4)
Deviation (%)
θ
M
Tc (°R)
pc (psia)
Vc (ft3/lb)
a
b
5
4.56730 10
24.27870
3.12281 109
7.52140 103
c
2.19620 1.0164
0.58848 0.3596
2.31250 2.3201
0.28960 0.7666
Average
Maximum
2.6
1.3
3.1
2.3
11.8
10.6
9.3
9.1
96 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where
θ ¼ any physical property
a–f ¼ constants for each property
Riazi and Daubert stated that θ1 and θ2 can be any two parameters capable of
characterizing the molecular forces and molecular size of a compound. They
identified (Tb,γ) and (M,γ) as appropriate pairs of input parameters in the
equation. The authors finally proposed the following two forms of the
generalized correlation.
In the first form, the boiling point, Tb, and the specific gravity, γ, of the
petroleum fraction are used as correlating parameters:
θ ¼ aTbb γ c exp ½dTb + eγ + fTb γ (2.5)
The constants a–f for each property θ are given in Table 2.6.
In the second form, the molecular weight, M, and specific gravity, γ, of
the component are used as correlating parameters. Their mathematical
expression has the following form:
θ ¼ aðMÞb γ c exp ½dM + eγ + f γM
(2.6)
In developing and obtaining the coefficients a–f, as given in Table 2.7, the
authors used data on the properties of 38 pure hydrocarbons in the carbon
number range 1–20, including paraffins, olefins, naphthenes, and aromatics
in the molecular weight range 70–300 and the boiling point range 80–650°F.
Table 2.6 Correlation Constants for Eq. (2.5)
θ
a
b
c
d
e
f
M
Tc (°R)
pc (psia)
Vc (ft3/lb)
581.96000
10.6443
6.16200 106
6.23300 104
0.97476
0.81067
0.48440
0.75060
6.51274
0.53691
4.08460
1.20280
5.43076 104
5.17470 104
4.72500 103
1.46790 103
9.53384
0.54444
4.80140
0.26404
1.11056 103
3.59950 104
3.19390 103
1.09500 103
Table 2.7 Correlation Constants for Eq. (2.6)
θ
Tc (°R)
Pc (psia)
Vc (ft3/lb)
Tb (°R)
a
544.40000
4.52030 104
1.20600 102
6.77857
b
0.299800
0.806300
0.203780
0.401673
c
1.05550
1.60150
1.30360
1.58262
d
4
1.34780 10
1.80780 103
2.65700 103
3.77409 103
e
f
0.616410
0.308400
0.528700
2.984036
0.00000
0.00000
2.60120 103
4.25288 103
Group 3. The Plus Fraction Characterization Methods 97
Cavett’s Correlations
Cavett (1962) proposed correlations for estimating the critical pressure and
temperature of hydrocarbon fractions. The correlations received wide
acceptance in the petroleum industry due to their reliability in extrapolating
conditions beyond those of the data used in developing the correlations. The
proposed correlations were expressed analytically as functions of the normal
boiling point, TbF, in °F and API gravity. Cavett proposed the following
expressions for estimating the critical temperature and pressure of petroleum fractions:
Tc ¼ a0 + a1 ðTbF Þ + a2 ðTbF Þ2 + a3 ðAPIÞðTbF Þ + a4 ðTbF Þ3 + a5 ðAPIÞðTbF Þ2
+ a6 ðAPIÞ2 ðTbF Þ2
(2.7)
log ðpc Þ ¼ b0 + b1 ðTbF Þ + b2 ðTbF Þ2 + b3 ðAPIÞðTbF Þ + b4 ðTbF Þ3 + b5 ðAPIÞðTbF Þ2
+ b6 ðAPIÞ2 ðTbF Þ + b7 ðAPIÞ2 ðTbF Þ2
(2.8)
where
Tc ¼ critical temperature, °R
pc ¼ critical pressure, psia
TbF ¼ normal boiling point, °F
API ¼ API gravity of the fraction
Note that the normal boiling point in the above relationships is expressed
in °F.
The coefficients of Eqs. (2.7) and (2.8) are tabulated in the table below.
However, Cavett presented these correlations without reference to the type
and source of data used for their development.
i
ai
bi
0
1
2
3
4
5
6
7
768.0712100000
1.7133693000
0.0010834003
0.0089212579
0.3889058400 106
0.5309492000 105
0.3271160000 107
2.82904060
0.94120109 103
0.30474749 105
0.20876110 104
0.15184103 108
0.11047899 107
0.48271599 107
0.13949619 109
Kesler and Lee Correlations
Kesler and Lee (1976) proposed a set of equations to estimate the critical
temperature, critical pressure, acentric factor, and molecular weight of
98 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
petroleum fractions. These relationships, as documented below, use specific
gravity “γ” and boiling point “Tb” as input parameters for their proposed
expressions:
0:0566
2:2898 0:11857
103 Tb
0:24244 +
+
γ
γ
γ2
3:648 0:47227
1:6977
7 2
10
1010 Tb3
+ 1:4685 +
T
0:42019
+
+
b
γ
γ2
γ2
(2.9)
lnðpc Þ ¼ 8:3634 Tc ¼ 341:7 + 811:1γ + ½0:4244 + 0:1174γ Tb +
½0:4669 3:26238γ 105
(2.10)
Tb
M ¼ 12272:6 + 9486:4γ + ½4:6523 3:3287γ Tb + 1 0:77084γ 0:02058γ 2
720:79 107
181:98 1012
+ 1 0:80882γ 0:02226γ 2 1:8828 1:3437 Tb
Tb
Tb
Tb3
(2.11)
The molecular weight equation was developed by regression analysis to
match laboratory data with molecular weights ranging from 60 to 650.
In addition, Kesler and Lee developed two expressions for estimating the
acentric factor that use the Watson characterization factor and the reduced
boiling point temperature as correlating parameters. The two correlating
parameters are defined by the following two parameters:
n
n
the Watson characterization factor “Kw”; ie, Kw ¼ (Tb)1/3/γ
the reduced boiling point “θ” as defined by θ ¼ Tb/Tc
where the boiling point Tb and critical temperature Tc are expressed in °R.
Kessler and Lee proposed the following two expressions for calculating the
acentric factor that are based on the value of the reduced boiling point:
For θ > 0:8 : ω ¼ 7:904 + 0:1352Kw 0:007456ðKw Þ2 + 8:359θ
+
ln
For θ < 0:8 : ω ¼
where
1:408 0:01063Kw
θ
(2.12)
pc
6:09648
+ 1:28862 ln ½θ 0:169347θ6
5:92714 +
θ
14:7
15:6875
15:2518 13:4721 ln ½θ + 0:43577θ6
θ
(2.13)
pc ¼ critical pressure, psia
Tc ¼ critical temperature, °R
Tb ¼ boiling point, °R
Group 3. The Plus Fraction Characterization Methods 99
ω ¼ acentric factor
M ¼ molecular weight
γ ¼ specific gravity
Kesler and Lee stated that Eqs. (2.9) and (2.10) give values for pc and Tc that
are nearly identical with those from the API data book up to a boiling point
of 1200°F.
Winn and Sim-Daubert Correlations
Winn (1957) developed convenient nomographs to estimate various physical properties including molecular weight and the pseudocritical pressure
for petroleum fractions. The input data in both nomographs are boiling point
(or Kw) and the specific gravity (or API gravity). Sim and Daubert (1980)
concluded that the Winn’s monograph is the most accurate approach for
characterizing petroleum fractions. Sim and Daubert developed analytical
relationships that closely matched the monograph graphical data. The
authors used specific gravity and boiling point as the correlating parameters
for calculating the critical pressure, critical temperature, and molecular
weight. Sim and Daubert relationships as given below can be used to estimate the pseudocritical properties of the undefined petroleum fraction:
pc ¼ 3:48242 109 Tb2:3177 γ 2:4853
(2.14)
Tc ¼ exp 3:9934718T 0:08615
γ 0:04614
b
(2.15)
M ¼ 1:4350476 105 Tb2:3776 γ 0:9371
(2.16)
where
pc ¼ critical pressure, psia
Tc ¼ critical temperature, °R
Tb ¼ boiling point, °R
Watansiri-Owens-Starling Correlations
Watansiri et al. (1985) developed a set of correlations to estimate the critical
properties and acentric factor of coal compounds and other undefined hydrocarbon components and their derivatives. The proposed correlations express
the critical and physical properties of the undefined fraction as a function of
the fraction normal boiling point, specific gravity, and molecular weight.
These relationships have the following forms:
lnðTc Þ ¼ 0:0650504 0:0005217Tb + 0:03095 ln ½M + 1:11067 lnðTb Þ
h
i
+ M 0:078154γ 1=2 0:061061γ 1=3 0:016943γ
(2.17)
100 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
lnðvc Þ ¼ 76:313887 129:8038γ + 63:1750γ 2 13:175γ 3 + 1:10108 ln ½M
+ 42:1958 ln ½γ (2.18)
0:8
Tc
M
Tb
0:08843889
8:712
lnðpc Þ ¼ 6:6418853 + 0:01617283
Vc
M
Tc
(2.19)
8
9
>
5:12316667 104 Tb + 0:281826667ðTb =MÞ + 382:904=M >
>
>
>
>
>
>
>
>
2
5
4
>
>
>
+ 0:074691 10 ðTb =γ Þ 0:12027778 10 ðTb ÞðMÞ >
>
>
>
>
>
>
>
>
2
4
>
>
>
>
< + 0:001261ðγ ÞðMÞ + 0:1265 10 ðMÞ
=5T b
ω¼
1=3
ð
T
Þ
>
>
9M
b
2
>
>
>
>
+ 0:2016 104 ðγ ÞðMÞ 66:29959
>
>
>
>
M
>
>
>
>
>
>
>
>
2=3
>
>
>
>
T
>
>
b
>
>
: 0:00255452 2
;
γ
(2.20)
where
ω ¼ acentric factor
pc ¼ critical pressure, psia
Tc ¼ critical temperature, °R
Tb ¼ normal boiling point, °R
vc ¼ critical volume, ft3/lb-mol
The proposed correlations produce an average absolute relative deviation of
1.2% for Tc, 3.8% for vc, 5.2% for pc, and 11.8% for ω.
Edmister’s Correlations
Edmister (1958) proposed a correlation for estimating the acentric factor, ω,
of pure fluids and petroleum fractions. The equation, widely used in the
petroleum industry, requires boiling point, critical temperature, and critical
pressure. The proposed expression is given by the following relationship:
ω¼
3½ log ðpc =14:70Þ
1
7½ðTc =Tb 1Þ
where
ω ¼ acentric factor
pc ¼ critical pressure, psia
Tc ¼ critical temperature, °R
Tb ¼ normal boiling point, °R
(2.21)
Group 3. The Plus Fraction Characterization Methods 101
If the acentric factor is available from another correlation, the Edmister
equation can be rearranged to solve for any of the three other properties
(providing the other two are known).
Critical Compressibility Factor Correlations
The critical compressibility factor is defined as the component compressibility factor calculated at its critical point by using the component critical
properties; ie, Tc, pc, and Vc. This property can be conveniently computed
by the real gas equation of state at the critical point, or
Zc ¼
p c vc
RTc
(2.22)
where
R ¼ universal gas constant, 10.73 psia ft3/lb-mol, °R
vc ¼ critical volume, ft3/lb-mol
If the critical volume, Vc, is given in ft3/lbm, Eq. (2.22) is written as
Zc ¼
pc Vc M
RTc
where
M ¼ molecule weight
Vc ¼ critical volume, ft3/lb
The accuracy of Eq. (2.22) depends on the accuracy of the values of pc, Tc,
and vc. The following table presents a summary of the critical compressibility estimation methods that have been published over the years:
Method
Haugen
Reid, Prausnitz,
and Sherwood
Salerno et al.
Nath
Year
Zc
Equation No.
1959
1977
Zc ¼ 1=ð1:28ω + 3:41Þ
Zc ¼ 0:291 0:080ω
(2.23)
(2.24)
1985
1985
Zc ¼ 0:291 0:080ω 0:016ω2
Zc ¼ 0:2918 0:0928ω
(2.25)
(2.26)
Rowe’s Characterization Method
Rowe (1978) proposed a set of correlations for estimating the normal boiling
point, the critical temperature, and the critical pressure of the heptanes-plus
fraction C7+. The prediction of the C7+ properties is based on the assumption
that the plus fraction behaves as a normal paraffin hydrocarbon fraction that
has a number of carbon atoms “n” with identical critical properties to that of
102 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
the plus fraction. Rowe proposed the following set of formulas for characterizing the C7+ fraction in terms of the critical temperature, critical pressure, and boiling point temperature that are equivalent to properties of a
normal paraffin hydrocarbon fraction with a number of carbon atoms “n”
that can be calculated from the molecular weight of the C7+ fraction by
the following relationship:
n¼
MC7 + 2:0
14
ðTc ÞC7 + ¼ 1:8½961 10a (2.27)
(2.28)
The coefficient “a” as defined by
a ¼ 2:95597 0:090597n2=3
The author correlated the critical pressure and the boiling point temperature
of the C7+ fraction in terms of the critical temperature and number of carbon
atoms by the following expressions:
ðpc ÞC7 + ¼
10ð4:89165 + Y Þ
ðTc ÞC7 +
(2.29)
with the parameter “Y” as given by
Y ¼ 0:0137726826 n + 0:6801481651
ðTb ÞC7 + ¼ 0:0004347ðTc Þ2C7 + + 265
(2.30)
where
ðTc ÞC7 + ¼ critical temperature of C7+, °R
ðTb ÞC7 + ¼ boiling point temperature of C7+, °R
ðpc ÞC7 + ¼ critical pressure of the C7+ in psia
MC7 + ¼ molecular weight of the heptanes-plus fraction
Soreide (1989) proposed an alternative methodology of calculating the boiling point that was generated from the analysis of 843 TBP fractions taken
from 68 reservoir C7+ samples. Soreide proposed the boiling point temperature correlation is expressed as a function of molecular weight and specific
gravity of the plus fraction:
Tb ¼ 1928:3 1:695 105 γ 3:266
exp 4:922 103 M 4:7685γ + 3:462 103 Mγ
M0:03522
where Tb is expressed in °R.
Group 3. The Plus Fraction Characterization Methods 103
Standing’s Correlations
Matthews et al. (1942) presented graphical correlations for determining the
critical temperature and pressure of the heptanes-plus fraction. Standing
(1977) expressed these graphical correlations more conveniently in mathematical forms as follows:
h
i h
i
ðTc ÞC7 + ¼ 608 + 364log ðMÞC7 + 71:2 + 2450log ðMÞC7 + 3800 log ðγ ÞC7 +
(2.31)
h
i
ðpc ÞC7 + ¼ 1188 431log ðMÞC7 + 61:1
n
h
ih
io
+ 2319 852 log ðMÞC7 + 53:7 ðγ ÞC7 + 0:8
(2.32)
where ðMÞC7 + and ðγ ÞC7 + are the molecular weight and specific gravity of
the C7+.
Willman-Teja Correlations
Willman and Teja (1987) proposed correlations for determining the critical
pressure and critical temperature of the n-alkane homologous series. The
authors used the normal boiling point and the number of carbon atoms of
the n-alkane as a correlating parameter. The authors used a nonlinear regression
model to generate a set of relationships that best match the critical properties
data reported by Bergman et al. (1977) and Whitson (1980). Willman and Teja
introduced the ECN parameter “n” into their proposed correlations and suggested that this correlating parameter can be calculated by matching the boiling
point of the undefined petroleum fraction. Ahmed (2014) suggested that the
ECN can be best estimated by regression (eg, Microsoft Solver) to match
the boiling point temperature “Tb” as given by following expression:
Tb ¼ 434:38878 + 50:125279n 0:9097293n2 + 7:0280657 103 n3
601:85651=n
where Tb is expressed in °R.
The critical temperature and pressure can then be calculated from the
following expressions:
h
i
Tc ¼ Tb 1 + ð1:25127 + 0:137252nÞ0:884540633
Pc ¼
339:0416805 + 1184:157759n
½0:873159 + 0:54285n1:9265669
where
n ¼ effective carbon number
Tb ¼ average boiling point of the undefined fraction, °R
(2.33)
(2.34)
104 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Tc ¼ critical temperature of the undefined fraction, °R
Pc ¼ critical pressure of the undefined fraction, psia
Hall and Yarborough Correlations
Hall and Yarborough (1971) proposed a simple expression to determine the
critical volume of a fraction from its molecular weight and specific gravity:
vc ¼
0:025M1:15
γ 0:7935
(2.35)
where vc is the critical volume as expressed in ft3/lb-mol. Note that, to
express the critical volume in ft3/lb, the relationship is given by
vc ¼ MVc
where
M ¼ molecular weight
Vc ¼ critical volume in ft3/lb
vc ¼ critical volume in ft3/lb-mol
The critical volume also can be calculated by applying the real gas equation
of state at the critical point of the component as
pV ¼ Z
m
M
RT
Applying real gas equation at the critical point, gives
Vc ¼
Zc RTc
pc M
Magoulas and Tassios Correlations
Magoulas and Tassios (1990) correlated the critical temperature, critical
pressure, and acentric factor with the specific gravity, γ, and molecular
weight, M, of the fraction as expressed by the following relationships:
Tc ¼ 1247:4 + 0:792M + 1971γ lnð pc Þ ¼ 0:01901 0:0048442M + 0:13239γ +
27000 707:4
+
M
γ
227 1:1663
+ 1:2702 lnðMÞ
M
γ
ω ¼ 0:64235 + 0:00014667M + 0:021876γ where
Tc ¼ critical temperature, °R
pc ¼ critical pressure, psia
4:559
+ 0:21699 ln ðMÞ
M
Group 3. The Plus Fraction Characterization Methods 105
Twu’s Correlations
Twu (1984) developed a suite of critical properties, based on perturbationexpansion theory with normal paraffins as the reference states, which can
be used for determining the critical and physical properties of undefined
hydrocarbon fractions, such as C7+. The methodology is based on selecting (finding) a normal paraffin fraction with a boiling temperature TbP
identical to that of the hydrocarbon-plus fraction “TbC + ,” such as C7+.
The methodology requires the availability of the boiling point temperature of the plus fraction “TbC + ,” molecular weight of the plus fraction
“MC + ,” as well as its specific gravity “γ C + .” If the boiling point temperature is not available, it can be estimated from the correlation proposed
by Soreide (1989):
a
TbC + ¼ a1 + a2 ðMC + Þa3 γ C + 4 exp a5 MC + + a6 γ C + + a7 MC + γ C +
where
a1 ¼ 1928.3
a2 ¼ 1.695(105)
a3 ¼ 0.03522
a4 ¼ 3.266
a5 ¼ 4.922(103)
a6 ¼ 4.7685
a7 ¼ 3.462(103)
The Twu approach is performed by applying the following two steps:
Step 1. Calculate the properties of the normal paraffins, ie, TcP, pcP, γ P, and
vcP, from the following set of working expressions:
n
Critical temperature of normal paraffins, TcP, in °R:
"
TcP ¼ TbC +
2
3
A1 + A2 TbC + + A3 TbC
+ A4 TbC
+
+
+
A5
ðA6 TbC + Þ13
where
A1 ¼ 0.533272
A2 ¼ 0.191017(103)
A3 ¼ 0.779681(107)
A4 ¼ 0.284376(1010)
A5 ¼ 0.959468(102)
A6 ¼ 0.01
n
Critical pressure of normal paraffins, pcP, in psia:
2
4
pcP ¼ A1 + A2 α0:5
i + A3 αi + A4 αi + A5 αi
2
#1
106 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
With the parameter αi as defined by the expression
αi ¼ 1 TbC +
TcP
where
A1 ¼ 3.83354
A2 ¼ 1.19629
A3 ¼ 34.8888
A4 ¼ 36.1952
A5 ¼ 104.193
n
Specific gravity of normal paraffins, γ P:
γ P ¼ A1 + A2 αi + A3 α3i + A4 α12
i
where
A1 ¼ 0.843593
A2 ¼ 0.128624
A3 ¼ 3.36159
A4 ¼ 13749.5
n
Critical volume of normal paraffins, vcP, in ft3/lbm-mol:
vcP ¼ 1 + A1 + A2 αi + A3 α3i + A4 α14
i
8
where
A1 ¼ 0.419869
A2 ¼ 0.505839
A3 ¼ 1.56436
A4 ¼ 9481.7
Step 2. Calculate the properties of the plus fraction by applying the
following relationships:
n
Critical temperature of the plus fraction, TC + , in °R:
TC + ¼ TcP
1 + 2fT 2
1 2fT
With the function “fT” as given by
fT ¼ exp 5 γ p γ C + 1
"
#
!
A1
A3 exp 5 γ p γ C + 1
+ A2 + 0:5
0:5
TbC
TbC +
+
where
A1 ¼ 0.362456
A2 ¼ 0.0398285
A3 ¼ 0.948125
Group 3. The Plus Fraction Characterization Methods 107
n
Critical volume of the plus fraction, vC + , in ft3/lbm-mol:
vC + ¼ vcP
1 + 2fv 2
1 2fv
with
!
"
#
h i
o A
h i
o
A3 n
1
2
2
2
2
exp 4 γ p γ C + 1
fv ¼ exp 4 γ p γ C + 1
+ A2 + 0:5
0:5
TbC
TbC +
+
n
where
A1 ¼ 0.466590
A2 ¼ 0.182421
A3 ¼ 3.01721
n
Critical pressure of the plus fraction, pC + , in psia:
pC + ¼ pcP
TC +
TcP
vcP
vC +
1 + 2fp 2
1 2fp
with
n
i
o
fp ¼ exp 0:5 γ p γ C + 1
n
h
h i
o
exp 0:5 γ p γ C + 1
!
A2
A1 + 0:5 + A3 TbC +
TbC +
!!
!
A5
+ A4 + 0:5 + A6 TbC +
TbC +
where
A1 ¼ 2.53262
A2 ¼ 46.19553
A3 ¼ 0.00127885
A4 ¼ 11.4277
A5 ¼ 252.14
A6 ¼ 0.00230535
Silva and Rodriguez Correlations
Silva and Rodriguez (1992) suggested the use of the following two expressions to estimate the boiling point temperature and specific gravity from the
molecular weight:
M
Tb ¼ 460 + 447:08723 ln
64:2576
Use the preceding calculated value of Tb to calculate the specific gravity of
the fraction from the following expression:
γ ¼ 0:132467 ln ðTb 460Þ + 0:0116483
where the boiling point temperature Tb is expressed in °R.
108 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Sancet Correlations
Sancet (2007) presented a set expressions to estimate the critical properties
and boiling point from the molecular weight “M”; these correlations have
the following forms:
pc ¼ 653 exp ð0:007427MÞ + 82:82
Tc ¼ ½383:5 ln ðM 4:075Þ 778:5
h
i
Tb ¼ 0:001241ðTc Þ1:869 + 194
where the boiling point temperature Tb and Tc are expressed in °R.
Comparison of Molecular Weight Methods Correlations
Mudgal (2014) compare several methodologies for predicting the molecular
weight from the boiling point temperature data. The author estimated the
molecular weight by using Lee-Kesler, Riazi-Daubert, Winn nomographs,
Goossens, and Twu methods. Mudgal concluded that Goossens and Twu
methods yielded appropriate estimations of the molecular weight. In addition, Ahmed (2014) applied the concept of the ECN and equivalent carbon
number “n” to match Mudgal’s boiling point temperature “Tb”; as given previously in the chapter by
Tb ¼ 427:2959078 + 50:08577848ðECNÞ 0:88693418ðECNÞ2
+ 0:00675667ðECNÞ3 551:2778516
ðECNÞ
with ‘n’ as given by
n ¼ INT½ðECNÞ + 1
The equivalent carbon number “n” is used to estimate the molecular weight
from
M ¼ 131:11375 + 24:96156n
0:34079022n2 + 0:002494118n3 + ð468:32575=nÞ
The above expression shows a good agreement with Twu’s methodology;
the tabulated comparison is shown in Table 2.8 and presented in a graphical
form in Fig. 2.8.
Group 3. The Plus Fraction Characterization Methods 109
Table 2.8 Comparison Molecular Weight Correlations
LeeKesler
RiazDaubert
Winn
Twu
Goossens
Ahmed
Fractions
Tb (°K)
Tb (°R)
M
M
M
M
M
n
M
1
2
3
4
5
6
7
8
9
10
Residue
363.15
403.15
453.15
473.15
498.15
548.15
583.15
593.15
613.15
644.35
764.83
653.67
725.67
815.67
851.67
896.67
986.67
1049.67
1067.67
1103.67
1159.83
1376.694
87.78
106.15
131.79
143.32
155.18
190.82
219.15
225.15
241.19
268.56
434.2
92.68
117.51
149.16
160.25
178.52
217.03
248.41
256.75
274.79
305.92
450.25
91.02
112.55
139.77
148.29
163.66
194.13
219.9
225.48
239.71
265.62
414.18
94.3
115.35
144.77
155.62
173.02
209.98
240.97
249.21
267.45
300.34
474.49
92
112
143.5
156
175
225.5
242
250
251
270
345
7
8.4
10.6
11.6
12.9
15.7
18
18.7
20.2
22.7
36.5
94.67761
111.7485
142.3398
156.8496
175.8379
216.263
248.3421
257.8502
277.7957
309.7129
460.0781
Molecular weight vs. boiling point
Lee-Kesler
Riaz-Daubert
Winn Nomogram
Twu
Goossens
Ahmed
Silva-Rodriguez
1200
1300
1400
600
500
400
M 300
200
100
0
600
700
800
900
n FIGURE 2.8 Comparison of molecular weight estimation methods.
1000
1100
Tb (°R)
1500
110 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
EXAMPLE 2.1
Estimate the critical properties and the acentric factor of the heptanes-plus
fraction, that is, C7+, with a measured molecular weight of 150 and specific
gravity of 0.78 by using:
n
n
the Riazi-Daubert equation ((2.6)) and Edmister’s equation for calculating
the acentric factor
Eqs. (2.3A)–(2.3E)
Solution
Applying the Riazi-Daubert equation:
n
θ ¼ aðMÞb γ c exp ½dM + eγ + f γM
Tc ¼ 544:2ð150Þ0:2998 ð0:78Þ1:0555 exp 1:3478 104 ð150Þ 0:61641ð0:78Þ + 0
¼ 1139:4°R
pc ¼ 4:5203 104 ð150Þ0:8063 ð0:78Þ1:6015
exp 1:8078 103 ð150Þ 0:3084ð0:78Þ + 0
¼ 320:3 psia
Vc ¼ 1:206 102 ð150Þ0:20378 ð0:78Þ1:3036 exp 2:657 103 ð150Þ
+ 0:5287ð0:78Þ2:6012 103 ð150Þð0:78Þ ¼ 0:06035 ft3 =lb
Tb ¼ 6:77857ð150Þ0:401673 ð0:78Þ1:58262 exp 3:77409 103 ð150Þ
+ 2:984036ð0:78Þ 4:25288 103 ð150Þð0:78Þ ¼ 825:26°R
Use Edmister’s equation (Eq. (2.21)) to estimate the acentric factor:
ω¼
3½ log ðpc =14:70Þ
1
7½Tc =Tb 1
ω¼
3½ log ð320:3=14:7Þ
1 ¼ 0:5067
7½1139:4=825:26 1
Applying Eqs. (2.3A)–(2.3E)
0:351579776
M
M
233:3891996 ¼ 1170°R
Tc ¼ 231:9733906
0:352767639
γ
γ
n
0:885326626
M
M
pc ¼ 31829
0:106467189
+ 49:62573013 ¼ 331:6 psi
γ
γ
M
M
+ 0:00141973
1:243207091 ¼ 0:499
ω ¼ 0:279354619 ln
γ
γ
0:984070268
M
vc ¼ 0:054703719 Mγ
+ 87:7041 106
γ
0:001007476 ¼ 9:69 ft3 =lbm mol ¼ 0:0646 ft3 =lbm
Group 3. The Plus Fraction Characterization Methods 111
0:664115311
M
M
379:4428973
Tb ¼ 33:72211
1:40066452
¼ 837°R
γ
γ
ðM=γ Þ
n
Silva and Rodriguez Correlations
Boiling point temperature and specific gravity of the cut are not available:
M
¼ 840°R
Tb ¼ 460 + 447:08723 ln
64:2576
γ ¼ 0:132467 ln ðTb 460Þ + 0:0116483 ¼ 0:7982
The following table summarizes the results for this example.
Method
Tc (°R)
pc (psia)
Vc (ft3/lb)
Tb (°R)
ω
Riazi-Daubert
Silva and Rodriguez
Sancet
Eqs. (2.3A)–(2.3E)
1139
–
1133
1170
320
–
297
331
0.0604
–
–
0.0646
825
840
828
837
0.507
–
–
0.499
EXAMPLE 2.2
Estimate the critical properties, molecular weight, and acentric factor of a
petroleum fraction with a boiling point of 198°F (658°R) and specific gravity of
0.7365 by using the following methods:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Riazi-Daubert (Eq. (2.5))
Riazi-Daubert (Eq. (2.5))
Cavett
Kesler-Lee
Winn and Sim-Daubert
Watansiri-Owens-Starling
Willman and Teja
Eqs. (2.3F)–(2.3J)
Solution
(a) Using Riazi-Daubert (Eq. (2.4))
θ ¼ aTbb γ c
M ¼ 4:5673 105 ð658Þ2:1962 ð0:7365Þ1:0164 ¼ 96:4
Tc ¼ 24:2787ð658Þ0:58848 ð0:7365Þ0:3596 ¼ 990:67°R
pc ¼ 3:12281 109 ð658Þ2:3125 ð0:7365Þ2:3201 ¼ 466:9 psia
Vc ¼ 7:5214 103 ð658Þ0:2896 ð0:7365Þ0:7666 ¼ 0:06227 ft3 =lb
112 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Solve for Zc by applying Eq. (2.22):
Zc ¼
pc Vc M ð466:9Þð0:06227Þð96:4Þ
¼
¼ 0:26365
RTc
ð10:73Þð990:67Þ
Solve for ω by applying Eq. (2.21):
ω¼
3½ log ðpc =14:70Þ
1
7½Tc =Tb 1
ω¼
3½ log ð466:9=14:7Þ
1 ¼ 0:2731
7½ð990:67=658Þ 1
(b) Riazi-Daubert (Eq. (2.5))
θ ¼ aTbb γ c exp ½dTb + eγ + fTb γ Applying the above expression and using the appropriate constants yields:
M ¼ 581:96ð658Þ0:97476 0:73656:51274 exp 5:4307 104 ð658Þ
+ 9:53384ð0:7365Þ + 1:11056 103 ð658Þð0:7365Þ M ¼ 96:911
Tc ¼ 10:6443ð658Þ0:810676 0:73650:53691 exp 5:1747 104 ð658Þ
0:54444ð0:7365Þ + 3:5995 104 ð658Þð0:7365Þ Tc ¼ 985:7°R
Similarly,
pc ¼ 465:83 psia
Vc ¼ 0:06257 ft3 =lb
The acentric factor and critical compressibility factor can be obtained by
applying Eqs. (2.21) and (2.22), respectively.
ω¼
3½ log ðpc =14:70Þ
3½ log ð465:83=14:70Þ
1¼
1 ¼ 0:2877
7½Tc =Tb 1
7½ð986:7=658Þ 1
Zc ¼
pc Vc M ð465:83Þð0:06257Þð96:911Þ
¼
¼ 0:2668
RTc
10:73ð986:7Þ
(c) Cavett’s Correlations:
Step 1. Solve for Tc using Eq. (2.7) with the coefficients
i
ai
bi
0
1
2
3
4
5
6
7
768.0712100000
1.7133693000
0.0010834003
0.0089212579
0.3889058400 106
0.5309492000 105
0.3271160000 107
2.82904060
0.94120109 103
0.30474749 105
0.20876110 104
0.15184103 108
0.11047899 107
0.48271599 107
0.13949619 109
Group 3. The Plus Fraction Characterization Methods 113
Tc ¼ a0 + a1 ðTb Þ + a2 ðTb Þ2 + a3 ðAPIÞðTb Þ + a4 ðTb Þ3 + a5 ðAPIÞðTb Þ2 + a6 ðEÞ2 ðTb Þ2
to give Tc ¼ 978.1°R
Step 2. Calculate pc with Eq. (2.8):
log ðpc Þ ¼ b0 + b1 ðTb Þ + b2 ðTb Þ2 + b3 ðAPIÞðTb Þ + b4 ðTb Þ3 + b5 ðAPIÞðTb Þ2
+ b6 ðAPIÞ2 ðTb Þ + b7 ðAPIÞ2 ðTb Þ2
to give pc ¼ 466.1 psia.
Step 3. Solve for the acentric factor by applying the Edmister correlation
(Eq. (2.21)):
ω¼
3½ log ð466:1=14:7Þ
1 ¼ 0:3147
7½ð980=658Þ 1
Step 4. Compute the critical compressibility by using Eq. (2.25):
Zc ¼ 0:291 0:080ω 0:016ω2
Zc ¼ 0:291 ð0:08Þð0:3147Þ 0:016ð0:3147Þ2 ¼ 0:2642
Step 5. Estimate vc from Eq. (2.22):
vc ¼
Zc RTc ð0:2642Þð10:731Þð980Þ
¼
¼ 5:9495 ft3 =lb mol
pc
466:1
Assume a molecular weight of 96; estimate the critical volume:
Vc ¼
5:9495
¼ 0:06197 ft3 =lb
96
(d) Kesler and Lee Correlations:
Step 1. Calculate pc from Eq. (2.9):
lnðpc Þ ¼ 8:3634 0:0566=γ 0:24244 + 2:2898=γ + 0:11857=γ 2 103 Tb
+ 1:4685 + 3:648=γ + 0:47227=γ 2 107 Tb2
0:42019 + 1:6977=γ 2 1010 Tb3
to give pc ¼ 470 psia.
Step 2. Solve for Tc by using Eq. (2.10); that is,
Tc ¼ 341:7 + 811:1γ + ½0:4244 + 0:1174γ Tb +
½0:4669 3:26238γ 105
Tb
to give Tc ¼ 980°R.
Step 3. Calculate the molecular weight, M, by using Eq. (2.11):
M ¼ 12,272:6 + 9,486:4γ + ½4:6523 3:3287γ Tb + 1 0:77084γ 0:02058γ 2
720:79 107
181:98 1012
1:3437 + 1 0:80882γ 0:02226γ 2 1:8828 Tb
Tb
Tb
Tb3
to give M ¼ 98.7
114 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 4. Compute the Watson characterization factor K and the parameter θ:
ð658Þ1=3
¼ 11:8
0:7365
658
θ¼
¼ 0:671
980
K¼
Step 5. Solve for acentric factor by applying Eq. (2.13):
ω¼
ln
h p i
6:09648
+ 1:28862 ln ½θ 0:169347θ6
θ
¼ 0:306
15:6875
15:2518 13:4721 ln ½θ + 0:43577θ6
θ
c
14:7
5:92714 +
Step 6. Estimate for the critical gas compressibility, Zc, by using
Eq. (2.26):
Zc ¼ 0:2918 0:0928ω
Zc ¼ 0:2918 ð0:0928Þð0:306Þ ¼ 0:2634
Step 7. Solve for Vc by applying Eq. (2.22):
Vc ¼
Zc RTc ð0:2634Þð10:73Þð980Þ
¼
¼ 0:0597 ft3 =lb
pc M
ð470Þð98:7Þ
(e) Winn-Sim-Daubert Correlations:
Step 1. Estimate pc from Eq. (2.14):
γ 2:4853
pc ¼ 3:48242 109 2:3177
Tb
pc ¼ 478:6 psia
Step 2. Solve for Tc by applying Eq. (2.15):
Tc ¼ exp 3:9934718Tb0:08615 γ 0:04614
Tc ¼ 979:2°R
Step 3. Calculate M from Eq. (2.16):
Tb2:3776
M ¼ 1:4350476 105 0:9371
γ
M ¼ 95:93
Step 4. Solve for the acentric factor from Eq. (2.21):
ω¼
3½ log ðpc =14:70Þ
1
7½ðTc =Tb 1Þ
ω ¼ 0.3280
Group 3. The Plus Fraction Characterization Methods 115
Solve for Zc by applying Eq. (2.24):
Zc ¼ 0:291 0:080ω
Zc ¼ 0:291 ð0:08Þð0:3280Þ ¼ 0:2648
Step 5. Calculate the critical volume Vc from Eq. (2.22):
Vc ¼
Vc ¼
Zc RTc
pc M
ð0:2648Þð10:731Þð979:2Þ
¼ 0:06059 ft3 =lb
ð478:6Þð95:93Þ
(f) Watansiri-Owens-Starling Correlations:
Step 1. Because Eqs. (2.17) through (2.19) require the molecular weight,
assume M ¼ 96.
Step 2. Calculate Tc from Eq. (2.17):
lnðTc Þ ¼ 0:0650504 0:00052217Tb + 0:03095 ln ½M + 1:11067 lnðTb Þ
h
i
+ M 0:078154γ 1=2 0:061061γ 1=3 0:016943γ Tc ¼ 980:0°R
Step 3. Determine the critical volume from Eq. (2.18) to give
lnðVc Þ ¼ 76:313887 129:8038γ + 63:1750γ 2 13:175γ 3
+ 1:10108 ln ½M + 42:1958 ln ½γ Vc ¼ 0:06548 ft3 =lb
Step 4. Solve for the critical pressure of the fraction by applying Eq. (2.19)
to produce
0:8
Tc
M
Tb
0:08843889
8:712
lnðpc Þ ¼ 6:6418853 + 0:01617283
Vc
M
Tc
pc ¼ 426:5 psia
Step 5. Calculate the acentric factor from Eq. (2.20)
8
9
5:12316667 104 Tb + 0:281826667ðTb =MÞ + 382:904=M >
>
>
>
>
>
>
>
2
>
>
5
4
>
>
ð
T
=γ
Þ
0:12027778
10
ð
T
Þ
ð
M
Þ
+
0:074691
10
>
>
b
b
>
>
>
>
>
>
2
4
>
>
>
>
ð
Þ
ð
Þ
ð
Þ
M
+
0:001261
γ
M
+
0:1265
10
<
=5T b
ω¼
1=3
ðTb Þ
>
>
9M
2
4
>
>
>
>
> + 0:2016 10 ðγ ÞðMÞ 66:29959 M
>
>
>
>
>
>
>
>
>
>
>
2=3
>
>
T
>
>
b
>
>
: 0:00255452 2
;
γ
to give ω ¼ 0.2222
116 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 6. Compute the critical compressibility factor by applying Eq. (2.24):
Zc ¼ 0:2918 0:0928ω ¼ 0:2918 0:0928ð0:2222Þ ¼ 0:27112
(g) Willman and Teja Correlations:
Step 1. Using Microsoft Solver, determine the ECN “n” by matching the
given boiling point “Tb”
Tb ¼ 434:38878 + 50:125279n 0:9097293n2 + 7:0280657 103 n3
601:85651=n
658 ¼ 434:38878 + 50:125279n 0:9097293n2 + 7:0280657 103 n3
601:85651=n
To give n ¼ 7
Step 2. Calculate Tc from Eq. (2.33):
h
i
Tc ¼ Tb 1 + ð1:25127 + 0:137252nÞ0:884540633
h
i
Tc ¼ Tb 1 + ð1:25127 + 0:137252ð7ÞÞ0:884540633 ¼ 983:7°R
Step 3. Calculate pc from Eq. (2.34):
Pc ¼
Pc ¼
339:0416805 + 1184:157759n
½0:873159 + 0:54285n1:9265669
339:0416805 + 1184:157759ð7Þ
½0:873159 + 0:54285ð7Þ1:9265669
¼ 441:8 psia
(h) Eqs. (2.3F) through (2.3J):
0:967105064
Tb
Tb
+ 0:010009512 ¼ 996°R
Tc ¼ 1:379242274
+ 0:012011487
γ
γ Tb
pc ¼ 2898:631565 exp ½0:002020245 ðTb =γ Þ 0:00867297
+ 0:099138742
γ
¼ 469 psi
1:250922701
Tb
Tb
ω ¼ 0:000115057
0:92098982 ¼ 0:26
+ 0:00068975
γ
γ
3
2
Tb
vc ¼ 8749:94 106 Tγb 9:75748 106 Tγb + 0:014653
γ
595:7922076
¼ 6:273 ft3 =lbm mol
Tb
γ
0:001617123Tb
M ¼ 52:11038728exp
132:1481762 ¼ 89
γ
4:60231684 Group 3. The Plus Fraction Characterization Methods 117
The following table summarizes the results for this example.
Tc (°R) pc (psia) Vc (ft3/lb) M
Method
Riazi-Daubert no. 1 990.67
Riazi-Daubert no. 2 986.70
Cavett
978.10
Kesler-Lee
980.00
Winn
979.20
Watansiri
980.00
Willman-Teja
983.7
Eqs. (2.3F)–(2.3J) 996
466.90
465.83
466.10
469.00
478.60
426.50
441.8
469
0.06227
0.06257
0.06197
0.05970
0.06059
0.06548
–
0.0689
96.400
96.911
–
98.700
95.930
–
–
89
ω
Zc
0.2731 0.26365
0.2877 0.26680
0.3147 0.26420
0.3060 0.26340
0.3280 0.26480
0.2222 0.27112
–
–
0.26
–
EXAMPLE 2.3
If the molecular weight and specific gravity of the heptanes-plus fraction are
216 and 0.8605, respectively, calculate the critical temperature and pressure by
using
(a) Rowe’s correlations.
(b) Standing’s correlations.
(c) Magoulas-Tassios’s correlations.
(d) Eqs. (2.3A) and (2.3B).
Solution
(a) Rowe’s correlations
Step 1. Calculate the number of carbon atoms of C7+ from Eq. (2.28) to give
n¼
MC7 + 2:0 216 2:0
¼
¼ 15:29
14
14
Step 2. Calculate the coefficient a:
a ¼ 2:95597 0:090597ðnÞ2=3
a ¼ 2:95597 0:090597ð15:29Þ2=3 ¼ 2:39786
Step 3. Solve for the critical temperature from Eq. (2.27) to yield
ðTc ÞC7 + ¼ 1:8½961 10a ðTc ÞC7 + ¼ 1:8 961 102:39786 ¼ 1279:8°R
Step 4. Calculate the coefficient Y:
Y ¼ 0:0137726826n + 0:6801481651
Y ¼ 0:0137726826ð2:39786Þ + 0:6801481651 ¼ 0:647123
118 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 5. Solve for the critical pressure from Eq. (2.29) to give
ðpc ÞC7 + ¼
10ð4:89165 + Y Þ 10ð4:89165 + 0:647123Þ
¼
¼ 270 psi
ðTc ÞC7 +
1279:8
(b) Standing’s correlations
Step 1. Solve for the critical temperature by using Eq. (2.31) to give
h
i h
i
ðTc ÞC7 + ¼ 608 + 364 log ðMÞC7 + 71:2 + 2450log ðMÞC7 + 3800 log ðγ ÞC7 +
ðTc ÞC7 + ¼ 1269:3°R
Step 2. Calculate the critical pressure from Eq. (2.32) to yield
h
i h
h
i
ðpc ÞC7 + ¼ 1188 431 log ðMÞC7 + 61:1 + 2319 852 log ðMÞC7 + 53:7
h
i
ðγ ÞC7 + 0:8 ðpc ÞC7 + ¼ 270 psia
(c) Magoulas-Tassios correlations
Tc ¼ 1247:4 + 0:792M + 1971γ 27,000 707:4
+
M
γ
27, 000 707:4
+
¼ 1317°R
216
0:8605
227 1:1663
+ 1:2702 lnðMÞ
lnðpc Þ ¼ 0:01901 0:0048442M + 0:13239γ +
M
γ
Tc ¼ 1247:4 + 0:792ð216Þ + 1971ð0:8605Þ lnðpc Þ ¼ 0:01901 0:0048442ð216Þ + 0:13239ð0:8605Þ
227 1:1663
+ 1:2702 ln ð216Þ ¼ 5:6098
216 0:8605
pc ¼ exp ð5:6098Þ ¼ 273 psi
+
(d) Eqs. (2.3A) and (2.3B)
0:351579776
M
M
Tc ¼ 231:9733906
0:352767639
233:3891996 ¼ 1294°R
γ
γ
pc ¼ 31829
0:885326626
M
M
0:106467189
+ 49:62573013 ¼ 262 psi
γ
γ
Summary of results is given below:
Method
Tc (°R)
pc (psia)
Rowe
Standing
Magoulas-Tassios
Eqs. (2.3A) and (2.3B)
1279
1269
1317
1294
270
270
273
262
Group 3. The Plus Fraction Characterization Methods 119
EXAMPLE 2.4
Calculate the critical properties and the acentric factor of C7+ with a measured
molecular weight of 198.71 and specific gravity of 0.8527. Employ the
following methods:
(a) Rowe’s correlations.
(b) Standing’s correlations.
(c) Riazi-Daubert’s correlations.
(d) Magoulas-Tassios’s correlations.
(e) Eqs. (2.3A) through (2.3E).
Solution
(a) Rowe’s correlations
Step 1. Calculate the number of carbon atoms, n, and the coefficient a of the
fraction to give
n¼
M 2 198:71 2
¼
¼ 14:0507
14
14
a ¼ 2:95597 0:090597n2=3
a ¼ 2:95597 0:090597ð14:0507Þ2=3 ¼ 2:42844
Step 2. Determine Tc from Eq. (2.28) to give
ðTc ÞC7 + ¼ 1:8½961 10a ðTc ÞC7 + ¼ 1:8 961 102:42844 ¼ 1247°R
Step 3. Calculate the coefficient Y:
Y ¼ 0:0137726826n + 0:6801481651
Y ¼ 0:0137726826ð2:42844Þ + 0:6801481651 ¼ 0:6467
Step 4. Compute pc from Eq. (2.29) to yield
ðpc ÞC7 + ¼
10ð4:89165 + 0:6467Þ
¼ 277 psi
1247
Step 5. Determine Tb by applying Eq. (2.30) to give
ðTb ÞC7 + ¼ 0:0004347ðTc Þ2C7 + + 265 ¼ 0:0004347ð1247Þ2 + 265 ¼ 941°R
Step 6. Solve for the acentric factor by applying Eq. (2.21) to give
ω¼
3½ log ðpc =14:70Þ
1
7½ðTc =Tb 1Þ
ω ¼ 0:6123
120 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
(b) Standing’s correlations
Step 1. Solve for the critical temperature of C7+ by using Eq. (2.31) to give
h
i h
i
ðTc ÞC7 + ¼ 608 + 364 log ðMÞC7 + 71:2 + 2450log ðMÞC7 + 3800 log ðγ ÞC7 +
ðTc ÞC7 + ¼ 1247:73°R
Step 2. Calculate the critical pressure from Eq. (2.32) to give
h
i h
h
i
ðpc ÞC7 + ¼ 1188 431 log ðMÞC7 + 61:1 + 2319 852 log ðMÞC7 + 53:7
h
i
ðγ ÞC7 + 0:8 ðpc ÞC7 + ¼ 291:41 psia
(c) Riazi-Daubert correlations
θ ¼ aðMÞb γ c exp ½dM + eγ + f γM
Tc ¼ 544:2ð198:71Þ0:2998 ð0:8577Þ1:0555
exp 1:3478 104 ð198:71Þ 0:61641ð0:8577Þ ¼ 1294°R
pc ¼ 4:5203 104 ð198:71Þ0:8061 ð0:8577Þ1:6015
exp 1:8078 103 ð198:71Þ 0:3084ð0:8577Þ ¼ 264 psi
Determine Tb by applying Eq. (2.6) to give Tb ¼ 958.5°R.
Solve for the acentric factor from Eq. (2.21) to give
ω¼
3½ log ðpc =14:70Þ
1
7½ðTc =Tb 1Þ
ω ¼ 0:5346
(d) Magoulas-Tassios correlations
Tc ¼ 1247:4 + 0:792M + 1971γ 27,000 707:4
+
M
γ
Tc ¼ 1247:4 + 0:792ð198:71Þ + 1971ð0:8527Þ lnðpc Þ ¼ 0:01901 0:0048442M + 0:13239γ +
27, 000 707:4
+
¼ 1284°R
198:71 0:8527
227 1:1663
+ 1:2702 lnðMÞ
M
γ
lnðpc Þ ¼ 0:01901 0:0048442ð198:71Þ + 0:13239ð0:8527Þ +
1:1663
+ 1:2702 ln ð198:71Þ ¼ 5:6656
0:8527
pc ¼ exp ð5:6656Þ ¼ 289 psi
227
198:71
Group 3. The Plus Fraction Characterization Methods 121
4:559
+ 0:21699 ln ðMÞ
M
4:559
ω ¼ 0:64235 + 0:0014667ð198:71Þ + 0:021876ð0:8527Þ
198:71
+ 0:21699 ln ð198:71Þ ¼ 0:531
ω ¼ 0:64235 + 0:0014667M + 0:21876γ
(e) Eqs. (2.3A) through (2.3E)
0:351579776
M
M
Tc ¼ 231:9733906
0:352767639
233:3891996 ¼ 1259°R
γ
γ
0:885326626
M
M
+ 49:62573013 ¼ 280 psi
0:106467189
γ
γ
M
M
ω ¼ 0:279354619 ln
+ 0:00141973
1:243207091 ¼ 0:61
γ
γ
pc ¼ 31829
0:984070268
M
M
+ 87:7041 106
vc ¼ 0:054703719
0:001007476
γ
γ
¼ 11:717 ft3 =lbm-mol ¼ 0:0589 ft3 =lbm
0:664115311
M
M
379:4428973
Tb ¼ 33:72211
1:40066452
¼ 931°R
γ
γ
ðM=γ Þ
The following table summarizes the results for this example.
Method
Tc (°R)
pc (psia)
Tb (°R)
ω
Rowe’s
Standing’s
Riazi-Daubert
Magoulas-Tassios
Eqs. (2.3A)–(2.3E)
1247
1247
1294
1284
1259
277
291
264
289
280
941
–
950
–
931
0.612
–
0.534
0.531
0.61
Recommended Plus Fraction Characterizations
As discussed in Chapter 5, numerous forms of cubic equations of state that
are widely used in the petroleum industry (eg, Peng-Robinson EOS)
require the critical properties and acentric factor for each of the defined
and undefined petroleum fractions in the hydrocarbon system. The characterizations of the undefined petroleum fractions rely on applying generalized correlations with their parameters, which were developed by
regression to match critical and physical properties of pure components.
Selected generalized correlations, often with significantly diverging
results among themselves, are used in equations of state applications that
122 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
commonly produce erroneous EOS results. As a result, equations of state
are not predictive and require adjusting or “tuning” the critical properties
of the undefined petroleum fractions by matching the available laboratory
PVT data. Recognizing the fact that tuning EOS requires the adjustment of
the critical properties of undefined hydrocarbon groups, the need is to
select a generalized property correlation (or correlations) that is capable
of producing a logical and consistent properties trend of the hydrocarbon
groups (eg, Tc and ω are increasing and pc is decreasing with increasing
molecular weight of the hydrocarbon group). The Riazi-Daubert critical
properties and Kesler-Lee acentric factor generalized correlations provide
the suggested required properties trend and, therefore, are recommended
for use in EOS applications.
As of the determination of the exact composition and critical characteristics
of the heavy-end plus fraction (eg, C7+) is a nearly impossible task, a useful
type of compositional analysis is to determine the relative amounts of paraffins “P,” naphthenes “N,” and aromatics “A” for pseudofractions distilled
by the TBP analysis. Detailed PNA analysis could provide accurate estimation of the physical and critical properties of these petroleum fractions. The
crude oil assay testing could include such measurements; however, analysis
could be costly and time consuming. This author agrees with Whitson and
Brule (2000) that PNA determination has a limited usefulness for improving
equations of state fluid characterizations. However, detail of the PNA
determination methodologies is summarized below.
PNA Determination
The vast number of hydrocarbon compounds making up naturally occurring
crude oil has been grouped chemically into several series of compounds.
Each series consists of those compounds similar in their molecular makeup
and characteristics. Within a given series, the compounds range from
extremely light, or chemically simple, to heavy, or chemically complex.
In general, it is assumed that the heavy (undefined) hydrocarbon fractions
are composed of three hydrocarbon groups:
n
n
n
paraffins (P)
naphthenes (N)
aromatics (A)
The PNA content of the plus fraction (the undefined hydrocarbon fraction)
can be estimated experimentally from distillation or a chromatographic analysis. Both types of analysis provide information valuable for use in characterizing the plus fractions. Bergman et al. (1977) outlined the chromatographic
analysis procedure by which distillation cuts are characterized by the density
Group 3. The Plus Fraction Characterization Methods 123
and molecular weight as well as by weight average boiling point (WABP).
It should be pointed out that when a single boiling point is given for a plus
fraction, it is given as its volume average boiling point (VABP).
Generally, five methods are used to define the normal boiling point for the
plus fraction (eg, C7+) these are
1. VABP — This is defined mathematically by the following expression:
X
Vi Tbi
VABP ¼
(2.36)
i
where
Tbi ¼ boiling point of the distillation cut i, °R
Vi ¼ volume fraction of the distillation cut i
2. WABP — This property is defined by the following expression:
WABP ¼
X
wi Tbi
(2.37)
i
where
wi ¼ weight fraction of the distillation cut i
3. Molar average boiling point (MABP) — The MABP is given by the
following relationship:
MABP ¼
X
xi Tbi
(2.38)
i
where
xi ¼ mole fraction of the distillation cut i
4. Cubic average boiling point (CABP) — The relationship is defined as
"
#3
X
1=3
CABP ¼
xi Tbi
(2.39)
i
5. Mean average boiling point (MeABP) — This is defined by
MeABP ¼
MABP + CABP
2
(2.40)
The MeABP (Tb) is recommended as the most representative boiling point
of the plus fraction because it allows better reproduction of the properties of
the entire mixture.
As indicated by Edmister and Lee (1984), the above five expressions for
calculating normal boiling points result in values that do not differ significantly from one another for narrow boiling petroleum fractions. The three
parameters that are commonly employed to estimate the PNA content of
the heavy hydrocarbon fraction include:
124 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
n
n
molecular weight
specific gravity
VABP or WABP
Hopke and Lin (1974), Erbar (1977), Bergman et al. (1977), and Robinson
and Peng (1978) used the above parameters and PNA concept to characterize the undefined hydrocarbon fractions in terms of their critical properties
and acentric factors. Two of the most widely used PNA based characterization methods are
n
n
Peng-Robinson Method
Bergman’s Method
Details of these two methods are summarized below
Peng-Robinson’s Method
Robinson and Peng (1978) proposed a detailed procedure for characterizing
heavy hydrocarbon fractions. The procedure is summarized in the following
steps.
Step 1. Calculate the PNA content (XP, XN, XA) of the undefined fraction by
solving the following three rigorously defined equations:
X
Xi ¼ 1
(2.41)
½Mi Tbi Xi ¼ ðMÞðWABPÞ
(2.42)
i¼P, N, A
X
i¼P, N, A
X
½Mi Xi ¼ M
(2.43)
i¼P, N, A
where
XP ¼ mole fraction of the paraffinic group in the undefined fraction
XN ¼ mole fraction of the naphthenic group in the undefined
fraction
XA ¼ mole fraction of the aromatic group in the undefined fraction
WABP ¼ weight average boiling point of the undefined fraction, °R
M ¼ molecular weight of the undefined fraction
(Mi) ¼ average molecular weight of each cut (ie, PNA)
(Tb)i ¼ boiling point of each cut, °R
Eqs. (2.41) through (2.43) can be written in a matrix form as follows:
2
1
1
1
3 2
XP
3
2
1
3
7
7 6 7
6
6
6 ½MTb P ½MTb N ½MTb A 7 6 XN 7 ¼ 6 MðWABPÞ 7
5
5 4 5
4
4
½M P
½M N
½M A
M
XA
(2.44)
Group 3. The Plus Fraction Characterization Methods 125
Robinson and Peng pointed out that it is possible to obtain negative
values for the PNA contents. To prevent these negative values, the
authors imposed the following constraints:
0 XP 0:90
XN 0:00
XA 0:00
Solving Eq. (2.44) for the PNA content requires the WABP and
molecular weight of the cut of the undefined hydrocarbon
fraction. Robinson and Peng proposed the following set of
working expressions to estimate the boiling point temperature
(Tb)P, (Tb)N, and (Tb)A and the molecular waiting (M)P, (M)N,
and (M)A:
Paraffinic group :
ln ðTb ÞP ¼ ln ð1:8Þ +
6 h
i
X
ai ðn 6Þi1
(2.45)
i¼1
Naphthenic group :
ln ðTb ÞN ¼ ln ð1:8Þ +
6 h
i
X
ai ðn 6Þi1
(2.46)
i¼1
Aromatic group :
ln ðTb ÞA ¼ ln ð1:8Þ +
6 h
i
X
ai ðn 6Þi1
(2.47)
i¼1
Paraffinic group : ðMÞP ¼ 14:026n + 2:016
(2.48)
Naphthenic group : ðMÞN ¼ 14:026n 14:026
(2.49)
Aromatic group : ðMÞA ¼ 14:026n 20:074
(2.50)
where
n ¼ number of carbon atoms in the undefined hydrocarbon
fraction, eg, 7, 8, …, etc.
ai ¼ coefficients of the equations, which are given below
Coefficient
Paraffin, P
Napthene, N
Aromatic, A
a1
a2
a3
a4
a5
a6
5.83451830
5.85793320
5.86717600
0.84909035 101 0.79805995 101 0.80436947 101
0.52635428 102 0.43098101 102 0.47136506 102
0.21252908 103 0.14783123 103 0.18233365 103
0.44933363 105 0.27095216 105 0.38327239 105
0.37285365 107 0.19907794 107 0.32550576 107
126 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 2. Having obtained the PNA content of the undefined hydrocarbon
fraction, as outlined in step 1, calculate the critical pressure of the
fraction by applying the following expression:
pc ¼ XP ðpc ÞP + XN ðpc ÞN + XA ðpc ÞA
(2.51)
where
pc ¼ critical pressure of the heavy hydrocarbon fraction, psia.
The critical pressure for each cut of the heavy fraction is calculated
according to the following equations:
Paraffinic group : ðpc ÞP ¼
Naphthenic group : ðpc ÞN ¼
206:126096n + 29:67136
ð0:227n + 0:340Þ2
206:126096n 206:126096
Aromatic group : ðpc ÞA ¼
ð0:227n 0:137Þ2
206:126096n 295:007504
ð0:227n 0:325Þ2
(2.52)
(2.53)
(2.54)
Step 3. Calculate the acentric factor of each cut of the undefined fraction by
using the following expressions:
Paraffinic group : ðωÞP ¼ 0:432n + 0:0457
(2.55)
Naphthenic group : ðωÞN ¼ 0:0432n 0:0880
(2.56)
Aromatic group : ðωÞA ¼ 0:0445n 0:0995
(2.57)
Step 4. Calculate the critical temperature of the fraction under consideration
by using the following relationship:
Tc ¼ XP ðTc ÞP + XN ðTc ÞN + XA ðTc ÞA
(2.58)
where Tc ¼ critical temperature of the fraction, °R.
The critical temperatures of the various cuts of the undefined
fractions are calculated from the following expressions:
(
)
3 log ðPc ÞP 3:501952
Paraffinic group : ðTc ÞP ¼ S 1 +
ð T b ÞP
7 1 + ðωÞP
(
Naphthenic group : ðTc ÞN ¼ S1
(
Aromatic group : ðTc ÞA ¼ S1
(2.59)
)
3 log ðPc ÞN 3:501952
1+
ðTb ÞN (2.60)
7 1 + ðωÞP
)
3 log ðPc ÞA 3:501952
1+
ðTb ÞA
7 1 + ðωÞA
(2.61)
Group 3. The Plus Fraction Characterization Methods 127
where the correction factors S and S1 are defined by the following
expressions:
S ¼ 0:99670400 + 0:00043155n
S1 ¼ 0:99627245 + 0:00043155n
Step 5. Calculate the acentric factor of the heavy hydrocarbon fraction
(residue) by using the Edmister’s correlation (Eq. (2.21)) to give
ω¼
3½ log ðPc =14:7Þ
1
7½ðTc =Tb Þ 1
(2.62)
where
ω ¼ acentric factor of the residue heavy fraction
Pc ¼ critical pressure of the heavy fraction, psia
Tc ¼ critical temperature of the heavy fraction, °R
Tb ¼ average weight boiling point, °R
EXAMPLE 2.5
Using the Peng and Robinson approach, calculate the critical pressure, critical
temperature, and acentric factor of an undefined hydrocarbon fraction that is
characterized by the laboratory measurements:
n
n
n
molecular weight ¼ 94
specific gravity ¼ 0.726133
WABP ¼ 195°F, ie, 655°R
Solution
Step 1. Calculate the boiling point of each cut by applying Eqs. (2.45) through
(2.47) to give
(
)
6 h
i
X
i1
¼ 666:58°R
ðTb ÞP ¼ exp ln ð1:8Þ +
ai ðn 6Þ
)
6 h
i
X
i1
¼ 630°R
ln ð1:8Þ +
ai ðn 6Þ
(
)
6 h
i
X
ln ð1:8Þ +
ai ðn 6Þi1 ¼ 635:85°R
ðTb ÞN ¼ exp
ðTb ÞA ¼ exp
i¼1
(
i¼1
i¼1
Step 2. Compute the molecular weight of various cuts by using Eqs. (2.48)
through (2.50) to give
Paraffinic group : ðMÞP ¼ 14:026n + 2:016
ðMÞP ¼ 14:026 7 + 2:016 ¼ 100:198
Naphthenic group : ðMÞN ¼ 14:026n 14:026
ðMÞN ¼ 14:0129 7 14:026 ¼ 84:156
128 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Aromatic group : ðMÞA ¼ 14:026n 20:074
ðMÞA ¼ 14:026 7 20:074 ¼ 78:180
Step 3. Solve Eq. (2.44) for XP, XN, and XA:
3
3 2 3
2
2
1
1
1
XP
1
7
7 6 7
6
6
6 ½MTb P ½MTb N ½MTb A 7 6 XN 7 ¼ 6 MðWABPÞ 7
5
5 4 5
4
4
½M P
½M N
½M A
M
XA
2
1
1
1
3 2
XP
3
2
1
3
7 6 7
7
6
6
6 66689:78 53018:28 49710:753 7 6 XN 7 ¼ 6 61570 7
5 4 5
5
4
4
199:198 84:156
78:180
94
XA
to give
XP ¼ 0:6313
XN ¼ 0:3262
XA ¼ 0:04250
Step 4. Calculate the critical pressure of each cut in the undefined fraction by
applying Eqs. (2.52) and (2.54):
ðpc ÞP ¼
ðpc ÞN ¼
ðpc ÞA ¼
206:126096 7 + 29:67136
ð0:227 7 + 0:340Þ2
¼ 395:70 psia
206:126096 7 206:126096
ð0:227 7 0:137Þ2
206:126096 7 295:007504
ð0:227 7 0:325Þ2
¼ 586:61
¼ 718:46
Step 5. Calculate the critical pressure of the heavy fraction from Eq. (2.51) to
give
pc ¼ XP ðpc ÞP + XN ðpc ÞN + XA ðpc ÞA
pc ¼ 0:6313ð395:70Þ + 0:3262ð586:61Þ + 0:0425ð718:46Þ ¼ 471 psia
Step 6. Compute the acentric factor for each cut in the fraction by using
Eqs. (2.55) through (2.57) to yield
ðωÞP ¼ 0:432 7 + 0:0457 ¼ 0:3481
ðωÞN ¼ 0:0432 7 0:0880 ¼ 0:2144
ðωÞA ¼ 0:0445 7 0:0995 ¼ 0:2120
Group 3. The Plus Fraction Characterization Methods 129
Step 7. Solve for (Tc)P, (Tc)N, and (Tc)A by using Eqs. (2.59) through (2.61)
to give
S ¼ 0:99670400 + 0:00043155 7 ¼ 0:99972
S1 ¼ 0:99627245 + 0:00043155 7 ¼ 0:99929
3 log ½395:7 3:501952
ðTc ÞP ¼ 0:99972 1 +
666:58 ¼ 969:4°R
7½1 + 0:3481
3 log ½586:61 3:501952
ðTc ÞN ¼ 0:99929 1 +
630 ¼ 974:3°R
7½1 + 0:2144
3 log ½718:46 3:501952
ðTc ÞA ¼ 0:99929 1 +
635:85 ¼ 1014:9°R
7½1 + 0:212
Step 8. Solve for (Tc) of the undefined fraction from Eq. (2.58):
Tc ¼ XP ðTc ÞP + XN ðTc ÞN + XA ðTc ÞA ¼ 964:1°R
Step 9. Calculate the acentric factor from Eq. (2.62) to give
ω¼
3½ log ð471:7=14:7Þ
1 ¼ 0:3680
7½ð964:1=655Þ 1
Bergman’s Method
Bergman et al. (1977) proposed a detailed procedure for characterizing the
undefined hydrocarbon fractions based on calculating the PNA content of
the fraction under consideration. The proposed procedure originated from
analyzing extensive experimental data on lean gases and condensate systems. In developing the correlation, the authors assumed that the paraffinic,
naphthenic, and aromatic groups have the same boiling point. The computational procedure is summarized in the following steps.
Step 1. Estimate the weight fraction of the aromatic content in the
undefined fraction by applying the following expression:
wA ¼ 8:47 0:7Kw
(2.63)
where
wA ¼ weight fraction of aromatics
Kw ¼ Watson characterization factor, defined mathematically by
the following expression
Kw ¼ ðTb Þ1=3 =γ
γ ¼ specific gravity of the undefined fraction
Tb ¼ WABP, °R
(2.64)
130 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Bergman et al. imposed the following constraint on the aromatic
content:
0:03 wA 0:35
Step 2. Bergman proposed the following set of expressions for estimating
specific gravities of the three groups, ie, γ P, γ N, and γ A
γ P ¼ 0:583286 + 0:00069481ðTb 460Þ
0:7572818 106 ðTb 460Þ2 + 0:3207736 109 ðTb 460Þ3
(2.65)
γ N ¼ 0:694208 + 0:0004909267ðTb 460Þ
0:659746 106 ðTb 460Þ2 + 0:330966 109 ðTb 460Þ3
(2.66)
γ A ¼ 0:916103 0:000250418ðTb 460Þ + 0:357967 106 ðTb 460Þ2
9 0:166318 10 ðTb 460Þ3
ð2:67Þ
Bergman suggested a minimum paraffin content of 0.20 was set by
Bergman et al. To ensure that this minimum value is met, the
estimated aromatic content that results in negative values of wP is
increased in increments of 0.03 up to a maximum of 15 times until
the paraffin content exceeds 0.20. They pointed out that this
procedure gives reasonable results for fractions up to C15.
Step 3. With the estimate of the aromatic content as outlined in step 1 and
specific gravity of the three PNA groups as calculated in step 2,
determine the weight fractions of the paraffinic and naphthenic cuts
by solving the following system of linear equations simultaneously:
wP + wN ¼ 1 wA
(2.68)
wP wN 1 wA
+
¼ γP γN γ γA
(2.69)
where
wP ¼ weight fraction of the paraffin cut
wN ¼ weight fraction of the naphthene cut
γ ¼ specific gravity of the undefined fraction
γ P, γ N, γ A ¼ specific gravity of the three groups at the WABP of
the undefined fraction
Combining Eq. (2.68) with (2.69) and solving for weight fraction of
the aromatic group gives
γ P γ N wA γ P γ N
ð1 wA Þγ P
γ
γA
wP
γN γP
wN ¼ 1 ðwA + wP Þ
Group 3. The Plus Fraction Characterization Methods 131
Step 4. Calculate the critical temperature, the critical pressure, and acentric
factor of each cut from the following expressions.
For paraffins,
ðTc ÞP ¼ 735:23 + 1:2061ðTb 460Þ 0:00032984ðTb 460Þ2
(2.70)
ðpc ÞP ¼ 573:011 1:13707ðTb 460Þ + 0:00131625ðTb 460Þ2
0:85103 106 ðTb 460Þ3
(2.71)
ðωÞP ¼ 0:14 + 0:0009ðTb 460Þ + 0:233 106 ðTb 460Þ2
(2.72)
For naphthenes,
ðTc ÞN ¼ 616:8906 + 2:6077ðTb 460Þ
0:003801ðTb 460Þ2 + 0:2544 105 ðTb 460Þ3
(2.73)
ðpc ÞN ¼ 726:414 1:3275ðTb 460Þ + 0:9846 103 ðTb 460Þ2
0:45169 106 ðTb 460Þ3
(2.74)
ðωÞN ¼ ðωÞP 0:075
(2.75)
Bergman et al. assigned the following special values of the acentric
factor to the C8, C9, and C10 naphthenes:
C8 ðωÞN ¼ 0:26
C9 ðωÞN ¼ 0:27
C10 ðωÞN ¼ 0:35
For the aromatics,
ðTc ÞA ¼ 749:535 + 1:7017ðTb 460Þ
0:0015843ðTb 460Þ2 + 0:82358 106 ðTb 460Þ3
(2.76)
ðpc ÞA ¼ 1184:514 3:44681ðTb 460Þ + 0:0045312ðTb 460Þ2
0:23416 105 ðTb 460Þ3
(2.77)
ðωÞA ¼ ðωÞP 0:1
(2.78)
Step 5. Calculate the critical pressure, the critical temperature, and acentric
factor of the undefined fraction from the following relationships:
pc ¼ wP ðpc ÞP + wN ðpc ÞN + wA ðpc ÞA
(2.79)
Tc ¼ wP ðTc ÞP + wN ðTc ÞN + wA ðTc ÞA
(2.80)
ω ¼ wP ðωÞP + wN ðωÞN + wA ðωÞA
(2.81)
Whitson (1984) suggested that the Peng-Robinson and Bergman
PNA methods are not recommended for characterizing
reservoir fluids containing fractions heavier than C20.
132 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
EXAMPLE 2.6
Using Bergman’s approach, calculate the critical pressure, critical temperature,
and acentric factor of an undefined hydrocarbon fraction that is
characterized by the laboratory measurements:
n
n
n
molecular weight ¼ 94
specific gravity ¼ 0.726133
WABP ¼ 195°F, ie, 655°R
Solution
Step 1. Calculate Watson characterization factor:
Kw ¼
ðTb Þ1=3 ð655Þ1=3
¼
¼ 11:96
γ
0:7261
Step 2. Estimate the weight fraction of the aromatic content from
wA ¼
0:847 0:847
¼
¼ 0:070819
Kw
11:96
Step 3. Estimate specific gravities of the three groups, ie, γ P, γ N, and γ A, by
applying Eqs. (2.65)–(2.67):
γ P ¼ 0:583286 + 0:00069481ð195Þ 0:7572818 106 ð195Þ2
+ 0:3207736 109 ð195Þ3 ¼ 0:692422
γ N ¼ 0:694208 + 0:0004909267ð195Þ 0:659746 106 ð195Þ2
+ 0:330966 109 ð195Þ3 ¼ 0:767306
γ A ¼ 0:916103 0:000250418ð195Þ + 0:357967 106 ð195Þ2
0:166318 109 ð195Þ3 ¼ 0:879652
Step 4. Estimate the weight fraction of the paraffinic and naphthenic contents
from
γ P γ N wA γ P γ N
ð1 wA Þγ P
γ
γA
wP
¼ 0:607933
γN γP
wN ¼ 1 ðwA + wP Þ ¼ 1 ð0:079819 + 0:607933Þ ¼ 0:321248
Step 5. Calculate the critical temperature, the critical pressure, and acentric
factor of each cut from the following expressions:
For paraffins,
ðTc ÞP ¼ 735:23 + 1:2061ð195Þ 0:00032984ð195Þ2 ¼ 957:9°R
ðpc ÞP ¼ 573:011 1:13707ð159Þ + 0:00131625ð195Þ2 0:85103 106 ð195Þ3
¼ 395 psi
ðωÞP ¼ 0:14 + 0:0009ð195Þ + 0:233 106 ð195Þ2 ¼ 0:3244
Group 3. The Plus Fraction Characterization Methods 133
For naphthenes,
ðTc ÞN ¼ 616:8906 + 2:6077ð195Þ 0:003801ð195Þ2 + 0:2544 105 ð195Þ3
¼ 999:8°R
ðpc ÞN ¼ 726:414 1:3275ð195Þ + 0:9846 103 ð195Þ2 0:45169 106 ð195Þ3
¼ 501:6 psi
ðωÞN ¼ ðωÞP 0:075 ¼ 0:2494
For the aromatics,
ðTc ÞA ¼ 749:535 + 1:7017ð195Þ 0:0015843ð195Þ2 + 0:82358 106 ð195Þ3
¼ 1027°R
ðpc ÞA ¼ 1184:514 3:44681ð195Þ + 0:0045312ð195Þ2 0:23416 105 ð195Þ3
¼ 667 psi
ðωÞA ¼ ðωÞP 0:1 ¼ 0:2244
Step 6. Calculate the critical pressure, the critical temperature, and acentric
factor of the undefined fraction from the following relationships:
pc ¼ wP ðpc ÞP + wN ðpc ÞN + wA ðpc ÞA ¼ 449 psi
Tc ¼ wP ðTc ÞP + wN ðTc ÞN + wA ðTc ÞA ¼ 976°R
ω ¼ wP ðωÞP + wN ðωÞN + wA ðωÞA ¼ 0:2932
Graphical Correlations
Several mathematical correlations for determining the physical and critical
properties of petroleum fractions have been presented. These correlations
are readily adapted to computer applications. However, it is important to
present the properties in graphical forms for a better understanding of the
behavior and interrelationships of the properties.
Boiling Points
Numerous graphical correlations have been proposed over the years for
determining the physical and critical properties of petroleum fractions. Most
of these correlations use the normal boiling point as one of the correlation
parameters. As stated previously, five methods are used to define the normal
boiling point:
134 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
1. VABP
2. WABP
3. MABP
4. CABP
5. MeABP.
Figure 2.91 schematically illustrates the conversions between the VABP and
the other four averaging types of boiling point temperatures. The following
steps summarize the procedure of using Fig. 2.9 in determining the desired
average boiling point temperature.
Step 1. On the basis of ASTM D-86 distillation data, calculate
the volumetric average boiling point from the following
expressions:
VABP ¼ ðt10 + t30 + t50 + t70 + t90 Þ=5
(2.82)
where t is the temperature in °F and the subscripts 10, 30, 50, 70,
and 90 refer to the volume percent recovered during the distillation.
Step 2. Calculate the 10–90% “slope” of the ASTM distillation curve
from the following expression:
Slope ¼ ðt90 t10 Þ=80
(2.83)
Enter the value of the slope in the graph and travel vertically to the
appropriate set for the type of boiling point desired.
Correction to be added to VABP to obtain
other boiling points
Slope =
2
1
3
2
t90 – t10
80
4
5
6
7
8
9
0
-2
-4
-6
-8
n FIGURE 2.9 Correction to volumetric average boiling points.
1
The original conversion chart can be found in the API Technical Data Book.
Group 3. The Plus Fraction Characterization Methods 135
Step 3. Read from the ordinate a correction factor for the VABP and
apply the relationship:
Desired boiling point ¼ VABP + correction factor
(2.84)
The use of the graph can best be illustrated by the following examples.
EXAMPLE 2.7
The ASTM distillation data for a 55° API gravity petroleum fraction is given
below. Calculate WAPB, MABP, CABP, and MeABP.
Cut
Distillation % Over
Temperature (°F)
1
2
3
4
5
6
7
8
9
10
Residue
IBP
10
20
30
40
50
60
70
80
90
EP
159
178
193
209
227
253
282
318
364
410
475
IBP, initial boiling point; EP, end point.
Solution
Step 1. Calculate VABP from Eq. (2.82):
VABP ¼ ð178 + 209 + 253 + 318 + 410Þ=5 ¼ 273°F
Step 2. Calculate the distillation curve slope from Eq. (2.83):
Slope ¼ ð410 178Þ=80 ¼ 2:9
Step 3. Enter the slope value of 2.9 in Fig. 2.9 and move down to the appropriate
set of boiling point curves. Read the corresponding correction factors
from the ordinate to give
Correction factors for WABP ¼ 6°F
Correction factors for CABP ¼ 7°F
Correction factors for MeABP ¼ 18°F
Correction factors for MABP ¼ 33°F
136 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 4. Calculate the desired boiling point by applying Eq. (2.84):
WABP ¼ 273 + 6 ¼ 279°F
CABP ¼ 273 7 ¼ 266°F
MeABP ¼ 273 18 ¼ 255°F
MABP ¼ 273 33 ¼ 240°F
Molecular Weight
Fig. 2.10 shows a convenient graphical correlation for determining the
molecular weight of petroleum fractions from their MeABP and API gravities. The following example illustrates the practical application of the
graphical method.
n FIGURE 2.10 Relationship between molecular weight, API gravity, and MeABPs. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering
Data Book, tenth edition, 1987.)
Group 3. The Plus Fraction Characterization Methods 137
EXAMPLE 2.8
Calculate the molecular weight of the petroleum fraction with an API gravity
and MeABP as given in Example 2.7, ie,
API ¼ 55°
MeABP ¼ 255°F
Solution
Enter these values in Fig. 2.10 to give MW ¼ 118.
Critical Temperature
The critical temperature of a petroleum fraction can be determined by using
the graphical correlation shown in Fig. 2.11. The required correlation
parameters are the API gravity and the MABP of the undefined fraction.
EXAMPLE 2.9
Calculate the critical temperature of the petroleum fraction with physical
properties as given in Example 2.7.
Solution
From Example 2.7,
API ¼ 55°
MABP ¼ 240°F
Enter the above values in Fig. 2.6 to give Tc ¼ 600°F.
Critical Pressure
Fig. 2.12 is a graphical correlation of the critical pressure of the undefined
petroleum fractions as a function of the MeABP and the API gravity. The
following example shows the practical use of the graphical correlation.
EXAMPLE 2.10
Calculate the critical pressure of the petroleum fraction from Example 2.7.
Solution
From Example 2.7,
API ¼ 55°
MeABP ¼ 255°F
Determine the critical pressure of the fraction from Fig. 2.12, to give
pc ¼ 428 psia.
138 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n FIGURE 2.11 Critical temperature as a function of API gravity and boiling points. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering
Data Book, tenth edition, 1987.)
Group 3. The Plus Fraction Characterization Methods 139
n FIGURE 2.12 Relationship between critical pressure, API gravity, and MeABPs. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering
Data Book, tenth edition, 1987.)
140 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
SPLITTING AND LUMPING SCHEMES
All forms of equations of state and their modifications require the critical
properties and acentric factor of each component in the hydrocarbon mixture. For pure components, these required properties are well defined; however, most hydrocarbon fluids contain hundreds of different components
that are difficult to identify and characterize by laboratory separation techniques. These large number of components heavier than C6 are traditionally
lumped together and categorized as the “plus fraction”; eg, heptanes-plus
“C7+” or undecanes-plus “C11+.” A significant portion of naturally occurring
hydrocarbon fluids contain these hydrocarbon-plus fractions, which could
create major problems when applying equations of state to model the volumetric behavior and predict the thermodynamic properties of hydrocarbon
fluids. These problems result from the difficulty of properly characterizing
the plus fractions (heavy ends) in terms of their critical properties and acentric factors. As presented earlier in this chapter, many correlations were
developed to generate the critical properties and acentric factor for the plus
fraction based on the molecular weight and specific gravity. However, routine laboratory experiments on the plus fraction (eg, C7+) provide description of the plus fraction in terms of its molecular weight and specific gravity,
noting that the measured molecular weight of the plus fraction can have an
error of as much as 20%.
In the absence of detailed analytical TBP distillation or chromatographic
analysis data for the plus fraction in a hydrocarbon mixture, erroneous predictions and conclusions can result if the plus fraction is categorized as one
component and directly used in EOS phase equilibria calculations. As an
example, errors could occur when calculating the saturation pressure for
a rich gas condensate sample. The EOS calculation will, at times, predict
a bubble point pressure instead of a dew point pressure. These problems
associated with treating the plus fraction as a single component can be
substantially reduced if the plus fraction is split into a manageable number
of pseudofractions. Splitting refers to the process of breaking down the
plus fraction into a certain number; an optimum number of SCN fractions.
Generally, with a sufficiently large number of pseudocomponents used in
characterizing the heavy fraction of a hydrocarbon mixture, a satisfactory
prediction of the PVT behavior by the equation of state can be obtained.
However, in compositional models, the cost and computing time can
increase significantly with the increased number of components in the system. Therefore, strict limitations are set on the maximum number of components that can be used in compositional models and the original
components have to be lumped into a smaller number of pseudocomponents
for equation-of-state calculations.
Splitting and Lumping Schemes 141
The characterization of the fraction (eg, C7+) generally consists of the following three steps:
(a) Splitting the plus fraction into pseudocomponents (eg, C7 to C45+)
(b) Lumping the generated pseudocomponents into an optimum number
of SCN fractions
(c) Characterizing the lumped fraction in terms of their critical properties
and acentric factors
The problem, then, is how to adequately split a C7+ fraction into a number of
pseudocomponents characterized by mole fraction, molecular weight, and
specific gravity. These characterization properties, when properly combined,
should match the measured plus fraction properties; that is, (M)7+ and (γ)7+.
Splitting Schemes
Splitting schemes refer to the procedures of dividing the heptanes-plus fractions into hydrocarbon groups with a SCN (C7, C8, C9, etc.), described by the
same physical properties used for pure components. It should be pointed out
that plotting the compositional analysis of any naturally occurring hydrocarbon system will exhibit a discrete compositional distribution for all components lighter than heptanes “C7” and show continuous compositional
distribution for all components heavier than hexane fraction, as shown in
Fig. 2.13.
Mole %
50.000
Discrete distribution
Continuous distribution
5.000
0.500
C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19
n FIGURE 2.13 Discrete and continuous compositional distribution.
142 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Based on the continuous compositional distribution observation, several
authors have proposed different schemes for extending the molar distribution behavior of C7+ in terms of mole fraction as a function of molecular
weight or number of carbon atoms. In general, the proposed schemes
are based on the observation that lighter systems, such as condensates, usually exhibit exponential molar distribution, while heavier systems
often show left-skewed distributions. This behavior is shown schematically in Fig. 2.14.
Three important requirements should be satisfied when applying any of the
proposed splitting models:
1. The sum of the mole fractions of the individual pseudocomponents is
equal to the mole fraction of C7+.
2. The sum of the products of the mole fraction and the molecular weight of
the individual pseudocomponents is equal to the product of the mole
fraction and molecular weight of C7+.
3. The sum of the product of the mole fraction and molecular weight
divided by the specific gravity of each individual component is equal to
that of C7+.
These requirements can be expressed mathematically by the following
relationship:
n+
X
z n ¼ z7 +
(2.85)
n¼7
Compositional distribution
6
5
Mole %
4
3
Norm Left-ske
w
al an
d he ed distrib
avy c
u
rude tion
oil sy
stem
Cond
Expo
s
n
ensa
te an ential dis
d ligh
tribu
tion
t hyd
roca
rbon
syste
ms
2
1
0
7
8
9
10
11
12 13 14 15 16
Number of carbon atoms
n FIGURE 2.14 The exponential and left-skewed distribution functions.
17
18
19
20
21
Splitting and Lumping Schemes 143
n+
X
½zn Mn ¼ z7 + M7 +
(2.86)
n¼7
n+
X
zn M n
n¼7
γn
¼
z7 + M7 +
γ7 +
(2.87)
Eqs. (2.86) and (2.87) can be solved for the molecular weight and specific
gravity of the last fraction after splitting, as
Mn + ¼
γn + ¼
z7 + M7 + Xðn + Þ1
n¼7
½ zn M n zn +
zn + M n +
z7 + M7 + Xðn + Þ1 zn Mn
n¼7
γ7 +
γn
where
z7+ ¼ mole fraction of C7+
n ¼ number of carbon atoms
n+ ¼ last hydrocarbon group in the C7+ with n carbon atoms, such as 20 +
zn ¼ mole fraction of pseudocomponent with n carbon atoms
M7+, γ 7+ ¼ measured molecular weight and specific gravity of C7+
Mn, γ n ¼ molecular weight and specific gravity of the pseudocomponent
with n carbon atoms
Several splitting schemes have been proposed. These schemes, as discussed
next, are used to predict the compositional distribution of the heavy plus
fraction.
Modified Katz’s Method
Katz (1983) presented an easy-to-use graphical correlation for breaking
down into pseudocomponents the C7+ fraction present in condensate systems. The method was originated by plotting the extended compositional
analysis of only six condensate systems as a function of the number of carbon atoms on a semilog scale and drawing a curve that best fit the data.
The limited amount of data used by Katz were not sufficient to develop a
generalized and relatively accurate expression that can be used to extend
the compositional analysis of the Heptanes-plus fraction from its mole
fraction; ie, z7+. To improve Katz graphical correlation, more experimental
144 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
compositional analysis was added to the Katz data to develop the following
expression:
zn ¼ 1:269831zC7 + exp ð0:26721nÞ + 0:0060884zC7 + + 10:4275 106 (2.88)
where
z7+ ¼ mole fraction of C7+ in the condensate system
n ¼ number of carbon atoms of the pseudocomponent
zn ¼ mole fraction of the pseudocomponent with the number of carbon
atoms of n
Eq. (2.88) is applied repeatedly until Eq. (2.85) is satisfied. The molecular
weight and specific gravity of the last pseudocomponent can be calculated
from Eqs. (2.86) and (2.87), respectively. The computational procedure of
using Eq. (2.88) is best explained through the following example.
EXAMPLE 2.11
A naturally occurring condensate-gas system has the following composition:
Component
zi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.7393
0.11938
0.04618
0.0124
0.01544
0.00759
0.00703
0.00996
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
Splitting and Lumping Schemes 145
Properties of C7+ and C20+ in terms of their molecular weights and specific
gravities are given below:
Group
Mole %
Weight %
MW
SG
Total fluid
C7+
C10+
100.000
4.271
1.830
100.000
22.690
13.346
27.74
147.39
202.31
0.412
0.789
0.832
Using the modified Katz splitting scheme as applied by Eq. (2.88), extend
the compositional distribution of C7+ to C20+ and calculate M, γ, Tb, pc, Tc, and ω
of C20+. Compare with laboratory extend compositional analysis
Solution
Applying Eq. (2.88) with z7+ ¼ 0.04271 gives the following results.
zn ¼ 1:269831zC7 + exp ð0:26721nÞ + 0:0060884zC7 + + 10:4275 106
zn ¼1:269831ð0:04271Þexp ð0:26721nÞ+0:0060884ð0:04271Þ+10:4275106
Component
zi
Eq. (2.88)
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
0.008229
0.006478
0.005104
0.004026
0.00318
0.002516
0.001995
0.001587
0.001267
0.001017
0.000822
0.000669
0.00055
0.00467a
a
This value was obtained by applying Eq. (2.85); ie,
0:04271 X19
z ¼ 0:00467
n¼7 n
The procedure for characterizing of C20+ is summarized in the following steps:
Step 1. Calculate the molecular weight and specific gravity of C20+ by solving
Eqs. (2.86) and (2.87) for these properties.
146 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
z006E
Mn
znMn
γn
znM/γ n
7
8
9
10
11
12
13
14
15
16
17
18
19
20+
P
0.00862
0.00667
0.00517
0.00402
0.00314
0.00247
0.00195
0.00156
0.00126
0.00102
0.00085
0.00071
0.00061
0.00467
96
107
121
134
147
161
175
190
206
222
237
251
263
–
0.827940631
0.713207474
0.625078138
0.538412919
0.461470288
0.397115715
0.341529967
0.295904773
0.258658497
0.227464594
0.200922779
0.17881299
0.160107188
–
5.226625952
0.727
0.749
0.768
0.782
0.793
0.804
0.815
0.826
0.836
0.843
0.851
0.856
0.861
–
1.138845434
0.952212916
0.813903826
0.688507569
0.581929746
0.493925019
0.419055174
0.358238224
0.309400117
0.269827514
0.236101973
0.208893679
0.185954922
–
6.656796111
Solving for the molecular weight of the C20+, gives
X19
z7 + M7 + ½Z M n¼7 n n
M20 + ¼
Z20 +
M20 + ¼
ð0:04271Þð147:39Þ 5:22662
¼ 228:67
0:00467
With a specific gravity of
γ 20 + ¼
γ 20 + ¼
z20 + M20 +
z7 + M7 + X19 zn Mn
n¼7 γ
γ7 +
n
ð0:00467Þð228:67Þ
¼ 0:808
½0:04271 147:39=0:789 6:656796
Step 2. Calculate the boiling points, critical pressure, and critical temperature of
C20+, using the Riazi-Daubert correlation (Eq. (2.6)), to give
θ ¼ aðMÞb γ c exp ½dMeγ + f γM
Tc ¼ 544:4ð228:67Þ0:2998 ð0:808Þ1:0555
exp 1:3478 104 ð228:67Þ 0:6164ð0:808Þ ¼ 1305°R
pc ¼ 4:5203 104 ð228:67Þ0:8063 ð0:808Þ1:6015
exp 1:8078 103 ð228:67Þ 0:3084ð0:808Þ ¼ 207 psi
Splitting and Lumping Schemes 147
Tb ¼ 6:77857ð228:67Þ0:401673 ð0:808Þ1:58262 exp
3:7709 103 ð228:67Þ 2:984036ð0:808Þ
4:25288 103 ð228:67Þð0:808Þ ¼ 1014°R
Step 3. Calculate the acentric factor of C20+ by applying the Edmister
correlation, to give solve for the acentric factor from Eq. (2.21)
to give
ω¼
ω¼
3½ log ðpc =14:70Þ
1
7½ðTc =Tb 1Þ
3 log ð207=14:7Þ
1 ¼ 0:712
7 ð1305=1014Þ 1
To ensure continuity and consistency in characterizing the heavy end (eg, C20+)
in terms of critical properties and acentric factor, the reader should consider
applying Eqs. (2.3A) through (2.3E), ie,
M 228:67
¼
¼ 283
γ
0:808
Tc ¼ 231:9733906
¼ 1355°R
0:351579776
M
M
0:352767639
233:3891996
γ
γ
0:885326626
M
M
0:106467189
+ 49:62573013 ¼ 234:3 psi
pc ¼ 31829
γ
γ
ω ¼ 0:279354619 ln
M
M
+ 0:00141973
1:243207091 ¼ 0:7357
γ
γ
Lohrenz’s Method
Lohrenz et al. (1964) proposed that the heptanes-plus fraction could be
divided into pseudocomponents with carbon number ranges from 7 to 40.
They mathematically stated that the mole fraction zn is related to its number
of carbon atoms n and the mole fraction of the Hexane fraction z6 by the
expression
2
zn ¼ z6 eAðn6Þ + Bðn6Þ
(2.89)
148 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where
z6 ¼ mole fraction of the Hexane component
n ¼ number of carbon atoms of the pseudocomponent
zn ¼ mole fraction of the pseudocomponent with number of carbon
atoms of n
A and B ¼ correlating parameters
Eq. (2.89) assumes that individual C7+ components are distributed from the
hexane mole fraction “C6” and tail off to an extremely small quantity of
heavy hydrocarbons. Lohrenz’s expression can be linearized to determine
the coefficient A and B; however, the method requires a partial extended
analysis of the hydrocarbon system (eg, C7 to C11+) to be able to determine
the values of the coefficients A and B. In addition, the validity and applicability of the equation could not be verified when testing to extend the molar
distribution on several condensate samples.
Pedersen’s Method
Pedersen et al. (1982) proposed that for naturally occurring hydrocarbon
mixtures, an exponential relationship exists between the mole fraction of
a component and the corresponding carbon number. They expressed this
relationship mathematically in the following exponential form:
zn ¼ eðAn + BÞ
(2.90)
where
A and B ¼ constants
n ¼ number of carbon atoms, eg, 7, 8, 9, …, etc.
For condensates and volatile oils, the authors suggested that A and B can
be determined by a least squares fit to the molar distribution of the
lighter fractions; that is, the methodology requires partial extended compositional analysis of the plus fraction. Pedersen’s expression can also be
expressed in the following linear form to determine the coefficients of the
equation, as
lnðzn Þ ¼ An + B
Eq. (2.90) can then be used to calculate the molar content of each of the
heavier fractions by extrapolation. The classical constraints as given by
Eqs. (2.85) through (2.87) also are imposed.
Splitting and Lumping Schemes 149
EXAMPLE 2.12
Rework Example 2.11 by using the Pedersen splitting correlation. Assume that
the partial molar distribution of C7+ is only available from C7 through C10.
Extend the analysis to C20+ and compare with the experimental data.
Component
zi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.7393
0.11938
0.04618
0.0124
0.01544
0.00759
0.00703
0.00996
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
Group
Mole %
Weight %
MW
SG
Total fluid
C7+
C10+
100.000
4.271
1.830
100.000
22.690
13.346
27.74
147.39
202.31
0.412
0.789
0.832
Solution
Step 1. Calculate the coefficients A and B graphically by plotting ln(zn) versus
“n” using the compositional analysis from C7 to C10 as shown in
Fig. 2.15 to give the slope “A” and intercept “B” as
A ¼ 0:252737
B ¼ 2:98744
Step 2. Predict the mole fraction of C10 through C20 by applying Eq. (2.90), as
shown below.
zn ¼ eðAn + BÞ
150 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
– 4.00000
– 4.20000
– 4.40000
y = −0.252737x − 2.987440
– 4.60000
Ln(zn)
– 4.80000
– 5.00000
– 5.20000
– 5.40000
– 5.60000
– 5.80000
– 6.00000
6
7
8
9
10
Number of carbon atoms “n”
11
12
n FIGURE 2.15 Calculating the coefficients A and B for Example 2.12.
Component
zi
Eq. (2.90)
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
0.00896
0.00953
0.00593
0.00405
0.00268
0.00246
0.00191
0.00149
0.00116
0.00090
0.00070
0.00054
0.00042
0.00199
Ahmed’s Method
Ahmed et al. (1985) devised a simplified method for splitting the C7+ fraction into pseudocomponents. The method originated from studying the
molar behavior of 34 condensate and crude oil systems through detailed laboratory compositional analysis of the heavy fractions. The only required
Splitting and Lumping Schemes 151
data for the proposed method are the molecular weight and the total mole
fraction of the heptanes-plus fraction.
The splitting scheme is based on calculating the mole fraction, zn, at a progressively higher number of carbon atoms. The extraction process continues
until the sum of the mole fraction of the pseudocomponents equals the total
mole fraction of the heptanes-plus (z7+).
z n ¼ zn +
Mðn + 1Þ + Mn +
Mð n + 1 Þ + M n
(2.91)
where
zn ¼ mole fraction of the pseudocomponent with a number of carbon
atoms of n
Mn ¼ molecular weight of the hydrocarbon group with n carbon atoms
Mn+ ¼ molecular weight of the n+ fraction as calculated by the following
expression:
Mðn + 1Þ + ¼ M7 + + Sðn 6Þ
(2.92)
where n is the number of carbon atoms and S is the coefficient of Eq. (2.92)
with the values given below.
No. of Carbon Atoms
Condensate Systems
Crude Oil Systems
n8
n>8
15.5
17.0
16.5
20.1
The stepwise calculation sequences of the proposed correlation are summarized in the following steps.
Step 1. According to the type of hydrocarbon system under investigation
(condensate or crude oil), select the appropriate values for the
coefficients.
Step 2. Knowing the molecular weight of C7+ fraction (M7+), calculate the
molecular weight of the octanes-plus fraction (M8+) by applying
Eq. (2.92).
Step 3. Calculate the mole fraction of the heptane fraction (z7) by using
Eq. (2.91).
Step 4. Repeat steps 2 and 3 for each component in the system (C8, C9, etc.)
until the sum of the calculated mole fractions is equal to the
mole fraction of C7+ of the system. The splitting scheme is best
explained through the following example.
152 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
EXAMPLE 2.13
Rework Example 2.12 using Ahmed’s splitting method.
Component
zi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.7393
0.11938
0.04618
0.0124
0.01544
0.00759
0.00703
0.00996
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
Group
Mole %
Weight %
MW
SG
Total fluid
C7+
C10+
100.000
4.271
1.830
100.000
22.690
13.346
27.74
147.39
202.31
0.412
0.789
0.832
Solution
Step 1. Extract the mole fraction of z7; ie, n ¼ 7
Mðn + 1Þ + Mn +
zn ¼ zn +
Mðn + 1Þ + Mn
M8 + M7 +
z7 ¼ z 7 +
M8 + M7
n
Calculate the molecular weight of C8+ by applying Eq. (2.92).
n¼7
Mðn + 1Þ + ¼ M7 + + Sðn 6Þ
M8 + ¼ 147:39 + 15:5ð7 6Þ ¼ 162:89
Splitting and Lumping Schemes 153
n
Solve for the mole fraction of heptane (z7) by applying Eq. (2.91).
Mðn + 1Þ + Mn +
z n ¼ zn +
Mð n + 1 Þ + M n
z 7 ¼ z7 +
M8 + M7 +
162:89 147:39
¼ 0:04271
¼ 0:0099
M8 + M7
162:89 96
Step 2. Extract the mole fraction of z8; ie, n ¼ 8
z n ¼ zn +
Mðn + 1Þ + Mn +
Mð n + 1 Þ + M n
M9 + M8 +
z8 ¼ z8 +
M9 + M8
n
Calculate the molecular weight of C9+ from Eq. (2.92).
n¼8
Mðn + 1Þ + ¼ M7 + + Sðn 6Þ
M9 + ¼ 147:39 + 15:5ð8 6Þ ¼ 178:39
n
Determine the mole fraction of C8+
z8 + ¼ z7 + z7 ¼ 0:04271 0:0099 ¼ 0:0328
n
Determine the mole fraction of C8 from Eq. (2.91).
z8 ¼ z8 + ½ðM9 + M8 + Þ=ðM9 + M8 Þ
z8 ¼ ð0:0328Þ½ð178:39 162:89Þ=ð178:39 107Þ ¼ 0:0074
Step 3. Extract the mole fraction of z9; ie, n ¼ 9
z n ¼ zn +
Mðn + 1Þ + Mn +
Mð n + 1 Þ + M n
M10 + M9 +
z9 ¼ z9 +
M10 + M9
n
Calculate the molecular weight of C9+ from Eq. (2.92).
n¼9
Mðn + 1Þ + ¼ M7 + + Sðn 6Þ
M10 + ¼ 147:39 + 17ð9 6Þ ¼ 198:39
n
Determine the mole fraction of z9+
z9 + ¼ z7 + z7 z8 ¼ 0:04271 0:0099 0074 ¼ 0:0254
154 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
n
Determine the mole fraction of C9 from Eq. (2.91).
Mðn + 1Þ + Mn +
zn ¼ zn +
Mðn + 1Þ + Mn
M10 + M9 +
198:29 178:39
¼ 0:0254
¼ 0:0065
z9 ¼ z9 +
M10 + M9
198:29 121
Step 4. Repeat the extracting method outlined in the preceding steps, to give the
following results.
N
Mn+
Mn
zn
7
8
9
10
11
12
13
14
15
16
17
18
19
20+
147.39
162.89
177.39
198.39
215.39
232.39
249.39
266.39
283.39
300.39
317.39
334.39
351.39
–
96
107
121
134
147
161
175
190
206
222
237
251
263
–
0.00990
0.00740
0.00650
0.00395
0.00298
0.00230
0.00180
0.00143
0.00116
0.00094
0.00076
0.00061
0.00048
0.00250
Step 5. The boiling point, critical properties, and the acentric factor of C20+ are
determined using the appropriate methods, to give
N
zn
Mn
znMn
γn
znM/γ n
7
8
9
10
11
12
13
14
15
16
17
18
19
20+
P
0.00990
0.00740
0.00650
0.00395
0.00298
0.00230
0.00180
0.00143
0.00116
0.00094
0.00076
0.00061
0.00048
0.00250
96
107
121
134
147
161
175
190
206
222
237
251
263
–
0.9504
0.7918
0.7865
0.529266249
0.437822572
0.371019007
0.315026462
0.272443701
0.239056697
0.209010771
0.179601656
0.152316618
0.126282707
–
5.360546442
0.727
0.749
0.768
0.782
0.793
0.804
0.815
0.826
0.836
0.843
0.851
0.856
0.861
–
1.307290234
1.057142857
1.024088542
0.67681106
0.55210917
0.461466427
0.386535537
0.32983499
0.285952987
0.247936858
0.211047775
0.177939974
0.146669811
–
6.864826221
Splitting and Lumping Schemes 155
Solving for the molecular weight of the C20+, gives
z7 + M7 + M20 + ¼
X19
n¼7
½Zn Mn Z20 +
M20 + ¼
ð0:04271Þð147:39Þ 5:36054
¼ 374
0:00250
With a specific gravity of
γ 20 + ¼
γ 20 + ¼
z20 + M20 +
z7 + M7 + X19 zn Mn
n¼7 γ
γ7 +
n
ð0:0025Þð374Þ
¼ 0:839
½0:04271x147:39=0:789 6:864826
Apply Eqs. (2.3A) through (2.3E), which gives
M 228:67
¼
¼ 283
γ
0:808
Tc ¼ 231:9733906
0:351579776
M
M
0:352767639
233:3891996 ¼1355°R
γ
γ
0:885326626
M
M
pc ¼ 31829
0:106467189
+ 49:62573013 ¼ 234 psi
γ
γ
ω ¼ 0:279354619 ln
M
M
+ 0:00141973
1:243207091 ¼ 0:736
γ
γ
Whitson’s Method
The most widely used distribution function is the three-parameter gamma
probability function as proposed by Whitson (1983). Unlike all previous splitting methods, the gamma function has the flexibility to describe a wider range
of distribution by adjusting its variance, which is left as an adjustable parameter. Whitson expressed the function in the following form:
pðMÞ ¼
ðM ηÞα1 exp f½M η=βg
β α Γ ð ηÞ
(2.93)
156 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where
Γ ¼ gamma function
η ¼ the minimum molecular weight included in the distribution; eg, MC7
α and β ¼ adjustable parameters that describe the shape of the
compositional distribution
M ¼ molecular weight
The key parameter defines the shape of the distribution “α” with its value
usually ranging from 0.5 to 2.5. Fig. 2.16 illustrates Whitson’s model for
several values of the parameter α:
n
n
n
n
For α ¼ 1, the distribution is exponential
For α < 1, the distribution model gives accelerated exponential
distribution
For α > 1, the distribution model gives left-skewed distributions
In the application of the gamma distribution to heavy oils, bitumen, and
petroleum residues, it is indicated that the upper limit for α is 25–30,
which statistically approaches a log-normal distribution
Whitson indicates that the parameter η can be physically interpreted
as the minimum molecular weight found in the Cn+ fraction. Whitson recommended η ¼ 92 as a good approximation if the C7+ is the plus fraction, for
other plus fractions (ie, Cn+) the following approximation is used:
η 14n 6
(2.94)
MCn + η
α
(2.95)
and
β¼
a =1
a = 10
a = 15
a = 25
zi
Molecular weight
n FIGURE 2.16 Gamma distributions for C7+.
Splitting and Lumping Schemes 157
It should be pointed out that when the parameter α ¼ 1, the compositional
distribution is exponential and probability function becomes
pðMÞ ¼
exp f½M η=βg
β
Rearranging, gives
exp ðη=βÞ
pðMÞ ¼ exp ðM=βÞ
β
(2.96)
with
β ¼ MC n + η
The cumulative frequency of occurrence “fi” for a SCN group, such as the C7
group, C8 group, C9 group, …, etc. having molecular weight boundaries
between Mi+1 and Mi is given by
fn ¼
ð Mn
PðMÞdM ¼ PðMn Þ PðMn1 Þ
Mn1
Integrating the above expression, gives
fn ¼ exp
η
Mn
Mn1
exp
exp
β
β
β
(2.97)
The mole fraction zn for each SCN group is given by multiplying the cumulative frequency of occurrence fn of the SCN times the mole fraction of the
plus fraction, that is,
zn ¼ zC7 + fn
Whitson and Brule indicated that, when characterizing multiple samples
simultaneously, the values of Mn, η, and β must be the same for all samples.
Individual sample values of MC7 + and α, however, can be different. The
result of this characterization is one set of molecular weights for the C7+
fractions, while each sample has different mole fractions zn (so that their
average molecular weights MC7 + are honored).
158 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
EXAMPLE 2.14
Rework Example 2.13 using Whitson’s exponential splitting method
Component
zi
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.01
0.009
0.0095
0.0059
0.0041
0.0027
0.0019
0.0017
0.0013
0.0011
0.0008
0.0007
0.0006
0.0006
0.0029
Solution
Step 1. Calculate η using
η 86
Step 2. Calculate β using
β ¼ MCn + η
β ¼ 147:39 86 ¼ 61:39
Step 3. Calculate the mole fraction zn for each SCN group:
η
Mn
Mn1
exp
exp
fn ¼ exp
β
β
β
86
Mn
Mn1
exp
fn ¼ exp
exp
61:39
61:39
61:39
zn ¼ zC7 + fn
86
Mn
Mn1
exp
zn ¼ zC7 + exp
exp
61:39
61:39
61:39
zn ¼ ð0:17335Þ
exp
Mn
Mn1
exp
61:39
61:39
Splitting and Lumping Schemes 159
Step 4. Calculate the mole fraction for each SCN group with values as tabulated
below:
N
Mn
zn
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 +
84
96
107
121
134
147
161
175
190
206
222
237
251
263
379
0.0078344
0.0059532
0.0061861
0.0046089
0.0037294
0.0032244
0.0025669
0.0021723
0.0018007
0.0013876
0.0010103
0.0007443
0.0005159
0.0009755
Group
Mole %
Weight %
MW
SG
C7+
C10+
4.271
1.83
22.69
13.346
147.39
202.31
0.789
0.832
Ahmed Modified Method
The gas rate from unconventional reservoirs typically exhibits a left-skewed
distribution; similar to what continues hydrocarbon compositional distribution. Ahmed (2014) adopted the methodology of describing flow-rate methodology to extend compositional analysis of the plus fraction. The
methodology, as illustrated by the following steps, requires partial extended
analysis of the plus fraction:
normalize the
Step 1. Given the partial extended compositional analysis “zn,”X
composition by calculating and tabulating the ratio zn = zn for each
component starting from the C7. Data below illustrates step 1:
n
zn
7
8
9
0.0583
0.0552
0.0374
X
zn
0.0583
0.1135
0.1509
zn =
X
zn
1
0.486344
0.247846
160 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
10
11
12
13
14
15
16
0.0338
0.0257
0.0202
0.0202
0.0165
0.0148
0.0116
0.1847
0.2104
0.2306
0.2508
0.2673
0.2821
0.2937
0.182999
0.122148
0.087598
0.080542
0.061728
0.052464
0.039496
X
Step 2. Plot the ratio zn = zn versus number of carbon atoms as defined by
“n 6” on a log-log scale, as shown in Fig. 2.17.
Step 3. Express the best fit of the data by a straight-line fit as described by
the power function form:
z
Xn ¼ aðn 6Þm
zn
(2.98)
And determine the parameters “m” and “a.”
The slope “m” must be treated as positive in all calculations.
Step 4. Locate the component with the highest mole fraction “zN” in
the observed extended molar analysis and “N” designated as its
number of carbon atoms. For example, if the observed extended
analysis shows that C8 has the highest mole fraction, then set
N ¼ 8.
10
1
a = 1.18799 and m = 1.43978
zn / Σzn
y = 1.18799x-1.43978
0.1
0.01
1
10
n–6
n FIGURE 2.17 The power function representation.
100
Splitting and Lumping Schemes 161
Step 5. Calculate zn as a function of “n” from the following expressions:
zn ¼ zN aðn 6Þm eX ðN 6Þ
(2.99)
a h
i
ðn 6Þ1m ðN 6Þ1m
1m
(2.100)
with
X¼
Step 6. Adjust the coefficients “a and m” to match the observed zn; use
Microsoft Solver to optimize the two coefficients.
Step 7. Using the regressed values of “a and m” from step 5, extend the
analysis by assuming different values of n in the above equation and
calculate the corresponding mole fraction of the component.
EXAMPLE 2.15
Rework Example 2.14 using the modified Ahmed splitting method. Use only the
data from C7 to C11.
Component
zi
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20+
0.009
0.0095
0.0059
0.0041
0.0027
0.0019
0.0017
0.0013
0.0011
0.0008
0.0007
0.0006
0.0006
0.0029
Solution
X
Step 1. Normalize compositional analysis “zn” by calculating the ratio zn = zn
for each component starting from C7 to C11.
n
zn
X
zn
zn =
n26
7
8
9
10
11
0.00896
0.00953
0.00593
0.00405
0.00268
0.00896
0.01849
0.02442
0.02847
0.03115
1
0.515414
0.242834
0.142255
0.086035
1
2
3
4
5
X
zn
162 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
10
y = 1.17556x-1.52272
a = 1.1755 and m = 1.52272
zn / Σzn
1
0.1
0.01
1
10
n–6
n FIGURE 2.18 Dimensionless ratio vs. number of carbon atoms.
X
Step 2. Plot zn = zn versus number of carbon atoms “n 6” on a log-log scale,
as shown in Fig. 2.18 and fit the data with power function to determine
the parameters “a” and “m,” to give a ¼ 1.175558 and m ¼ 1.52272
(notice the coefficient “m” must be positive).
Step 3. Identify the component C8 as the fraction with the highest mole fraction
in the partial analysis and set:
zN ¼ 0:00953 and N ¼ 8
Step 4. Apply Eqs. (2.99) and (2.11) and compare with laboratory data:
a h
i
ðn 6Þ1m ðN 6Þ1m
X¼
1m
i
1:175558 h
ðn 6Þ11:52272 ð8 6Þ11:52272
¼
1 1:52272
zn ¼ zN aðn 6Þm eX ðN 6Þ ¼ ð0:00953Þð1:175558Þðn 6Þm eX ð8 6Þ
N
Lab.
zn
a 5 1.175558
m 5 1.52272
X
zn
7
8
9
10
11
0.00896
0.00953
0.00593
0.00405
0.00268
0.68354
0.00000
0.29897
0.47579
0.59575
0.01131
0.00780
0.00567
0.00437
0.00351
Splitting and Lumping Schemes 163
Step 5. Regress on the coefficients a and m to match composition from C7
through C11 to give the following optimized values:
a ¼ 1:988198 and m ¼ 1:992053
N
Lab.
zn
a 5 1.988198
m 5 1.992053
X
zn
7
8
9
10
11
0.00896
0.00953
0.00593
0.00405
0.00268
0.99653
0.00000
0.33370
0.50102
0.60161
0.01399
0.00953
0.00593
0.00395
0.00280
Step 6. Extend the molar distribution to C12 and apply the equation through C19.
n
Lab.
zn
7
8
9
10
11
12
13
14
15
16
17
18
19
20 +
0.00896
0.00953
0.00593
0.00405
0.00268
0.00189
0.00168
0.00131
0.00107
0.00082
0.00073
0.00063
0.00055
0.00289
a 5 1.9881981
m 5 1.9920531
X
zn
0.66945
0.71789
0.75431
0.78271
0.80547
0.82412
0.83969
0.85289
-
0.00896
0.00953
0.00593
0.00405
0.00268
0.00208
0.00161
0.00128
0.00104
0.00086
0.00073
0.00062
0.00054
0.00281
Results of the methodology are shown graphically in Fig. 2.19.
Ahmed (2014) also suggested the use of the logistic growth function (LGF), as
given below, for extending the analysis of the C7+:
zn ¼
zC7 + m a nm1
½a + nm 2
Where the coefficients “a and m” are correlating, that must be adjusted to
match the observed partial extended analysis. The value of the coefficient “m”
will impact the shape of the compositional distribution, that is, exponential,
left-skewed, …, etc.
164 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
0.016
0.014
0.012
0.01
zi 0.008
0.006
0.004
0.002
0
6
8
10
12
14
16
18
20
n
n FIGURE 2.19 Example 2.16 using LGF.
Reworking Example 2.15 using LGF and matching the partial extended analysis
by adjusting the coefficients “a and m”, to give a ¼ 0.2292 and m ¼ 1.9144:
zn ¼
zC7 + m a nm1
½a + nm 2
¼
ð0:04272Þð1:9144Þð0:2292Þn1:91441
½0:2292 + n1:9144 2
The extended composition analysis is shown in Fig. 2.20.
0.01200
0.01000
0.00800
zi
0.00600
0.00400
0.00200
0.00000
7
8
9
10
11
12
13
n
n FIGURE 2.20 Mole fraction vs. number of carbon atoms.
14
15
16
17
18
19
Splitting and Lumping Schemes 165
Lumping Schemes
Equations-of-state calculations frequently are burdened by the large number
of components necessary to describe the hydrocarbon mixture for accurate
phase-behavior modeling. Often, the problem is either lumping together the
many experimentally determined fractions or modeling the hydrocarbon
system when the only experimental data available for the C7+ fraction are
the molecular weight and specific gravity.
The term lumping or pseudoization then denotes the reduction in the number
of components used in equations-of-state calculations for reservoir fluids.
This reduction is accomplished by employing the concept of the pseudocomponent. The pseudocomponent denotes a group of pure components
lumped together and represented by a single component with a SCN.
Essentially, two main problems are associated with “regrouping” the original components into a smaller number without losing the predicting power
of the equation of state:
1. How to select the groups of pure components to be represented by one
pseudocomponent each.
2. What mixing rules should be used for determining the physical
properties (eg, pc, Tc, M, γ, and ω) for the new lumped
pseudocomponents.
Several unique published techniques can be used to address these lumping
problems, notably the methods proposed by
n
n
n
n
n
n
n
Lee et al. (1979)
Whitson (1980)
Mehra et al. (1980)
Montel and Gouel (1984)
Schlijper (1984)
Behrens and Sandler (1986)
Gonzalez et al. (1986)
Several of these techniques are presented in the following discussion.
Whitson’s Lumping Scheme
Whitson (1980) proposed a regrouping scheme whereby the compositional
distribution of the C7+ fraction is reduced to only a few multiple carbon number (MCN) groups. Whitson suggested that the number of MCN groups necessary to describe the plus fraction is given by the following empirical rule:
Ng ¼ Int½1 + 3:3 log ðN nÞ
(2.101)
166 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where
Ng ¼ number of MCN groups
Int ¼ integer
N ¼ number of carbon atoms of the last component in the hydrocarbon
system
n ¼ number of carbon atoms of the first component in the plus fraction;
that is, n ¼ 7 for C7+
The integer function requires that the real expression evaluated inside the
brackets be rounded to the nearest integer.
The molecular weights separating each MCN group are calculated from the
following expression:
MN + I=Ng
MI ¼ MC7
MC7
where
(2.102)
(M)N+ ¼ molecular weight of the last reported component in the
extended analysis of the hydrocarbon system
MC7 ¼ molecular weight of C7
I ¼ 1, 2, …, Ng
For example, components with molecular weight of hydrocarbon groups
MI1 to MI falling within the boundaries of these molecular weight values
are included in the Ith MCN group. Example 2.16 illustrates the use of
Eqs. (2.101) and (2.102).
EXAMPLE 2.16
Given the following compositional analysis of the C7+ fraction in a condensate
system, determine the appropriate number of pseudocomponents forming in
the C7+:
Component
zi
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16+
M16+
0.00347
0.00268
0.00207
0.001596
0.00123
0.00095
0.00073
0.000566
0.000437
0.001671
259
Splitting and Lumping Schemes 167
Solution
Step 1. Determine the molecular weight of each component in the system.
Component
Zi
Mi
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16+
0.00347
0.00268
0.00207
0.001596
0.00123
0.00095
0.00073
0.000566
0.000437
0.001671
96
107
121
134
147
161
175
190
206
259
Step 2. Calculate the number of pseudocomponents from Eq. (2.102).
Ng ¼ Int½1 + 3:3 log ð16 7Þ
Ng ¼ Int½4:15
Ng ¼ 4
Step 3. Determine the molecular weights separating the hydrocarbon groups by
applying Eq. (2.109).
MI ¼ 96 259
96
I=4
MI ¼ 96½2:698I=4
where the values for each pseudocomponent, when MI ¼ 96[2.698]I/4, are
Pseudocomponent 1 ¼ 123
Pseudocomponent 2 ¼ 158
Pseudocomponent 3 ¼ 202
Pseudocomponent 4 ¼ 259
These are defined as
Pseudocomponent 1: The first pseudocomponent includes all components
with molecular weight in the range of 96–123. This group then includes C7,
C8, and C9.
Pseudocomponent 2: The second pseudocomponent contains all components
with a molecular weight higher than 123 to a molecular weight of 158. This
group includes C10 and C11.
Pseudocomponent 3: The third pseudocomponent includes components with
a molecular weight higher than 158 to a molecular weight of 202. Therefore,
this group includes C12, C13, and C14.
Pseudocomponent 4: This pseudocomponent includes all the remaining
components, that is, C15 and C16+.
168 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
These are summarized below.
Group I
Component
zi
zI
1
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16+
0.00347
0.00268
0.00207
0.001596
0.00123
0.00095
0.00073
0.000566
0.000437
0.001671
0.00822
2
3
4
0.002826
0.002246
0.002108
It is convenient at this stage to present the mixing rules that can be employed
to characterize the pseudocomponent in terms of its pseudophysical and
pseudocritical properties. Because the properties of the individual components can be mixed in numerous ways, each giving different properties
for the pseudocomponents, the choice of a correct mixing rule is as important as the lumping scheme. Some of these mixing rules are given next.
Hong’s Mixing Rules
Hong (1982) concluded that the weight fraction average, wi, is the best mixing
parameter in characterizing the C7+ fractions by the following mixing rules.
Defining the normalized weight fraction of a component, i, within the set of
the lumped fraction, that is, I 2 L, as
zi M i
w∗i ¼ XL
ZM
i2L i i
The proposed mixing rules are
Pseudocritical pressure PcL ¼
XL
w p
i2L i ci
XL
Pseudocritical temperature TcL ¼
w∗ T
i2L i ci
XL
Pseudocritical volume VcL ¼
w∗ v
i2L i ci
XL
Pseudoacentric factor ωL ¼
w∗ ω
i2L i i
XL
Pseudomolecular weight ML ¼
v∗ M
i2L i i
XL XL
Binary interaction coefficient kkL ¼ 1 w∗ w∗ 1 kij
i2L
j2L i j
Splitting and Lumping Schemes 169
where
wi* ¼ normalized weight fraction of component i in the lumped set
kkL ¼ binary interaction coefficient between the kth component and the
lumped fraction
The subscript L in this relationship denotes the lumped fraction.
Lee’s Mixing Rules
Lee et al. (1979), in their proposed regrouping model, employed Kay’s mixing rules as the characterizing approach for determining the properties of the
lumped fractions. Defining the normalized mole fraction of a component, i,
within the set of the lumped fraction, that is, I 2 L, as
zi
z∗i ¼ XL
z
i2L i
The following rules are proposed:
ML ¼
XL
z∗ M
i2L i i
XL
γ L ¼ ML =
VcL ¼
XL
pcL ¼
TcL ¼
ωL ¼
i2L
(2.103)
z∗i Mi =γ i
(2.104)
z∗i Mi Vci =ML
(2.105)
i2L
XL
i2L
XL
i2L
XL
i2L
z∗i pci
(2.106)
zi Tci
(2.107)
z∗i ωi
(2.108)
EXAMPLE 2.17
Using Lee’s mixing rules, determine the physical and critical properties of the
four pseudocomponents in Example 2.16.
Solution
Step 1. Assign the appropriate physical and critical properties to each
component, as shown below.
Group Component zi
1
C7
C8
C9
0.00347
0.00268
0.00207
zI
0.00822
Mi
γi
a
Vci
a
pci
a
Tci
a
ωi
a
96 0.272 0.06289 453 985 0.280a
107 0.749 0.06264 419 1036 0.312
121 0.768 0.06258 383 1058 0.348
170 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
2
3
4
C10
C11
C12
C13
C14
C15
C16+
0.001596 0.002826 134
0.00123
147
0.00095 0.002246 161
0.00073
175
0.000566
190
0.000437 0.002108 206
0.001671
259
0.782
0.793
0.804
0.815
0.826
0.826
0.908
0.06273
0.06291
0.06306
0.06311
0.06316
0.06325
0.0638b
351 1128
325 1166
302 1203
286 1236
270 1270
255 1304
215b 1467
0.385
0.419
0.454
0.484
0.516
0.550
0.68b
a
From Table 2.2.
From Table 2.2.
b
Step 2. Calculate the physical and critical properties of each group by applying
Eqs. (2.103) through (2.108) to give the results below.
Group
zI
ML
γL
VcL
pcL
TcL
ωL
1
2
3
4
0.00822
0.002826
0.002246
0.002108
105.9
139.7
172.9
248
0.746
0.787
0.814
0.892
0.0627
0.0628
0.0631
0.0637
424
339.7
288
223.3
1020
1144.5
1230.6
1433
0.3076
0.4000
0.4794
0.6531
Behrens and Sandler’s Lumping Scheme
Behrens and Sandler (1986) used the semicontinuous thermodynamic distribution theory to model the C7+ fraction for equation-of-state calculations.
The authors suggested that the heptanes-plus fraction can be fully described
with as few as two pseudocomponents.
A semicontinuous fluid mixture is defined as one in which the mole fractions of some components, such as C1 through C6, have discrete values,
while the concentrations of others, the unidentifiable components such as
C7+, are described as a continuous distribution function, F(I). This continuous distribution function F(I) describes the heavy fractions according to the
index I, chosen to be a property of individual components, such as the carbon number, boiling point, or molecular weight.
For a hydrocarbon system with k discrete components, the following relationship applies:
C6
X
zi + z7 + ¼ 1:0
i¼1
The mole fraction of C7+ in this equation is replaced with the selected distribution function, to give
ðB
C6
X
zi + FðIÞdI ¼ 1:0
i¼1
A
(2.109)
Splitting and Lumping Schemes 171
where
A ¼ lower limit of integration (beginning of the continuous distribution,
eg, C7)
B ¼ upper limit of integration (upper cutoff of the continuous
distribution, eg, C45)
This molar distribution behavior is shown schematically in Fig. 2.21. The
figure shows a semilog plot of the composition zi versus the carbon number
n of the individual components in a hydrocarbon system. The parameter A
can be determined from the plot or defaulted to C7; that is, A ¼ 7. The value
of the second parameter, B, ranges from 50 to infinity; that is, 50 B 1.
However, Behrens and Sandler pointed out that the exact choice of the cutoff
is not critical.
Selecting the index, I, of the distribution function F(I) to be the carbon number, n, Behrens and Sandler proposed the following exponential form of
F(I):
FðnÞ ¼ DðnÞeαn dn
(2.110)
with
AnB
in which the parameter α is given by the following function f(α):
1
½A BeBα
f ðαÞ ¼ cn + A Aα
¼0
α
e
eBα
(2.111)
Compositional distribution
Discrete distribution
Continuous distribution
Mole fraction
Beginning of the
continuous distribution
Upper cutoff of the
continuous distribution
Norm Left-skew
al an
e
d he d distrib
avy c
u
rude tion
oil sy
stem
s
Expo
Light nential d
istrib
hydr
ocar
bon ution
syste
ms
A
7
Number of carbon atoms
n FIGURE 2.21 Schematic illustration of the semicontinuous distribution model.
B
45
172 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
where cn is the average carbon number as defined by the relationship
cn ¼
MC7 + + 4
14
(2.112)
Eq. (2.111) can be solved for α iteratively by using the method of successive
substitutions or the Newton-Raphson method, with an initial value of α as
α ¼ ½1=
cn A
Substituting Eq. (2.112) into Eq. (2.111) yields
ðB
C6
X
zi + DðnÞeαn dn ¼ 1:0
A
i
or
z7 + ¼
ðB
DðnÞeαn dn
A
By a transformation of variables and changing the range of integration from
A and B to 0 and c, the equation becomes
z7 + ¼
ðc
DðrÞer dr
(2.113)
0
where
c ¼ ðB AÞα
r ¼ dummy variable of integration
(2.114)
The authors applied the “Gaussian quadrature numerical integration
method” with a two-point integration to evaluate Eq. (2.113), resulting in
z7 + ¼
2
X
Dðri Þwi ¼ Dðr1 Þw1 + Dðr2 Þw2
(2.115)
i¼1
where ri ¼ roots for quadrature of integrals after variable transformation and
wi ¼ weighting factor of Gaussian quadrature at point i. The values of r1, r2,
w1, and w2 are given in Table 2.9.
The computational sequences of the proposed method are summarized in the
following steps:
Step 1. Find the endpoints A and B of the distribution. As the endpoints are
assumed to start and end at the midpoint between the two carbon
numbers, the effective endpoints become
A ¼ ðstarting carbon numberÞ 0:5
(2.116)
B ¼ ðending carbon numberÞ + 0:5
(2.117)
Splitting and Lumping Schemes 173
Table 2.9 Behrens and Sandler Roots and Weights for Two-Point Integration
c
r1
r2
w1
w2
c
r1
r2
w1
w2
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
0.0615
0.0795
0.0977
0.1155
0.1326
0.1492
0.1652
0.1808
0.1959
0.2104
0.2245
0.2381
0.2512
0.2639
0.2763
0.2881
0.2996
0.3107
0.3215
0.3318
0.3418
0.3515
0.3608
0.3699
0.3786
0.3870
0.3951
0.4029
0.4104
0.4177
0.4247
0.4315
0.4380
0.4443
0.4504
0.4562
0.4618
0.4672
0.4724
0.4775
0.4823
0.2347
0.3101
0.3857
0.4607
0.5347
0.6082
0.6807
0.7524
0.8233
0.8933
0.9625
1.0307
1.0980
1.1644
1.2299
1.2944
1.3579
1.4204
1.4819
1.5424
1.6018
1.6602
1.7175
1.7738
1.8289
1.8830
1.9360
1.9878
2.0386
2.0882
2.1367
2.1840
2.2303
2.2754
2.3193
2.3621
2.4038
2.4444
2.4838
2.5221
2.5593
0.5324
0.5353
0.5431
0.5518
0.5601
0.5685
0.5767
0.5849
0.5932
0.6011
0.6091
0.6169
0.6245
0.6321
0.6395
0.6468
0.6539
0.6610
0.6678
0.6745
0.6810
0.6874
0.6937
0.6997
0.7056
0.7114
0.7170
0.7224
0.7277
0.7328
0.7378
0.7426
0.7472
0.7517
0.7561
0.7603
0.7644
0.7683
0.7721
0.7757
0.7792
0.4676
0.4647
0.4569
0.4482
0.4399
0.4315
0.4233
0.4151
0.4068
0.3989
0.3909
0.3831
0.3755
0.3679
0.3605
0.3532
0.3461
0.3390
0.3322
0.3255
0.3190
0.3126
0.3063
0.3003
0.2944
0.2886
0.2830
0.2776
0.2723
0.2672
0.2622
0.2574
0.2528
0.2483
0.2439
0.2397
0.2356
0.2317
0.2279
0.2243
0.2208
4.40
4.50
4.60
4.70
4.80
4.90
5.00
5.10
5.20
5.30
5.40
5.50
5.60
5.70
5.80
5.90
6.00
6.20
6.40
6.60
6.80
7.00
7.20
7.40
7.70
8.10
8.50
9.00
10.00
11.00
12.00
14.00
16.00
18.00
20.00
25.00
30.00
40.00
60.00
100.00
1
0.4869
0.4914
0.4957
0.4998
0.5038
0.5076
0.5112
0.5148
0.5181
0.5214
0.5245
0.5274
0.5303
0.5330
0.5356
0.5381
0.5405
0.5450
0.5491
0.5528
0.5562
0.5593
0.5621
0.5646
0.5680
0.5717
0.5748
0.5777
0.5816
0.5836
0.5847
0.5856
0.5857
0.5858
0.5858
0.5858
0.5858
0.5858
0.5858
0.5858
0.5858
2.5954
2.6304
2.6643
2.6971
2.7289
2.7596
2.7893
2.8179
2.8456
2.8722
2.8979
2.9226
2.9464
2.9693
2.9913
3.0124
3.0327
3.0707
3.1056
3.1375
3.1686
3.1930
3.2170
3.2388
3.2674
3.2992
3.3247
3.3494
3.3811
3.3978
3.4063
3.4125
3.4139
3.4141
3.4142
3.4142
3.4142
3.4142
3.4142
3.4142
3.4142
0.7826
0.7858
0.7890
0.7920
0.7949
0.7977
0.8003
0.8029
0.8054
0.8077
0.8100
0.8121
0.8142
0.8162
0.8181
0.8199
0.8216
0.8248
0.8278
0.8305
0.8329
0.8351
0.8371
0.8389
0.8413
0.8439
0.8460
0.8480
0.8507
0.8521
0.8529
0.8534
0.8535
0.8536
0.8536
0.8536
0.8536
0.8536
0.8536
0.8536
0.8536
0.2174
0.2142
0.2110
0.2080
0.2051
0.2023
0.1997
0.1971
0.1946
0.1923
0.1900
0.1879
0.1858
0.1838
0.1819
0.1801
0.1784
0.1754
0.1722
0.1695
0.1671
0.1649
0.1629
0.1611
0.1587
0.1561
0.1540
0.1520
0.1493
0.1479
0.1471
0.1466
0.1465
0.1464
0.1464
0.1464
0.1464
0.1464
0.1464
0.1464
0.1464
174 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Step 2. Calculate the value of the parameter α by solving Eq. (2.111)
iteratively.
Step 3. Determine the upper limit of integration c by applying Eq. (2.114).
Step 4. Find the integration points r1 and r2 and the weighting factors w1 and
w2 from Table 2.9.
Step 5. Find the pseudocomponent carbon numbers, ni, and mole fractions,
zi, from the following expressions:
For the first pseudocomponent,
r1
+A
α
z1 ¼ w1 z7 +
n1 ¼
(2.118)
For the second pseudocomponent,
r2
+A
α
z2 ¼ w2 z7 +
n2 ¼
(2.119)
Step 6. Assign the physical and critical properties of the two
pseudocomponents from Table 2.2.
EXAMPLE 2.18
A heptanes-plus fraction in a crude oil system has a mole fraction of 0.4608 with
a molecular weight of 226. Using the Behrens and Sandler lumping scheme,
characterize the C7+ by two pseudocomponents and calculate their mole
fractions.
Solution
Step 1. Assuming the starting and ending carbon numbers to be C7 and C50,
calculate A and B from Eqs. (2.116) and (2.117):
A ¼ ðstarting carbon numberÞ 0:5
A ¼ 7 0:5 ¼ 6:5
B ¼ ðending carbon numberÞ + 0:5
B ¼ 50 + 0:5 ¼ 50:5
Step 2. Calculate cn from Eq. (2.112):
MC7 + + 4
14
226 + 4
cn ¼
¼ 16:43
14
cn ¼
Step 3. Solve Eq. (2.111) iteratively for α, to give
1
½A BeBα
¼0
cn + A Aα
e
eBa
α
1
½6:5 50:5e50:5α
¼0
16:43 + 6:5 6:5α
e
e50:5α
α
Splitting and Lumping Schemes 175
Solving this expression iteratively for α gives α ¼ 0.0938967.
Step 4. Calculate the range of integration c from Eq. (2.114):
c ¼ ðB AÞα
c ¼ ð50:5 6:5Þ0:0938967 ¼ 4:13
Step 5. Find integration points ri and weights wi from Table 2.9:
r1 ¼ 0:4741
r2 ¼ 2:4965
w1 ¼ 0:7733
w2 ¼ 0:2267
Step 6. Find the pseudocomponent carbon numbers ni and mole fractions zi by
applying Eqs. (2.118) and (2.119).
For the first pseudocomponent,
r1
n1 ¼ + A
α
0:4741
n1 ¼
+ 6:5 ¼ 11:55
0:0938967
z1 ¼ w1 z7 +
z1 ¼ ð0:7733Þð0:4608Þ ¼ 0:3563
For the second pseudocomponent,
r2
n2 ¼ + A
α
2:4965
n2 ¼
+ 6:5 ¼ 33:08
0:0938967
z2 ¼ w2 z7 +
z2 ¼ ð0:2267Þð0:4608Þ ¼ 0:1045
The C7+ fraction is represented then by the two pseudocomponents below.
Pseudocomponent
Carbon Number
Mole Fraction
1
2
C11.55
C33.08
0.3563
0.1045
Step 7. Assign the physical properties of the two pseudocomponents according
to their number of carbon atoms using the Katz and Firoozabadi
generalized physical properties as given in Table 2.2 or by calculations
from Eq. (2.6). The assigned physical properties for the two fractions
are shown below.
Pseudocomponent n
1
2
Tb (°R) γ
11.55 848
33.08 1341
M
Tc (°R) pc (psia) ω
0.799 154 1185
0.915 426 1629
314
134
0.437
0.921
176 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Lee’s Lumping Scheme
Lee et al. (1979) devised a simple procedure for regrouping the oil fractions
into pseudocomponents. Lee and coworkers employed the physical reasoning
that crude oil fractions having relatively close physicochemical properties
(such as molecular weight and specific gravity) can be accurately represented
by a single fraction. Having observed that the closeness of these properties is
reflected by the slopes of curves when the properties are plotted against the
weight-averaged boiling point of each fraction, Lee et al. used the weighted
sum of the slopes of these curves as a criterion for lumping the crude oil fractions. The authors proposed the following computational steps:
Step 1. Plot the available physical and chemical properties of each original
fraction versus its weight-averaged boiling point.
Step 2. Calculate numerically the slope mij for each fraction at each WABP,
where
mij ¼ slope of the property curve versus boiling point
i ¼ l, …, nf
j ¼ l, …, np
nf ¼ number of original oil fractions
np ¼ number of available physicochemical properties
ij as defined:
Step 3. Compute the normalized absolute slope m
ij ¼
m
mij
max i¼1, …, nf mij
(2.120)
i for each fraction as follows:
Step 4. Compute the weighted sum of slopes M
Xnp
i ¼
M
j¼1
np
ij
m
(2.121)
i represents the averaged change of physicochemical
where M
properties of the crude oil fractions along the boiling point axis.
i for each fraction, group those
Step 5. Judging the numerical values of M
i values.
fractions that have similar M
Step 6. Using the mixing rules given by Eqs. (2.103) through (2.108),
calculate the physical properties of pseudocomponents.
EXAMPLE 2.19
The data in Table 2.10, as given by Hariu and Sage (1969), represent the average
boiling point molecular weight, specific gravity, and molar distribution of 15
pseudocomponents in a crude oil system. Lump the given data into an optimum
number of pseudocomponents using the Lee method.
Splitting and Lumping Schemes 177
Table 2.10 Characteristics of Pseudocomponents in Example 2.18
Pseudocomponent
Tb (°R)
M
γ
zi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
600
654
698
732
770
808
851
895
938
983
1052
1154
1257
1382
1540
95
101
108
116
126
139
154
173
191
215
248
322
415
540
700
0.680
0.710
0.732
0.750
0.767
0.781
0.793
0.800
0.826
0.836
0.850
0.883
0.910
0.940
0.975
0.0681
0.0686
0.0662
0.0631
0.0743
0.0686
0.0628
0.0564
0.0528
0.0474
0.0836
0.0669
0.0535
0.0425
0.0340
Solution
Step 1. Plot the molecular weights and specific gravities versus the boiling
points and calculate the slope mij for each pseudocomponent as in the
following table.
Pseudocomponent
Tb
mi1 5 @M/@Tb
mi2 5 @γ/@Tb
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
600
654
698
732
770
808
851
895
938
983
1052
1154
1257
1382
1540
0.1111
0.1327
0.1923
0.2500
0.3026
0.3457
0.3908
0.4253
0.4773
0.5000
0.6257
0.8146
0.9561
1.0071
1.0127
0.00056
0.00053
0.00051
0.00049
0.00041
0.00032
0.00022
0.00038
0.00041
0.00021
0.00027
0.00029
0.00025
0.00023
0.00022
178 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
i
Table 2.11 Absolute Slope mij and Weighted Slopes M
Pseudocomponent
i1 ¼ mij =1:0127
m
i2 ¼ mij =0:00056
m
i ¼ ðm
i1 + m
i2 Þ=2
M
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.1097
0.1310
0.1899
0.2469
0.2988
0.3414
0.3859
0.4200
0.4713
0.4937
0.6179
0.8044
0.9441
0.9945
1.0000
1.0000
0.9464
0.9107
0.8750
0.7321
0.5714
0.3929
0.6786
0.7321
0.3750
0.4821
0.5179
0.4464
0.4107
0.3929
0.55485
0.53870
0.55030
0.56100
0.51550
0.45640
0.38940
0.54930
0.60170
0.43440
0.55000
0.66120
0.69530
0.70260
0.69650
Step 2. Calculate the normalized absolute slope mij and the weighted sum of
i i using Eqs. (2.121) and (2.122), respectively. This is shown in
slopes M
Table 2.11. Note that the maximum value of mi1 is 1.0127 and for mi2 is
0.00056.
i , the pseudocomponents can be lumped into
Examining the values of M
three groups:
Group 1: Combine fractions 1–5 with a total mole fraction of 0.3403.
Group 2: Combine fractions 6–10 with a total mole fraction of
0.2880.
Group 3: Combine fractions 11–15 with a total mole fraction of
0.2805.
Step 3. Calculate the physical properties of each group. This can be achieved by
computing M and γ of each group by applying Eqs. (2.103) and (2.104),
respectively, followed by employing the Riazi-Daubert correlation
(Eq. (2.6)) to characterize each group. Results of the calculations are
shown below.
Group
I
Mole Fraction
(zi)
M
γ
Tb
Tc
pc
Vc
1
2
3
0.3403
0.2880
0.2805
109.4
171.0
396.5
0.7299
0.8073
0.9115
694
891
1383
1019
1224
1656
404
287
137
0.0637
0.0634
0.0656
Splitting and Lumping Schemes 179
The physical properties M, γ, and Tb reflect the chemical makeup of petroleum
fractions. Some of the methods used to characterize the pseudocomponents in
terms of their boiling point and specific gravity assume that a particular factor,
called the characterization factor, Cf, is constant for all the fractions that
constitute the C7+. Soreide (1989) developed an accurate correlation for
determining the specific gravity based on the analysis of 843 TBP fractions from
68 reservoir C7+ samples and introduced the characterization factor, Cf, into the
correlation, as given by the following relationship:
γ i ¼ 0:2855 + Cf ðMi 66Þ0:13
(2.122)
Characterization factor Cf is adjusted to satisfy the following relationship:
zC M C
γ C7 + exp ¼ C 7 + 7 +
(2.123)
N+
X
zi Mi
i¼C7
γi
where γ C7 + exp is the measured specific gravity of the C7+. Combining
Eq. (2.122) with Eq. (2.123) gives
zC7 + MC7 +
#
γ C7 + exp ¼ C "
N+
X
zi Mi
0:2855 + Cf ðMi 66Þ0:13
i¼C7
Rearranging,
f ðCf Þ ¼
CN +
X
"
#
zi Mi
0:13
i¼C7 0:2855 + Cf ðMi 66Þ
zC MC
7+ 7+ ¼ 0
γ C7 + exp
(2.124)
The optimum value of the characterization factor can be determined iteratively
by solving this expression for Cf. The Newton-Raphson method is an efficient
numerical technique that can be used conveniently to solve the preceding
nonlinear equation by employing the relationship
f Cnf
n+1
n
Cf ¼ Cf @f Cnf =@Cnf
with
2
CN +
X6
@f Cnf
¼
4
@Cnf
i¼C7
3
ðMi 66Þ0:13
0:2855 + Cnf ðMi 66Þ
0:13
7
2 5
The method is based on assuming a starting value Cnf and calculating an
improved value Cnf + 1 that can be used in the second iteration. The iteration
process continues until the difference between the two, Cnf + 1 and Cnf , is small,
say, 106. The calculated value of Cf then can be used in Eq. (2.122) to
determine specific gravity of any pseudocomponent γ i.
Soreide also developed the following boiling point temperature, Tb, correlation,
which is a function of the molecular weight and specific gravity of the fraction
180 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Tb ¼ 1928:3 1:695 105 Aγ 3:266
M0:03522
with
A ¼ exp ½0:003462Mγ 0:004922M 4:7685γ where
Tb ¼ boiling point temperature, °R
M ¼ molecular weight
EXAMPLE 2.20
The molar distribution of a heptanes-plus,
as given by Whitson and Brule, is
listed below, where MC7 + ¼ 143 and γ C7 + exp ¼ 0:795. Calculate the specific
gravity of the five pseudocomponents.
C7+ Fraction (i)
Mole Fraction (zi)
Molecular Weight (Mi)
1
2
3
4
5
2.4228
2.8921
1.2852
0.2367
0.0132
95.55
135.84
296.65
319.83
500.00
Solution
Step 1. Solve Eq. (2.124) for the characterization factor, Cf, by trial and error or
Newton-Raphson method, to give
Cf ¼ 0:26927
Step 2. Using Eq. (2.122), calculate the specific gravity of the five, shown in the
following table, where
γ i ¼ 0:2855 + Cf ðMi 66Þ0:13
γ i ¼ 0:2855 + 0:26927ðMi 66Þ0:13
C7+
Fraction (i)
Mole
Fraction (zi)
Molecular
Weight (Mi)
Specific gravity (γ i 5 0.2855
+ 0.26927(Mi 2 66)0.13)
1
2
3
4
5
2.4228
2.8921
1.2852
0.2367
0.0132
95.55
135.84
296.65
319.83
500.00
0.7407
0.7879
0.8358
0.8796
0.9226
Splitting and Lumping Schemes 181
Characterization of Multiple Hydrocarbon Samples
When characterizing multiple samples simultaneously for use in the equation of state applications, the following procedure is recommended:
1. Identify and select the main fluid sample among multiple samples from
the same reservoir. In general, the sample that dominates the simulation
process or has extensive laboratory data is chosen as the main sample.
2. Split the plus fractions of the main sample into several components using
Whitson’s method and determine the parameters MN, η, and β*, as
outlined by Eqs. (2.93) and (2.94).
3. Calculate the characterization factor, Cf, by solving Eq. (2.124) and
calculate specific gravities of the C7+ fractions by using Eq. (2.122).
4. Lump the produced splitting fractions of the main sample into a number
of pseudocomponents characterized by the physical properties pc, Tc, M,
γ, and ω.
5. Assign the gamma function parameters (MN, η, and β) of the main
sample to all the remaining samples; in addition, the characterization
factor, Cf, must be the same for all mixtures. However, each sample can
have a different MC7 + and α.
6. Using the same gamma function parameters as the main sample, split
each of the remaining samples into a number of components.
7. For each of the remaining samples, group or lump the fractions into
pseudocomponents with the condition that these pseudocomponents are
lumped in a way to give similar or the same molecular weights to those
of the lumped fractions of the main sample.
8. The results of this characterization procedure are
n
n
One set of molecular weights for the C7+ fractions
Each pseudocomponent has the same properties (ie, M, γ, Tb, pc, Tc,
and ω) as those of the main sample but with a different mole fraction
It should be pointed out that, when characterizing multiple samples, the outlier samples must be identified and treated separately. Assuming that these
multiple samples are obtained from different depths and each is described by
laboratory PVT measurements, the outlier samples can be identified by
making the following plots:
n
n
n
n
Saturation pressure versus depth
C1 mole % versus depth
C7+ mole % versus depth
Molecular weight of C7+ versus depth
Data deviating significantly from the general trend, as shown in Figs. 2.22–
2.25, should not be used in the multiple samples characterization procedure. Takahashi et al. (2002) point out that a single EOS model never
182 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
Saturation pressure
X
X
True vertical depth
X
X
X
X
X Outlier (removed) data
n FIGURE 2.22 Measured saturation pressure versus depth.
Mole % of methane
X
True vertical depth
X
X
X
X
X
X Outlier (removed) data
n FIGURE 2.23 Measured methane content versus depth.
Splitting and Lumping Schemes 183
Molecular weight of heptanes-plus fraction
True vertical depth
X
X
X
X Outlier (removed) data
n FIGURE 2.24 Measured C7+ molecular weight versus depth.
Mole % of heptanes-plus fraction
X
True vertical depth
X
X
X
X
n FIGURE 2.25 Measured mole % of C7+ versus depth.
X Outlier (removed) data
184 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
predicts such outlier behavior using a common characterization. Also note
that systematic deviation from a general trend may indicate a separate fluid
system, requiring a separate equation-of-state model, while randomlike
deviations from clear trends often are due to experimental data error or
inconsistencies in reported compositional data as compared with the actual
samples used in laboratory tests.
PROBLEMS
1. A heptanes-plus fraction with a molecular weight of 198 and a specific
gravity of 0.8135 presents a naturally occurring condensate system. The
reported mole fraction of the C7+ is 0.1145. Predict the molar
distribution of the plus fraction using
(a) Katz’s correlation
(b) Ahmed’s correlation
Characterize the last fraction in the predicted extended analysis in terms
of its physical and critical properties.
2. A naturally occurring crude oil system has a heptanes-plus fraction with
the following properties:
M7+ ¼ 213.0000
γ 7+ ¼ 0.8405
x7+ ¼ 0.3497
Extend the molar distribution of the plus fraction and determine the
critical properties and acentric factor of the last component.
3. A crude oil system has the following composition:
s
xi
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13+
0.3100
0.1042
0.1187
0.0732
0.0441
0.0255
0.0571
0.0472
0.0246
0.0233
0.0212
0.0169
0.1340
The molecular weight and specific gravity of C13+ are 325 and 0.842,
respectively. Calculate the appropriate number of pseudocomponents
necessary to adequately represent these components using
Problems 185
(a) Whitson’s lumping method
(b) Behrens-Sandler’s method
Characterize the resulting pseudocomponents in terms of their critical
properties.
4. If a petroleum fraction has a measured molecular weight of 190 and a
specific gravity of 0.8762, characterize this fraction by calculating the
boiling point, critical temperature, critical pressure, and critical volume
of the fraction. Use Riazi-Daubert’s correlation.
5. Calculate the acentric factor and critical compressibility factor of the
component in problem 4.
6. A petroleum fraction has the following physical properties:
API ¼ 50°
Tb ¼ 400°F
M ¼ 165
γ ¼ 0.79
Calculate pc, Tc, Vc, ω, and Zc using the following correlations:
(a) Cavett
(b) Kesler-Lee
(c) Winn-Sim-Daubert
(d) Watansiri-Owens-Starling
7. An undefined petroleum fraction with 10 carbon atoms has a measured
average boiling point of 791°R and a molecular weight of 134. If the
specific gravity of the fraction is 0.78, determine the critical pressure,
critical temperature, and acentric factor of the fraction using
(a) Robinson-Peng’s PNA method
(b) Bergman PNA’s method
(c) Riazi-Daubert’s method
(d) Cavett’s correlation
(e) Kesler-Lee’s correlation
(f) Willman-Teja’s correlation
8. A heptanes-plus fraction is characterized by a molecular weight of 200
and specific gravity of 0.810. Calculate pc, Tc, Tb, and acentric factor of
the plus fraction using
(a) Riazi-Daubert’s method
(b) Rowe’s correlation
(c) Standing’s correlation
9. Using the data given in problem 8 and the boiling point as calculated by
Riazi-Daubert’s correlation, determine the critical properties and
acentric factor employing
(a) Cavett’s correlation
(b) Kesler-Lee’s correlation
Compare the results with those obtained in problem 8.
186 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions
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Chapter
3
Natural Gas Properties
LAWS THAT DESCRIBE the behavior of reservoir gases in terms of pressure, p,
volume, V, and temperature, T, have been known for many years. These laws
are relatively simple for a hypothetical fluid known as a perfect or ideal gas.
This chapter reviews these laws and how they can be modified to describe
the behavior of reservoir gases, which may deviate significantly from these
simple laws.
Gas is defined as a homogeneous fluid of low viscosity and density, which
has no definite volume but expands to completely fill the vessel in which it is
placed. Generally, the natural gas is a mixture of hydrocarbon and nonhydrocarbon gases. The hydrocarbon gases normally found in a natural gas are
methane, ethane, propane, butanes, pentanes, and small amounts of hexanes
and heavier components. The nonhydrocarbon gases, that is, impurities,
include carbon dioxide, hydrogen sulfide, and nitrogen.
Knowledge of pressure-volume-temperature (PVT) relationships and other
physical and chemical properties of gases are essential for solving problems
in natural gas reservoir engineering. The properties of interest include
n
n
n
n
n
n
n
n
n
Apparent molecular weight, Ma.
Specific gravity, γ g.
Compressibility factor, Z.
Gas density, ρg.
Specific volume, υ.
Isothermal gas compressibility coefficient, cg.
Gas formation volume factor, Bg.
Gas expansion factor, Eg.
Viscosity, μg.
These gas properties may be obtained from direct laboratory measurements
or by prediction from generalized mathematical expressions. This chapter
reviews laws that describe the volumetric behavior of gases in terms of pressure and temperature and documents the mathematical correlations widely
used in determining the physical properties of natural gases.
Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00003-9
Copyright # 2016 Elsevier Inc. All rights reserved.
189
190 CHAPTER 3 Natural Gas Properties
BEHAVIOR OF IDEAL GASES
The kinetic theory of gases postulates that the gas is composed of a very
large number of particles called molecules. For an ideal gas, the volume
of these molecules is insignificant compared with the total volume occupied
by the gas. It is also assumed that these molecules have no attractive or
repulsive forces between them, and it is assumed that all collisions of molecules are perfectly elastic.
Based on this kinetic theory of gases, a mathematical equation, called equation of state, can be derived to express the relationship existing between
pressure, p, volume, V, and temperature, T, for a given quantity of moles
of gas, n. This relationship for perfect gases, called the ideal gas law, is
expressed mathematically by the following equation:
pV ¼ nRT
(3.1)
where
p ¼ absolute pressure, psia
V ¼ volume, ft3
T ¼ absolute temperature, °R
n ¼ number of moles of gas, lb-mol
R ¼ the universal gas constant that, for these units, has the value 10.73
psia ft3/lb-mol °R
The number of pound-moles of gas, that is, n, is defined as the weight of the
gas, m, divided by the molecular weight, M, or
n¼
m
M
(3.2)
Combining Eq. (3.1) with Eq. (3.2) gives
pV ¼
m
M
where
RT
(3.3)
m ¼ weight of gas, lb
M ¼ molecular weight, lb/lb-mol
Because the density is defined as the mass per unit volume of the substance,
Eq. (3.3) can be rearranged to estimate the gas density at any pressure and
temperature:
ρg ¼
m pM
¼
V RT
(3.4)
where ρg ¼ density of the gas, lb/ft3.
It should be pointed out that lb refers to pounds mass in all the subsequent
discussions of density in this text.
Behavior of Ideal Gases 191
EXAMPLE 3.1
Three pounds of n-butane are placed in a vessel at 120°F and 60 psia. Calculate
the volume of the gas assuming an ideal gas behavior.
Solution
Step 1. Determine the molecular weight of n-butane from Table 1.1 to give
M ¼ 58.123
Step 2. Solve Eq. (3.3) for the volume of gas:
m RT
V¼
M p
3
ð10:73Þð120 + 460Þ
V¼
¼ 5:35ft3
58:123
60
EXAMPLE 3.2
Using the data given in the preceding example, calculate the density of n-butane.
Solution
Solve for the density by applying Eq. (3.4):
m pM
¼
V RT
ð60Þð58:123Þ
ρg ¼
¼ 0:56lb=ft3
ð10:73Þð580Þ
ρg ¼
Petroleum engineers usually are interested in the behavior of mixtures and rarely
deal with pure component gases. Because natural gas is a mixture of
hydrocarbon components, the overall physical and chemical properties can be
determined from the physical properties of the individual components in the
mixture by using appropriate mixing rules.
The basic properties of gases commonly are expressed in terms of the
apparent molecular weight, standard volume, density, specific volume,
and specific gravity. These properties are defined in the following sections.
Apparent Molecular Weight, Ma
One of the main gas properties frequently of interest to engineers is the
apparent molecular weight. If yi represents the mole fraction of the ith component in a gas mixture, the apparent molecular weight is defined mathematically by the following equation:
Ma ¼
X
yi Mi
i¼1
(3.5)
192 CHAPTER 3 Natural Gas Properties
where
Ma ¼ apparent molecular weight of a gas mixture
Mi ¼ molecular weight of the ith component in the mixture
yi ¼ mole fraction of component i in the mixture
Conventionally, natural gas compositions can be expressed in three different
forms: mole fraction, yi, weight fraction, wi, and volume fraction, υi.
The mole fraction of a particular component, i, is defined as the number of
moles of that component, ni, divided by the total number of moles, n, of all
the components in the mixture:
yi ¼
ni
ni
¼X
n
ni
i
The weight fraction of a particular component, i, is defined as the weight of
that component, mi, divided by the total weight of m, the mixture:
wi ¼
mi
mi
¼X
m
mi
i
Similarly, the volume fraction of a particular component, vi, is defined as the
volume of that component, Vi, divided by the total volume of V, the mixture:
vi ¼
Vi
Vi
¼X
V
Vi
i
It is convenient in many engineering calculations to convert from mole fraction
to weight fraction and vice versa. The procedure is given in the following steps.
1. As the composition is one of the intensive properties
and independent of the quantity of the system, assume that the total
number of gas is 1; that is, n ¼ 1.
2. From the definitions of mole fraction and number of moles (see
Eq. (3.2)),
ni ni
¼ ¼ ni
n 1
mi ¼ ni Mi ¼ yi Mi
yi ¼
3. From the above two expressions, calculate the weight fraction to give
wi ¼
mi
mi
yi Mi
yi Mi
¼X ¼X
¼
m
mi
yi Mi Ma
i
4. Similarly,
i
wi =Mi
yi ¼ X
w =Mi
i i
Behavior of Ideal Gases 193
Standard Volume, Vsc
In many natural gas engineering calculations, it is convenient to measure the
volume occupied by l lb-mol of gas at a reference pressure and temperature.
These reference conditions are usually 14.7 psia and 60°F and are commonly referred to as standard conditions. The standard volume then is
defined as the volume of gas occupied by 1 lb-mol of gas at standard conditions. Applying these conditions to Eq. (3.1) and solving for the volume,
that is, the standard volume, gives
Vsc ¼
ð1ÞRTsc ð1Þð10:73Þð520Þ
¼
psc
14:7
(3.6)
Vsc ¼ 379:4scf=1b mol
where
Vsc ¼ standard volume, scf/lb-mol
scf ¼ standard cubic feet
Tsc ¼ standard temperature, °R
psc ¼ standard pressure, psia
Gas Density, ρg
The density of an ideal gas mixture is calculated by simply replacing the
molecular weight, M, of the pure component in Eq. (3.4) with the apparent
molecular weight, Ma, of the gas mixture to give
ρg ¼
pMa
RT
(3.7)
where ρg ¼ density of the gas mixture, lb/ft3, and Ma ¼ apparent molecular
weight.
Specific Volume, υ
The specific volume is defined as the volume occupied by a unit mass of the
gas. For an ideal gas, this property can be calculated by applying Eq. (3.3):
υ¼
V 1
¼
m ρg
Combining with Eq. (3.7) gives:
υ¼
RT
pMa
where υ ¼ specific volume, ft3/lb, and ρg ¼ gas density, lb/ft3.
(3.8)
194 CHAPTER 3 Natural Gas Properties
Specific Gravity, γ g
The specific gravity is defined as the ratio of the gas density to that of the air.
Both densities are measured or expressed at the same pressure and temperature. Commonly, the standard pressure, psc, and standard temperature, Tsc,
are used in defining the gas specific gravity.
γg ¼
gas density @ 14:7 and 60° ρg
¼
air density @ 14:7 and 60° ρair
(3.9)
Assuming that the behavior of both the gas mixture and the air is described
by the ideal gas equation, the specific gravity can be then expressed as
psc Ma
RTsc
γg ¼
psc Mair
RTsc
or
γg ¼
Ma
Ma
¼
Mair 28:96
(3.10)
where
γ g ¼ gas specific gravity, 60°/60°
ρair ¼ density of the air
Mair ¼ apparent molecular weight of the air ¼ 28.96
Ma ¼ apparent molecular weight of the gas
psc ¼ standard pressure, psia
Tsc ¼ standard temperature, °R
EXAMPLE 3.3
A gas well is producing gas with a specific gravity of 0.65 at a rate of 1.1 MMscf/
day. The average reservoir pressure and temperature are 1500 psi and 150°F.
Calculate
1. The apparent molecular weight of the gas.
2. The gas density at reservoir conditions.
3. The flow rate in lb/day.
Solution
From Eq. (3.10), solve for the apparent molecular weight:
Ma ¼ 28:96γ g
Ma ¼ ð28:96Þð0:65Þ ¼ 18:82
Apply Eq. (3.7) to determine gas density:
ρg ¼
pMa
RT
Behavior of Ideal Gases 195
ρg ¼
ð1500Þð18:82Þ
¼ 4:311b=ft3
ð10:73Þð610Þ
The following steps are used to calculate the flow rate.
Step 1. Because 1 lb-mol of any gas occupies 379.4 scf at standard conditions,
then the daily number of moles, n, that the gas well is producing is
n¼
ð1:1Þð10Þ6
¼ 28991b mol
379:4
Step 2. Determine the daily mass, m, of the gas produced from Eq. (2.2):
m ¼ nMa
m ¼ 2899 ð18:82Þ ¼ 54,5591b=day
EXAMPLE 3.4
A gas well is producing a natural gas with the following composition:
Component
yi
CO2
C1
C2
C3
0.05
0.90
0.03
0.02
Assuming an ideal gas behavior, calculate the apparent molecular weight,
specific gravity, gas density at 2000 psia and 150°F, and specific volume at 2000
psia and 150°F.
Solution
The table below describes the gas composition.
Component
yi
Mi
yiMi
CO2
C1
C2
C3
0.05
0.90
0.03
0.02
44.01
16.04
30.07
44.11
2.200
14.436
0.902
0.882
Ma ¼ 18.42
Apply Eq. (3.5) to calculate the apparent molecular weight:
X
yi Mi
Ma ¼
i¼1
Ma ¼ 18:42
Calculate the specific gravity by using Eq. (3.10):
γ g ¼ Ma =28:96 ¼ 18:42=28:96 ¼ 0:636
196 CHAPTER 3 Natural Gas Properties
Solve for the density by applying Eq. (3.7):
ρg ¼
ð2000Þð18:42Þ
¼ 5:6281b=ft3
ð10:73Þð610Þ
Determine the specific volume from Eq. (3.8):
υ¼
1
1
¼
¼ 0:178ft3 =1b
ρg 5:628
BEHAVIOR OF REAL GASES
In dealing with gases at a very low pressure, the ideal gas relationship is a
convenient and generally satisfactory tool. At higher pressures, the use of
the ideal gas equation of state may lead to errors as great as 500%, as compared to errors of 2–3% at atmospheric pressure.
Basically, the magnitude of deviations of real gases from the conditions of
the ideal gas law increases with increasing pressure and temperature and
varies widely with the composition of the gas. Real gases behave differently
than ideal gases. The reason for this is that the perfect gas law was derived
under the assumption that the volume of molecules is insignificant and neither molecular attraction nor repulsion exists between them. This is not the
case for real gases.
Numerous equations of state have been developed in the attempt to correlate
the PVT variables for real gases with experimental data. To express a more
exact relationship between the variables p, V, and T, a correction factor,
called the gas compressibility factor, gas deviation factor, or simply the
Z-factor, must be introduced into Eq. (3.1) to account for the departure of
gases from ideality. The equation has the following form:
pV ¼ ZnRT
(3.11)
where the gas compressibility factor, Z, is a dimensionless quantity, defined
as the ratio of the actual volume of n-moles of gas at T and p to the ideal
volume of the same number of moles at the same T and p:
Z¼
Vactual
V
¼
Videal ðnRT Þ=p
Studies of the gas compressibility factors for natural gases of various
compositions show that the compressibility factor can be generalized with
sufficient accuracy for most engineering purposes when they are expressed
in terms of the following two dimensionless properties: pseudoreduced
Behavior of Real Gases 197
pressure, ppr, and pseudoreduced temperature, Tpr. These dimensionless
terms are defined by the following expressions:
ppr ¼
p
ppc
(3.12)
Tpr ¼
T
Tpc
(3.13)
where
p ¼ system pressure, psia
ppr ¼ pseudoreduced pressure, dimensionless
T ¼ system temperature, °R
Tpr ¼ pseudoreduced temperature, dimensionless
ppc, Tpc ¼ pseudocritical pressure and temperature, respectively, defined
by the following relationships:
Ppc ¼
X
yi pci
(3.14)
X
yi Tci
(3.15)
i¼1
Tpc ¼
i¼1
Matthews et al. (1942) correlated the critical properties of the C7+ fraction as
a function of the molecular weight and specific gravity:
ðpc ÞC7 + ¼ 1188 431 log ðMC7 + 61:1Þ + ½2319 852log ðMC7 + 53:7Þ γ C7 + 0:8
ðTc ÞC7 + ¼ 608 + 364log ðMC7 + 71:2Þ + ½2450log ðMC7 + Þ 3800log γ C7 +
It should be pointed out that these pseudocritical properties, that is, ppc and
Tpc, do not represent the actual critical properties of the gas mixture. These
pseudoproperties are used as correlating parameters in generating gas
properties.
Based on the concept of pseudoreduced properties, Standing and Katz
(1942) presented a generalized gas compressibility factor chart as shown
in Fig. 3.1. The chart represents the compressibility factors of sweet natural gas as a function of ppr and Tpr. This chart is generally reliable for
natural gas with a minor amount of nonhydrocarbons. It is one of the most
widely accepted correlations in the oil and gas industry.
198 CHAPTER 3 Natural Gas Properties
n FIGURE 3.1 Standing and Katz compressibility factors chart. (Courtesy of the Gas Processors Suppliers Association, 1987. GPSA Engineering Data Book, 10th ed. Gas
Processors Suppliers Association, Tulsa, OK.)
EXAMPLE 3.5
A gas reservoir has the following gas composition:
Component
yi
CO2
N2
C1
0.02
0.01
0.85
Behavior of Real Gases 199
C2
C3
i-C4
n-C4
0.04
0.03
0.03
0.02
The initial reservoir pressure and temperature are 3000 psia and 180°F,
respectively. Calculate the gas compressibility factor under initial reservoir
conditions.
Solution
The table below describes the gas composition.
Component
yi
Tci (°R)
yiTci
pci
yipci
CO2
N2
C1
C2
C3
i-C4
n-C4
0.02
0.01
0.85
0.04
0.03
0.03
0.02
547.91
227.49
343.33
549.92
666.06
734.46
765.62
10.96
2.27
291.83
22.00
19.98
22.03
15.31
Tpc ¼ 383.38
1071
493.1
666.4
706.5
616.40
527.9
550.6
21.42
4.93
566.44
28.26
18.48
15.84
11.01
ppc ¼ 666.38
Step 1. Determine the pseudocritical pressure from Eq. (3.14):
X
ppc ¼
yi pci ¼ 666:38psi
Step 2. Calculate the pseudocritical temperature from Eq. (3.15):
Tpc ¼
X
yi Tci ¼ 383:38°R
Step 3. Calculate the pseudoreduced pressure and temperature by applying
Eqs. (3.12), (3.13), respectively:
ppr ¼
p
3000
¼
¼ 4:50
ppc 666:38
Tpc ¼
T
640
¼
¼ 1:67
Tpc 383:38
Step 4. Determine the Z-factor from Fig. 3.1, using the calculate values of ppr
and Tpr to give Z ¼ 0.85.
Eq. (3.11) can be written in terms of the apparent molecular weight, Ma, and the
weight of the gas, m:
m
RT
pV ¼ Z
Ma
200 CHAPTER 3 Natural Gas Properties
Solving this relationship for the gas’s specific volume and density give
V ZRT
¼
m pMa
(3.16)
1 pMa
ρg ¼ ¼
υ ZRT
(3.17)
υ¼
where
υ ¼ specific volume, ft3/lb
ρg ¼ density, lb/ft3
EXAMPLE 3.6
Using the data in Example 3.5 and assuming real gas behavior, calculate the
density of the gas phase under initial reservoir conditions. Compare the results
with that of ideal gas behavior.
Solution
The table below describes the gas composition.
Component yi
Mi
CO2
N2
C1
C2
C3
i-C4
n-C4
44.01 0.88 547.91
28.01 0.28 227.49
16.04 13.63 343.33
30.1
1.20 549.92
44.1
1.32 666.06
58.1
1.74 734.46
58.1
1.16 765.62
Ma ¼ 20.23
0.02
0.01
0.85
0.04
0.03
0.03
0.02
yiMi
Tci (°R) yiTci
pci
yipci
10.96 1071
21.42
2.27 493.1
4.93
291.83 666.4 566.44
22.00 706.5 28.26
19.98 616.40 18.48
22.03 527.9 15.84
15.31 550.6 11.01
Tpc ¼ 383.38
ppc ¼ 666.38
Step 1. Calculate the apparent molecular weight from Eq. (3.5):
X
Ma ¼
yi Mi ¼ 20:23
Step 2. Determine the pseudocritical pressure from Eq. (3.14):
X
yi pc i ¼ 666:38psi
ppc ¼
Step 3. Calculate the pseudocritical temperature from Eq. (3.15):
X
Tpc ¼
yi Tci ¼ 383:38°R
Step 4. Calculate the pseudoreduced pressure and temperature by applying
Eqs. (3.12), (3.13), respectively:
Behavior of Real Gases 201
p
3000
¼
¼ 450
ppc 666:38
T
640
¼
¼ 1:67
Tpr ¼
Tpc 383:38
ppr ¼
Step 5. Determine the Z-factor from Fig. 3.1 using the calculated values of ppr
and Tpr to give Z ¼ 0.85
Step 6. Calculate the density from Eq. (3.17).
pMa
ZRT
ð3000Þð20:23Þ
¼ 10:41b=ft3
ρg ¼
ð0:85Þð10:73Þð640Þ
ρg ¼
Step 7. Calculate the density of the gas, assuming ideal gas behavior, from
Eq. (3.7):
ð3000Þð20:23Þ
¼ 8:841b=ft3
ρg ¼
ð10:73Þð640Þ
The results of this example show that the ideal gas equation estimated the
gas density with an absolute error of 15% when compared with the density
value as predicted with the real gas equation.
In cases where the composition of a natural gas is not available, the pseudocritical properties, ppc and Tpc, can be predicted solely from the specific
gravity of the gas. Brown et al. (1948) presented a graphical method for a
convenient approximation of the pseudocritical pressure and pseudocritical
temperature of gases when only the specific gravity of the gas is available.
Standing (1977) expressed this graphical correlation in the following
mathematical forms.
For Case 1, natural gas systems,
Tpc ¼ 168 + 325γ g 12:5γ 2g
(3.18)
ppc ¼ 677 + 15:0γ 37:5γ 2g
(3.19)
For Case 2, wet gas systems,
Tpc ¼ 187 + 330γ 71:5γ 2g
(3.20)
ppc ¼ 706 51:7γ g 11:1γ 2g
(3.21)
where
Tpc ¼ pseudocritical temperature, °R
ppc ¼ pseudocritical pressure, psia
γ g ¼ specific gravity of the gas mixture
202 CHAPTER 3 Natural Gas Properties
EXAMPLE 3.7
Using Eqs. (3.18), (3.19), calculate the pseudocritical properties and rework
Example 3.5.
Solution
Step 1. Calculate the specific gravity of the gas:
γg ¼
Ma
20:23
¼
¼ 0:699
28:96 28:96
Step 2. Solve for the pseudocritical properties by applying Eqs. (3.18), (3.19):
2
Tpc ¼ 168 + 325γ g 12:5 γ g
Tpc ¼ 168 + 325ð0:699Þ 12:5ð0:699Þ2 ¼ 389:1°R
2
ppc ¼ 677 + 15γ g 37:5 γ g
ppc ¼ 677 + 15ð0:699Þ 37:5ð0:699Þ2 ¼ 669:2psi
Step 3. Calculate ppr and Tpr:
3000
¼ 4:48
669:2
640
¼ 1:64
Tpr ¼
389:1
ppr ¼
Step 4. Determine the gas compressibility factor from Fig. 3.1:
Z ¼ 0:824
Step 5. Calculate the density from Eq. (3.17):
pMa
ZRT
ð3000Þð20:23Þ
¼ 10:461b=ft3
ρg ¼
ð0:845Þð10:73Þð640Þ
ρg ¼
Effect of Nonhydrocarbon Components
on the Z-Factor
Natural gases frequently contain materials other than hydrocarbon components, such as nitrogen, carbon dioxide, and hydrogen sulfide. Hydrocarbon
gases are classified as sweet or sour depending on the hydrogen sulfide
content. Both sweet and sour gases may contain nitrogen, carbon dioxide,
or both. A hydrocarbon gas is termed a sour gas if it contains 1 grain of
H2S per 100 ft3.
The common occurrence of small percentages of nitrogen and carbon dioxide, in part, is considered in the correlations previously cited. Concentrations of up to 5% of these nonhydrocarbon components do not seriously
Nonhydrocarbon Adjustment Methods 203
affect accuracy. Errors in compressibility factor calculations as large as 10%
may occur in higher concentrations of nonhydrocarbon components in gas
mixtures.
NONHYDROCARBON ADJUSTMENT METHODS
Two methods were developed to adjust the pseudocritical properties of the
gases to account for the presence of the n-hydrocarbon components: the
Wichert-Aziz method and the Carr-Kobayashi-Burrows method.
Wichert-Aziz's Correction Method
Natural gases that contain H2S and/or CO2 frequently exhibit different compressibility factors behavior than sweet gases. Wichert and Aziz (1972)
developed a simple, easy-to-use calculation to account for these differences.
This method permits the use of the Standing-Katz Z-factor chart (Fig. 3.1),
by using a pseudocritical temperature adjustment factor, ε, which is a function of the concentration of CO2 and H2S in the sour gas. This correction
factor then is used to adjust the pseudocritical temperature and pressure
according to the following expressions:
0
Tpc
¼ Tpc ε
p0pc ¼
0
ppc Tpc
Tpc + Bð1 BÞε
(3.22)
(3.23)
where
Tpc ¼ pseudocritical temperature, °R
ppc ¼ pseudocritical pressure, psia
Tpc0 ¼ corrected pseudocritical temperature, °R
ppc0 ¼ corrected pseudocritical pressure, psia
B ¼ mole fraction of H2S in the gas mixture
ε ¼ pseudocritical temperature adjustment factor, defined
mathematically by the following expression:
ε ¼ 120 A0:9 A1:6 + 15 B0:5 B4:0
(3.24)
where the coefficient A is the sum of the mole fraction H2S and CO2 in the
gas mixture:
A ¼ yH2 S + yCO2
The computational steps of incorporating the adjustment factor ε into the
Z-factor calculations are summarized next.
204 CHAPTER 3 Natural Gas Properties
Step 1. Calculate the pseudocritical properties of the whole gas mixture by
applying Eqs. (3.18), (3.19) or Eqs. (3.20), (3.21).
Step 2. Calculate the adjustment factor, ε, from Eq. (3.24).
Step 3. Adjust the calculated ppc and Tpc (as computed in step 1) by applying
Eqs. (3.22), (3.23).
Step 4. Calculate the pseudoreduced properties, ppr and Tpr, from
Eqs. (3.11), (3.12).
Step 5. Read the compressibility factor from Fig. 3.1.
EXAMPLE 3.8
A sour natural gas has a specific gravity of 0.70. The compositional analysis
of the gas shows that it contains 5% CO2 and 10% H2S. Calculate the density
of the gas at 3500 psia and 160°F.
Solution
Step 1. Calculate the uncorrected pseudocritical properties of the gas from
Eqs. (3.18), (3.19):
2
Tpc ¼ 168 + 325γ g 12:5 γ g
Tpc ¼ 168 + 325ð0:7Þ 12:5ð0:7Þ2 ¼ 389:38°R
2
ppc ¼ 677 + 15γ g 37:5 γ g
ppc ¼ 677 + 15ð0:7Þ 37:5ð0:7Þ2 ¼ 669:1psi
Step 2. Calculate the pseudocritical temperature adjustment factor from
Eq. (3.24):
ε ¼ 120 0:150:9 0:151:6 + 15 0:50:5 0:14 ¼ 20:735
Step 3. Calculate the corrected pseudocritical temperature by applying
Eq. (3.22):
0
Tpc
¼ 389:38 20:735 ¼ 368:64
Step 4. Adjust the pseudocritical pressure ppc by applying Eq. (3.23):
p0pc ¼
ð669:1Þð368:64Þ
¼ 630:44psia
389:38 + 0:1ð1 0:1Þð20:635Þ
Step 5. Calculate ppr and Tpr:
ppr ¼
3500
¼ 5:55
630:44
Tpr ¼
160 + 460
¼ 1:68
368:64
Nonhydrocarbon Adjustment Methods 205
Step 6. Determine the Z-factor from Fig. 3.1:
Z ¼ 0:89
Step 7. Calculate the apparent molecular weight of the gas from Eq. (3.10):
Ma ¼ 28:96γ g ¼ 28:96ð0:7Þ 20:27
Step 8. Solve for the gas density by applying Eq. (3.17):
pMa
ZRT
ð3500Þð20:27Þ
¼ 11:98lb=ft3
ρg ¼
ð0:89Þð10:73Þð620Þ
ρg ¼
Whitson and Brule (2000) point out that, when only the gas specific gravity
and nonhydrocarbon content are known (including nitrogen, yN 2 ), the
following procedure is recommended:
n
Calculate hydrocarbon specific gravity γ gHC (excluding the
nonhydrocarbon components) from the following relationship:
γ gHC ¼
n
28:96γ g ðyN2 MN2 + yCO2 + MCO2 + yH2 S MH2 S Þ
28:96ð1 yN2 yCO2 yH2 S Þ
Using the calculated hydrocarbon specific gravity, γ gHC, determine
the pseudocritical properties from Eqs. (3.18), (3.19):
T
¼ 168 + 325γ gHC 12:5γ 2gHC
pc HC
ppc HC ¼ 677 + 150γ gHC 37:5γ 2gHC
n
Adjust these two values to account for the nonhydrocarbon components
by applying the following relationships:
ppc ¼ ð1 yN2 yCO2 yH2 S Þ ppc HC + yN2 ð pc ÞN2 + yCO2 ð pc ÞCO2 + yH2 S ðpc ÞH2 S
Tpc ¼ ð1 yN2 yCO2 yH2 S Þ Tpc HC + yN2 ðTc ÞN2 + yCO2 ðTc ÞCO2 + yH2 S ðTc ÞH2 S
n
Use the above calculated pseudocritical properties in Eqs. (3.22), (3.23)
to obtain the adjusted properties for the Wichert-Aziz method of
calculating the Z-factor.
Carr-Kobayashi-Burrows's Correction Method
Carr et al. (1954) proposed a simplified procedure to adjust the pseudocritical properties of natural gases when nonhydrocarbon components are
206 CHAPTER 3 Natural Gas Properties
present. The method can be used when the composition of the natural gas
is not available. The proposed procedure is summarized in the following
steps.
Step 1. Knowing the specific gravity of the natural gas, calculate the
pseudocritical temperature and pressure by applying Eqs. (3.18),
(3.19).
Step 2. Adjust the estimated pseudocritical properties by using the
following two expressions:
0
Tpc
¼ Tpc 80yCO2 + 130yH2 S 250yN2
(3.25)
p0pc ¼ ppc + 440yCO2 + 600yH2 S 170yN2
(3.26)
where
Tpc0 ¼ the adjusted pseudocritical temperature, °R
Tpc ¼ the unadjusted pseudocritical temperature, °R
yCO 2 ¼ mole fraction of CO2
yH 2 S ¼ mole fraction of H2S in the gas mixture
yN2 ¼ mole fraction of nitrogen
ppc0 ¼ the adjusted pseudocritical pressure, psia
ppc ¼ the unadjusted pseudocritical pressure, psia
Step 3. Use the adjusted pseudocritical temperature and pressure to
calculate the pseudoreduced properties.
Step 4. Calculate the Z-factor from Fig. 3.1.
EXAMPLE 3.9
Using the data in Example 3.8, calculate the density by employing the preceding
correction procedure.
Solution
Step 1. Determine the corrected pseudocritical properties from Eqs. (3.25),
(3.26).
0
Tpc
¼ 389:38 80ð0:05Þ + 130ð0:10Þ 250ð0Þ ¼ 398:38°R
0
ppc ¼ 669:1 + 440ð0:05Þ + 600ð0:01Þ 170ð0Þ ¼ 751:1psia
Step 2. Calculate ppr and Tpr:
3500
¼ 4:56
751:1
620
¼ 1:56
Tpr ¼
398:38
ppr ¼
Correction for High-Molecular Weight Gases 207
Step 3. Determine the gas compressibility factor from Fig. 3.1:
Z ¼ 0:820
Step 4. Calculate the gas density:
ρg ¼
pMa
ZRT
γg ¼
ð3500Þð20:27Þ
¼ 13:0 lb=ft3
ð0:82Þð10:73Þð620Þ
The gas density as calculated in Example 3.8 with a Z-factor of 0.89 is
ρg ¼
ð3500Þð20:27Þ
¼ 11:98lb=ft3
ð0:89Þð10:73Þð620Þ
CORRECTION FOR HIGH-MOLECULAR
WEIGHT GASES
It should be noted that the Standing and Katz Z-factor chart (Fig. 3.1)
was prepared from data on binary mixtures of methane with propane,
ethane, and butane and on natural gases, thus covering a wide range in
composition of hydrocarbon mixtures containing methane. No mixtures
having molecular weights in excess of 40 were included in preparing
this plot.
Sutton (1985) evaluated the accuracy of the Standing-Katz compressibility
factor chart using laboratory-measured gas compositions and Z-factors and
found that the chart provides satisfactory accuracy for engineering calculations. However, Kay’s mixing rules, that is, Eqs. (3.13), (3.14), or comparable gravity relationships for calculating pseudocritical pressure and
temperature, result in unsatisfactory Z-factors for high-molecular weight
reservoir gases. Sutton observed that large deviations occur to gases with
high heptanes-plus concentrations. He pointed out that Kay’s mixing rules
should not be used to determine the pseudocritical pressure and temperature
for reservoir gases with specific gravities greater than about 0.75. Sutton
proposed that this deviation can be minimized by utilizing the mixing rules
developed by Stewart et al. (1959), together with newly introduced empirical adjustment factors (FJ, EJ, and EK) to account for the presence of the
heptanes-plus fraction, C7+, in the gas mixture. The proposed approach is
outlined in the following steps.
208 CHAPTER 3 Natural Gas Properties
Step 1. Calculate the parameters J and K from the following relationships:
"
"
#
#2
1 X Tci
2 X Tci 0:5
+
J¼
yi
yi
pci
pci
3 i
3 i
K¼
X yi Tci
pffiffiffiffiffiffi
pci
i
(3.27)
(3.28)
where
J ¼ Stewart-Burkhardt-Voo correlating parameter, °R/psia
K ¼ Stewart-Burkhardt-Voo correlating parameter, °R/psia
yi ¼ mole fraction of component i in the gas mixture
Step 2. Calculate the adjustment parameters FJ, EJ, and EK from the
following expressions:
sffiffiffiffiffi!#2
" 1
Tc
2
Tc
+
FJ ¼ y
y
pc
pc
3
3
(3.29)
C7 +
EJ ¼ 0:6081FJ + 1:1325F2J 14:004FJ y2C7 +
h
i
T
0:3129yC7 + 4:8156ðyC7 + Þ2 + 27:3751ðyC7 + Þ3
EK ¼ pffiffiffiffiffi
p c C7 +
(3.30)
(3.31)
where
yC7 + ¼ mole fraction of the heptanes-plus component
ðTc ÞC7 + ¼ critical temperature of the C7+
ðpc ÞC7 + ¼ critical pressure of the C7+
Step 3. Adjust the parameters J and K by applying the adjustment factors EJ
and EK, according to these relationships:
J 0 ¼ J EJ
0
K ¼ K EK
(3.32)
(3.33)
where
J, K are calculated from Eqs. (3.27), (3.28)
EJ, EK are calculated from Eqs. (3.30), (3.31)
Step 4. Calculate the adjusted pseudocritical temperature and pressure from
the expressions
0
Tpc
¼
ðK 0 Þ2
J0
(3.34)
Correction for High-Molecular Weight Gases 209
p0pc ¼
0
Tpc
J0
(3.35)
Step 5. Having calculated the adjusted Tpc and ppc, the regular procedure of
calculating the compressibility factor from the Standing and Katz
chart is followed.
Sutton’s proposed mixing rules for calculating the pseudocritical properties
of high-molecular weight reservoir gases, γ g > 0.75, should significantly
improve the accuracy of the calculated Z-factor.
EXAMPLE 3.10
A hydrocarbon gas system has the following composition:
Component
yi
C1
C2
C3
n-C4
n-C5
C6
C7+
0.83
0.06
0.03
0.02
0.02
0.01
0.03
The heptanes-plus fraction is characterized by a molecular weight and
specific gravity of 161 and 0.81, respectively. Using Sutton’s methodology,
calculate the density of the gas 2000 psi and 150°F. Then, recalculate the
gas density without adjusting the pseudocritical properties.
Solution using Sutton's methodology
Step 1. Calculate the critical properties of the heptanes-plus fraction by the
Riazi and Daubert correlation [Chapter 2, Eq. (2.6)] and construct the
table as shown below:
θ ¼ aðMÞb γ c exp½d ðMÞ + eγ + f ðMÞγ ðTc ÞC7 + ¼ ð544:2Þ ð161Þ0:2998 ð0:81Þ1:0555 exp
h
i
1:3478ð10Þ4 ð150Þ 0:61641ð0:81Þ ¼ 1189°R
ðpc ÞC7 + ¼ 4:5203ð10Þ4 1610:8063 0:811:6015 exp
h
i
1:8078ð10Þ3 ð150Þ 0:3084ð0:81Þ ¼ 318:4psia
210 CHAPTER 3 Natural Gas Properties
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðTc =pc Þi
Component
yi
Mi
Tci
pci
yiMi
yi (Tci/pci)
yi
C1
C2
C3
n-C4
n-C5
C6
C7+
Total
0.83
0.06
0.03
0.02
0.02
0.01
0.03
16.0
30.1
44.1
58.1
72.2
84.0
161.0
343.33
549.92
666.06
765.62
845.60
923.00
1189.0
666.4
706.5
616.4
550.6
488.6
483.0
318.4
13.31
1.81
1.32
1.16
1.45
0.84
4.83
24.72
0.427
0.047
0.032
0.028
0.035
0.019
0.112
0.700
0.596
0.053
0.031
0.024
0.026
0.014
0.058
0.802
pffiffiffiffiffi
yi Tc = pc i
11.039
1.241
0.805
0.653
0.765
0.420
1.999
16.922
Step 2. Calculate the parameters J and K from Eqs. (3.27), (3.28) and using the
values from the above table.
1
2
J ¼ ð0:700Þ + ð0:802Þ2 ¼ 0:662
3
3
"
"
#
0:5 #2
1 X
Tci
2 X
Tci
+
yi
yi
J¼
pci
pci
3 i
3 i
K¼
X yi Tci
pffiffiffiffiffiffi ¼ 16:922
pci
i
Step 3. To account for the presence of the C7+ fraction, calculate the adjustment
factors FJ, EJ, and EK by applying Eqs. (3.29)–(3.31):
sffiffiffiffiffi!#2
" 1
Tc
2
Tc
FJ ¼ y
+
pc
pc
3
3
C7 +
1
2
FJ ¼ ½0:112 + ½0:0582 ¼ 0:0396
3
3
EJ ¼ 0:6081FJ + 1:1325F2J 14:004FJ yC7 + + 64:434FJ y2C7 +
EJ ¼ 0:6081ð0:04Þ + 1:0352ð0:04Þ2 14:004ð0:04Þð0:03Þ
+ 64:434ð0:04Þð0:3Þ2 ¼ 0:012
h i h
i
ffi
0:3129yC7 + 4:8156ðyC7 + Þ2 + 27:2751ð0:03Þ3
EK ¼ pTffiffiffi
pc
C7 +
h
i
EK ¼ 66:634 0:3129ð0:03Þ 4:8156ð0:03Þ2 + 27:3751ð0:03Þ3 ¼ 0:386
Step 4. Calculate the parameters J0 and K0 from Eqs. (3.34), (3.35):
J 0 ¼ J EJ ¼ 0:662 0:012 ¼ 0:650
K 0 ¼ K EK ¼ 16:922 0:386 ¼ 16:536
Correction for High-Molecular Weight Gases 211
Step 5. Determine the adjusted pseudocritical properties from Eqs. (3.34),
(3.35):
ðK 0 Þ2 ð16:536Þ2
¼
¼ 420:7
J0
0:65
0
Tpc
420:7
p0pc ¼ 0 ¼
¼ 647:2
J
0:65
0
¼
Tpc
Step 6. Calculate the pseudoreduced properties of the gas by applying
Eqs. (3.11), (3.12) to give
2000
¼ 3:09
647:2
610
¼ 1:45
Tpr ¼
420:7
ppr ¼
Step 7. Calculate the Z-factor from Fig. 3.1 to give
Z ¼ 0:745
Step 8. From Eq. (3.17), calculate the density of the gas:
pMa
ZRT
ð2000Þð24:73Þ
¼ 10:14lb=ft3
ρg ¼
ð10:73Þð610Þð0:745Þ
ρg ¼
Solution without adjusting the pseudocritical properties
Step 1. Calculate the specific gravity of the gas:
γg ¼
Ma
24:73
¼
¼ 0:854
28:96 28:96
Step 2. Solve for the pseudocritical properties by applying Eqs. (3.18), (3.19):
Tpc ¼ 168 + 325ð0:84Þ 12:5ð0:854Þ2 ¼ 436:4°R
ppc ¼ 677 + 15ð0:854Þ 37:5ð0:854Þ2 ¼ 662:5psia
Step 3. Calculate ppr and Tpr:
2000
¼ 3:02
662:5
610
¼ 1:40
Tpr ¼
436:4
ppr ¼
Step 4. Calculate the Z-factor from Fig. 3.1 to give
Z ¼ 0:710
Step 5. From Eq. (3.17), calculate the density of the gas:
pMa
ZRT
ð2000Þð24:73Þ
¼ 10:64lb=ft3
ρg ¼
ð10:73Þð610Þð0:710Þ
ρg ¼
212 CHAPTER 3 Natural Gas Properties
Direct Calculation of Compressibility Factors
After four decades of existence, the Standing-Katz Z-factor chart is still
widely used as a practical source of natural gas compressibility factors.
As a result, there was an apparent need for a simple mathematical description of that chart. Several empirical correlations for calculating Z-factors
have been developed over the years. Papay (1985) proposed a simple
expression for calculating the gas compressibility factor explicitly. Papay
correlated the Z-factor with pseudoreduced pressure ppr and temperature
Tpr as expressed next:
Z ¼1
0:274p2pr
3:53ppr
+ 0:8157T
0:9813Tpr
10
10
Tpr
For example, at ppr ¼ 3 and Tpr ¼ 2, the Z-factor from the preceding equation is
Z ¼1
0:274p2pr
3:53ppr
3:53ð3Þ
0:274ð3Þ2
+ 0:8157Tpr ¼ 1 0:9813ð2Þ + 0:8157ð2Þ ¼ 0:9422
0:9813Tpr
10
10
10
10
as compared with value obtained from the Standing and Katz chart of 0.954.
Numerous rigorous mathematical expressions have been proposed to
accurately reproduce the Standing and Katz Z-factor chart. Most of these
expressions are designed to solve for the gas compressibility factor at any
ppr and Tpr iteratively. Three of these empirical correlations are described
next: Hall-Yarborough, Dranchuk-Abu-Kassem, and Dranchuk-PurvisRobinson.
HALL-YARBOROUGH'S METHOD
Hall and Yarborough (1973) presented an equation of state that accurately
represents the Standing and Katz Z-factor chart. The proposed expression is
based on the Starling-Carnahan equation of state. The coefficients of the
correlation were determined by fitting them to data taken from the Standing
and Katz Z-factor chart. Hall and Yarborough proposed the following mathematical form:
Z¼
h
i
0:06125tppr
exp 1:2ð1 tÞ2
Y
(3.36)
where
ppr ¼ pseudoreduced pressure
t ¼ reciprocal of the pseudoreduced temperature (ie, Tpc/T)
Y ¼ the reduced density, which can be obtained as the solution of the
following equation:
Hall-Yarborough's Method 213
FðY Þ ¼ X1 +
where
Y + Y2 + Y3 Y4
ðX2 ÞY 2 + ðX3 ÞY X4 ¼ 0
ð1 Y Þ
(3.37)
h
i
X1 ¼ 0:06125ppr texp 1:2ð1 tÞ2
X2 ¼ 14:76t 9:76t2 + 4:58t3
X3 ¼ 90:7t 242:2t2 + 42:4t3
X4 ¼ ð2:18 + 2:82tÞ
Eq. (3.37) is a nonlinear equation and can be solved conveniently for the
reduced density Y by using the Newton-Raphson iteration technique. The
computational procedure of solving Eq. (3.37) at any specified pseudoreduced pressure, ppr, and temperature, Tpr, is summarized in the following
steps.
Step 1. Make an initial guess of the unknown parameter, Yk, where k is
an iteration counter. An appropriate initial guess of Y is given by
the following relationship:
h
i
Y k ¼ 0:0125ppr texp 1:2ð1 tÞ2
Step 2. Substitute this initial value in Eq. (3.37) and evaluate the nonlinear
function. Unless the correct value of Y has been initially selected,
Eq. (3.37) will have a nonzero value of f(Y).
Step 3. A new improved estimate of Y, that is, Yk+1, is calculated from the
following expression:
f Yk
Yk + 1 ¼ Yk 0 k
f ðY Þ
(3.38)
where f 0 (Yk) is obtained by evaluating the derivative of Eq. (3.37) at Yk, or
f 0 ðY Þ ¼
1 + 4Y + 4Y 2 4Y 3 + Y 4
ð1 Y Þ4
2ðX2 ÞY + ðX3 ÞðX4 ÞY ðX4 1Þ
(3.39)
Step 4. Steps 2 and 3 are repeated n times until the error; that is,
abs(Yk Yk+1) becomes smaller than a preset tolerance,
say 1012.
Step 5. The correct value of Y is then used to evaluate Eq. (3.36) for the
compressibility factor:
214 CHAPTER 3 Natural Gas Properties
Z¼
h
i
0:06125tppr
exp 1:2ð1 tÞ2
Y
Hall and Yarborough pointed out that the method is not recommended
for application if the pseudoreduced temperature is less than 1.
DRANCHUK AND ABU-KASSEM'S METHOD
Dranchuk and Abu-Kassem (1975) derived an analytical expression for
calculating the reduced gas density that can be used to estimate the gas compressibility factor. The reduced gas density ρr is defined as the ratio of the
gas density at a specified pressure and temperature to that of the gas at its
critical pressure or temperature:
ρr ¼
ρ
½pMa =ðZRT Þ
½p=ðZT Þ
¼
¼
ρc ½pc Ma =ðZc RTc Þ ½pc =ðZc Tc Þ
The critical gas compressibility factor Zc is approximately 0.27, which leads
to the following simplified expression for the reduced gas density as
expressed in terms of the reduced temperature Tr and reduced pressure pr:
ρr ¼
0:27ppr
ZTpr
(3.40)
The authors proposed the following 11-constant equation of state for calculating the reduced gas density:
f ðρr Þ ¼ ðR1 Þρr R2
+ ðR3 Þρ2r ðR4 Þρ5r + ðR5 Þ 1 + A11 ρ2r ρ2r exp A11 ρ2r + 1 ¼ 0
ρr
(3.41)
with the coefficients R1 through R5 as defined by the following relations:
R1 ¼ A1 +
A2 A3 Ar At
+ 3 + 4 + 5
Tpr Tpr
Tpr Tpr
0:27ppr
Tpr
A7 Ag
R3 ¼ A6 +
+ 2
Tpr Tpr
"
#
A7 A8
R4 ¼ A9
+ 2
Tpr Tpr
" #
A10
R5 ¼ 3
Tpr
R2 ¼
Dranchuk and Abu-Kassem's Method 215
The constants A1 through A11 were determined by fitting the equation, using
nonlinear regression models, to 1500 data points from the Standing and Katz
Z-factor chart. The coefficients have the following values:
A1 ¼ 0:3265
A4 ¼ 0:01569
A7 ¼ 0:7361
A10 ¼ 0:6134
A2 ¼ 1:0700
A3 ¼ 0:5339
A5 ¼ 0:05165
A6 ¼ 0:5475
A8 ¼ 0:1844
A9 ¼ 0:1056
A11 ¼ 0:7210
Eq. (3.41) can be solved for the reduced gas density ρr by applying the
Newton-Raphson iteration technique as summarized in the following steps.
Step 1. Make an initial guess of the unknown parameter, ρkr , where k is
an iteration counter. An appropriate initial guess of ρkr is given
by the following relationship:
ρr ¼
0:27ppr
Tpr
Step 2. Substitute this initial value in Eq. (3.41) and evaluate the nonlinear
function. Unless the correct value of ρkr has been initially selected,
Eq. (3.41) will have a nonzero value for the function f( ρkr ).
Step 3. A new improved estimate of ρr, that is, ρkr + 1 , is calculated from the
following expression:
f ρk
ρkr + 1 ¼ ρkr 0 rk f ρr
where
R2
+ 2ðR3 Þγ r 5ðR4 Þρ4r
ρ2r
+ 2ðR5 Þρr exp A11 ρ2r 1 + 2A11 ρ3r A11 ρ2r 1 + A11 ρ2r
f 0 ðρr Þ ¼ ðR1 Þ +
Step 4. Steps 2 and 3 are repeated n times, until the error, that is,
abs ρkr ρkr + 1 , becomes smaller than a preset tolerance, say, 1012.
Step 5. The correct value of ρr is then used to evaluate Eq. (3.40) for the
compressibility factor:
Z¼
0:27ppr
ρr Tpr
The proposed correlation was reported to duplicate compressibility factors
from the Standing and Katz chart with an average absolute error of 0.585%
and is applicable over the ranges
0:2 ppr < 30
1:0 < Tpr 3:0
216 CHAPTER 3 Natural Gas Properties
DRANCHUK-PURVIS-ROBINSON METHOD
Dranchuk et al. (1974) developed a correlation based on the BenedictWebb-Rubin type of equation of state. Fitting the equation to 1500 data
points from the Standing and Katz Z-factor chart optimized the eight
coefficients of the proposed equations. The equation has the following
form:
T5
1 + T1 ρr + T2 ρ2r + T3 ρ5r + T4 ρ2r 1 + A8 ρ2r exp A8 ρ2r ¼ 0
ρr
(3.42)
with
A2 A3
+ 3
Tpr Tpr
A5
T2 ¼ A4 +
Tpr
A5 A6
T3 ¼
Tpr
A7
T4 ¼ 3
Tpr
0:27ppr
T5 ¼
Tpr
T1 ¼ A1 +
where ρr is defined by Eq. (3.41) and the coefficients A1 through A8 have the
following values:
A1 ¼ 0:31506237
A5 ¼ 0:31506237
A2 ¼ 1:0467099
A6 ¼ 1:0467099
A3 ¼ 0:57832720
A7 ¼ 0:57832720
A4 ¼ 0:53530771
A8 ¼ 0:53530771
The solution procedure of Eq. (3.43) is similar to that of Dranchuk and AbuKassem.
The method is valid within the following ranges of pseudoreduced temperature and pressure:
1:05 Tpr < 3:0
0:2 ppr 3:0
Compressibility of Natural Gases
Knowledge of the variability of fluid compressibility with pressure and
temperature is essential in performing many reservoir engineering calculations. For a liquid phase, the compressibility is small and usually assumed to
Dranchuk-Purvis-Robinson Method 217
be constant. For a gas phase, the compressibility is neither small nor
constant.
By definition, the isothermal gas compressibility is the change in volume per
unit volume for a unit change in pressure, or in equation form,
cg ¼
1 @V
V @p T
(3.43)
where cg ¼ isothermal gas compressibility, 1/psi.
From the real gas equation of state,
V¼
nRTZ
p
Differentiating this equation with respect to pressure at constant temperature
T gives
@V
1 @Z
Z
¼ nRT
2
@p
p @p
p
Substituting into Eq. (3.43) produces the following generalized relationship:
1 1 @Z
cg ¼ p Z @p T
(3.44)
For an ideal gas, Z ¼ 1 and (@Z/@p)T ¼ 0; therefore,
cg ¼
1
p
(3.45)
It should be pointed out that Eq. (3.45) is useful in determining the expected
order of magnitude of the isothermal gas compressibility.
Eq. (3.44) can be conveniently expressed in terms of the pseudoreduced
pressure and temperature by simply replacing p with ( ppr ppc):
"
#
1
1
@Z
cg ¼
ppr ppc Z @ ppr ppc
Tpr
Multiplying this equation by ppc yields
cg ppc ¼ cpr ¼
1 1 @Z
ppr Z @ppr Tpr
(3.46)
218 CHAPTER 3 Natural Gas Properties
The term cpr is called the isothermal pseudoreduced compressibility,
defined by the relationship
cpr ¼ cg ppc
(3.47)
where
cpr ¼ isothermal pseudoreduced compressibility
cg ¼ isothermal gas compressibility, psi1
ppc ¼ pseudoreduced pressure, psi
Values of (@z/@ppr)Tpr can be calculated from the slope of the Tpr isotherm on
the Standing and Katz Z-factor chart.
EXAMPLE 3.11
A hydrocarbon gas mixture has a specific gravity of 0.72. Calculate the
isothermal gas compressibility coefficient at 2000 psia and 140°F assuming,
first, an ideal gas behavior, then a real gas behavior.
Solution
Assuming an ideal gas behavior, determine cg by applying Eq. (3.44)
cg ¼
1
¼ 500 106 psi1
2000
Assuming a real gas behavior, use the following steps.
Step 1. Calculate Tpc and ppc by applying Eqs. (3.17), (3.18):
Tpc ¼ 168 + 325ð0:72Þ 12:5ð0:72Þ2 ¼ 395:5°R
ppc ¼ 677 + 15ð0:72Þ 37:5ð0:72Þ2 ¼ 668:4 psia
Step 2. Compute ppr and Tpr from Eqs. (3.11), (3.12):
ppr ¼
2000
¼ 2:99
668:4
Tpr ¼
600
¼ 1:52
395:5
Step 3. Determine the Z-factor from Fig. 3.1:
Z ¼ 0:78
Step 4. Calculate the slope [dZ/dppr]Tpr ¼ 1.52:
@Z
¼ 0:022
@ppr Tpr
Dranchuk-Purvis-Robinson Method 219
Step 5. Solve for cpr by applying Eq. (3.47):
cpr ¼
1
1
½0:022 ¼ 0:3627
2:99 0:78
Step 6. Calculate cg from Eq. (3.48):
cpr ¼ cg ppc
cg ¼
cpr 0:327
¼
¼ 543 106 psi1
ppc 668:4
Trube (1957a,b) presented graphs from which the isothermal compressibility of
natural gases may be obtained. The graphs, Figs. 3.2 and 3.3, give the isothermal
pseudoreduced compressibility as a function of pseudoreduced pressure and
temperature.
n FIGURE 3.2 Trube's pseudoreduced compressibility for natural gases. (Permission to publish from the Society of Petroleum Engineers of the AIME. # SPE-AIME.)
220 CHAPTER 3 Natural Gas Properties
n FIGURE 3.3 Trube's pseudoreduced compressibility for natural gases. (Permission to publish from the Society of Petroleum Engineers of the AIME. # SPE-AIME.)
EXAMPLE 3.12
Using Trube’s generalized charts, rework Example 3.11.
Solution
Step 1. From Fig. 3.3, find cpr to give cpr ¼ 0.36
Step 2. Solve for cg by applying Eq. (3.49):
cg ¼
0:36
¼ 539 106 psi1
668:4
Mattar et al. (1975) presented an analytical technique for calculating the isothermal gas compressibility. The authors expressed cpr as a function of @Z/
@ρr rather than @Z/@ppr to give
2
3
@Z
0:27 6 ð@Z=@ρr ÞTpr 7
¼
5
4
@ppr Z2 Tpr 1 + ρr ð@Z=@ρ Þ
r Tpr
Z
(3.48)
Dranchuk-Purvis-Robinson Method 221
Eq. (3.48) may be substituted into Eq. (3.46) to express the pseudoreduced
compressibility as
3
2
1 0:27 6 ð@Z=@ρr ÞTpr 7
cpr ¼ 2 4 ρ
5
pr Z Tpr 1 + r ð@Z=@ρ Þ
r Tpr
Z
(3.49)
where ρr ¼ pseudoreduced gas density.
The partial derivative appearing in Eq. (3.49) is obtained from Eq. (3.42) as
@Z
¼ T1 + 2T2 ρr + 5T3 ρ4r + 2T4 ρr 1 + A8 ρ2r A28 ρ4r exp A8 ρ2r
@ρr Tpr
(3.50)
where the coefficients T1 through T4 and A1 through A8 are as defined previously by Eq. (3.42).
Gas Formation Volume Factor
The gas formation volume factor is used to relate the volume of gas, as
measured at reservoir conditions, to the volume of the gas as measured at
standard conditions; that is, 60°F and 14.7 psia. This gas property is then
defined as the actual volume occupied by a certain amount of gas at a
specified pressure and temperature, divided by the volume occupied by
the same amount of gas at standard conditions. In equation form, the
relationship is expressed as
Bg ¼
ðV Þp, T
Vsc
(3.51)
where
Bg ¼ gas formation volume factor, ft3/scf
Vp,T ¼ volume of gas at pressure p and temperature T, ft3
Vsc ¼ volume of gas at standard conditions
Applying the real gas equation of state, Eq. (3.11), and substituting for the
volume V, gives
ZnRT
psc ZT
p
Bg ¼
¼
Zsc nRTsc Tsc p
psc
where
Zsc ¼ Z-factor at standard conditions ¼ 1.0
psc, Tsc ¼ standard pressure and temperature
222 CHAPTER 3 Natural Gas Properties
Assuming that the standard conditions are represented by psc ¼ 14.7 psia and
Tsc ¼ 520, the preceding expression can be reduced to the following
relationship:
Bg ¼ 0:02827
ZT
p
(3.52)
where
Bg ¼ gas formation volume factor, ft3/scf
Z ¼ gas compressibility factor
T ¼ temperature, °R
In other field units, the gas formation volume factor can be expressed in bbl/
scf, to give
Bg ¼ 0:005035
ZT
p
(3.53)
Gas Expansion Factor
The reciprocal of the gas formation volume factor, called the gas expansion
factor, is designated by the symbol Eg:
Eg ¼
1
Bg
In terms of scf/ft3, the gas expansion factor is
p
, scf=ft3
ZT
(3.54)
p
, scf=bbl
ZT
(3.55)
Eg ¼ 35:37
In other units,
Eg ¼ 198:6
EXAMPLE 3.13
A gas well is producing at a rate of 15,000 ft3/day from a gas reservoir at an
average pressure of 2000 psia and a temperature of 120°F. The specific gravity is
0.72. Calculate the gas flow rate in scf/day.
Solution
Step 1. Calculate the pseudocritical properties from Eqs. (3.17), (3.18), to give
Tpc ¼ 168 + 325γ g 12:5γ 2g ¼ 395:5°R
ppc ¼ 677 + 15:0γ g 37:5γ 2g ¼ 668:4psia
Dranchuk-Purvis-Robinson Method 223
Step 2. Calculate the ppr and Tpr:
ppr ¼
p
2000
¼
¼ 2:99
ppc 668:4
Tpr ¼
T
600
¼
¼ 1:52
Tpc 395:5
Step 3. Determine the Z-factor from Fig. 3.1:
Z ¼ 0:78
Step 4. Calculate the gas expansion factor from Eq. (3.54):
Eg ¼ 35:37
p
2000
¼ 35:37
¼ 151:15scf=ft3
ZT
ð0:78Þð600Þ
Step 5. Calculate the gas flow rate in scf/day by multiplying the gas flow rate (in
ft3/day) by the gas expansion factor, Eg, as expressed in scf/ft3:
Gas flow rate ¼ ð151:15Þð15, 000Þ ¼ 2:267MMscf=day
It is also convenient in many engineering calculations to express the gas
density in terms of the Bg or Eg, from the definitions of gas density, gas formation volume factor, and the gas expansion factor as given by
Ma p R ZT
psc ZT
Bg ¼
Tsc P
Tsc p Eg ¼
psc ZT
ρg ¼
Combining the gas density equation with Bg and Eg equations gives
ρg ¼
psc Ma 1
0:002635Ma
¼
Tsc R Bg
Bg
ρg ¼
psc Ma Eg ¼ 0:002635Ma Eg
Tsc R
Gas Viscosity
The viscosity of a fluid is a measure of the internal fluid friction (resistance)
to flow. If the friction between layers of the fluid is small, that is, low viscosity, an applied shearing force will result in a large velocity gradient. As
the viscosity increases, each fluid layer exerts a larger frictional drag on the
adjacent layers and velocity gradient decreases.
224 CHAPTER 3 Natural Gas Properties
The viscosity of a fluid is generally defined as the ratio of the shear force
per unit area to the local velocity gradient. Viscosities are expressed in
terms of poises, centipoises, or micropoises. One poise equals a viscosity
of 1 dyn-s/cm2 and can be converted to other field units by the following
relationships:
1poise ¼ 100centipoises
¼ 1 106 micropoises
¼ 6:72 102 lbmass=ft s
¼ 20:9 103 lbf s=ft2
The gas viscosity is not commonly measured in the laboratory because it can be
estimated precisely from empirical correlations. Like all intensive properties,
viscosity of a natural gas is completely described by the following function:
μg ¼ ðp, T, yi Þ
where μg ¼ the viscosity of the gas phase. This relationship simply states that
the viscosity is a function of pressure, temperature, and composition. Many
of the widely used gas viscosity correlations may be viewed as modifications of that expression.
Two popular methods that are commonly used in the petroleum industry are
the Carr-Kobayashi-Burrows correlation and the Lee-Gonzalez-Eakin
method, which are described next.
CARR-KOBAYASHI-BURROWS'S METHOD
Carr et al. (1954) developed graphical correlations for estimating the viscosity of natural gas as a function of temperature, pressure, and gas gravity. The
computational procedure of applying the proposed correlations is summarized in the following steps.
Step 1. Calculate the pseudocritical pressure, pseudocritical temperature, and
apparent molecular weight from the specific gravity or the
composition of the natural gas. Corrections to these pseudocritical
properties for the presence of the nonhydrocarbon gases (CO2, N2,
and H2S) should be made if they are present in concentration greater
than 5 mol%.
Step 2. Obtain the viscosity of the natural gas at one atmosphere and the
temperature of interest from Fig. 3.4. This viscosity, as denoted by
μ1, must be corrected for the presence of nonhydrocarbon
components using the inserts of Fig. 3.4. The nonhydrocarbon
fractions tend to increase the viscosity of the gas phase. The effect
Carr-Kobayashi-Burrows's Method 225
n FIGURE 3.4 Carr et al.'s atmospheric gas viscosity correlation. (Carr, N., Kobayashi, R., Burrows, D., 1954. Viscosity of hydrocarbon gases under pressure. Trans. AIME 201, 270–275. Permission to publish
from the Society of Petroleum Engineers of the AIME. # SPE-AIME.)
226 CHAPTER 3 Natural Gas Properties
of nonhydrocarbon components on the viscosity of the natural gas
can be expressed mathematically by the following relationship:
μ1 ¼ ðμ1 Þuncorrected + ðΔμÞN2 + ðΔμÞCO2 + ðΔμÞH2 S
(3.56)
where
μ1 ¼ “corrected” gas viscosity at 1 atm and reservoir temperature, cp
ðΔμÞN2 ¼ viscosity corrections due to the presence of N2
ðΔμÞCO2 ¼ viscosity corrections due to the presence of CO2
ðΔμÞH2 S ¼ viscosity corrections due to the presence of H2S
(μ1)uncorrected ¼ uncorrected gas viscosity, cp
Step 3. Calculate the pseudoreduced pressure and temperature.
Step 4. From the pseudoreduced temperature and pressure, obtain the
viscosity ratio (μg/μ1) from Fig. 3.5. The term μg represents the
viscosity of the gas at the required conditions.
n FIGURE 3.5 Carr et al.'s viscosity ratio correlation. (Carr, N., Kobayashi, R., Burrows, D., 1954. Viscosity of hydrocarbon gases under pressure, Trans. AIME 201, 270–275.
Permission to publish from the Society of Petroleum Engineers of the AIME. # SPE-AIME.)
Carr-Kobayashi-Burrows's Method 227
Step 5. The gas viscosity, μg, at the pressure and temperature of interest, is
calculated by multiplying the viscosity at 1 atm and system
temperature, μ1, by the viscosity ratio. The following examples
illustrate the use of the proposed graphical correlations.
EXAMPLE 3.14
Using the data given in Example 3.13, calculate the viscosity of the gas.
Solution
Step 1. Calculate the apparent molecular weight of the gas:
Ma ¼ ð0:72Þð28:96Þ ¼ 20:85
Step 2. Determine the viscosity of the gas at 1 atm and 140°F from Fig. 3.4:
μ1 ¼ 0:0113
Step 3. Calculate ppr and Tpr:
ppr ¼ 2:99
Tpr ¼ 1:52
Step 4. Determine the viscosity rates from Fig. 3.5:
μg
¼1:5
μ1
Step 5. Solve for the viscosity of the natural gas:
μg
μg ¼ ðμ1 Þ ¼ ð1:5Þð0:0113Þ ¼ 0:01695cp
μ1
Standing (1977) proposed a convenient mathematical expression for
calculating the viscosity of the natural gas at atmospheric pressure and
reservoir temperature, μ1. Standing also presented equations for describing
the effects of N2, CO2, and H2S on μ1. The proposed relationships are
μ1 ¼ ðμ1 Þuncorrected + ðΔμÞN2 + ðΔμÞCO2 + ðΔμÞH2 S
ðμ1 Þuncorrected ¼ 8:118 103 6:15 103 log γ g
+ 1:709 105 2:062 106 γ g ðT 460Þ
ðΔμÞN2 ¼ yN2 8:48 103 log γ g + 9:59 103
ðΔμÞCO2 ¼ yCO2 9:08 103 log γ g + 6:24 103
ðΔμÞH2 S ¼ yH2 S 8:49 103 log γ g + 3:73 103
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
228 CHAPTER 3 Natural Gas Properties
where
μ1 ¼ viscosity of the gas at atmospheric pressure and reservoir
temperature, cp
T ¼ reservoir temperature, °R
γ g ¼ gas gravity; yN2 , yCO2 , yH2 S ¼ mole fraction of N2, CO2, and H2S,
respectively
Dempsey (1965) expressed the viscosity ratio μg/μ1 by the following
relationship:
ln Tpr
μg
¼ a0 + a1 ppr + a2 p2pr + a3 p3pr + Tpr a1 + a5 ppr + a6 p2pr + a7 p3pr
μ1
3
2
a8 + a9 ppr + a10 p2pr + a11 p3pr + Tpr
a12 + a13 ppr + a14 p2pr + a15 p3pr
+ Tpr
where
Tpr ¼ pseudoreduced temperature of the gas mixture
ppr ¼ pseudoreduced pressure of the gas mixture
a0, …, a17 ¼ coefficients of the equations, as follows:
a8 ¼ 7:93385648 101
a0 ¼ 2:46211820
a1 ¼ 2:970547414
1
a2 ¼ 2:86264054 10
a3 ¼ 8:05420522 103
a4 ¼ 2:80860949
a5 ¼ 3:49803305
a6 ¼ 3:60373020 101
a7 ¼ 1:044324 102
a9 ¼ 1:39643306
a10 ¼ 1:47144925 101
a11 ¼ 4:41015512 103
a12 ¼ 8:39387178 102
a13 ¼ 1:86408848 101
a14 ¼ 2:03367881 102
a15 ¼ 6:09579263 104
LEE-GONZALEZ-EAKIN'S METHOD
Lee et al. (1966) presented a semiempirical relationship for calculating the
viscosity of natural gases. The authors expressed the gas viscosity in terms
of the reservoir temperature, gas density, and the molecular weight of the
gas. Their proposed equation is given by
μg ¼ 104 K exp X
ρ Y
g
62:4
(3.62)
Lee-Gonzalez-Eakin's Method 229
where
K¼
ð9:4 + 0:02Ma ÞT 1:5
209 + 19Ma + T
(3.63)
986
+ 0:01Ma
T
(3.64)
X ¼ 3:5 +
Y ¼ 2:4 0:2X
(3.65)
ρ ¼ gas density at reservoir pressure and temperature, lb/ft3
T ¼ reservoir temperature, °R
Ma ¼ apparent molecular weight of the gas mixture
The proposed correlation can predict viscosity values with a standard deviation of 2.7% and a maximum deviation of 8.99%. The correlation is less
accurate for gases with higher specific gravities. The authors pointed out
that the method cannot be used for sour gases.
EXAMPLE 3.15
Rework Example 3.14 and calculate the gas viscosity by using the LeeGonzales-Eakin method.
Solution
Step 1. Calculate the gas density from Eq. (3.17):
pMa
ZRT
ð2000Þð20:85Þ
¼ 8:3 lb=ft3
ρg ¼
ð10:73Þð600Þð0:78Þ
ρg ¼
Step 2. Solve for the parameters K, X, and Y using Eqs. (3.63), (3.64), (3.65),
respectively:
K¼
ð9:4 + 0:02Ma ÞT 1:5
209 + 19Ma + T
½9:4 + 0:02ð20:85Þð600Þ1:5
¼ 119:72
209 + 19ð20:85Þ + 600
986
+ 0:01Ma
X ¼ 3:5 +
T
986
X ¼ 3:5 +
+ 0:01ð20:85Þ ¼ 5:35
600
Y ¼ 2:4 0:2X
K¼
Y ¼ 2:4 0:2ð5:35Þ ¼ 1:33
230 CHAPTER 3 Natural Gas Properties
Step 3. Calculate the viscosity from Eq. (3.62):
μg ¼ 104 K exp X
ρ Y
g
62:4
"
#
8:3 1:33
¼ 0:0173cp
μg ¼ 10 ð119:72Þexp 5:35
62:4
4
Specific Gravity Wet Gases
The specific gravity of a wet gas, γ g, is described by the weighted average of
the specific gravities of the separated gas from each separator. This
weighted average approach is based on the separator gas/oil ratio:
Xn γg ¼
γ sep i + Rst γ st
Rsep i + Rst
i¼1
R
sep
i¼1
Xn i
(3.66)
where
n ¼ number of separators
Rsep ¼ separator gas/oil ratio, scf/STB
γ sep ¼ separator gas gravity
Rst ¼ gas/oil ratio from the stock tank, scf/STB
γ st ¼ gas gravity from the stock tank
For wet gas reservoirs that produce liquid (condensate) at separator
conditions, the produced gas mixtures normally exist as a “single” gas
phase in the reservoir and production tubing. To determine the well-stream
specific gravity, the produced gas and condensate (liquid) must be
recombined in the correct ratio to find the average specific gravity of
the “single-phase” gas reservoir. To calculate the specific gravity of the
well-stream gas, let
γ w ¼ well-stream gas gravity
γ o ¼ condensate (oil) stock-tank gravity, 60°/60°
γ g ¼ average surface gas gravity as defined by Eq. (3.66)
Mo ¼ molecular weight of the stock-tank condensate (oil)
rp ¼ producing oil/gas ratio (reciprocal of the gas/oil ratio, Rs), STB/scf
The average specific gravity of the well stream is given by
γw ¼
γ g + 4580rp γ o
1 + 133, 000rp ðγ o =Mo Þ
(3.67)
Lee-Gonzalez-Eakin's Method 231
In terms of gas/oil ratio, Rs, Eq. (3.67) can be expressed as
γw ¼
γ g Rs + 4580γ o
Rs + 133, 000γ o =Mo
(3.68)
Standing (1974) suggested the following correlation for estimating the
molecular weight of the stock-tank condensate:
Mo ¼
6084
API 5:9
(3.69)
where API is the API gravity of the liquid as given by
API ¼
141:5
131:5
γo
Eq. (3.70) should be used only in the range 45° < API < 60°. Eilerts (1947)
proposed an expression for the ratio γ o/Mo as a function of the condensate
stock-tank API gravity:
γ o =Mo ¼ 0:001892 + 7:35 105 API 4:52 108 ðAPIÞ2
(3.70)
In retrograde and wet gas reservoirs calculations, it is convenient to express
the produced separated gas as a fraction of the total system produced.
This fraction fg can be expressed in terms of the separated moles of gas
and liquid as
fg ¼
ng
ng
¼
nt ng + nl
(3.71)
where
fg ¼ fraction of the separated gas produced in the entire system
ng ¼ number of moles of the separated gas
nl ¼ number of moles of the separated liquid
nt ¼ total number of moles of the well stream
For a total producing gas/oil ratio of Rs scf/STB, the equivalent number of
moles of gas as described by Eq. (3.6) is
ng ¼ Rs =379:4
(3.72)
The number of moles of 1 STB of the separated condensate is given by
no ¼
mass
ðvolumeÞðdensityÞ
¼
molecular weight
Mo
or
no ¼
ð1Þð5:615Þð62:4Þγ o 350:4γ o
¼
Mo
Mo
(3.73)
232 CHAPTER 3 Natural Gas Properties
Substituting Eqs. (3.72), (3.73) into Eq. (3.71) gives
fg ¼
Rs
Rs + 133, 000ðγ o =Mo Þ
(3.74)
When applying the material balance equation for a gas reservoir, it assumes
that a volume of gas in the reservoir will remain as a gas at surface conditions. When liquid is separated, the cumulative liquid volume must be converted into an equivalent gas volume, Veq, and added to the cumulative gas
production for use in the material balance equation. If Np STB of liquid
(condensate) has been produced, the equivalent number of moles of liquid
is given by Eq. (3.73) as
Np ð5:615Þð62:4Þγ o 350:4γ o Np
no ¼
¼
Mo
Mo
Expressing this number of moles of liquid as an equivalent gas volume at
standard conditions by applying the ideal gas equation of state gives
Veq ¼
350:4γ o Np ð10:73Þð520Þ
no RTsc
¼
psc
Mo
14:7
Simplifying the above expression to give Veq,
γ Np
Veq ¼ 133,000 o
Mo
More conveniently, the equivalent gas volume can be expressed in
scf/STB as
Veq ¼ 133, 000
γo
Mo
where
Veq ¼ equivalent gas volume, scf/STB
Np ¼ cumulative, or daily, liquid volume, STB
γ o ¼ specific gravity of the liquid, 60°/60°
Mo ¼ molecular weight of the liquid
EXAMPLE 3.16
The following data is available on a wet gas reservoir:
n
n
n
n
n
Initial reservoir pressure pi ¼ 3200 psia.
Reservoir temperature T ¼ 200°F.
Average porosity φ ¼ 18%.
Average connate water saturation Swi ¼ 30%.
Condensate daily flow rate Qo ¼ 400 STB/day.
(3.75)
Lee-Gonzalez-Eakin's Method 233
n
n
n
n
n
API gravity of the condensate ¼ 50°.
Daily separator gas rate Qgsep ¼ 4.20 MMscf/day.
Separator gas specific gravity γ sep ¼ 0.65.
Daily stock-tank gas rate Qgst ¼ 0.15 MMscf/day.
Stock-tank gas specific gravity γ gst ¼ 1.05.
Based on 1 ac-ft sand volume, calculate the initial oil (condensate) and gas in
place and the daily well-stream gas flow rate in scf/day.
Solution for initial oil and gas in place
Step 1. Use Eq. (3.66) to calculate the specific gravity of the separated gas:
Xn¼1 γg ¼
γg ¼
Rsep i γ sep i + Rst γ st
Rsep 1 γ sep 1 + Rst γ st
¼
Xn¼1 Rsep 1 + Rst
R
+
R
sep
st
i
i¼1
i¼1
ð4:2Þð0:65Þ + ð0:15Þð1:05Þ
¼ 0:664
4:2 + 0:15
Step 2. Calculate liquid specific gravity:
γo ¼
141:5
141:5
¼
¼ 0:780
API + 131:5 50 + 131:5
Step 3. Estimate the molecular weight of the liquid by applying Eq. (3.69):
6084
API 5:9
6084
Mo ¼
¼ 138lb=lb mol
50 5:9
Mo ¼
Step 4. Calculate the producing gas/oil ratio:
Qg ð4:2Þ 106 + ð0:15Þ 106
¼ 10, 875scf=STB
Rs ¼ ¼
Qo
400
Step 5. Calculate the well-stream specific gravity from Eq. (3.68):
γw ¼
ð0:664Þð10, 875Þ + ð4, 580Þð0:78Þ
¼ 0:928
10, 875 + ð133, 000Þð0:78=138Þ
Step 6. Solve for the pseudocritical properties of this wet gas using Eqs. (3.20),
(3.21):
Tpc ¼ 187 + 330ð0:928Þ 71:5ð0:928Þ2 ¼ 432°R
ppc ¼ 706 51:7ð0:928Þ 11:1ð0:928Þ2 ¼ 648 psi
Step 7. Calculate the pseudoreduced properties by applying Eqs. (3.12), (3.13):
3200
¼ 4:9
648
200 + 460
¼ 1:53
Tpr ¼
432
ppr ¼
234 CHAPTER 3 Natural Gas Properties
Step 8. Determine the Z-factor from Fig. 3.1 to give
Z ¼ 0:81
Step 9. Calculate the hydrocarbon space per acre-foot at reservoir conditions:
VHy ¼ 43, 560ðAbÞφð1 Swi Þ, ft3
VHy ¼ 43, 560ð1Þð0:18Þð1 0:2Þ ¼ 6273ft3 =ac ft
Step 10. Calculate total moles of gas per acre-foot at reservoir conditions by
applying the real gas equation of state, Eq. (3.11):
n¼
PV
ð3200Þð6273Þ
¼
¼ 3499mol=ac ft
ZRT ð0:81Þð10:73Þð660Þ
Step 11. Because 1 mol of gas occupies 379.4 scf (see Eq.( 3.6)) calculate the
total gas initially in place per acre-foot, to give
G ¼ ð379:4Þð3499Þ ¼ 1328 Mscf=ac ft
If all this gas quantity is produced at the surface, a portion of it will be produced
as stock-tank condensate.
Step 12. Calculate the fraction of the total that is produced as gas at the surface
from Eq. (3.75):
fg ¼
10,875
¼ 0:935
10,875 + 133, 000ð0:78=138Þ
Step 13. Calculate the initial gas and oil in place:
Initial gas in place ¼ Gfg ¼ ð1328Þð0:935Þ ¼ 1242Mscf=ac ft
G 1328 103
¼ 122STB=ac ft
Initial oil in place ¼ ¼
10, 875
Rs
Calculating Daily Well-Stream Gas Rate:
Step 1. Calculate the equivalent gas volume from Eq. (3.75):
γo
Mo
0:78
Veq ¼ 133, 000
¼ 752 scf=STB
138
Veq ¼ 133, 000
Step 2. Calculate the total well-stream gas flow rate:
Qg ¼ Total Qg sep + Veq Qo
Qg ¼ ð4:2 + 0:15Þ106 + ð752Þð400Þ ¼ 4:651MMscf=day
Problems 235
PROBLEMS
1. Assuming an ideal gas behavior, calculate the density of n-butane at
200°F and 50 psia. Recalculate the density assuming real gas behavior.
2. Show that
wi =Mi
yi ¼ X
ðwi =Mi Þ
i
3. Given the following weight fractions of a gas:
Component
yi
C1
C2
C3
n-C4
n-C5
0.60
0.17
0.13
0.06
0.04
Calculate
a. Mole fraction of the gas.
b. Apparent molecular weight.
c. Specific gravity, and specific volume at 300 psia and 130°F,
assuming an ideal gas behavior.
d. Density at 300 psia and 130°F, assuming real gas behavior.
e. Gas viscosity at 300 psia and 130°F.
4. An ideal gas mixture has a density of 2.15 lb/ft3 at 600 psia and 100°F.
Calculate the apparent molecular weight of the gas mixture.
5. Using the gas composition as given in problem 3 and assuming real gas
behavior, calculate
a. Gas density at 2500 psia and 160°F.
b. Specific volume at 2500 psia and 160°F.
c. Gas formation volume factor in scf/ft3.
d. Gas viscosity.
6. A natural gas with a specific gravity of 0.75 has a measured gas
formation volume factor of 0.00529 ft3/scf at the current reservoir
pressure and temperature. Calculate the density of the gas.
7. A natural gas has the following composition:
Component
yi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
0.80
0.07
0.03
0.04
0.03
0.02
0.01
236 CHAPTER 3 Natural Gas Properties
Reservoir conditions are 3500 psia and 200°F. Using, first, the CarrKobayashi-Burrows method, then the Lee-Gonzales-Eakin method,
calculate
a. Isothermal gas compressibility coefficient.
b. Gas viscosity.
8. Given the following gas composition:
Component
yi
CO2
N2
C1
C2
C3
n-C4
n-C5
0.06
0.003
0.80
0.05
0.03
0.02
0.01
If the reservoir pressure and temperature are 2500 psia and 175°F,
respectively, calculate
a. Gas density by accounting for the presence of nonhydrocarbon
components using the Wichert-Aziz method, then the CarrKobayashi-Burrows method.
b. Isothermal gas compressibility coefficient.
c. Gas viscosity using the Carr-Kobayashi-Burrows method, then the
Lee-Gonzales-Eakin method.
9. A high-pressure cell has a volume of 0.33 ft3 and contains gas at
2500 psia and 130°F, at which conditions its Z-factor is 0.75. When
43.6 scf of the gas are bled from the cell, the pressure drops to 1000 psia
while the temperature remains at 130°F. What is the gas deviation
factor at 1000 psia and 130°F?
10. A hydrocarbon gas mixture with a specific gravity of 0.65 has a density
of 9 lb/ft3 at the prevailing reservoir pressure and temperature.
Calculate the gas formation volume factor in bbl/scf.
11. A gas reservoir exists at a 150°F. The gas has the following composition:
C1 ¼ 89mol%
C2 ¼ 7mol%
C3 ¼ 4mol%
The gas expansion factory, Eg, was calculated as 210 scf/ft3 at the
existing reservoir pressure and temperature. Calculate the viscosity of
the gas.
12. A 20 ft3 tank at a pressure of 2500 psia and 200°F contains ethane gas.
How many pounds of ethane are in the tank?
References 237
13. A gas well is producing dry gas with a specific gravity of 0.60. The gas
flow rate is 1.2 MMft3/day at a bottom-hole flowing pressure of 1200
psi and temperature of 140°F. Calculate the gas flow rate in MMscf/day.
14. The following data are available on a wet gas reservoir:
Initial reservoir pressure, Pi ¼ 40,000 psia
Reservoir temperature, T ¼ 150°F
Average porosity, φ ¼ 15%
Average connate water saturation, Swi ¼ 25%
Condensate daily flow rate, Qo ¼ 350 STB/day
API gravity of the condensate ¼ 52°
Daily separator gas rate, Qgsep ¼ 3.80 MMscf/day
Separator gas specific gravity, γ sep ¼ 0.71
Daily stock-tank gas rate, Qgst ¼ 0.21 MMscf/day
Stock-tank gas specific gravity, γ gst ¼ 1.02
Based on 1 ac-ft sand volume, calculate
a. Initial oil (condensate) and gas in place.
b. Daily well-stream gas flow rate in scf/day.
REFERENCES
Brown, G.G., et al., 1948. Natural Gasoline and the Volatile Hydrocarbons. National Gas
Association of America, Tulsa, OK.
Carr, N., Kobayashi, R., Burrows, D., 1954. Viscosity of hydrocarbon gases under pressure. Trans. AIME 201, 270–275.
Dempsey, J.R., 1965. Computer routine treats gas viscosity as a variable. Oil Gas J.
141–143.
Dranchuk, P.M., Abu-Kassem, J.H., 1975. Calculate of Z-factors for natural gases using
equations-of-state. J. Can. Pet. Technol. 34–36.
Dranchuk, P.M., Purvis, R.A., Robinson, D.B., 1974. Computer Calculations of Natural
Gas Compressibility Factors Using the Standing and Katz Correlation. Technical
Series, no. IP 74–008. Institute of Petroleum, Alberta, Canada.
Eilerts, C., 1947. The reservoir fluid, its composition and phase behavior. Oil Gas J.
Hall, K.R., Yarborough, L., 1973. A new equation of state for Z-factor calculations. Oil
Gas J. 82–92.
Lee, A.L., Gonzalez, M.H., Eakin, B.E., 1966. The viscosity of natural gases. J. Petrol.
Technol. 997–1000.
Mattar, L.G., Brar, S., Aziz, K., 1975. Compressibility of natural gases. J. Can. Pet. Technol. 77–80.
Matthews, T., Roland, C., Katz, D., 1942. High pressure gas measurement. Proc. Nat. Gas
Assoc. AIME.
Papay, J.A., 1985. Termelestechnologiai Parameterek Valtozasa a Gazlelepk Muvelese
Soran. OGIL MUSZ, Tud, Kuzl. [Budapest], pp. 267–273.
Standing, M., 1974. Petroleum Engineering Data Book. Norwegian Institute of Technology, Trondheim.
Standing, M.B., 1977. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems.
Society of Petroleum Engineers, Dallas. pp. 125–126.
238 CHAPTER 3 Natural Gas Properties
Standing, M.B., Katz, D.L., 1942. Density of natural gases. Trans. AIME 146, 140–149.
Stewart, W.F., Burkhard, S.F., Voo, D., 1959. Prediction of Pseudo-Critical Parameters for
Mixtures. Paper Presented at the AIChE Meeting, Kansas City, MO.
Sutton, R.P.M., 1985. Compressibility factors for high-molecular-weight reservoir gases.
In: Paper SPE 14265, Presented at the 60th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Las Vegas.
Trube, A.S., 1957a. Compressibility of undersaturated hydrocarbon reservoir fluids. Trans.
AIME 210, 341–344.
Trube, A.S., 1957b. Compressibility of natural gases. Trans. AIME 210, 355–357.
Whitson, C.H., Brule, M.R., 2000. Phase Behavior. Society of Petroleum Engineers,
Richardson, TX.
Wichert, E., Aziz, K., 1972. Calculation of Z’s for sour gases. Hydrocarb. Process. 51 (5),
119–122.
Chapter
4
PVT Properties of Crude Oils
PETROLEUM (an equivalent term is crude oil) is a complex mixture consisting
predominantly of hydrocarbons and containing sulfur, nitrogen, oxygen, and
helium as minor constituents. The physical and chemical properties of crude
oils vary considerably and depend on the concentration of the various types
of hydrocarbons and minor constituents present.
An accurate description of physical properties of crude oils is of a considerable importance in the fields of both applied and theoretical science and
especially in the solution of petroleum reservoir engineering problems.
Physical properties of primary interest in petroleum engineering studies
include
1. Fluid gravity.
2. Specific gravity of the solution gas.
3. Oil density.
4. Gas solubility.
5. Bubble point pressure.
6. Oil formation volume factor (FVF).
7. Isothermal compressibility coefficient of undersaturated crude oils.
8. Undersaturated oil properties.
9. Total FVF.
10. Crude oil viscosity.
11. Surface tension.
Data on most of these fluid properties is usually determined by laboratory
experiments performed on samples of actual reservoir fluids. In the absence
of experimentally measured properties of crude oils, it is necessary for the
petroleum engineer to determine the properties from empirically derived
correlations.
CRUDE OIL API GRAVITY
The crude oil density is defined as the mass of a unit volume of the crude at a
specified pressure and temperature. It is usually expressed in pounds per
cubic foot. The specific gravity of a crude oil is defined as the ratio of
Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00004-0
Copyright # 2016 Elsevier Inc. All rights reserved.
239
240 CHAPTER 4 PVT Properties of Crude Oils
the density of the oil to that of water. Both densities are measured at 60°F
and atmospheric pressure:
γo ¼
ρo
ρw
(4.1)
where
γ o ¼ specific gravity of the oil
ρo ¼ density of the crude oil, lb/ft3
ρw ¼ density of the water, lb/ft3
It should be pointed out that the liquid specific gravity is dimensionless but
traditionally is given the units 60°/60° to emphasize that both densities are
measured at standard conditions (SC). The density of the water is approximately 62.4 lb/ft3, or
γo ¼
ρo
, 60°=60°
62:4
Although the density and specific gravity are used extensively in the
petroleum industry, the API gravity is the preferred gravity scale. This
gravity scale is precisely related to the specific gravity by the following
expression:
API ¼
141:5
131:5
γo
(4.2)
This equation can be also rearranged to express the oil specific gravity in
terms of the API gravity:
γo ¼
141:5
131:5 + API
The API gravities of crude oils usually range from 47°API for the lighter
crude oils to 10°API for the heavier asphaltic crude oils.
EXAMPLE 4.1
Calculate the specific gravity and the API gravity of a crude oil system with a
measured density of 53 lb/ft3 at SC.
Solution
Step 1. Calculate the specific gravity from Eq. (4.1):
ρ
γo ¼ o
ρw
γo ¼
53
¼ 0:849
62:4
Specific Gravity of the Solution Gas “γ g” 241
Step 2. Solve for the API gravity:
API ¼
141:5
131:5
γo
API ¼
141:5
131:5 ¼ 35:2°API
0:849
SPECIFIC GRAVITY OF THE SOLUTION GAS “γg”
The specific gravity of the solution gas, γ g, is described by the weighted average of the specific gravities of the separated gas from each separator. This
weighted average approach is based on the separator gas-oil ratio (GOR):
Xn Rsep i γ sep i + Rst γ st
i¼1
Xn γg ¼
Rsep i + Rst
i¼1
(4.3)
where
n ¼ number of separators
Rsep ¼ separator GOR, scf/STB
γ sep ¼ separator gas gravity
Rst ¼ GOR from the stock-tank, scf/STB
γ st ¼ gas gravity from the stock-tank
EXAMPLE 4.2
Separator tests were conducted on a crude oil sample. Results of the test in
terms of the separator GOR and specific gravity of the separated gas are given in
the table below. Calculate the specific gravity of the separated gas.
Separator
Pressure
(psig)
Temperature
(°F)
Gas-Oil Ratio
(scf/STB)
Gas Specific
Gravity
Primary
Intermediate
Stock-tank
660
75
0
150
110
60
724
202
58
0.743
0.956
1.296
Solution
Calculate the solution specific gravity by using Eq. (4.3):
Xn¼2 γg ¼
γg ¼
Rsep i γ sep i + Rst γ st
Xn¼2 Rsep i + Rst
i¼1
i¼1
ð724Þð0:743Þ + ð202Þð0:956Þ + ð58Þð1:296Þ
¼ 0:819
724 + 202 + 58
242 CHAPTER 4 PVT Properties of Crude Oils
Ostermann et al. (1987) proposed a correlation to account for the increase of the
solution gas gravity in a gas drive reservoir with decreasing the reservoir
pressure, p, below the bubble point pressure, pb. The authors expressed the
correlation in the following form:
(
γ g ¼ γ gi
a½1 ðp=pb Þb
1+
ðp=pb Þ
)
Parameters a and b are functions of the bubble point pressure, given by the
following two polynomials:
a ¼ 0:12087 0:06897pb + 0:01461ðpb Þ2
b ¼ 1:04223 + 2:17073pb 0:68851ðpb Þ2
γ g ¼ specific gravity of the solution gas at pressure p
γ gi ¼ initial specific gravity of the solution gas at the bubble point pressure pb
pb ¼ bubble point pressure, psia
p ¼ reservoir pressure, psia
CRUDE OIL DENSITY “ρO”
The crude oil density is defined as the mass of a unit volume of the crude
at a specified pressure and temperature, mass/volume. The density usually is expressed in pounds per cubic foot and it varies from 30 lb/ft3
for light volatile oil to 60 lb/ft3 for heavy crude oil with little or no
gas solubility. It is one of the most important oil properties, because
its value substantially affects crude oil volume calculations. This vital
oil property is traditionally measured in the laboratory as part of routine
pressure-volume-temperature (PVT) tests. When laboratory crude oil
density measurement is not available, correlations can be used to generate the required density data under reservoir pressure and temperature.
Numerous empirical correlations for calculating the density of liquids
have been proposed over the years. Based on the available limited measured data on the crude, the correlations can be divided into the following
two categories:
n
n
Correlations that use the crude oil composition to determine the
saturated oil density and temperature.
Correlations that use limited PVT data, such as gas gravity, oil gravity,
and GOR, as correlating parameters.
Crude Oil Density “ρo” 243
Density Correlations Based on the Oil Composition
Several reliable methods are available for determining the density of saturated crude oil mixtures from their compositions. The best known and most
widely used calculation methods are the following two:
n
n
Standing-Katz (1942)
Alani-Kennedy (1960)
These two methods were developed to calculate the saturated oil density.
Standing-Katz’s Method
Standing and Katz (1942) proposed a graphical correlation for determining
the density of hydrocarbon liquid mixtures. The authors developed the correlation from evaluating experimental, compositional, and density data on
15 crude oil samples containing up to 60 mol% methane. The proposed
method yielded an average error of 1.2% and maximum error of 4% for
the data on these crude oils. The original correlation had no procedure
for handling significant amounts of nonhydrocarbons.
The authors expressed the density of hydrocarbon liquid mixtures as a function of pressure and temperature by the following relationship:
ρo ¼ ρsc + Δρp ΔρT
(4.4)
where
ρo ¼ crude oil density at p and T, lb/ft3
ρsc ¼ crude oil density (with all the dissolved solution gas) at SC, that is,
14.7 psia and 60°F, lb/ft3
Δρp ¼ density correction for compressibility of oils, lb/ft3
ΔρT ¼ density correction for thermal expansion of oils, lb/ft3
Standing and Katz correlated graphically the liquid density at SC ρsc with
the density of the propane-plus fraction, ρC03 + ; the weight percent of methane
in the entire system, ðmC1 ÞC1 + ; and the weight percent of ethane in the
ethane-plus, ðmC2 ÞC2 + . This graphical correlation is shown in Fig. 4.1.
The proposed calculation procedure is performed on the basis of 1 lb-mol;
that is, nt ¼ 1, of the hydrocarbon system.
The following are the specific steps in the Standing and Katz procedure of
calculating the liquid density at a specified pressure and temperature.
Step 1. Calculate the total weight and the weight of each component in
1 lb-mol of the hydrocarbon mixture by applying the following
relationships:
244 CHAPTER 4 PVT Properties of Crude Oils
70
20
30
10
e pl
ane
in e
than
60
70
cen
t eth
per
0
40
60
10
e
ur
ixt
30
m
al
50
t
ne
ha
in
to
20
et
tm
en
40
c
er
p
ht
g
ei
W
30
30
20
Density of system including methane and ethane (lb/ft3)
50
We
ight
Density of propane-plus (lb/ft3)
us
40
0
50
10
n FIGURE 4.1 Standing and Katz density correlation. (Courtesy of the Gas Processors Suppliers Association, 1987. Engineering Data Book, 10th ed.)
mi ¼ X
xi Mi
mt ¼
x i Mi
where
mi ¼ weight of component i in the mixture, lb/lb-mol
xi ¼ mole fraction of component i in the mixture
Mi ¼ molecular weight of component i
mt ¼ total weight of 1 lb-mol of the mixture, lb/lb-mol
Crude Oil Density “ρo” 245
Step 2. Calculate the weight percent of methane in the entire system and the
weight percent of ethane in the ethane-plus from the following
expressions:
"
#
xC MC
m C1
100
ðmC1 ÞC1 + ¼ Xn1 1 100 ¼
mt
xM
i¼1 i i
(4.5)
xC2 MC2
mC2
100 ¼
100
mC2 +
mt mC1
(4.6)
and
ðmC2 ÞC2 + ¼
where
ðmC1 ÞC1 + ¼ weight percent of methane in the entire system
mC1 ¼ weight of methane in 1 lb-mol of the mixture, that is, xC1 MC1
ðmC2 ÞC2 + ¼ weight percent of ethane in ethane-plus
mC2 ¼ weight of ethane in 1 lb-mol of the mixture, that is, xC21 MC2
MC1 ¼ molecular weight of methane
MC2 ¼ molecular weight of ethane
Step 3. Calculate the density of the propane-plus fraction at SC by using the
following equations:
Xn
xi Mi
mC3 +
mt mC1 mC2
¼ Xn
ρC3 + ¼
¼ Xn i¼ C3
VC3 +
ððxi Mi Þ=ρoi Þ
ððxi Mi Þ=ρoi Þ
i¼C3
(4.7)
i¼C3
with
mC3 + ¼
X
xi Mi
(4.8)
i¼C3
VC3 + ¼
X
i¼ C3
Vi ¼
X mi
ρ
i¼ C3 oi
(4.9)
where
ρC3 + ¼ density of the propane and heavier components, lb/ft3
mC3 + ¼ weight of the propane and heavier components, lb/ft3
VC3 + ¼ volume of the propane-plus fraction, ft3/lb-mol
Vi ¼ volume of component i in 1 lb-mol of the mixture
mi ¼ weight of component i; that is, xiMi, lb/lb-mol
ρoi ¼ Density of component i at SC, lb/ft3 (density values for pure
components are tabulated in Table 1.1 in Chapter 1, but the
density of the plus fraction must be measured)
246 CHAPTER 4 PVT Properties of Crude Oils
Step 4. Using Fig. 4.1, enter the ρC3 + value into the left ordinate of the
chart and move horizontally to the line representing ðmC2 ÞC0 then
2+
drop vertically to the line representing ðmC1 ÞC1 + . The density of the
oil at SC is read on the right side of the chart. Standing (1977)
expressed the graphical correlation in the following mathematical
form:
h
i
ρsc ¼ ρC2 + 1 0:012ðmC1 ÞC1 + 0:000158ðmC1 Þ2C1 +
+ 0:0133ðmC1 ÞC1 + + 0:00058ðmC1 Þ2C2 +
(4.10)
with
h
i
ρC2 + ¼ ρC3 + 1 0:01386ðmC2 ÞC2 + 0:000082ðmC2 Þ2C2 +
+ 0:379ðmC2 ÞC2 + + 0:0042ðmC2 Þ2C2 +
(4.11)
where
ρC2 + ¼ density of the ethane-plus fraction.
Step 5. Correct the density at SC to the actual pressure by reading the
additive pressure correction factor, Δρp, from Fig. 4.2:
ρp, 60 ¼ ρsc + Δρp
The pressure correction term Δρp can be expressed mathematically by
Δρp ¼ 0:000167 + ð0:016181Þ100:0425ρsc p
8 10
0:299 + ð263Þ100:0603ρsc p2
(4.12)
Step 6. Correct the density at 60°F and pressure to the actual temperature
by reading the thermal expansion correction term, ΔρT, from
Fig. 4.3:
ρo ¼ ρp, 60 ΔρT
The thermal expansion correction term, ΔρT, can be expressed
mathematically by
h
2:45 i
ΔρT ¼ ðT 520Þ 0:0133 + 152:4 ρsc + Δρp
h i
ðT 520Þ2 8:1 106 ð0:0622Þ100:0764ðρsc + Δρp Þ (4.13)
where T is the system temperature in °R.
Crude Oil Density “ρo” 247
Density correction for compressibility
9
ia)
s
(p
re
su 00
es
Pr 15,0
8
7
0
,00
10
00
80
6
00
60
5
00
50
4
00
40
Density at pressure minus density at 60°F and 14.7 psia (lb/ft3)
10
3
30
00
2
20
00
1
0
25
1000
30
35
40
45
50
55
60
Density at 60°F and 14.7 psia (lb/ft3)
n FIGURE 4.2 Density correction for compressibility of crude oils. (Courtesy of the Gas Processors Suppliers
Association, 1987. Engineering Data Book, 10th ed.)
EXAMPLE 4.3
A crude oil system has the following composition:
Component
xi
C1
C2
C3
C4
C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
65
248 CHAPTER 4 PVT Properties of Crude Oils
Density correction for thermal expansion
16
16
250
Density at 60°F minus density at temperature (lb/ft3)
14
12
14
600°F
200
12
550
10
10
180
500
450
8
8
160
400
350
6
140
6
300
120
4
250
100
4
200
2
2
80
100
60°F
0
60
0
40
20
40
-2
20
-60
-80
0
-4
-100
-20
-6
0°F
-20
-40
-40
25
30
35
40
45
50
55
60
-2
-4
-6
Density at 60°F and pressure (lb/ft3)
n FIGURE 4.3 Density correction for isothermal expansion of crude oils. (Courtesy of the Gas Processors Suppliers Association, 1987. Engineering Data Book, 10th ed.)
If the molecular weight and specific gravity of the C7+ fractions are 215 and
0.87, respectively, calculate the density of the crude oil at 4000 psia and 160°F
using the Standing and Katz method.
Solution
The table below shows the values of the components.
mi 5 xiMi
Component
xi
Mi
C1
C2
C3
C4
C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
16.04
7.218
30.07
1.5035
44.09
2.2045
58.12
1.7436
72.15
0.7215
86.17
0.8617
215.0
86.00
P
¼ mt ¼ 100.253
a
From Table 1.1.
ρC7 + ¼ (0.87)(62.4) ¼ 54.288.
b
ρoi
(lb/ft3)a
–
–
31.64
35.71
39.08
41.36
54.288b
Vi 5 mi/ρoi
–
–
0.0697
0.0488
0.0185
0.0208
1.586
P
¼ VC3 + ¼ 1:7418
Crude Oil Density “ρo” 249
Step 1. Calculate the weight percent of C1 in the entire system and the weight
percent of C2 in the ethane-plus fraction:
"
#
xC1 MC1
m C1
100 ¼
100
ðmC1 ÞC1 + ¼ Xn
mt
xM
i¼1 i i
7:218
100 ¼ 7:2%
100:253
xC2 MC2
mC2
100 ¼
100
ðmC2 ÞC2 + ¼
mC2 +
mt mC1
1:5035
ðmC2 ÞC2 + ¼
100 ¼ 1:616%
100:253 7:218
ðmC1 ÞC1 + ¼
Step 2. Calculate the density of the propane and heavier components:
ρC3 + ¼
mC3 +
mt mC1 mC2
¼ Xn
VC3 +
ððxi Mi Þ=ρoi Þ
i¼C3
ρC3 + ¼
100:253 7:218 1:5035
¼ 52:55lb=ft3
1:7418
Step 3. Determine the density of the oil at SC from Fig. 4.1:
ρsc ¼ 47:5lb=ft3
Step 4. Correct for the pressure using Fig. 4.2:
Δρp ¼ 1:18lb=ft3
Density of the oil at 4000 psia and 60°F is then calculated by the
expression
ρp, 60 ¼ ρsc + Δρp ¼ 47:5 + 1:18 ¼ 48:68lb=ft3
Step 5. From Fig. 4.3, determine the thermal expansion correction factor:
ΔρT ¼ 2:45lb=ft3
Step 6. The required density at 4000 psia and 160°F is
ρo ¼ ρp, 60 ΔρT
ρo ¼ 48:68 2:45 ¼ 46:23lb=ft3
250 CHAPTER 4 PVT Properties of Crude Oils
Alani-Kennedy’s Method
Alani and Kennedy (1960) developed an equation to determine the molar
liquid volume, Vm, of pure hydrocarbons over a wide range of temperatures
and pressures. The equation then was adopted to apply to crude oils with
heavy hydrocarbons expressed as a heptanes-plus fraction; that is, C7+.
The Alani-Kennedy equation is similar in form to the Van der Waals equation, which takes the following form:
Vm3 RT
aVm ab
¼0
+ b Vm2 +
p
p
p
(4.14)
where
R ¼ gas constant, 10.73 psia ft3/lb-mol, °R
T ¼ temperature, °R
p ¼ pressure, psia
Vm ¼ molar volume, ft3/lb-mol
a, b ¼ constants for pure substances
Alani and Kennedy considered the constants a and b functions of temperature and proposed the expressions for calculating the two parameters:
a ¼ Ken=T
(4.15)
b ¼ mT + c
(4.16)
where K, n, m, and c are constants for each pure component in the mixture
and are tabulated in Table 4.1. The table does not contain constants from
which the values of the parameters a and b for heptanes-plus can be calculated. Therefore, Alani and Kennedy proposed the following equations for
determining a and b of C7+:
M
ln aC7 + ¼ 3:8405985 103 ðMÞC7 + 9:5638281 104
γ
+
C7 +
261:80818
T
+ 7:3104464 106 ðMÞ2C7 + + 10:753517
bC7 + ¼ 0:03499274ðMÞC7 + 7:2725403ðγ ÞC7 + + 2:232395 104 T
M
0:016322572
γ
+ 6:2256545
C7 +
Crude Oil Density “ρo” 251
Table 4.1 Alani and Kennedy Coefficients
Components
Temperature Range (°F)
K
n
m × 104
C1
C1
C2
C2
C3
i-C4
n-C4
i-C5
n-C5
n-C6
H2Sa
N2a
CO2a
70–300
301–460
100–249
250–460
9160.6413
147.47333
46,709.57
17,495.34
20,247.76
32,204.42
33,016.21
37,046.23
37,046.23
52,093.01
13,200.00
4300
8166
61.893223
3247.4533
404.48844
34.163551
190.2442
131.63171
146.15445
299.6263
299.6263
254.56097
0
2.293
126
3.3162472
14.072637
5.1520981
2.8201736
2.1586448
3.3862284
2.902157
2.1954785
2.1954785
3.6961858
17.9
4.49
1.818
a
Values for nonhydrocarbon components as proposed by Lohrenz et al. (1964).
where
MC7 + ¼ molecular weight of C7+
γ C7 + ¼ specific gravity of C7+
aC7 + , bC7 + ¼ constants of the heptanes-plus fraction
T ¼ temperature in °R
For hydrocarbon mixtures, the values of a and b of the mixture are calculated using the following mixing rules:
am ¼
C7 +
X
ai xi
(4.17)
i¼1
bm ¼
C7 +
X
bi xi
(4.18)
i¼1
where the coefficients ai and bi refer to the values of pure hydrocarbon, i, as
calculated from Eqs. (4.15), (4.16), at the existing temperature. xi is the mole
fraction of component i in the liquid phase. The values am and bm are then
used in Eq. (4.14) to solve for the molar volume Vm. The density of the mixture at the pressure and temperature of interest is determined from the following relationship:
ρo ¼
Ma
Vm
(4.19)
c
0.508743
1.832666
0.522397
0.623099
0.908325
1.101383
1.116814
1.436429
1.436429
1.592941
0.3945
0.3853
0.3872
252 CHAPTER 4 PVT Properties of Crude Oils
where
ρo ¼ density of the crude oil, lb/ft3
P
Ma ¼ apparent molecular weight; that is, Ma ¼ xiMi
Vm ¼ molar volume, ft3/lb-mol
The Alani and Kennedy method for calculating the density of liquids is summarized in the following steps.
Step 1. Calculate the constants a and b for each pure component from
Eqs. (4.15), (4.16):
a ¼ Ken=T
b ¼ mT + c
Step 2. Determine the C7+ parameters aC7 + and bC7 + .
Step 3. Calculate values of the mixture coefficients am and bm from
Eqs. (4.17), (4.18).
Step 4. Calculate the molar volume Vm by solving Eq. (4.14) for the
smallest real root. The equation can be solved iteratively by using
the Newton-Raphson iterative method. In applying this technique,
assume a starting value of Vm ¼ 2 and evaluate Eq. (4.14) using the
assumed value:
f ðVm Þ ¼ Vm3 RT
am Vm am bm
+ bm Vm2 +
p
p
p
If the absolute value of this function is smaller than a preset
tolerance, say, 1010, then the assumed value is the desired volume.
If not, a new assumed value of (Vm)new is used to evaluate the
function. The new value can be calculated from the following
expression:
f ðVm Þold
ðVm Þnew ¼ ðVm Þold 0
f ðVm Þold
where the derivative f´(Vm) is given by
RT
am
f 0 ðVm Þ ¼ 3Vm2 2
+ bm Vm +
p
p
Step 5. Compute the apparent molecular weight Ma from
Ma ¼
X
xi Mi
Step 6. Determine the density of the crude oil from
ρo ¼
Ma
Vm
Crude Oil Density “ρo” 253
EXAMPLE 4.4
A crude oil system has the composition:
Component
xi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0008
0.0164
0.2840
0.0716
0.1048
0.0420
0.0420
0.0191
0.0191
0.0405
0.3597
Given the following additional data:
MC7 + ¼ 252
γ C7 + ¼ 0:8424
Pressure ¼ 1708.7 psia
Temperature ¼ 591°R
Calculate the density of the crude oil.
Solution
Step 1. Calculate the parameters aC7 + and bC7 + :
M
lnðaC7 + Þ ¼ 3:8405985 103 MC7 + 9:5638281 104
γ
C7 +
261:80818
+ 7:3104464 106 ðMC7 + Þ2 + 10:753517
+
T
252
261:80818
+
lnðaC7 + Þ ¼ 3:8405985 103 ð252Þ 9:5638281 104
0:8424 C7 +
591
6 2
+ 7:3104464 10 ð252Þ + 10:753517 ¼ 12:34266
aC7 + ¼ exp ð12:34266Þ ¼ 229269:9
and
bC7 + ¼ 0:03499274MC7 + 7:2725403ðγ ÞC7 +
4 M
+ 2:232395 10 T 0:016322572
γ
+ 6:2256545
C7 +
bC7 + ¼ 0:03499274ð252Þ 7:2725403ð0:8424Þ
252
+ 6:2256545 ¼ 4:165811
+ 2:232395 104 591 0:016322572
0:8424
254 CHAPTER 4 PVT Properties of Crude Oils
Step 2. Calculate the mixture parameters am and bm:
am ¼
C7 +
X
ai xi ¼ 99111:71
i¼1
C7 +
X
bm ¼
bi xi ¼ 2:119383
i¼1
Step 3. Solve Eq. (4.14) iteratively for the molar volume:
RT
am Vm am bm
Vm3 ¼0
+ bm Vm2 +
p
p
p
10:73ð591Þ
99111:71Vm 99111:71ð2:119383Þ
Vm3 + 2:119383 Vm2 +
¼0
1708:7
1708:7
1708:7
Vm3 5:83064Vm2 + 58:004Vm 122:933 ¼ 0
Solving this expression iteratively for Vm as outlined previously gives
Vm ¼ 2:528417
Step 4. Determine the apparent molecular weight of this mixture:
X
xi Mi ¼ 113:5102
Ma ¼
Step 5. Compute the density of the oil system:
Ma
Vm
113:5102
ρo ¼
¼ 44:896lb=ft3
2:528417
ρo ¼
Density Correlations Based on Limited PVT Data
Several empirical correlations for calculating the density of liquids of unknown
compositional analysis have been proposed. The correlations employ limited
PVT data such as gas gravity, oil gravity, and gas solubility as correlating
parameters to estimate liquid density at the prevailing reservoir pressure and
temperature. Two methods are presented as representatives of this category:
n
n
Katz’s method
Standing’s method
Katz’s Method
Density, in general, can be defined as the mass of a unit volume of material
at a specified temperature and pressure. Accordingly, the density of a saturated crude oil (ie, with gas in solution) at SC can be defined mathematically
by the following relationship:
Crude Oil Density “ρo” 255
ρsc ¼
weight of stock tank oil + weight of solution gas
volume of stock tank oil + increase in stock tank volume due to solution gas
or
ρsc ¼
mo + mg
ðVo Þsc + ðΔVo Þsc
where
ρsc ¼ density of the oil at SC, lb/ft3
(Vo)sc ¼ volume of oil at SC, ft3/STB
mo ¼ total weight of one stock-tank barrel of oil, lb/STB
mg ¼ weight of the solution gas, lb/STB
(ΔVo)sc ¼ increase in stock-tank oil volume due to solution gas, ft3/STB
The procedure of calculating the density at SC is illustrated schematically in
Fig. 4.4. Katz (1942) expressed the increase in the volume of oil, (ΔVo)sc,
relative to the solution gas by introducing the apparent liquid density of the
dissolved gas density, ρga, to the preceding expression as
ðΔVo Þsc ¼
mg
Ega
Combining the preceding two expressions gives
ρsc ¼
mo + mg
mg
ðVo Þsc +
ρga
where ρga, as introduced by Katz, is called the apparent density of the liquefied dissolved gas at 60°F and 14.7 psia.
14.7 psia
60°F
(ΔVo)sc is the volume of the liquified
Rs scf of solution gas
14.7 psia
60°F
Rs, scf
(VO)sc = 1 STB
(ΔVo)sc
(VO)sc = 1 STB
n FIGURE 4.4 Schematic illustration of Katz’s density model at standard conditions.
256 CHAPTER 4 PVT Properties of Crude Oils
Katz correlated the apparent gas density, in lb/ft3, graphically with gas specific gravity, γ g; gas solubility (solution GOR), Rs; and specific gravity (or
the API gravity) of the stock-tank oil, γ o. This graphical correlation is presented in Fig. 4.5. The proposed method does not require the composition of
the crude oil.
Referring to the equation representing the surface density, that is,
ρsc ¼
mo + mg
mg
ðVo Þsc +
ρga
the weights of the solution gas and the stock-tank can be determined in terms
of the previous oil properties by the following relationships:
mg ¼
Rs
ð28:96Þ γ g , lb of solution gas=STB
379:4
mo ¼ ð5:615Þð62:4Þðγ o Þ, lb of oil=STB
Substituting these terms into Katz’s equation gives
Rs
ð5:615Þð62:4Þðγ o Þ +
ð28:96Þ γ g
379:4
ρsc ¼
Rs
5:615 +
ð28:96Þ γ g =ρga
379:4
Apparent liquid density of dissolved gas
At standard conditions (lb/ft3)
45
40
35
30
25
20
15
0.6
0.7
0.8
0.9
1.0
1.1
Gas gravity
n FIGURE 4.5 Apparent liquid density of natural gas.
1.2
1.3
1.4
Crude Oil Density “ρo” 257
or
Rs γ g
350:376γ o +
13:1
!
ρsc ¼
Rs γ g
5:615 +
13:1ρga
(4.20)
Standing (1981) showed that the apparent liquid density of the dissolved gas
as represented by Katz’s chart can be closely approximated by the following
relationship:
ρga ¼ ð38:52Þ100:00326API + ½94:75 33:93log ðAPIÞ log γ g
The pressure correction adjustment, Δρp, and the thermal expansion adjustment, ΔρT, for the calculated ρsc can be made using Figs. 4.2 and 4.3,
respectively.
EXAMPLE 4.5
A saturated crude oil exists at its bubble point pressure of 4000 psia and a
reservoir temperature of 180°F. Given
APIgravity ¼ 50°
Rs ¼ 650scf=STB
γ g ¼ 0:7
calculate the oil density at the specified pressure and temperature using
Katz’s method.
Solution
Step 1. From Fig. 4.5 or the preceding equation, determine the apparent liquid
density of dissolved gas:
ρga ¼ ð38:52Þ100:00326API + ½94:75 33:93log ðAPIÞ log γ g
ρga ¼ ð38:52Þ100:00326ð50ÞI + ½94:75 33:93 log ð50Þ log ð0:7Þ ¼ 20:7lb=ft3
Step 2. Calculate the stock-tank liquid gravity from Eq. (4.2):
γo ¼
141:5
141:5
¼
¼ 0:7796
API + 131:5 50 + 131:5
Step 3. Apply Eq. (4.20) to calculate the liquid density at SC:
Rs γ g
350:376γ o +
13:1
!
ρsc ¼
Rs γ g
5:615 +
13:1ρga
ð650Þð0:7Þ
ð350:376Þð0:7796Þ +
13:1 ¼ 42:12 lb=ft3
ρsc ¼
ð650Þð0:7Þ
5:615 +
ð13:1Þð20:7Þ
258 CHAPTER 4 PVT Properties of Crude Oils
Step 4. Determine the pressure correction factor from Fig. 4.2:
Δρp ¼ 1:55lb=ft3
Step 5. Adjust the oil density, as calculated at SC, to reservoir pressure:
ρp, 60° ¼ ρsc + Δρp
ρp, 60°F ¼ 42:12 + 1:55 ¼ 43:67lb=ft3
Step 6. Determine the isothermal adjustment factor from Fig. 4.3:
ΔρT ¼ 3:25lb=ft3
Step 7. Calculate the oil density at 4000 psia and 180°F:
ρo ¼ ρp, 60 ΔρT
ρo ¼ 43:67 3:25 ¼ 40:42lb=ft3
Standing’s Method
Standing (1981) proposed an empirical correlation for estimating the oil FVF
as a function of the gas solubility, Rs; the specific gravity of stock-tank oil, γ o;
the specific gravity of solution gas, γ g; and the system temperature, T. By coupling the mathematical definition of the oil FVF (as discussed in a later section) with Standing’s correlation, the density of a crude oil at a specified
pressure and temperature can be calculated from the following expression:
ρo ¼
62:4γ o + 0:0136Rs γ g
1:175
γ g 0:5
0:972 + 0:000147 Rs γ
+ 1:25ðT 460Þ
(4.21)
o
where
T ¼ system temperature, °R
γ o ¼ specific gravity of the stock-tank oil, 60°/60°
γ g ¼ specific gravity of the gas
Rs ¼ gas solubility, scf/STB
ρo ¼ oil density, lb/ft3
EXAMPLE 4.6
Rework Example 4.5 and solve for the density using Standing’s correlation.
Solution
ρo ¼
62:4γ o + 0:0136Rs γ g
1:175
γ g 0:5
0:972 + 0:000147 Rs γ
+ 1:25ðT 460Þ
o
62:4ð0:7796Þ + 0:0136ð650Þð0:7Þ
3
ρo ¼
h i1:175 ¼ 39:92lb=ft
0:7 0:5
+ 1:25ð180Þ
0:972 + 0:000147 650 0:7796
Crude Oil Density “ρo” 259
The obvious advantage of using Standing’s correlation is that it gives the density
of the oil at the temperature and pressure at which the gas solubility is measured,
and therefore, no further corrections are needed; that is, Δρp and ΔρT. Ahmed
(1988) proposed another approach for estimating the crude oil density at SC
based on the apparent molecular weight of the stock-tank oil. The correlation
calculates the apparent molecular weight of the oil from the readily available
PVT on the hydrocarbon system. Ahmed expressed the apparent molecular
weight of the crude oil with its dissolved solution gas by the following
relationship:
Ma ¼
0:0763Rs γ g Mst + 350:376γ o Mst
0:0026537Rs Mst + 350:376γ o
where
Ma ¼ apparent molecular weight of the oil
Mst ¼ molecular weight of the stock-tank oil and can be taken as the
molecular weight of the heptanes-plus fraction
γ o ¼ specific gravity of the stock-tank oil or the C7+ fraction, 60°/60°
The density of the oil at SC can then be determined from the expression
ρsc ¼
0:0763Rs γ g + 350:376γ o
199:71432
0:0026537Rs + γ o 5:615 +
Mst
If the molecular weight of the stock-tank oil is not available, the density of the oil
with its dissolved solution gas at SC can be estimated from the following
equation:
ρsc ¼
0:0763Rs γ g + 350:4γ o
0:0027Rs + 2:4893γ o + 3:491
The proposed approach requires the two additional steps of accounting for the
effect of reservoir pressure and temperature.
EXAMPLE 4.7
Using the data given in Example 4.5, calculate the density of the crude oil by
using the preceding simplified expression.
Solution
Step 1. Calculate the density of the crude oil at SC from
ρsc ¼
0:0763Rs γ g + 350:4γ o
0:0027Rs + 2:4893γ o + 3:491
ρsc ¼
0:0763ð650Þð0:7Þ + 350:4ð0:7796Þ
¼ 42:8lb=ft3
0:0027ð650Þ + 2:4893ð0:7796Þ + 3:491
Step 2. Determine Δρp from Fig. 4.2:
Δρp ¼ 1:5lb=ft3
260 CHAPTER 4 PVT Properties of Crude Oils
Step 3. Determine ΔρT from Fig. 4.3:
ΔρT ¼ 3:6lb=ft3
Step 4. Calculate po at 4000 psia and 180°F:
ρo ¼ ρsc + Δρp ΔρT
ρo ¼ 42:8 + 1:5 3:6 ¼ 40:7lb=ft3
The oil density can also be calculated rigorously from the experimental
measured PVT data at any specified pressure and temperature. The following
expression (as derived later in this chapter) relates the oil density, ρo, to the gas
solubility, Rs, specific gravity of the oil, gas gravity, and the oil FVF:
ρo ¼
62:4γ o + 0:0136Rs γ g
Bo
(4.22)
where
γ o ¼ specific gravity of the stock-tank oil, 60°/60°
Rs ¼ gas solubility, scf/STB
ρo ¼ oil density at pressure and temperature, lb/ft3
GAS SOLUBILITY “RS”
The gas solubility, Rs, is defined as the number of standard cubic feet of gas
that dissolve in one stock-tank barrel of crude oil at certain pressure and temperature. The solubility of a natural gas in a crude oil is a strong function of
the pressure, the temperature, the API gravity, and the gas gravity.
For a particular gas and crude oil to exist at a constant temperature, the
solubility increases with pressure until the saturation pressure is reached.
At the saturation pressure (bubble point pressure), all the available gases
are dissolved in the oil and the gas solubility reaches its maximum value.
Rather than measuring the amount of gas that dissolves in a given stocktank crude oil as the pressure is increased, it is customary to determine the
amount of gas that comes out of a sample of reservoir crude oil as pressure
decreases.
A typical gas solubility curve, as a function of pressure for an undersaturated
crude oil, is shown in Fig. 4.6. As the pressure is reduced from the initial
reservoir pressure, pi, to the bubble point pressure, pb, no gas evolves from
the oil and consequently the gas solubility remains constant at its maximum
value of Rsb. Below the bubble point pressure, the solution gas is liberated
and the value of Rs decreases with pressure.
In the absence of experimentally measured gas solubility on a crude oil
system, it is necessary to determine this property from empirically derived
correlations. Five empirical correlations for estimating the gas solubility are
Gas Solubility “Rs” 261
Rs = g g
p
18.2
1.2048
+ 1.4
10X
X = 0.0125 API − 0.00091(T−460)
Rsb
Rs
0
pb
0
pi
Pressure
n FIGURE 4.6 Typical gas solubility-pressure relationship.
n
n
n
n
n
Standing’s correlation
Vasquez-Beggs’s correlation
Glaso’s correlation
Marhoun’s correlation
Petrosky-Farshad’s correlation
Standing’s Correlation
Standing (1947) proposed a graphical correlation for determining the gas
solubility as a function of pressure, gas specific gravity, API gravity, and
system temperature. The correlation was developed from a total of 105
experimentally determined data points on 22 hydrocarbon mixtures from
California crude oils and natural gases. The proposed correlation has an
average error of 4.8%. In a mathematical form, Standing (1981) expressed
his proposed graphical correlation in the following more convenient mathematical form:
Rs ¼ γ g
h
i1:2048
p
+ 1:4 10x
18:2
with
x ¼ 0:0125API 0:00091ðT 460Þ
where
Rs ¼ gas solubility, scf/STB
T ¼ temperature, °R
(4.23)
262 CHAPTER 4 PVT Properties of Crude Oils
p ¼ system pressure, psia
γ g ¼ solution gas specific gravity
API ¼ oil gravity, °API
It should be noted that Standing’s equation is valid for applications at and
below the bubble point pressure of the crude oil.
EXAMPLE 4.8
The following table shows experimental PVT data on six crude oil systems.
Results are based on two-stage surface separation. Using Standing’s
correlation, estimate the gas solubility at the bubble point pressure and
compare with the experimental value in terms of the absolute average error
(AAE).
T
Oil (°F) pb
Rs
(scf/STB) Bo
1
250 2377 751
2
220 2620 768
3
260 2051 693
4
237 2884 968
5
218 3045 943
6
180 4239 807
ρo
(lb/ft3) co at p > pb
6
1.528 38.13 22.14 10 at
2689 psi
0.474 40.95 18.75 106 at
2810 psi
1.529 37.37 22.69 106 at
2526 psi
1.619 38.92 21.51 106 at
2942 psi
1.570 37.70 24.16 106 at
3273 psi
1.385 46.79 11.45 106 at
4370 psi
psep Tsep API γ g
150 60 47.1 0.851
100 75 40.7 0.855
100 72 48.6 0.911
60 120 40.5 0.898
200 60 44.2 0.781
85 173 27.3 0.848
T ¼ reservoir temperature, °F.
pb ¼ bubble point pressure, psig.
Bo ¼ oil FVF, bbl/STB.
psep ¼ separator pressure, psig.
Tsep ¼ separator temperature, °F.
co ¼ isothermal compressibility coefficient of the oil at a specified pressure, psi1.
Solution
Apply Eq. (4.23) to determine the gas solubility:
h p
i1:2048
Rs ¼ γ g
+ 1:4 10x
18:2
with
x ¼ 0:0125API 0:00091ðT 460Þ
Results of the calculations are given in the table below.
Gas Solubility “Rs” 263
Oil
T
(°F)
API
x
10x
Predicted Rs,
Eq. (4.23)
Measured
Rs
%
Error
1
2
3
4
5
6
250
220
260
237
218
180
47.1
40.7
48.6
40.5
44.2
27.3
0.361
0.309
0.371
0.291
0.322
0.177
2.297
2.035
2.349
1.952
2.097
1.505
838
817
774
914
1012
998
751
768
693
968
943
807
AAE
11.6
6.3
11.7
5.5
7.3
23.7
9.7%
x ¼ 0.0125API
460).
1:2048
p 0.00091(T
+ 1:4 10x
Rs ¼ γ g 18:2
.
Vasquez-Beggs’s Correlation
Vasquez and Beggs (1980) presented an improved empirical correlation for
estimating Rs. The correlation was obtained by regression analysis using
5008 measured gas solubility data points. Based on oil gravity, the measured
data were divided into two groups. This division was made at a value of oil
gravity of 30°API. The proposed equation has the following form:
API
Rs ¼ C1 γ gs pC2 exp C3
T
(4.24)
Values for the coefficients are as follows:
Coefficients
API 30
API > 30
C1
C2
C3
0.362
1.0937
25.7240
0.0178
1.1870
23.931
Realizing that the value of the specific gravity of the gas depends on the
conditions under which it is separated from the oil, Vasquez and Beggs proposed that the value of the gas specific gravity as obtained from a separator
pressure of 100 psig be used in this equation. The reference pressure was
chosen because it represents the average field separator conditions. The
authors proposed the following relationship for adjustment of the gas gravity, γ g, to the reference separator pressure:
h
γ gs ¼ γ g 1 + 5:912 105 ðAPIÞ Tsep 460 log
psep i
114:7
(4.25)
264 CHAPTER 4 PVT Properties of Crude Oils
where
γ gs ¼ gas gravity at the reference separator pressure
γ g ¼ gas gravity at the actual separator conditions of psep and Tsep
psep ¼ actual separator pressure, psia
Tsep ¼ actual separator temperature, °R
The gas gravity used to develop all the correlations reported by the authors
was that which would result from a two-stage separation. The first-stage
pressure was chosen as 100 psig and the second stage was the stock-tank.
If the separator conditions are unknown, the unadjusted gas gravity may
be used in Eq. (4.24).
An independent evaluation of this correlation by Sutton and Farshad (1984)
shows that the correlation is capable of predicting gas solubilities with an
average absolute error of 12.7%.
EXAMPLE 4.9
Using the PVT of the six crude oil systems of Example 4.8, solve for the gas
solubility using the Vasquez and Beggs method.
Solution
The results are shown in the following table.
Oil
γ gs, Eq. (4.25)
Predicted Rs, Eq. (4.24)
Measured Rs
% Error
1
2
3
4
5
6
0.8731
0.855
0.911
0.850
0.814
0.834
779
733
702
820
947
841
751
768
693
968
943
807
AAE
3.76
4.58
1.36
5.2
0.43
4.30
4.9%
h
psep i
γ gs ¼ γ g 1 + 5:912 105 ðAPI
Þ Tsep 460 log 114:7 N.
API
Rs ¼ C1 γ gs pC2 exp C3
.
T
Glaso’s Correlation
Glaso (1980) proposed a correlation for estimating the gas solubility as a
function of the API gravity, the pressure, the temperature, and the gas
specific gravity. The correlation was developed from studying 45 North
Sea crude oil samples. Glaso reported an average error of 1.28% with a
standard deviation of 6.98%. The proposed relationship has the following
form:
Gas Solubility “Rs” 265
("
Rs ¼ γ g
API0:989
ðT 460Þ
#
)1:2255
ðAÞ
0:172
(4.26)
The parameter A is a pressure-dependent coefficient defined by the following expression:
A ¼ 10 X
with the exponent X as given by
X ¼ 2:8869 ½14:1811 3:3093 log ðpÞ0:5
EXAMPLE 4.10
Rework Example 4.8 and solve for the gas solubility using Glaso’s correlation.
Solution
The results are shown in the table below.
Oil
p
X
A
Predicted Rs,
Eq. (4.26)
Measured
Rs
%
Error
1
2
3
4
5
6
2377
2620
2051
5884
3045
4239
1.155
1.196
1.095
1.237
1.260
1.413
14.286
15.687
12.450
17.243
18.210
25.883
737
714
686
843
868
842
751
768
693
968
943
807
AAE
1.84
6.92
0.90
12.92
7.94
4.34
5.8%
X ¼ 2:8869 ½14:1811 3:3093 log ðpÞ0:5 N.
A ¼ 10X .
h
i1:2255
API0:989
Rs ¼ γ g ðT 460
ð AÞ
.
Þ0:172
Marhoun’s Correlation
Marhoun (1988) developed an expression for estimating the saturation pressure of the Middle Eastern crude oil systems. The correlation originates
from 160 experimental saturation pressure data. The proposed correlation
can be rearranged and solved for the gas solubility to give
h
ie
Rs ¼ aγ bg γ co T d p
where
γ g ¼ gas specific gravity
γ o ¼ stock-tank oil gravity
(4.27)
266 CHAPTER 4 PVT Properties of Crude Oils
T ¼ temperature, °R
a–e ¼ coefficients of the above equation having these values
a ¼ 185.843208
b ¼ 1.877840
c ¼ 3.1437
d ¼ 1.32657
e ¼ 1.39844
EXAMPLE 4.11
Resolve Example 4.8 using Marhoun’s correlation.
Solution
The results are shown in the table below.
Oil
T (°F)
p
Predicted Rs,
Eq. (4.27)
1
2
3
4
5
6
250
220
260
237
218
180
2377
2620
2051
2884
3045
4239
740
792
729
1041
845
1186
h
ie
Rs ¼ aγ bg γ co T d p .
Measured Rs
% Error
751
768
693
968
943
807
AAE
1.43
3.09
5.21
7.55
10.37
47.03
12.4%
a ¼ 185:843208.
b ¼ 1:877840.
c ¼ 3:1437.
d ¼ 1:32657.
e ¼ 1:39844.
Petrosky-Farshad’s Correlation
Petrosky and Farshad (1993) used nonlinear multiple regression software
to develop a gas solubility correlation. The authors constructed a
PVT database from 81 laboratory analyses from the Gulf of Mexico
(GOM) crude oil system. Petrosky and Farshad proposed the following
expression:
Rs ¼
h
i1:73184
p
10x
+ 12:340 γ 0:8439
g
112:727
with
x ¼ 7:916 104 ðAPIÞ1:5410 4:561 105 ðT 460Þ1:3911
(4.28)
Gas Solubility “Rs” 267
where
Rs ¼ gas solubility, scf/STB
T ¼ temperature, °R
p ¼ system pressure, psia
γ g ¼ solution gas specific gravity
API ¼ oil gravity, °API
EXAMPLE 4.12
Test the predictive capability of the Petrosky and Farshad equation by resolving
Example 4.8.
Solution
The results are shown in the table below.
Oil
API
T (°F)
x
Predicted Rs,
Eq. (4.28)
1
2
3
4
5
6
47.1
40.7
48.6
40.5
44.2
27.3
250
220
260
237
218
180
0.2008
0.1566
0.2101
0.1579
0.1900
0.0667
772
726
758
875
865
900
x ¼ 7:916 104 ðAPIÞ1:5410 4:561 105 ðT 460Þ1:3911 .
h
i
1:73184
p
Rs ¼ 112:727
+ 12:340 γ 0:8439
10x
.
g
Measured Rs
% Error
751
768
693
968
943
807
AAE
2.86
5.46
9.32
9.57
8.28
11.57
7.84%
As derived later in this chapter from the material balance approach, the gas
solubility can also be calculated rigorously from the experimental, measured
PVT data at the specified pressure and temperature. The following expression
relates the gas solubility, Rs, to oil density, specific gravity of the oil, gas gravity,
and the oil FVF:
Rs ¼
Bo ρo 62:4γ o
0:0136γ g
(4.29)
where
ρo ¼ oil density at pressure and temperature, lb/ft3
Bo ¼ oil FVF, bbl/STB
γ o ¼ specific gravity of the stock-tank oil, 60°/60°
γ g ¼ specific gravity of the solution gas
McCain (1991) pointed out that the weight average of separator and stock-tank
gas specific gravities should be used for γ g. The error in calculating Rs by using
the preceding equation depends on the accuracy of only the available PVT data.
268 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.13
Using the data of Example 4.8, estimate Rs by applying Eq. (4.29).
Solution
The results are shown in the table below.
Oil
pb
Bo
ρo
API
γo
γg
Predicted Rs,
Eq. (4.29)
1
2
3
4
5
6
2377
2620
2051
2884
3045
4239
1.528
1.474
1.529
1.619
1.570
1.385
38.13
40.95
37.37
38.92
37.70
46.79
47.1
40.7
48.6
40.5
44.2
27.3
0.792
0.822
0.786
0.823
0.805
0.891
0.851
0.855
0.911
0.898
0.781
0.848
762
781
655
956
843
798
Measured Rs
% Error
751
768
693
968
943
807
AAE
1.53
1.73
5.51
1.23
10.57
1.13
3.61%
141:5
γ o ¼ 131:5
+ API.
ρo 62:4γ o
Rs ¼ Bo0:0136γ
.
g
BUBBLE POINT PRESSURE “pb”
The bubble point pressure, pb, of a hydrocarbon system is defined as the
highest pressure at which a bubble of gas is first liberated from the oil. This
important property can be measured experimentally for a crude oil system
by conducting a constant composition expansion (CCE) test.
In the absence of the experimentally measured bubble point pressure, it is
necessary for the engineer to make an estimate of this crude oil property
from the readily available measured producing parameters. Several graphical and mathematical correlations for determining pb have been proposed
during the last four decades. These correlations are essentially based on the
assumption that the bubble point pressure is a strong function of gas solubility, Rs; gas gravity, γ g; oil gravity, API; and temperature “T”:
pb ¼ f Rs , γ g , API, T
Techniques of combining these parameters in a graphical form or a mathematical expression were proposed by several authors, including:
n
n
n
n
n
Standing
Vasquez and Beggs
Glaso
Marhoun
Petrosky and Farshad
Bubble Point Pressure “pb” 269
The empirical correlations for estimating the bubble point pressure proposed
by these authors are detailed below.
Standing’s Correlation
Based on 105 experimentally measured bubble point pressures on 22 hydrocarbon systems from California oil fields, Standing (1947) proposed a
graphical correlation for determining the bubble point pressure of crude
oil systems. The correlating parameters in the proposed correlation are
the gas solubility, Rs; gas gravity, γ g; oil API gravity; and the system
temperature. The reported average error is 4.8%.
In a mathematical form, Standing (1981) expressed the graphical correlation
by the following expression:
h
i
0:83
ð10Þa 1:4
pb ¼ 18:2 Rs =γ g
(4.30)
a ¼ 0:00091ðT 460Þ 0:0125ðAPIÞ
(4.31)
with
where
Rs ¼ gas solubility, scf/STB
pb ¼ bubble point pressure, psia
T ¼ system temperature, °R
Standing’s correlation should be used with caution if nonhydrocarbon components are known to be present in the system.
Lasater (1958) used a different approach in predicting the bubble point pressure by introducing and using the mole fraction of the solution gas, ygas, in
the crude oil as a correlating parameter, defined by
ygas ¼
Mo R s
Mo Rs + 133, 000γ o
where Mo ¼ molecular weight of the stock-tank oil and γ o ¼ specific gravity
of the stock-tank oil, 60°/60°.
If the molecular weight is not available, it can be estimated from Cragoe
(1997) as
Mo ¼
6084
API 5:9
The bubble point pressure as proposed by Lasater is given by the following
relationship:
pb ¼
!
T
A
γg
270 CHAPTER 4 PVT Properties of Crude Oils
where T is the temperature in °R and A is a graphical correlating parameter
that is a function of the mole fraction of the solution gas, ygas. Whitson and
Brule (2000) described this correlating parameter by
A ¼ 0:83918 101:17664ygas y0:57246
, when ygas 0:6
gas
1:08000y 0:31109
gas
A ¼ 0:83918 10
ygas , when ygas > 0:6
EXAMPLE 4.14
The experimental data given in Example 4.8 are repeated below for
convenience. Predict the bubble point pressure using Standing’s correlation.
Oil
T (°F)
pb
Rs (scf/STB)
Bo
ρo (lb/ft3)
co at p > pb
psep
Tsep
API
γg
1
250
2377
751
1.528
38.13
150
60
47.1
0.851
2
220
2620
768
0.474
40.95
100
75
40.7
0.855
3
260
2051
693
1.529
37.37
100
72
48.6
0.911
4
237
2884
968
1.619
38.92
60
120
40.5
0.898
5
218
3045
943
1.570
37.70
200
60
44.2
0.781
6
180
4239
807
1.385
46.79
22.14 106
at 2689 psi
18.75 106
at 2810 psi
22.69 106
at 2526 psi
21.51 106
at 2942 psi
24.16 106
at 3273 psi
11.45 106
at 4370 psi
85
173
27.3
0.848
Solution
The results are shown in the following table.
Oil
API
γg
Coefficient
a, Eq. (4.31)
Predicted pb,
Eq. (4.30)
Measured
pb
% Error
1
2
3
4
5
6
47.1
40.7
48.6
40.5
44.2
27.3
0.851
0.855
0.911
0.898
0.781
0.848
0.3613
0.3086
0.3709
0.3115
0.3541
0.1775
2181
2503
1883
2896
2884
3561
2392
2635
2066
2899
3060
4254
AAE
8.8
5.0
8.8
0.1
5.7
16.3
7.4%
a ¼ 0.00091(T 460) 0.0125(API).
pb ¼ 18.2[(Rs /γ g)0.83(10)a 1.4].
McCain (1991) suggested that replacing the specific gravity of the gas in
Eq. (4.30) with that of the separator gas, that is, excluding the gas from the stocktank would improve the accuracy of the equation.
Bubble Point Pressure “pb” 271
EXAMPLE 4.15
Using the data from Example 4.14 and given the following first separator gas
gravities as observed by McCain, estimate the bubble point pressure by applying
Standing’s correlation.
Oil
First Separator Gas Gravity
1
2
3
4
5
6
0.755
0.786
0.801
0.888
0.705
0.813
Solution
The results are shown in the table below.
First
Separation Coefficient a, Predicted pb, Measured %
Oil API γ g
Eq. (4.31)
Eq. (4.30)
pb
Error
1
2
3
4
5
6
47.1 0.755
40.7 0.786
48.6 0.801
40.5 0.888
44.2 0.705
27.3 0.813
0.3613
0.3086
0.3709
0.3115
0.3541
0.1775
2411
2686
2098
2923
3143
3689
2392
2635
2066
2899
3060
4254
AAE
0.83
1.93
1.53
0.84
2.70
13.27
3.5%
a ¼ 0.00091(T 460) 0.0125(API).
pb ¼ 18.2[(Rs /γ g)0.83(10)a 1.4].
Vasquez-Beggs’s Correlation
Vasquez and Beggs’s gas solubility correlation as presented by Eq. (4.24)
can be solved for the bubble point pressure, pb, to give
"
pb ¼
!
# C2
Rs
a
ð10Þ
C1
γ gs
with exponent a as given by
a ¼ C3
(4.32)
API
T
where the temperature, T, is in °R. The coefficients C1, C2, and C3 of
Eq. (4.32) have the following values:
272 CHAPTER 4 PVT Properties of Crude Oils
Coefficient
API 30
API > 30
C1
C2
C3
27.624
10.914328
11.172
56.18
0.84246
10.393
The gas specific gravity γ gs at the reference separator pressure is defined by
Eq. (4.25) as
h
γ gs ¼ γ g 1 + 5:912 105 ðAPIÞ Tsep 460 log
psep i
114:7
EXAMPLE 4.16
Rework Example 4.14 by applying Eq. (4.32).
Solution
The results are shown in the table below.
Oil
T
Rs
API
γ gs, Eq. (4.25)
a
Predicted pb
Measured pb
% Error
1
2
3
4
5
6
250
220
260
237
218
180
751
768
693
968
943
807
47.1
40.7
48.6
40.5
44.2
27.3
0.873
0.855
0.911
0.850
0.814
0.834
0.689
0.622
0.702
0.625
0.678
0.477
2319
2741
2043
3331
3049
4093
2392
2635
2066
2899
3060
4254
AAE
3.07
4.03
1.14
14.91
0.36
3.78
4.5%
a ¼ C3
pb ¼
h
API
T
.
C1 γRs ð10Þa
gs
iC2
.
Glaso’s Correlation
Glaso (1980) used 45 oil samples, mostly from the North Sea hydrocarbon
system, to develop an accurate correlation for bubble point pressure prediction. Glaso proposed the following expression:
log ðpb Þ ¼ 1:7669 + 1:7447log ðAÞ 0:30218½ log ðAÞ2
(4.33)
The correlating parameter A in the equation is defined by the following
expression:
Bubble Point Pressure “pb” 273
A¼
Rs
γg
!0:816
ðT 460Þ0:172
(4.34)
ðAPIÞ0:989
where
Rs ¼ gas solubility, scf/STB
T ¼ system temperature, °R
γ g ¼ average specific gravity of the total surface gases
For volatile oils, Glaso recommends that the temperature exponent of
Eq. (4.34) be slightly changed from 0.172 to the value of 0.130, to give
A¼
Rs
γg
!0:816
ðT 460Þ0:1302
ðAPIÞ0:989
EXAMPLE 4.17
Resolve Example 4.14 using Glaso’s correlation.
Solution
The results are shown in the table below.
Oil
T
Rs
API
A,
Eq. (3.34)
pb,
Eq. (4.33)
Measured
pb
%
Error
1
2
3
4
5
6
250
220
260
237
218
180
751
768
693
968
943
807
47.1
40.7
48.6
40.5
44.2
27.3
14.51
16.63
12.54
19.30
19.48
25.00
2431
2797
2083
3240
3269
4125
2392
2635
2066
2899
3060
4254
AAE
1.62
6.14
0.82
11.75
6.83
3.04
5.03%
0:816
ðT 460Þ0:172
ðAPIÞ0:989 .
A¼
Rs
γg
log ðpb Þ ¼ 1:7669 + 1:7447 log ðAÞ 0:30218½ log ðAÞ2 .
Marhoun’s Correlation
Marhoun (1988) used 160 experimentally determined bubble point pressures from the PVT analysis of 69 Middle Eastern hydrocarbon mixtures
to develop a correlation for estimating pb. The author correlated the bubble
point pressure with the gas solubility, Rs, the temperature, T, and the specific
gravity of the oil and the gas. Marhoun proposed the following expression:
pb ¼ aRbs γ cg γ do T e
(4.35)
274 CHAPTER 4 PVT Properties of Crude Oils
where
T ¼ temperature, °R
Rs ¼ gas solubility, scf/STB
γ o ¼ stock-tank oil specific gravity
γ g ¼ gas specific gravity
a–e ¼ coefficients of the correlation having the following values
a ¼ 5.38088 103
b ¼ 0.715082
c ¼ 1.87784
d ¼ 3.1437
e ¼ 1.32657
The reported average absolute relative error for the correlation is 3.66%
when compared with the experimental data used to develop the correlation.
EXAMPLE 4.18
Using Eq. (4.35), rework Example 4.13.
Solution
The results are shown in the table below.
Oil
Predicted pb
Measured pb
% Error
1
2
3
4
5
6
2417
2578
1992
2752
3309
3229
2392
2635
2066
2899
3060
4254
1.03
2.16
3.57
5.07
8.14
24.09
AAE ¼ 7.3%
pb ¼ 0:00538033R0:715082
γ 1:87784
γ 3:1437
T 1:32657 .
s
g
o
Petrosky-Farshad’s Correlation
Petrosky and Farshad’s gas solubility equation, Eq. (4.28), can be solved for
the bubble point pressure to give
"
#
112:727R0:577421
s
1391:051
pb ¼
γ 0:8439
ð10Þx
g
(4.36)
where the correlating parameter, x, is as previously defined by the following
expression:
x ¼ 7:916 104 ðAPIÞ1:5410 4:561 105 ðT 460Þ1:3911
Oil FVF 275
where
Rs ¼ gas solubility, scf/STB
T ¼ temperature, °R
p ¼ system pressure, psia
γ g ¼ solution gas specific gravity
API ¼ oil gravity, °API
The authors concluded that the correlation predicts measured bubble point
pressures with an average absolute error of 3.28%.
EXAMPLE 4.19
Use the Petrosky and Farshad correlation to predict the bubble point pressure
data given in Example 4.14.
Solution
The results are shown in the table below.
Oil
T
Rs
API
x
pb,
Eq. (4.36)
Measured
pb
%
Error
1
2
3
4
5
6
250
220
260
237
218
180
751
768
693
968
943
807
47.1
40.7
48.6
40.5
44.2
27.3
0.2008
0.1566
0.2101
0.1579
0.1900
0.0667
2331
2768
1893
3156
3288
3908
2392
2635
2066
2899
3060
4254
AAE
2.55
5.04
8.39
8.86
7.44
8.13
6.74%
x ¼ 7:916 104 ðAPIÞ1:5410 4:561 105 ðT 460Þ1:3911 .
"
#
112:727R0:577421
s
1391:051.
pb ¼
x
γ 0:8439
ð10Þ
g
Using the neural network approach, Al-Shammasi (1999) proposed the
following expression that estimated the bubble point pressure with an average
absolute error of 18%:
0:783716
e1:841408½γo γg Rs Tγ g
pb ¼ γ 5:527215
o
where T is the temperature in °R.
OIL FVF
The oil FVF, Bo, is defined as the ratio of the volume of oil (plus the gas in
solution) at the prevailing reservoir temperature and pressure to the volume
276 CHAPTER 4 PVT Properties of Crude Oils
of oil at SC. Evidently, Bo always is greater than or equal to unity. The oil
FVF can be expressed mathematically as
Bo ¼
ðVo Þp, T
ðVo Þsc
(4.37)
where
Bo ¼ oil FVF, bbl/STB
(Vo)p,T ¼ volume of oil, in bbl, under reservoir pressure, p, and
temperature, T
(Vo)sc ¼ volume of oil is measured under SC, STB
A typical oil formation factor curve, as a function of pressure for an undersaturated crude oil (pi > pb), is shown in Fig. 4.7. As the pressure is reduced
below the initial reservoir pressure, pi, the oil volume increases due to the oil
expansion. This behavior results in an increase in the oil FVF and continues
until the bubble point pressure is reached. At pb, the oil reaches its maximum
expansion and consequently attains a maximum value of Bob for the oil FVF.
As the pressure is reduced below pb, volume of the oil and Bo are decreased
as the solution gas is liberated. When the pressure is reduced to atmospheric
pressure and the temperature to 60°F, the value of Bo is equal to 1.
Most of the published empirical Bo correlations utilize the following generalized relationship:
Bo ¼ f Rs , γ g , γ o , T
1.2
0.5
Bo 0.9759
0.000120 Rs
g
1.25 T
460
o
Bob
Bo
pb
1
0
Pressure
n FIGURE 4.7 Typical oil formation volume factor-pressure relationship.
Oil FVF 277
Six methods of predicting the oil FVF are presented, including:
n
n
n
n
n
n
Standing’s correlation
Vasquez and Beggs’s correlation
Glaso’s correlation
Marhoun’s correlation
Petrosky and Farshad’s correlation
The material balance equation
It should be noted that all the correlations could be used for any pressure
equal to or below the bubble point pressure.
Standing’s Correlation
Standing (1947) presented a graphical correlation for estimating the oil
FVF with the gas solubility, gas gravity, oil gravity, and reservoir
temperature as the correlating parameters. This graphical correlation
originated from examining 105 experimental data points on 22 California
hydrocarbon systems. An average error of 1.2% was reported for the
correlation.
Standing (1981) showed that the oil FVF can be expressed more conveniently in a mathematical form by the following equation:
" γg
γo
#1:2
0:5
Bo ¼ 0:9759 + 0:000120 Rs
+ 1:25ðT 460Þ
(4.38)
where
T ¼ temperature, °R
γ o ¼ specific gravity of the stock-tank oil, 60°/60°
γ g ¼ specific gravity of the solution gas
Vasquez-Beggs’s Correlation
Vasquez and Beggs (1980) developed a relationship for determining Bo as a
function of Rs, γ o, γ g, and T. The proposed correlation was based on 6000
measurements of Bo at various pressures. Using the regression analysis technique, Vasquez and Beggs found the following equation to be the best form
to reproduce the measured data:
!
API
Bo ¼ 1:0 + C1 Rs + ðT 520Þ
½C2 + C3 Rs γ gs
(4.39)
278 CHAPTER 4 PVT Properties of Crude Oils
where
R ¼ gas solubility, scf/STB
T ¼ temperature, °R
γ gs ¼ gas specific gravity as defined by Eq. (4.25):
h
γ gs ¼ γ g 1 + 5:912 105 ðAPIÞ Tsep 460 log
psep i
114:7
Values for the coefficients C1, C2, and C3 of Eq. (4.39) follow:
Coefficient
API 30
API > 30
4
4.677 10
1.751 105
1.811 108
C1
C2
C3
4.670 104
1.100 105
1.337 109
Vasquez and Beggs reported an average error of 4.7% for the proposed
correlation.
Glaso’s Correlation
Glaso (1980) proposed the following expressions for calculating the oil FVF:
Bo ¼ 1:0 + 10A
(4.40)
2
A ¼ 6:58511 + 2:91329log B∗ob 0:27683 log B∗ob
(4.41)
where
B∗ob is a “correlating number,” defined by the following equation:
B∗ob ¼ Rs
γg
γo
0:526
+ 0:968ðT 460Þ
(4.42)
where T ¼ temperature, °R, and γ o ¼ specific gravity of the stock-tank oil,
60°/60°. These correlations were originated from studying PVT data on
45 oil samples. The average error of the correlation was reported at
0.43% with a standard deviation of 2.18%.
Sutton and Farshad (1984) concluded that Glaso’s correlation offers the best
accuracy when compared with Standing’s and Vasquez-Beggs’s correlations. In general, Glaso’s correlation underpredicts FVF, Standing’s expression tends to overpredict oil FVFs, while Vasquez-Beggs’s correlation
typically overpredicts the oil FVF.
Oil FVF 279
Marhoun’s Correlation
Marhoun (1988) developed a correlation for determining the oil FVF as a
function of the gas solubility, stock-tank oil gravity, gas gravity, and temperature. The empirical equation was developed by use of the nonlinear multiple regression analysis on 160 experimental data points. The experimental
data were obtained from 69 Middle Eastern oil reserves. The author proposed the following expression:
Bo ¼ 0:497069 + 0:000862963T + 0:00182594F + 0:00000318099F2
(4.43)
with the correlating parameter F as defined by the following equation:
F ¼ Ras γ bg γ co
(4.44)
where T is the system temperature in °R and the coefficients a, b, and c have
the following values:
a ¼ 0:742390
b ¼ 0:323294
c ¼ 1:202040
Petrosky-Farshad’s Correlation
Petrosky and Farshad (1993) proposed a new expression for estimating Bo.
The proposed relationship is similar to the equation developed by Standing;
however, the equation introduces three additional fitting parameters to
increase the accuracy of the correlation.
The authors used a nonlinear regression model to match experimental
crude oil from the GOM hydrocarbon system. Their correlation has the
following form:
Bo ¼ 1:0113 + 7:2046 105 A
(4.45)
with the term A as given by
"
A¼
R0:3738
s
γ 0:2914
g
γ 0:6265
o
!
#3:0936
+ 0:24626ðT 460Þ
0:5371
where T ¼ temperature, °R, and γ o ¼ specific gravity of the stock-tank oil,
60°/60°.
280 CHAPTER 4 PVT Properties of Crude Oils
Material Balance Equation
From the definition of Bo as expressed mathematically by Eq. (4.37),
Bo ¼
ðVo Þp, T
ðVo Þsc
The oil volume under p and T can be replaced with total weight of the hydrocarbon system divided by the density at the prevailing pressure and
temperature:
mt
ρo
Bo ¼
ðVo Þsc
where the total weight of the hydrocarbon system is equal to the sum of the
stock-tank oil plus the weight of the solution gas:
mt ¼ mo + mg
or
Bo ¼
mo + mg
ρo ðVo Þsc
Given the gas solubility, Rs, per barrel of the stock-tank oil and the specific
gravity of the solution gas, the weight of Rs scf of the gas is calculated as
mg ¼
Rs
ð28:96Þ γ g
379:4
where mg ¼ weight of solution gas, lb of solution gas/STB.
The weight of one barrel of the stock-tank oil is calculated from its specific
gravity by the following relationship:
mo ¼ ð5:615Þð62:4Þðγ o Þ
Substituting for mo and mg,
Bo ¼
Rs
ð28:96Þγ g
379:4
5:615ρo
ð5:615Þð62:4Þγ o +
or
Bo ¼
62:4Eo + 0:0136Rs γ g
ρo
(4.46)
where ρo ¼ density of the oil at the specified pressure and temperature, lb/ft3.
The error in calculating Bo by using Eq. (4.46) depends on the accuracy of
only the input variables (Rs, γ g, and γ o) and the method of calculating ρo.
Oil FVF 281
EXAMPLE 4.20
The table below shows experimental PVT data on six crude oil systems.
Results are based on two-stage surface separation. Calculate the oil FVF at the
bubble point pressure using the preceding six different correlations. Compare
the results with the experimental values and calculate the AAE.
Oil
1
2
3
4
5
6
T
pb
250
220
260
237
218
180
Rs
2377
2620
2051
2884
3065
4239
ρo
Bo
751
768
693
968
943
807
1.528
1.474
0.529
1.619
0.570
0.385
co at p > pb
6
38.13
40.95
37.37
38.92
37.70
46.79
22.14 10 at 2689
18.75 106 at 2810
22.69 106 at 2526
21.51 106 at 2942
24.16 106 at 3273
11.65 106 at 4370
0:5
#1:2
Solution
For convenience, these six correlations follow:
" Method 1
γg
Bo ¼ 0:9759 + 0:000120 Rs
γo
+ 1:25ðT 460Þ
!
API
½C2 + C3 Rs Bo ¼ 1:0 + C1 Rs + ðT 520Þ
γ gs
Method 2
Method 3
Bo ¼ 1:0 + 10A
Method 4
Bo ¼ 0:497069 + 0:000862963T + 0:00182594F + 0:00000318099F2
Bo ¼ 1:0113 + 7:2046 105 A
Method 5
Method 6
Bo ¼
62:4γ o + 0:0136Rs γ g
ρo
Results of applying these correlations for calculating Bo are tabulated in the table
below.
Method
Crude Oil
1
2
3
4
5
Bo
1
2
3
4
5
6
1.528
1.474
1.529
1.619
1.570
1.506
1.487
1.495
1.618
1.571
1.474
1.450
1.451
1.542
1.546
1.473
1.459
1.461
1.589
1.541
1.516
1.477
1.511
1.575
1.554
1.552
1.508
1.556
1.632
1.584
1.525
1.470
1.542
1.623
1.599
psep
Tsep
API
γg
150
100
100
60
200
85
60
75
72
120
60
173
47.1
40.7
48.6
40.5
44.2
27.3
0.851
0.855
0.911
0.898
0.781
0.848
282 CHAPTER 4 PVT Properties of Crude Oils
6
%AAE
1.385
–
1.461
1.7
1.389
2.8
1.438
2.8
1.414
1.3
1.433
1.8
1.387
0.6
Method 1 ¼ Standing’s correlation.
Method 2 ¼ Vasquez and Beggs’s correlation.
Method 3 ¼ Glaso’s correlation.
Method 4 ¼ Marhoun’s correlation.
Method 5 ¼ Petrosky and Farshad’s correlation.
Method 6 ¼ Material balance equation.
Al-Shammasi (1999) used the neural network approach to generate an
expression for predicting Bo. The relationship, as given below, gives an average
absolute error of 1.81%:
Bo ¼ 1 + 5:53 107 ðT 520ÞRs + 0:000181ðRs =γ o Þ
+ ½0:000449ðT 520Þ=γ o + 0:000206Rs γ g =γ o
where T is the temperature in °R.
ISOTHERMAL COMPRESSIBILITY COEFFICIENT OF
CRUDE OIL “co”
The isothermal compressibility coefficient is defined as the rate of change in
volume with respect to pressure increase per unit volume, all variables other
than pressure being constant, including temperature. Mathematically, the
isothermal compressibility, c, of a substance is defined by the following
expression:
c¼ 1 @V
V @p
T
Isothermal compressibility coefficients are required in solving many reservoir engineering problems, including transient fluid flow problems; also,
they are required in the determination of the physical properties of the
undersaturated crude oil.
For a crude oil system, the isothermal compressibility coefficient of the oil
phase, co, is categorized into the following two types based on reservoir
pressure:
n
At reservoir pressures that are greater than or equal to the bubble point
pressure (p pb), the crude oil exists as a single phase with all its
dissolved gas still in solution. The isothermal compressibility coefficient
of the oil phase, co, above the bubble point reflects the changes in the
volume associated with oil expansion or compression of the single-phase
oil with changing the reservoir pressure. The oil compressibility in this
case is termed undersaturated isothermal compressibility coefficient.
Isothermal Compressibility Coefficient of Crude Oil “co” 283
n
Below the bubble point pressure, the solution gas is liberated with
decreasing reservoir pressure or redissolved with increasing the
pressure. The changes of the oil volume as the result of changing the gas
solubility must be considered when determining the isothermal
compressibility coefficient. The oil compressibility in this case is termed
saturated isothermal compressibility coefficient.
Undersaturated Isothermal Compressibility
Coefficient
Generally, isothermal compressibility coefficients of an undersaturated
crude oil are determined from a laboratory PVT study. A sample of the
crude oil is placed in a PVT cell at the reservoir temperature and a pressure
greater than the bubble point pressure of the crude oil. At these initial
conditions, the reservoir fluid exists as a single-phase liquid. The volume
of the oil is allowed to expand as its pressure declines. This volume is
recorded and plotted as a function of pressure. If the experimental
pressure-volume diagram for the oil is available, the instantaneous compressibility coefficient, co, at any pressure can be calculated by graphically
determining the volume, V, and the corresponding slope, (@V/@p)T, at this
pressure. At pressures above the bubble point, any of following equivalent
expressions is valid in defining co:
1 @V
co ¼ V @p T
1 @Bo
co ¼ Bo @p T
1 @ρo
co ¼
ρo @p T
(4.47)
where
co ¼ isothermal compressibility of the crude oil, psi1
@V
@p
¼ slope of the isothermal pressure=volume curve atp
T
ρo ¼ oil density, lb/ft3
Bo ¼ oil FVF, bbl/STB
Craft and Hawkins (1959) introduced the cumulative or average isothermal
compressibility coefficient, which defines the compressibility from the
initial reservoir pressure to current reservoir pressure. Craft and Hawkins’s
284 CHAPTER 4 PVT Properties of Crude Oils
average compressibility, as defined here, is used in all material balance
calculations:
ð pi
V
co ¼
co dp
p
V ½pi p
¼
V Vi
V ½pi p
where the subscript i represents initial condition. Equivalently, the
average compressibility can be expressed in terms of Bo and ρo by the
expressions
Bo Boi
Bo ½pi p
ρ ρo
co ¼ oi
ρo ½pi p
co ¼
The following four correlations for estimating the undersaturated oil
compressibility are presented below:
n
n
n
n
Trube’s correlation
Vasquez-Beggs’s correlation
Petrosky-Farshad’s correlation
Standing’s correlation
Trube’s Correlation
Trube (1957) introduced the concept of the isothermal pseudoreduced compressibility, cr, of undersaturated crude oils as defined by the relationship
cr ¼ co ppc
(4.48)
Trube correlated this property graphically with the pseudoreduced pressure and temperature, ppr and Tpr, as shown in Fig. 4.8. Additionally,
Trube presented two graphical correlations, as shown in Figs. 4.9 and
4.10, to estimate the pseudocritical properties of crude oils. The calculation procedure of the proposed method is summarized in the following
steps.
Step 1. From the bottom-hole pressure measurements and pressure gradient
data, calculate the average density of the undersaturated reservoir
oil, in g/cm3, from the following expression:
ðρo ÞT ¼
dp=dh
0:433
Isothermal Compressibility Coefficient of Crude Oil “co” 285
Ps
eu
do
re
du
ce
d
10
1.
Pseudoreduced compressibility
0.1
0.01
1.
00
0.
95
te
m
pe
ra
tu
re
0.
90
0.
80
0.
70
0.
60
0.
50
0.001
0.4
0
1
2
3
4
5
10
20
30
40 50
100
Pseudoreduced pressure
n FIGURE 4.8 Trube’s pseudoreduced compressibility of undersaturated crude oil. (Permission to publish by the Society of Petroleum Engineers of AIME. Copyright
SPE-AIME.)
where (ρo)T ¼ oil density at reservoir pressure and temperature T, g/
cm3, and dp/dh ¼ pressure gradient as obtained from a pressure
buildup test.
Step 2. Adjust the calculated undersaturated oil density to its value at 60°F
using the following equation:
ðρo Þ60 ¼ ðρo ÞT ¼ 0:00046ðT 520Þ
(4.49)
where (ρo)60 ¼ adjusted undersaturated oil density to 60°F, g/cm3,
and T ¼ reservoir temperature, °R.
286 CHAPTER 4 PVT Properties of Crude Oils
6000
5000
4000
0.
2000
66
0. 0.7
68 70 2
Specific gravity of
reservoir liquid at 60°F.
0.
6
0.8
4
0.8
2
1000
900
800
700
600
0.8
0
0.8
8
0.7
6
0.7
4
0.7
Bubblepoint pressure at 60°F (psia)
3000
500
400
300
200
100
700
800
900
1000
1100
Pseudocritical temperature (°R)
n FIGURE 4.9 Trube’s pseudocritical temperature correlation. (Permission to publish by the Society of Petroleum Engineers of AIME. Copyright SPE-AIME.)
Step 3. Determine the bubble point pressure, pb, of the crude oil at reservoir
temperature. If the bubble point pressure is not known, it can be
estimated from Standing (1981), Eq. (4.30), written in a compacted
form as
2
Rs
pb ¼ 18:24
γg
!0:83 100:00091ðT460Þ
°
100:0125 API
3
1:45
Step 4. Correct the calculated bubble point pressure, pb, at reservoir
temperature to its value at 60°F using the following equation as
proposed by Standing and Katz (1942):
ðpb Þ60 ¼
1:134pb
100:00091ðT460Þ
(4.50)
Isothermal Compressibility Coefficient of Crude Oil “co” 287
1300
1200
1000
Pseudocritical temperature
900
800
700
600
Pseudocritical pressure
500
Pseudocritical pressure
Pseudocritical temperature (°R)
1100
400
300
200
0.62
0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88
Specific gravity of undersaturated crude oil at reservoir pressure corrected to 60°F
n FIGURE 4.10 Trube’s pseudocritical properties correlation. (Permission to publish by the Society of Petroleum Engineers of AIME. Copyright SPE-AIME.)
where
(pb)60 ¼ bubble point pressure at 60°F, psi
pb ¼ bubble point pressure at reservoir temperature, psia
T ¼ reservoir temperature, °R
Step 5. Enter in Fig. 4.9 the values of (pb)60 and (ρo)60 and determine the
pseudocritical temperature, Tpc, of the crude.
Step 6. Enter the value of Tpc in Fig. 4.10 and determine the pseudocritical
pressure, ppc, of the crude.
Step 7. Calculate the pseudoreduced pressure, ppr, and temperature, Tpr,
from the following relationships:
T
Tpc
p
ppr ¼
ppc
Tpr ¼
288 CHAPTER 4 PVT Properties of Crude Oils
Step 8. Determine cr by entering into Fig. 4.8 the values of Tpr and ppr.
Step 9. Calculate co from Eq. (4.48):
co ¼
cr
ppc
Trube did not specify the data used to develop the correlation nor did he
allude to their accuracy, although the examples presented in his paper
showed an average absolute error of 7.9% between calculated and measured
values. Trube’s correlation can be best illustrated through the following
example.
EXAMPLE 4.21
Given the following data:
Oil gravity ¼ 45°
Gas solubility ¼ 600 scf/STB
Solution gas gravity ¼ 0.8
Reservoir temperature ¼ 212°F
Reservoir pressure ¼ 2000 psia
Pressure gradient of reservoir liquid at 2000 psia and 212°F ¼ 0.032 psi/ft
find co at 2000, 3000, and 4000 psia.
Solution
Step 1. Determine (ρo)T:
ðρo ÞT ¼
dp=dh 0:32
¼
¼ 0:739g=cm3
0:433 0:433
Step 2. Correct the calculated oil specific gravity to its value at 60°F by applying
Eq. (4.49):
ðρo Þ60 ¼ ðρo ÞT + 0:00046ðT 520Þ
ðρo Þ60 ¼ 0:739 + 0:00046ð152Þ ¼ 0:8089
Step 3. Calculate the bubble point pressure from Standing’s correlation
(Eq. (4.40)):
2
Rs
pb ¼ 18:24
γg
3
!0:83 100:00091ðT460Þ
1:45
100:0125°API
"
600
pb ¼ 18:2
0:8
0:83
#
100:0091ðT212Þ
1:4 ¼ 1866psia
100:0125ð45Þ
Isothermal Compressibility Coefficient of Crude Oil “co” 289
Step 4. Adjust pb to its value at 60°F by applying Eq. (4.50):
ðpb Þ60 ¼
1:134pb
100:00091 ðT 460Þ
ðpb Þ60 ¼
1:134ð1866Þ
¼ 1357:1psia
100:00091ð212Þ
Step 5. Estimate the pseudocritical temperature, Tpc, of the crude oil from
Fig. 4.9, to give
Tpc ¼ 840°R
Step 6. Estimate the pseudocritical pressure, ppc, of the crude oil from Fig. 4.10,
to give
ppc ¼ 500psia
Step 7. Calculate the pseudoreduced temperature:
Tpr ¼
672
¼ 0:8
840
Step 8. Calculate the pseudoreduced pressure, ppr, the pseudoreduced
isothermal compressibility coefficient, cr, and tabulate values of the
corresponding isothermal compressibility coefficient as shown in the
table below.
p
ppr 5 p/500
cr
co 5 cr/ppr1026 psia21
2000
3000
4000
4
6
8
0.0125
0.0089
0.0065
25.0
17.8
13.0
Vasquez-Beggs’s Correlation
From a total of 4036 experimental data points used in a linear regression
model, Vasquez and Beggs (1980) correlated the isothermal oil compressibility coefficients with Rs, T, °API, γ g, and p. They proposed the following
expression:
co ¼
1433 + 5Rsb + 17:2ðT 460Þ 1180γ gs + 12:61°API
105 p
where
T ¼ temperature, °R
p ¼ pressure above the bubble point pressure, psia
Rsb ¼ gas solubility at the bubble point pressure
γ gs ¼ corrected gas gravity as defined by Eq. (4.3)
(4.51)
290 CHAPTER 4 PVT Properties of Crude Oils
Petrosky-Farshad’s Correlation
Petrosky and Farshad (1993) proposed a relationship for determining the oil
compressibility for undersaturated hydrocarbon systems. The equation has
the following form:
co ¼ 1:705 107 R0:69357
γ 0:1885
API0:3272 ðT 460Þ0:6729 p0:5906
sb
g
(4.52)
where T ¼ temperature, °R, and Rsb ¼ gas solubility at the bubble point pressure, scf/STB.
Standing’s Correlation
Standing (1974) proposed a graphical correlation for determining the
oil compressibility for undersaturated hydrocarbon systems. Whitson and
Brule expressed his relationship in the following mathematical form:
6
co ¼ 10
ρob + 0:004347ðp pb Þ 79:1
exp
0:0007141ðp pb Þ 12:938
(4.53)
where
ρob ¼ oil density at the bubble point pressure, lb/ft3
pb ¼ bubble point pressure, psia
co ¼ oil compressibility, psia1
EXAMPLE 4.22
Using the experimental data given in Example 4.20, estimate the undersaturated
oil compressibility coefficient using the Vasquez-Beggs, Petrosky-Farshad, and
Standing correlations. The data given in Example 4.20 is reported in the
following table for convenience.
Oil
1
2
3
4
5
6
T
250
220
260
237
218
180
pb
2377
2620
2051
2884
3065
4239
Rs
751
768
693
968
943
807
ρo
Bo
1.528
1.474
0.529
1.619
0.570
0.385
38.13
40.95
37.37
38.92
37.70
46.79
co at p > pb
6
22.14 10 at 2689
18.75 106 at 2810
22.69 106 at 2526
21.51 106 at 2942
24.16 106 at 3273
11.65 106 at 4370
Solution
The results are shown in the table below.
psep
Tsep
API
γg
150
100
100
60
200
85
60
75
72
120
60
173
47.1
40.7
48.6
40.5
44.2
27.3
0.851
0.855
0.911
0.898
0.781
0.848
Isothermal Compressibility Coefficient of Crude Oil “co” 291
Measured co Standing’s Vasquez-Beggs Petrosky-Farshad
Oil Pressure 1026 psi
1026 psi 1026 psi
1026 psi
1
2
3
4
5
6
2689
2810
2526
2942
3273
4370
AAE
22.14
18.75
22.60
21.51
24.16
11.45
2.0%
Vasquez-Beggs : co ¼
22.54
18.46
23.30
22.11
23.72
11.84
6.18%
22.88
20.16
23.78
22.31
20.16
11.54
4.05%
22.24
19.27
22.92
21.78
20.39
11.77
1433 + 5Rsb + 17:2ðT 460Þ 1180γ gs + 12:61°API
105 p
:
Petrosky-Farshad :
co ¼ 1:705 107 R0:69357
γ 0:1885
API0:3272 ðT 460Þ0:6729 p0:5906 :
g
sb
Standing’s : co ¼ 106 exp
ρob + 0:004347ðp pb Þ 79:1
:
0:0007141ðp pb Þ 12:938
Saturated Isothermal Compressibility Coefficient
At pressures below the bubble point pressure, the isothermal compressibility coefficient of the oil must be modified to account for the shrinkage associated with the liberation of the solution gas with decreasing reservoir
pressure or swelling of the oil with repressurizing the reservoir and redissolving the solution gas. These changes in the oil volume as the result of changing the gas solubility must be considered when determining the isothermal
compressibility coefficient of the oil compressibility. Below the bubble
point pressure, co, is defined by the following expression:
co ¼
1@Bo Bg @Rs
+
Bo @p Bo @p
(4.54)
where Bg ¼ gas FVF, bbl/scf.
Analytically, Standing’s correlations for Rs (Eq. (4.23)) and Bo (Eq. (4.38))
can be differentiated with respect to the pressure, p, to give
@Rs
Rs
¼
@p 0:83p + 21:75
γg
@Bo
0:000144Rs
¼
@p
0:83p + 21:75 γ o
0:5
" γg
Rs
γo
#0:2
0:5
+ 1:25ðT 46Þ
292 CHAPTER 4 PVT Properties of Crude Oils
These two expressions can be substituted into Eq. (4.54) to give the following relationship:
(
)
0:2
rffiffiffiffiffi rffiffiffiffiffi
γg
γg
Rs
0:000144
Rs
+ 1:25ðT 460Þ
co ¼
Bg
Bo ð0:83p + 21:75Þ
γo
γo
(4.55)
where
p ¼ pressure, psia
T ¼ temperature, °R
Bg ¼ gas FVF at pressure p, bbl/scf
Rs ¼ gas solubility at pressure p, scf/STB
Bo ¼ oil FVF at p, bbl/STB
γ o ¼ specific gravity of the stock-tank oil
γ g ¼ specific gravity of the solution gas
McCain and coauthors (1988) correlated the oil compressibility with pressure, p, psia; oil API gravity; gas solubility at the bubble point, Rsb, in scf/
STB; and temperature, T, in °R. Their proposed relationship has the following form:
co ¼ exp ðAÞ
(4.56)
where the correlating parameter A is given by the following expression:
A ¼ 7:633 1:497 ln ðpÞ + 1:115 ln ðT Þ + 0:533 ln ðAPIÞ + 0:184 ln Rsp
(4.57)
The authors suggested that the accuracy of Eq. (4.56) can be substantially
improved if the bubble point pressure is known. They improved correlating
parameter, A, by including the bubble point pressure, pb, as one of the
parameters in the preceding equation, to give
A ¼ 7:573 1:45 ln ðpÞ
0:383 ln ðpb Þ + 1:402 ln ðT Þ + 0:256 ln ðAPIÞ + 0:449 ln Rsp
(4.58)
EXAMPLE 4.23
A crude oil system exists at 1650 psi and a temperature of 250°F. The system has
the following PVT properties:
API ¼ 47.1
pb ¼ 2377 psi
γ g ¼ 0.851
γ gs ¼ 0.873
Rsb ¼ 751 scf/STB
Bob ¼ 1.528 bbl/STB
Isothermal Compressibility Coefficient of Crude Oil “co” 293
The laboratory-measured oil PVT data at 1650 psig follow:
Bo ¼ 1.393 bbl/STB
Rs ¼ 515 scf/STB
Bg ¼ 0.001936 bbl/scf
co 324.8 106 psi1
Estimate the oil compressibility using McCain’s correlation, then Eq. (4.55).
Solution
Calculate the correlating parameter A by applying Eq. (4.58):
A ¼ 7:573 1:45 ln ðpÞ 0:383 ln ðpb Þ
+ 1:402 ln ðT Þ + 0:256 ln ðAPIÞ + 0:449 ln ðRsb ÞA
¼ 7:573 1:45 ln ð1665Þ 0:383 ln ð2392Þ
+ 1:402 ln ð710Þ + 0:256 ln ð47:1Þ + 0:449 ln ð451Þ
¼ 8:1445
Solve for co using Eq. (4.56):
co ¼ exp ðAÞ
co ¼ exp ð8:1445Þ ¼ 290:3 106 psi1
Solve for co using Eq. (4.55):
(
)
0:2
rffiffiffiffiffi rffiffiffiffiffi
γg
γg
Rs
co ¼
Bg
0:000144
Rs
+ 1:25ðT 460Þ
Bo ð0:83p + 21:75Þ
γo
γo
co ¼
515
1:393ð0:83ð1665Þ + 21:75Þ
8
9
#0:2
rffiffiffiffiffiffiffiffiffiffiffi"
rffiffiffiffiffiffiffiffiffiffiffi
<
=
0:851
0:851
0:001936
515
+ 1:25ð250Þ
0:000144
:
;
0:792
0:792
co ¼ 358 106 psi1
It should be pointed out that, when it is necessary to establish PVT relationships
for the hydrocarbon system through correlations or by extrapolation below
the bubble point pressure, care should be exercised to see that the PVT functions
are consistent. This consistency is assured if the increase in oil volume with
increasing pressure is less than the decrease in volume associated with the
gas going into solution. Because the oil compressibility coefficient, co, as
expressed by Eq. (4.54) must be positive, that leads to the following consistency
criteria:
@Bo
@Rs
< Bg
@p
@p
This consistency can easily be checked in the tabular form of PVT data. The
PVT consistency errors most frequently occur at higher pressures, where the gas
FVF, Bg, assumes relatively small values.
294 CHAPTER 4 PVT Properties of Crude Oils
UNDERSATURATED OIL PROPERTIES
Fig. 4.11 shows the volumetric behavior of the gas solubility, Rs, oil FVF,
Bo, and oil density, ρo, as a function of pressure. As previously defined, the
gas solubility, Rs, measures the tendency of the gas to dissolve in the oil at a
particular temperature with increasing pressure. This solubility tendency, as
measured in standard cubic feet of gas per one stock-tank barrel of oil,
increases with pressure until the bubble point pressure is reached. With
increasing of the gas solubility in the crude oil, the oil swells with the resulting increase in the oil FVF and an associated decrease in both its density and
viscosity, as shown in Fig. 4.11. With increasing the pressure above the bubble point pressure, pb, the gas solubility remains constant with a maximum
value that is denoted by Rsb (gas solubility at the bubble point pressure).
However, as the pressure increases above pb, the crude oil system experiences a reduction in its volume (Vo)p,T during this isothermal compression.
From the mathematical definition of Bo and ρo,
ðVo Þp, T
ðVo Þsc
mass
ρo ¼
ðVo Þp, T
Bo ¼
These two expressions indicate that, with increasing the pressure above pb,
the associated reduction in the volume of oil (Vo)p,T results in a decrease in
Bo and an increase in ρo.
Rs, Bo, ro, mo
Rs
Bo
Bo = B
ob exp [−
mo
ro
pb
0
Pressure
n FIGURE 4.11 Undersaturated fluid properties vs. pressure.
co (p −
pb )]
Undersaturated Oil Properties 295
The magnitude of the changes in the oil volume above the bubble point pressure is a function of its isothermal compressibility coefficient co. To account
for the effect of oil compression on the oil FVF and oil density, these two
properties are first calculated at the bubble point pressure as Bob and ρob
using any of the methods previously described and then adjusted to reflect
the increase in p over pb.
Undersaturated Oil FVF
The isothermal compressibility coefficient [as expressed mathematically by
Eq. (4.47)] can be equivalently written in terms of the oil FVF:
co ¼
1 @Vo
Vo @p
This relationship can be equivalently expressed in terms of Bo as
1 @ Vo =ðVo Þsc
1 @Bo
co ¼
¼
@p
Vo =ðVo Þsc
Bo @p
(4.59)
This relationship can be rearranged and integrated to produce
ðp
pb
co dp ¼
ð Bo
1
Bob Bo
dBo
(4.60)
Evaluating co at the arithmetic average pressure and concluding the integration procedure to give
Bo ¼ Bob exp ½co ðp pb Þ
(4.61)
where
Bo ¼ oil FVF at the pressure of interest, bbl/STB
Bob ¼ oil FVF at the bubble point pressure, bbl/STB
p ¼ pressure of interest, psia
pb ¼ bubble point pressure, psia
Replacing the isothermal compressibility coefficient in Eq. (4.60) with the
Vasquez-Beggs’s co expression, Eq. (4.51), and integrating the resulting
equation gives
p
Bo ¼ Bob exp A ln
pb
where
A ¼ 105 1433 + 5Rsb + 17:2ðT 460Þ 1180γ gs + 12:61API
(4.62)
296 CHAPTER 4 PVT Properties of Crude Oils
Similarly, replacing co in Eq. (4.60) with the Petrosky and Farshad expression (Eq. (4.52)) and integrating gives
Bo ¼ Bob exp A p0:4094 p0:4094
b
(4.63)
with the correlating parameter A as defined by
A ¼ 4:1646 107 R0:69357
γ 0:1885
ðAPIÞ0:3272 ðT 460Þ0:6729
sb
g
(4.64)
where
T ¼ temperature, °R
p ¼ pressure, psia
Rsb ¼ gas solubility at the bubble point pressure
EXAMPLE 4.24
Using the PVT data given in Example 4.23, calculate the oil FVF at 5000 psig
using Eq. (4.62) then Eq. (4.64). The experimental measured Bo is 1.457 bbl/STB.
Solution Using Eq. (4.62)
Step 1. Calculate parameter A of Eq. (4.62):
h
i
A ¼ 105 1433 + 5Rsb + 17:2ðT 460Þ 1180γ gs + 12:61API
A ¼ 105 ½1433 + 5ð751Þ + 17:2ð250Þ 1180ð0:873Þ + 12:61ð47:1Þ ¼ 0:061858
Step 2. Apply Eq. (4.62):
p
Bo ¼ Bob exp A ln
pb
5015
Bo ¼ 1:528 exp 0:061858 ln
¼ 1:459bbl=STB
2392
Solution Using Eq. (4.64)
Step 1. Calculate the correlating parameter, A, from Eq. (4.64):
γ 0:1885
ðAPIÞ0:3272 ðT 460Þ0:6729
A ¼ 4:1646 107 R0:69357
sb
g
A ¼ 4:1646 107 ð751Þ0:69357 ð0:851Þ0:1885 ð47:1Þ0:3272 ð250Þ0:6729 ¼ 0:005778
Step 2. Solve for Bo by applying Eq. (4.63):
Bo ¼ Bob exp A p0:4094 p0:4094
b
Bo ¼ ð1:528Þ exp 0:005778 50150:4094 23920:4096 ¼ 1:453bbl=STB
Undersaturated Oil Density
Oil density ρo is defined as mass, m, over volume, Vo, at any specified
pressure and temperature, or the volume can be expressed as
Undersaturated Oil Properties 297
Vo ¼
m
ρo
Differentiating density with respect to pressure gives
@Vo
@p
¼
T
m @ρo
ρ2o @p
Substituting these two relations into Eq. (4.47) gives
co ¼
1 @Vo
1 m @ρo
¼
Vo @p ðm=ρo Þ ρ2o @p
or, in terms of density,
co ¼
1 @ρo
ρo @p
Rearranging and integrating yields
ðp
ðp
dρo
pb
pb ρo
ρo
co ðp pb Þ ¼ ln
ρob
co dp ¼
Solving for the density of the oil at pressures above the bubble point pressure,
ρo ¼ ρob exp ½co ðp pb Þ
(4.65)
where
ρo ¼ density of the oil at pressure p, lb/ft3
ρob ¼ density of the oil at the bubble point pressure, lb/ft3
co ¼ isothermal compressibility coefficient at average pressure, psi1
Vasquez-Beggs’s oil compressibility correlation and Petrosky-Farshad’s co
expression can be incorporated in Eq. (4.65) to give the following.
For Vasquez-Beggs’s co equation,
p
ρo ¼ ρob exp A ln
pb
(4.66)
where
A ¼ 105 1433 + 5Rsb + 17:2ðT 460Þ 1180γ gs + 12:61°API
For Petrosky-Farshad’s co expression,
ρo ¼ ρob exp A p0:4094 p0:4094
b
with the correlating parameter A given by Eq. (4.64):
A ¼ 4:1646 107 R0:69357
γ 0:1885
ðAPIÞ0:3272 ðT 460Þ0:6729
sb
g
(4.67)
298 CHAPTER 4 PVT Properties of Crude Oils
TOTAL FVF “Bt”
To describe the pressure-volume relationship of hydrocarbon systems below
their bubble point pressure, it is convenient to express this relationship in
terms of the total FVF as a function of pressure. This property defines
the total volume of a system regardless of the number of phases present.
The total FVF, denoted Bt, is defined as the ratio of the total volume
of the hydrocarbon mixture; that is, oil and gas, if present, at the prevailing
pressure and temperature per unit volume of the stock-tank oil. Because naturally occurring hydrocarbon systems usually exist in either one or two
phases, the term two-phase FVF has become synonymous with the totalformation volume.
Mathematically, Bt is defined by the following relationship:
Bt ¼
ðVo Þp, T + Vg p, T
ðVo Þsc
(4.68)
where
Bt ¼ total FVF, bbl/STB
(Vo)p,T ¼ volume of the oil at p and T, bbl
(Vg)p,T ¼ volume of the liberated gas at p and T, bbl
(Vo)sc ¼ volume of the oil at SC, STB
Note that, when reservoir pressures are greater than or equal to the bubble
point pressure, pb, no free gas will exist in the reservoir and, therefore,
(Vg)p,T ¼ 0. At these conditions, Eq. (4.68) is reduced to the equation that
describes the oil FVF:
Bt ¼
ðVo Þp, T + 0 ðVo Þp, T
¼
¼ Bo
ðVo Þsc
ðVo Þsc
A typical plot of Bt as a function of pressure for an undersaturated crude oil
is shown in Fig. 4.12. The oil FVF curve is also included in the illustration.
As pointed out, Bo and Bt are identical at pressures above or equal to the
bubble point pressure because only one phase, the oil phase, exists at these
pressures. It should also be noted that at pressures below the bubble point
pressure, the difference in the values of the two oil properties represents
the volume of the evolved solution gas as measured at system conditions
per stock-tank barrel of oil.
Consider a crude oil sample placed in a PVT cell at its bubble point pressure,
pb, and reservoir temperature, as shown schematically in Fig. 4.13. Assume
that the volume of the oil sample is sufficient to yield one stock-tank barrel
of oil at SC. Let Rsb represent the gas solubility at pb. By lowering the cell
Total FVF “Bt” 299
Bo or Bt
(Rsb – Rs) Bg
Bt
Bt = B
o=B e
ob xp [−
co (p −
pb )]
Bo
pb
0
Pressure
n FIGURE 4.12 Bo and Bt vs. pressure.
Constant composition expansion
pb
First bubble of gas
p
Evolved solution gas:
(Vg)p,T = (Rsb–Rs) Bg
Oil
Rsb , Bob
14.7 psi
Total volume of
evolved gas at ST
Rsb scf
Bt
Oil
Rs , Bo
Hg
n FIGURE 4.13 The concept of the two-phase formation volume factor.
Hg
ST oil
Bo= 1 STB
Hg
300 CHAPTER 4 PVT Properties of Crude Oils
pressure to p, a portion of the solution gas is evolved and occupies a certain
volume of the PVT cell. Let Rs and Bo represent the corresponding gas solubility and oil FVF at p. Obviously, the term (Rsb Rs) represents the volume of the free gas as measured in scf per stock-tank barrel of the oil. The
volume of the free gas at the cell conditions is then
Vg p, T ¼ ðRsb Rs ÞBg
where (Vg)p,T ¼ volume of the free gas at p and T, bbl of gas/STB of oil, and
Bg ¼ gas FVF, bbl/scf.
The volume of the remaining oil at the cell condition is
ðV Þp, T ¼ Bo
From the definition of the two-phase FVF,
Bt ¼ Bo + ðRsb Rs ÞBg
(4.69)
where
Rsb ¼ gas solubility at the bubble point pressure, scf/STB
Rs ¼ gas solubility at any pressure, scf/STB
Bo ¼ oil FVF at any pressure, bbl/STB
Bg ¼ gas FVF, bbl/scf
It is important to note from the previously described laboratory procedure
that no gas or oil has been removed from the PVT cell with declining pressure; that is, no changes in the overall composition of the hydrocarbon mixture. As described later in this chapter, this is a property determined from the
constant composition expansion test. It should be pointed out that the accuracy of estimating Bt by Eq. (4.69) depends largely on the accuracy of the
other PVT parameters used in the equation: Bo, Rsb, Rs, and Bg.
Several correlations can be used to estimate the two-phase FVF when the
experimental data are not available; three of these methods are detailed below:
n
n
n
Standing’s correlation
Glaso’s correlation
Marhoun’s correlation
Standing’s Correlation
Standing (1947) used 387 experimental data points to develop a graphical
correlation for predicting the two-phase FVF with a reported average
error of 5%. The proposed correlation uses the following parameters for
estimating the two-phase FVF:
Total FVF “Bt” 301
n
n
n
n
n
The gas solubility at pressure of interest, Rs.
Solution gas gravity, γ g.
Oil gravity, γ o, 60°/60°.
Reservoir temperature, T.
Pressure of interest, p.
In developing his graphical correlation, Standing used a combined correlating parameter that is given by
"
log ðA∗Þ ¼ log Rs
# ðT 460Þ0:5 ðγ o ÞC
96:8
10:1 0:3
6:604 + log ðpÞ
γg
(4.70)
with the exponent C as defined by
C ¼ ð2:9Þ100:00027Rs
(4.71)
Whitson and Brule (2000) expressed Standing’s graphical correlation by the
following mathematical form:
log ðBt Þ ¼ 5:223 47:4
log ðA∗ Þ 12:22
(4.72)
Glaso’s Correlation
Experimental data on 45 crude oil samples from the North Sea were used
by Glaso (1980) in developing a generalized correlation for estimating
Bt. Glaso modified Standing’s correlating parameter A* as given by
Eq. (4.70) and used it in a regression analysis model to develop the
following expression for Bt:
log ðBt Þ ¼ 0:080135 + 0:47257log ðA∗Þ + 0:17351½ log ðA∗Þ2
(4.73)
The author included the pressure in Standing’s correlating parameter A*
to give:
"
#
Rs ðT 460Þ0:5 ðγ o ÞC 1:1089
p
A∗ ¼
0:3
γg
(4.74)
with the exponent C as given by Eq. (4.71):
C ¼ ð2:9Þ100:00027Rs
Glaso reported a standard deviation of 6.54% for the total FVF correlation.
302 CHAPTER 4 PVT Properties of Crude Oils
Marhoun’s Correlation
Based on 1556 experimentally determined total FVFs, Marhoun (1988) used
a nonlinear multiple regression model to develop a mathematical expression
for Bt. The empirical equation has the following form:
Bt ¼ 0:314693 + 0:106253 104 F + 0:18883 1010 F2
(4.75)
with the correlating parameter F as given by
F ¼ Ras γ bg γ co T d pe
where T is the temperature in °R and the coefficients a–e as given below:
a ¼ 0.644516
b ¼ 1.079340
c ¼ 0.724874
d ¼ 2.006210
e ¼ 0.761910
Marhoun reported an average absolute error of 4.11% with a standard deviation of 4.94% for the correlation.
EXAMPLE 4.25
Given the following PVT data:
pb ¼ 2744 psia
T ¼ 600°R
γ g ¼ 0.6744
Rs ¼ 444 scf/STB
Rsb ¼ 603 scf/STB
γ sb ¼ 0.843 60°/60°
p ¼ 2000 psia
Bo ¼ 1.1752 bbl/STB
calculate Bt at 2000.7 psia using Eq. (4.69), Standing’s correlation, Glaso’s
correlation, and Marhoun’s correlation.
Solution Using Eq. (4.69)
Step 1. Calculate Tpc and ppc of the solution gas from its specific gravity by
applying Eqs. (3.18), (3.19):
2
Tpc ¼ 168 + 325γ g 12:5 γ g
Tpc ¼ 168 + 325ð0:6744Þ 12:5ð0:6744Þ2 ¼ 381:49°R
2
ppc ¼ 677 + 15γ g 37:5 γ g ¼ 670:06psia
Step 2. Calculate ppr and Tpr:
ppr ¼
p
2000
¼
¼ 2:986
ppc 670:00
Tpr ¼
T
600
¼
¼ 1:57
Tpc 381:49
Total FVF “Bt” 303
Step 3. Determine the gas compressibility factor from Fig. 3.1:
Z ¼ 0:81
Step 4. Calculate Bg from Eq. (3.54):
Bg ¼ 0:00504
ð0:81Þð600Þ
¼ 0:001225bbl=scf
2000
Step 5. Solve for Bt by applying Eq. (4.69):
Bt ¼ Bo + ðRsb Rs ÞBg
Bt ¼ 1:1752 + 0:0001225ð603 444Þ ¼ 1:195bbl=STB
Solution Using Standing’s Correlation
Calculate the correlating parameters C and A* by applying Eqs. (4.71), (4.70),
respectively:
C ¼ ð2:9Þ100:00027Rs
C ¼ ð2:9Þ100:00027ð444Þ ¼ 2:20
2
3
96:8
6 ðT 460Þ ðγ o Þ 7
log ðA∗Þ ¼ log 4Rs
10:1 5
0:3
6:604 + log ðpÞ
γg
"
# ð140Þ0:5 ð0:843Þ2:2
96:8
¼ 3:281
log ðA∗Þ ¼ log ð444Þ
10:1 0:3
6:604
+
log ð2000Þ
ð0:6744Þ
0:5
C
Estimate Bt from Eq. (4.72):
log ðBt Þ ¼ 5:223 47:4
log ðA∗ Þ 12:22
log ðBt Þ ¼ 5:223 47:4
¼ 0:0792
3:281 12:22
to give
Bt ¼ 100:0792 ¼ 1:200bbl=STB
Solution Using Glaso’s Correlation
Step 1. Determine the coefficient C from Eq. (4.71):
C ¼ ð2:9Þ100:00027ð444Þ ¼ 2:2
Step 2. Calculate the correlating parameter A* from Eq. (4.74):
"
#
Rs ðT 460Þ0:5 ðγ o ÞC 1:1089
p
A∗ ¼
0:3
γg
"
#
ð444Þð140Þ0:5 ð0:843Þ2:2
20001:1089 ¼ 0:8873
A∗ ¼
ð0:6744Þ0:3
Step 3. Solve for Bt by applying Eq. (4.73) to yield
2
log ðBt Þ ¼ 0:080135 + 0:47257log A∗ + 0:17351 log A∗
log ðBt Þ ¼ 0:080135 + 0:47257log ð0:8873Þ + 0:17351½ log ð0:8873Þ2 ¼ 0:0561
304 CHAPTER 4 PVT Properties of Crude Oils
to give
Bt ¼ 100:0561 ¼ 1:138
Solution Using Marhoun’s Correlation
Step 1. Determine the correlating parameter F of Eq. (4.75) to give
F ¼ Ras γ bg γ co T d pe
F ¼ 4440:644516 0:67441:07934 0:8430:724874 6002:00621 20000:76191 ¼ 78590:6789
Step 2. Solve for Bt from Eq. (4.75):
Bt ¼ 0:314693 + 0:106253 104 F + 0:18883 1010 F2
Bt ¼ 0:314693 + 0:106253 104 ð78590:6789Þ + 0:18883 1010 ð78590:6789Þ2
Bt ¼ 1:2664bbl=STB
CRUDE OIL VISCOSITY “μO”
Crude oil viscosity is an important physical property that controls the flow
of oil through porous media and pipes. The viscosity, in general, is defined
as the internal resistance of the fluid to flow. It ranges from 0.1 cp for near
critical to over 100 cp for heavy oil. It is considered the most difficult oil
property to calculate with a reasonable accuracy from correlations.
Oil’s viscosity is a strong function of the temperature, pressure, oil gravity,
gas gravity, gas solubility, and composition of the crude oil. Whenever possible, oil viscosity should be determined by laboratory measurements at
reservoir temperature and pressure. The viscosity usually is reported in standard PVT analyses. If such laboratory data are not available, engineers
may refer to published correlations, which usually vary in complexity
and accuracy, depending on the available data on the crude oil. Based on
the available data on the oil mixture, correlations can be divided into the
following two types: correlations based on other measured PVT data, such
as API or Rs, and correlations based on oil composition. Depending on the
pressure, p, the viscosity of crude oils can be classified into three categories:
n
n
Dead oil viscosity, μod. The dead oil viscosity (oil with no gas in the
solution) is defined as the viscosity of crude oil at atmospheric
pressure and system temperature, T.
Saturated oil viscosity, μob. The saturated (bubble point) oil viscosity
is defined as the viscosity of the crude oil at any pressure less than
or equal to the bubble point pressure.
Crude Oil Viscosity “μo” 305
n
Undersaturated oil viscosity, μo. The undersaturated oil viscosity is
defined as the viscosity of the crude oil at a pressure above the bubble
point and reservoir temperature.
The definition of the three categories of oil viscosity is illustrated conceptually in Fig. 4.14. At atmospheric pressure and reservoir temperature, there
is no dissolved gas in the oil (ie, Rs ¼ 0) and therefore the oil has its highest
viscosity value of μod. As the pressure increases, the solubility of the gas
increases accordingly, with the resulting decrease in the oil viscosity. The
oil viscosity at any pressure pb is considered “saturated oil at this p.”
As the pressure reaches the bubble point pressure, the amount of gas in solution reaches its maximum at Rsb and the oil viscosity at its minimum of μob.
With increasing the pressure above pb, the viscosity of the undersaturated
crude oil μo increases with pressure due to the compression of the oil.
Based on these three categories, predicting the oil’s viscosity follows a
similar three-step procedure:
Step 1. Calculate the dead oil viscosity, μod, at the specified reservoir
temperature and atmospheric pressure without dissolved gas:
Rs ¼ 0.
Step 2. Adjust the dead oil viscosity to any specified reservoir pressure
(p pb) according to the gas solubility at p.
Step 3. For pressures above the bubble point pressure, a further adjustment
is made to μob to account for the compression of the oil above pb.
Crude oil viscosity
mod
mod = 0.32 +
A
1.8(107)
360
API1.53
T − 260
0.42 +
A 10
8.33
( API )
Rs
Rs and mo
mob
ob
a
b
od
a = 10.715 (Rs + 100)–0.515
b = 5.44 (Rs + 150)–0.338
pb
0
Pressure
n FIGURE 4.14 Oil viscosity as a function of Rs and p.
306 CHAPTER 4 PVT Properties of Crude Oils
Viscosity Correlations as Correlated with PVT Data
Dead Oil Correlations
Several empirical methods are proposed to estimate the viscosity of the dead
oil that are based on the API gravity of the crude oil and system temperature.
These correlations, as listed below, calculate the viscosity of the dead oil (ie,
no dissolved gas) at reservoir temperature “T ”:
n
n
n
Beal’s correlation
Beggs-Robinson’s correlation
Glaso’s correlation
Discussion of the above listed three methods follows.
Beal’s Correlation
From a total of 753 values for dead oil viscosity at and above 100°F, Beal
(1946) developed a graphical correlation for determining the viscosity of the
dead oil as a function of temperature and the API gravity of the crude.
Standing (1981) expressed the proposed graphical correlation in a mathematical relationship as follows:
18 107
360
μod ¼ 0:32 +
API4:53 T 260
A
(4.76)
with
A ¼ 100:42 + ð API Þ
8:33
where μod ¼ viscosity of the dead oil as measured at 14.7 psia and reservoir
temperature, cp, and T ¼ temperature, °R.
Beggs-Robinson’s Correlation
Beggs and Robinson (1975) developed an empirical correlation for determining the viscosity of the dead oil. The correlation originated from analyzing 460 dead oil viscosity measurements. The proposed relationship is
expressed mathematically as follows:
μod ¼ 10AðT460Þ
1:163
1:0
(4.77)
where
A ¼ 103.03240.02023 API
T ¼ temperature in °R
An average error of 0.64% with a standard deviation of 13.53% was
reported for the correlation when tested against the data used for its development. However, Sutton and Farshad (1984) reported an error of 114.3%
when the correlation was tested against 93 cases from the literature.
Crude Oil Viscosity “μo” 307
Glaso’s Correlation
Glaso (1980) proposed a generalized mathematical relationship for computing the dead oil viscosity. The relationship was developed from experimental measurements on 26 crude oil samples. The correlation has the following
form:
μod ¼ 3:141 1010 ðT 460Þ3:444 ½ log ðAPIÞA
(4.78)
The temperature T is expressed in °R and the coefficient A is given by
A ¼ 10:313½ log ðT 460Þ 36:447
This expression can be used within the range of 50–300°F for the system
temperature and 20–48° for the API gravity of the crude. Sutton and
Farshad (1984) concluded that Glaso’s correlation showed the best accuracy
of the three previous correlations.
Saturated Oil Viscosity
Having calculated the viscosity of the dead crude oil at reservoir temperature using one of the three above correlations, the dead oil viscosity
must be adjusted to account for the solution gas (ie, “Rs”). The following
three empirical methods are proposed to estimate the viscosity of the
saturated oil:
n
n
n
Chew-Connally correlation
Beggs-Robinson correlation
Abu-Khamsin and Al-Marhoun
These correlations are discussed next.
Chew-Connally Correlation
Chew and Connally (1959) presented a graphical correlation to account for
the reduction of the dead oil viscosity due to gas solubility. The proposed
correlation was developed from 457 crude oil samples. Standing (1981)
expressed the correlation in a mathematical form as follows:
μob ¼ ð10a Þðμod Þb
with
a ¼ Rs 2:2 107 Rs 7:4 104
b ¼ 0:68ð10Þc + 0:25ð10Þd + 0:062ð10Þe
c ¼ 0:0000862Rs
d ¼ 0:0011Rs
e ¼ 0:00374Rs
(4.79)
308 CHAPTER 4 PVT Properties of Crude Oils
where μob ¼ viscosity of the oil at the bubble point pressure, cp, and
μod ¼ viscosity of the dead oil at 147.7 psia and reservoir temperature, cp.
The experimental data used by Chew and Connally to develop their correlation
encompassed the following ranges of values for the independent variables:
Pressure: 132–5645 psia
Temperature: 72–292°F
Gas solubility: 51–3544 scf/STB
Dead oil viscosity: 0.377–50 cp
Beggs-Robinson Correlation
From 2073 saturated oil viscosity measurements, Beggs and Robinson
(1975) proposed an empirical correlation for estimating the saturated oil
viscosity. The proposed mathematical expression has the following form:
μob ¼ aðμod Þb
(4.80)
where
a ¼ 10.715(Rs + 100)0515
b ¼ 5.44(Rs + 150)0338
The reported accuracy of the correlation is 1.83% with a standard deviation of 27.25%. The ranges of the data used to develop Beggs and Robinson’s equation are:
Pressure: 132–5265 psia
Temperature: 70–295°F
API gravity: 16–58
Gas solubility: 20–2070 scf/STB
Abu-Khamsin and Al-Marhoun
An observation by Abu-Khamsin and Al-Marhoun (1991) suggests that the
saturated oil viscosity, μob, correlates very well with the saturated oil
density, ρob, as given by
lnðμob Þ ¼ 8:484462ρ4ob 2:652294
where the saturated oil density, ρob, is expressed in g/cm3; that is, ρob/62.4.
Viscosity of the Undersaturated Oil
Oil viscosity at pressures above the bubble point is estimated by first calculating the oil viscosity at its bubble point pressure and adjusted to higher
pressures. Three methods for estimating the oil viscosity at pressures above
saturation pressure are presented next:
Crude Oil Viscosity “μo” 309
n
n
n
Beal’s correlation
Khan’s correlation
Vasquez-Beggs correlation
Beal’s Correlation
Based on 52 viscosity measurements, Beal (1946) presented a graphical correlation for the variation of the undersaturated oil viscosity with pressure
where it has been curve fit by Standing (1981) by
0:56
μo ¼ μob + 0:001ðp pb Þ 0:024μ1:6
ob + 0:038μob
(4.81)
where μo ¼ undersaturated oil viscosity at pressure p and μob ¼ oil viscosity
at the bubble point pressure, cp. The reported average error for Beal’s
expression is 2.7%.
Khan’s Correlation
From 1500 experimental viscosity data points on Saudi Arabian crude oil
systems, Khan et al. (1987) developed the following equation with a
reported absolute average relative error of 2%:
μo ¼ μob exp 9:6 106 ðp pb Þ
(4.82)
Vasquez-Beggs’s Correlation
Vasquez and Beggs proposed a simple mathematical expression for estimating the viscosity of the oil above the bubble point pressure. From 3593 data
points, Vasquez and Beggs (1980) proposed the following expression for
estimating the viscosity of undersaturated crude oil:
μo ¼ μob
where
p
pb
m
(4.83)
A
m ¼ 2:6p1:187
10
A ¼ 3:9 105 p 5
The data used in developing the above correlation have the following
ranges:
Pressure: 141–9151 psia
Gas solubility: 9.3–2199 scf/STB
Viscosity: 0.117–148 cp
Gas gravity: 0.511–1.351
API gravity: 15.3–59.5
The average error of the viscosity correlation is reported as 7.54%.
310 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.26
The experimental PVT data given in Example 4.22 are repeated below. In
addition to this information, viscosity data are presented in the next table. Using
all the oil viscosity correlations discussed in this chapter, calculate μod, μob, and
the viscosity of the undersaturated oil.
Oil
1
2
3
4
5
6
T
250
220
260
237
218
180
pb
2377
2620
2051
2884
3065
4239
Rs
751
768
693
968
943
807
ρo
Bo
1.528
1.474
0.529
1.619
0.57
0.385
co at p > pb
6
22.14 10 at 2689
18.75 106 at 2810
22.69 106 at 2526
21.51 106 at 2942
24.16 106 at 3273
11.65 106 at 4370
38.13
40.95
37.37
38.92
37.7
46.79
psep
Tsep
API
γg
150
100
100
60
200
85
60
75
72
120
60
173
47.1
40.7
48.6
40.5
44.2
27.3
0.851
0.855
0.911
0.898
0.781
0.848
Oil
Dead Oil, μod, @ T
Saturated Oil, μob, cp
Undersaturated Oil, μo @ p
1
2
3
4
5
6
0.765 @ 250°F
1.286 @ 220°F
0.686 @ 260°F
1.014 @ 237°F
1.009 @ 218°F
4.166 @ 180°F
0.224
0.373
0.221
0.377
0.305
0.950
0.281 @ 5000 psi
0.450 @ 5000 psi
0.292 @ 5000 psi
0.414 @ 6000 psi
0.394 @ 6000 psi
1.008 @ 5000 psi
Solution
Dead Oil Viscosity
The results are shown in the table below.
Oil
Measured μod
Beal
Beggs-Robinson
Glaso
1
2
3
4
5
6
AAE
0.765
1.286
0.686
1.014
1.009
4.166
0.322
0.638
0.275
0.545
0.512
4.425
44.9%
0.568
1.020
0.493
0.917
0.829
4.246
17.32%
0.417
0.775
0.363
0.714
0.598
4.536
35.26%
Beal : μod ¼
0:32 +
1:8 107
360
T 260
API4:53
1:163
A
:
Beggs-Robinson : μod ¼ 10AðT460Þ
1:0:
10 3:444
Glaso : μod ¼ 3:141 10 ðT 460Þ
½ log ðAPIÞA :
Crude Oil Viscosity “μo” 311
Saturated Oil Viscosity
The results are shown in the table below.
Oil
1
2
3
4
5
6
AAE
Measured
ρob (g/cm3)
Measured
μob
ChewConnally
BeggsRobinson
AbuKhamsin and
Al-Marhoun
0.6111
0.6563
0.5989
0.6237
0.6042
0.7498
0.224
0.373
0.221
0.377
0.305
0.950
0.313a
0.426
0.308
0.311
0.316
0.842
21%
0.287a
0.377
0.279
0.297
0.300
0.689
17%
0.230
0.340
0.210
0.255
0.218
1.030
14%
Chew-Connally: μob ¼ (10)a (μod)b.
Beggs-Robinson: μob ¼ a(μod)b.
Abu-Khamsin and Al-Marhoun: lnðμob Þ ¼ 8:484462ρ4ob 2:652294.
a
Using the measured μod.
Undersaturated Oil Viscosity
The results are shown in the table below.
Oil
1
2
3
4
5
6
AAE
Measured μo
Beal
Vasquez-Beggs
a
0.281
0.45
0.292
0.414
0.396
1.008
0.273
0.437
0.275
0.434
0.373
0.945
3.8%
0.303a
0.485
0.318
0.472
0.417
1.016
7.5%
0:56
Beals: μo ¼ μob + 0:001ðp pb Þ 0:024μ1:6
ob + 0:038μob .
Vasquez-Beggs: μo ¼ μob ppb
m
.
Using the measured μob.
a
Viscosity Correlations From Oil Composition
Like all intensive properties, viscosity is a strong function of the pressure,
temperature, and composition of the oil, xi. In general, this relationship can
be expressed mathematically by the following function:
μo ¼ f ðp, T, x1 , …, xn Þ
In compositional reservoir simulation of miscible gas injection, the compositions of the reservoir gases and oils are calculated based on the equationof-state approach and used to perform the necessary volumetric behavior of
312 CHAPTER 4 PVT Properties of Crude Oils
the reservoir hydrocarbon system. The calculation of the viscosities of these
fluids from their compositions is required for a true and complete compositional material balance.
Two empirical correlations of calculating the viscosity of crude oils from
their compositions are widely used:
n
n
Lohrenz-Bray-Clark correlation
Little-Kennedy correlation
Lohrenz-Bray-Clark’s Correlation
Lohrenz, Bray, and Clark (1964) (LBC) developed an empirical correlation
for determining the viscosity of the saturated oil from its composition. The
proposed correlation has been enjoying great acceptance and application by
engineers in the petroleum industry. The authors proposed the following
generalized form of the equation:
a1 + a2 ρr + a3 ρ2r + a4 ρ3r + a5 ρ4r
μob ¼ μo +
ξm
4
0:0001
(4.84)
Coefficients a1 through a5 follow:
a1 ¼ 0.1023
a2 ¼ 0.023364
a3 ¼ 0.058533
a4 ¼ 0.040758
a5 ¼ 0.0093324
with the mixture viscosity parameter, ξm, the viscosity parameter, μo, and the
reduced density, ρr, given mathematically by
1=6
5:4402 Tpc
ξm ¼ pffiffiffiffiffiffi 2=3
Ma ppc
Xn
μo ¼
pffiffiffiffiffiffi
xi μi Mi
pffiffiffiffiffiffi
xi Mi
i¼ 1
i¼ 1
Xn
(4.86)
0
where
1
Xn
@
i ¼ 1 ½ðxi Mi Vci Þ + xC7 + VC7 + Aρo
i62C7 +
ρr ¼
Ma
(4.85)
μo ¼ oil viscosity parameter, cp
μi ¼ viscosity of component i, cp
Tpc ¼ pseudocritical temperatures of the crude, °R
(4.87)
Crude Oil Viscosity “μo” 313
ppc ¼ pseudocritical pressure of the crude, psia
Ma ¼ apparent molecular weight of the mixture
ρo ¼ oil density at the prevailing system condition, lb/ft3
xi ¼ mole fraction of component i
Mi ¼ molecular weight of component i
Vci ¼ critical volume of component i, ft3/lb
xC7 + ¼mole fraction of C7+
VC7 + ¼critical volume of C7+, ft3/lb-mol
n ¼ number of components in the mixture
The oil viscosity parameter, μo, essentially represents the oil viscosity at reservoir temperature and atmospheric pressure. The viscosity of the individual
component, μi, in the mixture is calculated from the following relationships:
34 105 ðTri Þ0:94
ξi
5 17:78 10 ½4:58Tri 1:670:625
when Tri 1:5 : μi ¼
ξi
when Tri 1:5 : μi ¼
(4.88)
(4.89)
where
Tri ¼ reduced temperature of component i; T=Tci
5:4402ðTci Þ1=6 (4.90)
ξi ¼ viscosity parameter of component i, given by ξi ¼ pffiffiffiffiffiffi
Mi ðpci Þ2=3
It should be noted that, when applying Eq. (4.86), the viscosity of the nth
component, C7+, must be calculated by any of the dead oil viscosity correlations discussed before.
Lohrenz and coworkers proposed the following expression for
calculating VC7 + :
VC7 + ¼ a1+ a2 MC7 + + a3 γ C7 + + a4 MC7 + γ C7 +
(4.91)
where
MC7 + ¼ molecular weight of C7+
γ C7 + ¼ specific gravity of C7+
a1 ¼ 21.572
a2 ¼ 0.015122
a3 ¼ 27.656
a4 ¼ 0.070615
Experience with the Lohrenz et al. equation has shown that the correlation is
extremely sensitive to the density of the oil and the critical volume of C7+.
When observed viscosity data are available, values of the coefficients a1–a5
314 CHAPTER 4 PVT Properties of Crude Oils
in Eq. (4.84) and the critical volume of C7+ usually are adjusted and used as
tuning parameters until a match with the experimental data is achieved.
Little-Kennedy’s Correlation
Little and Kennedy (1968) proposed an empirical equation for predicting the
viscosity of the saturated crude oil. The correlation was originated from
studying the behavior of 828 distinct crude oil systems representing 3349
viscosity measurements.
The equation is similar in form to van der Waals’s equation of state (EOS).
The authors expressed the equation in the following form:
μ3ob p 2
am
am bm
μ bm + μob +
¼0
T ob
T
T
(4.92)
where
μob ¼ viscosity of the saturated crude oil, cp
p ¼ system pressure, psia
T ¼ system temperature, °R
am, bm ¼ mixture coefficient parameters
Little and Kennedy proposed the following relationships for calculating the
parameters am and bm:
am ¼ exp ðAÞ
(4.93)
bm ¼ exp ðBÞ
(4.94)
The authors correlated the coefficients A and B with
Temperature, T.
Molecular weight of the C7+.
Specific gravity of the C7+.
Apparent molecular weight of the crude oil, Ma.
Density of the oil, ρo, at p and T.
The coefficients are given by the following equations:
A ¼ Ao +
A1
A3 MC7 + A4 ρo A5 ρ2o
+
+ A2 MC7 + +
+ 2 + A6 Ma + A7 Ma3
T
γ C7 +
T
T
+ A8 Ma ρo + A9 ðMa ρo Þ
3
(4.95)
+ A10 E2o
B5 M 4
B1 B2
B6 ρ4
+ 4 + B3 γ 3C7 + + B4 γ 4C7 + + 4 C7 + + 4 o + B7 Ma + B8 Ma ρo
T T
T
γ C7 +
+ B9 ðMa ρo Þ4 + B10 ρ3o + B11 ρ4o
(4.96)
B ¼ Bo +
Surface/Interfacial Tension “σ” 315
where
MC7 + ¼ molecular weight of C7+
γ C7 + ¼ specific gravity of C7+
ρo ¼ density of the saturated crude oil, lb/ft3
T ¼ temperature, °R
Ma ¼ apparent molecular weight of the oil
A0–A10, B0–B11 ¼ coefficients of Eqs. (4.95), (4.96) are tabulated below.
A0 ¼ 21:918581
B0 ¼ 2:6941621
A1 ¼ 16,815:621
B1 ¼ 3757:4919
A2 ¼ 0:0233159830
B2 ¼ 0:31409829 1012
A3 ¼ 0:0192189510
B3 ¼ 33:744827
A4 ¼ 479:783669
B4 ¼ 31:333913
A5 ¼ 719:808848
A6 ¼ 0:0968584490
A7 ¼ 0:5432455400 106
B5 ¼ 0:24400196 1010
B6 ¼ 4:632634
B7 ¼ 0:0370221950
A8 ¼ 0:0021040196
B8 ¼ 0:0011348044
A9 ¼ 0:4332274341 1011 B9 ¼ 0:0547665355 1015
3 A10 ¼ 0:0081362043
B10 ¼ 0:0893548761 10
B11 ¼ 2:0501808400 106
Eq. (4.92) can be solved for the viscosity of the saturated crude oil and the
pressure and temperature of interest by extracting the minimum real root of
the equation. An iterative technique, such as the Newton-Raphson iterative
method, can be employed to solve the proposed equation.
SURFACE/INTERFACIAL TENSION “σ”
The surface tension is defined as the force exerted on the boundary layer
between a liquid phase and a vapor phase per unit length. This force is
caused by differences between the molecular forces in the vapor phase
and those in the liquid phase and also by the imbalance of these forces at
the interface. The surface can be measured in the laboratory and usually
is expressed in dynes per centimeter. The surface tension is an important
property in reservoir engineering calculations and designing enhanced oil
recovery (EOR) projects.
Sugden (1924) suggested a relationship that correlates the surface tension of
a pure liquid in equilibrium with its own vapor. The correlating parameters
of the proposed relationship are the molecular weight, M, of the pure component; the densities of both phases; and a newly introduced temperature
316 CHAPTER 4 PVT Properties of Crude Oils
independent parameter, Pch. The relationship is expressed mathematically in
the following form:
σ¼
Pch ðρL ρv Þ 4
M
(4.97)
where σ is the surface tension and Pch is a temperature independent parameter, called the parachor.
The parachor is a dimensionless constant characteristic of a pure compound,
calculated by imposing experimentally measured surface tension and density data on Eq. (4.97) and solving for Pch. The parachor values for a selected
number of pure compounds are given in the table below, as reported by
Weinaug and Katz (1943).
Component
Parachor
Component
Parachor
CO2
N2
C1
C2
C3
i-C4
49.0
41.0
77.0
108.0
150.3
181.5
n-C4
i-C5
n-C5
n-C6
n-C7
n-C8
189.9
225.0
231.5
271.0
312.5
351.5
Fanchi (1985) correlated the parachor with molecular weight with a simple
linear equation. This linear equation is valid only for components heavier
than methane. Fanchi’s linear equation has the following form:
ðPch Þi ¼ 69:9 + 2:3Mi
(4.98)
where Mi ¼ molecular weight of component i and (Pch)i ¼ parachor of
component i.
Firoozabadi et al. (1988) developed a correlation that can be used to approximate the parachor of pure hydrocarbon fractions from C1 through C6 and for
C7+ fractions:
ðPch Þi ¼ 11:4 + 3:23Mi 0:0022ðMi Þ2
Katz and Saltman (1939) suggested the following expression for estimating
the parachor for C7+ fractions as
ðPch ÞC7 + ¼ 25:2 + 2:86MC7 +
For a complex hydrocarbon mixture, Weinaug and Katz (1943) employed the
Sugden correlation for mixtures by introducing the compositions of the two
phases into Eq. (4.97). The modified expression has the following form:
σ 1=4 ¼
n X
ðPch Þi ðAxi Byi Þ
i¼1
(4.99)
Surface/Interfacial Tension “σ” 317
with the parameters A and B defined by
ρo
62:4Mo
ρg
B¼
62:4Mg
A¼
where
ρo ¼ density of the oil phase, lb/ft3
Mo ¼ apparent molecular weight of the oil phase
ρg ¼ density of the gas phase, lb/ft3
Mg ¼ apparent molecular weight of the gas phase
xi ¼ mole fraction of component i in the oil phase
yi ¼ mole fraction of component i in the gas phase
n ¼ total number of components in the system
The interfacial tension varies between 72 dynes/cm for a water/gas system
to 20–30 dynes/cm for a water/oil system. Ramey (1973) proposed a graphical correlation for estimating water/hydrocarbon interfacial tension that
was curve fit by Whitson and Brule (2000) by the following expression:
σ wb ¼ 20 + 0:57692ðρw ρb Þ
where
σ w-h ¼ interfacial tension, dynes/cm
ρw ¼ density of the water phase, lb/ft3
ρh ¼ density of the hydrocarbon phase, lb/ft3
EXAMPLE 4.27
The composition of a crude oil and the associated equilibrium gas follows. The
reservoir pressure and temperature are 4000 psia and 160°F, respectively.
Component
xi
yi
C1
C2
C3
n-C4
n-C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
0.77
0.08
0.06
0.04
0.02
0.02
0.01
The following additional PVT data are available:
Oil density, ρo ¼ 46.23 lb/ft3
Gas density, ρg ¼ 18.21 lb/ft3
Molecular weight of C7+ ¼ 215
Calculate the surface tension.
318 CHAPTER 4 PVT Properties of Crude Oils
Solution
Step 1. Calculate the apparent molecular weight of the liquid and gas phases:
X
xi Mi ¼ 100:253
Mo ¼
X
Mg ¼
yi Mi ¼ 24:99
Step 2. Calculate the coefficients A and B:
A¼
ρo
46:23
¼
¼ 0:00739
62:4Mo ð62:4Þð100:253Þ
B¼
ρg
18:21
¼
¼ 0:01168
62:4Mg ð62:6Þð24:99Þ
Step 3. Calculate the parachor of C7+ from Eq. (4.98):
ðPch Þi ¼ 69:9 + 2:3Mi
ðPch ÞC7 + ¼ 69:9 + ð2:3Þð215Þ ¼ 564:4
Step 4. Construct a working table as shown in the table below.
Component
Pch
Axi 5 0.00739 xi
Byi 5 0.01169 yi
Pch(Axi 2 Byi)
C1
C2
C3
n-C4
n-C5
C6
C7+
77.0
108.0
150.3
189.9
231.5
271.0
564.4
0.00333
0.00037
0.00037
0.00022
0.00007
0.000074
0.00296
0.0090
0.00093
0.00070
0.00047
0.00023
0.00023
0.000117
0.4361
0.0605
0.0497
0.0475
0.0370
0.0423
P1.6046
¼ 0.9315
Step 5. Calculate the surface tension from Eq. (4.99):
σ 1=4 ¼
n X
ðPch Þi ðAxi Byi Þ ¼ 0:9315
i¼1
σ ¼ ð0:9315Þ4 ¼ 0:753dynes=cm
PVT CORRELATIONS FOR GOM OIL
Dindoruk and Christman (2004) developed a set of PVT correlations for
GOM crude oil systems. These correlations have been developed for the following properties:
n
n
n
Bubble point pressure.
Solution GOR at bubble point pressure.
Oil FVF at bubble point pressure.
PVT Correlations for GOM Oil 319
n
n
n
n
Undersaturated isothermal oil compressibility.
Dead oil viscosity.
Saturated oil viscosity.
Undersaturated oil viscosity.
In developing their correlations, the authors used the solver tool built into
Microsoft Excel to match data on more than 100 PVT reports from the
GOM. Dindoruk and Christman suggest that the proposed correlations
can be tuned for other basins and areas or certain classes of oils. The
authors’ correlations, as briefly summarized here, employ the following
field units:
Temperature, T, in 0176°R.
Pressure, p, in psia.
FVF, Bo, in bbl/STB.
Gas solubility, Rs, in scf/STB.
Viscosity, μo, in cp.
For the bubble point pressure correlation,
a9
R
pb ¼ a8 as10 10A + a11
γg
The correlating parameter A is given by
a1 T a2 + a3 APIa4
A¼ 2Ra6 2
as + as7
γg
The pb correlation coefficients a1–a11 are
a1 ¼ 1.42828(1010)
a2 ¼ 2.844591797
a3 ¼ 6.74896(104)
a4 ¼ 1.225226436
a5 ¼ 0.033383304
a6 ¼ 0.272945957
a7 ¼ 0.084226069
a8 ¼ 1.869979257
a9 ¼ 1.221486524
a10 ¼ 1.370508349
a11 ¼ 0.011688308
For the gas solubility correlation, Rsb,
Rsb ¼
a11
pb
+ a9 γ ag10 10A
a8
320 CHAPTER 4 PVT Properties of Crude Oils
with
A¼
a1 APIa2 + a3 T a4
a
6
a5 + 2API
pa7
2
b
The coefficients (Rsb correlation) a1–a11 are
a1 ¼ 4.86996(106)
a2 ¼ 5.730982539
a3 ¼ 9.92510(103)
a4 ¼ 1.776179364
a5 ¼ 44.25002680
a6 ¼ 2.702889206
a7 ¼ 0.744335673
a8 ¼ 3.359754970
a9 ¼ 28.10133245
a10 ¼ 1.579050160
a11 ¼ 0.928131344
For the oil FVF (Bob),
Bob ¼ a11 + a12 A + a13 A2 + a14 ðT 60Þ
API
γg
with the parameter A given by
h a1 a1
A¼
Rs γ g
a
γ o3
+ a4 ðT 60Þa5 + a6 Rs
h
i2
a9
s
a8 + 2R
ðT 60Þ
a
γ 10
ia7
g
The coefficients a1–a14 for the proposed Bob correlation are
a1 ¼ 2.510755(100)
a2 ¼ 4.852538(100)
a3 ¼ 1.183500(101)
a4 ¼ 1.365428(105)
a5 ¼ 2.252880(100)
a6 ¼ 1.007190(101)
a7 ¼ 4.450849(101)
a8 ¼ 5.352624(100)
a9 ¼ 6.308052(101)
a10 ¼ 9.000749(101)
a11 ¼ 9.871766(107)
a12 ¼ 7.865146(104)
a13 ¼ 2.689173(106)
a14 ¼ 1.100001(105)
PVT Correlations for GOM Oil 321
When the pressure is above the bubble point pressure, the authors proposed
the following expression for undersaturated isothermal oil compressibility:
co ¼ a11 + a12 A + a13 A2 106
h a1 a2
with
A¼
Rs γ g
a
γ o3
+ a4 ðT 60Þa5 + a6 Rs
2
a9
a8 + 2Ra10s ðT 60Þ
ia7
γg
The coefficients a1–a13 of the proposed co correlation are
a1 ¼ 0.980922372
a2 ¼ 0.021003077
a3 ¼ 0.338486128
a4 ¼ 20.00006368
a5 ¼ 0.300001059
a6 ¼ 0.876813622
a7 ¼ 1.759732076
a8 ¼ 2.749114986
a9 ¼ 1.713572145
a10 ¼ 9.999932841
a11 ¼ 4.487462368
a12 ¼ 0.005197040
a13 ¼ 0.000012580
Normally, the dead oil viscosity (μod) at reservoir temperature is expressed
in terms of reservoir temperature, T, and the oil API gravity. However, the
authors introduced two additional parameters: the bubble point pressure, pb,
and gas solubility, Rsb, at the bubble point pressure. Dindoruk and Christman justify this approach because the same amount of solution gas can cause
different levels of bubble point pressures for paraffinic and aromatic oils.
The authors point out that using this methodology will capture some information about the oil type without requiring additional data. The proposed
correlation has the following form:
μod ¼
a3 Ta4 ð log APIÞA
a5 pab6 + a7 Rasb8
with
A ¼ a1 log ðT Þ + a2
The coefficients (μod correlation) a1–a8 for the dead oil viscosity are
a1 ¼ 14.505357625
a2 ¼ 44.868655416
322 CHAPTER 4 PVT Properties of Crude Oils
a3 ¼ 9.36579(109)
a4 ¼ 4.194017808
a5 ¼ 3.1461171(109)
a6 ¼ 1.517652716
a7 ¼ 0.010433654
a8 ¼ 0.000776880
For the saturated oil viscosity (μob),
μob ¼ Aðμod ÞB
with the parameters A and B given by
a1
a3 Ras 4
+
exp ða2 Rs Þ exp ða5 Rs Þ
a6
a8 Ras 9
B¼
+
exp ða7 Rs Þ exp ða10 Rs Þ
A¼
with the following coefficients a1–a10 of the proposed saturated oil viscosity
correlation:
a1 ¼ 1.000000(100)
a2 ¼ 4.740729(104)
a3 ¼ 1.023451(102)
a4 ¼ 6.600358(101)
a5 ¼ 1.075080(103)
a6 ¼ 1.000000(100)
a7 ¼ 2.191172(105)
a8 ¼ 1.660981(102)
a9 ¼ 4.233179(101)
a10 ¼ 2.273945(104)
For the undersaturated oil viscosity (μo),
μo ¼ μob + a6 ðp pb Þ10A
with
A ¼ a1 + a2 logμob + a3 log ðRs Þ + a4 μob log ðRs Þ + a5 ðp pb Þ
The coefficients (μo correlation) a1–a6 are
a1 ¼ 0.776644115
a2 ¼ 0.987658646
a3 ¼ 0.190564677
a4 ¼ 0.009147711
a5 ¼ 0.0000191111
a6 ¼ 0.000063340
Properties of Reservoir Water 323
PROPERTIES OF RESERVOIR WATER
Like hydrocarbon systems, the properties of the formation water depend on
the pressure, temperature, and composition of the water. The compressibility of the connate water contributes materially in some cases to the production of undersaturated volumetric reservoirs and accounts for much of the
water influx in water drive reservoir. The formation water usually contains
salts, mainly sodium chloride (NaCl), and dissolved gas that affects its properties. Generally, formation water contains a much higher concentration of
NaCl, ranging from 10,000 to 300,000 ppm, than the seawater concentration
of approximately 30,000 ppm. Haas (1976) points out that there is a limit
on the maximum concentration of salt in the water that is given by
ðCsw Þmax ¼ 262:18 + 72:0T + 1:06T 2
where
(Csw)max ¼ maximum salt concentration, ppm by weight
T ¼ temperature, °C
In addition to describing the salinity of the water in terms of ppm by weight,
it can be also expressed in terms of the weight fraction, ws, or as ppm by
volume, Csv. The conversion rules between these expressions are
ρw Csw
62:4
Csw ¼ 106 ws
Csv ¼
where ρw is the density of the brine at SC in lb/ft3 and can be estimated from
the Rowe and Chou (1970) correlation as
ρw ¼
1
0:01604 0:011401ws + 0:0041755w2s
The solubility of gas in water is normally less than 30 scf/STB; however,
when combined with the concentration of salt, it can significantly affect
the properties of the formation water. Methods of estimating the PVT
properties, as documented next, are based on reservoir pressure, p, reservoir
temperature, T, and salt concentration, Csw.
Water FVF
The water FVF can be calculated by the following mathematical expression
(Hewlett-Packard, Petroleum Fluids PAC Manual, H.P. 41C, 1982):
Bw ¼ A1 + A2 p + A3 p2
(4.100)
where the coefficients A1–A3 are given by the following expression:
Ai ¼ a1 + a2 ðT 460Þ + a3 ðT 460Þ2
324 CHAPTER 4 PVT Properties of Crude Oils
with a1–a3 for gas-free and gas-saturated water given in the table below.
The temperature T in Eq. (4.100) is in °R.
Ai
a1
Gas-free water
0.9947
A1
4.228(106)
A2
1.3(1010)
A3
Gas-saturated water
0.9911
A1
1.093(106)
A2
5.0(1011)
A3
a2
a3
5.8(106)
1.8376(108)
1.3855(1012)
1.02(106)
6.77(1011)
4.285(1015)
6.35(105)
3.497(109)
6.429(1013)
8.5(107)
4.57(1012)
1.43(1015)
Source: Hewlett-Packard, Petroleum Fluids PAC Manual, H.P. 41C, 1982.
McCain et al. (1988) developed the following correlation for calculating Bw:
Bw ¼ ð1 + ΔVwt Þ 1 + ΔVwp
with
ΔVwt ¼ 0:01 + 1:33391 104 TF + 5:50654 107 TF2
ΔVwp ¼ pTF 1:95301 109 1:72834 1013 p
p 3:58922 107 + 2:25341 1010 p
where
TF ¼ temperature, °F
p ¼ pressure, psi
The correlation does not account for the salinity of the water; however,
McCain et al. observed that variations in salinity caused offsetting errors
in terms ΔVwt and ΔVwp.
Water Viscosity
The viscosity of the water is a function of the pressure, temperature, and
salinity. Ahmed (2014) proposed a pure water viscosity correlation that
accounts for both the effects of temperature and pressure. Ahmed proposed
the following expression for calculating pure water viscosity (salinity ¼ 0) at
1 atm and temperature “TF”:
0:763526623
μpure
+
w, 1 atm ¼ 30:86543156TF
0:07909888
0:234919042
TF2
To account for the impact of including the salinity on pure water viscosity,
the following simplified expression is proposed:
Properties of Reservoir Water 325
μbrine
w, 1 atm ¼
5167847:4μpure
w, 1 atm
0:234919042wS
517818:4 wS
where
(μw,1atm)pure ¼ pure water viscosity at 1 atm and TF, cp
(μw,1atm)brine ¼ brine water viscosity at 1 atm and TF, cp
wS ¼ concentration of NaCl as expressed in parts per million “ppm”
The water viscosity at pressure “p” is calculated from the following
relationship:
μw ¼ μbrine
w, 1 atm exp ½0:000055p
(4.101)
with
μwD ¼ A + B=TF
A ¼ 4:518 102 + 9:313 107 Y 3:93 1012 Y 2
B ¼ 70:634 + 9:576 1010 Y 2
where
μw ¼ water viscosity at p and T, cp
μw,1atm ¼ water viscosity at p ¼ 14.7 and reservoir temperature in °F, cp
μwD ¼ dead water viscosity, cp
p ¼ pressure of interest, psia
TF ¼ temperature of interest, °F
Brill and Beggs (1973) presented a simpler equation, which considers only
the temperature effects:
μw ¼ exp 1:003 0:01479TF + 1:982 105 TF2
(4.102)
McCain et al. (1988) accounted for the salinity and developed the following
correlation for estimating the water viscosity at atmospheric pressure,
reservoir temperature, and water salinity:
μw1 ¼ ATFB
with
A ¼ 109:574 0:0840564ws + 0:313314ðws Þ2 + 0:00872213ðws Þ3
B ¼ 1:12166 + 0:0263951ws 0:000679461ðws Þ2 0:0000547119ðws Þ3
+ 1:55586 106 ðws Þ4
where
TF ¼ temperature, °F
μwl ¼ water viscosity at 1 atm and reservoir temperature, cp
ws ¼ water salinity, percent by weight, solids
326 CHAPTER 4 PVT Properties of Crude Oils
McCain et al. adjusted μw1 to account for the increase of the pressure from
1 atm to p by the following correlation:
μw
¼ 0:9994 + 0:0000450295p + 3:1062 109 p2
μw1
Gas Solubility in Water
The gas solubility in pure saltwater (Rsw)pure, as expressed in scf/STB, can
be estimated from the following correlation:
ðRsw Þpure ¼ A + Bp + Cp2
(4.103)
where
A ¼ 2:12 + 3:45 103 TF 3:59 105 TF2
5 7 2
B ¼ 0:0107 5:26 10 TF + 1:48 10 TF
C ¼ 8:75 107 + 3:9 109 TF 1:02 1011 TF2
p ¼ pressure, psi
To account for the salinity of the water, which reduces the tendency of the
gas to dissolve in the water, McKetta and Wehe (1962) proposed the following adjustment:
Rsw ¼ D ðRsw Þpure
where
0:285854
D ¼ ws 100:0840655TF
Water Isothermal Compressibility
Brill and Beggs (1973) proposed the following equation for estimating water
isothermal compressibility, ignoring the corrections for dissolved gas and
solids:
cw ¼ C1 + C2 T + C3 TF2 106
where
C1 ¼ 3.8546 0.000134p
C2 ¼ 0.01052 + 4.77 107p
C3 ¼ 3.9267 105 8.8 1010p
p ¼ pressure in psia
cw ¼ water compressibility coefficient in psi1
(4.104)
Laboratory Analysis of Reservoir Fluids 327
McCain (1991) suggested the following approximation for water isothermal
compressibility:
cw ¼
1
7:033p + 541:55Csw 537:0TF + 403, 300
where Csw is the salinity of the water in mg/L.
LABORATORY ANALYSIS OF RESERVOIR FLUIDS
Accurate laboratory studies of PVT and phase-equilibria behavior of reservoir
fluids are necessary for characterizing these fluids and evaluating their volumetric performance at various pressure levels by conducting laboratory tests
on a reservoir fluid sample. The amount of data desired determines the number
of tests performed in the laboratory; however, for a successful PVT analysis it
requires fluid samples that represent the original hydrocarbon in the reservoir.
Fluid sampling for laboratory PVT analysis must be collected immediately
after the exploration wells are drilled to properly characterize the original
hydrocarbon in place. Sampling techniques depend on the reservoir.
This chapter focuses on discussing well conditioning procedures and fluid
sampling methods, reviewing the PVT laboratory tests and the proper use of
the information contained in PVT reports, and describing details of several
of the routine and special laboratory tests.
Well Conditioning and Fluid Sampling
Improved estimates of the PVT properties of the reservoir fluids at measure
in the laboratory can be made by obtaining samples that are representative of
the reservoir fluids. The proper sampling of the fluids is of the greatest
importance in securing accurate data. A prerequisite step for obtaining a representative fluid sample is a proper well conditioning.
Well Conditioning
Proper well conditioning is essential to obtain representative samples from
the reservoir. The best procedure is to use the lowest rate that results in smooth
well operation and the most dependable measurements of surface products.
Minimum drawdown of bottom-hole pressure during the conditioning period
is desirable and the produced gas-liquid ratio should remain constant (within
about 2%) for several days; less-permeable reservoirs require longer periods.
The farther the well deviates from the constant produced gas/liquid ratio, the
greater the likelihood that the samples will not be representative.
328 CHAPTER 4 PVT Properties of Crude Oils
The following procedure is recommended in conditioning of an oil well for
subsurface sampling. Before collecting fluid samples, the following guidelines and well condition consecutive steps must be considered:
1. The tested well should be allowed to produce for a sufficient amount of
time to remove the drilling fluids, acids, and other well stimulation
materials.
2. After the cleaning period, the flow rate should be reduced to one-half the
flow rate used during the cleaning period.
3. The well should be allowed to flow at this reduced rate for at least
24 h. However, for tight formation or viscous oil, the reduced flow-rate
period must be extended to 48 h or higher to stabilize the various
parameters monitored.
4. The pressure drawdown should be controlled and remain low to
ensure that bottom-hole pressure does not fall below the saturation
pressure.
5. During the reduced flow-rate period, the well should be closely
monitored to establish when the wellhead pressure, production rate, and
GOR have stabilized.
Fluid Sampling
The main objective of a successful sampling process is to obtain representative fluid samples for determining PVT properties. For proper definition
of the type of reservoir fluid and to perform a proper fluid study, the collection of reservoir fluid samples must occur before the reservoir pressure
is allowed to deplete below the saturation pressure of the reservoir fluid.
Therefore, it is essential that the reservoir fluid samples be collected immediately after the hydrocarbon discovery, and the only production before
sampling be as a result of well cleanout and to remove contamination.
The critical steps in any successful sampling program include
n
n
n
n
Avoiding two-phase flow in the reservoir; that is, bottom-hole pressure
below the saturation pressure.
Minimizing fluid contamination introduced by drilling and completion
fluids.
Obtaining adequate volumes.
Preserving sample integrity.
It should be pointed out that the resulting definition of the type of reservoir
fluid, the overall quality of the study, and the subsequent engineering calculations based on that study are no better than the quality of the fluid samples originally collected during the field sampling process.
Laboratory Analysis of Reservoir Fluids 329
There are three basic methods of sample collection:
n
n
n
Subsurface (bottom-hole) sampling
Surface (separator) sampling
Wellhead sampling
Surface sampling from the wellhead, separator, and stock-tank is performed
routinely during most well tests and is occasionally required from production process lines.
A thorough review of the equipment and techniques used to obtain these
different types of fluid samples is outside the scope of this book. Individuals
who are interested in more extensive information on sampling procedures
should refer to information available from equipment vendors and service
companies that specialize in sampling; for example, Schlumberger,
Haliburton, Baker Hughes, and others. However, a brief review of the most
common sampling techniques will be useful to establish the fundamental
principles that will be discussed later in this chapter.
Subsurface Samples
Subsurface sampling uses a sampler tool (eg, Schlumberger Repeat Formation Testing “RFT” or Modular Dynamic Testing tool “MDT”) that is run on
a wireline to the reservoir depth. After lowering the sampling chamber to the
top of the producing zone, the chamber is opened hydraulically or via an
electronic signal from the surface. The crude oil is allowed to flow at a very
low flow rate into the tool at constant pressure to avoid the liberation of the
solution gas. A piston seals the chamber and the tool is brought to the
surface.
It should be noted that sampling of saturated reservoirs with this technique
requires special care to obtain a representative sample because when the
flowing bottom-hole pressure is lower than the bubble point, the validity
of the sample remains doubtful. Multiple subsurface samples usually are
taken by running sample chambers in cycle or performing repeat runs.
The samples are checked for consistency by measuring their bubble point
pressure in surface temperatures. Samples whose bubble point lies within
2% of each other are considered reliable and may be sent to the laboratory
for PVT analysis. To obtain a bottom-hole sample, a period of reduced flow
rate that generally lasts from 2 to 4 days is required to remove drilling fluids
contamination and mud filtrates before sampling. After the reduced flowrate period, the well would be shut-in and allowed to reach static pressure.
This shut-in period generally lasts from a day up to a week of more, based
primarily on the permeability. The following guidelines are suggested:
330 CHAPTER 4 PVT Properties of Crude Oils
1. Bottom-hole sampling is recommended for all oil reservoirs, and may be
successful with rich gas-condensate fluids.
2. A minimum of three fluid samples of approximately 600 cm3 each
should be collected to ensure that at least one sample is valid.
3. Subsurface sampling generally is not recommended for gas-condensate
reservoirs or oil reservoirs producing substantial quantities of water.
4. A large water column in the tubing of a shut-in oil well prevents
sampling at the proper depth and usually creates a situation where the
collection of representative subsurface fluid is impossible.
5. A static pressure gradient should be run and interpreted to determine
the gas-oil level and oil-water level in the tubing. The sampler
should not be lowered below the oil/water interface.
6. A surface sample must be collected in the case of downhole twophase fluid.
It should be pointed out that wellbore frequently contains water even in
wells that do not experience water production. Age water column in the tubing of a shut-in well might prevent sampling and obtaining a representative
sample at the proper depth. For this reason, a static pressure gradient should
be performed and interpreted to determine levels of the gas-oil and oil-water
in the tubing, which is normally allowed to reach static pressure, as shown in
Fig. 4.15.
Pressure (psig)
2500
0
3000
3500
4000
4500
2000
4000
Depth
Oil level
6000
8000
Water level
10,000
12,000
n FIGURE 4.15 Static pressure gradient.
5000
Pressure Depth Gradient
(psig)
(ft)
(psi/ft)
2670
0
2720
2000
0.025
2770
4000
0.025
2830
5000
0.06
3165
6000
0.335
3497
7000
0.332
3827
8000
0.33
4157
9000
0.33
4291
9400
0.335
4370
9600
0.395
4458
9800
0.44
4547 10,000
0.445
Laboratory Analysis of Reservoir Fluids 331
Separator Samples
Separator recombination samples are often the only available representative
samples. In these circumstances, accurate measurement of hydrocarbon gas
and liquid production rates and stability of separation conditions are critical
because the laboratory tests will be based on fluid compositions recombined
in the same ratios as the hydrocarbon streams measured in the field.
The original reservoir fluid cannot be simulated in the laboratory unless
accurate field measurements of all the separator streams are taken. Surface
sampling involves taking samples of the gas and liquid flowing through the
surface separator, as conceptually shown in Fig. 4.16, and recombining the
two fluids in an appropriate ratio in such a way that the recombined sample
is representative of the overall reservoir fluid stream. Separator pressure and
temperatures should remain as constant as possible during the well conditioning period; this will help maintain constancy of the stream rates and thus
the observed hydrocarbon gas/liquid ratio.
GOR, scf/STB
Must be converted to:
GOR, scf/bbl
psep
psep
Gas
meter
Separator liquid
sample cylinder
Separator gas
Gas
sample sample cylinder
Liquid
sample
First stage separator
n FIGURE 4.16 Separator samples.
332 CHAPTER 4 PVT Properties of Crude Oils
The oil and gas samples are taken from the appropriate flow lines of
the same separator, whose pressure, temperature, and flow rate must
be carefully recorded to allow the recombination ratios to be calculated.
The oil and gas samples are sent separately to the laboratory, where
they are recombined before the PVT analysis is performed. Recombining separator samples requires thorough understanding of the relationship between gas solubility “Rs” and GOR. The surface GOR is
traditionally defined as the ratio of the instantaneous gas rate to that
of the oil rate, or
GOR ¼
total gas rate liberated solution gas rate + free gas rate
¼
oil rate
oil rate
GOR ¼
Qo Rs + Qg
Qg
¼ Rs +
Qo
Qo
Darcy’s equation for gas and oil flow rates is given by
Qo ¼
Qg ¼
0:00708hkkrog ðpe pwf Þ
re
μo Bo ln
rw
0:00708hkkrg ðpe pwf Þ
re
μg Bg ln
rw
To give the instantaneous GOR,
"
#
krg Bo μo
GOR ¼ Rs +
kro Bg μg
The above set of mathematical relationships describing the relationship
between GOR and gas solubility “Rs” suggest that a representative sample,
surface or bottom-hole sample, can be obtained if
The well is producing at a STABILIZED GOR; that is,
GOR ¼ constant
(b) The reservoir pressure and pwf are both pb, which suggests a single
phase is flowing; that is,
(a)
"
#
krg Bo μo
¼0
kro Bg μg
The above equality specifies the required setting for a single-phase flow,
which consequently satisfies the condition of representative samples.
Figs. 4.17 and 4.18 illustrate the impact and difference and between representative and unrepresentative samples.
Qg, scf/day
Qg, scf/day
GOR = Rs
GOR>>>Rs
Qo, STB/day
Qo, STB/day
Free gas
Oil
“Rs”
Oil
Oil
“Rs”
Representative samples
Observed
GOR
Unrepresentative samples
Stabilized GOR ≈ Rsb
Rsb
GORrecombine
Time
n FIGURE 4.17 Classification of surface samples.
GO
R
(GOR)current
Rsb
krg mo Bo
GOR = Rs +
kro mg Bg
(Rs)current
(GOR)current = (Rs)current +
Rs
krg mo Bo
kro mg Bg
current
pcurrent
(GOR)current >> (Rs)current
0
(pb)original
Pressure
0
Time
n FIGURE 4.18 Impact of recombining samples based on GOR below pb.
334 CHAPTER 4 PVT Properties of Crude Oils
A quality check (QC) on the sampling technique is that the bubble point of
the recombined sample at the temperature of the separator from which the
samples were taken should be equal to the separator pressure.
The advantages of surface sampling and recombination include:
1. Large gas and oil volumes can be collected.
2. Access to surface samples is readily available.
3. Stabilized conditions can be established over a number of hours prior to
sampling.
4. Cost of collecting surface samples is much lower than bottom-hole
samples.
5. The subsurface sampling requirements also apply to surface sampling:
If pwf is below pb, then it is probable that an excess volume of gas
will enter the wellbore and even good surface sampling practice will
not obtain a true reservoir fluid sample.
6. There is virtually no interruption of production during the sampling
period (although conditioning of the well prior to sampling is also
required).
The following guidelines for surface sampling are suggested:
1. Should be used routinely for all well tests where fluids are produced into
a separator.
2. It can be used as the principal method or as a backup or cross-check for
other methods.
3. Minimum of two gas samples of 20 L volume plus two liquid samples of
500 cm3 each are required.
4. The well must be producing under a constant GOR and A MINIMUM
24 h of stabilized GOR is recommended before obtaining a
representative sample.
5. Drawdown must be as low as possible; that is, low flow rate.
6. The produced GOR must be representative of the actual gas solubility
Rs; that is, GOR ¼ Rs.
Separator Shrinkage Factor “S”. The separator shrinkage factor,
expressed in STB/bbl, is obtained by flashing a known volume of separator
oil at separator conditions to that of stock-tank conditions. Mathematically,
the separator shrinkage factor is defined as the ratio of volume of oil at
stock-tank condition to that of separator conditions; that is,
S¼
ðvolume of oilÞST
; STB=bbl
ðvolume of oilÞsep
This shrinkage factor is then used to convert the known produced GOR
(expressed in scf/STB) to separator GOR (expressed in scf/bbl). Using
Laboratory Analysis of Reservoir Fluids 335
the separator GOR, the separator fluids are recombined in a PVT cell for
compositional and fluid analysis based on the volume ratio
ðGORÞscf=bbl ¼ ðSÞSTB=bbl ðGORÞscf=STB
Quality Checks of Separator Oil and Gas Samples. It should be pointed
out that when the opening pressures of the gas samples and bubble point
pressures of the liquid samples are similar to the reported field separator
pressure, the indication is that the reported separator is reasonably correct
and samples can be used for recombination and further PVT analysis. On
the other hand, if the bubble point pressures of the liquid samples are significantly lower than the first-stage separator pressure or lower than the
opening pressure of the separator gas samples it might indicate
(a) Leakage in the liquid cylinder
(b) The separator liquid sample had been collected from a secondary
separator, while the gas sample had been collected from the highpressure first-stage separator
Under the above two scenarios, this combination of separator products cannot be used to combine the two samples. In addition, it is critical that each
separator cylinder be checked for its integrity upon arriving in the laboratory. The “quality” checks of the various gas and liquid samples include
the following:
(A) Separator gas samples
n
Opening pressure of each gas sample cylinder.
All gas samples should be checked for opening pressure to verify
the integrity of the sample. Gas sample cylinders should open at a
pressure near the pressure of the separator in the field. The
cylinder(s) found to have such pressure are heated to a
temperature slightly higher than that of the separator in the field.
If the gas cylinder opening temperature Topen is different from the
separator temperature Tsep, the cylinder opening pressure should
be close to
popen ¼ psep
n
Topen
Tsep
Zsep
Zopen
If the opening pressure is much lower than the separator pressure,
it is highly probable that a leakage had occurred and the sample
should not be approved for PVT analysis.
Gas sample conditioning.
Before performing compositional analysis on a gas sample, the
sample is conditioned and held at a temperature higher than the
336 CHAPTER 4 PVT Properties of Crude Oils
(B)
sampling temperature for at least 24 h. The sample should be
rotated from time to time to evaporate any liquid present in the
cylinder. If liquid is present after the condition process, this could
be water of heavy-end hydrocarbons “carry over” during the
separator sampling process.
n
Existence of condensed liquid at separator temperature.
The gas sample cylinders that are found to contain no condensed
hydrocarbon liquids are selected for possible use in the
recombination and reservoir fluid study.
n
Air content determination.
The gas sample cylinder with the highest opening pressure not
containing liquid phase is analyzed for air. If the air content is
found to be less than 0.2 mol%, the sample is selected for
recombination of separator products and reservoir fluid study.
Separator liquid samples
n
Bubble point pressure.
The validity of the separator liquid cylinder samples is determined
by measuring their saturation pressures at ambient temperature;
all samples should show similar saturation pressures and be
compared to that of the field separator pressure. The selected
liquid sample is heated to slightly above the reservoir temperature
to ensure all waxes and resins are dissolved in solution.
n
Presence and quantity of water.
Once each separator liquid sample cylinder’s opening pressure is
measured, the bottom cylinder valve is opened slightly to check
for the presence of water. If water is found to be present, it is
drained and its volume measured. Often, the water recovered
from the cylinder is subjected to a water analysis to determine its
origin.
Quality Checks
(1) If the opening pressures of the gas samples and bubble point pressures
of the liquid samples are similar to the reported field separator
pressure, the indication is that the reported separator is reasonably
correct and samples can be used for recombination and further PVT
analysis. As shown in Table 4.2, samples with higher pressure are
usually selected for recombination.
(2) At times, there are QC results that indicate first-stage separator
gas had been collected with the separator liquid sample had been
collected from a secondary separator. In this situation, the separator
gas samples would open at a pressure near that of the first-stage
separator, and the bubble point pressures of the liquid samples would
Laboratory Analysis of Reservoir Fluids 337
Table 4.2 Samples Selection for Recombination
Summary of Samples Received in the Laboratory
Separator Gas
Sampling Conditions
Laboratory Opening Conditions
Cylinder
Number
Pressure
(psig)
Temperature (°F)
Pressure (psig)
Temperature (°F)
1357a
3571
1050
1050
90
90
1045
Acceptable samples pb within 2%
1043
90
90
Separator Liquid
Sampling Conditions
Laboratory Bubble Point Pressure
Cylinder Number
Pressure (psig)
Temperature (°F)
Pressure (psig)
Temperature (°F)
2468a
4682
1050
1050
90
90
937
Acceptable samples
929
70
70
a
Samples with higher pressure are selected for compositional and analysis and recombination.
#2006 Tarek Ahmed & Associates, Ltd. All Rights Reserved.
be significantly lower than the first-stage separator pressure. Under
no circumstances should this combination of separator products be
used to combine the two samples.
Wellhead Sampling
Wellhead sampling involves the collection of a fluid sample at the surface
from the wellhead itself or from the flow line or the upstream side of
the choke manifold, provided that the fluid is still in one-phase condition.
This option is restricted to wells producing dry gas, very low GOR oils
(ie, heavy oils), and highly undersaturated reservoir fluids. Dry gas wellhead
samples can be collected as for gas sampling from a separator, whereas wellhead of other or unknown fluids should be performed as for separator
liquids. The following wellhead sampling applicability and guidelines are
recommended:
1. For dry gases or highly undersaturated fluids wells.
2. This method can avoid bottom-hole sampling costs.
3. A backup surface samples should be taken as the state of the fluid is not
usually known with certainty; that is, in the case of two-phase flow at the
wellhead.
338 CHAPTER 4 PVT Properties of Crude Oils
4. All equipment and fluid sampling cylinders must be compatible with
maximum wellhead pressure.
5. Minimum of two fluid samples of approximately 600 cm3 for
undersaturated crude oils.
6. For a highly undersaturated retrograded or rich gas fluids, standard 20 L
gas bottles can be used.
Surface Samples Recombination Procedure
When p < pb
For a depleted reservoir (ie, p < pb), determining the original or current oil
composition from a current surface sample poses a special problem.
As shown in Fig. 4.19, when the reservoir or bottom-hole pressure is
below the bubble point pressure, the GOR is greater than Rs and Rsb.
The following procedure can be used to estimate the current or original
oil composition:
psep
Tsep
Gas
(GOR) current
Step 1
Oil
(pb) recombined
GOR
(GOR)current >>>Rsb
GORcurrent
Rsb = GOR
(Rs)current
R
s
krg
o Bo
kro
g Bg
Step 2
Reduce pressure to
pcurrent or (pb) original
Remove
all gas
Step 3
pi
Original (pb)
pcurrent
Pressure
Representative
sample
n FIGURE 4.19 Illustration of recombining surface samples when p < pb.
Step 4
Laboratory Analysis of Reservoir Fluids 339
(a) Place the separator gas and liquid samples in a PVT cell in a ratio
that is equivalent to that of the separator test GOR. It should be pointed
out that current (test) GOR > Rs.
(b) PVT cell is heated to reservoir temperature and the pressure is
increased to dissolve the gas to obtain the bubble point of the
recombined sample; that is, (pb)recombined ≫ (pb)original.
(c) The pressure in the PVT cell is slowly reduced to the current
reservoir pressure “pcurrent” and/or to original bubble point pressure
“pb.” The sample is then stabilized at that pressure for
approximately 6 h.
(d) After establishing complete equilibrium at the desired reservoir
pressure, the excess equilibrium gas phase is removed leaving behind a
saturated liquid that closely represents the composition at (pb)original or
pcurrent.
Fevang and Whitson (1994) provided an alternative approach to the above
methodology to estimate the original crude oil composition in a depleted
reservoir. The method is summarized in the following steps:
(a) Place the separator gas and liquid samples in a PVT cell in a ratio that
is equivalent to that of the separator test GOR.
(b) PVT cell is heated to reservoir temperature and brought to equilibrium
at the current average reservoir pressure “pcurrent.”
(c) After establishing complete equilibrium, remove ALL equilibrium gas
phase from the cell at constant pressure and transfer to a container,
leaving the equilibrium oil in the PVT cell.
(d) The removed gas is reinjected incrementally back to the PVT cell.
After each injection, the bubble point is measured and continues until
reaching the desired bubble point pressure. The mixture is considered
a good approximation of composition at the original bubble point
pressure.
Representative and Unrepresentative Samples
There are commonly three types of fluid sample classifications:
(a) Representative sample
(b) Unrepresentative sample
(c) In situ representative sample
The traditional definition of a "representative sample" is defined as any sample produced from a reservoir where its measured composition and PVT
properties describe the original “undersaturated or saturated” hydrocarbon
system; otherwise, the sample is traditionally labeled as “unrepresentative.”
340 CHAPTER 4 PVT Properties of Crude Oils
As defined by Fevang and Whitson (1994), an "in situ representative
sample" is a special case where the sample represents an in situ reservoir
composition at a specific depth (or an average composition for a depth
interval).
Using or accepting experimental PVT data that are performed on a hydrocarbon sample, regardless of its classification as a representative or unrepresentative sample, will depend entirely on the type of numerical simulator
used to model a reservoir with the most commonly used simulators being
the black oil and compositional simulators.
Black Oil Simulator
In the black oil formulation, it is assumed that the hydrocarbon fluids may be
sufficiently described by two components: a (pseudo) oil component and a
(pseudo) gas component. The main assumption when using a black oil simulator is that the overall hydrocarbon fluid composition will remain constant
during the simulation with all fluid PVT properties determined as a function
of the oil pressure. All mass transfer between the two components (ie, solubility and liberation of gas) is described by the gas solubility “Rs.” The
behavior of these two phases can then be sufficiently described in the black
oil simulator by a set of tabulated PVT data (ie, Bo, Rs, μo, etc.). These PVT
properties must be generated from a laboratory data on a “representative
hydrocarbon” sample with an acceptable and well-defined saturation pressure. It should be pointed out that the “black oil" approach uses a simple
interpolation of PVT properties as a function of pressure when modeling
the performance of a reservoir under different development scenarios,
including the natural depletion process of the reservoir, secondary recovery
process, and gas and water coning, among other development scenarios.
However, black oil simulators fail when they are used to model processes
such as the vaporization process by high-pressure lean gas/CO2 injection,
varying fluid volatility associated compositional gradient, retrograde gas
systems.
It should be noted the impact of accurate bubble point pressure can be significant when performing oil recovery calculations using a black oil simulator or the material balance approach. Fig. 4.20 shows the oil recovery
prediction results from material balance calculations using a representative
sample with a bubble point pressure of 2620 psig as compared with a bubble
point pressure of 2220 psig from an unrepresentative sample. Fig. 4.20 illustrates the impact of the saturation pressure on oil recovery. The impact is
clearly evident when comparing cumulative oil production using the PVT
characteristics of both samples. As the reservoir approaches the bubble point
Laboratory Analysis of Reservoir Fluids 341
5000
LEGEND
Measured fluid data — sample
is representative.
Measured fluid data — sample
bubblelpoint is 400 psi low.
Reservoir pressure
4000
Oil in place - 43.3 MMSTB
Closed reservoir solution 68s
drive recovery calculations
3000
Re
pre
oil senta
sam tiv
ple e
2000
Unr
epr
e
oil s senta
t
am
ple ive
1000
4.5% Low
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Cumulative oil (MMSTB )
n FIGURE 4.20 Oil recovery using representative and unrepresentative oil samples.
of the unrepresentative sample (ie, 2220 psig) the cumulative oil from the
representative sample case is 3.05 MMSTB as compared with 1.7 MMSTB
from the unrepresentative sample case.
Compositional Modeling
In cases where compositional effects are important, reliable simulation
results can only be obtained with a compositional simulator that is based
on a thermodynamically consistent model such as a cubic EOS with applications linked to
n
n
n
The reservoir type; for example, retrograde gas reservoir.
The type of the hydrocarbon system; for example, volatile oil.
The current and future field development processes; for example,
miscible gas injection.
Unlike the black oil simulator, a compositional simulator does not require a
tabulated PVT data, it only requires the optimized parameters of the selected
EOS; for example, Peng-Robinson EOS. These optimized parameters,
as detailed later in Chapter 5, result from matching a wide ranges of specialized laboratory data. A compositional simulator is used in a variety of
4.0
4.5
5.0
342 CHAPTER 4 PVT Properties of Crude Oils
applications in which a black oil simulator does not adequately describe the
fluid behavior experiencing compositional changes that can be attributed to
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Natural depletion behavior of the reservoir below saturation pressure
Mixing of the gas-cap gas with oil rim
Miscible or immiscible gas injection
Significant compositional gradient
Gas-condensate systems
Compositional changes due to crossflow, fluid mixing, and comingling
Other sources
To accurately simulate varying fluid combinations listed above and to
account for the dynamic compositional behavior of the existing reservoir
fluids will require special laboratory tests performed on representative
and unrepresentative hydrocarbon samples. For example, assuming
two of the collected crude oil samples from a reservoir were classified as
“unrepresentative” due to
n
n
The first sample collected below the saturation pressure.
The second sample was considered to be “contaminated” with gas from
the gas-cap.
Disregarding these two samples without performing PVT analysis is greatly
discouraged. These samples represent a natural reservoir depletion behavior
that should be accounted for by tuning EOS to match all experimental PVT
tests performed on all samples produced from the reservoir; independent of
whether samples are representative or unrepresentative. For an accurate
application of a compositional EOS simulator for field development and
forecast, any reservoir sample should be automatically considered as a representative of the reservoir fluid. The final EOS parameters characterizations should be optimized to match all accurate PVT measurements of all
samples produced from the reservoir.
PVT MEASUREMENTS
Black oil and compositional simulators are commonly used in the industry
when conducting quantitative simulation studies for field development and
optimization. Both simulators require an extensive collections of PVT data
that describe the volumetric phase behavior of the reservoir existing phases.
The “black oil” approach is based on a simple interpolation of PVT properties as a function of pressure. With the black oil approach, calculations
can be done quite simply, as the fluid behavior is less complex than that
with a compositional model. The compositional approach employs a thermodynamically consistent model such as a cubic EOS. Selection of a
PVT Measurements 343
compositional model is preferable where compositional effects are important, for example, in cases where the fluid composition varies significantly
in the reservoir, and/or the injected fluids are very different from the fluids
already present in the reservoir. In this case, reliable simulation results can
only be obtained with a compositional model. The compositional approach
is a more complex model and requires tuning EOS to match a variety of PVT
properties of oil and gas phases. The appropriate reservoir applications and
PVT data requirements of these two approaches are briefly outlined below.
(a) The black oil approach: The black oil approach is based on a simple
interpolation of PVT properties as a function of pressure. These PVT
properties are converted internally to a two-component model, defined
as surface oil and surface gas. The model is structured to use
laboratory-generated PVT data to describe the properties of each of the
existing reservoir fluids as a function of pressure and temperature.
These PVT data are generated by conducting a series of laboratory
tests to provide the required model with the required PVT data,
including
n
Oil, gas, and water FVFs “Bo, Bg, Bw”
n
Oil and water gas solubility “Rs, Rsw”
n
Oil, gas, and water viscosities “μo, μg, μw”
n
Solution oil-gas ratio “rs” if the reservoir gas contains
unsubstantial amounts of condensate in solution, with solution
oil-gas ratio rs 0
However, the black oil modeling lacks the ability to explicitly model
the volumetric fluid behavior associated with phase-equilibria
calculations. However, the black oil simulator does not consider
changes in composition of the hydrocarbons with the decline in the
reservoir pressure. Therefore, black oil simulators are used
in situations where recovery processes are insensitive to compositional
changes in the reservoir fluids.
(b) The compositional approach: Compositional simulators are used when
the reservoir depletion and development processes are sensitive to
compositional changes. The approach is the natural choice for
applications where complex phase behavior and compositionally
dependent properties are critical in predicting field recovery and
performance. The compositional approach employs a
thermodynamically consistent model such as a cubic EOS. An EOS
has the ability to provide accurate descriptions of the fluid
thermodynamic properties and volumetric behavior of hydrocarbon
mixtures in all type of reservoir classifications and developments,
including
344 CHAPTER 4 PVT Properties of Crude Oils
Gas condensate
n
Volatile oil systems
n
Miscible gas injection
An EOS requires a detailed knowledge of the molar composition of the
mixture as well as the critical properties and the acentric factor of all
the components. However, reservoir fluids contain several hundred
components that are impossible to identify to appropriately evaluate
the hydrocarbon fluid behavior using a thermodynamic-based EOS
model. As a result, a wide range of laboratory investigations are
performed to study the volumetric behavior of hydrocarbon reservoir
samples and to use the experimental results to calibrate/tune the
thermodynamic-based EOS model. These laboratory studies are
designed and performed on reservoir fluid samples to determine their
n
Compositions
n
Volumetric behavior as a function of pressure and temperature
n
Physical properties, such as density and viscosity
n
PVT tests are conducted in a “PVT CELL” and designed to study and quantify the phase behavior and properties of a reservoir fluid for the use in a
numerical simulator or performing material balance applications. Routine
laboratory tests are described as depletion experiments where the pressure
on a hydrocarbon sample is reduced in successive steps. Regardless of the
type of simulation approach, all aspects of PVT studies are carried out on a
complete range of reservoir fluid types, including
n
n
n
gas-condensate systems
crude oils
critical fluids
There are simple field (on-site) routine tests for reservoir fluid identification
and other data;
Including the following measurements:
n
n
n
n
n
n
Crude oil API gravity
Separator GOR
Tubing head pressure
Separator operating conditions
H2S content
Water analysis
In general, there are two types of laboratory tests that are traditionally performed on hydrocarbon samples:
1. Routine laboratory tests: Several laboratory tests are routinely
conducted to characterize the reservoir hydrocarbon fluid, including
PVT Measurements 345
Compositional analysis of the hydrocarbon system.
n
Constant composition expansion.
n
Differential liberation.
n
Separator tests.
n
Constant volume depletion.
2. Special laboratory PVT tests: These types of tests are performed for very
specific applications. If a reservoir is to be depleted under miscible gas
injection or gas cycling scheme, the following tests may be performed:
n
Slim-tube test
n
Swelling test
n
Multiple forward contact experiment
n
Multiple backward contact experiment
n
Flow assurance tests
n
Routine Laboratory PVT Tests
Compositional Analysis of the Reservoir Fluid
An important test on all reservoir samples is the determination of the fluid
composition. Compositional analysis generally refers to the measurement of
the distribution of hydrocarbons and other components present in oil and gas
samples. The components of petroleum and petroleum products number in
the tens of thousands. They range in molecular weight from methane (16) to
very large uncharacterized components with molecular weight in the thousands. Analysis of the hydrocarbon mixture is complicated not only by the
enormous number of components, but also by the large variation in compound concentrations. Using modern chromatography techniques, samples
are analyzed to determine the breakdown of the components in the sample.
Gas chromatography (GC) is a technique for separating and identifying
components of hydrocarbon mixtures.
GC, as shown schematically in Fig. 4.21, is a common type of chromatography used in analyzing hydrocarbon mixtures that can be vaporized without
decomposition. A small hydrocarbon sample of the hydrocarbon mixture is
injected into a heated entrance “port” of the GC by a microsyringe. The
heated port is set to a temperature higher than the components’ boiling point
to vaporize the injected sample inside the port. A carrier gas, such as helium,
flows through the port and displaces the gaseous components of the sample
into a capillary column. It is within the column that separation of the components takes place. The column is packed with a filling called the “stationary phase,” which has the ability to adsorb and release the various
components successively after certain retention time. Various released components travel through the stationary phase at different speeds, which causes
them to separate. Because different components emerge from the column at
346 CHAPTER 4 PVT Properties of Crude Oils
Injector
port
Flow
controller
Recorder
Column
Detector
Column oven
Carrier gas
n FIGURE 4.21 Schematic of GC main components.
different times, they can be identified by a detector at the outlet of the column. The detector is used to monitor the outlet stream from the column;
thus, the time at which each component reaches the outlet and the amount
of that component can be determined. The detector sends a signal to a chart
recorder that results in a peak on the chart paper. Each peak corresponds to a
different component, and the area under these peaks can be used to quantify
the mole fraction of each component. To quantify the composition in terms
of mole fractions, a calibration curve (peak area vs. concentration) is
obtained by performing GC analysis of samples of known concentrations.
With the development of more sophisticated equations of state to calculate
fluid properties, it is recommended that compositional analyses of the reservoir fluid include a separation of components through C20+ as a minimum.
Table 4.3 shows a chromatographic compositional analysis of surface separator oil and gas samples that were collected from the Big Butte Field. A
physical recombination is required to produce a representative reservoir
fluid. Following the compositional analyses of the two samples, separator
gas and separator liquid were physically recombined in a PVT cell at a stabilized producing GOR of 1597 scf/sep bbl. The recombined reservoir fluid
sample exhibited a bubble point of 3870 psia at 218°F.
Quality Assessment of the Laboratory Compositional Analysis
Obtaining an accurate fluid composition is perhaps the single most important laboratory measurement as compared with other PVT experimental
analyses. This is due to the fact that determining the compositional analysis
of the hydrocarbon sample precedes all subsequent PVT tests. Before
PVT Measurements 347
Table 4.3 Hydrocarbon Analysis of Recombined Reservoir Fluid Samples
Mathematical Recombination of Separator Products (Basis of Recombination: 1597 scf Separator Gas/bbl
Separator Liquid)
Component
(Symbol)
N2
CO2
H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
C29
C30+
Separator
Gas
(mol%)
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
0.096
1.376
0
77.234
14.157
4.58
0.74
1.036
0.281
0.223
0.155
0.074
0.037
0.009
0.002
0.016
0.53
0
14.666
11.076
9.218
2.702
5.103
2.998
3.631
5.538
6.109
7.419
5.413
4.022
2.922
2.257
2.187
1.859
1.631
1.282
1.183
1.066
0.983
0.755
0.645
0.592
0.503
0.434
0.453
0.312
0.296
0.26
0.223
1.717
0.004
0.228
0
2.297
3.254
3.968
1.533
2.896
2.111
2.558
4.659
5.67
7.747
6.276
5.266
4.193
3.548
3.736
3.448
3.279
2.779
2.737
2.613
2.523
2.026
1.832
1.764
1.563
1.401
1.526
1.092
1.079
0.983
0.875
8.54
Molecular
Weight
Specific
Gravity
(Water 5 1.0)
Reservoir
Fluid
(mol%)
Reservoir
Fluid
(wt.%)
28.01
44.01
34.08
16.04
30.07
44.1
58.12
58.12
72.15
72.15
86.18
95.05
106.95
118.76
134.14
147
161
175
190
206
222
237
251
263
275
291
305
318
331
345
359
374
388
402
509.6
0.809
0.818
0.801
0.3
0.356
0.507
0.563
0.584
0.624
0.631
0.664
0.71
0.74
0.773
0.779
0.79
0.801
0.812
0.823
0.833
0.84
0.848
0.853
08.58
0.863
0.868
0.873
0.878
0.882
0.886
0.89
0.894
0.897
0.9
0.922
0.067
1.066
0
54.289
13.027
6.281
1.459
2.528
1.277
1.473
2.129
2.287
2.744
1.991
1.476
1.071
0.828
0.802
0.682
0.598
0.47
0.434
0.391
0.36
0.277
0.237
0.217
0.185
0.159
0.166
0.114
0.108
0.095
0.082
0.63
0.037
0.923
0
17.133
7.706
5.448
1.669
2.89
1.813
2.091
3.609
4.276
5.773
4.651
3.89
3.098
2.622
2.761
2.548
2.423
2.054
2.023
1.931
1.864
1.497
1.354
1.304
1.155
1.036
1.128
0.807
0.798
0.727
0.647
6.311
(Continued)
348 CHAPTER 4 PVT Properties of Crude Oils
Table 4.3 Hydrocarbon Analysis of Recombined Reservoir Fluid Samples Continued
Mathematical Recombination of Separator Products (Basis of Recombination: 1597 scf Separator Gas/bbl
Separator Liquid)
Component
(Symbol)
Total
Molecular
weight
Density,
lb/ft3
Separator
Gas
(mol%)
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
100
20.95
100
102.44
100
Molecular
Weight
Specific
Gravity
(Water 5 1.0)
Reservoir
Fluid
(mol%)
Reservoir
Fluid
(wt.%)
100
50.80
100
43.0248
Compositional Groupings of Reservoir Fluid
Group
mol%
wt.%
MW
SG
Total fluid
C7+
C10+
C20+
C30+
100
16.404
9.382
2.27
0.63
100
56.682
41.981
16.763
6.311
50.83
175.65
227.46
375.43
509.6
0.557
0.816
0.846
0.895
0.922
instructing a commercial laboratory to proceed with other type of experiments, engineers should ensure the quality and reliability of the representative or recombined hydrocarbon samples. It should be pointed out that the
degree of accuracy of results from applying equations of state is strictly
linked to the accuracy of the original hydrocarbon composition as determined in the laboratory. There are four main QCs on the consistency of
the reported hydrocarbon compositions that should be performed:
n
n
n
n
Trend of equilibrium ratios “K-values”
Hoffman plot
Material balance test on overall composition
Material balance test on the GOR
Equilibrium Ratios Trend. As discussed in detail in Chapter 5, the equilibrium ratio, Ki, of a given component in a hydrocarbon mixture is defined
as the ratio of the mole fraction of the component in the gas phase, yi, to that
of the mole fraction of the component in the liquid phase, xi. Mathematically, the relationship is expressed as
Ki ¼
yi
xi
PVT Measurements 349
where
Ki ¼ equilibrium ratio of component i
yi ¼ mole fraction of component i in the gas phase
xi ¼ mole fraction of component i in the liquid phase
The K-value measures the tendency of the component (ie, component “i”),
to remain or escape to the gas phase. It is essentially a property that measures
the volatility of the component at a pressure and temperature. This tendency
suggests that in a multicomponent system, components’ Ki-value should follow the trend of their decreasing K-values with the increasing order of their
normal boiling points; that is,
KN2 > KCl > KCO2 > KC2 > KH2 S > KC3 > Ki-C4 > Kn-C4 > Ki-C5 > Kn-C5 > ⋯
To check the consistency of the laboratory recombined gas and liquid samples listed in Table 4.3 by applying the above K-value criteria, the equilibrium ratio for each component was calculated as shown in Fig. 4.22, which
indicated a consistency in the laboratory-reported compositional analysis.
Hoffman Plot. The Hoffman plot is a standard technique for evaluating the
consistency of the recombined fluid samples through a graphical technique.
The Hoffman plot is created by plotting the log(Kipsep) or alternately log(Ki)
versus the component characteristic factor “Fi” as defined by the following
relationship:
3
1
1
7
pci 6
6Tbi Tsep 7
Fi ¼ log
1 5
14:7 4 1
Tbi Tci
2
where
psep ¼ separator pressure, psia
Tsep ¼ separator temperature, °R
pci ¼ critical pressure of component “i,” psia
Tci ¼ critical temperature of component “i,” °R
Tbi ¼ boiling point temperature of component “i,” °R
When log(Kipsep) is plotted against Fi for each component on a Cartesian
scale, the resulting plot should be a nearly straight-line for the light hydrocarbon components; specifically for components C1–C6, components. However, some downward curvature could occur when plotting the heavy-end
components. A resulting straight line indicates that the liquid and vapor
samples are reasonably in equilibrium at the separator conditions, and the
measurement of the liquid and vapor compositions generally error free.
350 CHAPTER 4 PVT Properties of Crude Oils
Nitrogen
Methane
Carbon dioxide
Ethane
Propane
i-Butane
n-Butane
i-Pentane
n-Pentane
Hexane
Heptane
Octane
Nonane
Separator
Liquid “xi”
(mol%)
Separator
Gas “yi”
(mol%)
Ki = yi/xi
0.016
14.666
0.53
11.076
9.218
2.702
5.103
2.998
3.631
5.538
6.109
7.419
5.413
0.096
77.234
1.376
14.157
4.58
0.74
1.036
0.281
0.223
0.155
0.074
0.037
0.009
6
5.2661939
2.5962264
1.278169
0.496854
0.2738712
0.2030178
0.0937292
0.0614156
0.0279884
0.0121133
0.0049872
0.0016627
The K-value plot
7
6
5
Ki 4
3
2
1
ta
ne
i-P
en
ta
ne
nPe
nt
an
e
H
ex
an
e
H
ep
ta
ne
O
ct
an
e
N
on
an
e
e
Bu
an
n-
ut
i-B
pa
ne
ne
de
ha
Pr
o
Et
di
ox
i
ne
ha
bo
n
C
ar
M
et
N
itr
og
en
0
n FIGURE 4.22 The K-value consistency checks.
Usually the plus fraction (eg, heptane-plus (C7+)), is not included in
Hoffman consideration because it is generally not measured accurately.
Unless nonhydrocarbon components (particularly the nitrogen component)
they are usually excluded from the Hoffman plot. To further QC the laboratory recombined compositions listed in Table 4.3, the compositions were
used to generate the Hoffman plot as shown in Fig. 4.23, which indicates
consistency in the laboratory measurements.
The Material Balance Consistency Test on the Overall
Composition. The consistency test is derived from the component molal
balance criteria as given by the following relationship:
zi nt ¼ xi nL + yi nv
PVT Measurements 351
Methane
Ethane
Propane
i-Butane
n-Butane
i-Pentane
n-Pentane
Hexane
Heptane
Octane
Nonane
Separator
Liquid
(mol%)
14.666
11.076
9.218
2.702
5.103
2.998
3.631
5.538
6.109
7.419
5.413
Separator Pressure:
625 psia
Separator Temperature:
104°F
Separator
Gas
(mol%)
77.234
14.157
4.58
0.74
1.036
0.281
0.223
0.155
0.074
0.037
0.009
K = yi /xi
Tb
(°R)
200.95
332.21
415.94
470.45
490.75
541.76
556.56
615.37
668.74
717.84
763.07
log(K*P)
3.517377
2.902468
2.492109
2.233426
2.103414
1.767755
1.584159
1.242859
0.879142
0.493736
0.016684
5.266193918
1.278169014
0.496853981
0.273871207
0.203017833
0.093729153
0.061415588
0.027988443
0.012113275
0.004987195
0.001662664
Tc
(°R)
343.01
549.74
665.59
734.08
765.18
828.63
845.37
911.47
970.57
1023.17
1070.47
pc
(psia)
667
707.8
615
527.9
548.8
490.4
488.1
439.5
397.4
361.1
330.7
b
803.8875
1412.635
1798.201
2037.314
2151.155
2383.685
2478.166
2795.253
3079.198
3344.396
3592.934
b(1/Tb−1/T)
2.574268643
1.7460969
1.133056126
0.716190528
0.567065255
0.171024903
0.056164664
−0.416629931
−0.858290198
−1.274280291
−1.665655175
4.0
log(Ki psep) or log(Ki)
C1
3.0
C2
i-C4
2.0
i-C5
n-C5
C3
n-C4
C6
1.0
C7
C8
0.0
–2.0
C9
–1.0
0.0
1.0
bi (1/Tbi − 1/Tsep)
n FIGURE 4.23 Hoffman plot consistency check.
where
zi ¼ mole fraction of component i in the entire hydrocarbon mixture
xi ¼ mole fraction of component i in the liquid phase
yi ¼ mole fraction of component i in the gas phase
nt ¼ total moles of the hydrocarbon mixture, lb-mol
nL ¼ total moles of the liquid phase
nv ¼ total moles of the gas phase
zint ¼ total number of moles of component i in the system
xinL ¼ total number of moles of component i in the liquid phase
yinv ¼ total number of moles of component i in the vapor phase
If the laboratory-reported overall composition “zi” was recombined mathematically based on a separator GOR expressed in scf/bbl, the martial balance
can be used to identify discrepancies in laboratory recombined overall composition “zi.” This quality assessment can be performed by applying the following mathematical expression:
zi ¼
2130:331ρo xi + ML ðGORÞyi
2130:331ρo + ML ðGORÞ
2.0
3.0
352 CHAPTER 4 PVT Properties of Crude Oils
where
ML ¼ molecular weight of the separator liquid sample
ρo ¼ density of separator liquid sample at separator pressure and
temperature, lb/ft3
GOR ¼ recombined GOR, scf/bbl
xi ¼ mole fraction of component i in the separator liquid phase sample
yi ¼ mole fraction of component i in the separator gas phase sample
The laboratory gas and liquid samples listed in Table 4.3 were recombined at
GOR of 1597 scf/bbl. The separator liquid sample exhibits a molecular
weight “ML” of 102.44 and liquid density “ρo” of 43.0248 lb/ft3 at separator
pressure and temperature. To assess the quality of the reported compositional analysis given in Table 4.3, the molal material balance is applied
to give
zi ¼
zi ¼
2130:331ρo xi + ML ðGORÞyi
2130:331ρo + ML ðGORÞ
2130:331ð43:0248Þxi + ð102:44Þð1597Þyi
2130:331ð43:0248Þ + ð102:44Þð1597Þ
Applying the above relationship to assess the quality of mathematically
recombined overall composition given in Table 4.3, results in a very good
agreement with the laboratory data as shown in Table 4.4.
The Material Balance Test Consistency on the GOR. As shown in applying the component material balance (MB) to test and assess the quality
of the recombination of gas and liquid samples, a concise QC test on the
laboratory-reported GOR can also be performed by solving the component
MB relationship for the ratio “nL/nv” as follows:
zi nt ¼ xi nL + yi nv
yi nt nL xi
¼ zi nv nv zi
The above form of the component material balance suggests plotting “yi/
zi” versus “xi/zi” would produce a straight line with a negative slope
value of “nL/nv.” The GOR can then be calculated from the following
expression:
GOR ¼
2130:331ðρo Þpsep , Tsep
ðMWÞLiquid ðnL =nv Þ
; scf=bbl
Table 4.4 Material Balance Quality Test
Component
(Symbol)
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
0.096
1.376
77.234
14.157
4.58
0.74
1.036
0.281
0.223
0.155
0.074
0.037
0.009
0.002
0.016
0.53
14.666
11.076
9.218
2.702
5.103
2.998
3.631
5.538
6.109
7.419
5.413
4.022
2.922
2.257
2.187
1.859
1.631
1.282
1.183
1.066
0.983
0.755
0.645
0.592
0.503
0.004
0.228
2.297
3.251
3.968
1.533
2.896
2.111
2.558
4.659
5.67
7.747
6.276
5.266
4.193
3.548
3.736
3.448
3.279
2.779
2.737
2.613
2.523
2.026
1.832
1.764
1.563
Molecular
Weight
Specific
Gravity
(Water 5 1.0)
Reservoir
Fluid
(mol%)
Reservoir
Fluid
(wt.%)
Calculated zj
28.01
44.01
16.04
30.07
44.1
58.12
58.12
72.15
72.15
86.18
95.05
106.95
118.76
134.14
147
161
175
190
206
222
237
251
263
275
291
305
318
0.809
0.818
0.3
0.356
0.507
0.563
0.584
0.624
0.631
0.664
0.71
0.74
0.773
0.779
0.79
0.801
0.812
0.823
0.833
084
0.848
0.853
0.858
0.863
0.868
0.873
0.878
0.067
1.066
54.289
13.027
6.281
1.459
2.528
1.277
1.473
2.129
2.287
2.744
1.991
1.476
1.074
0.828
0.082
0.682
0.598
0.47
0.434
0.391
0.36
0.277
0.237
0.217
0.185
0.037
0.923
17.133
7.706
5.448
1.669
2.89
1.813
2.091
3.609
4.276
5.773
4.651
3.895
3.098
2.622
2.761
2.548
2.423
2.054
2.023
1.931
1.864
1.497
1.354
1.304
1.155
0.000664
0.010631
0.540952
0.130176
0.062952
0.014656
0.0254
0.012858
0.014833
0.021457
0.023059
0.02767
0.020075
0.014887
0.010806
0.008347
0.008088
0.006875
0.006032
0.004741
0.004375
0.003942
0.003635
0.002792
0.002385
0.002189
0.00186
(Continued)
PVT Measurements 353
N2
CO2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
Separator
Gas
(mol%)
Component
(Symbol)
C24
C25
C26
C27
C28
C29
C30+
Total
Molecular
weight
Density,
lb/ft3
Separator
Gas
(mol%)
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
100
20.95
0.434
0.453
0.312
0.296
0.26
0.223
1.717
100
102.44
1.401
1.526
1.092
1.079
0.983
0.875
8.54
100
43.0248
Molecular
Weight
331
345
359
374
388
402
509.6
Specific
Gravity
(Water 5 1.0)
0.882
0.886
0.89
0.894
0.897
0.9
0.922
Reservoir
Fluid
(mol%)
Reservoir
Fluid
(wt.%)
0.159
0.166
0.114
0.108
0.095
0.082
0.63
100
50.83
1.036
1.128
0.807
0.798
0.727
0.647
6.311
100
Calculated zj
0.001605
0.001675
0.001154
0.001095
0.000962
0.000825
0.00635
354 CHAPTER 4 PVT Properties of Crude Oils
Table 4.4 Material Balance Quality Test Continued
PVT Measurements 355
1.6
1.4
Component
N2
CO2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
1.2
yi /zi = –0.57797168(xi /zi ) + 1.57653829
yi /zi
1
0.8
Slope− = nL/nv
= 0.57797
0.6
0.4
0.2
0
0
0.5
1
1.5
xi /zi
2
2.5
n FIGURE 4.24 Recombined GOR consistency check.
It should be noted that the slope value of “nL/nv” must be treated as positive
in the above equation. Fig. 4.24 shows the straight line resulting from plotting “yi/zi” versus “xi/zi” using the laboratory gas and liquid samples listed in
Table 4.3. The slope of the line in the figure is 0.57797, which is converted
into GOR to give
GOR ¼
GOR ¼
2130:331ðρo Þpsep , Tsep
ðMWÞLiquid ðnL =nv Þ
2130:331ð43:0248Þ
¼ 1548scf=bbl
ð102:44Þð0:57797Þ
The calculated GOR value is about 3% lower than the reported laboratory
value of 1597 scf/bbl. The error is slightly higher than expected, usually
around 1%; however, considering the other three quality assessment tests
that were performed on the samples with good matches, the reported recombined GOR appears to be reasonable.
CCE Tests
CCE tests are conventionally performed on all types of reservoir fluids to
obtain a better understanding of the pressure-volume relationship of these
hydrocarbon systems. The CCE test can also be referred to by different
names, including pressure-volume relations, constant mass expansion, flash
liberation, flash vaporization, or flash expansion. The CCE test is performed
on crude oil systems for the purposes of determining
3
xi /zi
yi /zi
0.23881
0.49719
0.27015
0.85023
1.46760
1.85195
2.01859
2.34769
2.46504
2.60122
2.67118
2.70372
2.71873
1.43284
1.29081
1.42265
1.08674
0.72918
0.50720
0.40981
0.22005
0.15139
0.07280
0.03236
0.01348
0.00452
356 CHAPTER 4 PVT Properties of Crude Oils
n
n
n
n
n
n
n
n
Saturation pressure
Relative volume “Vrel”
Isothermal compressibility coefficient “Co” above pb
Oil density “ρo” at and above pb
The traditional Y-function
The extended Y-function “YEXT”
Oil FVF “Bo” above pb
Two-phase formation factor Bt below pb
The experimental procedure involves placing a hydrocarbon fluid sample in
a visual PVT cell at reservoir temperature that is held constant during the
experiment. To ensure that the hydrocarbon sample exits in a single phase,
the sample is pressurized to a much higher pressure than the initial reservoir
pressure. The pressure is reduced isothermally in steps by removing mercury
from the PVT cell and the total hydrocarbon volume “Vt” is measured at
each pressure. The combination of successive reductions in the pressure
and the measurement of total fluid volume “Vt” are performed at a CCE; that
is, no gas or liquid is removed from the PVT cell at any time throughout the
experiment. A schematic of CCE experiment is illustrated in Fig. 4.25. As
the pressure approaches the saturation pressure, the total volume continuously increases due to the expansion of the single-phase crude oil.
Reaching the bubble point pressure marks the appearance and start of liberation of the solution gas. The saturation pressure value is identified by visual
inspection and by the point of discontinuity on the plot “pressure-volume”
curve. As shown in Fig. 4.25, the saturation pressure and corresponding
Voil = 100%
100%
Oil
p3= pb
p2> > pb
p1> > pb
Vt1
oil
Voil < 100%
Voil ≈ 100%
Voil = 100%
Vt2
oil
Voil <<100%
p5<<<pb
p4<<pb
gas
gas
Vsat
oil
Vt4
oil
Hg
(a)
(b)
n FIGURE 4.25 Constant composition expansion test.
(c)
(d)
(e)
Vt5
PVT Measurements 357
volume are recorded and designated as “psat” and “Vsat,” respectively. The
volume at the saturation pressure “Vsat” is used as a reference volume and
the total volume “Vt” as a function of pressure is reported relative to Vsat as a
ratio of “Vt/Vsat.” This volume is termed the relative volume and is expressed
mathematically by the following equation:
Vrel ¼
Vt
Vsat
(4.105)
where
Vrel ¼ relative volume
Vt ¼ total hydrocarbon volume
Vsat ¼ volume at the saturation pressure
Table 4.5 shows the results of performing a CCE test on a recombined surface sample taken from the Big Butte crude oil system. The bubble point
pressure of the hydrocarbon system is 1930 psi at 247°F. The tabulated data
includes
n
n
n
n
Relative volume
Oil density at and above the saturation pressure
Average single-phase compressibility; that is, Co at p pb
The Y-function
Table 4.5 Constant Composition Expansion Data (Pressure/Volume
Relations at 247°F)
Pressure (psig)
Relative Volume
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
0.9371
0.9422
0.9475
0.9532
0.9592
0.9657
0.9728
0.9805
0.989
0.9909
0.9927
0.9947
0.9966
0.9987
Y-Function
Density (g/cm3)
0.6919
0.6882
0.6843
0.6802
0.6760
0.6714
0.6665
0.6613
0.6556
0.6544
0.6532
0.6519
0.6506
0.6492
(Continued)
358 CHAPTER 4 PVT Properties of Crude Oils
Table 4.5 Constant Composition Expansion Data (Pressure/Volume
Relations at 247°F) Continued
Pressure (psig)
Relative Volume
pb ≥ 1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
1
1.0014
1.0018
1.003
1.0042
1.0058
1.0139
1.0324
1.0625
1.1018
1.0611
1.2504
1.3694
1.502
1.9283
2.496
3.4464
Pressure Range (psig)
Y-Function
0.6484
2.108
2.044
1.965
1.874
1.784
1.71
1.56
1.453
1.356
Single-Phase Compressibility (psi21)
6500–6000
10.73(106)
6000–5500
11.31(106)
5500–5000
11.96(106)
5000–4500
12.70(106)
4500–4000
13.57(106)
4000–3500
14.61(106)
3500–3000
15.86(106)
3000–2500
17.43(106)
2500–2000
19.47(106)
2000–1936
20.79(106)
Saturation pressure (psat) ¼ 1936 psig
Vrel ¼
ρ¼
Vt
:
Vsat
ρsat
:
Vrel
1
ðVrel Þ1 ðVrel Þ2
:
ðVrel Þ1 + ðVrel Þ2
p1 p2
2
psat p
:
Y¼
pðVrel 1Þ
ðco Þaverage ¼
Density (g/cm3)
PVT Measurements 359
Traditionally, measured volumes from the CCE are expressed in terms of the
relative volume and plotted a function of pressure. The plot shows clearly
the transition from the single-phase system to the two-phase system. The
pressure at which the slope changes represents the bubble point pressure,
as shown in Fig. 4.26.
It should be pointed out the change in the slope is related and attributed to
the fluid compressibility resulting from the transition from the single phase
to the two-phase state; that is,
n
n
In the single liquid phase region, the liquid compressibility is relatively
small resulting in a steep decline in the pressure with a small increase
in the liquid volume.
As the pressure approaches the bubble point pressure and initiation
of liberation solution gas, the evolved gas is characterized by a high
compressibility that causes less reduction in the pressure resulting in
a sharp break in the slope between the single- and two-phase regions.
Single-phase expansion
4000
3500
3000
Pressure
2500
pb=1936 psi
2000
1500
Twophas
e exp
ansio
n
1000
500
0
0
0.5
1
1.5
2
Relative volume
n FIGURE 4.26 Pressure-volume relations from the CCE test.
2.5
3
3.5
4
p
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
Rel Vol
0.9371
0.9422
0.9475
0.9532
0.9592
0.9657
0.9728
0.9805
0.989
0.9909
0.9927
0.9947
0.9966
0.9987
1
1.0014
1.0018
1.003
1.0042
1.0058
1.0139
1.0324
1.0625
1.1018
1.0611
1.2504
1.3694
1.502
1.9283
2.496
3.4464
360 CHAPTER 4 PVT Properties of Crude Oils
It should be pointed out that in the case of light or volatile oils, the two slopes
are not clearly defined. The pressure-volume plot shows a continuous shape
with no clear discontinuity or break in volumetric behavior and consequently might produce an inaccurate bubble point pressure and corresponding volume “Vsat” measurements. Essentially, bubble point determination
relies instead on the visual observation of forming the first bubble of gas
that defines the bubble point. As outlined later in this chapter, Vsat is an
important property that will impact the accuracy of all other PVT properties
and, therefore, must be estimated accurately.
n
Oil density from the CCE test
As shown in Table 4.5, the density of the oil at the saturation pressure is
0.6484 g/cm3. The density is determined directly from mass “m” of the
fluid sample and fluid volume “V” measurements. Designated ρsat for the
fluid density at the saturation pressure, it can be calculated from
ρsat ¼
m
Vsat
Above the bubble point pressure, the density of the oil can be calculated
by using the recorded relative volume at any pressure, to give
ðm=Vsat Þ
ρ
¼ sat
ρ¼ Vrel
Vp, T =Vsat
(4.106)
where
ρ ¼ density at any pressure above the saturation pressure
ρsat ¼ density at the saturation pressure
Vrel ¼ relative volume at the pressure of interest
Vp,T ¼ volume at the pressure “p” and reservoir (cell) temperature
EXAMPLE 4.28
Given the experimental data in Table 4.5, verify the oil density values at 4000
and 6500 psi.
Solution
Applying Eq. (4.106) gives the following results:
At 4000 psi:
ρo ¼
ρsat 0:6484
¼
¼ 1:6714g=cm3
Vrel 0:9657
At 6500 psi:
ρo ¼
ρsat 0:6484
¼
¼ 0:6919
Vrel 0:9371
PVT Measurements 361
n
Isothermal compressibility coefficient from CCE test
The instantaneous isothermal compressibility coefficient of any substance
is defined as the rate of change in volume with respect to pressure per unit
volume at a constant temperature. Mathematically, the substance
isothermal compressibility “c” is defined by the following expression:
c¼ 1 @V
V @p
T
Applying the above expression for the crude oil instantaneous
isothermal compressibility gives
co ¼ 1 @V
V @p
T
And equivalently can be written in terms of the relative volume:
co ¼
1 @ ðV=Vsat Þ
ðV=Vsat Þ
@p
or
co ¼
1 @Vrel
Vrel @p
(4.107)
Commonly, the relative volume data above the bubble point pressure is
plotted as a function of pressure, as shown in Fig. 4.27. To evaluate
instantaneous co at any pressure p, it is necessary approximate the
1
0.99
0.98
(Vrel)1
(co)average =
–1
(Vrel)1 + (Vrel)2
(Vrel)1 – (Vrel)2
p1 – p 2
Relative volume
2
0.97
(Vrel)2
0.96
0.95
0.94
0.93
1500
2500
p1
3500
p2
4500
Pressure
n FIGURE 4.27 Relative volume data above the bubble point pressure.
5500
6500
7500
362 CHAPTER 4 PVT Properties of Crude Oils
derivative (@Vrel/@p) at the desired pressure. The pressure-volume curve
shown in Fig. 4.27 can be represented by the following third degree
polynomial:
Vrel ¼ 1:4701 1013 p3 + 2:95895 109 p2 30:124 106 p + 1:048212
with
@Vrel =@p ¼ 1:4701 1013 ð3Þp2 + 2:95895 109 ð2Þp 30:124 106
EXAMPLE 4.29
Using Fig. 4.27, evaluate co at 4500 psi.
Solution
n Relative volume at 4500 psi ¼ 0.9592
n
@Vrel
¼ 1:4701 1013 ð3Þð4500Þ2 + 2:95895 109 ð2Þ4500 30:124 106
@p
¼ 12:424 106
Apply Eq. (4.107) to give
1 @Vrel
1 ¼
co ¼
12:424 106 ¼ 12:953 106 psi1
Vrel @p
0:9592
Traditionally, the laboratory isothermal compressibility coefficients are reported
as average compressibility coefficients at several ranges of pressure as shown
Table 4.5. These values are determined by calculating the changes in the relative
volume at the indicated pressure interval and applying the following expression:
ðco Þaverage ¼
1
ðVrel Þ1 ðVrel Þ2
ðVrel Þ1 + ðVrel Þ2
p1 p2
2
(4.108)
The subscripts 1 and 2 represent the corresponding values at the lower and
higher pressure ranges, respectively.
EXAMPLE 4.30
Using the measured relative volume data in Table 4.5 for the Big Butte crude oil
system, calculate the average oil compressibility in the pressure range of 2500–
2000 psi.
Solution
Apply Eq. (4.108) to give
1
ðVrel Þ1 ðVrel Þ2
ðVrel Þ1 + ðVrel Þ2
p1 p2
2
1
0:9657 0:9592
ðco Þaverage ¼
¼ 13:51 106 psi1
0:9657 + 0:9592 4000 4500
2
ðco Þaverage ¼
PVT Measurements 363
n
The traditional Y-function
The relative volume data frequently require smoothing to correct for
laboratory inaccuracies in measuring the total hydrocarbon volume,
particularly below the saturation pressure and also at lower pressures.
A dimensionless compressibility function, commonly called the
Y-function, is used to smooth the values of the relative volume “Vrel”
below the saturation pressure. The Y-function in its mathematical
expression is ONLY defined below the saturation pressure; that is,
p < pb, and given by the following equation:
Y¼
psat p
pðVrel 1Þ
(4.109)
where
psat ¼ saturation pressure, psia
p ¼ pressure, psia
Vrel ¼ relative volume at pressure p
Table 4.5 lists the computed values of the Y-function as calculated
by using Eq. (4.109). To smooth the relative volume data below the
saturation pressure, the Y-function is plotted as a function of
pressure on a Cartesian scale. Depending on the classification of
fluid type, the Y-function could form a straight line or a line with
only a small curvature. Fig. 4.28 shows the Y-function versus
pressure for the Big Butte crude oil system. The figure illustrates
the erratic behavior of the laboratory-measured data near the
bubble point pressure.
The following steps summarize the simple procedure of smoothing
and correcting the relative volume data.
Step 1. Calculate the Y-function for all pressures below the
saturation pressure using Eq. (4.109).
Step 2. Plot the Y-function versus pressure on a Cartesian scale.
Step 3. Determine the coefficients of the best straight-line fit of the
Y-function data:
Y ¼ a + bp
(4.110)
where a and b are the intercept and slope of the lines,
respectively.
Step 4. Recalculate the relative volume at all pressures below the
saturation pressure from the following expression:
Vrel ¼ 1 +
psat p
pða + bpÞ
(4.111)
364 CHAPTER 4 PVT Properties of Crude Oils
erratic behavior of the
laboratory data
n FIGURE 4.28 Y-Function vs. pressure.
EXAMPLE 4.31
Smooth the laboratory recorded relative volume data of Table 4.5 with the best
straight-line fit of the Y-function as given by
Y ¼ a + bp
where
a ¼ 1:0981
b ¼ 0:000591
PVT Measurements 365
Notice that the laboratory report suggests that the bubble point pressure could
be greater than the reported value.
Solution
Applying Eq. (4.111) gives
Vrel ¼ 1 +
psat p
1950:7 p
¼ 1+
pða + bpÞ
pð1:0981 + 0:000591pÞ
Notice that the pressure must be expressed in psia. Results are shown in the table
below.
Pressure (psia)
Measured Vrel
Smoothed Vrel
1950.7
1944.7
1942.7
1937.7
1932.7
1925.7
1892.7
1822.7
1723.7
1614.7
1481.7
1327.7
1175.7
1049.7
796.7
614.7
451.7
–
–
–
–
–
–
–
–
1.0625
1.1018
1.1611
1.2504
1.3696
1.502
1.9283
2.496
3.4464
–
1.0014
1.0018
1.003
1.0042
1.0058
1.0139
1.0324
1.063
1.1028
1.1626
1.2532
1.3741
1.5091
1.9458
2.5328
3.529
As illustrated in the above example, erratic data from the CCE test were
obtained around the saturation pressure and also at lower pressures. To visually detect the appearance of the gas or first droplet of liquid that marks the
saturation pressure “psat” can be a tedious and difficult task, particularly
when conducting a CCE test on volatile oil and retrograde gas systems.
Erroneous measurement of the saturation pressure will lead to an incorrect
corresponding value of the reference volume Vsat, which consequently,
impacts the accuracy of the relative volume values. As detailed later in this
chapter, correct relative volume data are required to generate the PVT
properties of the crude oil system, including
n
n
Oil FVF “Bo” above the pb
The two-phase FVF “Bt”
366 CHAPTER 4 PVT Properties of Crude Oils
n
n
Isothermal compressibility coefficient of the oil “co”
The Extended Y-Function
Hosein et al. (2014) introduced the concept of the extended Y-function,
which is based on replacing the Vsat in the traditional Y-function with
that of the laboratory-measured value of initial volume “Vi” at the
highest pressure in the CCE test. Using Vi as a reference volume will
eliminate the tedious process of estimating Vsat, particularly when
performing laboratory PVT study on volatile oil and condensate systems.
Hosein and coauthors’ proposed approach offers a major advantage of
using Vi as a reference volume in that the Y-function calculations can be
extended above the saturation pressure; that is, removing the limitation
of Y-function applicability to only below the saturation pressure.
Applying the approach would produce two straight lines (above and
below the saturation pressure) that clearly identify the saturation
pressure by the intersection of the two lines, eliminating the need to
measure the saturation pressure by visually observing the appearance of
the first bubble of gas or droplet of liquid. This extended relative volume
“(Vrel)EXT” can be expressed by the following relationship:
ðVrel ÞEXT ¼
Vt
Vi
To distinguish between the traditional Y-function and the newly
introduced EXTENDED Y-function, Hosein and coauthors termed
the new Y-function as the YEXT function to signify its applicability
to applications above the saturation pressure. The YEXT function is
given by
"
#
pi p
pi p
¼ log YEXT ¼ log
pi ½ ðVt =Vi Þ 1
pi ðVrel ÞEXT 1
(4.111a)
Ahmed (2015) suggests that in addition to applying the above
expression, the “pi” can be also replaced with “p” and referring
to the function as “YEXTp”:
"
pi p
YEXTp ¼ log
p ðVrel ÞEXT 1
#
(4.111b)
The two expressions above should be plotted jointly for the purpose of
clearly identifying the saturation pressure “psat” and the corresponding
saturation volume “Vsat.” Ahmed (2015) proposed the following
modified procedure for analyzing the CCE laboratory data, which is
summarized in the following steps:
PVT Measurements 367
1. To convert laboratory relative volume “Vrel” data into “(Vrel)EXT,”
divide each Vrel value by the relative volume “ðVi =Vsat Þpmax ” that
corresponds to the highest pressure “pmax” in the CCE data set; that is,
ðVrel ÞEXT ¼
ðVt =Vsat Þ
Vt
¼
ðVi =Vsat Þpmax Vi
2. Calculate the extended Y-function values at each pressure using
Eqs. (4.111a), (4.111b); that is,
"
#
pi p
YEXT ¼ log pi ðVrel ÞEXT 1
"
#
pi p
YEXTp ¼ log p ðVrel ÞEXT 1
3. On a Cartesian (regular) scale, plot results from steps 1 and 2 as a
function of pressure on the same graph; that is, YEXT, YEXTp, and
(Vrel)EXT versus pressure.
4. The two extended Y-function plots will clearly confirm the
laboratory-reported saturation pressure or correctly identify a
correct saturation pressure “psat.”
5. From the “(Vrel)EXT” plot, identify relative volume value at the
saturation pressure and designate the value as Vpsat EXT , which
represents the correct ratio of “Vsat/Vi.”
6. Draw the best straight-line fit of the extended Y-function “YEXT” data
below the saturation pressure and determine the coefficients “a&b”
of the best fit to give
YEXT ¼ a + bp
7. Smooth the extended relative volume “(Vrel)EXT” data by
recalculating their values at all pressures below the saturation
pressure from the following expression:
ðVrel ÞEXT ¼ 1 +
p p
i
pi 10 a + b p
8. The traditional “smoothed” relative volume “Vrel” data below the
saturation pressure can then be generated as a function of pressure
from the following expression:
ðVrel ÞEXT
Vrel ¼ Vpsat EXT
The approach can be clarified further by reworking Example 4.31.
368 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.32
Smooth the laboratory recorded relative volume data of Table 4.5 for the
extended Y-function approach and verify the reported saturation pressure of
1936 psi. Notice that the laboratory report suggests that the bubble point pressure
could be greater than the reported value.
Solution
The application of the modified Y-function procedure are summarized in the
following steps:
1. Convert laboratory relative volume “Vrel” data into “(Vrel)EXT” by
dividing each Vrel by relative volume that corresponds to the highest
pressure “pmax” in the CCE data set; that is,
ðVrel ÞEXT ¼
ðVt =Vsat Þ
Vt
¼
ðVi =Vsat Þpmax Vi
p
Vrel
(Vrel)EXT
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
0.9371
0.9422
0.9475
0.9532
0.9592
0.9657
0.9728
0.9805
0.989
0.9909
0.9927
0.9947
0.9966
0.9987
1
1.0014
1.0018
1.003
1.0042
1.0058
1.0139
1.0324
1.0625
1.1018
1.0611
1.2504
1.3694
1.502
1.9283
2.496
3.4464
1
1.005442
1.0111098
1.017181
1.023583
1.03052
1.038096
1.046313
1.055384
1.057411
1.059332
1.061466
1.063494
1.065735
1.067122
1.068616
1.069043
1.070323
1.071604
1.073311
1.081955
1.101697
1.133817
1.175755
1.132323
1.334329
1.461317
1.602817
2.057731
2.663536
3.677729
PVT Measurements 369
2. Calculate the extended Y-function at pressure using Eqs. (4.111a), (4.111b).
"
#
pi p
pi p
¼ log pi ½ðVt =Vi Þ 1
pi ðVrel ÞEXT 1
YEXT ¼ log
"
#
pi p
YEXTp ¼ log p ðVrel ÞEXT 1
p
(Vrel)EXT
YEXT
YEXTp
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
1
1.005442
1.011098
1.017181
1.023583
1.03052
1.038096
1.046313
1.055384
1.057411
1.059332
1.061466
1.063494
1.065735
1.067122
1.068616
1.069043
1.070323
1.071604
1.073311
1.081955
1.101697
1.133817
1.175755
1.132323
1.334329
1.461317
1.602817
2.057731
2.663536
3.677729
1.150272
1.141839
1.128138
1.11551
1.100447
1.083326
1.065451
1.045765
1.040874
1.037047
1.031919
1.027808
1.022504
1.019568
1.010578
1.008075
1.000568
0.993205
0.983634
0.938341
0.851138
0.741001
0.632375
0.767278
0.377828
0.250547
0.144491
0.08004
0.26309
0.45799
1.185035
1.21439
1.242081
1.275211
1.3113
1.352171
1.401243
1.460739
1.473576
1.488233
1.502409
1.518502
1.534388
1.545576
1.537934
1.535881
1.529502
1.52327
1.515287
1.477559
1.406853
1.321173
1.241168
1.413761
1.072477
0.998628
0.942464
0.839662
0.771668
0.714439
3. On a Cartesian (regular) scale, plot results from steps 1 and 2 as a function of
pressure on the same graph; that is, YEXT, YEXTp, and (Vrel)EXT versus
pressure. The plot is shown in Fig. 4.29.
370 CHAPTER 4 PVT Properties of Crude Oils
1.4
2
YEXTp = log
pi – p
p (Vrel)EXT –1
1.5
1.3
YEXT = log
8P
0.5
pi – p
pi (Vrel)EXT –1
1.2
0.0
00
95
YEXT
1
42
6+
Vext
1.1
EX
T =
Y
–0
.84
0
–0.5
(Vrel)EXT =
Vt
Vi
–1
0
1000
2000
3000
4000
5000
6000
1
7000
n FIGURE 4.29 Extended Y-function vs. pressure.
4. The two extended Y-function plots show saturation pressure 1950 psia as
compared with the reported saturation pressure of 1936 psi.
5. From the “(Vrel)EXT” plot, identify
relative volume value at the “corrected”
saturation pressure to give: Vpsat EXT 1:066261. It should be noted that
the value of the (Vrel)EXT at laboratory-reported saturation pressure of 1936
is 1.067122.
6. Draw the best straight-line fit of the extended Y-function “YEXT” data
below the saturation pressure and determine the coefficients “a&b” of
the best fit to give
YEXT ¼ a + bp ¼ 0:84426 + 0:000958p
7. Smooth the extended relative volume “(Vrel)EXT” data by recalculating their
values at all pressures below the saturation pressure from the following
expression:
ðVrel ÞEXT ¼ 1 +
p p
6500 p
i
¼ 1+
pi 10a + bp
6500 100:84426 + 0:000958 p
PVT Measurements 371
p
Smoothed (Vrel)EXT
1950
1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
1.066261
1.068549
1.069554
1.069892
1.070745
1.071607
1.072833
1.078896
1.093463
1.118727
1.154434
1.21271
1.307899
1.443168
1.598972
2.095052
2.688102
3.485354
8. Generate the traditional “smoothed” relative volume “Vrel” data below the
saturation pressure from the following expression:
ðVrel ÞEXT
ðVrel ÞEXT
Vrel ¼ ¼
Vpsat EXT 1:066261
p
(Vrel)EXT
Smoothed Vrel
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
1950
1936
1930
1.000000
1.005442
1.011098
1.017181
1.023583
1.03052
1.038096
1.046313
1.055384
1.057411
1.059332
1.061466
1.063494
1.065735
1.066261
1.068549
1.069554
0.937857
0.942961
0.948265
0.95397
0.959975
0.96648
0.973586
0.981292
0.989799
0.9917
0.993502
0.995503
0.997405
0.999506
1
1.002146
1.003088
372 CHAPTER 4 PVT Properties of Crude Oils
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
1.069892
1.070745
1.071607
1.072833
1.078896
1.093463
1.118727
1.154434
1.21271
1.307899
1.443168
1.598972
2.095052
2.688102
3.485354
1.003406
1.004205
1.005014
1.006163
1.011849
1.025512
1.049206
1.082694
1.137348
1.226622
1.353485
1.499607
1.964859
2.521054
3.268762
The laboratory density of 0.6484 g/cm3 measured at 1936 psi should be
corrected to the new saturation pressure of 1950 psi to give
ðρsat Þ1950psi ¼
ðVrel ÞEXT @ 1936psi
Vrel ÞEXT @ 1950psi
ρsat @1936psi ¼
1:07122
0:6468
1:066261
¼ 0:65142g=cm3
with all densities at higher pressure than 1950 psi given by
ρ¼
ρsat 0:65142
¼
Vrel
Vrel
p
(Vrel)EXT
Vrel
Density
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
1950
1
1.005442
1.011098
1.017181
1.023583
1.03052
1.038096
1.046313
1.055384
1.057411
1.059332
1.061466
1.063494
1.065735
1.066261
0.937857
0.942961
0.948265
0.95397
0.959975
0.96648
0.973586
0.981292
0.989799
0.9917
0.993502
0.995503
0.997405
0.999506
1
0.694579
0.690819
0.686955
0.682847
0.678576
0.674009
0.669089
0.663835
0.658129
0.656868
0.655676
0.654358
0.653111
0.651737
0.651416
PVT Measurements 373
Differential Liberation Test
The differential liberation (DL) test is perhaps the most common laboratory
test that is conducted on crude oil samples. The DL process is summarized in
the following steps and is illustrated schematically in Fig. 4.30:
n
n
n
n
n
n
A crude oil sample is placed in a visual PVT cell at its bubble point
pressure and reservoir temperature.
The cell pressure is reduced in stages, usually 10–15 pressure
increments, from the saturation pressure to atmospheric pressure.
The liberated gas at each pressure stage is allowed to reach equilibrium
with the cell remaining oil.
The volume of the two phases (ie, remaining oil volume “VL” and liberated
gas volume “Vgas”) are measured and recorded at each pressure level.
The liberated gas is then displaced at constant cell pressure to a metering
device (eg, gasometer) and the measured volume is corrected to SC and
designated to as “(Vgas)sc.”
The above depletion process is repeated at constant reservoir
temperature until a pressure close to atmospheric pressure is reached.
Having arrived at the last stage, the volume of the residual oil is first
measured at last cell conditions, then corrected to SC (ie, 14.7 psia and
60°F), and designated as “VLsc.”
pb
VL
p1<pb
p2<p1
p1 Gas off
(V gas)sc
Vgas
Vsat
VL1
Vt
Hg
p sc Gas off
(V gas) sc
Gas off
(V gas) sc
Vgas
VL2
Vt
VL1
Hg
p2
VLsc
VL2
Hg
= total volume at p&T
Vt
VL
= remaining oil volume at p&T
V Lsc
= remaining oil volume at p sc&T sc
V gas
= liberated gas volume at p&T
(Vgas)sc = liberated gas volume at psc&Tsc
n FIGURE 4.30 Differential vaporization test at constant temperature “T.”
Hg
Hg
Hg
14.7 psia
60°F
374 CHAPTER 4 PVT Properties of Crude Oils
The following nomenclatures are used to describe the various hydrocarbon
volume measurements associated with each pressure-depletion stage during
the DL experiment:
p ¼ cell pressure
T ¼ cell temperature
Vt ¼ total combined volume of the liberated gas and remaining
oil at p&T
VL ¼ remaining oil volume at p&T
VLsc ¼ remaining oil volume at psc&Tsc
Vgas ¼ liberated gas volume at p&T
(Vgas)sc ¼ liberated gas volume at psc&Tsc
It should be noted that the remaining oil at each depletion stage is subjected to continual compositional changes as it becomes progressively
richer in the heavier components. In addition, the above described type
of DL is characterized by a varying composition of the total hydrocarbon system; whereas the total system composition is kept unchanged
during the CCE test. The combined data from the DL and CCE tests
are considered sufficient to simulate the separation process taking place
in the reservoir. Consider the illustration in Fig. 4.31 which depicts a
possible depletion scenario of an oil reservoir and pressure profile
around the production well. In region “A”, the reservoir pressure “pr”
is greater than the bubble point pressure “pb” indicating an undersaturated oil system with the oil as the only moving phase, which is migrating (possibly by expansion) to region “B”. In region “B,” the reservoir
pressure is slightly below the bubble point pressure, which signifies the
start of the liberation of solution gas. However, the gas saturation “Sg” is
presumably below the critical gas saturation “Sgc” indicating that the gas
is immobile and will remain in contact with the oil, which resembles the
CCE separation process. Therefore, region “B” is appropriately simulated by the data obtained from the flash liberation test (CCE test).
Region “C” represents the pressure profile near the wellbore where
the pressure is approaching the bottom-hole flowing pressure “pwf.”
A large pressure sink occurs in region “B” causing the liberated gas
to exceed Sgc allowing the gas to flow. Because the gas has higher
mobility than oil, the liberated gas begins to flow leaving behind the
oil that originally contained it. Consequently, this behavior follows
the DL sequence. However, it should be pointed out that in the tubing
and surface separation facilities, the gas and oil flow simultaneously in
equilibrium resembling the flash expansion process. In general, the reservoir fluid flow dynamic is modeled by the combined flash and
DL tests.
PVT Measurements 375
Flash separation
Oil
Gas
Well
pr <<pb
pr ≤ pb
pr > pb
Sg > Sgc
Sg ≤ Sgc
Sg = 0
Only oil is
flowing
Only oil is
flowing
Oil+gas
are flowing
Pressure
C
A
B
pb
Oil+gas
are flowing
Only oil is
flowing
Only oil is
flowing
pwf
Distance from wellbore
Pressure profile around wellbore
n FIGURE 4.31 Reservoir phase distribution in an oil reservoir.
The experimental data obtained from the DL test include
n
n
n
n
n
n
n
n
The differential oil FVF “Bod”
The differential gas solubility; that is, solution GOR, “Rsd”
Gas FVF “Bg” in ft3/scf
Gas compressibility factor “Z”
Total FVF “Btd”
Composition of the liberated gas
Gas specific gravity
Density of the remaining oil as a function of pressure
Differential Oil FVFs
The differential oil FVFs, Bod (commonly called the relative oil volume
factors), are calculated at all pressure levels by dividing the recorded oil
volumes, VL, by the volume of residual oil, VLsc:
Bod ¼
VL
VLsc
(4.112)
376 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.33
A DL test was conducted on a crude oil sample at 180°F. The sample exhibited a
bubble point pressure of 3565 psig. The residual (remaining) oil volume was
corrected to SC of 14.7 psia and 60°F to give 60 cm3. The following
measurements were obtained from a differential expansion (DE) test:
Pressure
(psig)
Total Volume
(cm3)
Liquid Volume
(cm3)
Gas Removed
(scf)
3565
3000
2400
1800
1200
600
200
0
99.29514779
105.1180611
100.8688861
98.43652432
100.8219751
118.801706
168.9280437
63.85313132
99.29514779
92.66229773
87.02573049
82.03645568
77.70667102
73.45739134
70.05558226
63.85313132
0
0.086792948
0.07660421
0.067170194
0.061887145
0.059245621
0.043396474
0.061509785
Bold number represents boiling point pressure.
Using the above reported DE measurements data, calculate the differential oil
FVF “Bod”; that is, relative oil volume factor.
Solution
Calculate Bod by applying Eq. (4.112) to give
Bod ¼
VL
VL
¼
VLsc 60
Pressure
(psig)
Total Volume
(cm3)
Liquid
Volume (cm3)
3565
3000
2400
1800
1200
600
200
0
0 psig, 60°F
99.29514779
105.1180611
100.8688861
98.43652432
100.8219751
118.801706
168.9280437
63.85313132
99.29514779
92.66229773
87.02573049
82.03645568
77.70667102
73.45739134
70.05558226
63.85313132
60
Gas
Removed
(scf)
0
0.086792948
0.07660421
0.067170194
0.061887145
0.059245621
0.043396474
0.061509785
Bod (bbl/
STB)
1.6549
1.7520
1.6811
1.6406
1.6804
1.9800
2.8155
1.0642
1.0000
Bold number represents boiling point pressure.
Differential Gas Solubility “Rsd”
The gas solubility (ie, differential solution GOR, “Rsd”) at any pressure
is defined as the volume of gas that is dissolved (remaining) in solution
at the pressure per stock-tank barrel of the corrected residual oil.
PVT Measurements 377
Mathematically, the differential solution GOR, Rsd, is calculated by dividing the volume of gas remaining (dissolved) in solution at the specified
pressure by the residual oil volume.
EXAMPLE 4.34
Using the DE measurements data in Example 4.33, calculate the differential
solution GOR “Rsd.”
Solution
Step 1. Convert the volume of the corrected residual oil volume as expressed in
cubic centimeter to stock-tank barrel:
Stock tank volume ¼
60
ð30:48Þ3 5:615
¼ 0:000377STB
Step 2. Calculate TOTAL volume of the dissolved gas:
Total volume of the dissolved gas ¼ 0.456606 scf
Step 3. Calculate Rsd by dividing the volume of the dissolved gas at each
pressure by 0.000377, to give
Pressure
(psig)
Total Volume
(cm3)
Liquid Volume
(cm3)
Gas Removed (scf)
3565
3000
2400
1800
1200
600
200
0
0 psig, 60°F
99.29514779
105.1180611
100.8688861
98.43652432
100.8219751
118.801706
168.9280437
63.85313132
99.29514779
92.66229773
87.02573049
82.03645568
77.70667102
73.45739134
70.05558226
63.85313132
60
0
0.086792948
0.07660421
0.067170194
0.061887145
0.059245621
0.043396474
0.061509785
Total ¼ 0.456606 scf
Bold number represents boiling point pressure.
Gas FVF “Bg”
The gas FVF is the defined ratio of the volume “(Vgas)p,T” occupied by n
moles of gas at a specified pressure and temperature to the volume
“(Vgas)sc” occupied by the same number of moles (ie, n moles) at SC. Mathematically, this is defined by
Vgas p, T 3
; ft =scf
Bg ¼ Vgas sc
(4.113)
Dissolved
(Gas scf)
Rsd
(scf/STB)
0.4566
0.3698
0.2932
0.2260
0.1642
0.1049
0.0615
0.0000
1210
980
777
599
435
278
163
0
378 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.35
Using the DE measurements data in Example 4.33, calculate the gas FVF.
Solution
As an Example of calculating Bg at 3000 psig:
n
Volume of the free gas at 3000 psig ¼ total volume-volume of the liquid
Vgas 3000, 180 ¼ ð105:1180611 92:66229773Þ=ð30:48Þ3 ¼ 0:00044ft3
n
Calculate Bg at 3000 psig and 180°F by applying the definition of this gas
property:
Bg ¼ 0:00044=0:086792948 ¼ 0:005068ft3 =scf
n
Pressure
(psig)
Total
Volume (cm3)
3565
3000
2400
1800
1200
600
200
0
0 psig, 60°F
99.29514779
105.1180611
100.8688861
98.43652432
100.8219751
118.801706
168.9280437
63.85313132
The tabulated values of Bg are given below:
Liquid Volume (cm3)
Gas Removed (scf)
Free Gas (ft3)
Bg (ft3/scf)
99.29514779
92.66229773
87.02573049
82.03645568
77.70667102
73.45739134
70.05558226
63.85313132
60
0
0.086792948
0.07660421
0.067170194
0.061887145
0.059245621
0.043396474
0.061509785
Total ¼ 0.456606 scf
No free gas
0.00044
0.00049
0.00058
0.00082
0.00160
0.00349
0.005068
0.006382
0.008622
0.01319
0.027028
0.080459
Bold number represents boiling point pressure.
Gas Deviation Factor “Z”
Using the volume of the free gas at any pressure and the corresponding
volume at SC, the gas deviation factor can be calculated by applying the
gas ESO at both conditions; that is,
p Vgas p, T ¼ ZnRT to give : n ¼ p Vgas p, T =ðZRT Þ
psc Vgas sc ¼ Zsc nRTsc to give : n ¼ p Vgas sc =ðZsc RTsc Þ
where
Zsc ¼ 1
Tsc ¼520°R
T ¼ 640°R
psc ¼ 14.7 psia
PVT Measurements 379
Equating the above two expressions and solving for Z, gives
Z¼
p Vgas p, T
T
T
sc psc Vgas sc
EXAMPLE 4.36
Using the DE measurements data in Example 4.33, calculate the gas FVF.
Solution
As a sample of the Z-factor calculations at 3000 psig:
p Vgas p, T
T
sc Z¼
T
psc Vgas sc
Z¼
ð3000 + 14:7Þð0:00044Þ
520
¼ 0:8445
180 + 460
14:7ð0:08679Þ
The tabulated values of Z as a function of pressure are given below:
Pressure
(psig)
Total Volume
(cm3)
Liquid Volume
(cm3)
Gas Removed (scf)
Free Gas (ft3)
Z
3565
3000
2400
1800
1200
600
200
0
0 psig, 60°F
99.29514779
105.1180611
100.8688861
98.43652432
100.8219751
118.801706
168.9280437
63.85313132
99.29514779
92.66229773
87.02573049
82.03645568
77.70667102
73.45739134
70.05558226
63.85313132
60
0
0.086792948
0.07660421
0.067170194
0.061887145
0.059245621
0.043396474
0.061509785
Total ¼ 0.456606 scf
No free gas
0.00044
0.00049
0.00058
0.00082
0.00160
0.00349
0.844483
0.85174
0.864839
0.885583
0.918313
0.954804
Bold number represents boiling point pressure.
Total FVF “Btd” From DE Test
The two-phase (total) FVF from DE test is determined by applying the
definition of this property as expressed mathematically by Eq. (4.69).
In terms of differential data, Btd is given by
where
Btd ¼ Bod + ðRsbd Rsd ÞBg
Rsbd ¼ gas solubility at the bubble point pressure, scf/STB
Rsd ¼ gas solubility at any pressure, scf/STB
Bod ¼ oil FVF at any pressure, bbl/STB
Bg ¼ gas FVF, bbl/scf
The generated PVT data in Examples 4.33–4.35 are used to calculate B.
380 CHAPTER 4 PVT Properties of Crude Oils
EXAMPLE 4.37
Using solution data from Examples 4.33–4.35, calculate the total FVF.
Solution
Applying Eq. (4.115), to calculated Btd gives
Btd ¼ Bod + ðRsbd Rsd ÞBg
Btd ¼ Bod + ð1210 Rsd ÞBg
Notice, Bg must be expressed in bbl/scf where 1 bbl ¼ 5.615. For example,
at 3000 psig
Btd ¼ 1:544 + ð1210 980Þ ð0:00507=5:615Þ ¼ 1:752bbl=STB
Pressure
(psig)
Rsd (scf/STB)
Bod (bbl/STB)
Bg (ft3/scf)
Btd (bbl/STB)
3565
3000
2400
1800
1200
600
200
0
1210
980
777
599
435
278
163
0
1.655
1.544
1.450
1.367
1.295
1.224
1.168
1.064
0.00507
0.00638
0.00862
0.01319
0.02703
0.08046
1.655
1.752
1.943
2.306
3.116
5.711
16.170
Bold number represents boiling point pressure.
Table 4.6 shows results of the DL test conducted on a crude oil bottom-hole
sample taken from the Big Butte Field in Montana. The graphical presentation of
the data in terms of the relative oil volume and differential GOR are presented in
Figs. 4.32 and 4.33, respectively. The test indicates that the differential GOR and
differential relative oil volume at the bubble point pressure are 933 scf/STB and
1.730 bbl/STB, respectively. The symbols Rsdb and Bodb are used to represent
these two values:
Rsdb ¼ 933 scf=STB
Bodb ¼ 1:730 bbl=STB
Column 4 of Table 4.6 shows the relative total volume, Btd, from DL as
calculated from the following expression:
Btd ¼ Bod + ðRsdb Rsd ÞBg
PVT Measurements 381
Table 4.6 Differential Liberation Data (Differential Vaporization at 247°F)
Pressure (psig)
1
Rsda
2
Bodb
3
Btdc
4
ρ (g/cm3)
5
Z
6
Bgd
7
Incremental γ g
8
pb > 1936
1700
1500
1300
1100
900
700
500
300
185
120
0
933
841
766
693
622
551
479
400
309
242
195
0
1.73
1.679
1.639
1.6
1.563
1.525
1.486
1.44
1.382
1.335
1.298
1.099
1.73
1.846
1.982
2.171
2.444
2.862
3.557
4.881
8.138
13.302
20.439
0.7745
0.6484
0.6577
0.665
0.672
0.679
0.6863
0.6944
0.7039
0.7161
0.7256
0.7328
2.375
0.864
0.869
0.876
0.885
0.898
0.913
0.932
0.955
0.97
0.979
0.01009
0.01149
0.01334
0.01591
0.01965
0.02559
0.03626
0.06075
0.09727
0.14562
0.885
0.894
0.901
0.909
0.927
0.966
1.051
1.23
1.423
1.593
At 60°F ¼ 1.000.
Gravity of residual oil ¼ 34.6°API at 60°F.
Density of residual oil ¼ 0.8511 g/cm3 at 60°F.
a
Cubic feet of gas at 14.73 psia and 60°F per barrel of residual oil at 60°F.
b
Barrels of oil at indicated pressure and temperature per barrel of residual oil at 60°F.
c
Barrels of oil and liberated gas at indicated pressure and temperature per barrel of residual oil at 60°F.
d
Cubic feet of gas at indicated pressure and temperature per cubic feet at 14.73 psia and at 60°F.
n FIGURE 4.32 Relative volume vs. pressure.
382 CHAPTER 4 PVT Properties of Crude Oils
Solution GOR “Rsd” at 247°F
1000
Rsbd
Solution GOR (scf/STB)
“Rsd”
800
600
R sd
400
200
0
0
500
1000
1500
2000
Pressure
n FIGURE 4.33 Solution gas-oil ratio vs. pressure.
The gas deviation factor, Z, listed in column 6 represents the Z-factor of the
liberated (removed) solution gas at the specific pressure. These values are
calculated from the recorded gas volume measurements as follows:
Z¼
!
Vgas p, T p
T
Tsc
!
Vgas sc psc
Column 7 of Table 4.6 that contains the gas FVF, Bg, which can be also
determined in terms of the Z-factor by applying Eq. (3.52); that is,
psc ZT
Bg ¼
Tsc p
Quality Checks of DE Data
n
Oil density from DE and CCE tests
Plotting the saturated oil density data (ie, (p pb) from the DL test and
those of the undersaturated density from the CCE should exhibit a
smooth transition from the saturated to the undersaturated density
values, as shown schematically in Fig. 4.34. A steep increase or erratic
density values around the saturation pressure might indicate flow
assurance problems; for example, asphaltene precipitation.
PVT Measurements 383
Pressure-Volume Relations
(at 247 °F)
Differential Vaporization
(at 247 °F)
Pressure
(psig)
b>>1936
1700
1500
1300
1100
900
700
500
300
185
120
0
Relative
Oil
Volume
Bod (B)
Solution
Gas/Oil
Ratio
Rsd (A)
933
1.730
841
1.679
766
1.639
693
1.600
622
1.563
551
1.525
479
1.486
400
1.440
309
1.382
242
1.335
195
1.298
0
1.099
@ 60⬚F = 1.000
Relative
Total
Volume
Btd (C)
1.730
1.846
1.982
2.171
2.444
2.862
3.557
4.881
8.138
13.302
20.439
Gas
Formation Incremental
Gas
Oil
Deviation Volume
Gravity
Factor
Density
Factor
3
(Air = 1.000)
(D)
(g/cm )
Z
0.6484
0.6577
0.6650
0.6720
0.6790
0.6863
0.6944
0.7039
0.7161
0.7256
0.7328
0.7745
0.01009
0.01149
0.01334
0.01591
0.01965
0.02559
0.03626
0.06075
0.09727
0.14562
0.864
0.869
0.876
0.885
0.898
0.913
0.932
0.955
0.970
0.979
0.885
0.894
0.901
0.909
0.927
0.966
1.051
1.230
1.423
1.593
2.375
De
Density
ns
ity
e?
en
fro
m
Pressure
(psig)
6500
6000
5500
5000
4500
4000
3500
3000
2500
2400
2300
2200
2100
2000
b>>1936
1930
1928
1923
1918
1911
1878
1808
1709
1600
1467
1313
1161
1035
782
600
437
DE
lt
ha
p
As
rob
y from
Densit
pb
CCE
Pressure
n FIGURE 4.34 Quality checks of oil density from DE and CCE.
n
Quality checks of saturated density
The saturated density from any particular separation test should be
calculated and compared with the reported laboratory data. All the
reported and calculated saturated density values should fall within a
maximum of 1% of each other; otherwise, the validity of the test should
be reconsidered. The following mathematical expression resulting from
the mass material balance on the crude oil is used to calculate the
saturated oil density:
"
#
n
5 X
1
141:5
ρodb ¼
+ 1:483 10 psc
γ gi + 1 ðRsd Þi ðRsd Þi + 1
Bodb 131:5 + °APIRO
i¼1
(4.114)
where
ρodb ¼ oil density in g/cm3 at pb from DE test, g/cm3
APIRO ¼ residual oil API gravity from DE test
Relative
Volume (A)
0.9371
0.9422
0.9475
0.9532
0.9592
0.9657
0.9728
0.9805
0.9890
0.9909
0.9927
0.9947
0.9966
0.9987
1.0000
1.0014
1.0018
1.0030
1.0042
1.0058
1.0139
1.0324
1.0625
1.1018
1.1611
1.2504
1.3694
1.5020
1.9283
2.4960
3.4464
Y-Function
(B)
Density
(g/cm3)
0.6919
0.6882
0.6843
0.6803
0.6760
0.6714
0.6665
0.6613
0.6556
0.6544
0.6531
0.6519
0.6506
0.6493
0.6484
2.108
2.044
1.965
1.874
1.784
1.710
1.560
1.453
1.356
384 CHAPTER 4 PVT Properties of Crude Oils
Bodb ¼ oil FVF at pb from DE test, bbl/STB
Rsd ¼ gas solubility from DE test, scf/STB
psc ¼ standard pressure (eg, 14.7), psia
EXAMPLE 4.38
Using the following DE test data listed in Table 4.7, validate the saturated oil
density.
Solution
Calculate the saturated density by applying Eq. (4.114), to give
"
#
n
5 X
1
141:5
+ 1:483 10 psc
γ gi + 1 ðRsd Þi ðRsd Þi + 1
ρodb ¼
Bodb 131:5 + °APIRO
i¼ 1
ρodb ¼
1
141:5
+ 1:483 105 ð14:7Þ½0:7559ð1050 924Þ
1:602 131:5 + 34:9
+ 0:7376ð924 737Þ + 0:7272ð737 547Þ + 0:7448ð547 359Þ
+ 0:8649ð359 195Þ + 1:9116ð195 0Þ ¼ 0:6704g=cm3
The absolute error is 0.02% indicating a reliable data set.
n
Quality checks of the total FVF from DE and CCE tests
The two-phase FVF “Bt” as a function of pressure can be calculated and
compared graphically with those generated from the DE test to validate the
quality of these tests. The two-phase FVF from the CCE test can be
calculated below the saturation pressure from the following expression:
ðBt ÞCCE ¼ Bodb Vrel
(4.115)
Table 4.7 DE Test Data for Example 4.38
Differential Liberation at 182°F
Physical Properties
i
psia
Bo
Rs
Z
Bg
Sp.gr.
Density (g/cm3)
1
2
3
4
5
6
7
3009
2610
2012
1408
807
310
15
1.602
1.55
1.475
1.4
1.324
1.242
1.063
1050
924
737
547
359
195
0
0.777
0.779
0.802
0.855
0.929
1
0.00538
0.00699
0.01026
0.01891
0.05206
0.7559
0.7376
0.7274
0.7448
0.8649
1.9116
0.6699
0.6789
0.6929
0.7083
0.7265
0.7487
0.7994
Residual oil gravity at STP: 0.8494 [34.9°API].
Bold numbers represent boiling point pressure.
PVT Measurements 385
Plotting (Bt)CCE and Btd as a function of pressure on the same plot is used to
evaluate the quality of both tests. By visual inspection, if the two curves are
overlapped it will indicate acceptable CCE and DE tests.
EXAMPLE 4.39
Perform QCs on the total FVF using the DE and CCE data listed in Table 4.8.
Table 4.8 DE and CCE Test Data for Example 4.39
DE Test Data
p
Rsd
Bod
Btd
3565
3000
2400
1800
1200
600
200
0
0
1210
980
777
599
435
278
163
0
0
1.655
1.655
1.544
1.752
1.450
1.943
1.367
2.306
1.295
3.116
1.224
5.713
1.168
16.180
1.064
1.064219
Bold number represents boiling point pressure.
CCE Test Data
Density
0.6397
0.6614
0.6823
0.7035
0.7231
0.7437
0.7610
0.7794
0.7794
Z
Bg
Sp.gr
0.843
0.850
0.863
0.884
0.916
0.953
0.00507
0.00638
0.00862
0.01319
0.02703
0.08046
0.741
0.720
0.710
0.712
0.757
0.876
1.662
1.661785
Pressure
(psig)
Relative
Volume
Density
(g/cm3)
6500
6475
6023
5524
5019
4528
4027
3722
3669
3626
3604
3585
3565
3559
3540
3519
3480
3445
3356
3197
2938
2556
2078
1554
1151
876
709
565
0.9677
0.9679
0.9714
0.9759
0.9810
0.9866
0.9931
0.9976
0.9984
0.9990
0.9994
0.9997
1.0000
1.0005
1.0019
1.0036
1.0068
1.0098
1.0179
1.0341
1.0664
1.1328
1.2656
1.5353
1.9425
2.4526
2.9633
3.6454
0.6611
0.6609
0.6585
0.6555
0.6521
0.6484
0.6441
0.6413
0.6407
0.6403
0.6401
0.6399
0.6397
386 CHAPTER 4 PVT Properties of Crude Oils
18.000
16.000
14.000
12.000
10.000
Bt
8.000
DE Bt
• CCE Bt
6.000
4.000
2.000
0.000
0
1000
2000
3000
Pressure
4000
Pressure
Relative
Density
CCE Bt
(psig)
6500
6475
6023
5524
5019
4528
4027
3722
3669
3626
3604
3585
3565
3559
3540
3519
3480
3445
3356
3197
2938
2556
2078
1554
1151
876
709
565
Volume
0.9677
0.9679
0.9714
0.9759
0.9810
0.9866
0.9931
0.9976
0.9984
0.9990
0.9994
0.9997
1.0000
1.0005
1.0019
1.0036
1.0068
1.0098
1.0179
1.0341
1.0664
1.1328
1.2656
1.5353
1.9425
2.4526
2.9633
3.6454
(g/cm3)
0.6611
0.6609
0.6585
0.6555
0.6521
0.6484
0.6441
0.6413
0.6407
0.6403
0.6401
0.6399
0.6397
(bbl/STB)
1.6014
1.60172
1.60767
1.61505
1.62349
1.6328
1.64358
1.65091
1.65223
1.65333
1.6539
1.6544
1.65492
1.6557
1.65811
1.6609
1.66623
1.67116
1.68453
1.71132
1.76487
1.87462
2.09449
2.54087
3.21465
4.05887
4.90396
6.0328
n FIGURE 4.35 Total formation volume factor from DE and CCE test data for Example 4.39.
Solution
n Calculate the Bt using the CCE test data by applying Eq. (4.115).
ðBt ÞCCE ¼ Bodb Vrel
ðBt ÞCCE ¼ 1:655Vrel
Validate both tests by plotting (Bt)CCE and (Bt)DE as a function of pressure as
shown in Fig. 4.35.
n Fig. 4.35 shows an excellent match indicating quality laboratory data.
n
Separator Tests
It should be noted that the plots presented in Figs. 4.32 and 4.33 for Bod and
Rsd data as a function of pressure resemble the traditional appearance of the
oil FVF “Bo” and the solution gas solubility “Rs” curves, leading to their misuse in reservoir calculations. The DL test simulates the behavior of the crude
oil and the liberation of the solution gas in the reservoir; however, in the
applications of the material balance equation and other reservoir engineering
mathematical tools require properties of the produced fluid as related to the
surface separation facilities, not those of the DE test. It should be noted that
the values of Rsd and Bod from the DE test are determined based on using the
PVT Measurements 387
n FIGURE 4.36 Impact of surface facilities on DE data.
reservoir residual oil volume as a reference volume, whereas the oil FVF
“Bo” and gas solubility “Rs” are based on using the volume of the residual
oil at separator stock-tank conditions as the reference volume. As shown
in Fig. 4.36, the process of producing the hydrocarbon system through the
surface facilities must be included and accounted for to generate the required
PVT properties of all aspects of reservoir engineering calculations.
Separator tests are conducted to determine the changes in the volumetric
behavior of the reservoir fluid as the fluid passes through the separator
(or separators) and then into the stock-tank. The resulting volumetric
behavior is influenced to a large extent by the operating conditions of
the surface separation facilities in terms of their pressures and temperatures. The primary objective of conducting separator tests, therefore, is
to provide the essential laboratory information necessary for determining
the optimum surface separation conditions, which in turn maximize the
stock-tank oil production. The laboratory flash separator unit is designed
to simulate a full-scale field separation equipment. The laboratory unit
usually operates from atmospheric pressure up to 2000 psig and temperatures from 40 to 250°F. The lab separation unit is used to determine the
388 CHAPTER 4 PVT Properties of Crude Oils
optimum conditions for field separation by calculating the changes in FVF
and GORs with relation to changes in separator pressure and temperature.
In addition, it provides
n
n
n
volume of gas liberated at each stage,
composition and specific gravity of the liberated gas from each
stage, and
composition of the stock-tank liquid and specific gravity.
Results of the flash separation test, when appropriately combined with the
DL test data, provide a means of obtaining the PVT parameters (Bo, Rs, and
Bt) required for petroleum engineering calculations.
The test involves placing a hydrocarbon sample at its saturation pressure and
reservoir temperature in a PVT cell. The volume of the sample is measured
as Vsat. The hydrocarbon sample then is displaced and flashed through a laboratory multistage separator system, commonly one to three stages. The
pressure and temperature of these stages are set to represent the desired
or actual surface separation facilities. The gas liberated from each stage
is removed to measure its specific gravity, composition, and volume at
SC. The volume of the remaining oil in the last stage (representing the
stock-tank condition) is measured and recorded as (Vo)st. These experimental measured data can then be used to determine the oil FVF and gas solubility at the bubble point pressure as follows:
Vsat
ðVo Þst
Vg sc
Rsfb ¼
ðVo Þst
Bofb ¼
(4.116)
(4.117)
where
Bofb ¼ bubble point oil FVF, as measured by flash liberation, bbl
of the bubble point oil/STB
Rsfb ¼ bubble point solution GOR as measured by flash liberation, scf/STB
(Vg)sc ¼ total volume of gas removed from separators, scf
This laboratory procedure is repeated at a series of different separator pressures and a fixed temperature. It usually is recommended that four of these
tests be used to determine the optimum separator pressure, which usually is
considered the separator pressure that results in minimum oil FVF. At the
same pressure, the stock-tank oil gravity is at a maximum and the total
evolved gas (ie, the separator gas and the stock-tank gas) is at a minimum.
A typical example of a set of separator tests for a two-stage separation system is shown in Table 4.9. The tabulated values in Table 4.9 indicate that
PVT Measurements 389
Table 4.9 Separator Tests
Separator
Pressure (psig)
Temperature
(°F)
GOR,a
Rsfb
°API at
60°F
FVF,b
Bofb
50
0
Total
100
0
Total
200
0
Total
300
0
Total
75
75
737
41
778
676
92
768
602
178
780
549
246
795
40.5
1.481
40.7
1.474
40.4
1.483
40.1
1.495
75
75
75
75
75
75
a
GOR in cubic feet of gas at 17.65 psia and 60°F per barrel of stock-tank oil at 60°F.
FVF is barrels of saturated oil at 2.620 psig and 220°F per barrel of stock-tank oil at 60°F.
b
optimum separator pressure is around 100 psia and is considered to be the
separator pressure that results in the minimum oil FVF. It is important to
note that the oil FVF varies from 1.474 to 1.495 bbl/STB, while the gas solubility ranges from 768 to 795 scf/STB. Table 4.9 indicates that the values of
the crude oil PVT data depend on the method of surface separation.
Table 4.10 presents the results of performing a separator test on the Big
Butte crude oil. The DL data, as expressed in Table 4.6, shows that the solution GOR at the bubble point is 933 scf/STB, as compared with the measured
value of 646 scf/STB from the separator test. This significant difference, as
also shown in Figs. 4.37 and 4.38, is attributed to the fact that the processes
of obtaining residual oil and stock-tank oil from bubble point oil are different. The DL is considered a multiple series of flashes at the elevated reservoir temperatures; the separator test generally is a one- or two-stage flash at
low pressure and low temperature. The quantity of gas released is different,
and the quantity of final liquid is different. Again, it should be pointed out
that oil FVF, as expressed by Eq. (4.116), is defined as “the volume of oil at
reservoir pressure and temperature divided by the resulting stock-tank oil
volume after it passes through the surface separators.”
Adjustment of DE Data to Separator Conditions
The material balance calculations and black oil reservoir simulation
require the PVT data of the crude oil including the oil FVF “Bo” and
gas solubility “Rs” as a function of the pressure. Amyx et al. (1960)
Table 4.10 Separator Test Data, Separator Flash Analysis
Separator Flash Analysis
Gas/Oil
Ratio
(scf/bbl)
Flash
Conditions
(psig)
(°F)
(A)
Gas/Oil
Ratio
(scf/STB)
(B)
StockTank Oil
Gravity at
60°F
(°API)
Formation
Volume
Factor
Bofb (C)
Separator
Volume
Factor
Specific
Gravity of
Flashed Gas
(D)
(Air 5
1.000)
1936 247
28
0
130
80
Oil
Phase
Density
0.6484
593
13
Rsfb
632
13
38.8
1.527
Rsfb = 646
GOR1 = 632
1.066
1.010
1.132*
**
Bofb
GOR2 = 13
Rsfb = GOR1 + GOR2 = 646 scf/STB
Pb
Bofb =
(V)pb,T
VST
T, pb
Tsep = 130°F
psep = 28 psig
n FIGURE 4.37 A comparison between Bodb and Bofb.
Tsc = 80
psc = 0 psig
(V)pb,T
(V)ST
= 1.527 bbl / STB
0.7823
0.8220
PVT Measurements 391
n FIGURE 4.38 A comparison between Rsdb and Rsfb.
proposed a procedure for constructing the oil FVF and gas solubility
curves using the DL data in conjunction with the experimental separator
flash data for a given set of separator conditions. The method is summarized in the following steps.
Step 1. Calculate the differential shrinkage factors at various pressures by
dividing each relative oil volume factors, Bod, by the relative oil
volume factor at the bubble point, Bodb:
Sod ¼
Bod
Bodb
(4.118)
where
Bod ¼ differential relative oil volume factor at pressure p, bbl/STB
Bodb ¼ differential relative oil volume factor at the bubble point
pressure pb, psia, bbl/STB
Sod ¼ differential oil shrinkage factor, bbl/bbl of bubble point oil
The differential oil shrinkage factor has a value of 1 at the
bubble point and a value less than 1 at subsequent pressures
below pb.
Step 2. Adjust the relative volume data by multiplying the separator (flash)
FVF at the bubble point, Bofb, by the differential oil shrinkage
factor, Sod, at each pressures. Mathematically, this relationship is
expressed as follows:
392 CHAPTER 4 PVT Properties of Crude Oils
Bo ¼ Bofb
Bod
¼ Bofb Sod
Bodb
(4.119)
where
Bo ¼ oil FVF, bbl/STB
Bofb ¼ oil FVF at pb from separator test
Sod ¼ differential oil shrinkage factor, bbl/bbl of bubble point oil
Step 3. Calculate the oil FVF at pressures above the bubble point pressure
by multiplying the relative oil volume data, Vrel, as generated from
the CCE test, by Bofb:
Bo ¼ ðVrel ÞðBofb Þ
(4.120)
where
Bo ¼ oil FVF above the bubble point pressure, bbl/STB
Vrel ¼ relative oil volume, bbl/bbl.
Step 4. Adjust the differential gas solubility data, Rsd, to give the required
gas solubility factor, Rs:
Rs ¼ Rsfb ðRsdb Rsd Þ
Bofb
Bodb
(4.121)
where
Rs ¼ gas solubility, scf/STB
Rsfb ¼ bubble point solution GOR from the separator test, scf/STB
Rsdb ¼ solution GOR at the bubble point pressure as measured by
the DL test, scf/STB
Rsd ¼ solution GOR at various pressure levels as measured by the
DL test, scf/STB
It should be pointed out the Eqs. (4.119), (4.121) usually produce
values less than 1 for Bo and negative values for Rs at low pressures.
The calculated curves of Bo and Rs versus pressures must be
manually drawn to Bo ¼ 1.0 and Rs ¼ 0 at atmospheric pressure.
McCain (2002) suggested the following simple expression for
adjusting the differential gas solubility data, Rsd, to give the
required gas solubility factor, Rs:
Rs ¼ Rsd
Rsfb
Rsdb
Step 5. Obtain the two-phase (total) FVF, Bt, by multiplying values of
the relative oil volume, Vrel, below the bubble point pressure
by Bofb:
Bt ¼ ðBofb ÞðVrel Þ
(4.122)
PVT Measurements 393
where
Bt ¼ two-phase FVF, bbl/STB
Vrel ¼ relative oil volume below the pb, bbl/bbl.
Similar values for Bt can be obtained from the DL test by
multiplying the relative total volume, Btd by Bofb:
Bt ¼ ðBtd ÞðBofb Þ=Bodb
(4.123)
EXAMPLE 4.40
The CCE test, DL test, and separator test for the Big Butte crude oil system
are given in Tables 4.5, 4.6, and 4.10, respectively. Calculate the oil FVF at 4000
and 1100 psi, the gas solubility at 1100 psi, and the two-phase FVF at 1300 psi.
Solution
Step 1. Determine Bodb, Rsdb, Bofb, and Rsfb from Tables 4.9 and 4.10:
Bodb ¼ 1:730 bbl=STB
Rsdb ¼ 933 scf=STB
Bofb ¼ 1:527 bbl=STB
Rsfb ¼ 646 scf=STB
Step 2. Calculate Bo at 4000 by applying Eq. (4.120):
Bo ¼ ðVrel ÞðBofb Þ
Bo ¼ ð0:9657Þð1:57Þ ¼ 1:4746 bbl=STB
Step 3. Calculate Bo at 1100 psi by applying Eqs. (4.118), (4.119):
Sod ¼
Bod
Bodb
Sod ¼
1:563
0:9035
1:730
and
Bo ¼ Bofb Sod
Bo ¼ ð0:9035Þð1:5277Þ ¼ 1:379 bbl=STB
Step 4. Calculate the gas solubility at 1100 psi using Eq. (4.121):
Bofb
Bodb
1:527
Rs ¼ 646 ð933 622Þ
¼ 371 scf=STB
1:730
Rs ¼ Rsfb ðRsdb Rsd Þ
Note: Using the correction proposed by McCain gives
Rsfb
646
¼ 622
¼ 431 scf=STB
Rs ¼ Rsd
Rsdb
933
394 CHAPTER 4 PVT Properties of Crude Oils
Step 5. From the pressure-volume relations (ie, constant composition data) of
Table 4.5, the relative volume at 1300 psi is 1.2579 bbl/bbl. Using
Eq. (4.122), calculate Bt to give
Bt ¼ ðBofb ÞðVrel Þ
Bt ¼ ð1:527Þð1:2579Þ ¼ 1:921 bbl=STB
Equivalently applying Eq. (4.123) gives
Bt ¼ ðBtd ÞðBofb Þ=Bodb
Bt ¼ ð2:171Þð1:527Þ=1:73 ¼ 1:916 bbl=STB
Table 4.11 presents a complete documentation of the adjusted differential
vaporization data for the Big Butte crude oil system. Figs. 4.39 and 4.40
compare graphically the adjusted values of Rs and Bo with those of unadjusted values. Note that no adjustments are needed for the gas FVF, oil density, or viscosity data.
Dodson et al. (1953) proposed perhaps the most ideal laboratory procedure
for obtaining crude oil PVT data. The methodology is commonly called the
Dodson method or, alternatively, the composite liberation test. The proposed procedure is summarized below and is schematically illustrated in
Fig. 4.41:
n
n
n
n
n
A large crude oil sample is placed in a PVT cell at reservoir temperature
and bubble point pressure.
A small sample of the crude oil is withdrawn to conduct a separator test
and generate Bofb and Rsfb.
The pressure is reduced below the saturation pressure, the evolved gas is
removed at constant pressure, and its composition and properties are
determined.
A small oil sample is again withdrawn from the remaining oil and
pumped through the separators to obtain Bo and Rs and other properties
at this particular reduced pressure.
The above process should be repeated at several progressively lower
reservoir pressures until complete curves of Bo and Rs versus reservoir
pressure have been obtained.
It should be noted that the Dodson procedure is an expensive laboratory procedure and, therefore, is rarely conducted.
Quality Checks of Separator Test Data. The saturated density as obtained
from the separator test should be calculated and compared with the reported
PVT Measurements 395
Table 4.11 Differentiation Liberation Data Adjusted to Separator Conditions
Pressure (psig)
Rsa
Bob
6500
6000
5500
5000
4500
4000
3500
3000
2500
2100
2300
2200
2100
2000
b > 1936
1700
1500
1300
1100
900
700
500
300
185
120
0
646
646
646
646
646
646
646
646
646
646
646
646
646
646
646
564
498
434
371
309
244
175
95
36
1.431
1.439
1.447
1.456
1.465
1.475
1.486
1.497
1.51
1.513
1.516
1.519
1.522
1.525
1.527
1.482
1.446
1.412
1.379
1.346
1.311
1.271
1.22
1.178
1.146
Gas FVFc
0.01009
0.01149
0.01334
0.01591
0.01965
0.02559
0.03626
0.06075
0.09727
0.14562
Oil Density (g/cm3)
0.6919
0.6882
0.6843
0.6803
0.676
0.6714
0.6665
0.6613
0.6556
0.6544
0.6531
0.6519
0.6506
0.6493
0.6484
0.6577
0.665
0.672
0.679
0.6863
0.6944
0.7039
0.7161
0.7256
0.7328
0.7745
Separator conditions:
First stage ¼ 28 psig at 130°F.
Stock-tank ¼ 0 psig at 80°F.
a
Cubic feet of gas at 14.73 psia and 60°F per barrel of stock-tank oil at 60°F.
b
Barrel of oil at indicated pressure and temperature per barrel of stock-tank oil at 60°F.
c
Cubic feet of gas at indicated pressure and temperature per cubic feet at 14.73 psia and 60°F.
laboratory data. The reported and calculated saturated density value should
fall within a maximum of 1% of each other; otherwise, the validity of the test
should be reconsidered. The following mathematical expression resulting
from the mass material balance on the crude oil is used to calculate the saturated oil density:
ρob ¼
"
#
n X
1
141:5
+ 1:483 105 psc
γ g GOR sep
Bofb 131:5 + °APIST
sep¼1
Oil/Gas Viscosity Ratio
19
21.3
23.8
26.6
29.8
33.7
38.6
46
52.8
58.4
396 CHAPTER 4 PVT Properties of Crude Oils
n FIGURE 4.39 Adjusted gas solubility vs. pressure.
n FIGURE 4.40 Adjusted oil formation volume factor vs. pressure.
where
ρob ¼ oil density in g/cm3
APIST ¼ stock-tank API gravity
Bofb ¼ oil FVF at pb from the separator test bbl/STB
(GOR)sep ¼ separator GOR, scf/STB
γ g ¼ liberated gas gravity from each separator stage
PVT Measurements 397
n FIGURE 4.41 Composite DE test “Dodson Procedure.”
EXAMPLE 4.41
Table 4.12 results from a separator test performed on a bottom-hole crude
oil sample. The test was conducted using a three-stage plus stock-tank
laboratory separation arrangement. The test shows a saturated oil density of
0.669 g/cm3 at the bubble point pressure of 3009 psi and 182°F. The following
additional measured data were obtained:
n
n
Bofb ¼ 1.465 bbl/STB
Rsfb ¼ 839 scf/STB
The residual oil specific gravity and API gravity at stock-tank conditions are
0.8313 and 38.72, respectively. Validate the saturated oil density.
Solution
Calculate the saturated density from the following density expression:
"
#
n 5 X
1
141:5
ρob ¼
+ 1:483 10 psc
γ g GOR sep
Bofb 131:5 + °APIST
sep¼1
to give
ρob ¼
1
141:5
+ 1:483 105 ð14:7Þ½587 0:757
1:465 131:5 + 38:72
+ 114 0:76 + 104 0:997 + 34 1:395 ¼ 0:669
398 CHAPTER 4 PVT Properties of Crude Oils
Table 4.12 Separator Test Data, Separator Flash Analysis
Stage
Pressure
Temperature
Rs (scf/STB)
Bo
Gas Gravity
ρo (g/cm3)
1
2
3
Stock-tank
3009
796
317
65
15
182
134
80
80
60
839
587
114
104
34
1.465
1.226
1.139
1.082
1
0.757
0.76
0.997
1.395
0.669
0.7206
0.7587
0.7781
0.8313
The calculated density is an exact match with the reported laboratory value,
confirming the validity of the experimental saturated oil density.
Extrapolation of Reservoir Fluid Data
In partially depleted reservoirs or fields that originally existed at the bubble
point pressure, it is difficult to obtain a fluid sample that represents the oil in
the reservoir at the time of discovery. Also, in collecting fluid samples from
oil wells, the possibility exists of obtaining samples with a saturation pressure
that might be lower than or higher than the actual saturation pressure of the
reservoir. In these cases, it is necessary to correct or adjust the laboratory PVT
measured data to reflect the actual or desired saturation pressure, including
n
n
n
n
constant composition expansion
DE test
oil viscosity data
separator test data
Correcting CCE Data. The correction procedure summarized in the following steps is based on calculating the Y-function value for each point
below the “old” saturation pressure.
Step 1.
n
Calculate the Y-function, as expressed by Eq. (4.107), for each
point by using the old saturation pressure:
Y¼
n
n
pold
sat p
pðVrel 1Þ
Plot the values of the Y-function versus pressure on a Cartesian
scale and draw the best straight line. Notice that points in the
neighborhood of the saturation pressure may be erratic and need
not be used.
Calculate the coefficients a and b of the straight-line equation:
Y ¼ a + bp
PVT Measurements 399
n
Recalculate the relative volume, Vrel, values by applying
Eq. (4.109) and using the “new” saturation pressure and the
coefficients “a&b” from step 3:
Vrel ¼ 1 +
pnew
sat p
pða + bpÞ
(4.124)
Step 2.
To determine points above the “new” saturation pressure, proceed by
the following steps:
n
Plot the “old” relative volume values above the “old” saturation
pressure versus pressure on a regular scale and draw a smooth
curve connecting points. IF a straight line is obtained, draw the
best straight line through these points and determine the slope
of line S. Note that the slope is negative; that is, S < 0.
n
Draw a smooth curve or straight line that passes through the point
with coordinate Vrel ¼ 1, pnew
and parallel to the line/curve
sat
of previous step.
n
Relative volume data above the new saturation pressure are
read from the curve. If a straight line, generate the new relative
volume data from the following expression at any pressure p:
Vrel ¼ 1 S pnew
sat p
(4.125)
where S ¼ slope of the line and p ¼ pressure.
EXAMPLE 4.42
The pressure-volume relations of the Big Butte crude oil system are given in
Table 4.5. The test indicates that the oil has a bubble point pressure of 1930 psig at
247°F. The Y-function for the oil system is expressed by the following linear equation:
Y ¼ 1:0981 + 0:000591p
Above the bubble point pressure, the relative volume data versus pressure
exhibits a straight-line relationship with a slope, S, of 0.0000138. The field
surface production and well testing data indicate that the actual bubble point
pressure is approximately 2500 psig. Please reconstruct the pressure-volume
data using the new reported saturation pressure.
Solution
Use Eqs. (4.124), (4.125):
pnew
2500 p
sat p
¼ 1+
pða + bpÞ
pð1:0981 + 0:000591pÞ
new
Vrel ¼ 1 S psat p ¼ 1 ð0:0000138Þð2500 pÞ
Vrel ¼ 1 +
to give the results in the table below.
400 CHAPTER 4 PVT Properties of Crude Oils
Pressure (psig)
Old Vrel
New Vrel
Comments
6500
6000
5000
4000
3000
(pb)new 5 2500
2000
(pb)old 5 1936
1911
1808
1600
600
437
0.9371
0.9422
0.9532
0.9657
0.9805
0.9890
0.9987
1.0000
1.0058
1.0324
1.1018
2.4960
3.4404
0.9448
0.9517
0.9655
0.9793
0.9931
1.0000
1.1096
1.1299
1.1384
1.1767
1.1018
2.4960
3.4404
Eq. (4.125)
Eq. (4.125)
Eq. (4.125)
Eq. (4.125)
Eq. (4.125)
Eq. (4.124)
Eq. (4.124)
Eq. (4.124)
Eq. (4.124)
Eq. (4.124)
Eq. (4.124)
Eq. (4.124)
Bold number represents boiling point pressure.
Correcting DL Data. The laboratory-measured relative oil volume Bod
data must be corrected to account for the new bubble point pressure pnew
b .
The proposed procedure is summarized in the following steps:
Step 1. Plot the Bod data versus gauge pressure on a regular scale.
Step 2. Draw the best straight line through the middle pressure range of
30–90% pb.
Step 3. Extend the straight line to the new bubble point pressure, as shown
schematically in Fig. 4.42.
Step 4. Transfer any curvature at the end of the original curve, that is,
ΔBo1 at pold
b , to the new bubble point pressure by placing ΔBo1
above or below the straight line at pnew
b .
Step 5. Select any differential pressure Δp below the pold
b and transfer the
corresponding curvature ΔBo1 to the pressure (pnew
b –Δp).
Step 6. Repeat this process and draw a curve that connects the generated Bod
points with the original curve at the point of intersection with the
straight line. Below this point no change is needed.
The correction procedure for the solution to the GOR, Rsd, data is identical to
that of the relative oil volume data.
Correcting Oil Viscosity Data. The oil viscosity data can be extrapolated
to a new, higher bubble point pressure by applying the following steps.
Step 1. Defining the “fluidity” as the reciprocal of the oil viscosity (ie, 1/
μo), calculate the fluidity for each point below the original saturation
pressure.
Step 2. Plot fluidity versus pressure on a Cartesian scale (see Fig. 4.43).
PVT Measurements 401
New (Bod)@ new pb
Original (Bod)@ old pb + ΔB1
ΔB1
Original (Bod)@ old pb
ΔB2
Original Bo curve
ΔB1
ΔB2
Bod
Δp
Δp
0.9 pb
0.3 pb
(pb)old
n FIGURE 4.42 Adjusting Bod curve to reflect new bubble point pressure.
.
1/m o
. .
.
mo
.
(m ob)old
(m ob)new
0
n FIGURE 4.43 Extrapolation of oil viscosity to new pb.
Pressure
(pb)old
(pb)new
(pb)new
402 CHAPTER 4 PVT Properties of Crude Oils
mo
A
(m ob)old
B
(m ob)new
0
Pressure
(pb)old
(pb)new
n FIGURE 4.44 Extrapolation of oil viscosity above the new pb.
Step 3. Draw the best straight line through the points and extend it to the
new saturation pressure, pold
b .
Step 4. New oil viscosity values above pold
b are read from the straight line.
To obtain the oil viscosity for pressures above the new bubble point pressure
follow the following steps:
pnew
b
Step 1. Plot the viscosity values for all points above the old saturation
pressure on a Cartesian coordinate, as shown schematically in
Fig. 4.44, and draw the best straight line through them, line A.
draw
Step 2. Through the point on the extended viscosity curve at pnew
b
a straight line (line B) parallel to A.
Step 3. Viscosities above the new saturation pressure then are read from
line A.
Correcting the Separator Tests Data
n
Stock-tank GOR and gravity
No corrections are needed for the stock-tank GOR and the stock-tank
API gravity.
n
Separator GOR
The gas solubility at the new bubble point pressure (Rsfb)new is changed
in the same proportion as the differential ratio was changed; that is,
old
Rnew
sfb ¼ Rsfb
new
Rsfb
Rold
sdb
(4.126)
PVT Measurements 403
n
The separator GOR then is the difference between the new (corrected)
gas solubility Rnew
sfb and the unchanged stock-tank GOR.
Formation volume factor
The oil FVF at the new bubble point pressure “Bofb” is adjusted in the
same proportion as the DL value:
old
Bnew
ofb ¼ Bofb
new
Bodb
Bold
odb
(4.127)
EXAMPLE 4.43
Results of the DL and the separator tests on the Big Butte crude oil system are
given in Tables 4.6 and 4.10, respectively. The field data indicates that the actual
bubble point value is 2000 psi. The correction procedure for Bod and Rsd as
described previously was applied, to give the following values at the new bubble
point of 2000 psi:
Bnew
odb ¼ 2:013bbl=STB
Rnew
sbd ¼ 1134scf=STB
Using the separator test data given in Table 4.10, calculate the gas solubility and
the oil FVF at the new bubble point pressure.
Solution
Gas solubility is found from Eq. (4.126):
new
1134
old Rsdb
¼
646
¼
R
¼ 785scf=STB
Rnew
sfb
sfb
933
Rold
sdb
Separator GOR ¼ 785 13 ¼ 772scf=STB
The oil FVF is found by applying Eq. (4.127):
new
old Bodb
¼
B
Bnew
ofb
ofb
Bold
odb
2:013
Bob ¼ 1:527
¼ 1:777bbl=STB
1:730
Laboratory Analysis of Gas-Condensate Systems
A standard laboratory analysis on a gas-condensate sample consists of
n
n
n
recombination and analysis of the separator samples,
measuring the pressure-volume relationship; that is, CCE test, and
constant volume depletion (CVD) test.
The above listed routine laboratory tests are detailed next.
404 CHAPTER 4 PVT Properties of Crude Oils
Recombination of Separator Samples
Obtaining a representative sample of the reservoir fluid is considerably more
difficult for a gas-condensate fluid than for a conventional black oil reservoir. The principal reason for this difficulty is that the liquid may condense
from the reservoir fluid during the sampling process, and if representative
proportions of both liquid and gas are not recovered, then an erroneous composition is calculated. Because of the possibility of erroneous compositions
and the limited volumes obtainable, subsurface sampling is seldom used in
gas-condensate reservoirs. Instead, surface sampling techniques are used,
and samples are obtained only after long-stabilized flow periods. During this
stabilized flow period, volumes of liquid and gas produced in the surface
separation facilities are accurately measured, and the fluid samples are then
recombined in these proportions. The hydrocarbon composition of separator
samples also are determined by chromatography, low-temperature
fractional distillation, or a combination of both. Table 4.13 shows the
Table 4.13 Reservoir Fluid Composition
Basis of Recombination: 7407 scf Separator Gas/Barrel Separator Liquid
Component
Symbol
Name
N2
CO2
H2S
Nitrogen
Carbon dioxide
Hydrogen
sulfide
Methane
Ethane
Propane
i-Butane
n-Butane
i-Pentane
n-Pentane
Hexanes
Heptanes
Octanes
Nonanes
Decanes
Undecanes
Dodecanes
Tridecanes
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
Reservoir
Fluid
(mol%)
Separator
Gas
(mol%)
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
Molecular
Weight
Specific
Gravity
(Water 5 1.0)
0.063
1.154
0.000
0.002
0.470
0.000
0.001
0.242
0.000
28.01
44.01
34.08
0.809
0.818
0.801
0.055
1.069
0.000
80.655
11.141
3.935
0.941
1.087
0.385
0.234
0.191
0.090
0.074
0.040
0.010
16.006
8.871
7.900
3.501
5.491
4.330
4.112
7.754
7.754
9.629
5.953
4.110
2.824
2.053
1.819
3.002
3.118
4.073
2.379
3.731
3.652
3.468
7.812
9.272
12.057
8.325
6.445
4.854
3.864
3.722
16.04
30.07
44.10
58.12
58.12
72.15
72.15
86.18
95.90
107.07
119.51
134.17
147.00
161.00
175.00
0.300
0.356
0.507
0.563
0.584
0.624
0.631
0.664
0.706
0.738
0.766
0.779
0.790
0.801
0.812
72.624
10.859
4.428
1.259
1.634
0.875
0.716
1.130
1.105
1.261
0.775
0.519
0.351
0.255
0.226
(Continued)
PVT Measurements 405
Table 4.13 Reservoir Fluid Composition Continued
Basis of Recombination: 7407 scf Separator Gas/Barrel Separator Liquid
Component
Separator
Gas
(mol%)
Symbol
Name
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
C29
C30+
Tetradecanes
Pentadecanes
Hexadecanes
Heptadecanes
Octadecanes
Nonadecanes
Eicosanes
Heneicosanes
Docosanes
Tricosanes
Tetracosanes
Pentacosanes
Hexacosanes
Heptacosanes
Octacosanes
Nonacosanes
Triacontanes
plus
Total
Molecular
weight
100.000
20.56
Separator
Liquid
(mol%)
Separator
Liquid
(wt.%)
1.417
1.120
0.833
0.713
0.576
0.484
0.339
0.280
0.239
0.186
0.147
0.128
0.096
0.075
0.060
0.047
0.171
3.148
2.698
2.161
1.976
1.691
1.489
1.091
0.951
0.854
0.691
0.569
0.518
0.401
0.329
0.271
0.219
0.929
100.000
85.54
100.00
Molecular
Weight
Specific
Gravity
(Water 5 1.0)
190.00
206.00
222.00
237.00
251.00
263.00
275.00
291.00
305.00
318.00
331.00
345.00
359.00
374.00
388.00
402.00
465.22
0.823
0.833
0.840
0.848
0.853
0.858
0.863
0.868
0.873
0.878
0.882
0.886
0.890
0.894
0.897
0.900
0.914
Reservoir
Fluid
(mol%)
0.176
0.139
0.103
0.089
0.072
0.060
0.042
0.035
0.030
0.023
0.018
0.016
0.012
0.009
0.007
0.006
0.021
100.000
28.64
Compositional Groupings of Reservoir Fluid
Group
mol%
wt.%
MW
SG
Tb
Total fluid
C7+
C10+
C20+
C30+
100.000
5.350
2.210
0.220
0.021
100.000
26.113
14.464
2.531
0.345
28.64
139.76
187.45
330.17
465.22
0.781
0.822
0.882
0.914
N/A
N/A
N/A
1232
1410
Bold number represents boiling point pressure.
hydrocarbon analyses of the separator liquid and gas samples taken from the
Nameless field. The samples were combined with a GOR of 7407 scf per
barrel of separator liquid at separator conditions of 640 psia and 100°F.
The separator liquid density is 0.683 g/cm3 or 42.62 lb/ft3. The laboratory
recombined data indicates that the overall well stream system contains
72.624 mol% methane and 5.35 mol% heptanes-plus.
406 CHAPTER 4 PVT Properties of Crude Oils
The Consistency Test on the Recombined GOR
As detailed earlier, a concise QC test on the laboratory-reported GOR can be
performed by solving the component material balance relationship for the
ratio “nL/nv” from
yi nt nL xi
¼ zi nv nv zi
which suggests plotting “yi/zi” versus “xi/zi” would produce a straight line
with a negative slope value of “nL/nv.” The GOR can then be calculated
from the following expression:
GOR ¼
2130:331ðρo Þpsep , Tsep
ðMWÞLiquid ðnL =nv Þ
; scf=bbl
Fig. 4.45 shows the straight line resulting from plotting “yi/zi” versus “xi/zi”
using the laboratory gas and liquid samples listed in Table 4.13. The slope of
the line in the figure is 0.1421, which is converted into GOR, to give
GOR ¼
GOR ¼
2130:331ðρo Þpsep , Tsep
ðMWÞLiquid ðnL =nv Þ
2130:331ð43:62Þ
¼ 7467 scf=bbl
ð85:54Þð0:1421Þ
which shows a very acceptable match with the reported GOR of 7407 scf/bbl
with an absolute error of 0.8%.
yi / zi
Symbol
N2
CO2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
xi / zi
n FIGURE 4.45 Recombined GOR consistency check.
Molecular
weight
Density,
g/cm3
Separator
Gas yi
Separator
Liquid xi
Reservoir
Fluid zi
(mol %)
(mol %)
(mol %)
0.063
1.154
80.655
11.141
3.935
0.941
1.087
0.385
0.234
0.191
0.090
0.074
0.040
0.010
0.002
0.470
16.006
8.871
7.900
3.501
5.491
4.330
4.112
7.754
8.264
9.629
5.953
4.110
2.824
2.053
1.819
0.055
1.069
72.624
10.859
4.428
1.259
1.634
0.875
0.716
1.130
1.105
1.261
0.775
0.519
0.351
0.255
0.226
34.13
85.54
0.683
xi /zi
yi /zi
0.03636
0.43966
0.22040
0.81693
1.78410
2.78078
3.36047
4.94857
5.74302
6.86195
7.47873
7.63600
7.68129
7.91908
1.14545
1.07951
1.11058
1.02597
0.88866
0.74742
0.66524
0.44000
0.32682
0.16903
0.08145
0.05868
0.05161
0.01927
PVT Measurements 407
Separator Pressure:
Separator Temperature:
Separator
Components Liquid
(mol%)
Methane
Ethane
Propane
i-Butane
n-Butane
i-Pentane
n-Pentane
Hexane
Heptane
Octane
Nonane
640
100
Separator
Gas
(mol%)
16.006
8.871
7.900
3.501
5.491
4.330
4.112
7.754
8.264
9.629
5.953
80.655
11.141
3.935
0.941
1.087
0.385
0.234
0.191
0.090
0.074
0.040
psia
°F
K = yi /xi log(K*p)
Tb
(°R)
5.039
1.256
0.498
0.269
0.198
0.089
0.057
0.025
0.011
0.008
0.007
200.95 343.01
332.21 549.74
415.94 665.59
470.45 734.08
490.75 765.18
541.76 828.63
556.56 845.37
615.37 911.47
668.74 970.57
717.84 1023.17
763.07 1070.47
3.509
2.905
2.503
2.236
2.103
1.755
1.561
1.198
0.843
0.692
0.634
Tc
(°R)
pc
(psia)
b
b(1/Tb-1/T)
667.0
707.8
615.0
527.9
548.8
490.4
488.1
439.5
397.4
361.1
330.7
803.9
1412.6
1798.2
2037.3
2151.2
2383.7
2478.2
2795.3
3079.2
3344.4
3592.9
3.2694
2.9676
2.6880
2.4779
2.4272
2.2323
2.1991
2.0005
1.8044
1.6177
1.4412
4.0
C1
3.0
log(k*p)
C2
C3
i-C4
n-C4
2.0
i-C5
n-C5
1.0
C6
C7
0.0
–2.5
C8
C9
–2.0
–1.5
–1.0
–0.5
0.0
0.5
b(1/Tb - 1/T)
1.0
n FIGURE 4.46 Hoffman plot consistency check on gas-condensate system.
The Hoffman plot for the separator samples should be made to evaluate the
consistency of the recombined fluid samples. Using the data in Table 4.13,
the Hoffman plot shown in Fig. 4.46 indicates good consistency in the laboratory measurement data.
Constant Composition Test
The experimental test procedure, as shown schematically in Fig. 4.46, is
similar to that conducted on a crude oil system that involves measuring
the pressure-volume relations of the reservoir fluid at reservoir temperature
with a PVT visual cell. The PVT cell allows the visual observation of the
1.5
2.0
2.5
3.0
408 CHAPTER 4 PVT Properties of Crude Oils
condensation process and the measurement of liquid dropout (LDO) volume
as the cell pressure is reduced below the saturation pressure. In general, the
CCE test on a gas-condensate sample is performed to provide the engineer
with the fundamental pressure-volume relationship of a retrograde gas system and other properties, including
n
n
The dew point pressure “pd.”
The relative volume “Vrel,” which is defined as the ratio of the total
measured volume of the hydrocarbon system “Vt” to that of the volume at
the dew point pressure “Vsat.” This property is defined by Eq. (4.105) as
Vrel ¼
n
Vt
Vsat
The liquid volume “VL” at each pressure below the dew point pressure
is reported into two different dimensionless forms. The first form
is based on the reference volume Vsat and other is based on Vt; that is,
ðLDOÞCCE
Vsat ¼
VL
Vsat
(4.128)
ðLDOÞCCE
Vt ¼
VL
Vt
(4.129)
It should be pointed out the above two expressions are interrelated
through the relative volume by the following relationship:
CCE
ðLDOÞCCE
Vsat ¼ Vrel ðLDOÞVt
n
(4.130)
The calculation of the traditional Y-function is not valid or applicable
using the relative volume from the CCE test on retrograde gas. However,
the two forms of extended Y-function along with the relative volume
can be used to confirm the reported dew point pressure. The two
extended Y-functions were given earlier by Eqs. (4.111a), (4.111b) as
"
#
pi p
YEXT ¼ log
pi ðVrel ÞEXT 1
"
#
pi p
YEXTp ¼ log p ðVrel ÞEXT 1
where
ðVrel ÞEXT ¼
n
Vrel
ðVrel Þpmax
The gas compressibility factor and gas density at the dew point pressure
are calculated from the volumetric, mass, and number of moles using
the following basic relationships:
Zd ¼
pd Vsat
nRT
PVT Measurements 409
ρd ¼
m
pd Ma
¼
Vsat Zd RT
The Z-factor and gas density at pressures above the saturation pressure
are calculated from
Z ¼ Zd
p
ðVrel Þ
pd
ρ¼
ρd
Vsat
where
Zd ¼ gas compressibility factor at the dew point pressure, pd
pd ¼ dew point pressure, psia
m ¼ weight of the sample
Ma ¼ apparent molecular weight of the gas sample
p ¼ pressure, psia
It should be noted that the above relationships are only valid for p pd.
Table 4.14 shows the dew point determination and the pressure-volume relations of the Nameless field. The dew point pressure of the system is reported
as 4100 psi at 320°F.
Table 4.14 Pressure-Volume Relations “CCE” of a Gas-Condensate
System at 320°F
Pressure
(psia)
Relative
Volume
(V/Vsat)
Gas
Density
(g/cm3)
11,150
10,000
9000
8000
7000
6000
5000
4250
4100
4000
3800
3600
3500
3000
2500
2000
0.616
0.641
0.666
0.698
0.739
0.795
0.880
0.978
1.000
1.023
1.064
1.125
1.153
1.302
1.538
1.919
0.389
0.374
0.360
0.344
0.325
0.302
0.273
0.245
0.240
Relative
Liquid
Volume (%)
0.00
0.08
0.28
0.52
0.65
1.42
2.23
2.98
Deviation
Factor (z)
1.572
1.466
1.371
1.276
1.183
1.090
1.006
0.950
0.938
(Continued)
410 CHAPTER 4 PVT Properties of Crude Oils
Table 4.14 Pressure-Volume Relations “CCE” of a Gas-Condensate
System at 320°F Continued
Pressure
(psia)
Relative
Volume
(V/Vsat)
1500
1100
2.587
3.614
Gas
Density
(g/cm3)
Relative
Liquid
Volume (%)
Deviation
Factor (z)
3.74
4.33
Notes:
Relative volume ( V/ Vsat) is the fluid volume at the indicated pressure and temperature relative to
the saturated fluid volume.
Specific volume (ft3/lb) ¼ 1/density (g/cm3) 62.428.
Density (lb/ft3) ¼ Density (g/cm3) 62.428.
Relative liquid volume % is the volume of liquid at indicated p and T/total volume at saturation pressure.
Bold numbers represent boiling point pressure.
EXAMPLE 4.44
Using the data in Table 4.14, confirm the reported dew point pressure of 4100 psi.
Solution
Step 1. Calculate (Vrel)EXT, (Yrel)EXT, and (Yrel)EXTp with results as tabulated below.
Pressure
(psia)
Relative Volume
(Vt/Vsat)
Gas Density
(g/cm3)
11,150
10,000
9000
8000
7000
6000
5000
4250
4100
4000
3800
3600
3500
3000
2500
2000
1500
1100
0.616
0.641
0.666
0.698
0.739
0.795
0.880
0.978
1.000
1.023
1.064
1.125
1.153
1.302
1.538
1.919
2.587
3.614
0.389
0.374
0.360
0.344
0.325
0.302
0.273
0.245
0.240
h
ðLDOÞVsat
0.00
0.08
0.28
0.52
0.65
1.42
2.23
2.98
3.74
4.33
iCCE
(%)
Z-Factor
(Vrel)EXT
YEXT
YEXTp
1.572
1.466
1.371
1.276
1.183
1.090
1.006
0.950
0.938
1.00000
1.04058
1.08117
1.13312
1.19968
1.29058
1.42857
1.58766
1.62338
1.66071
1.72727
1.82630
1.87175
2.11364
2.49675
3.11526
4.19968
5.86688
1.39164
1.09061
0.87577
0.69968
0.53673
0.36798
0.23087
0.20525
0.17999
0.13830
0.08286
0.05961
0.04674
0.17515
0.32536
0.50511
0.68725
0.45234
0.46881
0.47099
0.47263
0.47038
0.45788
0.44133
0.44065
0.43223
0.42481
0.40451
0.39920
0.38729
0.36393
0.33503
0.30333
0.27352
Bold number represents boiling point pressure.
Step 2. Plot Vrel, (Vrel)EXT, (Yrel)EXT, and (Yrel)EXTp as a function of pressure, as
shown in Figs. 4.47 and 4.48, to give a dew point pressure of 4100 psi,
confirming the reported value.
PVT Measurements 411
100%
gas
p < pd
pd
pi > pd
Vt
Vsat
Vt
VL
Hg
Vrel =
Vt
Vsat
(LDO)Vsat =
VL
Vsat
(LDO)VTot =
VL
Vt
n FIGURE 4.47 A schematic illustration of constant composition expansion (CCE) test.
n FIGURE 4.48 Dew point pressure from CCE test data for Example 4.44.
p << pd
p << pd
Vt
VL
Vt
VL
412 CHAPTER 4 PVT Properties of Crude Oils
CVD Test
CVD experiments are performed on gas condensates and volatile oils to simulate reservoir depletion performance and the compositional variations in
the producing gas. The test is designed to simulate LDO behavior in the pore
as pressure drops below the saturation pressure. The basic assumption in the
CVD test is that retrograde LDO will not have a sufficient liquid saturation
that is greater than critical saturation to become mobile. In addition, it is also
assumed that the reservoir pore volume will not change or impact the results
of the test; hence, the name “constant volume depletion test.” The test
provides a variety of useful and important information that is used in reservoir engineering and field processing calculations. The laboratory procedure
of the test is shown schematically in Fig. 4.49 and summarized in the
following steps:
Step 1. A measured amount of a representative sample of the original
reservoir fluid with a known overall composition of zi is charged to a
visual PVT cell at the dew point pressure, as shown in Fig. 4.49A.
The temperature of the PVT cell is maintained at the reservoir
temperature, T, throughout the experiment. The initial volume at the
saturation pressure “Vsat” is measured and used as a reference
volume. It should be pointed out that the CVD test procedure
n FIGURE 4.49 A schematic illustration of the constant volume depletion “CVD” test.
PVT Measurements 413
can also be conducted on a volatile oil sample. In this case, the PVT
cell initially contains liquid, instead of gas, at its bubble point
pressure.
Step 2. The initial gas compressibility factor is calculated from the real gas
equation:
Zd ¼
pd Vsat
ni RT
where
pd ¼ dew point pressure, psia
Vsat ¼ initial gas volume at dew point pressure, ft3
ni ¼ initial number of moles of the gas ¼ m/Ma
R ¼ gas constant, 10.73
T ¼ temperature, °R
Zd ¼ compressibility factor at dew point pressure
m ¼ weight of the gas sample, lb
Ma ¼ apparent molecular weight of the gas sample
Step 3. The cell pressure is reduced from the saturation pressure to
a predetermined level “p1” below the dew point pressure.
This reduction in the pressure is achieved by withdrawing
mercury from the cell, as illustrated in Fig. 4.49B. During
this process and as the pressure drops below the dew point
pressure, a second phase (retrograde liquid) is formed.
The fluid in the cell is brought to equilibrium at p1 and
the total volume (gas + liquid) is measured and recorded
as “Vt1.”
Step 4. Mercury is reinjected into the PVT cell at constant pressure “p1”
while an equivalent volume of remaining equilibrium gas is
simultaneously removed from the cell. When the initial volume
“Vsat” is reached, mercury injection ceases. The volume of the
retrograde liquid “VL” is visually measured as illustrated in
Fig. 4.49C while the volume of the removed gas is measured
and recorded as
Vg1 ¼ Vt1 Vsat
This step is designed to resemble the characteristics of the pressuredepletion process in retrograde gas reservoirs, whereas the
retrograde liquid remaining immobile in the reservoir and the
equilibrium gas is the only producing phase.
Step 5. The removed gas is charged at constant pressure of p1 into
a gasometer, where its volume is measured at SC and
414 CHAPTER 4 PVT Properties of Crude Oils
recorded as (Vgp)sc. A gas chromatographic analysis is
also performed to determine the composition of the removed gas, yi.
The number of moles of gas produced is calculated from the
expression
np ¼
psc Vgp sc
RTsc
where
np ¼ moles of gas produced
(Vgp)sc ¼ volume of gas produced measured at SC, scf
Tsc ¼ standard temperature, °R
psc ¼ standard pressure, psia
R ¼ 10.73
Step 6. The gas compressibility factor at cell pressure of p1 and temperature
is calculated from the real gas EOS as follows:
Z¼
p1 ðVt1 Vsat Þ
np RT
Step 7. Volume of the retrograde liquid “VL” is reported as a fraction of Vsat
and referred to as liquid dropout and given by
LDO ¼
VL
Vsat
where
VL ¼ liquid volume in the PVT cell at any depletion pressure
Vsat ¼ volume at dew point pressure
Step 8. Another important property that is routinely reported is termed
the two-phase compressibility factor “Ztwo-phase.” The two-phase
compressibility factor represents the total compressibility
of all the remaining fluids (gas and retrograde liquid) in the
cell and is computed at pressure “p” from the real gas
law as
Ztwo-phase ¼ pVsat
ni np RT
(4.131)
where
(ni np) ¼ represents the remaining moles of fluid in the cell at
pressure p
ni ¼ initial moles in the cell
np ¼ cumulative moles of gas removed
Step 9. The volume of gas produced is usually expressed as a
percentage of gas initially in place, and is calculated by dividing
PVT Measurements 415
the cumulative volume of the produced gas by the gas initially
in place, both at SC. It can be treated as the recovery
factor “RF.”
%Gp ¼
" X
#
np
ðni Þoriginal
100
Because 1 mol occupies 379.4 scf at SC, percentage of cumulative
gas produced can be expressed as
%Gp ¼
" X
#
379:4np
379:4ðni Þoriginal
100 ¼
"X #
Vgp sc
GIIP
100
or commonly as
RF ¼
Gp
100
GIIP
where
Gp ¼ cumulative gas removed (produced) from the PVT cell
pressure p, scf
GIIP ¼ volume of gas initially placed in the PVT at saturation
pressure, scf
RF ¼ recovery factor, %
It should be pointed out that Eq. (4.131) can be expressed in a
more convenient form by replacing moles of gas (ie, ni and np)
with their equivalent gas volumes:
Zd
p
Ztwo-phase ¼
pd 1 ðRFÞ
(4.132)
where
Zd ¼ gas deviation factor at the dew point pressure
pd ¼ dew point pressure, psia
p ¼ reservoir pressure, psia
RF ¼ gas recovery factor
Step 10. This experimental procedure is repeated over several
pressure increments until a minimum test pressure is reached, after
which the quantity and composition of the gas and retrograde liquid
remaining in the cell are determined.
416 CHAPTER 4 PVT Properties of Crude Oils
DISCUSSION OF RETROGRADE GAS
LABORATORY DATA
Results from CCE and CVD laboratory tests on a surface sample taken from
the Nameless field are presented in Tables 4.15 and 4.16, and Fig. 4.50.
A detailed description of these tests are given below.
n
The CCE test data presented in Table 4.15 indicates a dew point pressure
of 5022 psig at 218°F. Notice that column 4 lists the measured liquid
volumes from the CCE test as expressed with reference to Vsat as given
by Eq. (4.128); that is,
ðLDOÞCCE
Vsat ¼
VL
Vsat
Table 4.15 Constant Composition Expansion at 218°F
1
2
Pressure Vrel ¼ VVsatt
8000
7500
7000
6500
6000
5659 pres
5337
5022 psat
4713
4454
4034
3712
3328
3032
2660
2383
1987
1714
1511
1353
1226
839
0.87823
0.89166
0.90677
0.92442
0.94518
0.96149
0.97930
1.00000
1.02746
1.05552
1.11360
1.17160
1.26275
1.35464
1.50981
1.66647
1.98235
2.30023
2.61926
2.93902
3.25925
4.86448
3
Density
(g/cm3)
0.44768
0.44094
0.43359
0.42531
0.41597
0.40891
0.40148
0.39317
5
6
4
ðLDOÞVsat ¼ VL =Vsat Z-Factor Eg (Mscf/bbl)
0.00%
14.25%
20.65%
25.53%
27.33%
28.35%
28.70%
28.32%
28.32%
27.33%
26.44%
25.71%
25.20%
24.70%
22.86%
LDO is reported with Vsat as the reference volume.
1.42422
1.35579
1.28702
1.21855
1.15029
1.10381
1.06045
1.01912
1.63789
1.61302
1.58592
1.55540
1.52095
1.49492
1.46750
1.43688
Discussion of Retrograde Gas Laboratory Data 417
Table 4.16 Depletion Study at 218°F (Hydrocarbon Analyses of Produced Well Stream, mol%)
Reservoir Pressure (psig)
Well Stream Components
(D.P.) 5022
mol%
4100
mol%
3200
mol%
2300
mol%
1400
mol%
500
mol%
0
mol%
Hydrogen sulfide
Nitrogen
Carbon dioxide
Methane
Ethane
Propane
iso-Butane
n-Butane
iso-Pentane
n-Pentane
Hexanes
Heptanes-plus
Totals
0.002
0.084
1.163
65.410
12.107
5.329
1.280
2.039
0.911
0.902
1.247
9.526
100.000
0.000
0.091
1.201
69.857
12.331
5.256
1.196
1.862
0.760
0.760
1.023
5.662
100.000
0.000
0.095
1.231
72.736
12.474
5.005
1.121
1.721
0.690
0.690
0.887
3.350
100.000
0.000
0.097
1.263
74.286
12.942
4.895
1.046
1.567
0.635
0.598
0.760
1.910
100.000
0.000
0.096
1.295
74.907
13.243
5.033
1.049
1.557
0.570
0.537
0.675
1.038
100.000
0.000
0.087
1.307
72.742
13.552
6.053
1.318
1.983
0.706
0650
0.805
0.796
100.000
0.000
0.061
1.047
56.939
12.190
8.511
2.698
4.519
2.254
2.148
1.633
8.000
100.000
Heptanes-plus (C7+) fraction characteristics
1 Molecular weight
2 Specific gravity
154.383
0.8003
145.652
0.7917
143.256
0.7886
142.671
0.7874
141.163
0.7859
137.776
0.7830
133.233
1.2591
0.000
0.000
24.284
167.771
27.681
191.238
26.030
179.833
22.856
157.903
18.951
130.928
16.507
114.041
1.0191
1.0191
0.8922
0.9181
0.8411
0.8426
0.8354
0.7858
0.8586
0.7295
0.9207
0.5610
N/A
N/A
9.316
22.803
40.399
60.762
81.436
92.789
6.662
5.224
4.251
93.16
Mscf
5.054
3.684
2.780
228.03
Mscf
4.019
2.679
1.848
403.99
Mscf
3.471
2.093
1.264
607.62
Mscf
3.969
2.312
1.262
814.36
Mscf
11.189
8.860
6.567
927.89
Mscf
Condensed retrograde liquid volume
3 HC pore volume %
4 Bbls/MMscf of DP gas
Gas deviation factor
5 Equilibrium gas
6 Two-phase
Cumulative produced well stream volume
7 Vol% of initial DP Gas
0.000
GPM from CVD well stream compositions
8 Propane plus (C3+)
Butanes plus (C4+)
Pentanes-plus (C5+)
9 Cumulative recovery Gp
based on GIIP ¼ 1000 Mscf
n
9.471
8.013
6.957
0 Mscf
Table 4.16 contains the gas chromatographic analysis of the produced
gas stream from the CVD test, which clearly shows the expected
reduction trend in the concentration of C7+ and the associated increase in
the methane fraction concentration with pressure depletion. Rows 1 and
418 CHAPTER 4 PVT Properties of Crude Oils
30.0%
25.0%
LDO
20.0%
Pressure
(psig)
Liquid Dropout
%
5022
4100
3200
2300
1400
500
0
0.000
24.284
27.681
26.030
22.856
18.951
16.507
15.0%
10.0%
5.0%
0.0%
0
1000
2000
3000
4000
5000
6000
Pressure (psig)
n FIGURE 4.50 Retrograde condensation during gas depletion at 218°F.
n
n
n
2 of Table 4.16 show the molecular weights and specific gravities of the
C7+ in the produced well stream.
With pressure depletion, the concentrations of intermediate components
(ie, C2 through C6, in the produced gas stream) are seen to decrease as
they condense forming part of the LDO in the PVT cell. At lower
pressures, however, their concentrations in the produced gas increase as
they revaporize back to the gas phase, resulting in a reduction in the
volume of the LDO.
Row 3 in Table 4.16 lists the retrograde liquid dropout “LDO” as a
function of pressure. The LDO is expressed as a percentage of initial
volume “Vsat” at the dew point pressure.
Fig. 4.50 shows the traditional graphical presentation of the LDO as
a function of pressure. The cell volume at the dew point pressure is
used as a reference volume that essentially represents the reservoir
pore volume. The figure shows the dropped-out liquid is vaporized
back into the gas phase at lower pressure conditions. This behavior
occurs as a result of the transfer of components between the LDO
phase and gas phase. Some gas-condensate systems exhibit what is
referred to as a tail, where the liquid drops out very slowly before
increasing toward a maximum. The initial gradual and small buildup
of the condensate phase has been attributed to the contamination of
collected samples with hydraulic fluids from various sources during
drilling, production, and sampling. There is no firm evidence,
Discussion of Retrograde Gas Laboratory Data 419
however, that the trailing cannot be the true characteristic of a real
reservoir fluid.
The CVD measurements indicate that the maximum LDO of 27.7%
occurs around 3200 psi. For most retrograde reservoirs, the maximum
LDO occurs near 2000–2500 psi. Cho et al. (1985) correlated the
maximum LDO (expressed in %) with reservoir temperature “T” (in °R)
and the overall mole fraction of the heptanes-plus fraction “zC7 + ” in the
dew point mixture by the following expression:
ðLDOÞmax ¼ 93:404 + 479:9zC7 + 19:73 ln ðT 460Þ
The above expression can only be used to approximate the maximum
LDO. Using the key data from Table 4.16 applicable to the above
relationship, gives
ðLDOÞmax ¼ 93:404 + 479:9ð0:09526Þ 19:73 ln ð218Þ ¼ 32:9%
n
It should be noted that the CVD test is performed to simulate the LDO
occurring in the reservoir porous media as the pressure drops below the
dew point pressure. This retrograde condensation process is routinely
performed on a hydrocarbon sample placed in a PVT cell representing
the “pore volume”; however, without including initial reservoir water.
The LDO can be converted into oil saturation by accounting for the
presence of the initial water saturation, to give
So ¼ ðLDOÞð1 Swi Þ
where
So ¼ retrograde liquid (oil) saturation, %
LDO ¼ liquid dropout, %
Swi ¼ initial water saturation, fraction
Using the LDO data in Table 4.16 and assuming an initial water
saturation of 30%, the calculated values of the oil saturation as a
function of pressure can be tabulated from the data below.
n
p
LDO
So
5022
4100
3200
2300
1400
500
0
0
24.284
27.681
26.03
22.856
18.951
16.507
0
16.9988
19.3767
18.221
15.9992
13.2657
11.5549
Rows 5 and 6 in Table 4.16 display the equilibrium gas Z-factor and the
two-phase Z-factor, respectively, resulting from the laboratory CVD
420 CHAPTER 4 PVT Properties of Crude Oils
n
test. It should pointed out that when applying the traditional gas material
balance in terms of plotting (p/Z) versus Gp, the two-phase
compressibility factor must be used instead of single-phase Z-factor
when applying this analysis.
Row 7 in Table 4.16 represents cumulative “wellstream produced as a
percentage of initial gas-in-place at the dew point pressure.” This is the
total recovery factor of the well stream; that is,
RF ¼
n
Gp
100
GIIP
The recovery factor data from the CVD test can be used directly with
field production data to calculate future field performance and reserves
if the hydrocarbon pore volume does not change with declining reservoir
pressure.
Frequently, the surface gas is processed to remove and liquify all
hydrocarbon components that are heavier than methane, such as ethane,
propane, and heavier gases. These liquefied components are called
plant products or the liquid yield. The quantities of liquid products
are expressed in gallons of liquid per thousand standard cubic feet
of gas processed “gal/Mscf” and usually denoted as “GPM.” The
GPM is commonly reported for C3+ through C5+ groups in the
produced well streams at each pressure-depletion stage, as shown
in Row 8 of Table 4.16. The following expression can be used
for calculating the anticipated GPM for each component in the
gas phase:
psc
yi Mi
GPMi ¼ 11:173
Tsc
γ oi
Assuming psc ¼ 14.7 psia and Tsc ¼ 520°R, the above relationship is
reduced to
yi Mi
GPMi ¼ 0:31585
γ oi
with a total plant product GPM as given by the summation of GPM of
individual components:
GPM ¼
X
GPMi
i
For example, the liquid yield of the C3+ group at CVD depletion
pressure stage k is given by
ðGPMÞC3 + ¼
C7 +
X
i¼C3
ðGPMi Þk ¼ 0:31585
C7 + X
yi Mi
i¼ C3
γ oi
k
Discussion of Retrograde Gas Laboratory Data 421
where
yi ¼ mole fraction of component i in the gas phase
Mi ¼ molecular weight of component i
k ¼ pressure-depletion stage
γ oi ¼ specific gravity of component i as a liquid at SC (Chapter 1,
Table 1.1, column E)
The plant products, or the Yield “Y,” can also be expressed in terms of
STB/MMscf, to give
yi Mi
YSTB=MMscf i ¼ 7:5224
γ oi
It should be pointed out that liquefying and recovery of these products
depend on the plant separation efficiency “EP.” Because a complete
recovery of these products is not feasible, a rule of thumb indicates 5–
25% of ethane, 80–90% of the propane, 95% or more of the butanes, and
100% of the heavier components can be recovered from a simple surface
facility. Including the plant separation efficiency “EP” in calculating the
plant GPM gives
ðGPMÞC3 + ¼
C7 +
X
i¼C3
n
ðGPMi Þk ¼ 0:31585
C7 + X
yi Mi
i¼C3
γ oi
EP
(4.133)
k
Row 9 in Table 4.16 displays the cumulative recovery during depletion,
assuming 1000 Mscf of gas initially in place at the dew point pressure
(ie, G ¼ 1000 Mscf). The listed cumulative gas “Gp” is essentially the
product of GIIP times RF at each depletion pressure. For example,
cumulative gas at a depletion pressure of 3200 psi is
Gp ¼ GðRFÞ
Gp ¼ 1000ð0:22803Þ ¼ 228:03Mscf
It should be pointed out that all cumulative gas production listed in Row
9 in Table 4.16 assumes that the reservoir pressure is initially at the dew
point. Whitson and Brule (2000) point out that, if the initial reservoir
pressure “pi” does not equal the dew point pressure, the assumed gas-inplace “G” of 1000 Mscf and tabulated cumulative gas values (ie, Row 9)
can be adjusted by using the following methodology.
Step 1. Calculate the adjusted gas-in-place (GIIP)adj at the initial reservoir
pressure pi as given by
ðGIIPÞadj ¼ ðGÞpsat + ΔG
with the added correction to gas-in-place as given by
422 CHAPTER 4 PVT Properties of Crude Oils
ΔG ¼ G
ðpi =Zi Þi
1
ðpd =Zd Þ
where
ΔG ¼ additional GIIP
G ¼ gas initially in place at the dewpoint; that is, 100 Mscf
(p/Z)i ¼ value of (p/Z) at initial reservoir pressure “pi”
(p/Z)d ¼ value of (p/Z) at dew point pressure “pd”
For example, Table 4.15 indicates that the initial reservoir pressure
is 5659 psig with a Z-factor of 1.10381 with a dew point pressure of
5022 psig. Applying the additional ΔG gives
½ð5659 + 14:7Þ=1:10381
ΔG ¼ 106
1 ¼ 40Mscf
½ð5022 + 14:7Þ=1:0191
Step 2. Calculate wet gas initially in place at initial reservoir pressure
from
GIIP ¼ G + ΔG
to give
GIIP ¼ 1000 + 40 ¼ 1040Mscf
Step 3. Based on initial reservoir pressure, pi, adjust the tabulated
cumulative gas production data by applying the following
expressions:
For pressures p > pd
Gp ¼
ðGIIPÞ pi pd
ðpd =Zd Þ Zi Zd
For pressures p pd
Gp adj ¼ Gp + ΔG
When p < pd, the above expression suggests that the entire
tabulated cumulative gas production (Row 9) for all depletion
pressures should be increased by ΔG. For example, using the
CCE data in Tables 4.15 and 4.16 to calculate the cumulative
well stream gas production at 5337, 5022, and 3200 psig:
n
ΔG ¼ 40 Mscf and GIIP ¼ 1040 Mscf
n
At 5337 psig, the Z-factor from Table 4.15 is 1.06045, then
Gp ¼
ðGIIPÞ pi pd
ðpd =Zd Þ Zi Zd
Discussion of Retrograde Gas Laboratory Data 423
Gp ¼
n
n
1040
5673:7
5351:7
¼ 20 Mscf
ð5036:7=1:01912Þ 1:10381 1:06045
At 5022 psi, the cumulative gas production Gp ¼ 0 + 40 ¼ 40 Mscf
At 3200 psig, the cumulative gas production from Table 4.16 is
228.03 Mscf, the adjusted cumulative gas:
Gp adj ¼ Gp + ΔG
Gp adj ¼ 228:03 + 40 ¼ 268:03 Mscf
EXAMPLE 4.45
Table 4.16 shows the well stream compositional analysis of the Nameless field.
Using Eq. (4.133), calculate the maximum plant liquefied components recovery
ðGPMÞC3 + using the following plant recovery efficiency “EP”:
n
n
n
n
5% for ethane
80% for propane
95% for butanes
100% for heavier components
Solution
Using the composition of well stream at the dew point pressure, calculate the
GPM for each component by applying Eq. (4.133), to give
psc
yi Mi
GPMi ¼ 11:173
EP
Tsc
γ oi
15:025 yi Mi
yi Mi
EP ¼ 0:3228
EP
GPMi ¼ 11:173
γ oi
γ oi
520
Construct the following working table:
Component
yi
Mi
γ oi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
65.41
12.107
5.329
1.28
2.039
0.911
0.902
1.247
9.526
30.07
44.097
58.123
58.123
72.15
72.15
86.177
154.383
0.35619
0.50699
0.56287
0.58401
0.6247
0.63112
0.66383
0.8003
EP
EP (GPM)i
0.05
0.165
0.8
1.1973
0.95
0.4055
0.95
0.6225
1
0.3397
1
0.3330
1
0.5227
1
5.9337
ðGPMÞC3 + ¼ 9:5194
The calculated value of ðGPMÞC3 + is within 2.3% of the reported laboratory
value of 9.471 GPM.
424 CHAPTER 4 PVT Properties of Crude Oils
CONDENSATE GOR AND CONDENSATE YIELD
As the reservoir pressure drops below the dew point pressure of a retrograde
gas system, a large amount of the heavy hydrocarbon components constituting most of the LDO is left immobile in the reservoir rather than produced at
the surface. The liquid phase accumulating in the reservoir is generally
richer in these heavy ends while the produced gas is depleted of heavy ends
resulting in less condensate yield at the surface. The increase in the reservoir
LDO and the corresponding decrease in the condensate yield continues
with decreasing pressure until the maximum LDO is reached. Reaching
the pressure at which the maximum LDO occurs signifies the start of the
vaporization of the LDO components to the gas phase. This reverse process
is illustrated in Fig. 4.51.
It is important to differentiate between the condensate yield “Y” and condensate-gas-ratio “CGR” and also to develop an expression to describe the relationship between these two field measurements. Based on the illustration
presented in Fig. 4.52, the following relationships apply:
280
30.0%
O
LD
200
20.0%
LDO (%)
CGR
= Q /Q
o
gas
160
15.0%
Y=Q
o /Qs
tream
10.0%
120
5.0%
80
0.0%
0
1000
2000
3000
Pressure (psig)
4000
n FIGURE 4.51 LDO, condensate yield, and condensate-gas-oil ratio during gas depletion at 218°F.
5000
40
6000
Y & CGR (STB/MMscf)
240
25.0%
Condensate GOR and Condensate Yield 425
Qg
The stream from a
“Well” or “PVT cell”
CGR =
Qstream
Qo
Qg
Qo
Y < CGR
n FIGURE 4.52 LDO, condensate yield, and condensate-gas-oil ratio relationship.
Condensate yield:
Y¼
Qo
; STB=MMscf
Qstream
Condensate GOR:
CGR ¼
Qo
; STB=MMscf
Qg
Gas equivalent volume:
Veq ¼ 0:133000
γo
; MMscf=STB
Mo
Total volume of gas stream:
Qstream ¼ Veq Qo + Qg ; MMscf
Combining the above three relationships gives
Y¼
Qo
Veq Qo + Qg
426 CHAPTER 4 PVT Properties of Crude Oils
or equivalently as
Y¼
CGR
Veq CGR + 1
The above expression indicates that Y < CGR.
THE MODIFIED MBE FOR RETROGRADE GAS
RESERVOIRS
It should be pointed that the volumetric material balance for conventional
gas reservoirs, commonly called the tank model, is based on the single-phase
Z-factor. This simple tank model is formulated by applying the following
gas mole balance:
np ¼ ni nr
where
np ¼ moles of gas produced
ni ¼ moles of gas initially in the reservoir
nr ¼ moles of gas remaining in the reservoir
the gas moles can be replaced by their equivalents using the real gas law, to
give the following conventional material balance equation “MBE”:
psc Gp
pi V
pV
¼
RTsc Zi RT Zg RT
However, for condensate reservoirs, the remaining number of models must
expressed in terms of the two-phase compressibility factor instead of the
single-phase Z-factor “Zg”; that is,
psc Gp
pi V
pV
¼
RTsc Zi RT Ztwo-phase RT
Arranging this equation gives
p
Ztwo-phase
¼
pi
psc T
Gp
Zi
Tsc V
The initial reservoir gas volume “V” can be expressed in terms of the volume
of gas at SC by
V ¼ Bg G ¼
psc Zi T
G
Tsc pi
The Modified MBE for Retrograde Gas Reservoirs 427
Combining the above relationships gives
p
pi
¼ Ztwo-phase Zi
pi 1
Gp
Zi G
The above relationship suggests that a plot of (p/Z) versus Gp would result in
a straight line with a slope of m ¼ [(pi/Zi)(1/G)]. Expressing the above material balance in terms of recovery factor “RF” gives
p
Ztwo-phase
¼
pi
pi
RF
Zi
Zi
(4.134)
where
pi ¼ initial reservoir pressure
Gp ¼ cumulative gas production, scf
p ¼ current reservoir pressure
V ¼ original gas volume, ft3
Zi ¼ gas deviation factor at pi
Zg ¼ single-phase gas deviation factor at p
T ¼ temperature, °R
Ahmed (2015) developed the following expression to calculated the liquid
Z-factor “ZL” using the CVD data at any pressure:
ZL ¼
Ztwo-phase Zg ðLDOÞ
Zg + Ztwo-phase ðLDO 1Þ
To account for the initial water saturation, the above relationship can be
modified to give
ZL ¼
Ztwo-phase Zg ðLDOÞð1 Swi Þ
Zg + Ztwo-phase ½LDOð1 Swi Þ 1
The material balance can be expressed in terms of remaining gas moles “nrg”
and remaining oil moles “nrL” as
np ¼ ni nrg nrL
to give
1 ðLDOÞ ðLDOÞ
pi
pi
¼
Gp
p
Zi
GZi
Zg
ZL
or in terms of RF,
1 ðLDOÞ ðLDOÞ
pi
pi
p
¼
RF
Zi
Zi
Zg
ZL
(4.135)
428 CHAPTER 4 PVT Properties of Crude Oils
The above modified material balance equation can be used to accurately calculate the remaining fluid in the reservoir at abandonment and also to investigate the impact of initial water saturation on recovery.
Consistency Tests on the CCE and CVD Data
Tables 4.17 and 4.18 shows detailed CCE and CVD laboratory data performed on a recombined separator samples from a retrograde gas field.
The CCE test shows a dew point pressure of 3763 psia with a saturated
gas density and Z-factor of 0.3986 g/cm3 and 0.79416, respectively. There
are several simple measures that can be used to check the quality of the CCE
and CVD laboratory data. Some of these measures are applied next using the
reported values in Tables 4.17 and 4.18.
Table 4.17 CCE and CVD Depletion Test Data
CCE Test Data
p (psia)
Rel. Vol., Vrel
Z
5022
4660
4498
4348
4211
4085
3942
3763
3618
3475
3348
3236
3135
3044
2960
2884
2744
2623
2515
2415
2208
2039
1773
0.89862
0.92153
0.93309
0.94462
0.95594
0.96724
0.98105
1
1.02283
1.04834
1.07387
1.09894
1.12388
1.14846
1.17312
1.19724
1.24672
1.29555
1.34469
1.3956
1.52101
1.64873
1.91461
0.95243
0.9063
0.88577
0.86681
0.84956
0.83388
0.81617
0.79416
LDO 5 VL/Vsat (%)
0
11.4454677
20.5684308
25.0748645
26.9899664
28.3105372
29.2283070
29.9966784
30.5535648
31.1680000
31.5725535
31.7212371
31.7778120
31.7738989
31.5566922
30.9209515
Gas Density (g/cm3)
0.424
0.4135
0.4083
0.4034
0.3986
0.3939
0.3884
0.381
LDO 5 VL/Vt (%)
0
11.19
19.62
23.35
24.56
25.19
25.45
25.57
25.52
25.00
24.37
23.59
22.77
20.89
19.14
16.15
(Continued)
The Modified MBE for Retrograde Gas Reservoirs 429
Table 4.17 CCE and CVD Depletion Test Data Continued
CVD Test Data
Pressure
LDO
Z-Factor
Ztwo-phase
%
3763
3413
3113
2813
2513
2213
1913
1613
1313
1013
513
0
22.35
27.26
29.22
29.82
29.43
28.68
27.67
26.62
25.48
23.49
0.7942
0.7887
0.7884
0.793
0.7977
0.8071
0.8192
0.835
0.8552
0.8884
0.9374
0.7942
0.7672
0.7531
0.7351
0.7191
0.704
0.6875
0.6638
0.6353
0.5991
0.4604
0
6.1707
12.7807
19.2219
26.1969
33.568
41.145
48.55
56.1684
64.0568
76.1755
Table 4.18 Density of CO2 as a Function of Pressure and Temperature
CO2 Density (g/cm3), Pressure (psi)
Temperature (°F)
362.594
725.189
1087.78
1450.38
2175.57
2900.75
3625.94
4351.13
68
86
104
122
140
158
176
194
212
230
248
266
284
302
320
0.0527
0.0499
0.0476
0.0456
0.0437
0.0421
0.0406
0.0391
0.0378
0.0366
0.0354
0.0344
0.0334
0.0325
0.0316
0.1423
0.1251
0.1135
0.1052
0.0984
0.093
0.0883
0.0845
0.081
0.0778
0.0749
0.0722
0.0697
0.0674
0.0653
0.81
0.655
0.2305
0.1932
0.1726
0.1584
0.1469
0.1381
0.1305
0.1239
0.1187
0.1141
0.1094
0.1054
0.1018
0.855
0.782
0.638
0.3901
0.2868
0.2478
0.2215
0.2019
0.1877
0.1765
0.1673
0.1595
0.1525
0.1461
0.1403
0.901
0.85
0.785
0.705
0.604
0.504
0.43
0.373
0.333
0.304
0.28
0.262
0.2465
0.2337
0.2229
0.9335
0.8887
0.8415
0.7855
0.724
0.6605
0.5935
0.5325
0.4815
0.4378
0.4015
0.3718
0.347
0.3267
0.3089
0.96
0.919
0.8771
0.8347
0.7889
0.7379
0.6872
0.6359
0.588
0.5443
0.5053
0.4718
0.4419
0.4151
0.3918
0.9832
0.946
0.9077
0.8687
0.8292
0.7882
0.7466
0.704
0.663
0.623
0.5855
0.5517
0.52
0.4925
0.468
430 CHAPTER 4 PVT Properties of Crude Oils
n
The listed Z-factor at any pressure in the CCE test data must satisfy the
following relationship:
Z¼
pVrel
Zdew
pdew
For example, at 4211 psia
Z-factor ¼ 0.8495
Relative volume “Vrel” ¼ 0.95593
Gas density ¼ 03986 g/cm3
Applying the above relation gives
Z¼
n
ð4211Þð0:95594Þ
0:79415 ¼ 0:8495
3763
an exact match with the reported value.
Similarly, the gas density at any pressure in the CCE test data must
satisfy the following relationship:
ρ¼
ρsat
Vrel
Applying the above relation gives the exact reported value:
ρ¼
n
0:381
¼ 0:3986g=cm3
0:95594
Below the dew point pressure, the following expression provides the
link that exists between relative volume and both reported LDO as
given by
ðLDOÞCCE
Vt ¼
ðLDOÞCCE
Vsat
Vrel
At 2884 psia, the CCE test data shows
Vrel ¼ 1.19724
ðLDOÞVsat ¼ 30.55356
ðLDOÞVt ¼ 25.52
Applying the above expression to give the exact match:
ðLDOÞCCE
Vt ¼
n
ðLDOÞCCE
30:55356
Vsat
¼
¼ 25:52
Vrel
1:19724
The values of the two-phase Z-factor in the CVD test data can be
checked for accuracy by comparing laboratory data against values as
calculated from the following relationship:
Ztwo-phase ¼
Zd
p
pd 1 ðRFÞ
Liquid Blockage in Gas-Condensate Reservoirs 431
n FIGURE 4.53 Liquid dropout from CCE and CVD tests.
For example, at 2513 psia:
Reported Ztwo-phase ¼ 0.7191
Recovery factor ¼ 26.1969%
Applying the above equation gives
Ztwo-phase ¼
n
Zd
p
0:7942
2513
¼
¼ 0:7186
pd 1 ðRFÞ
3763
1 0:261969
A match with less than 1%.
A plot of the LDO obtained from the CVD data and those of two
LDOs from the CCE test, as shown in Figs. 4.53 and 4.54 should
reveal the following consistency characteristics as observed by
Lawrence and Gupta (2009):
n
The CVD LDO should all be between the two CCE LDOs.
n
The CVD maximum LDO should occur at lower pressure than
the CCE maximum LDO.
LIQUID BLOCKAGE IN GAS-CONDENSATE RESERVOIRS
Well productivity is a key parameter in the development of gas-condensate
reservoirs. However, accurate prediction of well productivity can be difficult because of the need to understand and account for the complex flow
behavior occurring in both.
432 CHAPTER 4 PVT Properties of Crude Oils
p*
CVD
Region 1
Region 2
Gas
Gas
oil
oil
Gas
Gas
Region 3
Gas
Gas
Gas
oil
oil
0.6
Pressure
Oil saturation
0.8
CCE
oil
Flowing
oil + gas
Oil
0.4
sat
Flowing
gas
ura
tion
pro
Flowing
gas
pe
Pressure profile
pd
file
0.2
SO>Soc & kro> 0
SO ≤ Soc & kro= 0
So= 0 & kro= 0
pwf
0
rw
rdew
Radius (ft)
re
n FIGURE 4.54 Condensate liquid blockage.
n
n
The extended well flow lines from wellbore to surface facilities.
The behavior of the retrograde gas system with declining reservoir
pressure.
A brief discussion of the above issues are presented next.
Behavior of Gas-Condensate Wells
In conventional dry or wet gas reservoirs, the calculation of the bottom-hole
flowing pressure “pwf” as a function of the wellhead pressure and the pressure loss in the tubing is a relatively simple task — resulting in an accurate
calculation of the gas rate as given by Darcy’s equation:
Qg ¼
ðp
kk h
rg
re
1422T ln
+ s 0:75 pwf
rw
or in a traditional form as
!
2p
dp
μg Z
(4.136)
Liquid Blockage in Gas-Condensate Reservoirs 433
Qg ¼
kkrg h½mðpe Þ mðpwf Þ
re
+ s 0:75
1422T ln
rw
with
mðpe Þ ¼
ð pe
o
!
2p
dp
μg Z
where
Qg ¼ gas flow rate, Mscf/day
T ¼ reservoir temperature, °R
s ¼ skin factor
k ¼ absolute permeability, md
krg ¼ relative permeability
h ¼ thickness, ft
re ¼ drainage radius, ft
rw ¼ wellbore radius, ft
m( pe) ¼ real gas potential as evaluated from “0 to pe,” psi2/cp
m(pwf) ¼ real gas potential as evaluated from “0 to pwf,” psi2/cp
In retrograde gas reservoirs, however, wells could experience a significant
decrease in the well productivity when the pressure drops below the dew
point pressure in any segment of flow line extended from wellbore to surface
facilities. The gas rate calculation requires an accurate estimation of the
wellbore flowing pressure, which is impacted by the pressure loss in the tubing and the additional wellhead back pressure. Estimating the bottom-hole
flowing pressure is considered a difficult task to perform for several reasons,
including:
n
n
n
n
n
n
n
Multiphase flow could occur in the tubing based on the pressure
gradient.
Well temperature distribution in the tubing will impact the dew point
pressure of the flowing fluid, which, in turn, will impact pwf. In general,
the dew point pressure is a strong function of temperature; for example,
for a fixed overall composition “zi,” the dew point will decrease with
decreasing temperature.
Effect of liquid loading on bottom-hole flowing pressure.
Different flow regimes as the pressure drops below the dew point
pressure.
Flow regime transitions in the tubing.
Physical phenomena, such as turbulence flow.
Thermodynamic phase changes with pressure and temperature
occurring as the condensate gas leaves the wellbore to surface
separation facilities.
434 CHAPTER 4 PVT Properties of Crude Oils
Many studies have investigated wellbore pressure drop in gas-condensate
wells based on well pressure gradient condensate wells. The pressure gradient may be divided into three classes:
1. Wellhead pressure and bottom-hole flowing pressure are both above
dew point, indicating a single gas phase flow in the tubing. The
condensate yield and condensate-gas ratio will remain constant as long
as the separation conditions of surface facilities remain unchanged.
2. Wellhead pressure is below the fluid dew point pressure at wellhead
temperature while the bottom-hole flowing pressure is above dew point.
In this case, the condensate yield and CGR will remain relatively
constant.
3. Wellhead pressure as well as the bottom-hole flowing pressure are both
below the fluid dew point pressure, indicating two-phase flow.
Behavior of Retrograde Gas Reservoirs
When the bottom-hole flowing pressure drops below the dewpoint, a region
of high liquid saturation might build up around the wellbore, causing a “liquid blockage” that reduces the gas flow and well productivity. Ahmed
(2000) studied the effectiveness of lean gas, N2, and CO2 injection in reducing the liquid blockage. The author suggested the use of the Huff-n-Puff
injection to vaporize the LDO, providing that the reservoir pressure is above
the dew point pressure. The gas cycling, on the other hand, is a common and
viable approach when the bottom-hole flowing pressure and reservoir pressure are both below the dew point pressure. Results of Huff-n-Puff study
indicated the importance of selecting the volume of the injected gas with
the conclusion that the injected gas volume and success of the injection
is impacted to a large degree with stage of reservoir depletion at the start
of the injection. Fevang and Whitson (1994) presented a comprehensive
study on the performance of retrograde gas reservoirs through the identification of the following three regions:
Region 1
As most of the pressure drop is located within a short distance from the wellbore, it will create a large pressure sink around the wellbore. A substantial
accumulation of the LDO could occur near the wellbore when the bottomhole flowing pressure drops below the dew point pressure. As shown schematically in Figs. 4.53 and 4.54, this inner near-wellbore zone is identified
as Region 1 and characterized by the simultaneous flow of oil and gas. The
richness of the flowing well stream and the PVT properties of the reservoir
gas will determine the amount of liquid saturation in Region 1. It should be
pointed out that the assumption of the two-phase flow could only occur if
Liquid Blockage in Gas-Condensate Reservoirs 435
n
n
n
The retrograde gas is classified as rich retrograde system.
The pressure is below the dew point pressure.
The LDO saturation is above the critical saturation; that is, So > Soc to
allow for the liquid to flow with the gas.
As documented by several reservoir simulation studies, most of the retrograde
gas well deliverability loss is caused by the reduction of the gas relative permeability “krg” due to the condensate buildup in Region 1. The simultaneous flow of
gas and oil in Region 1, presumably flowing in equilibrium, resembles the prevailing equilibrium state of the CCE test condition. Therefore, the data obtained
from the CCE test can be manipulated and combined to generate the relative permeability ratio “krg/kro” as a function of pressure. As detailed in the earlier discussion of the CCE test, the following three measured pressure-volume
relationships are reported mathematically by Eqs. (4.105), (4.128), (4.129) as
Vrel ¼
Vg + VL
Vt
¼
Vsat
Vsat
ðLDOÞCCE
Vsat ¼
ðLDOÞCCE
Vt ¼
VL
Vsat
VL
VL
¼
Vt Vg + VL
Combining the above expressions gives
Vg
1
¼
1
VL ðLDOÞCCE
V
t
Assuming the simultaneous flow if Region 1 is governed by the Darcy’s
steady state flow, to give
krg khΔp =½ug ð ln ðre =rw Þ krg uo
Vg
¼
¼
VL ðkro khΔpÞ=½uo ð ln ðre =rw Þ kro ug
Combining the above two expressions, gives
"
#
μg
krg
1
¼
1
CCE
kro
μo
ðLDOÞVt
(4.137)
It should be pointed out the relative permeability ratio “krg/kro” as given by the
above relationship can only be generated as a function of pressure by using PVT
software as the viscosity ratio “μg/μo” cannot be obtained from the CCE process.
Region 2
As shown in Figs. 4.53 and 4.54, Region 2 is the intermediate zone
where condensate dropout originates. Based on the pressure profile
around the wellbore that is bounded between the flowing bottom-hole
436 CHAPTER 4 PVT Properties of Crude Oils
pressure and reservoir pressure, the dew point exits at a distance “rdew”
from the wellbore. The dew point pressure at the radius rdew marks the
boundary between Regions 3 and 2. As the single-phase original reservoir gas from Region 3 enters Region 2, the first droplets of liquid
condense at the boundary. In Region 2, contrary to Region 1, the saturation “So” of the net condensate accumulation is considered below the
critical saturation and, therefore, immobile. Region 2 is then characterized by a single-phase flow with the gas as the only flowing phase while
the condensate liquid remains immobile. Because the gas is considered
the only moving “removing phase” from Region 2 to Region 1, the
pressure-saturation relationship in Region 2 is similar to that of the
CVD process. The saturation pressure can then be obtained directly from
the CVD experiment data after correcting for the initial water saturation
“Swi” as
So ¼ ð1 Swi ÞLDO
It should be noted that the overall composition “zi” of the fluid in Region 1 is
equivalent to that of the flowing gas from Region 2; in addition, the dew
point pressure of the total (combined) fluid in Region 1 is identical to that
of flowing gas from Region 2 as represented by p*.
Region 3
This zone consists of single-phase gas, as the pressure in this zone is above
the dewpoint pressure, with no hydrocarbon liquid in the region.
The single-phase gas rate can be calculated accurately by from Eq. (4.136);
that is,
Qg ¼
ð pe
kk h
rg
re
+ s 0:75 pwf
1422T ln
rw
!
2p
dp
μg Z
However, the above expression must be modified to account for the impact
of the above three regions on the gas rate. Eq. (4.136) can be modified to
give the gas rate as
#
ð pe "
krg
kh
kro
dp
Rs +
Qg ¼
re
μo Bo
μg Bgd
0:75 + s pwf
1422 ln
rw
(4.138)
or
#
ð pe "
krg
kro
dp
Rs +
Qg ¼ C
μo Bo
μg BE
pwf
(4.139)
Liquid Blockage in Gas-Condensate Reservoirs 437
where the performance coefficient C is defined by
C¼
kh
re
0:75 + s
1422 ln
rw
where
Qg ¼ gas flow rate, Mscf/day
k ¼ absolute permeability, md
h ¼ thickness, ft
re ¼ drainage radius
rw ¼ well base radius
pe ¼ reservoir pressure at the external well drainage radius, psi
p* ¼ dew point pressure of the gas entering Region 1 from Region 2
pwf ¼ bottom-hole flow pressure
BgD ¼ dry gas FVF, bbl/scf
Bo ¼ oil FVF of the condensate liquid, bbl/STB
s ¼ skin factor
The integral part of Eq. (4.139) that represents the pressure drop from pe to
pwf is divided into three portions to describe the individual pressure drop in
each region as
ð pe
pwf
!
krg
kro
+
Rs dp ¼ ðΔpÞRegion1 + ðΔpÞRegion2 + ðΔpÞRegion3
Bgd μg Bo μo
where
ðΔpÞRegion1 ¼
ðp
pwf
ðΔpÞRegion2 ¼
ðΔpÞRegion3 ¼
!
krg
kro
+
Rs dp
Bgd μg Bgd μg
ð pd
p
ð pe
pd
!
krg
dp
Bgd μg
!
1
dp
Bgd μg
Fevang and Whitson (1994) presented a comprehensive treatment and
detailed discussion of the methodology of implementing the above treatment of the three regions in reservoir simulation.
Special Laboratory PVT Tests
In addition to the previously described routine laboratory tests, a number
of other projects may be performed for very specific applications. If a
438 CHAPTER 4 PVT Properties of Crude Oils
reservoir is to be developed under gas injection, miscible gas injection, or a
dry gas cycling scheme, a number of specialized tests must be conducted,
including
n
n
n
n
n
the swelling test
slim-tube test
rising bubble test
core flood
other tests.
The above tests are designed to provide the essential “optimized” key data
that can be used when planning reservoir developing scenarios with miscible
gas displacement. The displacement efficiency of reservoir oil by gas injection is highly pressure dependent and miscible displacement is only
achieved at pressures greater than a certain minimum pressure, termed
the minimum miscibility pressure “MMP.” The minimum miscibility pressure is defined as the “lowest pressure for which a given injected gas composition can develop miscibility through a multicontact process with a given
reservoir oil at reservoir temperature.”
The reservoir to which the process is applied must be operated at or above
the MMP to develop miscibility. Reservoir pressures below the MMP result
in immiscible displacements and consequently lower oil recoveries. It
should be pointed out that the minimum miscibility pressure is completely
independent of the reservoir heterogeneity; however, the MMP is a strong
function of
n
n
n
Oil composition
Composition of the injected gas
Reservoir temperature
Swelling Test
As shown schematically in Fig. 4.55, a gas with a known composition (usually similar to proposed injection gas) is added to the original reservoir oil at
varying proportions in a series of steps. After each addition of a certain measured volume of gas, the overall mixture is quantified in terms of the molar
percentage of the injection gas, such as 10 mol% of N2. The PVT cell then is
pressured to the saturation pressure of the new mixture (only one phase is
present). The gas addition starts at the saturation pressure of the reservoir
fluid (bubble point pressure if an oil sample or dew point pressure if a
gas sample) and the addition of the gas continues to perhaps 80 mol%
injected gas in the fluid sample. It should be pointed out that the saturation
pressure may change from bubble point to dewpoint after significant
Liquid Blockage in Gas-Condensate Reservoirs 439
n FIGURE 4.55 Schematic diagram of the swelling test.
volumes of gas injection. As shown in Fig. 4.56, results from the test are
expressed in terms of
n
n
saturation pressure as a function of volume of gas injected per barrel of
the original fluid (ie, scf/bbl).
The swollen volume as a function of injected gas expressed in scf/bbl.
The swollen volume is expressed as the ratio of volume of the saturated
fluid mixture to the volume of the original saturated reservoir oil.
The data collected from the swelling test is essential as it reveals the ability
of the injected gas to:
n
n
n
n
dissolve in the reservoir fluid
swell the hydrocarbon system
vaporize the hydrocarbon
form multiphases in case of CO2 injection.
Results from the swelling test must be used to tune EOS parameters for compositional modeling.
440 CHAPTER 4 PVT Properties of Crude Oils
2.00
swelling ratio
(Vsat)new/(Vsat)orig
psat
1.50
Original psat
1.00
0
500
1000
1500
2000
scf of injected gas / bbl of original fluid
2500
3000
n FIGURE 4.56 Saturation pressure and swelling ration as a function of volume of gas injected.
Slim Tube
Gas injection is considered to be one of the most effective EOR methods,
particularly in developing light oil reservoirs. Depending on the injection
pressure, there are three classifications of gas injection:
n
n
n
Miscible gas injection
Partial miscible gas injection
Immiscible gas injection
The key parameter in the above classifications is based on the minimum miscibility pressure “MMP,” which is considered the most important parameter
for assessing the applicability of any miscible gas flood for an oil reservoir.
The classical thermodynamics definition of miscibility is the condition of
pressure and temperature at which two fluids, when mixed in any proportion, form a single phase. For example, kerosene and crude oil are miscible
at room conditions, whereas stock-tank and water are clearly immiscible. At
miscible condition, the displacement efficiency exceeds 90%; however, if
the flow is one-dimensional with no dispersive mixing, the displacement
efficiency could reach 100%. A slim-tube miscibility test is performed to
determine the MMP by using the equipment setup as shown schematically
in Fig. 4.57. Typically the test is performed in a 40-ft coiled stainless steel
tubing prepacked with glass beads or sand and is filled with oil at constant
Liquid Blockage in Gas-Condensate Reservoirs 441
Coiled stainless steel tubing
prepacked with glass beads
Oil
Inj.
gas
Sight glass
Back pressure
regulator
Pump
Gas meter
Gas
chromatograph
n FIGURE 4.57 Schematic diagram of slim-tube apparatus.
temperature. The coiled tube is characterized by a very high porosity and
permeability. Unlike the previously described laboratory tests that concentrate specifically on fluid property variations with pressure and composition,
the slim-tube test is specially designed to examine the flushing efficiency
and fluid mixing during a miscible displacement process for a given gas
composition. The slim-tube equipment and test procedure are designed to
eliminate the effect of the following parameters:
n
n
n
Reservoir heterogeneities
Impact of reservoir water
Gravity effect
The displacement of the oil from the slim tube is normally repeated four to
six times, each at sequentially increasing pressure. The test is terminated
after the injection of 1.2 pore volume of gas injected at each pressure.
The fraction of the initial oil-in-place (ie, in the stainless steel coil) recovered after each injection is recorded and plotted as a function of pressure as
shown in Fig. 4.58. Typically, pressure-recovery increases rapidly with
increasing pressure and then starts to flatten and eventually forms a straight
line starting at certain pressure that is defined as minimum miscibility
pressure.
It should be pointed out that different recovery levels, such as 80% at the gas
breakthrough, or 90–95% ultimate recovery have also been suggested as the
criteria for miscible displacement. The oil recovery, however, depends on
the tube design and operating conditions. A slim tube may deliver only
80% oil recovery at miscible conditions. The evaluation of recovery changes
Separator
442 CHAPTER 4 PVT Properties of Crude Oils
3
2
Lean Gas Injection
1
Recovery
factor
RF after 1.2 PV injected
Test #
1
2
3
4
5
6
p
4000
4400
4600
4700
4900
5200
RF, %
76
85
93
96
99
99
Cumulative gas injected, PV
Recovery
factor
MMP
Pressure
n FIGURE 4.58 Determination of MMP from the slim-tube oil recovery data.
with displacement pressure, or gas enrichment, is more appropriate to determine miscibility conditions than searching for a high recovery. The most
acceptable definition is the pressure, or enrichment level, at the break-over
point of the ultimate oil recovery, as shown in Fig. 4.58. The recovery is
expected to increase by increasing the displacement pressure, but the additional recovery above MMP is generally minimal. The effluent liquid is usually monitored and flashed to atmospheric conditions and produced gases
are passed through GC for compositional analysis.
It should be pointed out that static tests, where the injection gas and the reservoir oil are equilibrated in a cell, also are conducted to determine the mixture phase behavior. Although these tests do not closely simulate the
dynamic reservoir conditions, they provide accurate, well-controlled data
that are quite valuable, particularly for tuning the EOS used in modeling
the process.
Liquid Blockage in Gas-Condensate Reservoirs 443
It should be noted that the pressure-recovery factor curve shows a pressure
that is lower than the MMP and designated as the vaporization pressure
“pvap.” This is the pressure at which the recovery factor curve starts to rise
with increasing pressure until it reaches the MMP. For all pressures below
the vaporization pressure, the displacement is classified as immiscible. The
following expression, called the miscibility function “α,” can be used to
define and classify the type of gas injection:
α¼
p pvap
MMP pvap
where “p” is the injected gas pressure in the reservoir. The above function
suggests if
(1) p pvap; indicates immiscible displacement and miscibility function is
set to zero, α ¼ 0.
(2) pvap p MMP; indicates partial miscible displacement with
0 < α < 1.
(3) p MMP; indicates miscible displacement with α ¼ 1.
The miscibility function is routinely used in black oil simulator, as an
alternative to compositional simulator, to simulate miscible displacement
by gas injection. The miscibility function is basically used to adjust the
viscosity of the displacing gas and the displaced oil to account for gas solubility in oil or vaporization of oil components to the gas phase. In addition, α is used to adjust relative permeability curves based on type of
displacement.
Viscosity Adjustment
The viscosity adjustment procedure is based on the following fluid mixing
and property characteristics:
n
Mobility ratio: The mobility ratio “M,” which is defined as the ratio of
the oil viscosity “μo” to that of injected gas or the solvent “μs”; that is,
M¼
n
μo
μs
Viscosity mixing parameter: This parameter is represented by ω and
given by the following expression:
ω¼ 1
h
i
4 log 0:78 + 0:22ðMÞ1=4
log ðMÞ
It should be noted that for complete miscible displacement, M ¼ 1 and
ω ¼ 1.
444 CHAPTER 4 PVT Properties of Crude Oils
n
Mixture viscosity: The mixture is correlated with saturation of all
phased and viscosity of oil and solvent (gas) by the following
expression:
μm ¼
1 Sw
So
s
+ μS0:25
0:25
μ
o
n
!4
s
Mixed oil viscosity: The mixed oil viscosity is given by
μom ¼ μ1ω
μωm
o
n
Mixed solvent (gas) viscosity: The mixed oil viscosity is given by
ω
μSm ¼ μ1ω
S μm
The above defined mixing parameters are used to the calculate the effective
oil and solvent viscosity by applying the following relationships:
μse ¼ ð1 αÞμs + αμsm
μoe ¼ ð1 αÞμo + αμom
Relative Permeability Adjustment
The relative permeability data is obtained from a laboratory test conducted
on a reservoir core saturated with crude oil and initial connate water. The
crude is displaced from the core by immiscible gas. In reservoir simulation
using a black oil simulator, the laboratory-generated relative permeability
must be adjusted to account for the interaction and mixing that are taking
place between the injected gas and oil. The adjustment is based calculating
the mixture relative permeability parameter “krm” as given by
Sg
So Sorm
krow +
krg
krm ¼
1 Sw Sorm
1 Sw Sorm
where Sorm is the residual oil saturation to miscible displacement, usually set
between 5% and 10%.
The effective (adjusted) relative permeability of the gas and oil are then
given by
So Sorm
krm
1 Sw Sorm
Sg
krm
krg eff ¼ ð1 αÞkrg +
1 Sw Sorm
ðkro Þeff ¼ ð1 αÞkro + α
Notice that no adjustment is needed if α ¼ 0.
Liquid Blockage in Gas-Condensate Reservoirs 445
Rising Bubble Experiment
Rising bubble apparatus is employed in another experimental technique,
which is commonly used for quick and reasonable estimates of gas-oil miscibility. In this method, the miscibility is determined from the visual observations of changes in the shape and appearance of bubbles of injected gas as
they rise through a visual high-pressure cell filled with the reservoir crude
oil. A series of tests are conducted at different pressures or enrichment levels
of the injected gas, and the bubble shape is continuously monitored to determine miscibility. This test is qualitative in nature, as miscibility is inferred
from visual observations. Hence, some subjectivity is associated with the
miscibility interpretation of this technique.
Therefore, the results obtained from this test are somewhat arbitrary; however, this test is quite rapid and requires less than 2 h to determine miscibility. This method also is cheaper and requires smaller quantities of fluids
compared to the slim tube. The subjective interpretations of miscibility from
visual observations, lack of quantitative information to support the results,
and some arbitrariness associated with miscibility interpretation are some of
the disadvantages of this technique. Also, no strong theoretical background
appears to be associated with this technique and it provides only reasonable
estimates of gas-oil miscibility conditions.
Minimum Miscibility Pressure Correlations
The displacement efficiency of oil by gas depends highly on pressure, and
miscible displacement is achieved only at pressures greater than a certain
minimum, the minimum miscibility pressure. Slim-tube displacement tests
are commonly used to determine the MMP for a given crude oil. The minimum miscibility pressure is defined as the pressure at which the oil recovery versus pressure curve (as generated from the slim-tube test) shows a
sharp change in slope, the inflection point.
There are several factors that affect MMP, including
n
n
n
n
n
n
Reservoir temperature.
Oil characteristics and properties, including the API gravity.
Injected gas composition.
Concentration of C1 and N2 in the crude oil.
Oil molecular weight.
Concentration of intermediate components (C2–C5) in the oil phase.
There exists a distinct possibility between reservoir temperature and MMP
because MMP increases as the reservoir temperature increases. The MMP
also increases with high-molecular-weight oil and oils containing higher
446 CHAPTER 4 PVT Properties of Crude Oils
concentrations of C1 and N2. However, the MMP decreases with increasing
the percentage of the intermediates in the oil phase.
To facilitate screening procedures and gain insight into the miscible displacement process, many correlations relating the MMP to the physical
properties of the oil and the displacing gas have been proposed. It should
be noted that, ideally, any MMP correlation should
n
n
n
Account for each parameter known to affect the MMP.
Be based on thermodynamic or physical principles that affect the
miscibility of fluids.
Be directly related to the multiple-contact miscibility process.
A variety of correlations or estimations of the MMP developed from regressions of slim-tube data. Although less accurate, these correlations are quick
and easy to use and generally require only a few input parameters. Hence,
they are very useful for a fast screening of a reservoir for various types of gas
injection. They also are useful when detailed fluid characterizations are not
available. One significant disadvantage of current MMP correlations is that
the regressions use MMP from slim-tube data, which themselves are
uncertain.
Some MMP correlations require only the input of reservoir temperature and
the API gravity of the reservoir fluid. Other, more accurate, correlations
require reservoir temperature and the total C2–C6 content of the reservoir
fluid. In nearly all the correlations, the methane content of the oil is assumed
to not affect the MMP significantly.
In general, the MMP correlations fall into the following two categories:
n
n
Those dedicated to pure and impure CO2.
Those that examine the MMP of other gases.
The following section briefly reviews the correlations associated with each
of these categories.
Pure and Impure CO2 MMP
It is well documented that the development of miscibility in a CO2 crude oil
displacement is the result of extraction of some hydrocarbons from the oil by
dense CO2. Many authors found considerable evidence that the extraction of
hydrocarbons from a crude oil is strongly influenced by the density of CO2.
Improvement of extraction with the increase in CO2 density that accompanies increasing pressure accounts for the development of miscibility.
The presence of impurities can affect the pressure required to achieve
Liquid Blockage in Gas-Condensate Reservoirs 447
miscible displacement. Some of the widely used correlations are outlined
next, including the following:
n
n
n
n
n
n
n
Orr and Silva correlation
Extrapolated vapor pressure (EVP) method
Yellig and Metcalfe correlation
Alston’s correlation
National Petroleum Council (NPC) method
Cronquist’s correlation
Yuan’s correlation
Orr and Silva Correlation. Orr and Silva (1987) developed a methodology
for determining the MMP for pure and contaminated CO2/crude oil systems.
Orr and Silva pointed out that the distribution of molecular sizes present in a
crude oil has a significantly larger impact on the MMP than variations in
hydrocarbon structure. The carbon-number distributions of the crude oil system are the only data needed to use the correlation. The authors introduced a
weighted-composition parameter, F, based on a normalized partition coefficient, Ki, for the components C2 through C37 fractions. The specific steps
of the method are summarized next.
(1) From the chromatographic or the simulated compositional distribution
of the crude oil, omit C1 and all the nonhydrocarbon components (ie,
CO2, N2, and H2S) from the oil composition. Normalize the weight
fractions (wi) of the remaining components (C2–C37).
(2) Calculate the partition coefficient Ki for each component in the
remaining oil fractions; that is, C2–C37, using the following equation:
log ðKi Þ ¼ 0:761 0:04175Ci
where Ci is the number of carbon atoms of component i.
(3) Evaluate the weighted-composition parameter, F, from
F¼
37
X
Ki wi
2
(4) Calculate the density of CO2 required to achieve miscibility from the
following expressions:
ρmmp ¼ 1:189 0:542F, for F < 1:467
ρmmp ¼ 0:42, for F > 1:467
(5) Given the reservoir temperature, find the pressure, which is assigned to
the MMP, at which the CO2 density is equal to the required ρmmp.
Kennedy (1954) presented the relationship of the density of CO2 as a
448 CHAPTER 4 PVT Properties of Crude Oils
function of pressure and temperature in a tabulated as shown in
Table 4.18. The tabulated values of CO2 density as a function of
pressure and temperature can be used to estimate the MMP by entering
the table with the calculated value of ρmmp and reservoir temperature to
find the pressure that corresponds to the MMP.
EVP Method. Orr and Jensen (1986) suggested that the vapor pressure
curve of CO2 can be extrapolated and equated with the minimum miscibility
pressure to estimate the MMP for low-temperature reservoirs (T < 120°F). A
convenient equation for the vapor pressure has been given by Newitt et al.
(1996):
EVP ¼ 14:7exp 10:91 2015
255:372 + 0:5556ðT 460Þ
with the EVP in psia and the temperature, T, in °R.
EXAMPLE 4.46
Estimate the MMP for pure CO2 at 610°R using the EVP method.
Solution
Estimate the MMP for pure CO2 at 610°R using the EVP method:
2015
EVP ¼ 14:7exp 10:91 2098 psi
255:372 + 0:5556ð610 460Þ
Researchers in the Petroleum Recovery Institute suggest equating the MMP with
the vapor pressure of CO2 when the system temperature is below the critical
temperature, Tc, of CO2. For temperatures greater than Tc, they proposed the
following expression:
MMP ¼ 1071:82893 10b
with the coefficient b as defined by
b ¼ ½2:772 ð1519=T Þ
where MMP is in psia and T in °R.
EXAMPLE 4.47
Estimate the MMP for pure CO2 at 610°R using the method proposed by the
Petroleum Recovery Institute.
Liquid Blockage in Gas-Condensate Reservoirs 449
Solution
Step 1. Calculate the coefficient b:
b ¼ ½2:772 ð1519=T Þ
b ¼ ½2:772 ð1519=610Þ ¼ 0:28184
Step 2. Calculate MMP from
MMP ¼ 1071:82893 10b
MMP ¼ 1071:82893 100:28184 ¼ 2051 psi
Yellig and Metcalfe Correlation. From their experimental study, Yellig
and Metcalfe (1980) proposed a correlation for predicating the CO2 MMPs
that use the temperature, T, in °R, as the only correlating parameter. The
proposed expression follows:
MMP ¼ 1833:7217 + 2:2518055ðT 460Þ + 0:01800674ðT 460Þ2
103949:93
T 460
Yellig and Metcalfe pointed out that, if the bubble point pressure of the oil is
greater than the predicted MMP, then the CO2 MMP is set equal to the bubble point pressure.
EXAMPLE 4.48
Estimate the MMP for pure CO2 at 610°R using the Yellig and Metcalfe method.
Solution
MMP ¼ 1833:7217 + 2:2518055ðT 460Þ + 0:01800674ðT 460Þ2 103949:93
T 460
MMP ¼ 1833:7217 + 2:2518055ð610 460Þ + 0:01800674ð610 460Þ2
103949:93
¼ 1884psi
T 460
Alston’s Correlation. Alston et al. (1985) developed an empirically derived
correlation for estimating the MMPs for pure or impure CO2/oil systems.
Alston and coworkers used the temperature, oil C5+ molecular weight, volatile
oil fraction, intermediate oil fraction, and the composition of the CO2 stream as
the correlating parameters. The MMP for pure CO2/oil systems is given by
MMP ¼ 0:000878ðT 460Þ1:06 ðMC5 + Þ1:78
Xvol 0:136
Xint
450 CHAPTER 4 PVT Properties of Crude Oils
where
T ¼ system temperature, °R
MC5 + ¼ molecular weight of pentane and heavier fractions in the
oil phase
Xint ¼ mole fraction of intermediate oil components (C2–C4, CO2,
and H2S)
Xvol ¼ mole fraction of the volatile (C1 and N2) oil components
Contamination of CO2 with N2 or C1 has been shown to adversely affect the
minimum miscibility pressure of carbon dioxide. Conversely, the addition
of C2, C3, C4, or H2S to CO2 has been shown to lower the MMP. To account
for the effects of the presence of contaminants in the injected CO2, the
authors correlated impure CO2 MMP with the weighted average pseudocritical temperature, Tpc, of the injected gas and the pure CO2 MMP by the
following expression:
168:893
87:8 Tpc 460
MMPimp ¼ MMP
Tpc
The pseudocritical temperature of the injected gas is given by
Tpc ¼
X
wi Tci 460
where
MMP ¼ minimum miscibility pressure of pure CO2
MMPimp ¼ minimum miscibility pressure of the contaminated CO2
wi ¼ weight fraction of component i in the injected gas
Tci ¼ critical temperatures of component i in the injected gas, in °R
T ¼ system temperature, in °R
It should be pointed out that the authors assigned a uniform critical temperature value of 585°R for H2S and C2 in the injected gas.
Sebastian’s Method. Sebastian et al. (1985) proposed a similar corrective
step that adjusts the pure CO2 MMP by an amount related to the mole average critical temperature, Tcm, of the injected gas by
MMPimp ¼ ½CMMP
where the correction parameter, C, is given by
C ¼ 1:0 A 0:0213 0:000251A 2:35 107 A2
with
A ¼ ½Tcm 87:89=1:8
X
yi ðTci 460Þ
Tcm ¼
Liquid Blockage in Gas-Condensate Reservoirs 451
where
MMP ¼ minimum miscibility pressure of pure CO2
MMPimp ¼ minimum miscibility pressure of the contaminated CO2
yi ¼ mole fraction of component i in the injected gas
Tci ¼ critical temperature of component i in the injected gas, °R
To give a better fit to their data, the authors adjusted Tc of H2S from 212 to 125°F.
NPC Method. The NPC proposed an empirical correlation that provides
rough estimates of the pure CO2 MMPs. The correlation uses the API gravity
and the temperature as the correlating parameters, as follows:
Gravity (API)
MMP (psi)
<27°
27–30°
>30°
4000
3000
1200
With reservoir temperature correction as given below:
T (°F)
Additional Pressure (psi)
<120
120–150
150–200
200–250
0
200
350
500
Cronquist’s Correlation. Cronquist (1978) proposed an empirical equation that was generated from a regression fit on 58 data points. Cronquist
characterizes the miscibility pressure as a function of T, molecular weight
of the oil pentanes-plus fraction, and the mole percentage of methane and
nitrogen. The correlation has the following form:
MMP ¼ 15:988ðT 460ÞA
with
A ¼ 0:744206 + 0:0011038MC5 + + 0:0015279yc1 N2
where T ¼ reservoir temperature, °R, and yc1 N2 ¼ mole percentage of methane and nitrogen in the injected gas.
Yuan’s Correlation. Yuan et al. (2005) developed a correlation for predicting minimum miscibility pressure for pure CO2 based on matching
41 experimental slim-tube MMP values. The authors correlated the MMP
with the molecular weight of the heptanes-plus fraction, MC7 + ; temperature
T, in °R; and mole percent of the intermediate components, CM; that is, C2–C6.
452 CHAPTER 4 PVT Properties of Crude Oils
The correlation has the following form:
"
ðMMPÞpureCO2 ¼ a1 + a2 MC7 + + a3 CM + a4 + a5 MC7 + +
a6 CM
ðM C 7 + Þ2
#
ðT 460Þ
+ a7 + a8 MC7 + + a9 ðMC7 + Þ2 + a10 CM ðT 460Þ2
with the coefficients as follows:
a1 ¼ 1463.4
a2 ¼ 6.612
a3 ¼ 44.979
a4 ¼ 21.39
a5 ¼ 0.11667
a6 ¼ 8166.1
a7 ¼ 0.12258
a8 ¼ 0.0012283
a9 ¼ 4.052(106)
a10 ¼ 9.2577(104)
The authors proposed an additional correlation to account for the contamination of CO2 with methane. The correlation is valid for methane contents
in the gas only up to 40%. The proposed expression takes the form
ðMMPÞimpure
ðMMPÞpureCO2
¼ 1 + mðyCO2 100Þ
with
"
a6 CM
#
ðT 460Þ
ðMC7 + Þ2
h
i
+ a7 + a8 MC7 + + a9 ðMC7 + Þ2 + a10 CM ðT 460Þ2
m ¼ a1 + a2 MC7 + + a3 CM + a4 + a5 MC7 + +
with the coefficients as follows:
a1 ¼ 0.065996
a2 ¼ 1.524(104)
a3 ¼ 0.0013807
a4 ¼ 6.2384(104)
a5 ¼ 6.7725(107)
a6 ¼ 0.027344
a7 ¼ 2.6953(106)
a8 ¼ 1.7279(108)
a9 ¼ 43.1436 (1011)
a10 ¼ 1.9566 (108)
Lean Gas and Nitrogen Miscibility Correlations
High-pressure lean gases and N2 injection have been successfully used as
displacing fluids for EOR projects and also are widely used in gas cycling
and pressure maintenance. In general, miscibility pressure with lean gas
Liquid Blockage in Gas-Condensate Reservoirs 453
(eg, methane) increases with increasing temperature because the solubility
of methane in hydrocarbons decreases with increasing temperature, which
in turn causes the size of the two-phase region to increase. However, miscibility pressure with nitrogen decreases with increasing temperatures
because the solubility of nitrogen in hydrocarbons increases. Firoozabadi
and Aziz documented the successful use of lean gas/N2 as a high-pressure
miscible gas injection in several oil fields. The are several correlations that
can be used to estimate the MMP with lean gas, including the following:
n
n
n
n
n
Firoozabadi and Aziz’s correlation
Hudgins’s correlation
Glaso’s correlation
Sebastian and Lawrence’s correlation
Lange’s correlation
Firoozabadi and Aziz’s Correlation
Firoozabadi and Aziz (1986) proposed a generalized correlation that can
predict MMP for N2 and lean gases. They used the concentration of oil intermediate components, temperature, and the molecular weight of C7+ as the
correlating parameters. The authors defined the intermediate components as
the sum of the mole % of C2–C5, CO2, and H2S in the crude oil. The correlation has the following form:
MMP ¼ 9433 188 103 F + 1430 103 F2
with
F¼
I
MC7 + ðT 460Þ0:25
I xC2 C5 + xCO2 + xH2 S
where
I ¼ concentration of intermediates in the oil phase, mol%
T ¼ temperature, °R
MC7 + ¼ molecular weight of C7+
Conrard (1987) points out that most nitrogen and lean gas miscibility
correlations overestimate the MMP as compared with experimental
MMP. The author attributed this departure to the difference between
MMP and saturation pressure and suggested the following expression
to improve the predictive capability of the Firoozabadi and Aziz correlation:
MMP ¼ 0:6909ðMMPÞFA + 0:3091pb
454 CHAPTER 4 PVT Properties of Crude Oils
where
MMP ¼ corrected minimum miscibility pressure, psi
MMPF–A ¼ Firoozabadi and Aziz MMP, psi
pb ¼ bubble point pressure, psi
A very simplified expression that uses the Firoozabadi and Aziz form is
given below for estimating MMP for a nitrogen-crude oil system:
1:8I
MMP ¼ 75:652
TMC7 +
0:5236
Hudgins’s Correlation
Hudgins et al. (1990) performed a comprehensive laboratory study of N2
miscible flooding for enhanced recovery of light crude oil. They stated that
the reservoir fluid composition, especially the amounts of C1 through C5
fractions in the oil phase, is the major determining factor for miscibility.
For pure N2 the authors proposed the following expression:
MMP ¼ 5568 eR1 + 3641 eR2
with
R1 ¼
792:06xC2 C5
MC7 + T 0:25
2:158 106 ðC1 Þ5:632
R2 ¼
MC7 + T 0:25
where
T ¼ temperature, °F
C1 ¼ mole fraction of methane
xC2 C5 ¼ sum of the mole fraction of C2–C5 in the oil phase
Glaso’s Correlation
Glaso (1990) investigated the effect of reservoir fluid composition, displacement velocity, column length of the slim tube, and temperature on
slim-tube oil recovery with N2. Glaso used the following correlating parameters in developing his MMP relationship:
n
n
n
n
Molecular weight of C7+; that is, MC7 + .
Temperature T, in °R.
Sum of the mole % intermediates (C2–C6) in the oil phase; that is, xC2 C5 .
Mole % of methane in the oil phase; that is, xC1 .
Liquid Blockage in Gas-Condensate Reservoirs 455
Glaso proposed the following relationships:
For API < 40:
MMP ¼ 80:14 + 35:35H + 0:76H 2
where
H¼
MC0:88
ðT 460Þ0:11
7+
ðxC2 C6 Þ0:64 ðxC1 Þ0:33
For API > 40:
MMP ¼ 648:5 + 2619:5H + 1347:6H2
where
H¼
MC0:88
ðT 460Þ0:25
7+
ðxC2 C6 Þ0:12 ðxC1 Þ0:42
Sebastian and Lawrence’s Correlation
Sebastian and Lawrence (1992) developed a correlation for predicting the
MMP for nitrogen, N2. The authors used the following variables to develop
their correlation:
n
n
n
n
Molecular weight of C7+ in the oil phase, MC7 + .
Mole fraction of methane in the oil phase, xC1 .
Mole fraction of intermediate components (C2–C6 and CO2) in the oil
phase, xm.
Reservoir temperature, T, in °R.
The proposed correlation has the following form:
$
%
3283xC1 T 4:776ðxC1 Þ2 T 2 + 4:008xm T 2
MMPN2 ¼ 4603 MC7 +
+ 2:05MC7 + + 7:541T
Lange’s Correlation
Lange (1998) developed a generalized MMP correlation that can be used for a
wide range of injected gases, crude oils, temperatures, and pressures. The correlation is based on representation of the physical and chemical properties of
crude oil and injected gas through the well-known Hildebrand solubility
parameters. The solubility parameter concept is widely used to judge the
“goodness” of a solvent for a solute, such that a small difference in solubility
456 CHAPTER 4 PVT Properties of Crude Oils
parameters between a solute and solvent suggests that the solute may dissolve
in the solvent. The solubility parameter of a high-pressure gas depends primarily on its reduced density, ρr, which can be estimated with an equationof-state calculation and is expressed by the following relationship:
pffiffiffiffiffiffi
δgi ¼ 0:122ρri pci
where
ρri ¼ reduced density of component i in the injected gas, lb/ft3
pci ¼ critical pressure of component i in the injected gas, psia
δgi ¼ solubility parameter of component i in the injected gas, (cal/cm3)0.5
The reduced density as defined previously is given by
ρr ¼ ρi =ρci
or
δgi ¼ 0:122
ρi pffiffiffiffiffiffi
pci
ρci
The solubility parameter of the injected gas, δgas, can be calculated by using
the following mixing rules, based on an individual component’s solubility,
δgi, and its volume fraction in the injected gas:
δgas ¼
N X
υi δgi
(4.140)
i¼1
where
δgas ¼ solubility parameter of the injected gas (cal/cm3)0.5
δgi ¼ solubility parameter of component i in the injected gas (cal/cm3)0.5
vi ¼ volume fraction of component i in the injected gas, psia
pci ¼ critical pressure of component i in the injected gas, psia
ρi ¼ density of component i in the injected gas at injection pressure and
temperature, lb/ft3
ρci ¼ critical density of component i in the injected gas, lb/ft3
The solubility of a crude oil depends primarily on its molecular weight and
can be approximated by the following expression:
δoil ¼ ð6:97Þ0:01M 0:00556ðT 460Þ
(4.141)
where M ¼ molecular weight of the reservoir oil and T ¼ reservoir temperature, °R.
Lange points out that miscibility occurs when the solubility difference
between the oil injected gas is around 3(cal/cm3)0.5, or
Miscibilitycriterion : jδoil δgas j 3
0:4
Problems 457
The process of determining the MMP using Lange’s correlation is essentially a trial-and-error approach. A pressure is assumed and δgas is calculated; if the miscibility criterion, as just defined, is met, then the assumed
pressure is the MMP; otherwise the process is repeated.
Lange also developed an expression for estimating the residual oil saturation
to miscible displacement, Sorm, which is very useful as a screening parameter. The correlation is based on data from a large pool of core floods with
the following form:
Sorm ¼ Sorw 0:12 δoil δgas 0:11
where Sorw is residual oil saturation to water flood.
PROBLEMS
1. Tables 4.19–4.21 show the experimental results performed on a crude
oil sample taken from the MTech field. The results include the CCE,
DE, and separator tests.
a. Select the optimum separator conditions and generate Bo, Rs, and Bt
values for the crude oil system. Plot your results and compare to the
unadjusted values.
b. Assume that new field indicates that the bubble point pressure is
better described by a value of 2500 psi. Adjust the PVT to reflect
for the new bubble point pressure.
2. A crude oil system exists at its bubble point pressure of 1708.7 psia and
a temperature of 131°F. Given the following data:
API ¼ 40°
Average specific gravity of separator gas ¼ 0.85
Separator pressure ¼ 100 psig
a. Calculate Rsb using
1. Standing’s correlation.
2. Vasquez and Beggs’s method.
3. Glaso’s correlation.
4. Marhoun’s equation.
5. Petrosky and Farshad’s correlation.
b. Calculate Bob by applying methods listed in part a.
3. Estimate the bubble point pressure of a crude oil system with the
following limited
PVT data:
API ¼ 35°
T ¼ 160°F
Rsb ¼ 700 scf/STB
γ g ¼ 0.75
Use the five different methods listed in problem 2, part a.
458 CHAPTER 4 PVT Properties of Crude Oils
Table 4.19 Pressure-Volume Relations of a Reservoir Fluid at 260°F
(Constant Composition Expansion)
Pressure (psig)
Relative Volume
5000
4500
4000
3500
3000
2500
2300
2200
2100
2051
2047
2041
2024
2002
1933
1843
1742
1612
1467
1297
1102
863
653
482
0.946
0.953
0.9607
0.9691
0.9785
0.989
0.9938
0.9962
0.9987
1
1.001
1.0025
1.0069
1.0127
1.032
1.0602
1.0966
1.1524
1.2299
1.3431
1.5325
1.8992
2.4711
3.405
Table 4.20 Differential Vaporization at 260°
p
Rsd
Btd
Bod
2051
1900
1700
1500
1300
1100
900
700
500
300
170
0
1004
930
838
757
678
601
529
456
379
291
223
0
1.808
1.764
1.708
1.66
1.612
1.566
1.521
1.476
1.424
1.362
1.309
1.11
1.808
1.887
2.017
2.185
2.413
2.743
3.229
4.029
5.537
9.214
16.246
ρ (g/cm3)
0.6063
0.6165
0.6253
0.6348
0.644
0.6536
0.6635
0.6755
0.6896
0.702
0.7298
Z
Bg (bbl/scf)
γg
0.5989
0.88
0.884
0.887
0.892
0.899
0.906
0.917
0.933
0.955
0.974
0.00937
0.01052
0.01194
0.01384
0.01644
0.02019
0.02616
0.03695
0.06183
0.10738
0.843
0.84
0.844
0.857
0.876
0.901
0.948
0.018
1.373
2.23
Problems 459
Table 4.21 Separator Tests of Reservoir Fluid Sample
Separator
Pressure (psig)
Separator
Temperature
Rs
(scf/bbl)
Rs
(scf/STB)
200–0
71
71
72
72
71
71
70
70
431a
222
522
126
607
54
669
25
490b
223
566
127
632
54
682
25
100–0
50–0
25–0
API
Bo
48.2
1.549
48.6
1.529
48.6
1.532
48.4
1.558
a
c
Sep Vol
Factor
Gas
Sp.gr.
1.138d
1.006
1.083
1.006
1.041
1.006
1.02
1.006
0.739e
1.367
0.801e
1.402
0.869e
1.398
0.923e
1.34
Gas-oil ratio in cubic feet of gas at 60°F and 14.75 psi absolute per barrel of oil at indicated pressure and temperature.
Gas-oil ratio in cubic feet of gas at 60°F and 14.75 psi absolute per barrel of stock-tank oil at 60°F.
Formation volume factor in barrels of saturated oil at 2051 psi gauge and 260°F per barrel of stock-tank oil at 60°F.
d
Separator volume factor in barrels of oil at indicated pressure and temperature per barrel of stock-tank oil at 60°F.
e
Collected and analyzed in the laboratory.
b
c
4. A crude oil system exists at an initial reservoir pressure of 3000 psi and
185°F. The bubble point pressure is estimated at 2109 psi. The oil
properties at the bubble point pressure are as follows:
Bob ¼ 1.406 bbl/STB
Rsb ¼ 692 scf/STB
γ g ¼ 0.876
API ¼ 41.9°
Calculate
a. Oil density at the bubble point pressure.
b. Oil density at 3000 psi.
c. Bo at 3000 psi.
5. The PVT data as shown in Table 4.22 were obtained on a crude oil
sample taken from the Nameless field. The initial reservoir pressure
was 3600 psia at 160°F. The average specific gravity of the solution gas
is 0.65. The reservoir contains 250 MMbbl of oil initially in place. The
oil has a bubble point pressure of 2500 psi.
a. Calculate the two-phase oil FVF at
1. 3200 psia.
2. 2800 psia.
3. 1800 psia.
b. Calculate the initial oil in place as expressed in MMSTB.
c. What is the initial volume of dissolved gas in the reservoir?
d. Calculate the oil compressibility coefficient at 3200 psia.
460 CHAPTER 4 PVT Properties of Crude Oils
Table 4.22 PVT Data for Problem 5
Pressure (psia)
3600
3200
2800
2500
2400
1800
1200
600
200
Rs (scf/STB)
Bo (bbl STB)
567
554
436
337
223
143
1.31
1.317
1.325
1.333
1.31
1.263
1.21
1.14
1.07
6. The PVT data below was obtained from the analysis of a bottom-hole
sample.
p (psia)
Relative Volume, V/Vsat
3000
2927
2703
2199
1610
1206
999
1.0000
1.0063
1.0286
1.1043
1.2786
1.5243
1.7399
a. Plot the Y-function versus pressure on rectangular coordinate paper.
b. Determine the constants in the equation Y ¼ mp + b using the method
of least squares.
c. Recalculate relative oil volume from the equation.
7. A 295 cm3 of a crude oil sample was placed in a PVT at an initial
pressure of 3500 psi. The cell temperature was held at a constant
temperature of 220°F. A DL test was then performed on the crude oil
sample with the recorded measurements as given in Table 4.23. Using
the recorded measurements and assuming an oil gravity of 40°API,
calculate the following PVT properties:
a. Oil FVF at 3500 psi.
b. Gas solubility at 3500 psi.
c. Oil viscosity at 3500 psi.
d. Isothermal compressibility coefficient at 3300 psi.
e. Oil density at 1000 psi.
8. Experiments were made on a bottom-hole crude oil sample taken from
the North Grieve field to determine the gas solubility and oil FVF as a
Problems 461
Table 4.23 Crude Oil Sample for Problem 7
p (psi)
T (°F)
Total
Volume
(cm3)
3500
3300
3000a
2000
1000
14.7
220
220
220
220
220
60
290
294
300
323.2
375.2
–
Volume of
Liquids (cm3)
Vol.
Liberated
Gas (cm3)
Gas
Gravity
290
294
300
286.4
271.5
179.53
0
0
0
0.1627
0.184
0.5488
–
–
–
0.823
0.823
0.823
a
Bubble point pressure.
function of pressure. The initial reservoir pressure was recorded as 3600
psia and reservoir temperature was 130°F. The data in Table 4.24 were
obtained from the measurements. At the end of the experiments, the API
gravity of the oil was measured as 40°. If the average specific gravity of
the solution gas is 0.7, calculate
a. Total FVF at 3200 psia.
b. Oil viscosity at 3200 psia.
c. Isothermal compressibility coefficient at 1800 psia.
9. You are producing a 35°API crude oil from a reservoir at 5000 psia and
140°F. The bubble point pressure of the reservoir liquids is 4000 psia at
140°F. Gas with a gravity of 0.7 is produced with the oil at a rate of 900
scf/STB. Calculate
a. Density of the oil at 5000 psia and 140°F.
b. Total FVF at 5000 psia and 140°F.
10. An undersaturated oil reservoir exists at an initial reservoir pressure
3112 psia and a reservoir temperature of 125°F. The bubble point of
Table 4.24 Data for Problem 8
Pressure (psia)
Rs (scf/STB)
Bo (bbl/STB)
3600
3200
2800
2500
2400
1800
1200
600
200
567
567
567
567
554
436
337
223
143
1.31
1.317
1.325
1.333
1.31
1.263
1.21
1.14
1.07
462 CHAPTER 4 PVT Properties of Crude Oils
Table 4.25 Pressure vs. Oil Relationship for Problem 10
Pressure (psia)
Bo (bbl/STB)
3112
2800
2400
2000
1725
1700
1600
1500
1400
1.4235
1.429
1.437
1.4446
1.4509
1.4468
1.4303
1.4139
1.3978
the oil is 1725 psia. The crude oil has the pressure versus oil FVF
relationship found in Table 4.25. The API gravity of the crude oil
and the specific gravity of the solution gas are 40° and 0.65,
respectively. Calculate the density of the crude oil at 3112 psia
and 125°F.
11. A PVT cell contains 320 cm3 of oil and its bubble point pressure of 2500
psia and 200°F. When the pressure was reduced to 2000 psia, the
volume increased to 335.2 cm3. The gas was bled off and found to
occupy a volume of 0.145 scf. The volume of the oil was recorded as
303 cm3. The pressure was reduced to 14.7 psia and the temperature to
60°F while 0.58 scf of gas was evolved, leaving 230 cm3 of oil with a
gravity of 42°API. Calculate
a. Gas compressibility factor at 2000 psia.
b. Gas solubility at 2000 psia.
12. The composition of a crude oil and the associated equilibrium gas is
given below. The reservoir pressure and temperature are 3000 psia and
140°F, respectively.
Component
xi
yi
C1
C2
C3
n-C4
n-C5
C6
C7+
0.40
0.08
0.07
0.03
0.01
0.01
0.40
0.79
0.06
0.05
0.04
0.02
0.02
0.02
References 463
The following additional PVT data are available:
Molecular weight of C7+ ¼ 215
Specific gravity of C7+ ¼ 0.77
Calculate the surface tension
13. Estimate the MMP for pure CO2 at 630°R using the following methods:
a. EVP method.
b. Petroleum Recovery Institute method.
c. Yellig and Metcalfe’s method.
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466 CHAPTER 4 PVT Properties of Crude Oils
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Chapter
5
Equations of State and Phase Equilibria
A phase is the part of a system that is uniform in physical and chemical properties, homogeneous in composition, and separated from other, coexisting
phases by definite boundary surfaces. The most important phases occurring
in petroleum production are the hydrocarbon liquid phase and gas phase.
Water is also commonly present as an additional liquid phase. These can
coexist in equilibrium when the variables describing change in the entire
system remain constant in time and position. The chief variables that determine the state of equilibrium are system temperature, system pressure, and
composition.
The conditions under which these different phases can exist are a matter of
considerable practical importance in designing surface separation facilities
and developing compositional models. These types of calculations are based
on the following two concepts, equilibrium ratios and flash calculations,
which are discussed next.
EQUILIBRIUM RATIOS
As indicated in Chapter 1, a system that contains only one component is considered the simplest type of hydrocarbon system. The qualitative understanding of the relationship that exists between temperature, T, pressure,
p, and volume, V, of pure components can provide an excellent basis for
understanding the phase behavior of complex hydrocarbon mixtures.
In a multicomponent system, the partitioning and the tendency of components to escape between the liquid and gas phase are described by the equilibrium ratio “Ki” of a given component. The equilibrium ratio is defined as
the ratio of the mole fraction of the component in the gas phase “yi” to the
mole fraction of the component in the liquid phase “xi”. Mathematically, the
relationship is expressed as
Ki ¼
yi
xi
Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00005-2
Copyright # 2016 Elsevier Inc. All rights reserved.
(5.1)
467
468 CHAPTER 5 Equations of State and Phase Equilibria
where
Ki ¼ equilibrium ratio of component i
yi ¼ mole fraction of component i in the gas phase
xi ¼ mole fraction of component i in the liquid phase
The above expression indicates that when the Ki-value for a given component is greater than 1, it signifies that the component tends to concentrate in
the gas phase. Phase equilibria and other phase behavior calculations require
accurate estimation equilibrium ratio values. At pressures below 100 psia,
Raoult’s and Dalton’s laws for ideal solutions provide a simplified means
of predicting equilibrium ratios. Raoult’s law states that the partial pressure,
pi, of a component in a multicomponent system is the product of its mole
fraction in the liquid phase, xi, and the vapor pressure of the component, pvi:
pi ¼ xi pvi
(5.2)
where
pi ¼ partial pressure of a component i, psia
pvi ¼ vapor pressure of component i, psia
xi ¼ mole fraction of component i in the liquid phase
Dalton’s law states that the partial pressure of a component is the product of
its mole fraction in the gas phase “yi” and system pressure “p”; that is,
p i ¼ yi p
(5.3)
where p ¼ system pressure, psia.
At equilibrium and in accordance with the previously cited laws, the partial
pressure exerted by a component in the gas phase must be equal to the partial
pressure exerted by the same component in the liquid phase. Therefore,
equating the equations describing the two laws, gives
xi pvi ¼ yi p
By rearranging the preceding relationship and introducing the concept of the
equilibrium ratio gives
yi pvi
¼ ¼ Ki
xi p
(5.4)
Eq. (5.4) suggests that for an “ideal solution” with an overall composition of
“zi” that exist at low pressure and temperature exhibits the following two
properties:
n
n
The value of equilibrium ratio “Ki” for any component is independent of
the overall composition.
As shown in Fig. 5.1, the vapor pressure “pvi” is only a function of
temperature, which implies that the K-value is only a function of system
pressure “p” and temperature “T”.
Equilibrium Ratios 469
n FIGURE 5.1 Vapor pressure chart for hydrocarbon components. Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering Data Book, Tenth Edition, 1987.
470 CHAPTER 5 Equations of State and Phase Equilibria
Taking the logarithm of both sides of Eq. (5.4) and rearranging gives
log ðKi Þ ¼ log ðpvi Þ log ðpÞ
This relationship indicates that for a constant temperature, T (implying a
constant pvi), the component equilibrium ratio “Ki” is a linear function of
pressure with a slope of 1 when plotted on a log-log scale. It is appropriate
at this stage to introduce and define the following nomenclatures:
zi ¼ mole fraction of component i in the entire hydrocarbon mixture
nt ¼ total number of moles of the hydrocarbon mixture, lb-mol
nL ¼ total number of moles in the liquid phase
nυ ¼ total number of moles in the vapor (gas) phase
By definition,
nt ¼ nL + nυ
(5.5)
Eq. (5.5) indicates that the total number of moles in the system is equal to the
total number of moles in the liquid phase plus the total number of moles in
the vapor phase. A material balance on the ith component results in
zi nt ¼ xi nL + yi nv
(5.6)
where
zint ¼ total number of moles of component i in the system
xinL ¼ total number of moles of component i in the liquid phase
yinv ¼ total number of moles of component i in the vapor phase
Also, by the definition of the total mole fraction in a hydrocarbon system,
we may write
X
zi ¼ 1
(5.7)
i
X
xi ¼ 1
(5.8)
i
X
yi ¼ 1
(5.9)
i
It is convenient to perform all the phase equilibria calculations on the basis
of 1 mol of the hydrocarbon mixture; that is, nt ¼ 1. That assumption reduces
Eqs. (5.5), (5.6) to
nL + nυ ¼ 1
(5.10)
xi nL + yi nυ ¼ zi
(5.11)
Combining Eqs. (5.4), (5.11) to eliminate yi from Eq. (5.11) gives
xi nL + ðxi Ki Þnυ ¼ z
Flash Calculations 471
Solving for xi yields
zi
nL + nυ Ki
xi ¼
(5.12)
Eq. (5.11) can also be solved for yi by combining it with Eq. (5.4) to eliminate xi, which gives
yi ¼
zi Ki
¼ xi Ki
nL + nυ Ki
(5.13)
Combining Eq. (5.12) with Eq. (5.8) and Eq. (5.13) with Eq. (5.9) results in
X
X
xi ¼
i
X
i
yi ¼
i
X
i
zi
¼1
nL + nυ Ki
(5.14)
zi Ki
¼1
nL + nυ Ki
(5.15)
As
X
i
yi X
xi ¼ 0
i
then
X
i
X
zi Ki
zi
¼0
nL + nυ Ki
n
+
L nυ Ki
i
Rearranging the above expression to give
X zi ðKi 1Þ
i
nL + nυ Ki
¼0
Replacing nL in the above relationship with (1 nυ) yields
f ðnυ Þ ¼
X zi ðKi 1Þ
¼0
nυ ðKi 1Þ + 1
i
(5.16)
Eqs. (5.12)–(5.16) provide the necessary relationships to perform volumetric and compositional calculations on a hydrocarbon system. These types of
calculation are referred to as flash calculations. Details of performing flash
calculations are described next.
FLASH CALCULATIONS
Flash calculations are an integral part of all reservoir and process engineering calculations. They are required whenever it is desirable to know the
amounts (in moles) of hydrocarbon liquid and gas coexisting in a reservoir
472 CHAPTER 5 Equations of State and Phase Equilibria
or a vessel at a given pressure and temperature. These calculations also are
performed to determine the composition of the existing hydrocarbon phases.
Given the overall composition of a hydrocarbon system at a specified pressure and temperature, flash calculations are performed to determine the
moles of the gas phase, nυ, moles of the liquid phase, nL, composition of
the liquid phase, xi, and composition of the gas phase, yi. The computational
steps for determining nL, nv, yi, and xi of a hydrocarbon mixture with a
known overall composition of zi and characterized by a set of equilibrium
ratios, Ki, are summarized in the following steps:
Step 1. Calculate number of moles of gas phase “nυ”:
Eq. (5.16) can be solved for the number of moles of the vapor phase
nυ by using the Newton-Raphson iteration technique. In applying
this iterative technique, assume any arbitrary value of nυ between
0 and 1, such as nυ ¼ 0.5. A good assumed value may be calculated
from the following relationship:
nυ ¼ A=ðA + BÞ
where
A¼
X
½zi ðKi 1Þ
i
X 1
zi
1
B¼
Ki
i
Using the assumed value of nv, evaluate the function f(nυ) as given
by Eq. (5.16); that is,
f ðnυ Þ ¼
X zi ðKi 1Þ
nv ðKi 1Þ + 1
i
If the absolute value of the function f(nυ) is smaller than a preset
tolerance, such as 106, then the assumed value of nυ is the desired
solution. If the absolute value of f(nυ) is greater than the preset
tolerance, then a new value of (nυ)new is calculated from the
following expression:
f ðnυ Þ
ðnυ Þnew ¼ nυ 0
f ðnυ Þ
with the derivative f 0 (nυ) as given by
f 0 ðnυ Þ ¼ (
X
zi ðKi 1Þ2
i
½nυ ðKi 1Þ + 12
)
Flash Calculations 473
where (nυ)new is the new value of nυ to be used for the next iteration.
This procedure is repeated with the new value of nυ until
convergence is achieved; that is, when
jf ðnυ Þj eps or alternatively jðnυ Þnew ðnυ Þj eps
in which eps is a selected tolerance, such as eps ¼ 106.
Step 2. Calculate number of moles of liquid phase “nL”:
The number of moles of the liquid phase “nL” can be determined
from Eq. (5.10) to give
nL ¼ 1 nυ
Step 3. Calculate the composition of the liquid phase “xi”:
Given nv and nL, calculate the composition of the liquid phase by
applying Eq. (5.12):
xi ¼
zi
nL + nυ Ki
Step 4. Calculate the composition of the gas phase “yi”:
Determine the composition of the gas phase from Eq. (5.13):
yi ¼
zi Ki
¼ xi Ki
nL + nυ Ki
EXAMPLE 5.1
A hydrocarbon mixture with the following overall composition is flashed in a
separator at 50 psia and 100°F:
Component
zi
C3
i-C4
n-C4
i-C5
n-C5
C6
0.20
0.10
0.10
0.20
0.20
0.20
Assuming an ideal solution behavior, perform flash calculations.
Solution
Step 1. Determine the vapor pressure pvi from the Cox chart (Fig. 5.1) and
calculate the equilibrium ratios using Eq. (5.4). The results are shown
below.
474 CHAPTER 5 Equations of State and Phase Equilibria
Component
zi
pvi at 100°F
Ki 5 pvi/50
C3
i-C4
n-C4
i-C5
n-C5
C6
0.20
0.10
0.10
0.20
0.20
0.20
190
72.2
51.6
20.44
15.57
4.956
3.80
1.444
1.032
0.4088
0.3114
0.09912
Step 2. Solve Eq. (5.16) for nυ using the Newton-Raphson method:
f ðnυ Þ
ðnυ Þn ¼ nυ 0
f ðnυ Þ
Iteration
nυ
f(nv)
0
1
2
3
4
0.08196579
0.1079687
0.1086363
0.1086368
0.1086368
3.073(102)
8.894(104)
7.60(107)
1.49(108)
0.0
to give nυ ¼ 0.1086368.
Step 3. Solve for nL:
nL ¼ 1 nυ
nL ¼ 1 0:1086368 ¼ 0:8913631
Step 4. Solve for xi and yi to yield
xi ¼
zi
nL + nυ Ki
yi ¼ xi Ki
The component results are shown below.
Component
zi
Ki
xi 5 zi/(0.8914 + 0.1086 Ki)
yi 5 xi Ki
C3
i-C4
n-C4
i-C5
n-C5
C6
0.20
0.10
0.10
0.20
0.20
0.20
3.80
1.444
1.032
0.4088
0.3114
0.09912
0.1534
0.0954
0.0997
0.2137
0.2162
0.2216
0.5829
0.1378
0.1029
0.0874
0.0673
0.0220
Flash Calculations 475
Binary System
Note that, for a binary system (ie, a two-component system), flash calculations can be performed without restoring to the preceding iterative technique. Flash calculations can be performed by applying the following steps.
Step 1. Solve for the composition of the liquid phase, xi. For a twocomponent system, Eqs. (5.8), (5.9) can be expanded as
X
xi ¼ x1 + x2 ¼ 1
i
X
yi ¼ y1 + y2 ¼ K1 x1 + K2 x2 ¼ 1
i
Solving these expressions for the liquid composition, x1 and x2,
gives
x1 ¼
1 K2
K1 K2
and
x2 ¼ 1 x1
where
x1 ¼ mole fraction of the first component in the liquid phase
x2 ¼ mole fraction of the second component in the liquid phase
K1 ¼ equilibrium ratio of the first component
K2 ¼ equilibrium ratio of the second component
Step 2. Solve for the composition of the gas phase, yi. From the definition of
the equilibrium ratio, calculate the composition of the liquid as
follows:
y1 ¼ x1 K1
y2 ¼ x2 K2 ¼ 1 y1
Step 3. Solve for the number of moles of the vapor phase, nυ, and liquid
phase, nL. Arrange Eq. (5.12) to solve for nυ by using the mole
fraction and K-value of one of the two components to give
nυ ¼
z1 x 1
x1 ðK1 1Þ
and
nL ¼ 1 nυ
Exact results will be obtained if selecting the second component; that is,
nυ ¼
z2 x 2
x2 ðK2 1Þ
476 CHAPTER 5 Equations of State and Phase Equilibria
and
nL ¼ 1 nυ
where
z1 ¼ mole fraction of the first component in the binary system
x1 ¼ mole fraction of the first component in the liquid phase
z2 ¼ mole fraction of the second component in the binary system
x2 ¼ mole fraction of the second component in the liquid phase
K1 ¼ equilibrium ratio of the first component
K2 ¼ equilibrium ratio of the second component
The equilibrium ratio, as calculated by Eq. (5.4) in terms of vapor pressure
and system pressure, proved inadequate. Eq. (5.4) was developed based on
the following assumptions:
n
n
The vapor phase is an ideal gas as described by Dalton’s law.
The liquid phase is an ideal solution as described by Raoult’s law.
The combination of the above two assumptions is unrealistic and results in
inaccurate predictions of equilibrium ratios at high pressures.
EQUILIBRIUM RATIOS FOR REAL SOLUTIONS
For a real solution, the equilibrium ratios are no longer a function of the
pressure and temperature alone but also the composition of the hydrocarbon
mixture. This observation can be stated mathematically as
Ki ¼ K ðp, T, zi Þ
Numerous methods have been proposed for predicting the equilibrium ratios
of hydrocarbon mixtures. These correlations range from a simple mathematical expression to a complicated expression containing several compositional dependent variables. The following methods are presented: Wilson’s
correlation, Standing’s correlation, the convergence pressure method, and
Whitson and Torp’s correlation.
Wilson’s Correlation
Wilson (1968) proposed a simplified thermodynamic expression for estimating K-values. The proposed expression has the following form:
Ki ¼
pci
Tci
exp 5:37ð1 + ωi Þ 1 p
T
(5.17)
Equilibrium Ratios for Real Solutions 477
where
pci ¼ critical pressure of component i, psia
p ¼ system pressure, psia
Tci ¼ critical temperature of component i, °R
T ¼ system temperature, °R
This relationship generates reasonable values for the equilibrium ratio
when applied at low pressures. The main advantage of using Wilson’s
expression to estimate the equilibrium ratios is in its ability to provide a
consistent set of K-values at a specified pressure and temperature; that
is, KC1 > KC2 > KC3 >, and so on. As detailed later in this chapter, equations of state rely on iterative technique to calculate the K-value with stating
values based on Wilson’s equation.
Standing’s Correlation
Several authors, including Hoffmann et al. (1953), Brinkman and Sicking
(1960), Kehn (1964), and Dykstra and Mueller (1965), suggested that any
hydrocarbon or nonhydrocarbon component could be uniquely characterized by combining its boiling point temperature, critical temperature, and
critical pressure into a characterization parameter, which is defined by
the following expression:
Fi ¼ bi ½1=Tbi 1=T (5.18)
log ðpci =14:7Þ
½1=Tbi 1=Tci (5.19)
with
bi ¼
where
Fi ¼ component characterization factor
Tbi ¼ normal boiling point of component i, °R.
Based on the characterization parameter “Fi,” Standing (1979) derived a set
of equations that fit the graphical presentations of the experimentally determined equilibrium ratio of Katz and Hachmuth (1937). The proposed form
of the correlation is based on an observation that plots of log(Ki p) versus Fi
at a given pressure often form straight lines with a slope of “c” and intercept
of “a”. The basic equation of the straight-line relationship is given by
log ðKi pÞ ¼ a + cFi
Solving for the equilibrium ratio, Ki, gives
478 CHAPTER 5 Equations of State and Phase Equilibria
1
Ki ¼ 10ða + cFi Þ
p
(5.20)
where the coefficients “a” and “c” in the relationship are the intercept and
the slope of the line, respectively.
From six isobar plots of log(Ki p) versus Fi for 18 sets of equilibrium ratio
values, Standing correlated the coefficients “a” and “c” with the pressure, to
give
a ¼ 1:2 + 0:00045p + 15 108 p2
c ¼ 0:89 0:00017p 3:5 108 p2
(5.21)
(5.22)
Standing pointed out that the predicted values of the equilibrium ratios of
N2, CO2, H2S, and C1–C6 can be improved considerably by changing
n
n
the correlating parameter “bi” of Eq. (5.19)
the boiling point “Tbi” of these components.
The author proposed the following optimized modified values:
Component
bi
Tbi (°R)
N2
CO2
H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
n-C6
n-C7
n-C8
n-C9
n-C10
470
652
1136
300
1145
1799
2037
2153
2368
2480
2738
2780
3068
3335
3590
3828
109
194
331
94
303
416
471
491
542
557
610
616
616
718
763
805
K-Value of the Plus Fraction
When making flash calculations, the equilibrium ratio for each component
including the plus fraction (eg, C7+) must be calculated at the prevailing
pressure and temperature. Katz and Hachmuth (1937) proposed a
Equilibrium Ratios for Real Solutions 479
“rule-of-thumb” that might be applied for light hydrocarbon mixtures that
suggest the K-value of C7+ can be taken as 15% of the K of C7; that is,
KC7 + ¼ 0:15KC7
Standing offered an alternative approach for determining the K-value of the
heptanes and heavier fractions. By imposing experimental equilibrium ratio
values for C7+ on Eq. (5.20), Standing calculated the corresponding characterization factors, Fi, for the plus fraction; that is,
log
FC7 + ¼
KC7 +
p
c
a
The calculated FC7 + values were used to specify or FIND the pure normal
paraffin hydrocarbon having the K-value equivalent to that of the C7+
fraction.
Standing suggested the following computational steps for determining the
parameters b and Tb of the heptanes-plus fraction.
Step 1. Determine from the following relationship the number of carbon
atoms “n” of the normal paraffin hydrocarbon that have the K-value
of the C7+ fraction:
n ¼ 7:30 + 0:0075ðT 460Þ + 0:0016p
(5.23)
Step 2. Calculate the correlating parameter “b” and the boiling point “Tb” of
that normal paraffin component from the following expression:
b ¼ 1013 + 324n 4:256n2
(5.24)
Tb ¼ 301 + 59:85n 0:971n2
(5.25)
Values of “b” and “Tb” can then be used to calculate the
characterization parameter “Fi” of that particular normal paraffin
hydrocarbon and be used to calculate its K-value, which is assigned
to the C7+.
It is interesting to note that numerous experimental phase
equilibria data suggest that the equilibrium ratio for carbon dioxide
can be closely approximated by the following relationship:
KCO2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
KC1 KC2
where
KCO2 ¼ equilibrium ratio of CO2 at system pressure p and
temperature T
KC1 ¼ equilibrium ratio of methane at p and T
KC2 ¼ equilibrium ratio of ethane at p and T
480 CHAPTER 5 Equations of State and Phase Equilibria
It should be pointed out that Standing’s proposed correlations are valid for
pressures and temperatures that are less than 1000 psia and 200°F, respectively. This limitation on the use of Standing method lends itself appropriately for performing separation calculations, which usually operate within
the range of Standing’s limit of applications.
A Simplified K-Value Method
For light hydrocarbon mixtures that exist at pressure below 1000 psia, the
K-value for the methane fraction can be estimated from the following
relationship:
B
exp A T
KC1 ¼
p
with
A ¼ 2:0 107 p2 0:0005p + 9:5073
B ¼ 0:0001p2 0:456p + 856
where
p ¼ pressure, psia
T ¼ temperature, °R.
It is possible to correlate the equilibrium ratios of C2–C7 with that of the Kvalue of C1 by the following relationship:
Ki ¼
KC1 Ri
lnðpKC1 Þ
with the component characterization parameter Ri for component “i” as
given by
Ri ¼ ai T bi
The temperature, T, is in °R. Values of the coefficients ai and bi for C2–C7
are tabulated below:
Component
ai
bi
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
0.00665
0.00431
0.00327
0.00245
0.00203
0.00147
0.00094
0.00091
1.29114
1.269876
1.206566
1.1002
0.910736
0.737
0.528
0.516263
Equilibrium Ratios for Real Solutions 481
This correlation provides a very rough estimate of the K-values for
the above listed components at pressures below 1000 psia. The K-value
for the C7+ can be estimated from the “rule-of-thumb” equation
(ie, KC7 + ¼ 0:15KC7 ).
EXAMPLE 5.2
A hydrocarbon mixture with the following composition is flashed at 1000 psia
and 150°F.
Component
zi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.009
0.003
0.535
0.115
0.088
0.023
0.023
0.015
0.015
0.015
0.159
If the molecular weight and specific gravity of C7+ are 150.0 and 0.78,
respectively, calculate the equilibrium ratios using:
n
n
n
Wilson’s correlation
Standing’s correlation
Simplified method
Solution
Using Wilson’s method
Step 1. Calculate the critical pressure, critical temperature, and acentric factor
of C7+ by using the characterization method of Riazi and Daubert
discussed in Chapter 2 and Example 2.1, to give
Tc ¼ 1139:4°R
pc ¼ 320:3 psia
ω ¼ 0:5067
Step 2. Apply Eq. (5.17) to get the results shown below:
Ki 5
pci
Tci
exp 5:37ð1 + ωi Þ 1
1000
610
482 CHAPTER 5 Equations of State and Phase Equilibria
Component
P0 c (psia)
T0 c (°R)
ω
Ki
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
1071
493
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
320.3
547.9
227.6
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1139.4
0.225
0.040
0.0104
0.0986
0.1524
0.1848
0.2010
0.2223
0.2539
0.3007
0.5069
2.0923
16.343
7.155
1.263
0.349
0.144
0.106
0.046
0.036
0.013
0.00029
Using Standing’s method
Step 1. Calculate the coefficients a and c from Eqs. (5.21), (5.22) to give
a ¼ 1:2 + 0:00045p + 15 108 p2
a ¼ 1:2 + 0:00045 ð1000Þ + 15 108 ð1000Þ2 ¼ 1:80
c ¼ 0:89 0:00017p 3:5 108 p2
c ¼ 0:89 0:00017 ð1000Þ 3:5 108 ð1000Þ2 ¼ 0:685
Step 2. Calculate the number of carbon atoms, n, from Eq. (5.23) to give
n ¼ 7:30 + 0:0075 ðT 460Þ + 0:0016p
n ¼ 7:3 + 0:0075 ð150Þ + 0:0016 ð1000Þ ¼ 10:025
Step 3. Determine the parameter b and the boiling point, Tb, for the hydrocarbon
component with n carbon atoms by using Eqs. (5.24), (5.25):
b ¼ 1013 + 324n 4:256n2
b ¼ 1013 + 324ð10:025Þ 4:256ð10:025Þ2 ¼ 3833:369
Tb ¼ 301 + 59:85n 0:971n2
Tb ¼ 301 + 59:85ð10:025Þ 0:971ð10:025Þ2 ¼ 803:41°R
Step 4. Apply Eqs. (5.18), (5.20) to give the results shown below:
Fi ¼ bi ½1=Tbi 1=T 1
Ki ¼ 10ða + cFi Þ
p
Equilibrium Ratios for Real Solutions 483
Component
bi
Tbi
Fi
Ki
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
652
470
300
1145
1799
2037
2153
2368
2480
2738
3833.369
194
109
94
303
416
471
491
542
557
610
803.41
2.292
3.541
2.700
1.902
1.375
0.985
0.855
0.487
0.387
0
1.513
2.344
16.811
4.462
1.267
0.552
0.298
0.243
0.136
0.116
0.063
0.0058
The Convergence Pressure Method
Early high-pressure phase equilibria studies revealed that, when a hydrocarbon mixture of a fixed overall composition is held at a constant temperature
as the pressure increases, the equilibrium values of all components converge
toward a common value of unity at certain pressure. This pressure is termed
the convergence pressure, pk, of the hydrocarbon mixture. The convergence
pressure essentially is used to correlate the effect of the composition on
equilibrium ratios.
The concept of the convergence pressure can be better explained by examining Fig. 5.2. The figure is a schematic diagram of a typical set of equilibrium
ratios plotted versus pressure on log-log paper for a hydrocarbon mixture held
at a constant temperature. The illustration shows a tendency of the equilibrium ratios to converge isothermally to a value of Ki ¼ 1 for all components
at a specific pressure; that is, convergence pressure. A different hydrocarbon
mixture may exhibit a different convergence pressure.
Standing (1977) suggested that the convergence pressure could be roughly
correlated linearly with the molecular weight of the heptanes-plus fraction.
Whitson and Torp (1981) expressed this relationship by the following
equation:
pk ¼ 60MC7 + 4200
where MC7 + is the molecular weight of the heptanes-plus fraction.
(5.26)
484 CHAPTER 5 Equations of State and Phase Equilibria
Constant temperature
Me
tha
ne
10
Eth
an
Pro
e
Equilibrium ratio
“K-value”
pa
ne
Bu
tan
1.0
es
Pen
tan
es
He
xan
es
0.1
Hep
tan
es
1000
Pressure
100
PK
10,000
n FIGURE 5.2 Equilibrium ratios for a hydrocarbon system.
Whitson and Torp reformulated Wilson’s equation [Eq. (5.17)] to yield
more accurate results at higher pressures by incorporating the convergence
pressure into the correlation, to give
Ki ¼
A1 pci
pci
Tci
exp 5:37Að1 + ωi Þ 1 pk
p
T
(5.27)
with the exponent “A” roughly approximated by the following expression:
A¼ 1
0:7
p
pk
where
p ¼ system pressure, psig
pk ¼ convergence pressure, psig
T ¼ system temperature, °R
ωi ¼ acentric factor of component i
Equilibrium Ratios for the Plus Fractions 485
EXAMPLE 5.3
Rework Example 5.2 and calculate the equilibrium ratios by using Whitson and
Torp’s method.
Solution
Step 1. Determine the convergence pressure from Eq. (5.26) to give
pk ¼ 60MC7 + 4200 ¼ 9474
Step 2. Calculate the coefficient A:
0:7
p
A¼1
pk
A¼1
1000 0:7
¼ 0:793
9474
Step 3. Calculate the equilibrium ratios from Eq. (5.27) to give the results shown
below:
pci 0:7931 pci
Tci
Ki ¼
exp 5:37Að1 + ωi Þ 1 9474
1000
610
Component
pc
T0 c (°R)
ω
Ki
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
1071
493
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
320.3
547.9
227.6
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1139.4
0.225
0.040
0.0104
0.0986
0.1524
0.1848
0.2010
0.2223
0.2539
0.3007
0.5069
2.9
14.6
7.6
2.1
0.7
0.42
0.332
0.1794
0.150
0.0719
0.683(103)
EQUILIBRIUM RATIOS FOR THE PLUS FRACTIONS
The equilibrium ratios of the plus fractions often behave in a manner
different from the other components of a system. This is because the plus
fraction, in itself, is a mixture of components. Several techniques have been
proposed for estimating the K-value of the plus fractions. Three methods are
presented next.
486 CHAPTER 5 Equations of State and Phase Equilibria
n
n
n
Campbell’s method
Winn’s method
Katz’s method
Campbell’s Method
Campbell (1976) proposed that the plot of log(Ki) versus (Tci)2 for each
component is a linear relationship for any hydrocarbon system. Campbell
suggested that, by drawing the best straight line through the points for propane through hexane components, the resulting line can be extrapolated to
obtain the K-value of the plus fraction. Alternatively, a plot of log (Ki) versus (1=Tbi ) of each fraction in the mixture could also produce a straight line
that can be extrapolated to obtain the equilibrium ratio of the plus fraction
from the reciprocal of its average boiling point.
Winn’s Method
Winn (1954) proposed the following expression for determining the equilibrium ratio of heavy fractions with a boiling point above 210°F:
KC + ¼
KC7
ðKC2 =KC7 Þb
(5.28)
where
KC + ¼value of the plus fraction
KC7 ¼ K-value of n-heptane at system pressure, temperature, and
convergence pressure
KC2 ¼ K-value of ethane
b ¼ volatility exponent
Winn correlated, graphically, the volatility component, b, of the heavy fraction, with the atmosphere boiling point, as shown in Fig. 5.3. This graphical
correlation can be expressed mathematically by the following equation:
b ¼ a1 + a2 ðTb 460Þ + a3 ðTb 460Þ2 + a4 ðTb 460Þ3 + a5 =ðT 460Þ (5.29)
where
Tb ¼ boiling point, °R
a1–a5 ¼ coefficients with the following values:
a1 ¼ 1:6744337
a2 ¼ 3:4563079 103
a3 ¼ 6:1764103 106
a4 ¼ 2:4406839 109
a5 ¼ 2:9289623 102
Equilibrium Ratios for the Plus Fractions 487
n FIGURE 5.3 Volatility exponent “b”.
Katz’s Method
Katz et al. (1959) suggested that a factor of 0.15 times the equilibrium ratio
for the heptane component gives a reasonably close approximation to the
equilibrium ratio for heptanes and heavier. This suggestion is expressed
mathematically by the following equation:
KC7 + ¼ 0:15KC7
(5.30)
K-values for Nonhydrocarbon Components
Lohrenze et al. (1963) developed the following set of correlations that
express the K-values of H2S, N2, and CO2 as a function of pressure, temperature, and the convergence pressure, pk.
For H2 S :
"
p 0:8
1399:2204
6:3992127 +
lnðKH2 S Þ ¼ 1 pk
T
#
18:215052
1112446:2
ln ðpÞ 0:76885112 +
T
T2
488 CHAPTER 5 Equations of State and Phase Equilibria
For N2 : p 0:4
1184:2409
11:294748 0:90459907 ln ðpÞ
ln KN 2 ¼ 1 pk
T
For CO2 : p 0:6
152:7291
7:0201913 ln ðpÞ
ln KCO 2 ¼ 1 pk
T
#
1719:2856 644740:69
1:8896974 +
T
T2
where
T ¼ temperature, °R
p ¼ pressure, psia
pk ¼ convergence pressure, psia
VAPOR-LIQUID EQUILIBRIUM CALCULATIONS
The vast amount of experimental and theoretical work performed on equilibrium ratio studies indicates their importance in solving phase equilibrium
problems in reservoir and process engineering. The basic phase equilibrium
calculations include:
n
n
n
the dew point pressure “Pd”
bubble point pressure “Pb”
separator calculations.
The use of the equilibrium ratio in performing these applications is
discussed next.
Dew Point Pressure
The dew point pressure, pd, of a hydrocarbon system is defined as the pressure at which an infinitesimal quantity of liquid is in equilibrium with a large
quantity of gas. For a total of 1 lb-mol of a hydrocarbon mixture (ie, nt ¼ 1),
the following conditions are applied at the dew point pressure:
nL 0
nv 1
yi ¼ zi
Imposing the above constraints on Eq. (5.14), gives
X
i
X
X zi
zi
zi
¼
¼
¼1
nL + nυ Ki
0 + ð1:0ÞKi
Ki
i
i
where zi is the overall composition of the hydrocarbon system.
(5.31)
Vapor-Liquid Equilibrium Calculations 489
The solution of Eq. (5.31) for the dew point pressure, pd, involves a trialand-error process, as summarized in the following steps:
Step 1. Assume a trial value of pd. A good starting value can be obtained
by combining Wilson’s equation (ie, Eq. (5.17)) with Eq. (5.31)
to give
X zi
i
Ki
¼
8
>
<
X>
9
>
>
=
zi
¼ 1
>
>
p
T
i >
: ci exp 5:37ð1 + ωi Þ 1 ci >
;
pd
T
(5.32)
Solving the above expression for pd to give an initial estimate of pd
as
pd ¼
1
8
>
<
X>
9
>
>
=
zi
>
>
T
i >
:pci exp 5:37ð1 + ωi Þ 1 ci >
;
T
(5.33)
Step 2. Using the assumed dew point pressure, calculate the equilibrium
ratio, Ki, for each component at the system temperature.
Step 3. Calculate the summation of Eq. (5.31) using the K-values of step 2
X
zi =Ki
i
Step 4. The correct value of the dew point pressure is obtained when the
sum 1; however, if
n
the sum ≪ 1, repeat steps 2 and 3 at a higher initial value
of pressure
n
the sum ≫ 1, repeat the calculations with a lower initial value
of pd.
The above outlined procedure using Wilson’s correlation is intended to
only provide approximations of the K-values and Pd. These approximations are used as initial values in equations of state application. However, using Eq. (5.33) will generate a very low value for the dew
point pressure; perhaps close to the lower dew point value. The main
reason is that when applying Eq. (5.33), the term “zi/[Pc exp(5.37(1 + ω)(1 Tc/T))]” for the plus fraction is much greater than the nearest component
value by over 40 times, resulting in a large value of the denominator of
Eq. (5.33). Because it’s only an approximation, the plus fraction should
be excluded.
490 CHAPTER 5 Equations of State and Phase Equilibria
EXAMPLE 5.4
A natural gas reservoir has the following composition:
Component
zi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.7778
0.0858
0.0394
0.0083
0.016
0.0064
0.0068
0.0078
0.0517
The heptanes-plus fraction is characterized by a molecular weight and specific
gravity of 150 and 0.78, respectively. The reservoir exists at 250°F; estimate the
dew point pressure of the gas mixture.
Solution
Step 1. Calculate the critical properties of the C7+ using the Riazi-Daubert
equation:
θ ¼ a ðMÞb γ c exp½d M + eγ + f γ M
Tc ¼ 544:2ð150Þ0:2998 ð0:78Þ1:0555 exp 1:3478 104 ð150Þ 0:61641ð0:78Þ + 0
¼ 1139:4°R
pc ¼ 4:5203 104 ð150Þ0:8063 ð0:78Þ1:6015 exp 1:8078 103 ð150Þ 0:3084ð0:78Þ + 0
¼ 320:3 psia
Tb ¼ 6:77857ð150Þ0:401673 ð0:78Þ1:58262 exp 3:77409 103 ð150Þ 2:984036ð0:78Þ
4:25288 103 ð150Þð0:78Þ ¼ 825:26°R
Use Edmister’s Eq. (2.21) to estimate the acentric factor:
3½log ðpc =14:70Þ
1
7½Tc =Tb 1
3½log ð320:3=14:7Þ
ω¼
1 ¼ 0:5067
7½1139:4=825:26 1
ω¼
Step 2. Construct the following working table and approximate the dew point
pressure by applying Eq. (5.33):
pd ¼
8
>
<
X>
1
9
>
>
=
zi
>
T
>
:pci exp 5:37ð1 + ωi Þ 1 ci >
;
T
i>
Vapor-Liquid Equilibrium Calculations 491
Component
zi
Pci
Tci
ωi
Sum
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.7778
0.0858
0.0394
0.0083
0.016
0.0064
0.0068
0.0078
0.0517
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
320.3
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1139.4
0.0104
0.0986
0.1524
0.1848
0.201
0.2223
0.2539
0.3007
0.5067
7.0699E-05
3.2101E-05
4.357E-05
1.9623E-05
4.8166E-05
3.9247E-05
5.0404E-05
0.00013244
0.02153184
n
n
Total sum of the last column ¼ 0.02197 to give a dew point
pressure ¼ 1/0.02197 ¼ 45 psi
Excluding the C7+, sum ¼ 0.000436 to give a dew point
pressure ¼ 1/0.000436 ¼ 2292 psi
Bubble Point Pressure
At the bubble point pb, the hydrocarbon system is essentially liquid, except
for an infinitesimal amount of vapor. For a total of 1 lb-mol of the hydrocarbon mixture, the following conditions are applied at the bubble point
pressure:
nL 1
nv 0
xi ¼ zi
Obviously, under these conditions, xi ¼ zi. Imposing the above constraints to
Eq. (5.15) yields
X
i
X zi K i
X
zi K i
¼
¼
ðzi K i Þ ¼ 1
nL + nv Ki
1 + ð0ÞKi
i
i
(5.34)
Following the procedure outlined for the dew point pressure determination,
Eq. (5.34) is solved for the bubble point pressure, pb, by assuming various
pressures and determining the pressure that produces K-values satisfying
Eq. (5.34). During the iterative process, if
X
ðzi Ki Þ < 1, then the assumed pressure is high
i
X
ðzi Ki Þ > 1, then the assumed pressure is low
i
Wilson’s equation can be used to give an estimate of the bubble point pressure for equations of state application:
492 CHAPTER 5 Equations of State and Phase Equilibria
X
zi
i
pci
Tci
¼1
exp 5:37ð1 + ωÞ 1 pb
T
Solving for the bubble point pressure gives
pb X
i
Tci
zi pci exp 5:37ð1 + ωÞ 1 T
(5.35)
EXAMPLE 5.5
A crude oil reservoir exits at 200°F with a composition and associated
component properties as tabulated below:
Component
zi
Pci
Tci
ωi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.42
0.05
0.05
0.03
0.02
0.01
0.01
0.01
0.40
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
230.4
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1279.8
0.0104
0.0986
0.1524
0.1848
0.2010
0.2223
0.2539
0.3007
1.0653
The heptanes-plus fraction is characterized by the following measured
properties:
ðMÞC7 + ¼ 216:0
ðγ ÞC7 + ¼ 0:8605
ðTb ÞC7 + ¼ 977°R
Estimate the bubble point pressure.
Solution
Step 1. Calculate the critical pressure and temperature of the C7+ by the RiaziDaubert equation, Eq. (2.4), to give
pc ¼ 3:12281 109 ð977Þ2:3125 ð0:8605Þ2:3201 ¼ 230:4psi
Tc ¼ 24:27870ð977Þ0:58848 ð0:8605Þ0:3596 ¼ 1279:8°R
Step 2. Calculate the acentric factor by employing the Edmister correlation
[Eq. (2.21)] to yield
ω¼
3½ log ðpc =14:70Þ
¼ 1:0653
7½Tc =Tb 1
Vapor-Liquid Equilibrium Calculations 493
Step 3. Construct the following table:
Component
zi
Pci
Tci
ωi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.42
0.05
0.05
0.03
0.02
0.01
0.01
0.01
0.40
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
230.4
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1279.8
0.0104
0.0986
0.1524
0.1848
0.2010
0.2223
0.2539
0.3007
1.0653
Step 4. Estimate the bubble point pressure from Eq. (5.35) as follows:
X
Tci
zi pci exp 5:37ð1 + ωÞ 1 pb T
i
Component
zi
Pci
Tci
ωi
Eq. (5.35)
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C
P7+
0.42
0.05
0.05
0.03
0.02
0.01
0.01
0.01
0.4
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
320.3
343.37
550.09
666.01
734.98
765.65
829.1
845.7
913.7
1279.8
0.0104
0.0986
0.1524
0.1848
0.201
0.2223
0.2539
0.3007
1.0633
4620.64266
133.639009
45.2146052
12.6894565
6.6436649
1.630693
1.34909342
0.58895645
0.01761403
4822.41575
The estimated bubble point pressure is 4822 psia; notice the impact of the
methane on the estimated saturation pressure.
Separator Calculations
Produced reservoir fluids are complex mixtures of different physical characteristics. As a well stream flows from the high-temperature, high-pressure
petroleum reservoir, it experiences pressure and temperature reductions.
Gases evolve from the liquids and the well stream changes in character.
The physical separation of these phases is by far the most common of all
field-processing operations and one of the most critical. The manner in
which the hydrocarbon phases are separated at the surface influences the
stock-tank oil recovery. The principal means of surface separation of gas
and oil is the conventional stage separation.
494 CHAPTER 5 Equations of State and Phase Equilibria
Stage separation is a process in which gaseous and liquid hydrocarbons are
flashed (separated) into vapor and liquid phases by two or more separators.
These separators usually are operated in a series at consecutively lower pressures. Each condition of pressure and temperature at which hydrocarbon
phases are flashed is called a stage of separation. Traditionally, the stock
tank is considered a separate stage of separation. Example of two-stage separation processes are shown in Fig. 5.4.
To describe the liquid-gas separation process, it is convenient to group the
composition of a hydrocarbon mixture into three groups of components:
n
n
n
Volatile (light) components: The group includes nitrogen, methane, and
ethane.
Intermediate components: This group comprises components with
intermediate volatility such as propane through hexane and CO2.
Heavy components: This group contains components of less volatility
such as heptane and heavier components.
Mechanically, there are two types of gas-oil separation
a. differential separation
b. flash separation.
n FIGURE 5.4 Schematic drawing of two-stage separation processes.
Vapor-Liquid Equilibrium Calculations 495
Differential Separation Process
In this process, the pressure is reduced in stages and the liberated gas is
removed from contact with the oil as soon as the pressure is reduced. Initially, the liberated gas is composed mainly of lighter components with
intermediate and heavy components remaining in the liquid. When the
gas is separated in this manner, the maximum amount of heavy and intermediate components remain in the liquid resulting in a minimum shrinkage
of the oil. A greater stock-tank oil recovery will be obtained from the differential liberation process due to the fact that the lighter components liberated earlier at higher pressures and is not present at lower pressures to
attract the intermediate and heavy components and pull them into the
gas phase.
Flash Separation
In this flash liberation process, the liberated gas remains in contact with the
oil until its instantaneous removal at the final separation pressure. A maximum proportion of intermediate and heavy components are attracted into
the gas phase by this process resulting in a maximum oil shrinkage and,
therefore, lower oil recovery.
It should be pointed out that the separation process of the solution gas taking
place in the reservoir as the pressure drops below the bubble point pressure
can be described experimentally by the gas differential liberation process.
The main reason for this classification is that due to the high mobility of
the liberated gas, the gas moves faster than the oil toward the producing
well, a process similarly simulated in the laboratory through a differential
liberation test by removing the liberated gas as soon as it forms. However,
there is a brief period when the liberated gas saturation is less that its critical
saturation; that is, gas is immobile and remains in contact with the oil, which
can be described by flash liberation.
The flow from the wellbore through surface facilities can be described by a
series of flash separations. However, it is believed that the produced hydrocarbon system undergoes a flash liberation in the primary separator followed
by a differential process as the actual separation occurs where gas and liquid
are removed from contact to a subsequent stage of separations. As the number of stages increases, the differential aspect of the overall separation
becomes greater.
The purpose of stage separation then is to reduce the pressure on the produced oil in steps to maximize the stock-tank oil recovery results. Separator
calculations are basically performed to determine
496 CHAPTER 5 Equations of State and Phase Equilibria
a. Optimum separation conditions in terms of separator pressure and
temperature.
b. Composition of the separated gas and oil phases.
c. Oil formation volume factor.
d. Producing gas-oil ratio.
e. API gravity of the stock-tank oil.
Stage separation of oil and gas is accomplished with a series of separator
stages operating at sequentially reduced pressures when the liquid is discharged from a higher pressure separator into a lower pressure separator.
The objective is to select the optimum set of working separators
pressures that yield the maximum liquid recovery of liquid at stock-tank
conditions. The optimization problem is constrained because the pressure of the first stage (primary separator) must be less and also impacted
by the wellhead pressure and the pressure loss occurring between the two
nodes. Most of optimization is performed on the subsequent separation
stages, particularly the second stage. The optimization procedure is
based on and supported by the following field and phase behavior
observations:
n
n
n
When the separator pressure is high, large amounts of light components
remain in the liquid phase; however, they are liberated at the stock-tank
conditions along with other valuable components (intermediate and
heavy components) that attracted to the to light-end molecules. This
process, when it occurs, will result in a large amount of the stock-tank oil
in the gas phase at the stock tank.
If the pressure is too low, large amounts of light components are
separated from the liquid, and they attract substantial quantities of
intermediate and heavier components resulting in a substantial stocktank oil recovery.
An intermediate pressure, called the optimum separator pressure, should
be selected to maximize the oil volume accumulation in the stock tank.
Selecting the optimum pressure will yield
1. A maximum in the stock-tank API gravity.
2. A minimum in the oil formation volume factor (ie, less oil
shrinkage).
3. A minimum in the producing gas-oil ratio (gas solubility).
This separator pressure optimization can be performed by conducting a separator test. Typical results of the test are tabulated below with the concept of
determining the optimum separator pressure is shown in Fig. 5.5 by graphically plotting the API gravity, Bo, and Rs as a function of pressure, which
indicates an optimum separator pressure of 100 psig.
Vapor-Liquid Equilibrium Calculations 497
Rs
API
Rs
Bo
Optimum separator pressure
API
First stage separator pressure
n FIGURE 5.5 Effect of separator pressure on API, Bo, and GOR.
Psep (psig)
Tsep (°F)
Rsfb
°API
Bofb
50
to 0
Total
100
to 0
Total
200
to 0
Total
300
to 0
Total
75
75
737
41
778
676
92
768
602
178
780
549
246
795
40.5
1.481
40.7
1.474
40.4
1.483
40.1
1.495
75
75
75
75
75
75
The computational steps of the “separator calculations” are described by the
following steps and in conjunction with Fig. 5.6, which schematically shows
a bubble point reservoir flowing into a surface separation unit consisting of
n-stages operating at successively lower pressures.
Step 1. Calculate the volume of oil occupied by 1 lb-mol of the crude
oil with a given overall composition “zi” at the reservoir pressure
498 CHAPTER 5 Equations of State and Phase Equilibria
Gas removed 1
Gas removed 2
(nv)1
yi
(nv)2
yi
xi
(nL)1
zi
xi
(nL)3 Gas removed
Gas removed 3
(nv)3
yi
xi
(nL)2
Separator 1
Separator 2
Stage 1
Stage 2
n
(nL)st =
(nL)i
i =1
Separator 3
Stage 3
Stock-tank
Well stream
“zi”
Pb
1 mol
Vo, ft3
n FIGURE 5.6 Schematic illustration of n-separation stages.
(or bubble point pressure) and temperature. This volume, denoted
as Vo, is calculated by recalling and applying the equation that
defines the number of moles; that is,
n¼
m ρo Vo
¼
¼1
Ma
Ma
(5.36)
Solving for the oil volume gives
X
Vo ¼
Ma
¼
ρo
ðzi M i Þ
i
ρo
(5.37)
where
m ¼ total weight of 1 lb-mol of the crude oil, lb/mol
Vo ¼ volume of 1 lb-mol of the crude oil at reservoir conditions,
ft3/mol
Ma ¼ apparent molecular weight
Mi ¼ molecular weight of component i
ρo ¼ density of the reservoir oil at the bubble point pressure, lb/ft3
It should be pointed out that the density can be calculated from the
crude oil composition using the Standing-Katz or Alani-Kennedy
method.
Vapor-Liquid Equilibrium Calculations 499
Step 2. Given the composition of the hydrocarbon stream “zi” to the
first separator and the operating conditions of the separator,
calculate a set of equilibrium ratios that describes the hydrocarbon
mixture.
Step 3. Assuming a total of 1 mol of the feed (ie, nt ¼ 1) entering the first
separator, perform flash calculations to obtain
n
compositions of the separated gas “yi” and liquid “xi”
n
number of moles of the gas “nv” and the liquid “nL”.
Designating these moles as (nL)1 and (nυ)1, the actual number of
moles of the gas and the liquid leaving the first separation stage are
½nυ1 a ¼ ðnt Þðnυ Þ1 ¼ ð1Þðnυ Þ1
½nL1 a ¼ ðnt ÞðnL Þ1 ¼ ð1ÞðnL Þ1
where
[nv1]a ¼ actual number of moles of vapor leaving the first
separator
[nL1]a ¼ actual number of moles of liquid leaving the first
separator.
Step 4. Using the composition of the liquid “xi” leaving the first separator
as the feed for the second separator “zi ¼ xi” and calculate the
equilibrium ratios of the hydrocarbon mixture at the prevailing
pressure and temperature of the second-stage separator.
Step 5. Assume 1 mol of the feed from first-stage separator to the second
stage, perform flash calculations to determine the compositions
and quantities of the gas and liquid leaving the second separation
stage. The actual number of moles of the two phases then is
calculated from
½nυ2 a ¼ ½nL1 a ðnυ Þ2 ¼ ð1ÞðnL Þ1 ðnυ Þ2
½nL2 a ¼ ½nL1 a ðnL Þ2 ¼ ð1ÞðnL Þ1 ðnL Þ2
where
[nv2]a, [nL2]a ¼ actual moles of gas and liquid leaving
separator 2
(nv)2, (nL)2 ¼ moles of gas and liquid as determined from flash
calculations
Step 6. The previously outlined procedure is repeated for each separation
stage, including the stock-tank storage, and the calculated moles
and compositions are recorded. The total number of moles of
liberated solution gas from all stages are calculated as
ðnν Þt ¼
n
X
ðnva Þi ¼ ðnν Þ1 + ðnL Þ1 ðnν Þ2 + ðnL Þ1 ðnL Þ2 ðnν Þ3 + ⋯ + ðnL Þ1 …ðnL Þn1 ðnν Þn
i¼1
500 CHAPTER 5 Equations of State and Phase Equilibria
In a more compact form, this expression can be written as
ðnν Þt ¼ ðnν Þ1 +
"
#
n
i1
Y
X
ðnν Þi ðnL Þj
i¼2
(5.38)
j¼1
where
(nv)t ¼ total moles of liberated gas from all stages, lb-mol/mol
of feed
n ¼ number of separation stages.
The total moles of liquid remaining in the stock tank can also be
calculated as
ðnL Þst ¼ nL1 nL2 …nLn
or
ðnL Þst ¼
n
Y
ðnL Þi
(5.39)
i¼1
where (nL)st ¼ number of moles of liquid remaining in the
stock tank.
Step 7. Calculate the volume, in scf, of all the liberated solution gas from
Vg ¼ 379:4ðnυ Þt
(5.40)
where
Vg ¼ total volume of the liberated solution gas scf/mol of feed.
Step 8. Determine the volume of stock-tank oil occupied by (nL)st moles of
liquid from
ðVo Þst ¼
ðnL Þst ðMa Þst
ðρo Þst
(5.41)
where
(Vo)st ¼ volume of stock-tank oil, ft3/mol of feed
(Ma)st ¼ apparent molecular weight of the stock-tank oil
(ρo)st ¼ density of the stock-tank oil, lb/ft3. This desnsity is
calculated from the oil composition at stock-tank
Step 9. Calculate the specific gravity and the API gravity of the stock-tank
oil by applying the expression
γo ¼
ðρo Þst
62:4
(5.42)
Vapor-Liquid Equilibrium Calculations 501
Step 10. Calculate the total gas-oil ratio (gas solubility) Rs:
GOR ¼
Vg
ð5:615Þð379:4Þðnv Þt
¼
ðVo Þst =5:615
ðnL Þst ðMÞst =ðρo Þst
GOR ¼
2130:331ðnv Þt ðρo Þst
ðnL Þst ðMÞst
(5.43)
where GOR ¼ gas-oil ratio, scf/STB.
Step 11. Calculate the oil formation volume factor from the relationship
Bo ¼
Vo
ðVo Þst
Substituting Eqs. (5.36), (5.41) with equivalent term in the above
expression gives
Bo ¼
Ma ðρo Þst
ρo ðnL Þst ðMa Þst
(5.44)
where
Bo ¼ oil formation volume factor, bbl/STB
Ma ¼ apparent molecular weight of the feed
(Ma)st ¼ apparent molecular weight of the stock-tank oil
ρo ¼ density of crude oil at reservoir conditions, lb/ft3
The separator pressure can be optimized by calculating the API gravity,
GOR, and Bo in the manner just outlined at different assumed pressures.
The optimum pressure corresponds to a maximum in the API gravity and
a minimum in gas-oil ratio and oil formation volume factor.
EXAMPLE 5.6
A crude oil, with the composition as follows, exists at its bubble point pressure of
1708.7 psia and a temperature of 131°F. The surface separation facilities include
two separators and stock tank with the following operating conditions:
Stage #
Pressure (psia)
Temperature (°F)
1
2
Stock tank
400
350
14.7
72
72
60
502 CHAPTER 5 Equations of State and Phase Equilibria
The composition of the crude oil “feed” to the primary separator is tabulated
below:
Mi
Component
zi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0008
0.0164
0.2840
0.0716
0.1048
0.0420
0.0420
0.0191
0.1912
0.0405
0.3597
44.0
28.0
164.0
30.0
44.0
584.0
58.0
72.0
72.0
86.0
252
Given that the molecular weight and specific gravity of C7+ are 252 and 0.8429,
calculate
n
n
n
Oil formation volume factor at the bubble point pressure Bofb
Gas solubility at the bubble point pressure Rsfb.
Stock-tank density and the API gravity of the hydrocarbon system.
Solution
Step 1. Calculate the apparent molecular weight of the crude oil to give
X
Ma ¼
zi Mi ¼ 113:5102
Step 2. Calculate the oil density at bubble point pressure by using the Standing
and Katz correlations to yield
ρob ¼ 44:794 lb=ft3
Step 3. Using the pressure and temperature of the primary separator (ie, 400
psia and 72°F) calculate the K-value for each component by applying
the Standing’s K-value, to give
1
Ki ¼ 10ða + cFi Þ
p
Component
zi
Ki
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0008
0.0164
0.2840
0.0716
0.1048
0.0420
0.0420
0.0191
0.0191
0.0405
0.3597
3.509
39.90
8.850
1.349
0.373
0.161
0.120
0.054
0.043
0.018
0.0021
Vapor-Liquid Equilibrium Calculations 503
Step 4. Perform flash calculations on the hydrocarbon stream to the first
separator to give
nL ¼ 0:7209
nv ¼ 0:29791
Step 5. Solve for xi and yi to give
xi ¼
zi
nL + nv Ki
yi ¼ xi Ki
Component
zi
Ki
xi
yi
Mi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0008
0.0164
0.2840
0.0716
0.1048
0.0420
0.0420
0.0191
0.0191
0.0405
0.3597
3.509
39.90
8.850
1.349
0.373
0.161
0.120
0.054
0.043
0.018
0.0021
0.0005
0.0014
0.089
0.0652
0.1270
0.0548
0.0557
0.0259
0.0261
0.0558
0.4986
0.0018
0.0552
0.7877
0.0880
0.0474
0.0088
0.0067
0.0014
0.0011
0.0010
0.0009
44.0
28.0
16.0
30.0
44.0
58.0
58.0
72.0
72.0
86.0
252
Step 6. Use the calculated liquid composition as the feed for the second
separator or flash the composition at the operating condition of the
separator. The results are shown in the table below, with nL ¼ 0.9851
and nυ ¼ 0.0149.
Component
zi
Ki
xi
yi
Mi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0005
0.0014
0.089
0.0652
0.1270
0.0548
0.0557
0.0259
0.0261
0.0558
0.4986
3.944
46.18
10.06
1.499
0.4082
0.1744
0.1291
0.0581
0.0456
0.0194
0.00228
0.0005
0.0008
0.0786
0.0648
0.1282
0.0555
0.0564
0.0263
0.0264
0.0566
0.5061
0.0018
0.0382
0.7877
0.0971
0.0523
0.0097
0.0072
0.0015
0.0012
0.0011
0.0012
44.0
28.0
16.0
30.0
44.0
58.0
58.0
72.0
72.0
86.0
252
Step 7. Repeat the calculation for the stock-tank stage, to give the results in the
following table, with nL ¼ 0.6837 and nυ ¼ 0.3163.
504 CHAPTER 5 Equations of State and Phase Equilibria
Component
zi
Ki
xi
yi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0005
0.0008
0.0784
0.0648
0.1282
0.0555
0.0564
0.0263
0.0264
0.0566
0.5061
81.14
1159
229
27.47
6.411
2.518
1.805
0.7504
0.573
0.2238
0.03613
0000
0000
0.0011
0.0069
0.0473
0.0375
0.0450
0.0286
0.02306
0.0750
0.7281
0.0014
0.026
0.2455
0.1898
0.3030
0.0945
0.0812
0.0214
0.0175
0.0168
0.0263
Step 8. Calculate the actual number of moles of the liquid phase at the stocktank conditions from Eq. (5.39):
ðnL Þst ¼
n
Y
ðnL Þi
i¼1
ðnL Þst ¼ ð1Þð0:7209Þð0:9851Þð0:6837Þ ¼ 0:48554
Step 9. Calculate the total number of moles of the liberated gas from the entire
surface separation system:
nυ ¼ 1 ðnL Þst ¼ 1 0:48554 ¼ 0:51446
Step 10. Calculate apparent molecular weight of the stock-tank oil from its
composition, to give
ðMa Þst ¼ ΣMi xi ¼ 200:6
Step 11. Using the composition of the stock-tank oil, calculate the density of the
stock-tank oil by using Standing’s correlation, to give
ðρo Þst ¼ 50:920
Step 12. Calculate the API gravity of the stock-tank oil from the specific
gravity:
γ ¼ ρo =62:4 ¼ 50:920=62:4 ¼ 0:81660°=60°
API ¼ ð141:5=γ Þ 131:5 ¼ ð141:5=0:816Þ 131:5 ¼ 41:9
Step 13. Calculate the gas solubility from Eq. (5.43) to give
Rs ¼
2130:331ðnv Þt ðρo Þst
ðnL Þst ðMÞst
Rs ¼
2130:331ð0:51446Þð50:92Þ
¼ 573:0 scf=STB
0:48554ð200:6Þ
Step 14. Calculate Bo from Eq. (5.44) to give
Equations of State 505
Bo ¼
Ma ðρo Þst
ρo ðnL Þst ðMa Þst
Bo ¼
ð113:5102Þð50:92Þ
¼ 1:325bbl=STB
ð44:794Þð0:48554Þð200:6Þ
To optimize the operating pressure of the separator, these steps should be
repeated several times under different assumed pressures and the results, in
terms of API, Bo, and Rs, should be expressed graphically and used to determine
the optimum pressure.
EQUATIONS OF STATE
An equation of state (EOS) is an analytical expression relating the pressure,
p, to the temperature, T, and the volume, V. A proper description of this PVT
relationship for real hydrocarbon fluids is essential in determining the volumetric and phase behavior of petroleum reservoir fluids and predicting the
performance of surface separation facilities; these can be described accurately by equations of state. In general, most equations of state require only
the critical properties and acentric factor of individual components. The
main advantage of using an EOS is that the same equation can be used to
model the behavior of all phases, thereby assuring consistency when performing phase equilibria calculations.
The best known and the simplest example of an equation of state is the ideal
gas equation, expressed mathematically by the expression
p¼
RT
V
(5.45)
where
V ¼ gas volume in ft3 per 1 mol of gas, ft3/mol.
The PVT relationship is used to describe the volumetric behavior of hydrocarbon gases at pressures close to the atmospheric pressure for which it was
experimentally derived. However, the extreme limitations of the applicability of Eq. (5.45) prompted numerous attempts to develop an equation of state
suitable for describing the behavior of real fluids at extended ranges of pressures and temperatures.
A review of recent developments and advances in the field of empirical cubic
equations of state are presented next, along with samples of their applications
in petroleum engineering. Four equations of state are discussed here:
n
n
n
n
van der Waals
Redlich-Kwong
Soave-Redlich-Kwong
Peng-Robinson
506 CHAPTER 5 Equations of State and Phase Equilibria
The Van der Waals Equation of State “VdW EOS”
In developing the ideal gas EOS, two assumptions were made:
n
n
The volume of the gas molecules is insignificant compared to the total
volume and distance between the molecules.
There are no attractive or repulsive forces between the molecules.
Van der Waals (1873) attempted to eliminate these two assumptions in
developing an empirical equation of state for real gases. In his attempt to
eliminate the first assumption, van der Waals pointed out that the gas molecules occupy a significant fraction of the volume at higher pressures and
proposed that the volume of the molecules, denoted by the parameter b,
be subtracted from the actual molar volume, V, in Eq. (5.45), to give
p¼
RT
V b
where the parameter b is known as the covolume and considered to reflect
the volume of molecules. The variable V represents the actual volume in ft3
per 1 mol of gas.
To eliminate the second assumption, van der Waals subtracted a corrective
term, denoted by a/V2, from this equation to account for the attractive forces
between molecules. In a mathematical form, van der Waals proposed the
following expression:
p¼
RT
a
V b V2
(5.46)
where
p ¼ system pressure, psia
T ¼ system temperature, °R
R ¼ gas constant, 10.73 psi-ft3/lb-mol, °R
V ¼ volume, ft3/mol
a ¼ “attraction” parameter
b ¼ “repulsion” parameter
The two parameters, a and b, are constants characterizing the molecular
properties of the individual components. The symbol a is considered a measure of the intermolecular attractive forces between the molecules.
Eq. (5.46) shows the following important characteristics:
1. At low pressures, the volume of the gas phase is large in comparison
with the volume of the molecules. The parameter b becomes negligible
in comparison with V and the attractive forces term, a/V2, becomes
insignificant; therefore, the van der Waals equation reduces to the ideal
gas equation [Eq. (5.45)].
Equations of State 507
2. At high pressure (ie, p ! 1) the volume, V, becomes very small and
approaches the value b, which is the actual molecular volume, as
mathematically given by
lim V ðpÞ ¼ b
p!1
The van der Waals or any other equation of state can be expressed in a more
generalized form as follows:
p ¼ prepulsion pattraction
(5.47)
where the repulsion pressure term, prepulsion, is represented by the term
RT/(V b), and the attraction pressure term, pattraction, is described by a/V2.
In determining the values of the two constants, a and b, for any pure substance, van der Waals observed that the critical isotherm has a horizontal
slope and an inflection point at the critical point, as shown in Fig. 5.7. This
observation can be expressed mathematically as follows:
@p
¼0
@V Tc, pc
2 @ p
¼0
@V 2 Tc, pc
(5.48)
Differentiating Eq. (5.46) with respect to the volume at the critical point
results in
@p
RTc
2a
¼
+
¼0
@V Tc , pc ðVc bÞ2 Vc3
Liquid
Gas
H
VdW EOS critical point condition
[
Pressure
(5.49)
∂p
∂2p
] T , p ,V = 0 & [ 2 ] T , p ,V = 0
c
c
c
∂V
∂V c c c
T1 < T2 < T3 < Tc < T4
C
Pc
Liquid – Vapor
G
F
B
Vc
A
Volume V
n FIGURE 5.7 An idealized pressure-volume relationship for a pure component.
T4
Tc
T3
T
E 2
T1
508 CHAPTER 5 Equations of State and Phase Equilibria
2 @ p
2RTc
6a
¼
+
¼0
@V 2 Tc , pc ðVc bÞ3 Vc4
(5.50)
Solving Eqs. (5.49), (5.50) simultaneously for the parameters a and b gives
1
Vc
3
8
a¼
RTc Vc
9
b¼
(5.51)
(5.52)
Eq. (5.51) suggests that the covolume “b” is approximately 0.333 of the critical volume “Vc” of the substance. Numerous experimental studies revealed
that the co-volume “b” is in the range 0.24–0.28 of the critical volume and
pure component.
Applying Eq. (5.48) to the critical point (i.e., by setting T ¼ Tc, p ¼ pc, and
V ¼ Vc) and combining with Eqs. (5.51), (5.52) yields
pc Vc ¼ ð0:375ÞRTc
(5.53)
Eq. (5.53) indicates that regardless of the type of the substance, the Van der
Waals EOS gives a universal critical gas compressibility factor “Zc” of
0.375. Experimental studies show that Zc values for substances are ranging
between 0.23 and 0.31.
Eq. (5.53) can be combined with Eqs. (5.51), (5.52) to give more convenient
and traditional expressions for calculating the parameters a and b, to yield
a ¼ Ωa
R2 Tc2
pc
(5.54)
RTc
pc
(5.55)
b ¼ Ωb
where
R ¼ gas constant, 10.73 psia-ft3/lb-mol-°R
pc ¼ critical pressure, psia
Tc ¼ critical temperature, °R
Ωa ¼ 0.421875
Ωb ¼ 0.125
Eq. (5.46) can also be expressed in a cubic form in terms of the volume “V”
as follows:
p¼
RT
a
2
V b V
Equations of State 509
Rearranging gives
RT 2
a
ab
V b+
V +
V
¼0
p
p
p
3
(5.56)
Eq. (5.56) is referred to as the van der Waals two-parameter cubic equation
of state. The term two-parameter refers to parameters a and b. The term
cubic equation of state implies that the equation have three possible values
for the volume “V,” at least one of which is real.
Perhaps the most significant features of Eq. (5.56) are that it describes the
liquid-condensation phenomenon and the passage from the gas to the liquid
phase as the gas is compressed. These important features of the van der
Waals EOS are discussed next in conjunction with Fig. 5.8.
Consider a pure substance with a p-V behavior as shown in Fig. 5.8. Assume
that the substance is kept at a constant temperature, T, below its critical temperature. At this temperature, Eq. (5.56) has three real roots (volumes) for
each specified pressure, p. A typical solution of Eq. (5.56) at constant temperature, T, is shown graphically by the dashed isotherm, the constant temperature curve DWEZB, in Fig. 5.8. The three values of V are the
intersections B, E, and D on the horizontal line, corresponding to a fixed
value of the pressure. This dashed calculated line (DWEZB) then appears
to give a continuous transition from the gaseous phase to the liquid phase,
but in reality, the transition is abrupt and discontinuous, with both liquid and
vapor existing along the straight horizontal line DB. Examining the graphical solution of Eq. (5.56) shows that the largest root (volume), indicated by
point D, corresponds to the volume of the saturated vapor, while the smallest
positive volume, indicated by point B, corresponds to the volume of the
C
Pressure
Pure component isotherm
as calculated from VdW EOS
W
D
P
B
E
Z
Volume
n FIGURE 5.8 Pressure-volume diagram for a pure component.
T
510 CHAPTER 5 Equations of State and Phase Equilibria
saturated liquid. The third root, point E, has no physical meaning. Note that
these values become identical as the temperature approaches the critical
temperature, Tc, of the substance.
Eq. (5.56) can be expressed in a more practical form in terms of the compressibility factor, Z. Replacing the molar volume, V, in Eq. (5.56) with
ZRT/p gives
RT 2
a
ab
V3 ¼ b +
V +
V
¼0
p
p
p
2 RT
ZRT
a ZRT
ab
+
¼0
V3 b +
p
p
p
p
p
or
Z 3 ð1 + BÞZ2 + AZ AB ¼ 0
(5.57)
where
A¼
ap
R2 T 2
(5.58)
bp
RT
(5.59)
B¼
Z ¼ compressibility factor
p ¼ system pressure, psia
T ¼ system temperature, °R
Eq. (5.57) yields one real root1 in the one-phase region and three real roots in
the two-phase region (where system pressure equals the vapor pressure of
the substance). In the two-phase region, the largest positive root corresponds to the compressibility factor of the vapor phase, Zv, while the smallest positive root corresponds to that of the liquid phase, ZL.
An important practical application of Eq. (5.57) is density calculations, as
illustrated in the following example.
EXAMPLE 5.7
A pure propane is held in a closed container at 100°F. Both gas and liquid are
present. Calculate, using the van der Waals EOS, the density of the gas and
liquid phases.
1
In some super critical regions, Eq. (5.57) can yield three real roots for Z. From the three
real roots, the largest root is the value of the compressibility with physical meaning.
Equations of State 511
Solution
Step 1. Determine the vapor pressure, pv, of the propane from the Cox chart.
This is the only pressure at which two phases can exist at the specified
temperature:
pv ¼ 185 psi
Step 2. Calculate parameters a and b from Eqs. (5.54), (5.55), respectively:
a ¼ Ωa
R2 Tc2
ð10:73Þ2 ð666Þ2
¼ 0:421875
¼ 34,957:4
pc
616:3
and
b ¼ Ωb
RTc
10:73ð666Þ
¼ 0:125
¼ 1:4494
616:3
pc
Step 3. Compute the coefficients A and B by applying Eqs. (5.58), (5.59),
respectively:
A¼
ap
ð34, 957:4Þð185Þ
¼
¼ 0:179122
R2 T 2 ð10:73Þ2 ð560Þ2
B¼
bp ð1:4494Þð185Þ
¼
¼ 0:044625
RT ð10:73Þð560Þ
Step 4. Substitute the values of A and B into Eq. (5.57), to give
Z 3 ð1 + BÞZ2 + AZ AB ¼ 0
Z 3 1:044625Z2 + 0:179122Z 0:007993 ¼ 0
Step 5. Solve the above third-degree polynomial by extracting the largest and
smallest roots of the polynomial using the appropriate direct or iterative
method, to give
Zv ¼ 0:72365
ZL ¼ 0:07534
Step 6. Solve for the density of the gas and liquid phases using Eq. (2.17):
ρg ¼
pM
ð185Þð44:0Þ
¼
¼ 1:87 lb=ft3
Z v RT ð0:72365Þð10:73Þð560Þ
and
ρL ¼
pM
ð185Þð44Þ
¼
¼ 17:98 lb=ft3
ZL RT ð0:07534Þð10:73Þð560Þ
The van der Waals equation of state, despite its simplicity, provides a correct
description, at least qualitatively, of the PVT behavior of substances in the liquid
and gaseous states. Yet, it is not accurate enough to be suitable for design
purposes.
With the rapid development of computers, the equation-of-state approach for
calculating physical properties and phase equilibria proved to be a powerful tool,
and much energy was devoted to the development of new and accurate equations
512 CHAPTER 5 Equations of State and Phase Equilibria
of state. Many of these equations are a modification of the van der Waals EOS
and range in complexity from simple expressions containing 2 or 3 parameters to
complicated forms containing more than 50 parameters. Although the
complexity of any equation of state presents no computational problem, most
authors prefer to retain the simplicity found in the van der Waals cubic equation
while improving its accuracy through modifications.
All equations of state generally are developed for pure fluids first and then
extended to mixtures through the use of mixing rules.
Redlich-Kwong’s Equation of State “RK EOS”
Redlich and Kwong (1949) observed that the van der Waals a/V2 term does
not contain the system temperature to account for its impact on the intermolecular attractive force between the molecules. Redlich and Kwong demonstrated that by a simple adjustment of the van der Waals’ a/V2 term to
explicitly include the system temperature could considerably improve the
prediction of the volumetric and physical properties of the vapor phase. Redlich and Kwong replaced the attraction pressure term with a generalized
temperature dependence term, as given in the following form:
p¼
RT
a
pffiffiffi
V b V ðV + bÞ T
(5.60)
in which T is the system temperature, in °R.
Redlich and Kwong noted that, as the system pressure becomes very large
(ie, p ! 1) the molar volume, V, of the substance shrinks to about 26% of its
critical volume, Vc, regardless of the system temperature. Accordingly, they
constructed Eq. (5.60) to satisfy the following condition:
b ¼ 0:26Vc
(5.61)
Imposing the critical point conditions (as expressed by Eq. (5.48)) on
Eq. (5.60) and solving the resulting two equations simultaneously, gives
a ¼ Ωa
R2 Tc2:5
pc
(5.62)
RTc
pc
(5.63)
b ¼ Ωb
where
Ωa ¼ 0.42747
Ωb ¼ 0.08664.
Equations of State 513
It should be noted that by equating Eq. (5.63) with (5.61) gives
0:26Vc ¼ Ωb
RTc
pc
Arranging the above expression:
pc Vc ¼ 0:33RTc
(5.64)
Eq. (5.64) shows that the RK EOS produces a universal critical compressibility factor (Zc) of 0.333 for all substances. As indicated earlier, the critical
gas compressibility ranges from 0.23 to 0.31 for most of the substances, with
an average value of 0.27.
Replacing the molar volume, V, in Eq. (5.60) with ZRT/p and rearranging
gives
Z3 Z2 + A B B2 Z AB ¼ 0
(5.65)
where
A¼
ap
R2 T 2:5
(5.66)
bp
RT
(5.67)
B¼
As in the van der Waals EOS, Eq. (5.65) yields one real root in the one-phase
region (gas-phase region or liquid-phase region) and three real roots in the
two-phase region. In the latter case, the largest root corresponds to the compressibility factor of the gas phase, Zv, while the smallest positive root corresponds to that of the liquid, ZL.
EXAMPLE 5.8
Rework Example 5.7 using the Redlich-Kwong equation of state.
Solution
Step 1. Calculate the parameters a, b, A, and B.
a ¼ Ωa
R2 Tc2:5
ð10:73Þ2 ð666Þ2:5
¼ 0:4247
¼ 914, 1101:1
pc
616:3
b ¼ Ωb
RTc
ð10:73Þð666Þ
¼ 0:08664
¼ 1:0046
pc
616:3
A¼
ð914, 110:1Þð185Þ
ap
¼
¼ 0:197925
R2 T 2:5 ð10:73Þ2 ð560Þ2:5
B¼
bp ð1:0046Þð185Þ
¼
¼ 0:03093
RT ð10:73Þð560Þ
Step 2. Substitute the parameters A and B into Eq. (5.65) and extract the largest
and the smallest roots, to give
514 CHAPTER 5 Equations of State and Phase Equilibria
Z3 Z2 + A B B2 Z AB ¼ 0
Z 3 Z2 + 0:1660384Z 0:0061218 ¼ 0
Largest root : Z v ¼ 0:802641
Smallest root : Z L ¼ 0:0527377
Step 3. Solve for the density of the liquid phase and gas phase:
ρL ¼
pM
ð185Þð44Þ
¼
¼ 25:7 lb=ft3
ZL RT ð0:0527377Þð10:73Þð560Þ
ρv ¼
pM
ð185Þð44Þ
¼
¼ 1:688 lb=ft3
Zv RT ð0:0802641Þð10:73Þð560Þ
Redlich and Kwong extended the application of their equation to hydrocarbon
liquid and gas mixtures by employing the following mixing rules:
"
#2
n
X
pffiffiffiffi
am ¼
xi ai
(5.68)
i¼1
bm ¼
n
X
½xi bi (5.69)
i¼1
where
n ¼ number of components in the mixture
ai ¼ Redlich-Kwong a parameter for the ith component as given by Eq. (5.62)
bi ¼ Redlich-Kwong b parameter for the ith component as given by Eq. (5.63)
am ¼ parameter a for mixture
bm ¼ parameter b for mixture
xi ¼ mole fraction of component i in the liquid phase
To calculate am and bm for a hydrocarbon gas mixture with a composition of yi,
use Eqs. (5.68), (5.69) and replace xi with yi:
"
#2
n
X
pffiffiffiffi
am ¼
yi ai
i¼1
n
X
bm ¼
½yi bi i¼1
Eq. (5.65) gives the compressibility factor of the gas phase or the liquid with the
coefficients A and B as defined by Eqs. (5.66), (5.67) as
A¼
am p
R2 T 2:5
B¼
bm p
RT
The application of the Redlich and Kwong equation of state for hydrocarbon
mixtures can be best illustrated through the following two examples (Examples
5.9 and 5.10).
Equations of State 515
EXAMPLE 5.9
Calculate the density of a crude oil with the composition at 4000 psia and 160°F
given in the table below. Use the Redlich-Kwong EOS.
Component
xi
M
pc
Tc
C1
C2
C3
n-C4
n-C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
16.043
30.070
44.097
58.123
72.150
84.00
215
666.4
706.5
616.0
527.9
488.6
453
285
343.33
549.92
666.06
765.62
845.8
923
1287
Solution
Step 1. Determine the parameters ai and bi for each component using
Eqs. (5.62), (5.63):
ai ¼ 0:47274
R2 Tc2:5
pc
bi ¼ 0:08664
RTc
pc
The results are shown in the table below.
Component xi
C1
C2
C3
n-C4
n-C5
C6
C7+
M
pc
0.45 16.043
0.05 30.070
0.05 44.097
0.03 58.123
0.01 72.150
0.01 84.00
0.40 215
Tc
666.4 343.33
706.5 549.92
616.0 666.06
527.9 765.62
488.6 845.8
453
923
285 1287
ai
bi
161,044.3
0.4780514
493,582.7
0.7225732
914,314.8
1.004725
1,449,929
1.292629
2,095,431
1.609242
2,845,191
1.945712
1.022348 (107) 4.191958
Step 2. Calculate the mixture parameters, am and bm, from Eqs. (5.68), (5.69), to
give
"
#2
n
X
pffiffiffiffi
am ¼
xi ai ¼ 2,591, 967
i¼1
and
bm ¼
n
X
½xi bi ¼ 2:0526
i¼1
Step 3. Compute the coefficients A and B by using Eqs. (5.66), (5.67), to
produce
516 CHAPTER 5 Equations of State and Phase Equilibria
A¼
am p
2,591, 967ð4000Þ
¼
¼ 9:406539
R2 T 2:5
10:732 ð620Þ2:5
B¼
bm p 2:0526ð4000Þ
¼
¼ 1:234049
RT
10:73ð620Þ
Step 4. Solve Eq. (5.65) for the largest positive root, to yield
Z3 Z2 + 6:934845Z 11:60813 ¼ 0
ZL ¼ 1:548126
Step 5. Calculate the apparent molecular weight of the crude oil:
X
xi Mi ¼ 110:2547
Ma ¼
Step 6. Solve for the density of the crude oil:
ρL ¼
pMa
ð4000Þð100:2547Þ
¼
¼ 38:93 lb=ft3
ZL RT ð10:73Þð620Þð1:548120Þ
Note that calculating the liquid density as by Standing’s method gives a value of
46.23 lb/ft3.
EXAMPLE 5.10
Using the the RK EOS, calculate the density of a gas phase with the composition
at 4000 psia and 160°F shown in the following table.
Component
yi
M
pc
Tc
C1
C2
C3
C4
C5
C6
C7+
0.86
0.05
0.05
0.02
0.01
0.005
0.005
16.043
30.070
44.097
58.123
72.150
84.00
215
666.4
706.5
616.0
527.9
488.6
453
285
343.33
549.92
666.06
765.62
845.8
923
1287
Solution
Step 1. Determine the parameters ai and bi for each component using
Eqs. (5.62), (5.63):
ai ¼ 0:47274
R2 Tc2:5
pc
bi ¼ 0:08664
RTc
pc
The results are shown in the table below.
Equations of State 517
Component yi
C1
C2
C3
C4
C5
C6
C7+
M
pc
0.86
16.043
0.05
30.070
0.05
44.097
0.02
58.123
0.01
72.150
0.005 84.00
0.005 215
Tc
666.4 343.33
706.5 549.92
616.0 666.06
527.9 765.62
488.6 845.8
453
923
285 1287
ai
bi
161,044.3
0.4780514
493,582.7
0.7225732
914,314.8
1.004725
1,449,929
1.292629
2,095,431
1.609242
2,845,191
1.945712
1.022348(107) 4.191958
Step 2. Calculate am and bm using Eqs. (5.68), (5.69), to give
"
#2
n
X
pffiffiffiffi
am ¼
yi ai ¼ 241,118
i¼1
X
bi xi ¼ 0:5701225
bm ¼
Step 3. Calculate the coefficients A and B by applying Eqs. (5.66), (5.67), to
yield
A¼
am p
241,118ð4000Þ
¼
¼ 0:8750
R2 T 2:5 10:733 ð620Þ2:5
B¼
bm p 0:5701225ð4000Þ
¼
¼ 0:3428
RT
10:73ð620Þ
Step 4. Solve Eq. (5.65) for Zv, to give
Z 3 Z 2 + 0:414688Z 0:29995 ¼ 0
Z v ¼ 0:907
Step 5. Calculate the apparent molecular weight and density of the gas mixture:
P
Ma ¼ yi Mi ¼ 20:89
ρv ¼
pMa
ð4000Þð20:89Þ
¼
¼ 13:85 lb=ft3
Zv RT ð10:73Þð620Þð0:907Þ
Soave-Redlich-Kwong Equation of State “SRK EOS”
One of the most significant milestones in the development of cubic equations of state is accredited to Soave (1972) with his proposed modification
of the RK EOS attraction pressure term. Soave replaced the explicit temperature term (a/T0.5) in Eq. (5.60) with a more generalized temperaturedependent term, denoted by [a α (T)]; that is,
p¼
RT
aαðT Þ
V b V ðV + bÞ
(5.70)
518 CHAPTER 5 Equations of State and Phase Equilibria
where α(T) is a dimensionless parameter that is defined by the following
expression:
h
pffiffiffiffiffiffiffiffiffiffi i2
αðT Þ ¼ 1 + m 1 T=Tc
Or equivalently in terms of reduced temperature “Tr”:
pffiffiffiffiffi 2
α ðT Þ ¼ 1 + m 1 T r
(5.71)
Soave imposed the following conditions on the a dimensionless
parameter α(T):
n
n
α(T) ¼ 1 when the reduced temperature Tr, ¼ 1; that is, α(T ¼ Tc) ¼ 1
when T/Tc ¼ 1
At temperatures other than the critical temperature, Soave regressed on
the vapor pressure of pure components to develop a temperature
correction parameter “m” that correlated with the acentric factor “ω” by
the following relationship:
m ¼ 0:480 + 1:74ω 0:176ω2
(5.72)
where
Tr ¼ reduced temperature, T/Tc
ω ¼ acentric factor of the substance
T ¼ system temperature, °R
For simplicity and convenience, the temperature-dependent term “α(T)” is
replaced by the symbol “α” for the remainder of this chapter.
For any pure component, the constants “a” and “b” in Eq. (5.70) are found
by imposing the classical van der Waals’s critical point constraints on
Eq. (5.70) and solving the resulting two equations, to give
a ¼ Ωa
R2 Tc2
pc
(5.73)
RTc
pc
(5.74)
b ¼ Ωb
where
Ωa ¼ 0:42747
Ωb ¼ 0:08664
Edmister and Lee (1986) pointed out that the following constrain satisfies
the critical isotherm as given by
ðV Vc Þ3 ¼ V 3 ½3Vc V 2 + 3Vc2 V Vc3 ¼ 0
(5.75)
Equations of State 519
Eq. (5.70) also can be expressed into a cubic form, to give
V3 RT 2 aα bRT
ðaαÞb
V +
b2 V ¼0
p
p
p
p
(5.76)
At the critical point, Eqs. (5.75), (5.76) are identical with α ¼ 1. Equating the
corresponding coefficients in both equations gives
RTc
pc
(5.77)
a bRTc
b2
pc
pc
(5.78)
ab
pc
(5.79)
3Vc ¼
3Vc2 ¼
Vc3 ¼
Solving these equations simultaneously for parameters a and b will give
relationships identical to those given by Eqs. (5.73), (5.74), in addition:
n
rearranging Eq. (5.77) gives
1
pc Vc ¼ RTc
3
which indicates that the SRK EOS gives a universal critical gas compressibility factor “Zc” of 0.333.
n
combining Eq. (5.74) with (5.77), gives the expected value of the
covolume of 26% of the critical volume; that is,
b ¼ 0:26Vc
The compressibility factor “Z” can be introduced into Eq. (5.76) by replacing the molar volume “V” with (ZRT/p) and rearranging to give
Z3 Z2 + A B B2 Z AB ¼ 0
(5.80)
with
A¼
ðaαÞp
ðRT Þ2
B¼
bp
RT
(5.81)
(5.82)
520 CHAPTER 5 Equations of State and Phase Equilibria
where
p ¼ system pressure, psia
T ¼ system temperature, °R
R ¼ 10.730 psia ft3/lb-mol °R
EXAMPLE 5.11
Rework Example 5.7 and solve for the density of the two phases using the
SRK EOS.
Solution
Step 1. Determine the critical pressure, critical temperature, and acentric factor
from Table 1.1 of Chapter 1, to give
Tc ¼ 666:01°R
pc ¼ 616:3 psia
ω ¼ 0:1524
Step 2. Calculate the reduced temperature:
Tr ¼ T=Tc ¼ 560=666:01 ¼ 0:8408
Step 3. Calculate the parameter m by applying Eq. (5.72), to yield
m ¼ 0:480 + 1:574ω 0:176ω2
m ¼ 0:480 + 1:574ð0:1524Þ 0:176ð1:524Þ2
Step 4. Solve for parameter α using Eq. (5.71), to give
pffiffiffiffiffi 2
α ¼ 1 + m 1 Tr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
α ¼ 1 + 0:7052 1 0:8408 ¼ 1:120518
Step 5. Compute the coefficients a and b by applying Eqs. (5.73), (5.74), to yield
α ¼ Ωa
R2 Tc2
10:732 ð666:01Þ2
¼ 35, 427:6
¼ 0:42747
pc
616:3
α ¼ Ωb
RTc
10:73ð666:01Þ
¼ 0:08664
¼ 1:00471
616:3
pc
Step 6. Calculate the coefficients A and B from Eqs. (5.81), (5.82) to produce
A¼
ðaαÞp ð35, 427:6Þð1:120518Þ185
¼
¼ 0:203365
R2 T 2
10:732 ð560Þ2
B¼
bp ð1:00471Þð185Þ
¼
¼ 0:034658
RT
ð10:73Þð560Þ
Step 7. Solve Eq. (5.80) for ZL and Zv:
Z 3 Z2 + A B B2 Z + AB ¼ 0
Z 3 Z2 + 0:203365 0:034658 0:0346582 Z + ð0:203365Þð0:034658Þ ¼ 0
Solving this third-degree polynomial gives
ZL ¼ 0:06729
Zv ¼ 0:80212
Equations of State 521
Step 8. Calculate the gas and liquid density, to give
ρv ¼
pM
ð185Þð44:0Þ
¼ 1:6887 lb=ft3
¼
Z v RT ð0:802121Þð10:73Þð560Þ
ρL ¼
pM
ð185Þð44:0Þ
¼
¼ 20:13 lb=ft3
L
Z RT ð0:06729Þð10:73Þð560Þ
The Mixture Mixing Rule
To use Eq. (5.80) with mixtures, mixing rules are required to determine the
terms (aα) and b for the mixtures. Soave adopted the following mixing rules:
ðaαÞm ¼
XX
i
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi xj ai aj αi αj 1 kij
(5.83)
X
½xi bi (5.84)
j
bm ¼
i
with
A¼
ðaαÞm p
ðRT Þ2
(5.85)
and
B¼
bm p
RT
(5.86)
The parameter kij is an empirically determined correction factor called the
binary interaction coefficient (BIC), which is included to characterize any
binary system formed by components i and j in the hydrocarbon mixture.
These binary interaction coefficients are used to model the intermolecular
interaction through empirical adjustment of the (aα)m term as represented
mathematically by Eq. (5.83). They are dependent on the difference in
molecular size of components in a binary system and they are characterized
by the following properties:
n
The interaction between hydrocarbon components increases as the
relative difference between their molecular weights increases:
ki, j + 1 > ki, j
n
Hydrocarbon components with the same molecular weight have a binary
interaction coefficient of zero:
ki, j ¼ 0
522 CHAPTER 5 Equations of State and Phase Equilibria
n
The binary interaction coefficient matrix is symmetric:
ki, j ¼ kj, i
Slot-Petersen (1987) and Vidal and Daubert (1978) presented theoretical
background to the meaning of the interaction coefficient and techniques
for determining their values. Graboski and Daubert (1978) and Soave
(1972) suggested that no binary interaction coefficients are required for
hydrocarbon-hydrocarbon pairs (eg, kij ¼ 0), except between C1 and C7+.
Whiston and Brule (2000) proposed the following expression for calculating
the binary interaction coefficient between methane and the heavy fractions:
kC1 C7 + ¼ 0:18 h
16:668 vci
1:1311 + ðvci Þ1=3
i6
where vci is the heavy fraction critical volume, which can be estimated from
vci ¼ 0:4804 + 0:06011 Mi + 0:00001076 ðMi Þ2
where
vci ¼ critical volume of the C7+, or its lumped component, ft3/lbm
Mi ¼ molecular weight
If nonhydrocarbons are present in the mixture, including binary interaction
parameters can greatly improve the volumetric and phase behavior predictions
of the mixture by the SRK EOS. Reid et al. (1987) proposed the following tabulated binary interaction coefficients for applications with the SRK EOS:
Component
N2
CO2
H2S
N2
CO2
H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7 +
0
0
0.12
0.02
0.06
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0
0
0.12
0.12
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0
0
0
0.08
0.07
0.07
0.06
0.06
0.06
0.06
0.05
0.03
In applying Eq. (5.75) for the compressibility factor of the liquid phase “ZL”
the composition of the liquid “xi” is used to calculate the coefficients A and B
Equations of State 523
of Eqs. (5.85), (5.86) through the use of the mixing rules as described by
Eqs. (5.83), (5.84). For determining the compressibility factor of the gas
phase, Zv, the previously outlined procedure is used with composition of
the gas phase, yi, replacing xi.
EXAMPLE 5.12
A two-phase hydrocarbon system exists in equilibrium at 4000 psia and 160°F.
The system has the composition shown in the following table.
Component
xi
yi
pc
Tc
ωi
C1
C2
C3
C4
C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
0.86
0.05
0.05
0.02
0.01
0.005
0.0005
666.4
706.5
616.0
527.9
488.6
453
285
343.33
549.92
666.06
765.62
845.8
923
1160
0.0104
0.0979
0.1522
0.1852
0.2280
0.2500
0.5200
The heptanes-plus fraction has the following properties:
m ¼ 215
Pc ¼ 285 psia
Tc ¼ 700°F
ω ¼ 0:52
Assuming kij ¼ 0, calculate the density of each phase by using the SRK EOS.
Solution
Step 1. Calculate the parameters α, a, and b by applying Eqs. (5.71), (5.73),
(5.74):
pffiffiffiffiffi 2
α ¼ 1 + m 1 Tr
R2 Tc2
pc
RTc
b ¼ 0:08664
pc
a ¼ 0:42747
The results are shown in the table below.
Component
pc
Tc
ωi
αi
ai
bi
C1
C2
C3
C4
C5
C6
C7+
666.4
706.5
616.0
527.9
488.6
453
285
343.33
549.92
666.06
765.62
845.8
923
1160
0.0104
0.0979
0.1522
0.1852
0.2280
0.2500
0.5200
0.6869
0.9248
1.0502
1.1616
1.2639
1.3547
1.7859
8689.3
21,040.8
35,422.1
52,390.3
72,041.7
94,108.4
232,367.9
0.4780
0.7225
1.0046
1.2925
1.6091
1.9455
3.7838
524 CHAPTER 5 Equations of State and Phase Equilibria
Step 2. Calculate the mixture parameters (aα)m and bm for the gas phase and
liquid phase by applying Eqs. (5.83), (5.84), to give the following.
For the gas phase, using yi:
XX
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi yj ai aj αi αj 1 kij ¼ 9219:3
ðaαÞm ¼
X i j
bm ¼
½yi bi ¼ 0:5680
i
For the liquid phase, using xi:
XX
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi xj ai aj αi αj 1 kij ¼ 104,362:9
ðaαÞm ¼
X i j
bm ¼
½xi bi ¼ 1:8893
i
Step 3. Calculate the coefficients A and B for each phase by applying
Eqs. (5.85), (5.86), to yield the following.
For the gas phase,
A¼
ðaαÞm p ð9219:3Þð4000Þ
¼
¼ 0:8332
R2 T 2
ð10:73Þ2 ð620Þ2
B¼
bm p ð0:5680Þð4000Þ
¼
¼ 0:3415
RT
ð10:73Þð620Þ
For the liquid phase,
A¼
ðaαÞm p ð104, 362:9Þð4000Þ
¼
¼ 9:4324
R2 T 2
ð10:73Þ2 ð620Þ2
B¼
bm p ð1:8893Þð4000Þ
¼
¼ 1:136
RT
ð10:73Þð620Þ
Step 4. Solve Eq. (5.80) for the compressibility factor of the gas phase, to
produce
Z3 Z2 + A B B2 Z + AB ¼ 0
Z 3 Z2 + ð0:8332 0:3415 0:34152ÞZ + ð0:8332Þð0:3415Þ ¼ 0
Solving this polynomial for the largest root gives
Zv ¼ 0:9267
Step 5. Solve Eq. (5.80) for the compressibility factor of the liquid phase, to
produce
Z3 Z 2 + A B B2 Z + AB ¼ 0
Z 3 Z 2 + ð9:4324 1:136 1:1362ÞZ + ð9:4324Þð1:136Þ ¼ 0
Solving the polynomial for the smallest root gives
ZL ¼ 1:4121
Step 6. Calculate the apparent molecular weight of the gas phase and liquid
phase from their composition, to yield the following:
Equations of State 525
For the gas phase: Ma ¼
X
For the liquid phase: Ma ¼
yi Mi ¼ 20:89
X
xi Mi ¼ 100:25
Step 7. Calculate the density of each phase.
ρ¼
pMa
RTZ
For the gas phase: ρv ¼
ð4000Þð20:89Þ
¼ 13:556 lb=ft3
ð10:73Þð620Þð0:9267Þ
For the liquid phase: ρL ¼
ð4000Þð100:25Þ
¼ 42:68 lb=ft3
ð10:73Þð620Þð1:4121Þ
It is appropriate at this time to introduce and define the concept of the fugacity and the fugacity coefficient of the component. The fugacity “f” is a measure of the molar Gibbs energy of a real gas. It is evident from the definition
that the fugacity has the units of pressure; in fact, the fugacity may be looked
on as a vapor pressure modified to correctly represent the tendency of the
molecules from one phase to escape into the other. In a mathematical form,
the fugacity of a pure component is defined by the following expression:
f o ¼ p exp
Z p Z1
dp
p
o
(5.87)
where
fo ¼ fugacity of a pure component, psia
p ¼ pressure, psia
Z ¼ compressibility factor
The ratio of the fugacity to the pressure, f/p, is called the fugacity coefficient
“Φ”. This dimensionless property is calculated from Eq. (5.87) as
fo
¼ Φ ¼ exp
p
Z p Z1
dp
p
o
Soave applied this generalized thermodynamic relationship to Eq. (5.70) to
determine the fugacity coefficient of a pure component, to give
o
f
A
Z+B
¼ ln ðΦÞ ¼ Z 1 ln ðZ BÞ ln
ln
p
B
Z
(5.88)
For a hydrocarbon multicomponent mixture that exits in the two-phase
region, the component fugacity in each phase is introduced to develop a criterion for thermodynamic equilibrium. Physically, the fugacity of a
526 CHAPTER 5 Equations of State and Phase Equilibria
component i in one phase with respect to the fugacity of the component in a
second phase is a measure of the potential for transfer of the component
between phases. The phase with the lower component fugacity accepts
the component from the phase with a higher component fugacity. Equal
fugacities of a component in the two phases results in a zero net transfer.
A zero transfer for all components implies that the hydrocarbon system is
in thermodynamic equilibrium. Therefore, the condition of the thermodynamic equilibrium can be expressed mathematically by
fiv ¼ fiL , 1 i n
(5.89)
where
f vi ¼ fugacity of component i in the gas phase, psi
f Li ¼ fugacity of component i in the liquid phase, psi
n ¼ number of components in the system
The fugacity coefficient of component i in a hydrocarbon liquid mixture or
hydrocarbon gas mixture is a function of the system pressure, mole fraction,
and fugacity of the component. The fugacity coefficient is defined as
fv
For a component i in the gas phase: Φvi ¼ i
yi p
(5.90)
fL
For a component i in the liquid phase: ΦLi ¼ i
xi p
(5.91)
where
Φvi ¼ fugacity coefficient of component i in the vapor phase
ΦLi ¼ fugacity coefficient of component i in the liquid phase.
Equilibrium Ratio “Ki”
The equilibrium ratio is defined mathematically by Eq. (5.1) as
Ki ¼ yi =xi
When two phases exist in equilibrium, it suggests that fiL ¼ fiv , which leads
to redefining the K-value in terms of the fugacity of components as
Ki ¼
fiL =ðxi pÞ
ΦLi
¼
Φvi
fiv ðyi pÞ
(5.92)
Reid et al. (1987) defined the fugacity coefficient of component i in a hydrocarbon mixture by the following generalized thermodynamic relationship:
1
lnðΦi Þ ¼
RT
Z 1 V
@p RT
@ni V
dV ln ðZ Þ
(5.93)
Equations of State 527
where
V ¼ total volume of n moles of the mixture
ni ¼ number of moles of component i
Z ¼ compressibility factor of the hydrocarbon mixture
By combining the above thermodynamic definition of the fugacity with the
SRK EOS [Eq. (5.70)], Soave proposed the following expression for the
fugacity coefficient of component i in the liquid phase:
ln
L
bi ðZL 1Þ
fi
A
2Ψ i
bi
B
¼ ln ΦLi
ln 1 + L
ln ZL B xi p
bm
B ðaαÞm bm
Z
(5.94)
where
Ψi ¼
X
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xj ai aj αi αj 1 kij
(5.95)
j
ðaαÞm ¼
XX
i
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi xj ai aj αi αj 1 kij
(5.96)
j
Eq. (5.94) is also used to determine the fugacity coefficient of component in
the gas phase, Φvi , by using the composition of the gas phase, yi, in calculating A, B, Zv, and other composition-dependent terms; or,
bi ðZv 1Þ
A
2Ψ i
bi
B
ln 1 + v
ln Φvi ¼
lnðZv BÞ bm
B ðaαÞm bm
Z
where
X
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yj ai aj αi αj 1 kij
j XX
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðaαÞm ¼
yi yj ai aj αi αj 1 kij
Ψi ¼
i
j
Gibbs Energy and Selecting the Correct Z-Factor
In solving the cubic Z-factor equation for hydrocarbon mixtures, one or
three real roots may exist. When three roots exist with different values
designated as
n
n
n
Largest root: ZLargest
Middle root: Zmiddle
Smallest root: ZSmallest,
The middle root ZMiddle, always is discarded as it has nonphysical value or
meaning. However, selecting of the correct root between the remaining two
528 CHAPTER 5 Equations of State and Phase Equilibria
roots is important to avoid phase equilibria convergence problems, which
will lead the EOS to generate unrealistic results. The correct root is chosen
as the one with the lowest normalized Gibbs energy function, g*. The
normalized Gibbs energy function is defined in terms of the composition
of the hydrocarbon phase and the individual component fugacity in that
phase. Mathematically, the normalized Gibbs energy function is defined
by the following expressions:
Normalized Gibbs energy for gas phase: ggas ¼
n
X
yi ln fiv
i¼1
Normalized Gibbs energy for liquid phase: gLiquid ¼
n
X
xi ln fiL
i¼1
Therefore, assume that a liquid phase with a composition of xi has three
Z-factor roots; the middle root is discarded automatically and the remaining
two are designated ZL1 and ZL2 . To select the correct root, calculate the
normalized Gibbs energy function using the two remaining roots:
gZL ¼
1
n
X
xi ln fiL
i¼1
n
X
xi ln fiL
gZL ¼
2
i¼1
in which the fugacity fLi is determined from Eq. (5.94) using the two remaining roots, ZL1 and ZL2 , as
L
bi ðZL1 1Þ
A
2Ψ i
bi
B
ln 1 +
fi zL1 ¼ xi p + exp
lnðZL1 BÞ bm
B ðaαÞm bm
ZL1
L
bi ðZL2 1Þ
A
2Ψ i
bi
B
ln 1 +
lnðZL2 BÞ fi zL2 ¼ xi + exp
bm
B ðaαÞm bm
ZL2
If gZL1 <gZL2 then ZL1 is chosen as the correct root; otherwise, ZL2 is chosen.
It should be pointed out that the above procedure is applied for the gas phase
if three Z-factor values are obtained with largest as referred to as Zg1 and
smallest as Zg2; with the normalized Gibbs energy function as given by
gZg1 ¼
n
X
yi ln fiv
gZg2 ¼
n
X
yi ln fiv
i¼1
i¼1
Equations of State 529
in which the fugacity f vi is determined by using the two remaining roots, Zg1
and Zg2, as
v
bi Zg1 1
A
2Ψ i
bi
B
ln 1 +
fi zg1 ¼ xi p + exp
ln Zg1 B bm
B ðaαÞm bm
Zg1
v
bi Zg2 1
A
2Ψ i
bi
B
ln 1 +
fi zg2 ¼ xi + exp
ln Zg2 B bm
B ðaαÞm bm
Zg2
If gZg1 < gZg2 then Zg1 is chosen as the correct root; otherwise, Zg2 is chosen.
Chemical Potential
The chemical potential “μi” is a quantity that measures how much energy a
component brings to a mixture as the component transfers from one phase to
another. In a mixture, the chemical potential is defined as the slope of the
free energy “G” of the system with respect to a change in the number of
moles of that specific component. Thus, it is the partial derivative of the
free energy with respect to the amount of that specific component, which
is given the name chemical potential:
@G
μi ¼
@ni
p, T , nj
In a mixture, the normalized Gibbs free energy “g” is not the same in both
phases because their compositions are different. Instead, the criterion for
equilibrium is defined mathematically by
i
@G h gas
¼0
¼ μi μliquid
i
@ni
This criterion implies that, at equilibrium, the chemical potential of each
component μi has the same value in each phase:
liquid
μgas
i ¼ μi
The chemical potential of component i is related to the activity a^i by
μi ¼ μoi + RT lnða^i Þ
with
fi
a^i ¼ o
fi
530 CHAPTER 5 Equations of State and Phase Equilibria
where
a^i ¼ activity of component i in mixture at p and T
μoi ¼ chemical potential of pure of component i at p and T
f oi ¼ fugacity of pure component i at p and T
fi ¼ fugacity of component i in mixture at p and T
MODIFICATIONS OF THE SRK EOS
To improve the pure component vapor pressure predictions by the SRK
equation of state, Groboski and Daubert (1978) proposed a new expression
for calculating parameter m of Eq. (5.72). The proposed relationship, which
originated from analyzing extensive experimental data for pure hydrocarbons, has the following form:
m ¼ 0:48508 + 1:55171ω 0:15613ω2
(5.97)
Sim and Daubert (1980) pointed out that, because the coefficients of
Eq. (5.97) were determined by analyzing vapor pressure data of lowmolecular-weight hydrocarbons, it is unlikely that Eq. (5.97) will suffice
for high-molecular-weight petroleum fractions. Realizing that the acentric
factors for the heavy petroleum fractions are calculated from an equation
such as the Edmister correlation or the Lee and Kesler correlation, the
authors proposed the following expressions for determining parameter m:
n
If the acentric factor is determined using the Edmister correlation, then
m ¼ 0:431 + 1:57ωi 0:161ðωÞ2i
n
(5.98)
If the acentric factor is determined using the Lee and Kesler
correction, then
m ¼ 0:315 + 1:60ωi 0:166ω2i
(5.99)
Elliot and Daubert (1985) stated that the optimal binary interaction coefficient, kij, would minimize the error in the representation of all the thermodynamic properties of a mixture. Properties of particular interest in phase
equilibrium calculations include bubble point pressure, dew point pressure,
and equilibrium ratios. The authors proposed a set of relationships for determining interaction coefficients for asymmetric mixtures2 that contain
2
An asymmetric mixture is defined as one in which two of the components are considerably different in their chemical behavior. Mixtures of methane with hydrocarbons of 10 or
more carbon atoms can be considered asymmetric. Mixtures containing gases such as
nitrogen or hydrogen are asymmetric.
Modifications of the SRK EOS 531
methane, N2, CO2, and H2S. Referring to the principal component as i and
other fractions as j, Elliot and Daubert proposed the following expressions:
Binary Interaction between N2 – Components
kij ¼ 0:107089 + 2:9776kij1
(5.100)
Binary Interaction between CO2 – Components
kij ¼ 0:08058 0:77215kij1 1:8404 kij1
(5.101)
Binary Interaction between H2S – Components
kij ¼ 0:07654 + 0:017921kij1
(5.102)
Methane with components > C9 (Nonane)
kij ¼ 0:17985 + 2:6958kij1 + 10:853 kij1
2
(5.103)
where
kij1 ¼
2
εi εj
2εi εj
(5.104)
and
εi ¼
pffiffiffiffi
0:480453 ai
bi
(5.105)
The two parameters, ai and bi, of Eq. (5.105) were defined previously by
Eqs. (5.73), (5.74); that is,
a ¼ Ωa
R2 Tc2
pc
b ¼ Ωb
RTc
pc
Volume Translation (Volume Shift) Parameter
The major drawback in the SRK EOS is that the critical compressibility
factor takes on the unrealistic universal critical compressibility of 0.333
for all substances. Consequently, the molar volumes typically are overestimated; that is, densities are underestimated.
Peneloux et al. (1982) developed a procedure for improving the volumetric
predictions of the SRK EOS by introducing a volume correction parameter,
ci, into the equation. This third parameter does not change the vapor/liquid
532 CHAPTER 5 Equations of State and Phase Equilibria
equilibrium conditions determined by the unmodified SRK equation (ie,
the equilibrium ratio Ki) but it modifies the liquid and gas volumes. The
proposed methodology, known as the volume translation method, uses
the following expressions:
L
Vcorr
¼ VL X
ðxi ci Þ
(5.106)
i
v
Vcorr
¼ Vv X
ðy i c i Þ
(5.107)
i
where
VL ¼ uncorrected liquid molar volume (ie, VL ¼ ZLRT/p), ft3/mol
Vv ¼ uncorrected gas molar volume Vv ¼ ZvRT/p, ft3/mol
VLcorr ¼ corrected liquid molar volume, ft3/mol
Vvcorr ¼ corrected gas molar volume, ft3/mol
xi ¼ mole fraction of component i in the liquid phase
yi ¼ mole fraction of component i in the gas phase
The authors proposed six schemes for calculating the correction factor, ci,
for each component. For petroleum fluids and heavy hydrocarbons, Peneloux and coworkers suggested that the best correlating parameter for the volume correction factor ci is the Rackett compressibility factor, ZRA. The
correction factor then is defined mathematically by the following
relationship:
ci ¼ 4:437978ð0:29441 ZRA ÞTci =pci
(5.108)
where
ci ¼ volume shift coefficient for component i, ft3/lb-mol
Tci ¼ critical temperature of component i, °R
pci ¼ critical pressure of component i, psia
The parameter ZRA is a unique constant for each compound. The values of
ZRA, in general, are not much different from those of the critical compressibility factors, Zc. If their values are not available, Peneloux et al. proposed
the following correlation for calculating ci:
ci ¼ ð0:0115831168 + 0:411844152ωi Þ
where
ωi ¼ acentric factor of component i.
Tci
pci
(5.109)
Modifications of the SRK EOS 533
EXAMPLE 5.13
Rework Example 5.12 by incorporating the Peneloux et al. volume correction
approach in the solution. Key information from Example 5.12 includes:
For gas: Zv ¼ 0:9267, Ma ¼ 20:89
For liquid: Z L ¼ 1:4212, Ma ¼ 100:25
T ¼ 160°F, and p ¼ 4000psi
Solution
Step 1. Calculate the correction factor ci using Eq. (5.109):
Tci
ci ¼ ð0:0115831168 + 0:411844152ωi Þ
pci
The results are shown in the table below.
Component xi
C1
C2
C3
C4
C5
C6
C
P7+
pc
Tc
0.45 666.4 343.33
0.05 706.5 549.92
0.05 616.0 666.06
0.03 527.9 765.62
0.01 488.6 845.8
0.01 453
923
0.40 285 1160
ωi
ci
cixi
yi
ciyi
0.0104 0.00839 0.003776 0.86 0.00722
0.0979 0.03807 0.001903 0.05 0.00190
0.1522 0.07729 0.003861 0.05 0.00386
0.1852 0.1265 0.00379 0.02 0.00253
0.2280 0.19897 0.001989 0.01 0.00198
0.2500 0.2791 0.00279 0.005 0.00139
0.5200 0.91881 0.36752 0.005 0.00459
0.38564
0.02349
Step 2. Calculate the uncorrected volume of the gas and liquid phase using the
compressibility factors as calculated in Example 5.12.
Vv ¼
RTZv ð10:73Þð620Þð0:9267Þ
¼
¼ 1:54119ft3 =mol
p
4000
VL ¼
RTZL ð10:73Þð620Þð1:4121Þ
¼
¼ 2:3485ft3 =mol
p
4000
Step 3. Calculate the corrected gas and liquid volumes by applying Eqs. (5.106),
(5.107).
X
L
¼ VL ðxi ci Þ ¼ 2:3485 0:38564 ¼ 1:962927ft3 =mol
Vcorr
i
v
Vcorr
¼ Vv X
ðyi ci Þ ¼ 1:54119 0:02394 ¼ 1:5177 ft3 =mol
i
Step 4. Calculate the corrected compressibility factors.
v
Zcorr
¼
ð4000Þð1:5177Þ
¼ 0:91254
ð10:73Þð620Þ
L
¼
Zcorr
ð4000Þð1:962927Þ
¼ 1:18025
ð10:73Þð620Þ
534 CHAPTER 5 Equations of State and Phase Equilibria
Step 5. Determine the corrected densities of both phases.
ρ¼
pMa
RTZ
ð4000Þð20:89Þ
¼ 13:767lb=ft3
ð10:73Þð620Þð0:91254Þ
ð4000Þð100:25Þ
ρL ¼
¼ 51:07lb=ft3
ð10:73Þð620Þð1:18025Þ
ρv ¼
As pointed out by Whitson and Brule (2000), when the volume shift is introduced to the EOS for mixtures, the resulting expressions for fugacity are
h
L
pi
fi modified ¼ fiL original exp ci
RT
and
h
v
pi
fi modified ¼ fiv original exp ci
RT
This implies that the fugacity ratios are unchanged by the volume shift:
L
L
fi
f
iv modified ¼ v original
fi modified
fi modified
The Peng-Robinson Equation of State
Peng and Robinson (1976a) conducted a comprehensive study to evaluate
the use of the SRK equation of state for predicting the behavior of naturally
occurring hydrocarbon systems. They illustrated the need for an improvement in the ability of the equation of state to predict liquid densities and
other fluid properties, particularly in the vicinity of the critical region. As
a basis for creating an improved model, Peng and Robinson (PR) proposed
the following expression:
p¼
RT
aα
V b ðV + bÞ2 cb2
where a, b, and α have the same significance as they have in the SRK model,
and parameter c is a whole number optimized by analyzing the values of the
terms Zc and b/Vc as obtained from the equation. It is generally accepted that
Zc should be close to 0.28 and b/Vc should be approximately 0.26. An optimized value of c ¼ 2 gives Zc ¼ 0.307 and (b/Vc) ¼ 0.253. Based on this value
Modifications of the SRK EOS 535
of c, Peng and Robinson proposed the following equation of state (commonly referred to as PR EOS):
p¼
RT
aα
V b V ðV + bÞ + bðV bÞ
(5.110)
Imposing the classical critical point conditions [Eq. (5.48)] on Eq. (5.110)
and solving for parameters a and b yields
a ¼ Ωa
R2 Tc2
pc
(5.111)
RTc
pc
(5.112)
b ¼ Ωb
where
Ωa ¼ 0:45724
Ωb ¼ 0:07780
This equation predicts a universal critical gas compressibility factor, Zc, of
0.307, compared to 0.333 for the SRK model. Peng and Robinson also
adopted Soave’s approach for calculating parameter α:
pffiffiffiffiffi 2
α ¼ 1 + m 1 Tr
(5.113)
where m ¼ 0.3796 + 1.54226ω 0.2699ω2.
Peng and Robinson (1978) proposed the following modified expression for m
that is recommended for heavier components with acentric values ω > 0.49:
m ¼ 0:379642 + 1:48503ω 0:1644ω2 + 0:016667ω3
(5.114)
Rearranging Eq. (5.110) into the compressibility factor form, gives
Z3 + ðB 1ÞZ 2 + A 3B2 2B Z AB B2 B3 ¼ 0
(5.115)
where A and B are given by Eqs. (5.81), (5.82) for pure component and by
Eqs. (5.85), (5.86) for mixtures; that is,
A¼
B¼
with
ðaαÞm ¼
bm ¼
XX
ðRT Þ2
bm p
RT
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi xj ai aj αi αj 1 kij
i
j
X
½xi bi i
ðaαÞm p
536 CHAPTER 5 Equations of State and Phase Equilibria
EXAMPLE 5.14
Using the composition given in Example 5.12, calculate the density of the gas
phase and liquid phase by using the Peng-Robinson EOS. Assume kij ¼ 0. The
components are described in the following table.
Component
xi
yi
pc
Tc
ωi
C1
C2
C3
C4
C5
C6
C7+
0.45
0.05
0.05
0.03
0.01
0.01
0.40
0.86
0.05
0.05
0.02
0.01
0.005
0.0005
666.4
706.5
616.0
527.9
488.6
453
285
343.33
549.92
666.06
765.62
845.8
923
1160
0.0104
0.0979
0.1522
0.1852
0.2280
0.2500
0.5200
The heptanes-plus fraction has the following properties:
M ¼ 215
pc ¼ 285 psia
Tc ¼ 700°F
ω ¼ 0:582
Solution
Step 1. Calculate the parameters a, b, and α for each component in the system by
applying Eqs. (5.111)–(5.113):
R2 Tc2
pc
RTc
b ¼ 0:07780
pc
pffiffiffiffiffi 2
α ¼ 1 + m 1 Tr
a ¼ 0:45724
The results are shown in the table below.
Component pc
C1
C2
C3
C4
C5
C6
C7+
Tc
666.4 343.33
706.5 549.92
616.0 666.06
527.9 765.62
488.6 845.8
453
923
285 1160
ωi
m
αi
ai
0.0104 1.8058 0.7465
9294.4
0.0979 1.1274 0.9358 22,506.1
0.1522 0.9308 1.0433 37,889.0
0.1852 0.80980 1.1357 56,038.9
0.2280 0.73303 1.2170 77,058.9
0.2500 0.67172 1.2882 100,662.3
0.5200 0.53448 1.6851 248,550.5
bi
0.4347
0.6571
0.9136
1.1755
1.6091
1.4635
3.4414
Step 2. Calculate the mixture parameters (aα)m and bm for the gas and liquid
phase, to give the following.
Modifications of the SRK EOS 537
For the gas phase,
XX
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi yj ai aj αi αj 1 kij ¼ 10,423:54
ðaαÞm ¼
i
j
X
bm ¼
ðyi bi Þ ¼ 0:862528
i
For the liquid phase,
XX
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi xj ai aj αi αj 1 kij ¼ 107,325:4
ðaαÞm ¼
i
j
X
bm ¼
ðxi bi Þ ¼ 1:696543
i
Step 3. Calculate the coefficients A and B, to give the following.
For the gas phase,
A¼
ðaαÞm p ð10, 423:54Þð4000Þ
¼
¼ 0:94209
R2 T 2
ð10:73Þ2 ð620Þ2
B¼
bm p ð0:862528Þð4000Þ
¼
¼ 0:30669
RT
ð10:73Þð620Þ
For the liquid phase,
A¼
ðaαÞm p ð107, 325:4Þð4000Þ
¼
¼ 9:700183
R2 T 2
ð10:73Þ2 ð620Þ2
B¼
bm p ð1:696543Þð4000Þ
¼
¼ 1:020078
RT
ð10:73Þð620Þ
Step 4. Solve Eq. (5.115) for the compressibility factor of the gas and liquid
phases to give
Z3 + ðB 1ÞZ 2 + A 3B2 2B Z AB B2 B3 ¼ 0
For the gas phase:
Substituting for A ¼ 0.94209 and B ¼ 0.30669 in the above equation gives
Z 3 + ðB 1ÞZ2 + A 3B2 2B2 2B Z AB B2 B3 ¼ 0
h
i
Z 3 ¼ ð0:30669 1ÞZ 2 + 0:94209 3ð0:30669Þ2 2ð0:30669Þ Z
h
i
ð0:94209Þð0:30669Þ ð0:30669Þ2 ð0:30669Þ3 ¼ 0
Solving for Z gives
Z v ¼ 0:8625
For the liquid phase:
Substituting A ¼ 9.700183 and B ¼ 1.020078 in Eq. (5.115) gives
Z + ðB 1ÞZ 2 + A 3B2 2B Z AB B2 B3 ¼ 0
h
i
Z3 ¼ ð1:020078 1ÞZ2 + ð9:700183Þ 3ð1:020078Þ2 2ð1:020078Þ Z
h
i
ð9:700183Þð1:020078Þ ð1:020078Þ2 ð1:020078Þ3 ¼ 0
3
538 CHAPTER 5 Equations of State and Phase Equilibria
Solving for Z gives
ZL ¼ 1:2645
Step 5. Calculate the density of both phases.
Density of the vapor phase:
pMa
Zv RT
ð4000Þð20:89Þ
ρv ¼
¼ 14:566lb=ft3
ð10:73Þð620Þð0:8625Þ
ρv ¼
Density of the liquid phase:
pMa
Zv RT
ð4000Þð100:25Þ
¼ 47:67lb=ft3
ρL ¼
ð10:73Þð620Þð1:2645Þ
ρL ¼
Applying the thermodynamic relationship, as given by Eq. (5.88), to
Eq. (5.111) yields the following expression for the fugacity of a pure
component:
pffiffiffi #
"
Z+ 1+ 2 B
f
A
pffiffiffi
ln
¼ lnðΦÞ ¼ Z 1 lnðZ BÞ pffiffiffiffiffiffi ln
p
Z 1 2 B
2 2B
(5.116)
The fugacity coefficient of component i in a hydrocarbon liquid mixture is
calculated from the following expression:
L
bi ðZL 1Þ
f
ln
ln ZL B
¼ ln ΦLi ¼
xi p
bm
" L pffiffiffi #
Z + 1 + 2B
A
2Ψ i
bi
pffiffiffi ln
pffiffiffi
ð
aα
Þ
b
Z L + 1 2B
2 2B
m
m
(5.117)
where the mixture parameters bm, B, A, Ψ i, and (aα)m are defined
previously.
Eq. (5.117) also is used to determine the fugacity coefficient of any component in the gas phase by replacing the composition of the liquid phase, xi,
with the composition of the gas phase, yi, in calculating the compositiondependent terms of the equation:
ln
v
bi ðZ v 1Þ
f
A
2Ψ i
bi
lnðZ v BÞ pffiffiffi
¼ ln ΦLi ¼
yi p
bm
2 2B ðaαÞm bm
"
pffiffiffi #
v
Z + 1 + 2B
pffiffiffi ln
Z v + 1 2B
Modifications of the SRK EOS 539
The table below shows the set of binary interaction coefficient, kij, traditionally used when predicting the volumetric behavior of hydrocarbon mixture
with the Peng-Robinson equation of state.
Component CO2 N2 H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
CO2
N2
H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
0.105
0.025
0.070
0
0.130
0.010
0.085
0.005
0
0.125
0.090
0.080
0.010
0.005
0
0.120
0.095
0.075
0.035
0.005
0.000
0
0.115
0.095
0.075
0.025
0.010
0.000
0.005
0
0.115
0.100
0.070
0.050
0.020
0.015
0.005
0.005
0
0.115
0.100
0.070
0.030
0.020
0.015
0.005
0.005
0.000
0
0.115
0.110
0.070
0.030
0.020
0.010
0.005
0.005
0.000
0.000
0
0.115
0.115
0.060
0.035
0.020
0.005
0.005
0.005
0.000
0.000
0.000
0
0.115
0.120
0.060
0.040
0.020
0.005
0.005
0.005
0.000
0.000
0.000
0.000
0
0.115
0.120
0.060
0.040
0.020
0.005
0.005
0.005
0.000
0.000
0.000
0.000
0.000
0
0.115
0.125
0.055
0.045
0.020
0.005
0.005
0.005
0.000
0.000
0.000
0.000
0.000
0.00
0
0
0
0
0.135
0.130
0
Note that kij ¼ kij.
To improve the predictive capability of the PR EOS when applying for mixtures containing nonhydrocarbon components (eg, N2, CO2, and CH4),
Nikos et al. (1986) proposed a generalized correlation for generating the
binary interaction coefficient, kij. The authors correlated these coefficients
with system pressure, temperature, and the acentric factor. These generalized correlations originated with all the binary experimental data available
in the literature. The authors proposed the following generalized form for kij:
kij ¼ λ2 Trj2 + λ1 Trj + λ0
(5.118)
where i refers to the principal component, N2, CO2, or CH4, and j refers to
the other hydrocarbon component of the binary. The acentric-factordependent coefficients, λ0, λ1, and λ2 are determined for each set of binaries
by applying the following expressions:
For nitrogen-hydrocarbons,
2
λ0 ¼ 0:1751787 0:7043 log ωj 0:862066 log ωj
(5.119)
2
λ1 ¼ 0:584474 1:328 log ωj + 2:035767 log ωj
(5.120)
540 CHAPTER 5 Equations of State and Phase Equilibria
2
λ2 ¼ 2:257079 + 7:869765 log ωj + 13:50466 log ωj + 8:3864½ log ðωÞ3
(5.121)
Nikos et al. also suggested the following pressure correction:
kij0 ¼ kij 1:04 4:2 105 p
(5.122)
where p is the pressure in psi.
For methane-hydrocarbons,
2
λ0 ¼ 0:01664 0:37283 log ωj + 1:31757 log ωj
(5.123)
λ1 ¼ 0:48147 + 3:35342 log ðωi Þ 1:0783½ log ðωi Þ2
(5.124)
λ2 ¼ 0:4114 3:5072 log ðωi Þ 0:78798½ log ðωi Þ2
(5.125)
For CO2-hydrocarbons,
λ0 ¼ 0:4025636 + 0:1748927 log ωj
λ1 ¼ 0:94812 0:6009864 log ωj
(5.127)
λ2 ¼ 0:741843368 + 0:441775 log ðωi Þ
(5.128)
(5.126)
For the CO2 interaction parameters, the following pressure correction is
suggested:
kij0 ¼ kij 1:044269 4:375 105 p
(5.129)
Stryjek and Vera (1986) proposed an improvement in the reproduction of
vapor pressures of a pure component by the PR EOS in the reduced temperature range from 0.7 to 1.0, by replacing the m term in Eq. (5.113) with the
following expression:
mo ¼ 0:378893 + 1:4897153 0:17131848ω2 + 0:0196554ω3
(5.130)
To reproduce vapor pressures at reduced temperatures below 0.7, Stryjek
and Vera further modified the m parameter in the PR equation by introducing an adjustable parameter, m1, characteristic of each compound to
Eq. (5.113). They proposed the following generalized relationship for the
parameter m:
pffiffiffiffiffi
m ¼ mo + m1 1 + Tr ð0:7 Tr Þ
where
Tr ¼ reduced temperature of the pure component
m0 ¼ defined by Eq. (5.130)
m1 ¼ adjustable parameter
(5.131)
Modifications of the SRK EOS 541
For all components with a reduced temperature above 0.7, Stryjek and Vera
recommended setting m1 ¼ 0. For components with a reduced temperature
greater than 0.7, the optimum values of m1 for compounds of industrial interest are tabulated below,
Component
m1
Nitrogen
Carbon dioxide
Water
Methane
Ethane
Propane
Butane
Pentane
Hexane
Heptane
Octane
Nonane
Decane
Undecane
Dodecane
Tridecane
Tetradecane
Pentadecane
Hexadecane
0.01996
0.04285
0.0664
0.0016
0.02669
0.03136
0.03443
0.03946
0.05104
0.04648
0.04464
0.04104
0.0451
0.02919
0.05426
0.04157
0.02686
0.01892
0.02665
Due to the totally empirical nature of the parameter m1, Stryjek and Vera could
not find a generalized correlation for m1 in terms of pure component parameters. They pointed out that these values of m1 should be used without changes.
Jhaveri and Youngren (1984) pointed out that, when applying the PengRobinson equation of state to reservoir fluids, the error associated with
the equation in the prediction of gas-phase Z-factors ranged from 3% to
5% and the error in the liquid density predictions ranged from 6% to
12%. Following the procedure proposed by Peneloux and coworkers (see
the SRK EOS), Jhaveri and Youngren introduced the volume correction
parameter, ci, to the PR EOS. This third parameter has the same units as
the second parameter, bi, of the unmodified PR equation and is defined
by the following relationship:
ci ¼ Si bi
(5.132)
where Si is a dimensionless parameter, called the shift parameter, and bi is
the Peng-Robinson covolume, as given by Eq. (5.112).
542 CHAPTER 5 Equations of State and Phase Equilibria
As in the SRK EOS, the volume correction parameter, ci, does not change
the vapor/liquid equilibrium conditions, equilibrium ratio Ki. The corrected
hydrocarbon phase volumes are given by the following expressions:
L
Vcorr
¼ VL X
ðxi ci Þ
v
¼ Vv Vcorr
X
ðy i c i Þ
i¼1
i¼1
where
VL, Vv ¼ volumes of the liquid phase and gas phase as calculated by the
unmodified PR EOS, ft3/mol
VLcorr, Vvcorr ¼ corrected volumes of the liquid and gas phase, ft3/mol
Whitson and Brule (2000) point out that the volume translation (correction)
concept can be applied to any two-constant cubic equation, thereby eliminating the volumetric deficiency associated with the application of EOS.
Whitson and Brule extended the work of Jhaveri and Youngren and tabulated
the shift parameters, Si, for a selected number of pure components. These tabulated values, which follow, are used in Eq. (5.132) to calculate the volume
correction parameter, ci, for the Peng-Robinson and SRK equations of state.
Component
PR EOS
N2
CO2
H2S
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
n-C6
n-C7
n-C8
n-C9
n-C10
0.1927
0.0817
0.1288
0.1595
0.1134
0.0863
0.0844
0.0675
0.0608
0.0390
0.0080
0.0033
0.0314
0.0408
0.0655
SRK EOS
0.0079
0.0833
0.0466
0.0234
0.0605
0.0825
0.0830
0.0975
0.1022
0.1209
0.1467
0.1554
0.1794
0.1868
0.2080
Jhaveri and Youngren proposed the following expression for calculating the
shift parameter for the C7+:
SC7 + ¼ 1 d
ð M Þe
Modifications of the SRK EOS 543
where
M ¼ molecular weight of the heptanes-plus fraction
d, e ¼ positive correlation coefficients
The authors proposed that, in the absence of the experimental information
needed for calculating e and d, the power coefficient e could be set equal to
0.2051 and the coefficient d adjusted to match the C7+ density with the
values of d ranging from 2.2 to 3.2. In general, the following values may
be used for C7+ fractions, by hydrocarbon family:
Hydrocarbon family
d
e
Paraffins
Naphthenes
Aromatics
2.258
3.044
2.516
0.1823
0.2324
0.2008
To use the Peng-Robinson equation of state to predict the phase and volumetric behavior of mixtures, one must be able to provide the critical pressure, the critical temperature, and the acentric factor for each component in
the mixture. For pure compounds, the required properties are well defined
and known. Nearly all naturally occurring petroleum fluids contain a quantity of heavy fractions that are not well defined. These heavy fractions often
are lumped together as a heptanes-plus fraction. The problem of how to adequately characterize the C7+ fractions in terms of their critical properties and
acentric factors has been long recognized in the petroleum industry. Changing the characterization of C7+ fractions present in even small amounts can
have a profound effect on the PVT properties and the phase equilibria of a
hydrocarbon system as predicted by the Peng-Robinson equation of state.
The usual approach for such situations is to “tune” the parameters in the
EOS in an attempt to improve the accuracy of prediction. During the tuning
process, the critical properties of the heptanes-plus fraction and the binary
interaction coefficients are adjusted to obtain a reasonable match with
experimental data available on the hydrocarbon mixture.
Recognizing that the inadequacy of the predictive capability of the PR EOS
lies with the improper procedure for calculating the parameters a, b, and α of
the equation for the C7+ fraction, Ahmed (1991) devised an approach for
determining these parameters from the following two readily measured physical properties of C7+: the molecular weight, M7+, and the specific gravity, γ 7+.
The approach is based on generating 49 density values for the C7+ by applying the Riazi and Daubert correlation. These values were subsequently subjected to 10 temperature and 10 pressure values in the range of 60–300°F
and 14.7–7000 psia, respectively. The Peng-Robinson EOS was then
544 CHAPTER 5 Equations of State and Phase Equilibria
applied to match the 4900 generated density values by optimizing the
parameters a, b, and α using a nonlinear regression model. The optimized
parameters for the heptanes-plus fraction are given by the following
expressions.
For the parameter α of C7+,
"
rffiffiffiffiffiffiffiffi!#2
520
α¼ 1+m 1
T
(5.133)
with m as defined by
m¼
D
A4
A7
+ A2 M7 + + A3 M72 + +
+ A5 γ 7 + + A6 γ 27 + +
M7 +
γ7 +
A0 + A1 D
(5.134)
with parameter D as defined by the ratio of the molecular weight to the specific gravity of the heptanes-plus fraction:
D¼
M7 +
γ7 +
where
M7+ ¼ molecular weight of C7+
γ 7+ ¼ specific gravity of C7+
A0-A7 ¼ coefficients as given in the table below
Coefficient a
A0
A1
A2
A3
A4
A5
A6
A7
b
m
2.433525 10
6.8453198
36.91776
8.3201587 103
1.730243 102
5.2393763 102
2
6
0.18444102 10 6.2055064 10
1.7316235 102
2
9
3.6003101 10
9.0910383 10
1.3743308 105
7
3.4992796 10
13.378898
12.718844
2.838756 107
7.9492922
10.246122
1.1325365 107
3.1779077
7.6697942
6.418828 106
1.7190311
2.6078099
7
For parameters a and b of C7+, the following generalized correlation is
proposed:
"
a or b ¼
3 X
i¼0
Ai D
i
#
"
#
6 X
A4
A7
i4
+
+
Ai γ 7 + +
D
γ
7+
i¼5
(5.135)
The coefficients A0–A7 are included in the table above. To further improve
the predictive capability of the Peng-Robinson EOS, Ahmed (1991)
Modifications of the SRK EOS 545
optimized the coefficients a, b, and m for nitrogen, CO2, and methane by
matching 100 Z-factor values for each of these components. Using a
nonlinear regression model, the optimized values in the table below are
recommended.
Component
a
CO2
N2
C1
1.499914 10
4.5693589 103
7.709708 103
4
b
m (Eq. (5.133))
0.41503575
0.4682582
0.46749727
0.73605717
0.97962859
0.549765
To provide the modified PR EOS with a consistent procedure for determining the binary interaction coefficient, kij, the following computational steps
are proposed.
Step 1. Calculate the binary interaction coefficient between methane and
the heptanes-plus fraction from
kC1 C7 + ¼ 0:00189T 1:167059
where the temperature, T, is in °R.
Step 2. Set
kCO2 -N2 ¼ 0:12
KCO2 -hydrocarbon ¼ 0:10
kN2 -hydrocarbon ¼ 0:10
Step 3. Adopting the procedure recommended by Pedersen, Thomassen,
and Fredenslund (1989), calculate the binary interaction
coefficients between components heavier than methane (ie, C2, C3,
etc.) and the heptanes-plus fraction from
kCn C7 + ¼ 0:8kCðn1Þ C7 +
where n is the number of carbon atoms of component Cn; for
example,
Binary interaction coefficient between C2 and C7 + : kC2 C7 + ¼ 0:8kC1 C7 +
Binary interaction coefficient between C3 and C7 + : kC3 C7 + ¼ 0:8kC2 C7 +
Step 4. Determine the remaining kij from
" kij kiC7 +
Mj
5
ðM i Þ5
ðMC7 + Þ5 ðMi Þ5
#
546 CHAPTER 5 Equations of State and Phase Equilibria
where M is the molecular weight of any specified component. For example,
the binary interaction coefficient between propane, C3, and butane, C4, is
"
kC3 C4 ¼ kC3 C7 +
ðM C 4 Þ5 ðM C 3 Þ5
#
ðMC7 + Þ5 ðMC3 Þ5
Summary of Equations of State
A summary of the different equations of state discussed in this chapter are
listed in the table below in the form of
p ¼ prepulsion pattraction
EOS
prepulsion
pattraction
a
b
Ideal
RT
V
RT
V b
RT
V b
RT
V b
RT
V b
0
0
0
a
V2
R2 Tc2
pc
R2 Tc2:5
Ωa
pc
R2 Tc2
Ωa
pc
R2 Tc2
Ωa
pc
Ωb
vdW
RK
SRK
PR
a
pffiffiffi
V ð V + bÞ T
aαðT Þ
V ð V + bÞ
aαðT Þ
V ðV + bÞ + bðV bÞ
Ωa
RTc
pc
RTc
Ωb
pc
RTc
Ωb
pc
RTc
Ωb
pc
EQUATION-OF-STATE APPLICATIONS
Most petroleum engineering applications rely on the use of the equation of
state due to its simplicity, consistency, and accuracy when properly tuned.
Cubic equations of state have found widespread acceptance as tools that permit the convenient and flexible calculation of the complex phase behavior of
reservoir fluids. Some of these applications are presented in this section,
including:
n
n
n
n
n
n
Equilibrium ratio “K-value”
Dew point pressure “pd”
Bubble point pressure “pb”
Three-phase flash calculations
Vapor pressure “pv”
Simulating PVT laboratory experiments.
Equilibrium Ratio
A flow diagram is presented in Fig. 5.9 to illustrate the procedure of determining equilibrium ratios of a hydrocarbon mixture. For this type of calculation, the system temperature, T, the system pressure, p, and the overall
Equation-of-State Applications 547
Given:
zi, P, T
Assume KAi
KA
i = Ki
yi, nV
ZV, FVi
Xi, nL
Perform flash calculations
Z L, F Li
Calculate Ki
e > 0.0001
No, set
Tci
pci
)]
Ki = p exp[5.37(1 + wi)(1−
T
n
KA
i = Ki
Test for
convergence
e ≤ 0.0001
Ki
−1 ≤ e
KA
i
where e ≈ 0.0001
i=1
Yes
Solution gives:
Ki, xi, yi, nL, nV, Z L, Z V
n FIGURE 5.9 Flow diagram of equilibrium ratio determination by EOS.
composition of the mixture, zi, must be known. The procedure is summarized in the following steps in conjunction with Fig. 5.9.
Step 1. Assume an initial value of the equilibrium ratio for each component
in the mixture at the specified system pressure and temperature.
Wilson’s equation can provide the starting values of Ki:
KiA ¼
pci
exp½5:37ð1 + ωi Þð1 Tci =T Þ
p
where KA
i is an assumed equilibrium ratio for component i.
Step 2. Using the overall composition and the assumed K-values, perform
flash calculations to determine xi, yi, nL, and nυ.
Step 3. Using the calculated composition of the liquid phase, xi, determine
the fugacity coefficient ΦLi for each component in the liquid phase
by applying Eq. (5.117):
ln ΦLi
" L pffiffiffi #
L
Z + 1+ 2 B
bi ðZL 1Þ
A
2Ψ i
bi
pffiffiffi
ln
ln Z B pffiffiffi
¼
bm
ZL 1 2 B
2 2B ðaαÞm bm
548 CHAPTER 5 Equations of State and Phase Equilibria
Step 4. Repeat step 3 using the calculated composition of the gas phase, yi,
to determine Φvi .
" v pffiffiffi #
L bi ðZv 1Þ
Z + 1+ 2 B
A
2Ψ i
bi
v
pffiffiffi
ln
ln Φi ¼
lnðZ BÞ pffiffiffi
bm
2 2B ðaαÞm bm
Zv 1 2 B
Step 5. Calculate the new set of equilibrium ratios from
Ki ¼
ΦLi
Φvi
Step 6. Check for the solution by applying the following constraint:
n
X
2
Ki =KiA 1 ε
i¼1
where
ε ¼ preset error tolerance, such as 0.0001
n ¼ number of components in the system.
If these conditions are satisfied, then the solution has been reached. If not,
steps 1–6 are repeated using the calculated equilibrium ratios as initial
values.
Dew Point Pressure
A saturated vapor exists for a given temperature at the pressure at which an
infinitesimal amount of liquid first appears. This pressure is referred to as
the dew point pressure, pd. The dew point pressure of a mixture is described
mathematically by the following two conditions:
yi ¼ zi , for 1 i n and nv ¼ 1
n X
zi
¼1
K
i
i1
(5.136)
(5.137)
Applying the definition of Ki in terms of the fugacity coefficient to
Eq. (5.137) gives
n X
zi
i¼1
ki
"
n
X
¼
i¼1
#
X zi f v zi
i
¼1
¼
ΦLi zi pd
ΦLi =Φvi
or
pd ¼
n v
X
f
i
i¼1
ΦLi
Equation-of-State Applications 549
This above equation is arranged to give
f ðpd Þ ¼
n v
X
f
i
i¼1
ΦLi
pd ¼ 0
(5.138)
where
pd ¼ dew point pressure, psia
f vi ¼ fugacity of component i in the vapor phase, psia
ΦLi ¼ fugacity coefficient of component i in the liquid phase
Eq. (5.35) can be solved for the dew point pressure by using the NewtonRaphson iterative method. To use the iterative method, the derivative of
Eq. (5.138) with respect to the dew point pressure, pd, is required. This
derivative is given by the following expression:
" #
n
ΦLi @fiv =@pd fiv @ΦLi =@pd
@f X
1
¼
L 2
@pd i¼1
Φ
(5.139)
i
The two derivatives in this equation can be approximated numerically as
follows:
v
@f v
f ðpd + Δpd Þ fiv ðpd Δpd Þ
¼ i
@pd
2Δpd
(5.140)
L
@ΦLi
Φ ðpd + Δpd Þ ΦLi ðpd Δpd Þ
¼ i
@pd
2Δpd
(5.141)
and
where
Δpd ¼ pressure increment (eg, 5 psia)
fiv ðpd + Δpd Þ ¼ fugacity of component i at (pd + Δpd)
f vi (pd–Δpd) ¼ fugacity of component i at (pd Δpd)
ΦLi ðpd + Δpd Þ ¼ fugacity coefficient of component i at (pd + Δpd)
ΦLi (pd–Δpd) ¼ fugacity coefficient of component i at (pd Δpd)
ΦLi ¼ fugacity coefficient of component i at pd
The computational procedure of determining pd is summarized in the following steps.
Step 1. Assume an initial value for the dew point pressure, pA
d.
Step 2. Using the assumed value of pA
d , calculate a set of equilibrium ratios
for the mixture by using any of the previous correlations, such as
Wilson’s correlation.
550 CHAPTER 5 Equations of State and Phase Equilibria
Step 3. Calculate the composition of the liquid phase, that is, the
composition of the droplets of liquid, by applying the mathematical
definition of Ki, to give
xi ¼
zi
Ki
It should be noted that yi ¼ zi.
L
Step 4. Calculate pA
d using the composition of the gas phase, zi and Φi , using
the composition of liquid phase, xi, at the following three pressures:
n
n
n
pA
d
pA
d + Δpd
pA
d –Δpd
where pA
d is the assumed dew point pressure and Δpd is a selected
pressure increment of 5–10 psi.
Step 5. Evaluate the function f(pd) [ie, Eq. (5.138)] and its derivative using
Eqs. (5.139)–(5.141).
Step 6. Using the values of the function f(pd) and the derivative @f/@pd as
determined in step 5, calculate a new dew point pressure by
applying the Newton-Raphson formula:
pd ¼ pA
d f ðpd Þ=½@f =@pd (5.142)
Step 7. The calculated value of pd is checked numerically against the
assumed value by applying the following condition:
jpd pA
dj5
If this condition is met, then the correct dew point pressure, pd, has been
found. Otherwise, steps 3–6 are repeated using the calculated pd as the
new value for the next iteration. A set of equilibrium ratios must be calculated at the new assumed dew point pressure from
ki ¼
ΦLi
Φvi
Bubble Point Pressure
The bubble point pressure, pb, is defined as the pressure at which the first
bubble of gas is formed. Accordingly, the bubble point pressure is defined
mathematically by the following equations:
yi ¼ zi and nL ¼ 1:0
(5.143)
n
X
½zi Ki ¼ 1
(5.144)
i¼1
Equation-of-State Applications 551
Introducing the concept of the fugacity coefficient into Eq. (5.144) gives
2 L 3
fi
n
n 6
X
X
7
ΦLi
z
6zi i pb 7 ¼ 1
zi v ¼
v 5
4
Φi
Φi
i¼1
i¼1
Rearranging,
pb ¼
n L
X
f
i
Φvi
i¼1
or
f ðpb Þ ¼
n L
X
f
i
i¼1
p
b ¼0
v
Φi
(5.145)
The iteration sequence for calculation of pb from the preceding function is
similar to that of the de-point pressure, which requires differentiating that
function with respect to the bubble point pressure:
" #
n
Φvi @fiL =@pb fiL @Φvi =@pb
@f X
1
¼
v 2
@pb i¼1
Φ
(5.146)
i
The phase envelope can be constructed by either calculating the saturation
pressures, pb and pd, as a function of temperature or calculating the bubble
point and dew point temperatures at selected pressures. The selection to calculate a pressure or temperature depends on how steep or flat the phase
envelope is at a given point. In terms of the slope of the envelope at a point,
the criteria are
n
Calculate the bubble point or dew point pressure if
@ lnðpÞ lnðp2 Þ lnðp1 Þ @ lnðT Þ lnðT Þ lnðT Þ < 20
2
1
n
Calculate the bubble point or dew point temperature if
@ lnðpÞ lnðp2 Þ lnðp1 Þ @ lnðT Þ lnðT Þ lnðT Þ > 20
2
1
Three-Phase Equilibrium Calculations
Two- and three-phase equilibria occur frequently during the processing of
hydrocarbon and related systems. Peng and Robinson (1976b) proposed a
three-phase equilibrium calculation scheme of systems that exhibit a
water-rich liquid phase, a hydrocarbon-rich liquid phase, and a vapor phase.
552 CHAPTER 5 Equations of State and Phase Equilibria
In applying the principle of mass conversation to 1 mol of a waterhydrocarbon in a three-phase state of thermodynamic equilibrium at a fixed
temperature, T, and pressure, p, gives
nL + nw + nv ¼ 1
(5.147)
nL xi + nw xwi + nv yi ¼ zi
(5.148)
n
n
n
n
X
X
X
X
xi ¼
xwi ¼
yi ¼
zi ¼ 1
(5.149)
i
i¼1
i¼1
i¼1
where
nL, nw, nυ ¼ number of moles of the hydrocarbon-rich liquid, the waterrich liquid, and the vapor, respectively
xi, xwi, yi ¼ mole fraction of component i in the hydrocarbon-rich liquid,
the water rich liquid, and the vapor, respectively
The equilibrium relations between the compositions of each phase are
defined by the following expressions:
Ki ¼
yi ΦLi
¼
xi Φvi
(5.150)
Kwi ¼
yi Φw
¼ i
xwi Φvi
(5.151)
and
where
Ki ¼ equilibrium ratio of component i between vapor and hydrocarbonrich liquid
Kwi ¼ equilibrium ratio of component i between the vapor and waterrich liquid
ΦLi ¼ fugacity coefficient of component i in the hydrocarbon-rich liquid
Φvi ¼ fugacity coefficient of component i in the vapor phase
Φw
i ¼ fugacity coefficient of component i in the water-rich liquid
Combining Eqs. (5.147)–(5.151) gives the following conventional nonlinear
equations:
3
2
X
X6
6
xi ¼
4
i¼1
i¼1
7
7¼1
5
Ki
Ki + Ki
Kwi
zi
nL ð1 Ki Þ + nw
(5.152)
Equation-of-State Applications 553
3
2
X
X6
7
zi ðKi =Kwi Þ
7¼1
6
xwi ¼
5
4
K
i
i¼1
i¼1 n ð1 K Þ + n
Ki + Ki
L
i
w
Kwi
3
2
X
yi ¼
i¼1
X6
6
4
i¼1
7
7¼1
5
Ki
Ki + Ki
Kwi
zi Ki
nL ð1 Ki Þ + nw
(5.153)
(5.154)
Assuming that the equilibrium ratios between phases can be calculated,
these equations are combined to solve for the two unknowns, nL and nυ,
and hence xi, xwi, and yi. The nature of the specific equilibrium calculation
is what determines the appropriate combination of Eqs. (5.152)–(5.154).
The combination of these three expressions then can be used to determine
the phase and volumetric properties of the three-phase system.
There are essentially three types of phase behavior calculations for the threephase system: bubble point prediction, dew point prediction, and flash calculation. Peng and Robinson (1980) proposed the following combination
schemes of Eqs. (5.152)–(5.154).
For the bubble point pressure determination,
X
X
xi xwi ¼ 0
i
i
and
"
X
#
yi 1 ¼ 0
i
Substituting Eqs. (5.152)–(5.154) in this relationships gives
f ðnL , nw Þ ¼
X
zi ð1 Ki =Kwi Þ
¼0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.155)
zi K i
1¼0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.156)
i
and
gðnL , nw Þ ¼
X
i
For the dew point pressure,
X
X
xwi yi ¼ 0
"i
# i
X
xi 1 ¼ 0
i
554 CHAPTER 5 Equations of State and Phase Equilibria
Combining with Eqs. (5.152)–(5.154) yields
f ðnL , nw Þ ¼
X
zi Ki ð1=Kwi 1Þ
¼0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.157)
zi
1¼0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.158)
i
and
gðnL , nw Þ ¼
X
i
For flash calculations,
X
X
xi yi ¼ 0
i
i
"
#
X
xwi 1 ¼ 0
i
or
f ðnL , nw Þ ¼
X
zi ð1 Ki Þ
¼0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.159)
zi Ki =Kwi
1:0 ¼ 0
nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki
(5.160)
i
and
gðnL , nw Þ ¼
X
i
Note that, in performing any of these property predictions, we always have two
unknown variables, nL and nw, and between them, two equations. Providing
that the equilibrium ratios and the overall composition are known, the equations
can be solved simultaneously using the appropriate iterative technique, such as
the Newton-Raphson method. The application of this iterative technique for
solving Eqs. (5.159), (5.160) is summarized in the following steps.
Step 1. Assume initial values for the unknown variables nL and nw.
Step 2. Calculate new values of nL and nw by solving the following two
linear equations:
nL
nw
new
1 f ðnL , nw Þ
@f =@nL @f =@nw
nL
¼
gðnL , nw Þ
nw
@g=@nL @g=@nw
where f(nL, nw) ¼ value of the function f(nL, nw) as expressed by
Eq. (5.159) and g(nL, nw) ¼ value of the function g(nL, nw) as
expressed by Eq. (5.160).
The first derivative of these functions with respect to nL and nw is
given by the following expressions:
Equation-of-State Applications 555
ð@f =@nL Þ ¼
(
X
zi ð1 Ki Þ2
)
½nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki 2
(
)
X
zi ð1 Ki ÞðKi =Kwi Ki Þ
i¼1
ð@f =@nw Þ ¼
i¼1
ð@g=@nL Þ ¼
(
X
½nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki 2
zi ðKi =Kwi Þð1 Ki Þ
)
½nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki 2
(
)
X
zi ðKi Kwi ÞðKi =Kwi Ki Þ
i¼1
ð@g=@nw Þ ¼
½nL ð1 Ki Þ + nw ðKi =Kwi Ki Þ + Ki 2
i¼1
Step 3. The new calculated values of nL and nw then are compared with the
initial values. If no changes in the values are observed, then the
correct values of nL and nw have been obtained. Otherwise, the steps
are repeated with the new calculated values used as initial values.
Peng and Robinson (1980) proposed two modifications when using their
equation of state for three-phase equilibrium calculations:
n
The first modification concerns the use of the parameter α as expressed
by Eq. (5.113) for the water compound. Peng and Robinson suggested
that, when the reduced temperature of this compound is less than 0.85,
the following equation should be applied:
0:5 2
αw ¼ 1:0085677 + 0:82154 1 Trw
(5.161)
where
n
Trw is the reduced temperature of the water component; that is, Trw ¼ T/
Tcw.
The second important modification of the PR EOS is the introduction of
alternative mixing rules and new interaction coefficients for the
parameter (aα). A temperature-dependent binary interaction coefficient
was introduced into the equation, to give
ðaαÞwater ¼
XXh
i
0:5 i
xwi xwj ai aj αi αj
1 τij
(5.162)
j
where τij ¼ temperature-dependent binary interaction coefficient and
xwi ¼ mole fraction of component i in the water phase.
Peng and Robinson proposed graphical correlations for determining this
parameter for each aqueous binary pair. Lim and Ahmed (1984) expressed
these graphical correlations mathematically by the following generalized
equation:
556 CHAPTER 5 Equations of State and Phase Equilibria
τij ¼ a1
T 2 pci 2
T pci
+ a3
+ a2
pcj
Tci
Tci pcj
(5.163)
where
T ¼ system temperature, °R
Tci ¼ critical temperature of the component of interest, °R
pci ¼ critical pressure of the component of interest, psia
pcj ¼ critical pressure of the water compound, psia
Values of the coefficients a1, a2, and a3 of this polynomial are given in the
table below for selected binaries.
Component i
a1
a2
a3
C1
C2
C3
n-C4
n-C6
0
0
18.032
0
49.472
1.659
2.109
9.441
2.800
5.783
0.761
0.607
1.208
0.488
0.152
For selected nonhydrocarbon components, values of interaction parameters
are given by the following expressions:
For N2-H2O binary,
τij ¼ 0:402ðT=Tci Þ 1:586
(5.164)
where
τij ¼ binary parameter between nitrogen and the water compound
Tci ¼ critical temperature of nitrogen, °R.
For CO2-H2O binary,
2
T
T
0:503
τij ¼ 0:074
+ 0:478
Tci
Tci
(5.165)
where Tci is the critical temperature of CO2.
In the course of making phase equilibrium calculations, it is always desirable to provide initial values for the equilibrium ratios so the iterative procedure can proceed as reliably and rapidly as possible. Peng and Robinson
adopted Wilson’s equilibrium ratio correlation to provide initial K-values
for the hydrocarbon/vapor phase:
Ki ¼
pci
Tci
exp 5:3727ð1 + ωi Þ 1 p
T
Equation-of-State Applications 557
While for the water-vapor phase, Peng and Robinson proposed the following
expression for Kwi:
Kwi ¼ 106 ½pci T=ðTci pÞ
Vapor Pressure
The calculation of the vapor pressure of a pure component through an equation of state usually is made by the same trial-and-error algorithms used to
calculate vapor-liquid equilibria of mixtures. Soave (1972) suggests that the
van der Waals (vdW), Soave-Redlich-Kwong (SRK), and the PengRobinson (PR) equations of state can be written in the following generalized
form:
p¼
RT
aα
V b V 2 + μVb + wb2
(5.166)
with
R2 Tc2
pc
RTc
b ¼ Ωb
pc
a ¼ Ωa
where the values of u, w, Ωa, and Ωb for three different equations of state are
given in the table below.
EOS
u
w
Ωa
Ωb
vdW
SRK
PR
0
1
2
0
0
1
0.421875
0.42748
0.45724
0.125
0.08664
0.07780
Soave introduced the reduced pressure, pr, and reduced temperature, Tr, to
these equations, to give
aαp
αpr
¼ Ωa
Tr
R2 T 2
(5.167)
bp
pr
¼ Ωb
Tr
RT
(5.168)
A Ωa α
¼
B Ωb Tr
(5.169)
A¼
B¼
and
where
pr ¼ p=pc
Tr ¼ T=Tc
558 CHAPTER 5 Equations of State and Phase Equilibria
In the cubic form and in terms of the Z-factor, the three equations of state can
be written as
vdW : Z3 Z2 ð1 +BÞ + ZA AB
¼0
SRK : Z 3 Z2 + Z A B B2 AB ¼ 0
PR : Z 3 Z2 ð1 BÞ + Z A 3B2 2B AB B2 B2 ¼ 0
And the pure component fugacity coefficient is given by
o
f
A
vdW : ln
¼ Z 1 lnðZ BÞ p
Z
o
f
A
B
¼ Z 1 lnðZ BÞ SRK : ln
ln 1 +
p
B
Z
pffiffiffi #
o
"
Z+ 1+ 2 B
f
A
pffiffiffi
¼ Z 1 lnðZ BÞ pffiffiffi ln
PR : ln
p
Z 1 2 B
2 2B
A typical iterative procedure for the calculation of pure component vapor
pressure at any temperature, T, through one of these EOS is summarized
next.
Step 1. Calculate the reduced temperature; that is, Tr ¼ T/Tc.
Step 2. Calculate the ratio A/B from Eq. (5.169).
Step 3. Assume a value for B.
Step 4. Solve the selected equation of state, such as SRK EOS, to obtain ZL
and Z (ie, smallest and largest roots) for both phases.
Step 5. Substitute ZL and Zv into the pure component fugacity coefficient
and obtain ln(fo/p) for both phases.
Step 6. Compare the two values of f/p. If the isofugacity condition is not
satisfied, assume a new value of B and repeat steps 3–6.
Step 7. From the final value of B, obtain the vapor pressure from
Eq. (5.168):
B ¼ Ωb
ðpv =pc Þ
Tr
pv ¼
BTr pc
Ωb
Solving for pv gives
SIMULATING OF LABORATORY PVT DATA BY
EQUATIONS OF STATE
Before attempting to use an EOS-based compositional model in a full-field
simulation study, it is essential the selected equation of state is capable of
achieving a satisfactory match between EOS results and all the available
Simulating of Laboratory PVT Data by Equations of State 559
PVT test data. It should be pointed out that an equation of state generally is
not predictive without tuning its parameters to match the relevant experimental data. Several PVT laboratory tests were presented in Chapter 4
and are repeated here for convenience to illustrate the procedure of applying
an EOS to simulate these experiments, including
n
n
n
n
n
n
n
n
Constant volume depletion.
Constant composition expansion.
Differential liberation.
Flash separation tests.
Composite liberation.
Swelling tests.
Compositional gradients.
Minimum miscibility pressure.
Constant Volume Depletion Test
A reliable prediction of the pressure depletion performance of a gascondensate reservoir is necessary in determining reserves and evaluating
field-separation methods. The predicted performance is also used in planning future operations and studying the economics of projects for increasing
liquid recovery by gas cycling. Such predictions can be performed with aid
of the experimental data collected by conducting constant volume depletion
(CVD) tests on gas condensates. These tests are performed on a reservoir
fluid sample in such a manner as to simulate depletion of the actual reservoir, assuming that retrograde liquid appearing during production remains
immobile in the reservoir.
The CVD test provides five important laboratory measurements that can be
used in a variety of reservoir engineering predictions:
n
n
n
n
n
Dew point pressure.
Composition changes of the gas phase with pressure depletion.
Compressibility factor at reservoir pressure and temperature.
Recovery of original in-place hydrocarbons at any pressure.
Retrograde condensate accumulation; that is, liquid saturation.
The laboratory procedure of the CVD test (with immobile condensate), as
shown schematically in Fig. 5.10, is summarized in the following steps.
Step 1. A measured amount, m, of a representative sample of the original
reservoir fluid with a known overall composition of zi is charged to a
visual PVT cell at the dew point pressure, pd (section A). The
temperature of the PVT cell is maintained at the reservoir
560 CHAPTER 5 Equations of State and Phase Equilibria
Constant volume depletion (CVD)
Gas
Pd
Vi
Original gas
@ Pd
P
P
Gas
Gas
Vt
Vi
Condensate
Condensate
Hg
Hg
Hg
A
B
C
n FIGURE 5.10 Schematic illustration of the CVD test.
temperature, T, throughout the experiment. The initial volume, Vi,
of the saturated fluid is used as a reference volume.
Step 2. The initial gas compressibility factor is calculated from the real gas
equation:
Zdew ¼
pd Vi
ni RT
ni ¼
m
Ma
(5.170)
with
where
pd ¼ dew point pressure, psia
Vi ¼ initial gas volume, ft3
ni ¼ initial number of moles of the gas
Ma ¼ apparent molecular weight, lb/lb-mol
m ¼ mass of the initial gas in place, lb
R ¼ gas constant, 10.73
T ¼ temperature, °R
zd ¼ compressibility factor at dew-point pressure
with the gas initially in place as expressed in standard units, scf,
given by
G ¼ 379:4ni
(5.171)
Simulating of Laboratory PVT Data by Equations of State 561
Step 3. The cell pressure is reduced from the saturation pressure to a
predetermined level, p. This can be achieved by withdrawing
mercury from the cell, as illustrated in Fig. 5.10, section B. During
the process, a second phase (retrograde liquid) is formed. The fluid
in the cell is brought to equilibrium and the total volume, Vt, and
volume of the retrograde liquid, VL, are visually measured.
This retrograde volume is reported as a percent of the initial
volume, Vi:
SL ¼
VL
100
Vi
This value can be translated into condensate saturation in the presence of water saturation by
So ¼ ð1 Sw ÞSL =100
Step 4. Mercury is reinjected into the PVT cell at constant pressure p while
an equivalent volume of gas is removed. When the initial volume,
Vi, is reached, mercury injection ceases, as illustrated in Fig. 5.10,
section C. The volume of the removed gas is measured at the cell
conditions and recorded as (Vgp)p,T. This step simulates a reservoir
producing only gas, with retrograde liquid remaining immobile in
the reservoir.
Step 5. The removed gas is charged to analytical equipment, where its
composition, yi, is determined and its volume is measured at
standard conditions and recorded as (Vgp)sc. The corresponding
moles of gas produced can be calculated from the expression
Vgp sc
np ¼
379:4
(5.172)
where
np ¼ moles of gas produced
(Vgp)sc ¼ volume of gas produced measured at standard
conditions, scf.
Step 6. The compressibility factor of the gas phase at cell pressure and
temperature is calculated by
Vgp p, T
Vactual
Z¼
¼
Videal
Videal
where
Videal ¼
RTnp
p
(5.173)
562 CHAPTER 5 Equations of State and Phase Equilibria
Combining Eqs. (5.172), (5.173) and solving for the compressibility
factor Z gives
379:4p Vgp p, T
Z¼
RT Vgp sc
(5.174)
Another property, the two-phase compressibility factor, also is calculated. The two-phase Z-factor represents the total compressibility
of all the remaining fluid (gas and retrograde liquid) in the cell and is
computed from the real gas law as
Ztwo-phase ¼ pVi
ni np ðRT Þ
(5.175)
where
(ni np) ¼ represents the remaining moles of fluid in the cell
ni ¼ initial moles in the cell
np ¼ cumulative moles of gas removed
Step 7. The volume of gas produced as a percentage of gas initially in place
is calculated by dividing the cumulative volume of the produced gas
by the gas initially in place, both at standard conditions:
"X
%Gp ¼
#
Vgp sc
G
100
(5.176)
or equivalently as
" X
#
np
100
%Gp ¼
ðni Þoriginal
This experimental procedure is repeated several times, until a minimum test
pressure is reached, after which the quantity and composition of the gas and
retrograde liquid remaining in the cell are determined.
This test procedure also can be conducted on a volatile oil sample. In this
case, the PVT cell initially contains liquid, instead of gas, at its bubble point
pressure.
Simulating of the CVD Test by EOS
In the absence of the CVD test data on a specific gas-condensate system,
predictions of pressure-depletion behavior can be obtained by using any
of the well-established equations of state to compute the phase behavior
when the composition of the total gas-condensate system is known. The
stepwise computational procedure using the Peng-Robinson EOS as a
Simulating of Laboratory PVT Data by Equations of State 563
representative equation of state now is summarized in conjunction with the
flow diagram shown in Fig. 5.11.
Step 1. Assume that the original hydrocarbon system with a total
composition zi occupies an initial volume of 1 ft3 at the dew point
pressure pd and system temperature T:
Vi ¼ 1
Step 2. Using the initial gas composition, calculate the gas compressibility
factor Zdew at the dew point pressure from Eq. (5.115):
Z3 + ðB 1ÞZ 2 + A 3B2 2B Z AB B2 B3 ¼ 0
Given:
zi, Pd, T
Calculate Zdew from:
Z3 + (B − 1) Z2 + (A − 3 B2 − 2B)Z − (AB − B2 − B3) = 0
Assume
Vi =1 ft3
Vp
Calculate ni
ni =
ZRT
=
(1) (pd)
Zdew TR
Assume depletion
pressure “P”
Calculate Ki for zi @ P
see Fig. 15-11
VL =
(nL)actual ZL RT
P
Solution; gives:
Ki, xi, yi, nL, nV, ZL, ZV
Calculate volume of
condensate VL
Calculate volume
of gas Vg
Calculate TOTAL volume
Vt = Vg + VL
Determine moles of gas
removed nP & remaining nr
xi (nL)actual + yi nr
zi =
(nL)actual + nr
n FIGURE 5.11 Flow diagram for EOS simulating CVD test.
Combine the remaining phase
and determine new Zi
Vg =
(nV)actual ZV RT
(Vg)P = Vt − 1
np =
P(Vg)P
ZL RT
nr = (nV)actual − np
P
564 CHAPTER 5 Equations of State and Phase Equilibria
Step 3. Calculate the initial moles in place by applying the real gas law:
ni ¼
ð1Þðpd Þ
Zdew RT
(5.177)
where
pd ¼ dew point pressure, psi
Zdew ¼ gas compressibility factor at the dew point pressure.
Step 4. Reduce the pressure to a predetermined value p. At this pressure
level, the equilibrium ratios (K-values) are calculated from EOS.
Results of the calculation at this stage include
n
n
n
n
n
n
n
Equilibrium ratios (K-values).
Composition of the liquid phase (ie, retrograde liquid), xi.
Moles of the liquid phase, nL.
Composition of the gas phase, yi.
Moles of the gas phase, nυ.
Compressibility factor of the liquid phase, ZL.
Compressibility factor of the gas phase, Zv.
The composition of the gas phase, yi, should reasonably match the
experimental gas composition at pressure, p.
Step 5. Because flash calculations, as part of the K-value results, are
usually performed assuming the total moles are equal to 1,
calculate the actual moles of the liquid and gas phases from
ðnL Þactual ¼ ni nL
(5.178)
ðnυ Þactual ¼ ni nυ
(5.179)
where nL and nυ are moles of liquid and gas, respectively, as determined from flash calculations.
Step 6. Calculate the volume of each hydrocarbon phase by applying the
expression
VL ¼
ðnL Þactual ZL RT
p
(5.180)
Vg ¼
ðnv Þactual Zv RT
p
(5.181)
where
VL ¼ volume of the retrograde liquid, ft3/ft3
Vg ¼ volume of the gas phase, ft3/ft3
Simulating of Laboratory PVT Data by Equations of State 565
ZL, Zv ¼ compressibility factors of the liquid and gas phase
T ¼ cell (reservoir) temperature, °R
As Vi ¼ 1, then
SL ¼ ðVL Þ100
This value should match the experimental value if the equation of
state is properly tuned.
Step 7. Calculate the total volume of fluid in the cell:
Vt ¼ VL + Vg
(5.182)
where Vt ¼ total volume of the fluid, ft3.
Step 8. As the volume of the cell is constant at 1 ft3, remove the following
excess gas volume from the cell:
Vgp p, T ¼ Vt 1
(5.183)
Step 9. Calculate the number of moles of gas removed:
np ¼
p Vgp p, T
(5.184)
Zv RT
Step 10. Calculate cumulative gas produced as a percentage of gas initially
P
in place by dividing cumulative moles of gas removed, np, by
the original moles in place:
%Gp ¼
" X #
np
ðni Þoriginal
100
Step 11. Calculate the two-phase gas deviation factor from the
relationship:
Ztwo-phase ¼ ðpÞð1Þ
ni np RT
(5.185)
Step 12. Calculate the remaining moles of gas (nυ)r by subtracting the
moles produced (np) from the actual number of moles of the gas
phase (nυ)actual:
ðnυ Þr ¼ ðnυ Þactual np
(5.186)
566 CHAPTER 5 Equations of State and Phase Equilibria
Step 13. Calculate the new total moles and new composition remaining in
the cell by applying the molal and component balances,
respectively:
ni ¼ ðnL Þactual + ðnυ Þr ;
(5.187)
xi ðnL Þactual + yi ðnυ Þr
ni
(5.188)
zi ¼
Step 14. Consider a new lower pressure and repeat steps 4–13.
Constant Composition Expansion Test
Constant composition expansion experiments, commonly called pressure/
volume tests, are performed on gas condensates or crude oil to simulate
the pressure/volume relations of these hydrocarbon systems. The test is conducted for the purposes of determining
Saturation pressure (bubble point or dew-point pressure).
Isothermal compressibility coefficients of the single-phase fluid in
excess of saturation pressure.
Compressibility factors of the gas phase.
Total hydrocarbon volume as a function of pressure.
n
n
n
n
The experimental procedure, shown schematically in Fig. 5.12, involves
placing a hydrocarbon fluid sample (oil or gas) in a visual PVT cell at
Voil = 100%
P2 > Pb
P1 >> Pb
100%
Oil
Voil ≈ 100%
Voil = 100%
Vt1
Oil
Voil < 100%
P3 = Pb
Vt2
Oil
Voil << 100%
P4 << Pb
P5 <<< Pb
Gas
Gas
Vsat
Oil
Vt4
Oil
Hg
(a)
(b)
n FIGURE 5.12 Constant composition expansion test.
(c)
(d)
(e)
Vt5
Simulating of Laboratory PVT Data by Equations of State 567
reservoir temperature and a pressure in excess of the initial reservoir pressure (Fig. 5.12, section A). The pressure is reduced in steps at constant
temperature by removing mercury from the cell, and the change in the
hydrocarbon volume, V, is measured for each pressure increment. The saturation pressure (bubble point or dew point pressure) and the corresponding volume are observed and recorded (Fig. 5.12, section B). The volume
at the saturation pressure is used as a reference volume. At pressure levels
higher than the saturation pressure, the volume of the hydrocarbon system
is recorded as a ratio of the reference volume. This volume, commonly
termed the relative volume, is expressed mathematically by the following
equation:
Vrel ¼
V
Vsat
(5.189)
where
Vrel ¼ relative volume
V ¼ volume of the hydrocarbon system
Vsat ¼ volume at the saturation pressure
Also, above the saturation pressure, the isothermal compressibility coefficient of the single-phase fluid usually is determined from the expression
c¼
1 @Vrel
Vrel @p T
(5.190)
where
c ¼ isothermal compressibility coefficient, psi1.
For gas-condensate systems, the gas compressibility factor, Z, is determined
in addition to the preceding experimentally determined properties.
Below the saturation pressure, the two-phase volume, Vt, is measured relative to the volume at saturation pressure and expressed as
Relativetotalvolume ¼
Vt
Vsat
(5.191)
where
Vt ¼ total hydrocarbon volume.
Note that no hydrocarbon material is removed from the cell; therefore, the
composition of the total hydrocarbon mixture in the cell remains fixed at
the original composition.
568 CHAPTER 5 Equations of State and Phase Equilibria
Simulating of the CCE Test by EOS
The simulation procedure utilizing the Peng-Robinson equation of state is
illustrated in the following steps.
Step 1. Given the total composition of the hydrocarbon system, zi, and
saturation pressure (pb for oil systems, pd for gas systems), calculate
the total volume occupied by 1 mol of the system. This volume
corresponds to the reference volume Vsat (volume at the saturation
pressure). Mathematically, the volume is calculated from the
relationship
Vsat ¼
ð1ÞZRT
psat
(5.192)
where
Vsat ¼ volume of saturation pressure, ft3/mol
psat ¼ saturation pressure (dew point or bubble point pressure), psia
T ¼ system temperature, °R
Z ¼ compressibility factor, ZL or Zv depending on the type of
system
Step 2. The pressure is increased in steps above the saturation pressure,
where the single phase still exists. At each pressure, the
compressibility factor, ZL or Zv, is calculated by solving Eq. (5.115)
and used to determine the fluid volume.
V¼
ð1ÞZRT
p
where
V ¼ compressed liquid or gas volume at the pressure level p, ft3/mol
Z ¼ compressibility factor of the compressed liquid or gas, ZL or Zv
p ¼ system pressure, p > psat, psia
R ¼ gas constant; 10.73 psi ft3/lb-mol °R
The corresponding relative phase volume, Vrel, is calculated from
the expression
Vrel ¼
V
Vsat
Step 3. The pressure is then reduced in steps below the saturation pressure
psat, where two phases are formed. The equilibrium ratios are
calculated and flash calculations are performed at each pressure
level. Results of the calculation at each pressure level include Ki, xi,
yi, nL, nv, Zv, and ZL. Because no hydrocarbon material is removed
during pressure depletion, the original moles (ni ¼ 1) and
Simulating of Laboratory PVT Data by Equations of State 569
composition, zi, remain constant. The volumes of the liquid and gas
phases can then be calculated from the expressions
VL ¼
ð1ÞðnL ÞZL RT
p
(5.193)
Vg ¼
ð1Þðnυ ÞZ v RT
p
(5.194)
and
Vt ¼ VL + Vg
where
nL, nυ ¼ moles of liquid and gas as calculated from flash
calculations
ZL, Zv ¼ liquid and gas compressibility factors
Vt ¼ total volume of the hydrocarbon system
Step 4. Calculate the relative total volume from the following expression:
Relativetotalvolume ¼
Vt
Vsat
Differential Liberation Test
The test is carried out on reservoir oil samples and involves charging a visual
PVT cell with a liquid sample at the bubble point pressure and reservoir temperature, as described in detail in Chapter 4. The pressure is reduced in steps,
usually 10–15 pressure levels, and all the liberated gas is removed and its
volume, Gp, is measured under standard conditions. The volume of oil
remaining, VL, also is measured at each pressure level. Note that the remaining oil is subjected to continual compositional changes as it becomes progressively richer in the heavier components.
This procedure is continued to atmospheric pressure, where the volume of
the residual (remaining) oil is measured and converted to a volume at 60°F,
Vsc. The differential oil formation volume factors Bod (commonly called the
relative oil volume factors) at all the various pressure levels are calculated
by dividing the recorded oil volumes, VL, by the volume of residual oil, Vsc:
Bod ¼
VL
Vsc
(5.195)
The differential solution gas-oil ratio Rsd is calculated by dividing the volume of gas in the solution by the residual oil volume. Typical table and in
Fig. 5.13.
570 CHAPTER 5 Equations of State and Phase Equilibria
Bod
Rsd
B od
R sd
Pressure
n FIGURE 5.13 Bod and Rsd versus pressure.
Pressure (psig)
Rsda
Bodb
Shrinkage (bbl/bbl)
2620
2350
2100
1850
1600
1350
1100
850
600
350
159
0
0
854
763
684
612
544
479
416
354
292
223
157
0
0
1.6
1.554
1.515
1.479
1.445
1.412
1.382
1.351
1.32
1.283
1.244
1.075
at 60°F ¼ 1.000
1.0000
0.9713
0.9469
0.9244
0.9031
0.8825
0.8638
0.8444
0.8250
0.8019
0.7775
0.6250
a
b
Cubic feet of gas at 14.65 psia and 60°F per barrel of residual oil at 60°F.
Barrels of oil at indicated pressure and temperature per barrel of residual oil at 60°F.
It should be pointed out that reporting the experimental data relative to the
residual oil volume at 60°F gives the relative oil volume curve the appearance of the formation volume factor curve, leading to its misuse in reservoir
calculations. A better method of reporting these data is in the form of a
shrinkage curve, as shown in Fig. 5.14. The relative-oil volume data in
Fig. 5.13 can be converted to a shrinkage curve by dividing each
Sod
Simulating of Laboratory PVT Data by Equations of State 571
Pressure
n FIGURE 5.14 Oil shrinkage curve versus pressure.
relative-oil volume factor Bod by the relative-oil volume factor at the bubble
point, Bodb:
Sod ¼
Bod
Bodb
(5.196)
where
Bod ¼ differential relative-oil volume factor at pressure p, bbl/STB
Bodb ¼ differential relative-oil volume factor at the bubble point
pressure, pb, psia, bbl/STB
Sod ¼ differential oil shrinkage factor, bbl/bbl, of bubble point oil
The shrinkage curve has a value of 1 at the bubble point and a value less than
1 at subsequent pressures below pb. In this suggested form, the shrinkage
describes the changes in the volume of the bubble point oil as reservoir pressure declines.
The results of the differential liberation test, when combined with the results
of the flash separation test, provide a means of calculating the oil formation
volume factors and gas solubility as a function of reservoir pressure. These
calculations are outlined in the next section.
Simulating the Differential Liberation Test by EOS
The simulation procedure of the test by the Peng-Robinson EOS is summarized in the following steps and in conjunction with the flow diagram shown
in Fig. 5.15.
572 CHAPTER 5 Equations of State and Phase Equilibria
Given:
zi, Pb, T
Calculate ZL from:
Z3 + (B − 1) Z2 + (A − 3B2 − 2B)Z − (AB − B2 − B3) = 0
Assume
ni = 1 mole
Calculate:
Vsat
(1) Z LRT
nZRT
Vsat =
P
=
Pb
Assume depletion
pressure P < Pb
Calculate Ki for zi @ P
See Fig. 15-11
VL =
(nL)actual ZLRT
P
Calculate volume of
condensate VL
Solution gives:
Ki, xi, yi, nL, nV, ZL, ZV
Calculate volume
of gas Vg
(nV)actual ZVRT
P
GP = 379.4 (nV)actual
(GP)total =
Set:
zi = xi and ni = nL
No
Vg =
ΣG
p
i=1
Last pressure ?
Calculate Bod & Rsd
n FIGURE 5.15 Flow diagram for EOS simulating DE test.
Step 1. Starting with the saturation pressure, psat, and reservoir
temperature, T, calculate the volume occupied by a total of 1 mol
(ie, ni ¼ 1) of the hydrocarbon system with an overall composition of
zi. This volume, Vsat, is calculated by applying Eq. (5.192):
Vsat ¼
ð1ÞZRT
psat
Step 2. Reduce the pressure to a predetermined value of p at which the
equilibrium ratios are calculated and used in performing flash
calculations. The actual number of moles of the liquid phase, with a
composition of xi, and the actual number of moles of the gas phase,
with a composition of yi, then are calculated from the expressions
Simulating of Laboratory PVT Data by Equations of State 573
ðnL Þactual ¼ ni nL
ðnυ Þactual ¼ ni nυ
where
(nL)actual ¼ actual number of moles of the liquid phase
(nυ)actual ¼ actual number of moles of the gas phase
nL, nυ ¼ number of moles of the liquid and gas phases as
computed by performing flash calculations
Step 3. Determine the volume of the liquid and gas phase from
VL ¼
ZL RT ðnL Þactual
p
(5.197)
Vg ¼
Zv RT ðnυ Þactual
p
(5.198)
where
VL, Vg ¼ volumes of the liquid and gas phases, ft3
ZL, Zv ¼ compressibility factors of the liquid and gas phases.
The volume of the produced (liberated) solution gas as measured at
standard conditions is determined from the relationship
Gp ¼ 379:4ðnυ Þactual
(5.199)
where
Gp ¼ gas produced during depletion pressure p, scf.
The total cumulative gas produced at any depletion pressure, p, is the
cumulative gas liberated from the crude oil sample during the
pressure reduction process (sum of all the liberated gas from previous
pressures and current pressure) as calculated from the expression
p
X
Gp p ¼
Gp
psat
where (Gp)p is the cumulative produced solution gas at pressure level p.
Step 4. Assume that all the equilibrium gas is removed from contact with
the oil. This can be achieved mathematically by setting the overall
composition, zi, equal to the composition of the liquid phase, xi, and
setting the total moles equal to the liquid phase:
zi ¼ x i
ni ¼ ðnL Þactual
Step 5. Using the new overall composition and total moles, steps 2–4 are
repeated. When the depletion pressure reaches the atmospheric
pressure, the temperature is changed to 60°F and the residual-oil
574 CHAPTER 5 Equations of State and Phase Equilibria
volume is calculated. This residual-oil volume, calculated by
Eq. (5.197), is referred to as Vsc. The total volume of gas that
evolved from the oil and produced (Gp)Total is the sum of the
all-liberated gas including that at atmospheric pressure:
14:7
X
Gp Total ¼
Gp
psat
Step 6. The calculated volumes of the oil and removed gas then are divided
by the residual-oil volume to calculate the relative-oil volumes (Bod)
and the solution GOR at all the selected pressure levels from
Bod ¼
VL
Vsc
h i
ð5:615Þðremaining gas in solutionÞ ð5:615Þ Gp Total Gp p
Rsd ¼
¼
Vsc
Vsc
(5.200)
Flash Separation Tests
Flash separation tests, commonly called separator tests, are conducted to
determine the changes in the volumetric behavior of the reservoir fluid as
the fluid passes through the separator (or separators) and then into the stock
tank. The resulting volumetric behavior is influenced to a large extent by
the operating conditions; that is, pressures and temperatures of the surface
separation facilities. The primary objective of conducting separator tests,
therefore, is to provide the essential laboratory information necessary for
determining the optimum surface separation conditions, which in turn maximize the stock-tank oil production. In addition, the results of the test, when
appropriately combined with the differential liberation test data, provide a
means of obtaining the PVT parameters (Bo, Rs, and Bt) required for petroleum engineering calculations. The laboratory procedure of the flash separation test involves the following steps.
Step 1. The oil sample is charged into a PVT cell at reservoir temperature
and its bubble point pressure.
Step 2. A small amount of the oil sample, with a volume of (Vo)pb, is
removed from the PVT cell at constant pressure and flashed through
a multistage separator system, with each separator at a fixed
pressure and temperature. The gas liberated from each stage
(separator) is removed and its volume and specific gravity
measured. The volume of the remaining oil in the last stage
(stock-tank condition) is recorded as (Vo)st.
Simulating of Laboratory PVT Data by Equations of State 575
Step 3. The total solution gas-oil ratio and the oil formation volume factor at
the bubble point pressure are then calculated:
Bofb ¼ ðVo Þpb =ðVo Þst
Rsfb ¼ Vg sc =ðVo Þst
(5.201)
(5.202)
where
Bofb ¼ bubble point oil formation volume factor, as measured by
flash liberation, bbl of the bubble point oil/STB
Rsfb ¼ bubble point solution gas-oil ratio, as measured by flash
liberation, scf/STB
(Vg)sc ¼ total volume of gas removed from separators, scf
Step 4. Repeat steps 1–3.
A typical example of a set of separator tests for a two-stage separation is
shown in the table below.
Separator
pressure (psig)
Temperature (°F)
Rsfba
°API
Bofbb
50
to 0
75
75
40.5
1.481
100
to 0
75
75
40.7
1.474
200
to 0
75
75
40.4
1.483
300
to 0
75
75
737
41
P
¼ 778
676
92
P
¼ 768
602
178
P
¼ 780
549
246
P
¼ 795
40.1
1.495
a
GOR in cubic feet of gas at 14.65 psia and 60°F per barrel of stock-tank oil at 60°F.
FVF in barrels of saturated oil at 2620 psig and 220°F per barrel of stock-tank oil at 60°F.
b
The differential liberation data must be adjusted to account for the separation process taking place at the separator condition. The procedure, as discussed in Chapter 4, is based on multiplying the flash oil formation factor at
the bubble point, Bofb, by the differential oil shrinkage factor Sod (as defined
by Eq. (5.196)) at various reservoir pressures. Mathematically, this relationship is expressed as follows:
Bo ¼ Bofb Sod
(5.203)
576 CHAPTER 5 Equations of State and Phase Equilibria
where
Bo ¼ oil formation volume factor, bbl/STB
Bofb ¼ bubble point oil formation volume factor, bbl of the bubble point
oil/STB
Sod ¼ differential oil shrinkage factor, bbl/bbl of bubble point oil
The adjusted differential gas solubility data are given by
Rs ¼ Rsfb ðRsdb Rsd Þ
where
Bofb
Bodb
(5.204)
Rs ¼ gas solubility, scf/STB
Rsfb ¼ bubble point solution gas-oil ratio as defined by Eq. (5.202), scf/
STB
Rsdb ¼ solution gas-oil ratio at the bubble point pressure as measured by
the differential liberation test, scf/STB
Rsd ¼ solution gas-oil ratio at various pressure levels as measured by the
differential liberation test, scf/STB
These adjustments typically produce lower formation volume factors and
gas solubilities than the differential liberation data. This adjustment procedure is illustrated graphically in Figs. 5.16 and 5.17.
n FIGURE 5.16 Adjustment of oil-volume curve to separate conditions.
Simulating of Laboratory PVT Data by Equations of State 577
n FIGURE 5.17 Adjustment of gas-in-solution curve to separator conditions.
Composite Liberation Test
The laboratory procedures for conducting the differential liberation and
flash liberation tests for the purpose of generating Bo and Rs versus p relationships are considered approximations to the actual relationships. Another
type of test, called a composite liberation test, has been suggested by
Dodson et al. (1953) and represents a combination of differential and flash
liberation processes. The laboratory test, commonly called the Dodson test,
provides a better means of describing the PVT relationships. The experimental procedure for this composite liberation, as proposed by Dodson
et al., is summarized in the following steps.
Step 1. A large representative fluid sample is placed into a cell at a pressure
higher than the bubble point pressure of the sample. The
temperature of the cell then is raised to reservoir temperature.
Step 2. The pressure is reduced in steps by removing mercury from the cell,
and the change in the oil volume is recorded. The process is repeated
until the bubble point of the hydrocarbon is reached.
578 CHAPTER 5 Equations of State and Phase Equilibria
Step 3. A carefully measured small volume of oil is removed from the cell at
constant pressure and flashed at temperatures and pressures equal to
those in the surface separators and stock tank. The liberated gas
volume and stock-tank oil volume are measured. The oil formation
volume factor, Bo, and gas solubility, Rs, then are calculated from
the measured volumes:
Bo ¼ ðVo Þp, T =ðVo Þst
Rs ¼ Vg sc =ðVo Þst
(5.205)
(5.206)
where
(Vo)p,T ¼ volume of oil removed from the cell at constant pressure p
(Vg)sc ¼ total volume of the liberated gas as measured under
standard conditions
(Vo)st ¼ volume of the stock-tank oil
Step 4. The volume of the oil remaining in the cell is allowed to expand
through a pressure decrement and the gas evolved is removed, as in
the differential liberation.
Step 5. Following the gas removal, step 3 is repeated and Bo and Rs
calculated.
Step 6. Steps 3–5 are repeated at several progressively lower reservoir
pressures to secure a complete PVT relationship.
Note that this type of test, while more accurately representing the PVT
behavior of complex hydrocarbon systems, is more difficult and costly to
perform than other liberation tests. Consequently, these experiments usually
are not included in a routine fluid property analysis.
Simulating of the Composite Liberation Test by EOS
The stepwise computational procedure for simulating the composite liberation test using the Peng-Robinson EOS is summarized next in conjunction
with the flow chart shown in Fig. 5.18.
Step 1. Assume a large reservoir fluid sample with an overall
composition of zi is placed in a cell at the bubble point pressure, pb,
and reservoir temperature, T.
Step 2 Remove 1 lb-mol (ie, ni ¼ 1) of the liquid from the cell and calculate
the corresponding volume at pb and T by applying the PengRobinson EOS to estimate the liquid compressibility factor, ZL:
ðVo Þp, T ¼
ð1ÞZL RT
pb
(5.207)
Simulating of Laboratory PVT Data by Equations of State 579
Given:
zi, Pb, T
Assume
n = 1 mole
Calculate:
VP,T
Flash zi at surface separation
conditions. See Fig.15-11
Calculate Z L from:
Z3 + (B − 1) Z2 + (A − 3B2 − 2B)Z − (AB − B2 − B3) = 0
(1) ZLRT
nZRT
VP,T =
P
=
P
Solution gives:
Ki, xi, yi, nL, nV, ZL, ZV
Calculate
B0 and Rs
Assume P < Pb and
Perform flash calculations
Set:
zi = xi and ni = 1
P
No
Last pressure ?
Yes
Results Bo & Rs
n FIGURE 5.18 Flow diagram for EOS simulating composite liberation test.
where
(Vo)p,T ¼ volume of 1 mol of the oil, ft3.
Step 3. Flash the resulting volume (step 2) at temperatures and pressures
equal to those in the surface separators and stock tank. Designating
the resulting total liberated gas volume (Vg)sc and the volume of the
stock-tank oil (Vo)st, calculate the Bo and Rs by applying
Eqs. (5.205), (5.206), respectively:
Bo ¼ ðVo Þp, T =ðVo Þst
Rs ¼ Vg sc =ðVo Þst
Step 4. Set the cell pressure to a lower pressure level, p, and using the
original composition, zi, generate the Ki values through the
application of the PR EOS.
580 CHAPTER 5 Equations of State and Phase Equilibria
Step 5. Perform flash calculations based on zi and the calculated ki values.
Step 6. To simulate the differential liberation step of removing the
equilibrium liberated gas from the cell at constant pressure, simply
set the overall composition, zi, equal to the mole fraction of the
equilibrium liquid, x, and the bubble point pressure, pb, equal to the
new pressure level, p. Mathematically, this step is summarized by
the following relationships:
zi ¼ x i
pb ¼ p
Step 7. Repeat steps 2–6.
Swelling Tests
Swelling tests should be performed if a reservoir is to be depleted under gas
injection or a dry gas cycling scheme. The swelling test can be conducted on
gas-condensate or crude oil samples. The purpose of this laboratory experiment is to determine the degree to which the proposed injection gas will
dissolve in the hydrocarbon mixture. The data that can be obtained during
a test of this type include
n
n
The relationship of saturation pressure and volume of gas injected.
The volume of the saturated fluid mixture compared to the volume of the
original saturated fluid.
The test procedure, as shown schematically in Fig. 5.19, is summarized in
the following steps.
Step 1. A representative sample of the hydrocarbon mixture with a known
overall composition, zi, is charged to a visual PVT cell at saturation
pressure and reservoir temperature. The volume at the saturation
pressure is recorded and designated (Vsat)orig.
Step 2. A predetermined volume of gas with a composition similar to the
proposed injection gas is added to the hydrocarbon mixture. The
cell is pressured up until only one phase is present. The new
saturation pressure and volume are recorded and designated ps and
Vsat, respectively. The original saturation volume is used as a
reference value and the results are presented as relative total
volumes:
Vrel ¼ Vsat =ðVsat Þorig
where
Vrel ¼ relative total volume
(Vsat)orig ¼ original saturation volume.
Simulating of Laboratory PVT Data by Equations of State 581
Swelling test
Injection gas
Psat
(Vsat)orig
Original fluid
(gas or oil) @
saturation
pressure Psat
(Psat)new
ie Pb or Pd
P
Vt
Hg
Original fluid
+
injection gas
B
n FIGURE 5.19 Schematic illustration of the Swelling test.
Step 3. Repeat step 2 until the mole percent of the injected gas in the fluid
sample reaches a preset value (around 80%).
Typical laboratory results of a gas-condensate swelling test with lean gas of
a composition are shown Figs. 5.20 and 5.21. These two figures show the
gradual increase in the saturation pressure and the swollen volume with each
addition of the injection lean gas. As shown in Figs. 5.20 and 5.21, the dew
point increased from 3428 to 4880 psig and relative total volume from 1 to
2.5043 after the final addition.
Simulating of the Swelling Test by EOS
The simulation procedure of the swelling test by the PR EOS is summarized
in the following steps.
Step 1. Assume the hydrocarbon system occupies a total volume of 1 bbl at
the saturation pressure, ps, and reservoir temperature, T. Calculate
the initial moles of the hydrocarbon system from the following
expression:
ni ¼
5:615psat
ZRT
Original fluid
+
injection gas
Hg
Hg
A
(Vsat)new
C
582 CHAPTER 5 Equations of State and Phase Equilibria
5000
Injected gas
4800
New dewpoint pressure
4600
Component
yi
CO2
0
N2
0
4400
C1
0.9468
4200
C2
0.0527
C3
0.0005
4000
3800
Cum. gas inj,
scf/bbl
3600
0
1
3428
1.1224
3635
572
1.3542
4015
1523
1.9248
4610
2467
2.5043
4880
3200
0
500
1000
1500
2000
2500
3000
Dewpoint
pressure, psig
190
3400
3000
Swollen
volume,
bbl/bbl
scf/bbl of dewpoint gas
n FIGURE 5.20 Dew point pressure during swelling of reservoir gas with lean gas at 200°F.
Relative volume vs. volume of gas injected
2.6
Injected gas
Swelling ratio, (Vsat)new/(Vsat)orig
2.4
2.2
2
1.8
Component
yi
CO2
0
N2
0
C1
0.9468
C2
0.0527
C3
0.0005
1.6
Cum. gas inj,
scf/bbl
1.4
1.2
1
0
500
1000
1500
2000
scf/bbl of the dewpoint gas
2500
3000
Swollen
volume,
bbl/bbl
Dewpoint
pressure, psig
0
1
3428
190
1.1224
3635
572
1.3542
4015
1523
1.9248
4610
2467
2.5043
4880
n FIGURE 5.21 Reservoir gas swelling with lean gas at 200°F as expressed in terms.
where
ni ¼ initial moles of the hydrocarbon system
psat ¼ saturation pressure “pb or pd” depending on the type of the
hydrocarbon system, psia
Z ¼ gas or liquid compressibility factor
Step 2. Given the composition yinj of the proposed injection gas, add a
predetermined volume (as measured in scf) of the injection gas to
Simulating of Laboratory PVT Data by Equations of State 583
the original hydrocarbon system and calculate the new overall
composition by applying the molal and component balances,
respectively:
nt ¼ ni + ninj ,
zi ¼
yinj ninj + ðzsat Þi ni
nt
with
ninj ¼
Vinj
379:4
where
ninj ¼ total moles of the injection gas
Vinj ¼ volume of the injection gas, scf
(zsat)i ¼ mole fraction of component i in the saturated hydrocarbon
system
Step 3. Using the new composition, zi, calculate the new saturation
pressure, pb or pd, using the Peng-Robinson EOS as outlined in this
chapter.
Step 4. Calculate the relative total volume (swollen volume) by applying
the relationship
Vrel ¼
Znt RT
5:615psat
where
Vrel ¼ swollen volume
psat ¼ saturation pressure.
Step 5. Steps 2–4 are repeated until the last predetermined volume of the
lean gas is combined with the original hydrocarbon system.
Compositional Gradients
Vertical compositional variation within a reservoir is expected during early
reservoir life. One might expect the reservoir fluids to have attained equilibrium at maturity due to molecular diffusion and mixing over geological
time. However, the diffusive mixing may require many tens of million years
to eliminate compositional heterogeneities. When the reservoir is considered mature, it often is assumed that fluids are at equilibrium at uniform temperature and pressure and the thermal and gravitational equilibrium are
established, which lead to the uniformity of each component fugacity
throughout all the coexisting phases. For a single phase, the uniformity of
fugacity is equivalent to the uniformity of concentration.
584 CHAPTER 5 Equations of State and Phase Equilibria
The pressure and temperature, however, are not uniform throughout a reservoir. The temperature increases with depth with a gradient of about
1°F/100 ft. The pressure also changes according to the hydrostatic head
of fluid in the formation. Therefore, compositional variations within a reservoir, particularly those with a tall column of fluid, should be expected.
Table 5.1 shows the fluid compositional variation of a North Sea reservoir
at different depths.
Note that the methane concentration decreases from 72.30 to 54.92 mol%
over a depth interval of only 81 m, indicating a major change in composition that must be considered when estimating reserves and production
planning.
In general, the hydrocarbon mixture is expected to get richer in heavier compounds and contains less of light components (such as methane) with
increasing depth. The variations in the composition and temperature of fluid
with depth will result in changes of saturation pressure as a function of
depth. In the oil column, the bubble point pressure generally decreases with
Table 5.1 Variations in Fluid Composition with Depth in a Reservoir
Fluid
D, Well 1
C, Well 2
B, Well 2
A, Well 2
Depth (meters subsea)
Nitrogen
Carbon dioxide
Methane
Ethane
Propane
i-Butane
n-Butane
n-Pentane
Hexanes
Heptanes
Octanes
Nonanes
Decanes
Undecanes-plus
Molecular weight
Undecanes-plus
characteristics
Molecular weight
Specific gravity
3136
0.65
2.56
72.3
8.79
4.83
0.61
1.79
0.75
0.86
1.13
0.92
0.54
0.28
3.49
33.1
3156
0.59
2.48
64.18
8.85
5.6
0.68
2.07
0.94
1.24
2.14
2.18
1.51
0.91
6
43.6
3181
0.6
2.46
59.12
8.18
5.5
0.66
2.09
1.09
1.49
3.18
2.75
1.88
1.08
9.25
55.4
3217
0.53
2.44
54.92
9.02
6.04
0.74
2.47
1.33
1.71
3.15
2.96
2.03
1.22
10.62
61
260
0.848
267
0.8625
285
0.8722
290
0.8768
Simulating of Laboratory PVT Data by Equations of State 585
7500.
De
wp
oin
o
Reserv
tp
7600.
re
ss
ur
e
ir press
7700.
p
g
144
g
7800.
dient
Depth/ft
; psi / ft
ure gra
D
7900.
Sat P
GOC
Res P
pd = pb = pi
8000.
8100.
3300
3400
p
o
D
144
3500
o
; psi / ft
3600
t
oin
-p e
ble sur
b
s
Bu pre
3700
3800
3900
Pressure/psia
n FIGURE 5.22 Determination of GOC from pressure gradients.
decreasing methane concentration, whereas the gas condensate dew point
pressure increases with increasing heavy fractions, as shown in Fig. 5.22.
As illustrated in Fig. 5.22, the GOC is defined as the depth at which the fluid
system changes from a mixture with a dew point to a mixture with a bubble
point. This may occur at a saturated condition, in which the GOC “gas” is in
thermodynamic equilibrium with the GOC “oil” at the gas-oil contact reservoir pressure. By this definition, the following equality applies:
pb ¼ pd ¼ pGOC
where
pb ¼ bubble point pressure
pd ¼ dew point pressure
pGOC ¼ reservoir pressure at GOC
Fig. 5.23 shows an undersaturated GOC that describes the transition from
dew point gas to bubble point oil through a “critical” or “new-critical” mixture with critical temperatures that equal the reservoir temperature. However, the critical pressure of this critical GOC mixture is lower than
reservoir pressure.
586 CHAPTER 5 Equations of State and Phase Equilibria
Pressure & temperature
in
Dewpo
t press
Retrograde gas-cap
ure
Depth
re
pres
oint
blep
Bub
on
ure
carb
dro
press
peratu
oir tem
f hy
TCo
rvoir
Reserv
Oil rim
Rese
sure
A compositional transition zone that contains a near critical
hydrocarbon mixture that separate gas-cap gas from oil
n FIGURE 5.23 Concept of undersaturated GOC; near critical mixture separates gas-cap gas from oil rim.
Minimum Miscibility Pressure
Several schemes for predicting MMP based on the equation-of-state
approach have been proposed. Benmekki and Mansoori (1988) applied
the PR EOS with different mixing rules that were joined with a newly formulated expression for the unlike three-body interactions between the injection gas and the reservoir fluid. Several authors used upward pressure
increments to locate the MMP, taking advantage through extrapolation of
the fact the K-values approach unity as the limiting coincident tie line is
approached.
A simplified scheme proposed by Ahmed (1997) is based on the fact that the
MMP for a solvent/oil mixture generally is considered to correspond to the
critical point at which the K-values for all components converge to unity.
The following function, miscibility function, is found to provide accurate
miscibility criteria as the solvent/hydrocarbon mixture approaches the critical point. This miscibility function, as defined next, approaches 0 or a very
small value as the pressure reaches the MMP:
FM ¼ "
!#
n
X
½Φðyi Þvi
0
zi 1 ½Φðzi ÞLi
i¼1
where FM ¼ miscibility function and Φ ¼ fugacity coefficient.
This expression shows that, as the overall composition, zi, changes by gas
injection and reaches the critical composition, the miscibility function is
Tuning EOS Parameters 587
monotonically decreasing and approaching 0 or a negative value. The specific steps of applying the miscibility function follow.
Step 1. Select a convenient reference volume (eg, one barrel) of the crude
oil with a specified initial composition and temperature.
Step 2. Combine an incremental volume of the injection gas with the crude
oil system and determine the overall composition, zi.
Step 3. Calculate the bubble point pressure, pb, of the new composition by
applying the PR EOS (as given by Eq. (5.145))
Step 4. Evaluate the miscibility function, FM, at this pressure using the
calculated values Φ(zi), Φ(yi), and the overall composition at the
bubble point pressure.
Step 5. If the value of the miscibility function is small, then the calculated pb
is approximately equal to the MMP. Otherwise, repeat steps 2–7.
TUNING EOS PARAMETERS
Equations-of-state calculations are frequently burdened by the large number
of fractions necessary to describe the hydrocarbon mixture for accurate
phase behavior simulation. However, the cost and computer resources
required for compositional reservoir simulation increase considerably with
the number of components used in describing the reservoir fluid. For example, solving a two-phase compositional simulation problem of a reservoir
system that is described by Nc components requires the simultaneous solution of 2Nc + 4 equations for each grid block. Therefore, reducing the number of components through lumping has been a common industry practice in
compositional simulation to significantly speed up the simulation process.
As discussed previously in this chapter, calculating EOS parameters (ie, ai,
bi, and αi) requires component properties such as Tc, pc, and ω. These
properties generally are well defined for pure components; however, determining these properties for the heavy fractions and lumped components rely
on empirical correlations and the use of mixing rules. Inevitably, these
empirical correlations and the selected mixing rules introduce uncertainties
and errors in equation-of-state predictions.
Tuning the selected equation of state is an important prerequisite to provide
meaningful and reliable predictions. Tuning an equation of state refers to
adjusting the parameters of the selected EOS to achieve a satisfactory match
between the laboratory fluid PVT data and EOS results. The experimental
data used should be closely relevant to the reservoir fluid and other recovery
processes implemented in the field.
The main objective of tuning EOS is to match all the available laboratory
data, including
588 CHAPTER 5 Equations of State and Phase Equilibria
n
n
n
n
n
n
n
Saturation pressures; dew point and bubblepoint pressure
DE data (Bod, Rsd, Bgd, ρ, etc.)
CCE data (Y-function, Co, μ, ρ, etc.)
CVD data (LDO, RF, Z-factor, etc.)
Swelling tests data
Separator tests data
MMP
Manual adjustments through trial and error or by an automatic nonlinear
regression algorithm are used to adjust the EOS parameters to achieve a
match between laboratory and EOS results. The regression variables are
based on selecting a number of EOS parameters that may be adjusted or
tuned to achieve a match between the experimental and EOS predictions.
The following EOS parameters are commonly used as regression variables:
n
n
n
Properties of the undefined fractions that include Tc, pc, and ω or
alternatively through the adjustments of Ωa and Ωb of these fractions.
The Ω modifications should be interpreted as modifications of the
critical properties, as they are related by the expressions
a ¼ Ωa
R2 Tc2
pc
b ¼ Ωb
RTc
pc
Binary interaction coefficient, kij, between the methane and C7+
fractions.
When an injection gas contains significant amounts of nonhydrocarbon
components, CO2 and N2, the binary interaction, kij, between these
fractions and methane also should be modified.
The regression is a nonlinear mathematical model that places global upper
and lower limits on each regression variable, ΩaC1 , ΩbC1 ; and so forth. The
nonlinear mathematical model determines the optimum values of these
regression variables that minimize the objective function, defined as
exp
Nexp
E Ecal X
j
j F¼
Wj Eexp
j
j
where
Nexp ¼ total number of experimental data points
¼ experimental (laboratory) value of observation j, such as pd, pb,
Eexp
j
or ρob
Tuning EOS Parameters 589
Ecal
j ¼ EOC-calculated value of observation j, such as pd, pb, or ρob
Wj ¼ weight factor on observation j
The values of the weight factors in the objective function are internally
set with default or user override values. The default values are 1.0
with the exceptions of values of 40 and 20 for saturation pressure and
density, respectively. Note that the calculated value of any observation,
such as Ecal
j , is a function of the all-regression variables:
Ecal
j ¼ f ðpC1 ,TC1 ,…Þ
The stepwise regression procedure in tuning EOS parameters is summarized
next.
Step 1. Split the plus fraction to C35 + or C45 +
Step 2. Lump the extended compositional analysis into an optimum number
of pseudocomponents; for example, F1, F2, F3:
Comp Mol%
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
F1
F2
F3
MW
0.25
44.01
0.88
28.01
23.94
16.04
11.67
30.07
9.36
44.1
1.39
58.12
4.61
58.12
1.5
72.15
2.48
72.15
3.26
84
19.8198 111.46
13.5902 161.57
6.63966 241.13
Sp.gr
Pc
Tc
0.818 1071
547.9
0.8094 493.1 227.49
0.3
666.4 343.33
0.3562 706.5 549.92
0.507
616
666.06
0.5629 527.9 734.46
0.584
550.6 765.62
0.6247 490.4 829.1
0.6312 489.6 845.8
0.6781 457.1 910.1
0.7542 409.4 1047.55
0.8061 308 1199.96
0.8526 229.5 1363.62
Vc
ω
Tb
0.0342
0.0514
0.0991
0.0788
0.0737
0.0724
0.0702
0.0679
0.0675
0.064
0.06272
0.06299
0.06355
0.231
0.0372
0.0105
0.0992
0.1523
0.1852
0.1995
0.228
0.2514
0.2806
0.3653
0.4718
0.6199
351
140
201
333
416
471
491
542
557
607
714.42
864.2
1036.9
Step 3. Match PVT data by regressing on the following EOS parameters:
•
•
•
•
•
BIC: “kij”
Critical pressure: Pc
Critical temperature: Tc
Shift volume: C
Acentric factor: ω
A summary of the first is described in the following tabulated form with the
EOS regression parameters represented by “x”:
590 CHAPTER 5 Equations of State and Phase Equilibria
Component
N2
CO2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
F1
F2
F3
kij
pc
Tc
c
ω
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Step 4. Having achieved a satisfactory match with the experimental data
and obtained the “optimum” values of EOS parameters for C1, F1,
F2, F3, and F4, further reduce the number of fractions representing
the system by groping N2 with C1 and CO2 with C2. Match the
available PVT data by only regression on the parameters of the
newly combined groups, as shown below by “x”. Compare results
with those obtained in step 2. If satisfactory, proceed to match with
further grouping as shown in step 4.
Component
kij
Pc
Tc
c
ω
N2 + C1
CO2 + C2
C3
i-C4
n-C4
i-C5
n-C5
C6
F1
F2
F3
x
x
x
x
x
x
x
x
x
x
x
x
Step 5. Continue the regression process on EOS parameters on each newly
lumped groups without adjusting EOS parameters formed in the
previous step. This step is summarized in the following two
tabulated forms:
Tuning EOS Parameters 591
Component
kij
N2 + C1
CO2 + C2
C3
i-C4 n-C4
i-C5 n-C5
C6
F1
F2
F3
x
Component
kij
N2 + C1
CO2 + C2
C3+ i-C4 n-C4
i-C5 n-C5+ C6
F1
F2
F3
x
Pc
Tc
c
ω
x
x
x
x
x
x
x
x
Pc
Tc
c
ω
x
x
x
x
x
x
x
x
x
x
x
x
x
x
As pointed out in Chapter 4, the Lohrenz, Bray, and Clark (LBC) viscosity
correlation has become a standard in compositional reservoir simulation.
When observed viscosity data are available, values of the coefficients a1–
a5 in Eq. (4.86) and the critical volume of C7+ are adjusted and used as
tuning parameters until a match with the experimental data is achieved.
These adjustments are performed independent of the process of tuning
equation-of-state parameters to match other PVT data. The key relationships
of the LBC viscosity model follow:
μob ¼ μo +
0
a1 + a2 ρr + a3 ρ2r + a4 ρ3r + a5 ρ4r
ξm
1
n
C
B X
C
B
½ðxi Mi Vci Þ + xC7 + VC7 + Cρo
B
A
@
i¼1
i62C7 +
ρr ¼
Ma
where
a1 ¼ 0.1023
a2 ¼ 0.023364
a3 ¼ 0.058533
a4 ¼ 0.040758
a5 ¼ 0.0093324
4
0:0001
592 CHAPTER 5 Equations of State and Phase Equilibria
PROBLEMS
1. A hydrocarbon system has the following overall composition:
Component
zi
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.30
0.10
0.05
0.03
0.03
0.02
0.02
0.05
0.40
given
System pressure ¼ 2100 psia
System temperature ¼ 150°F
Specific gravity of C7+ ¼ 0.80
Molecular weight of C7+ ¼ 140
Calculate the equilibrium ratios of this system assuming both an
ideal solution and real solution behavior.
2. A well is producing oil and gas with the following compositions at a gasoil ratio of 500 scf/STB:
Component
xi
yi
C1
C2
C3
n-C4
n-C5
C6
C7+
0.35
0.08
0.07
0.06
0.05
0.05
0.34
0.60
0.10
0.10
0.07
0.05
0.05
0.03
given
Current reservoir pressure ¼ 3000 psia
Bubble point pressure ¼ 2800 psia
Reservoir temperature ¼ 120°F
Molecular weight of C7+ ¼ 160
Specific gravity of C7+ ¼ 0.823
Problems 593
Calculate the overall composition of the system; that is, zi.
3. The hydrocarbon mixture with the following composition exists in a
reservoir at 234°F and 3500 psig:
Component
zi
C1
C2
C3
C4
C5
C6
C7+
0.3805
0.0933
0.0885
0.0600
0.0378
0.0356
0.3043
The C7+ has a molecular weight of 200 and a specific gravity of 0.8366.
Calculate
a. The bubble point pressure of the mixture.
b. The compositions of the two phases if the mixture is flashed at 500
psia and 150°F.
c. The density of the resulting liquid phase.
d. The density of the resulting gas phase.
e. The compositions of the two phases if the liquid from the first
separator is further flashed at 14.7 psia and 60°F.
f. The oil formation volume factor at the bubble point pressure.
g. The original gas solubility.
h. The oil viscosity at the bubble point pressure.
4. A crude oil exists in a reservoir at its bubble point pressure of 2520 psig
and a temperature of 180°F. The oil has the following composition:
Component
xi
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
0.0044
0.0045
0.3505
0.0464
0.0246
0.0683
0.0083
0.0080
0.0080
0.0546
0.4224
The molecular weight and specific gravity of C7+ are 225 and 0.8364.
The reservoir initially contains 122 MMbbl of oil. The surface facilities
594 CHAPTER 5 Equations of State and Phase Equilibria
consist of two separation stages connected in series. The first separation
stage operates at 500 psig and 100°F. The second stage operates under
standard conditions.
a. Characterize C7+ in terms of its critical properties, boiling point, and
acentric factor.
b. Calculate the initial oil in place in STB.
c. Calculate the standard cubic feet of gas initially in solution.
d. Calculate the composition of the free gas and the composition of the
remaining oil at 2495 psig, assuming the overall composition of
the system remains constant.
5. A pure n-butane exists in the two-phase region at 120°F. Calculate the
density of the coexisting phase using the following equations of state:
a. van der Waals.
b. Redlich-Kwong.
c. Soave-Redlich-Kwong.
d. Peng-Robinson.
6. A crude oil system with the following composition exists at its bubble
point pressure of 3250 psia and 155°F.
Component
xi
C1
C2
C3
C4
C5
C6
C7+
0.42
0.08
0.06
0.02
0.01
0.04
0.37
If the molecular weight and specific gravity of the heptanes-plus fraction
are 225 and 0.823, calculate the density of the crude using the
a. Standing-Katz density correlation.
b. Alani-Kennedy density correlation.
c. van der Waals correlation.
d. Redlich-Kwong EOS.
e. Soave-Redlich-Kwong correlation.
f. Soave-Redlich-Kwong correlation with the shift parameter.
g. Peneloux volume correction.
h. Peng-Robinson EOS.
References 595
7. The following reservoir oil composition is available:
Component
Mol% Oil
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7+
40.0
8.0
5.0
1.0
3.0
1.0
1.5
6.0
34.5
The system exists at 150°F. Estimate the MMP if the oil is to be
displaced by
a. Methane.
b. Nitrogen.
c. CO2.
d. 90% CO2 and 10% N2.
e. 90% CO2 and 10% C1.
8. The compositional gradients of the Nameless field are as follows:
C1 + N2
C7+
C2–C6
TVD, ft
69.57
67.44
65.61
63.04
62.11
61.23
58.87
54.08
6.89
8.58
10.16
12.54
13.44
14.32
16.75
22.02
23.54
23.98
24.23
24.42
24.45
24.45
24.38
23.9
4500
4640
4700
4740
4750
4760
4800
5000
Estimate the location of the GOC.
REFERENCES
Ahmed, T., 1991. A practical equation of state. (translation). SPE Reserv. Eng. 6 (1),
137–146.
Ahmed, T., 1997. A generalized methodology for minimum miscibility pressure. In: Paper
SPE 39034 presented at the Latin American Petroleum Conference, Rio de Janeiro,
Brazil, August 30-September 3, 1997.
596 CHAPTER 5 Equations of State and Phase Equilibria
Benmekki, E., Mansoori, G., 1988. Minimum miscibility pressure prediction with EOS.
SPE Reserv. Eng.
Brinkman, F.H., Sicking, J.N., 1960. Equilibrium ratios for reservoir studies. Trans. AIME
219, 313–319.
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Index
Note: Page numbers followed by f indicate figures, t indicate tables, and b indicate boxes.
A
Abu-Khamsin and Al-Marhoun
correlation, 308
Acentric factor, of propane, 13–14
Acid number, 73
Ahmed modified method, 159–164, 161b
Ahmed’s splitting method, 150–155, 152b
Alani-Kennedy method, 498
crude oil density calculation, 250–254,
251t, 253b
n-Alkanes
boiling point data, 89t
GC data calibration, 88–89, 88f
Alston’s correlation, MMP, 449–450
American Petroleum Institute (API)
gravity, 71
crude oil, 239–241, 240b
Aniline point, 73
Apparent molecular weight, 191–192
ASTM distillation curve, 134–135
B
Beal’s correlation
dead oil viscosity, 306
undersaturated oil viscosity, 309
Beggs-Robinson correlation
dead oil viscosity, 306
saturated oil viscosity, 308
Behavior
gas-condensate wells, 432–434
retrograde gas reservoirs, 434–437
Behrens and Sandler’s lumping scheme,
170–175, 173t
Bergman’s method, 129–133, 132b
Binary interaction coefficient (BIC),
521–522
Binary system, 475–476
characteristic of, 22
pressure-composition diagram for,
25–30, 26–27f
pressure-volume diagram for, 23f
properties, 521–522
two-component system, 22
Black oil
approach, 342–344
simulator, 340–341
Boiling point
curve, 4, 78
graphical correlation, 133–136
range, 78
temperature, 4
Bubble point curve, 3, 34
two-component system, 23
Bubble point pressure, 2, 268–275
equation of state, 550–551
Glaso’s correlation, 272–273, 273b
Marhoun’s correlation, 273–274, 274b
Petrosky-Farshad’s correlation, 274–275,
275b
Standing’s correlation, 269–271,
270–271b
two-component system, 22–23
vapor-liquid equilibrium, 491–493,
492b
Vasquez-Beggs’s correlation, 271–272,
272b
n-Butane, saturated liquid and gas
density, 17b
C
Campbell’s method, 486
Carbon atoms
critical properties and acentric factor,
119b
measured mole, 183f
methane content vs. depth, 182f
molecular weight vs. depth, 183f
saturation pressure vs. depth, 182f
Carr-Kobayashi-Burrows’s correction
method, 205–207, 224–228
Cavett’s correlations, 97
petroleum fraction, 112
CCE test. See Constant composition
expansion (CCE) test
Chemical potential, mixture mixing rule,
529–530
Chew-Connally correlation, saturated oil
viscosity, 307–308
Cloud point, 72
Composite liberation test
EOS, 578–580, 579f
experimental procedure, 577–578
Compositional approach, 343
Compositional gradients, EOS, 583–585,
584t, 585–586f
Compositional simulators, 343
Compressibility factors. See also Gas
compressibility factor
direct calculation of, 212
Standing-Katz, 207–209
Condensate GOR, and condensate yield,
424–426, 424–425f
Conradson carbon residue (CCR), 73
Constant composition expansion (CCE)
test, 268, 355–372, 356f, 357t,
359f, 566f
EOS, 568–569
experimental procedure, 566–567
gas-condensate system, 567
laboratory analysis, 407–411, 409t
isothermal compressibility coefficient
from, 361, 361f
oil density from, 360, 382, 383f
pressure-volume relations from, 359f
quality checks of total FVF from, 384,
385b
Constant volume depletion (CVD)
test, 560f
EOS, 562–566, 563f
gas-condensate systems laboratory
analysis, 412–415, 412f
laboratory procedure, 559–562
reservoir engineering predictions, 559
Continuous distribution function, 170
Convergence pressure method, 483–485
Cox chart, 11, 13b
Cricondenbar, 33
Cricondentherm, 33
Critical compressibility factor
correlations, 101
Critical density, 14–16
Critical point, 3, 33
Critical pressure, 3
graphical correlations, 137–139, 139f
Critical temperature, 3
graphical correlation, 137, 138f
Critical volume, 3
Cronquist’s correlation, MMP, 451
599
600 Index
Crude oil
API gravity, 239–241, 240b
assay
molecular and chemical
characteristics, 71–73
plus fraction characterization methods,
93–139
stimulated distillation by GC, 87–93
TBP, 77–87
undefined components, 77
well-defined components, 73–76
characteristic properties of, 39–41
classification, 35–43
heptanes-plus fraction in, 174b
liquid shrinkage for, 43f
low-shrinkage oil, 39
near-critical crude oil, 41
ordinary black oil, 37
volatile crude oil, 39
Crude oil density, 239–240, 242–260
Alani-Kennedy’s method, 250–254, 251t,
253b
Standing-Katz’s method, 243–249, 244f,
247b, 247–248f
Crude oil viscosity, 304–315, 305f
dead oil correlations, 306, 310b
Beal’s correlation, 306
Beggs-Robinson’s correlation, 306
Glaso’s correlation, 307
saturated oil viscosity, 307, 310b
Abu-Khamsin and Al-Marhoun, 308
Beggs-Robinson correlation, 308
Chew-Connally correlation, 307–308
undersaturated oil viscosity, 308–309,
310b
Beal’s correlation, 309
Khan’s correlation, 309
Vasquez-Beggs’s correlation, 309–311
Cubic average boiling point (CABP), 123
CVD test. See Constant volume depletion
(CVD) test
D
Dalton’s law, 468
Darcy’s equation, for gas and oil flow rates,
332
Dead oil viscosity, 304, 306, 310b
Beal’s correlation, 306
Beggs-Robinson’s correlation, 306
Glaso’s correlation, 307
Density, 11
crude oil, 239–240, 242–260
Alani-Kennedy’s method, 250–254,
251t, 253b
Standing-Katz’s method, 243–249,
244f, 247b, 247–248f
liquid, estimation
Katz’s method, 254–258, 255–256f,
257b, 259b
Standing’s method, 258–260, 258b
saturated oil
DE test, 384b
quality checks of, 383, 384b
undersaturated oil, 296–297
Density-temperature diagram
n-butane, 17
propane, 14–16, 15f
DE test. See Differential expansion (DE)
test
Dew point curve, 3, 34
two-component system, 23
Dew point pressure, 2
equation of state, 548–550
swelling tests, 582f
two-component system, 22–23
vapor-liquid equilibrium, 488–491, 490b
Differential expansion (DE) test
Dodson procedure, 394, 397f
oil density from, 382, 383f
saturated oil density, 384b
total FVF from, 379–382, 380b, 385b, 386f
quality check, 384, 385b
Differential liberation (DL) test, 373–386,
373f, 376b
differential gas solubility, 376–377, 377b
differential oil FVF, 375–376, 376b
differential solution gas-oil ratio,
569–570, 570f
EOS, 571–574, 572f
gas deviation factor “Z”, 378–379
gas FVF, 377–378, 378–379b
quality checks of DE data, 382–386
relative-oil volume factor, 569–570, 570f
Differential separation process, 495
Distillation process
GC, 87–93
TBP analysis, 78–80, 79f
DL test. See Differential liberation (DL) test
Dodson test, 394, 397f, 577–578
Dranchuk and Abu-Kassem’s method,
214–215
Dranchuk-Purvis-Robinson method,
216–224
Dry gas reservoirs, 50–59, 51f
E
ECN. See Effective carbon numbers (ECN)
Edmister correlation, 100–101, 530
Edmister’s equation, 110
Effective carbon numbers (ECN), 90–93
Enhanced oil recovery (EOR) projects, 315,
452–453
Equation of state (EOS), 505–530
applications, 546
bubble point pressure, 550–551
cubic, 341–344
definition, 190, 505
dew point pressure, 548–550
equilibrium ratio, 546–548, 547f
model, thermodynamic-based, 344
parameters, 342
Peng-Robinson, 341–342, 534–546, 536b
PVT test data, 558–587
composite liberation test, 577–580,
579f
compositional gradients, 583–585,
584t, 585–586f
constant composition expansion test,
566–569, 566f
constant volume depletion test,
559–566, 560f, 563f
differential liberation test, 569–574,
572f
flash separation tests, 574–576
minimum miscibility pressure,
586–587
swelling tests, 580–583, 581–582f
real gas, 414
Redlich-Kwong, 512–517, 513b,
515–516b
simulator, compositional, 342
Soave-Redlich-Kwong, 517–521
example, 520b, 523b
mixture mixing rule, 521–530
modifications, 530–546
volume translation method, 531–534,
533b
Starling-Carnahan, 212–213
three-phase equilibrium calculations,
551–557
tuning, 342–344, 442
tuning parameters, 587–591
two-parameter cubic, 509
van der Waals, 506–512, 510b
vapor pressure, 557–558
Equilibrium ratios, 467–471
development, 476
Index 601
equation of state, 546–548, 547f
for hydrocarbon system, 484f
mixture mixing rule, 526–527
for plus fractions, 485–488
Campbell’s method, 486
Katz’s method, 487
K-values, 487–488
Winn’s method, 486–487
for real solutions, 476–485
convergence pressure method, 483–485
example, 481b
K-value of plus fraction, 478–480
simplified K-value method, 480–483
Standing’s correlation, 477–483
Wilson’s correlation, 476–477
Whitson and Torp’s method, 485b
Extended Y-function, 366, 368b, 370f
Extrapolated vapor pressure (EVP) method,
MMP correlation, 448–449, 448b
F
Fanchi’s linear equation, 316
Final boiling point (FBP), 78
Firoozabadi and Aziz’s correlation, MMP,
453–454
Flash calculation, 471–476, 473b
Flash liberation test, 374
Flash point, 72
Flash separation, 495–505
Flash separation test, 574–576
gas-in-solution curve, 577f
laboratory procedure, 574–575
oil-volume curve, 576f
Fluid density, 16–17, 16f
Formation volume factor (FVF).
See also Oil formation volume
factor (FVF); Total formation
volume factor (FVF)
gas, 377–378, 378–379b
water, 323–324
Freeze point, 73
Fugacity, 525–526
Fusion, 10
FVF. See Formation volume factor (FVF)
G
Gas
composition, 198b
definition, 189
deviation factor “Z”, 378–379
expansion factor, 222–223
formation volume factor, 221–222
FVF, 377–378, 378–379b
high-molecular weight, 207–212
properties, 189, 191
sample conditioning, 335
separator, 335–336
specific gravity of, 241–242, 241b
Gas-cap reservoir, 35
classification, 36f
dry, 53f
retrograde, 53f
Gas chromatography (GC), 345–346, 346f
stimulated distillation by, 87–93, 90f
Gas compressibility factor, 196
calculation, 212
chart, 197–201, 198f
for natural gases, 196–197
Gas-condensate reservoir, liquid blockage
in, 431–457
Gas-condensate systems laboratory
analysis, 254
constant composition test, 407–411, 409t
CVD test, 412–415, 412f
retrograde gas laboratory data,
416–423
recombination of separator samples,
404–407, 416–417t, 418f, 423b
consistency test on recombined GOR,
404t, 406–407, 406–407f, 409t
Gas-condensate wells, behavior of, 432–434
Gas density, 193, 204b
of n-butane, 17b
Gas-down-to (GDT) level, 60–61
Gas flow rate, 222b
Darcy’s equation for, 332
Gas-oil contact (GOC), 59–64
compositional changes, 55f
determination, 55f
discovery well in gas column, 60–62f
locating, 54t
reservoir pressure, 585, 585–586f
true vertical depth, 63b
undersaturated, 65, 65f
well-bore and stock-tank density, 56f
Gas-oil ratio (GOR), 37–43, 241–242
consistency check, 355f
consistency test on recombined, 404t,
406–407, 406–407f, 409t
material balance consistency test on,
352–355
surface, 332
Gas-oil separation
differential separation process, 495
flash separation, 495–505
types of, 494–495
Gas reservoir, 34
categories, 43
compositions, 52f
dry gas reservoirs, 50–59
MBE for retrograde, 426–431
consistency tests on CCE and CVD
Data, 428–431, 428–429t,
431–432f
near-critical gas-condensate reservoirs, 46
retrograde gas reservoirs, 44–46
wet gas reservoirs, 47–50
Gas solubility, 260–268, 261f, 268b
differential, 376–377, 377b
Glaso’s correlation, 264–265, 265b
Marhoun’s correlation, 265–266, 266b
Petrosky-Farshad’s correlation, 266–268,
267b
Standing’s correlation, 261–263, 262b
Vasquez-Beggs’s correlation, 263–264,
264b
water, 326
Gas viscosity, 223–224, 227b
atmospheric correlation, 225f
example, 227b
Gas-water contact (GWC), 61
Generalized correlations, 94–122
Gibbs energy, mixture mixing rule, 527–529
Glaso’s correlation
bubble point pressure, 272–273, 273b
dead oil viscosity, 307
gas solubility, 264–265, 265b
MMP correlation, 454–455
oil FVF, 278
total FVF, 301, 302b
GOC. See Gas-oil contact (GOC)
GOM crude oil system. See Gulf of Mexico
(GOM) crude oil system
GOR. See Gas-oil ratio (GOR)
Graphical correlations
boiling points, 133–136
critical pressure, 137–139, 139f
critical temperature, 137, 138f
molecular weight, 136–137
Gross heat of combustion (high heating
value), 73
Gulf of Mexico (GOM) crude oil system,
266–268
hydrocarbon system, 279
PVT correlations for, 318–322
GWC. See Gas-water contact (GWC)
602 Index
H
I
Hall and Yarborough correlations, 104
Hall-Yarborough’s method, 212–214
Heptanes-plus fraction, 78, 110b,
479–480
bubble point pressure, 492
in crude oil system, 174b
dew point pressure, 490
molar distribution of, 180b
molecular weight and specific gravity,
117b
Hoffman plot, 349–350, 351f
Hong’s mixing rules, 168–169
Hudgins’s correlation, MMP, 454
Hydrocarbon
analysis of recombined reservoir fluid
samples, 347t
in petroleum reservoirs, 1
vapor pressure chart for, 12f
weights and volumes calculation, 18b
Hydrocarbon phase behavior
crude oil system, 35–43
gas-oil contact, 59–64
gas reservoirs, 43–59
multicomponent systems, 32
oil reservoirs, 34–35
phase rule, 65–67
pressure-temperature diagram, 32–34
pure component saturated liquid density,
19–21
pure component volumetric properties,
11–19
p-x diagram, 25–30
reservoirs and reservoir fluids, 34–59
single-component systems, 1–21
three-component systems, 30–31
two-component systems, 22–30
undersaturated GOC, 65
Hydrocarbon-plus fractions
crude oil assay, 71–77
lumping schemes, 165–180
PNA determination, 122–133
splitting schemes, 141–164
undefined components, 77
well-defined components, 73–76
Hydrocarbon system. See also specific types
of hydrocarbon system
components, vapor pressure, 469f
equilibrium ratios for, 484f
gas composition, 209b
total mole fraction in, 470
IBP. See Initial boiling point (IBP)
Ideal gas. See also Natural gas
apparent molecular weight, 191–192
behavior of, 190–196, 200b
definition, 189
density, 193
example, 191b, 195b
specific gravity, 194–196, 194b
specific volume, 193
standard volume, 193
Ideal gas law, 190
Ideal solution, 468–470
Initial boiling point (IBP), 78
Interfacial tension, 315–318
Inverse lever rule, 28–30
Isobutene, liquid and vapor calculation, 29b
Isothermal compressibility coefficient
from CCE test, 361, 361f
of crude oil, 282–293
saturated, 283, 291–293, 292b
undersaturated, 282–291
K
Katz’s method, 487
liquid density estimation, 254–258,
255–256f, 257b, 259b
Kesler and Lee correlations, 97–99
petroleum fraction, 113
Khan’s correlation, undersaturated oil
viscosity, 309
K-values
for nonhydrocarbon components,
487–488
of plus fraction, 478–480
simplified method, 480–483
L
Lange’s correlation, MMP, 455–457
LBC viscosity correlation. See Lohrenz,
Bray, and Clark (LBC) viscosity
correlation
Lean gas and nitrogen miscibility
correlation, MMP, 452–457
Lee and Kesler correlation, 530
Lee-Gonzalez-Eakin’s method, 228–234,
229b
Lee’s lumping scheme, 176–180
Lee’s mixing rules, 169–170
Lever rule, 31
LGF. See Logistic growth function (LGF)
Liquid blockage, in gas-condensate
reservoirs, 431–457
Liquid density estimation
Katz’s method, 254–258, 255–256f,
257b, 259b
Standing’s method, 258–260, 258b
Liquid-gas separation process, 494
Logistic growth function (LGF), 163–164,
164f
Log-log scale, 470
Lohrenz, Bray, and Clark (LBC) viscosity
correlation, 591–592
Lohrenz’s splitting method, 147–148
Low-shrinkage oil, 39
oil-shrinkage curve, 40f
phase diagram for, 39f
Lumping schemes
Behrens and Sandler’s lumping scheme,
170–175, 173t
Hong’s mixing rules, 168–169
Lee’s lumping scheme, 176–180
Lee’s mixing rules, 169–170
Whitson’s lumping scheme, 165–168
M
MABP. See Molar average boiling point
(MABP)
Magoulas and Tassios correlations, 104
carbon atoms, 120
heptanes-plus fraction, 118
Marhoun’s correlation
bubble point pressure, 273–274, 274b
gas solubility, 265–266, 266b
oil FVF, 279
total FVF, 302–304, 302b
Material balance (MB)
consistency test, 350–352
on GOR, 352–355
quality test, 353t
Material balance equation (MBE)
oil FVF, 280–282
for retrograde gas reservoirs, 426–431
consistency tests on CCE and CVD
data, 428–431, 428–429t, 431–432f
MB. See Material balance (MB)
MBE. See Material balance equation (MBE)
MDT tool. See Modular dynamic testing
(MDT) tool
Mean average boiling point (MeABP), 123
Melting curve, 10
Melting-point temperature, 10
Index 603
Microsoft Solver, 90–92
Minimum miscibility pressure (MMP), 438,
441, 586–587
determination, 442f
pure and impure CO2, 446–452, 448b
Minimum miscibility pressure (MMP)
correlations, 445–452
Alston’s correlation, 449–450
Cronquist’s correlation, 451
EVP method, 448–449, 448b
Firoozabadi and Aziz’s correlation,
453–454
Glaso’s correlation, 454–455
Hudgins’s correlation, 454
Lange’s correlation, 455–457
lean gas and nitrogen miscibility
correlations, 452–457
NPC method, 451
Orr and Silva correlation, 447–448
pure and impure CO2 MMP, 446–452,
448b
Sebastian and Lawrence’s correlation,
455
Sebastian’s method, 450–451
Yellig and Metcalfe correlation,
449, 449b
Yuan’s correlation, 451–452
Mixture mixing rule, SRK EOS, 521–530
chemical potential, 529–530
correct Z-factor selection, 527–529
equilibrium ratio, 526–527
Gibbs energy, 527–529
MMP. See Minimum miscibility pressure
(MMP)
Modified Katz’s splitting method, 143–147
Modular dynamic testing (MDT) tool, 329
Molar average boiling point (MABP), 123
Molar density, 12
Molar volume, 12
Molecular weight
graphical correlation, 136–137
methods correlations, 108–121, 109t, 109f
Mole fraction, 192–193
Multicomponent hydrocarbon systems, 32
pressure-temperature diagram,
32–34, 33f
Multiple hydrocarbon samples, 181–184
N
National Petroleum Council (NPC) method,
MMP correlation, 451
Natural gas, 189. See also Ideal gas
Carr-Kobayashi-Burrows’s correction
method, 205–207, 224–228
compressibility of, 216–221
Dranchuk and Abu-Kassem’s method,
214–215
Dranchuk-Purvis-Robinson method,
216–224
gas compressibility factor, 196–197
Hall-Yarborough’s method, 212–214
hydrocarbon components, 202
Lee-Gonzalez-Eakin’s method, 228–234,
229b
Newton-Raphson iteration technique, 215
physical properties, 189
Trube’s pseudoreduced compressibility,
219–220f
viscosity, 223–224, 227b
Wichert-Aziz’s correction method,
203–205
Near-critical crude oil, 41
liquid-shrinkage curve for, 42f
phase diagram for, 42f
Near-critical gas-condensate reservoirs, 46,
46–47f
Near-critical reservoirs, 34
Net heat of combustion (lower heating
value), 73
Newton-Raphson iteration technique, 215,
472
Newton-Raphson method, 554
Nonhydrocarbon adjustment methods
Carr-Kobayashi-Burrows’s correction
method, 205–207
Wichert-Aziz’s correction method,
203–205
Nonhydrocarbon components, Z-factor
effect, 196–203
NPC method. See National Petroleum
Council (NPC) method
O
Oil
flow rates, Darcy’s equation, 332
properties, undersaturated, 294–297,
294f
reservoir phase distribution in, 375f
Oil density. See also Crude oil density
from CCE test, 360
undersaturated, 296–297
Oil-down-to (ODT) level, 59–60, 59f
Oil formation volume factor (FVF),
275–282, 281b
differential, 375–376, 376b
Glaso’s correlation, 278
Marhoun’s correlation, 279
material balance equation, 280–282
Petrosky-Farshad’s correlation, 279
pressure relationship, 276, 276f
Standing’s correlation, 277
undersaturated, 295–296, 296b
Vasquez-Beggs’s correlation, 277–278
Oil reservoir, 34
classification of, 34–35
phase distribution, 375f
Optimum separator pressure, 496
Ordinary black oil, 37
liquid-shrinkage curve for, 38f
p-T diagram for, 38f
Orr and Silva correlation, MMP, 447–448
P
Parachor, 315–316
Pedersen’s splitting method, 148–150, 149b
Peng-Robinson equation of state (PR EOS),
534–546, 536b
Peng-Robinson’s method, 124–129, 127b
Perfect gas, 189
Perfect gas law, 196
Petroleum fraction
acentric factor, 111b
critical properties, 111b
molecular weight, 111b
Petrosky-Farshad’s correlation
bubble point pressure, 274–275, 275b
gas solubility, 266–268, 267b
oil FVF, 279
undersaturated oil compressibility
estimation, 290, 290b
Phase behavior, 1
Phase diagram, 1, 32
of binary mixture, 25f
dry gas, 51f
low-shrinkage oil, 39f
near-critical crude oil, 42f
near-critical gas-condensate
reservoir, 46f
retrograde system, 44f
ternary, 32f
wet gas, 47f
Phase envelope, 3, 33
impact of composition, 36f
604 Index
Phase equilibria, 468
calculations, 470, 488–505
Phase rule, 65–67
Plus fraction
equilibrium ratios for, 485–488
Campbell’s method, 486
Katz’s method, 487
Winn’s method, 486–487
K-value of, 478–480, 487–488
Plus fraction characterization methods
Cavett’s correlations, 97
critical compressibility factor
correlations, 101
Edmister’s correlations, 100–101
generalized correlations, 94–122
Hall and Yarborough correlations, 104
Kesler and Lee correlations, 97–99
Magoulas and Tassios correlations, 104
molecular weight methods correlations,
108–121
properties, 93–94
recommended characterizations,
121–122
Riazi and Daubert generalized
correlations, 95–96
Rowe’s characterization method,
101–102
Sancet correlations, 108
Silva and Rodriguez correlations, 107
Standing’s correlations, 103
Twu’s correlations, 105–107
Watansiri-Owens-Starling correlations,
99–100
Willman-Teja correlations, 103–104
Winn and Sim-Daubert correlations, 99
PNA determination, 122–133
Pour point, 72
Power function form, 160, 160f
PR EOS. See Peng-Robinson equation of
state (PR EOS)
Pressure-composition diagram, for binary
systems, 25–30, 26–27f
Pressure-temperature diagram
multicomponent systems, 32–34, 33f
ordinary black oil, 38f
single-component system, 4, 10f, 11
two-component systems, 24f
Pressure-volume diagram
for binary system, 23f
pure component, 2–3, 2f
Pressure-volume relation, from CCE test,
359f
Pressure-volume-temperature (PVT)
correlations for GOM crude oil system,
318–322
test data, EOS, 558–587
composite liberation test, 577–580,
579f
compositional gradients, 583–585,
584t, 585–586f
constant composition expansion test,
566–569, 566f
constant volume depletion test,
559–566, 560f, 563f
differential liberation test, 569–574,
572f
flash separation tests, 574–576
minimum miscibility pressure,
586–587
swelling tests, 580–583, 581–582f
Pressure-volume-temperature (PVT)
measurements, 342–415
differential liberation test, 373–386, 373f,
376b
differential gas solubility, 376–377,
377b
differential oil FVF, 375–376, 376b
gas deviation factor “Z”, 378–379
gas FVF, 377–378, 378–379b
quality checks of DE data, 382–386
routine laboratory PVT tests
compositional analysis of reservoir
fluid, 345–355, 346f, 347t
quality assessment of laboratory
compositional analysis, 346–355,
350f
separator tests, 386–403, 387f, 389–390t,
390–391f
adjustment of DE data to separator
conditions, 389–398, 395t,
396–397f, 398t
extrapolation of reservoir fluid data,
398–403
Pressure/volume tests. See Constant
composition expansion (CCE) test
Propane
existence state, 13b
saturated densities of, 19
saturated liquid density of, 21b
in underground cavern, 18b
vapor pressure of, 14b
Pseudocomponent, from TBP test, 78–79
Pseudofractions, 74
Pseudoization, 165
Pure component
physical properties for, 5t
pressure-volume phase diagram, 2–3, 2f
pressure-volume relationship, 507f, 509f
Rackett’s compressibility factor, 20,
20t, 21b
saturated liquid density, 19–21
vapor pressure, 557–558
volumetric properties, 11–19
PVT. See Pressure-volume-temperature
(PVT)
Q
Quality lines, 24, 33
R
Rackett compressibility factor, 20, 20t,
21b, 532
Raoult’s law, 468
Real gases, behavior of, 196–203
Real solutions, equilibrium ratios for,
476–485
convergence pressure method, 483–485
example, 481b
K-value of plus fraction, 478–480
simplified K-value method, 480–483
Standing’s correlation, 477–483
Wilson’s correlation, 476–477
Recombined GOR, consistency test on,
404t, 406–407, 406–407f, 409t
Redlich-Kwong equation of state (RK
EOS), 512–517, 513b, 515–516b
Reduced temperature, 13
Refractive index (RI), 72
Reid vapor pressure (RVP), 72
Repeat formation tester (RFT) tool, 51–52,
56–57
Reservoir classification, 34–59, 35f
Reservoir fluids
classification of, 34–59
laboratory analysis, 327
black oil simulator, 340–341
compositional modeling, 341–342
fluid sampling, 328–338
representative sample, 339–340, 341f
separator samples, 331–337, 331f,
333f, 337t
in situ representative sample, 339–340
subsurface sampling, 329–330,
330f, 334
unrepresentative sample, 339–340, 341f
well conditioning, 327–328
wellhead sampling, 337–338
Index 605
Reservoir water properties, 323–327
gas solubility in water, 326
water FVF, 323–324
water isothermal compressibility,
326–327
water viscosity, 324–326
Retention time, 87
Retrograde gas reservoir, 44–46, 44f
behavior of, 434–437
MBE for, 426–431
Riazi and Daubert generalized correlations,
95–96
Riazi-Daubert correlation, carbon
atoms, 120
Riazi-Daubert equation, 110
petroleum fraction, 111
Rising bubble test, 445
RK EOS. See Redlich-Kwong equation of
state (RK EOS)
Routine laboratory PVT tests
compositional analysis of reservoir fluid,
345–355, 346f, 347t
quality assessment of laboratory
compositional analysis,
346–355, 350f
Rowe’s characterization method, 101–102
Rowe’s correlations
carbon atoms, 119
heptanes-plus fraction, 117
RVP. See Reid vapor pressure (RVP)
S
Sancet correlations, 108
Saturated gas, 3
Saturated liquid, 3
of n-butane, 17b
density curve, 16–17
Saturated oil
density
DE test, 384b
quality checks of, 383, 384b
isothermal compressibility coefficient,
283, 291–293, 292b
reservoir, 35
viscosity, 304, 307, 310b
Abu-Khamsin and Al-Marhoun
correlation, 308
Beggs-Robinson correlation, 308
Chew-Connally correlation, 307–308
Saturated vapor density curve, 16–17
Schlumberger Repeat Formation Testing
(RFT), 329
Sebastian and Lawrence’s correlation,
MMP, 455
Sebastian’s method, MMP correlation,
450–451
Separation process
differential, 495
flash, 495–505
gas-in-solution curve, 577f
gas-oil separation, 494–495
liquid-gas, 494
n-separation stages, 498f
oil-volume curve, 576f
two-stage, 494, 494f
vapor-liquid equilibrium calculations,
493–505
Separator gas, 335–336
Separator liquid samples, 336
Separator shrinkage factor, 334
Separator test. See also Flash separation test
PVT measurement, 386–403, 387f,
389–390t, 390–391f
adjustment of DE data to separator
conditions, 389–398, 395t,
396–397f, 398t
extrapolation of reservoir fluid data,
398–403
Silva and Rodriguez correlations, 107, 111
Simulated distillation (SD), GC, 87–93, 90f
Simulator
black oil, 340–341
compositional, 343
Single carbon number (SCN), 74
coefficients of equation, 77t
concept of, 75t
generalized physical properties, 76t
Single-component hydrocarbon system, 1–21
degrees of freedom, 66b
physical properties, 5t
pressure-temperature diagram of, 4,
10f, 11
pressure-volume diagram, 2f
saturated liquid density, 19–21
volumetric properties, 11–19
Slim tube test, 440–444, 441–442f
relative permeability adjustment, 444
viscosity adjustment, 443–444
Smoke point, 73
Soave-Redlich-Kwong equation of state
(SRK EOS), 517–521
example, 520b, 523b
mixture mixing rule, 521–530
chemical potential, 529–530
correct Z-factor selection, 527–529
equilibrium ratio, 526–527
Gibbs energy, 527–529
modifications, 530–546
volume translation method, 531–534,
533b
Solution gas, specific gravity of, 240b,
241–242
Special laboratory PVT tests, 437–457
minimum miscibility pressure
correlations, 445–452
Alston’s correlation, 449–450
Cronquist’s correlation, 451
EVP method, 448–449, 448b
Firoozabadi and Aziz’s correlation,
453–454
Glaso’s correlation, 454–455
Hudgins’s correlation, 454
Lange’s correlation, 455–457
lean gas and nitrogen miscibility
correlations, 452–457
NPC method, 451
Orr and Silva correlation, 447–448
pure and impure CO2 MMP,
446–452, 448b
Sebastian and Lawrence’s
correlation, 455
Sebastian’s method, 450–451
Yellig and Metcalfe correlation,
449, 449b
Yuan’s correlation, 451–452
rising bubble test, 445
slim tube test, 440–444, 441–442f
relative permeability adjustment, 444
viscosity adjustment, 443–444
swelling test, 438–439, 439–440f
Specific gravity, 194–196
example, 194b
wet gases, 230–234, 232b
Specific volume, 12, 193
Splitting schemes, 141–164
Ahmed modified method, 159–164, 161b
Ahmed’s method, 150–155, 152b
condensate-gas system, 144b
discrete and continuous compositional
distribution, 141f
exponential and left-skewed distribution
functions, 142f
Lohrenz’s method, 147–148
modified Katz’s method, 143–147
Pedersen’s method, 148–150, 149b
Whitson’s method, 155–159, 158b
606 Index
SRK EOS. See Soave-Redlich-Kwong
equation of state (SRK EOS)
Standard volume, 193
Standing-Katz chart
compressibility factor, 207–209
Z-factor, 212
Standing-Katz’s method, 498
crude oil density calculation, 243–249,
244f, 247b, 247–248f
Standing’s correlation, 103, 477–483
bubble point pressure, 269–271,
270–271b
carbon atoms, 120
gas solubility, 261–263, 262b
heptanes-plus fraction, 118
oil FVF, 277
total FVF, 300–301, 302b
undersaturated oil compressibility
estimation, 290–291, 290b
Standing’s method, liquid density
estimation, 258–260, 258b
Starling-Carnahan equation of state,
212–213
Straight-line relationship, 15, 15f
Sublimation pressure curve, 10
Sulfur content, 72
Surface GOR, 332
Surface samples recombination procedure,
338–339, 338f
Surface tension, 315–318
Swelling test, 438–439, 439–440f, 581f
dew point pressure, 582f
equation of state, 581–583
procedure, 580–581
reservoir gas, 582f
T
Three-component hydrocarbon systems,
30–31
at constant temperature and pressure, 32f
degrees of freedom, 67b
features of, 31
properties of, 31f
Three-parameter gamma probability
function, 155–156, 156f
Three-phase equilibrium calculation, EOS,
551–557
Tie line, 28–31
Total formation volume factor (FVF),
298–304, 299f
CCE test, quality check, 384, 385b
Glaso’s correlation, 301, 302b
Marhoun’s correlation, 302–304, 302b
Standing’s correlation, 300–301, 302b
Total mole fraction, in hydrocarbon, 470
Traditional Y-function, 363, 364f, 364b
Trial-and-error process, 489
Trube’s correlation, undersaturated oil
compressibility estimation,
284–289, 285–287f, 288b
True boiling point (TBP), 77–87
condensate-gas specific gravity vs.
molecular weight, 83f
distillation process, 78–80, 79f
mathematical representation of, 86f, 87
pseudocomponent from, 78–79
stock-tank oil sample, 80f
volatile oil specific gravity vs. molecular
weight, 82f
Watson characterization factor, 80–82, 84
Tuning equation of state parameters, 543,
587–591
Two-component hydrocarbon system, 22–30
amounts of liquid and vapor, 28, 29f
degrees of freedom, 67b
isobutene, liquid and vapor calculation, 29b
nomenclatures, 27–28
pressure-composition diagram, 25–30,
26–27f
pressure-temperature diagram, 24f
pressure-volume isotherm, 22f
volatile component, 28
Two-parameter cubic equation of state, 509
Two-phase region, 3
Two-stage separation processes, 494, 494f
Twu’s correlations, 105–107
U
Undefined petroleum fractions, 77
Undersaturated GOC, 65, 65f
Undersaturated oil
compressibility estimation
Petrosky-Farshad’s correlation, 290,
290b
Standing’s correlation, 290–291,
290b
Trube’s correlation, 284–289,
285–287f, 288b
Vasquez-Beggs’s correlation, 289, 290b
density, 296–297
FVF, 295–296, 296b
isothermal compressibility coefficient,
282–291
properties, 294–297, 294f
reservoir, 34
viscosity, 305, 308–309, 310b
Beal’s correlation, 309
Khan’s correlation, 309
Vasquez-Beggs’s correlation, 309–311
Universal oil products, 80–81
V
VABP. See Volume average boiling point
(VABP)
Van der Waals equation of state (VdW
EOS), 506–512, 510b
Vapor-liquid equilibrium
bubble point pressure, 491–493, 492b
dew point pressure, 488–491, 490b
separator calculations, 493–505
Vapor phase, weights and volumes of, 18b
Vapor pressure, 4
chart for hydrocarbon components, 12f
curve, 4–10
equation of state, 557–558
hydrocarbon components, 469f
Vasquez-Beggs’s correlation
bubble point pressure, 271–272, 272b
gas solubility, 263–264, 264b
oil FVF, 277–278
undersaturated oil compressibility
estimation, 289, 290b
undersaturated oil viscosity, 309–311
VdW EOS. See Van der Waals equation of
state (VdW EOS)
Viscosity. See also specific types of viscosity
correlations from oil composition,
311–315, 317b
Little-Kennedy’s correlation, 314–315
Lohrenz-Bray-Clark’s (LBC)
correlation, 312–314
of gas, 223–224, 227b
slim tube test, 443–444
water, 324–326
Volatile crude oil, 39
liquid-shrinkage curve for, 41f
p-T diagram for, 40f
Volume average boiling point (VABP),
122–123, 134–135
Volume fraction, 192
Volume translation method, SRK EOS,
531–534, 533b
Index 607
W
Watansiri-Owens-Starling correlations,
99–100
petroleum fraction, 115
Water-oil contact (WOC), 56
Watson characterization factor, and true
boiling point, 80–82, 84
Weight average boiling point (WABP),
122–123
Weight fraction, 192–193
Well-defined petroleum fractions, 73–76
Wet gas
reservoirs, 47–50, 47f, 50b
specific gravity, 230–234, 232b
Whitson’s splitting method, 155–159,
158b
Wichert-Aziz’s correction method,
203–205
Willman and Teja correlations, 103–104
petroleum fraction, 116
Wilson’s correlation, 476–477
Winn and Sim-Daubert correlation, 99
petroleum fraction, 114
Winn’s method, 486–487
WOC. See Water-oil contact (WOC)
Y
Yellig and Metcalfe correlation, MMP, 449,
449b
Y-function
extended, 366, 368b, 370f
traditional, 363, 364f, 364b
Yuan’s correlation, MMP, 451–452
Z
Z-factor
nonhydrocarbon components effect,
196–203
selection, 527–529
Standing-Katz chart, 212