Phase Behavior [Whitson,2000]

PHASE BEHAVIOR
CURTIS H. WHITSON AND MICHAEL R. BRULÉ
MONOGRAPH VOLUME 20
SPE
HENRY L. DOHERTY SERIES
PHASE BEHAVIOR
Curtis H. Whitson
Professor of Petroleum Engineering
U. Trondheim, NTH
and
Founder
PERA a/s
and
Michael R. Brulé
President and Chief Executive Officer
Technomation Systems Inc.
First Printing
Henry L. Doherty Memorial Fund of AIME
Society of Petroleum Engineers Inc.
Richardson, Texas
2000
i
SPE Monograph Series
The Monograph Series of the Society of Petroleum Engineers was established in 1965 by
action of the SPE Board of Directors. The Series is intended to provide authoritative,
up-to-date treatment of the fundamental principles and state of the art in selected fields of
technology. The Series is directed by the Society’s Monograph Committee. A committee
member designated as Monograph Editor provides technical evaluation with the aid of the
Review Committee. Below is a listing of those who have been most closely involved with the
preparation of this monograph.
Monograph Review Committee
Peter G. Christman, Shell Intl. E&P B.V., Monograph Editor
David F. Bergman, Amoco Production Co.
W. David Constant, Louisiana State U.
A.S. Cullick, Landmark Graphics Corp.
Gustave A. Mistrot III, Mistrot & Assocs.
Teresa G. Monger-McClure, Marathon Oil Co.
Franklin M. Orr Jr., Stanford U.
Robert R. Wood, Shell Intl. E&P B.V.
Aaron A. Zick, Zick Technologies
Monograph Committee (2000)
Mary Jane Wilson, WZI, Chairperson
Jesse Frederick, WZI
Russell T. Johns, U. of Texas, Austin
Medhat Kamal, Arco E&P Technology
Mark Miller, U. of Texas, Austin
Ken Newman, CTES L.S.
Dan O’Meara Jr., U. of Oklahoma
David Underdown, Chevron Production Technology Co.
Acknowledgments
Many people contributed to the production of this monograph. It is first and foremost the
product of the authors. I am sure that the effort was more significant than either author had
anticipated, but they persevered and should be proud of the book they wrote. I want to thank
R.R. Wood, who initiated the project, chose the authors, and formed a distinguished review
committee. I succeeded Rob in 1990 and coordinated the efforts of A.A. Zick, G.A. Mistrot,
T.G. Monger-McClure, D.F. Bergman, A.S. Cullick, and W.D. Constant, who reviewed every
chapter from their own unique perspectives. F.M. Orr contributed significant reviews on
selected chapters. It was a pleasure to work with such a talented group of engineers. I am
confident that we kept the focus of the monograph on use by the working engineer. The book
is meant to serve as a reference. As such, I hope it will be a valuable addition to the library of
every petroleum engineer working in phase behavior.
Peter G. Christman
Copyright 2000 by the Society of Petroleum Engineers Inc.
Printed in the United States of America. All rights reserved.
This book, or any part thereof, cannot be reproduced in any
form without written consent of the publisher.
ISBN 1-55563-087-1
ii
Dedication
To Morris Muskat, a pioneer in the field of reservoir engineering,
who made important contributions in the area of phase behavior.
iii
Acknowledgments
We thank the SPE editorial staff, the Monograph Review Committee members, our
professional colleagues, our students, and the petroleum industry at large for valuable
assistance and input toward the completion of this monograph. In particular, we thank
the two technical editors, Rob R. Wood and Peter G. Christman, and our staff editor,
Flora Cohen.
We have been strongly influenced by the pioneering phase-behavior research of
Donald Katz, Muz Standing, and Ken Starling and the many others who have made
invaluable contributions to the field. The scientific contributions of these engineers and
their coworkers, together with contributions from the community of petroleum and
chemical engineers, have laid the foundation for the material selected, synthesized, and
presented in this monograph. We hope that all contributors have been correctly cited
and given due credit for their contributions.
We are confident that the material contained herein is valuable for dealing with
engineering problems affected by phase behavior, both today and in the future. We use
the technology presented in this monograph daily to solve problems for the industry
and as the basis of our long-term research.
Curtis H. Whitson
Michael R. Brulé
iv
Table of Contents
Chapter 1—Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
1.2
1.3
1.4
1.5
Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scope and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nomenclature and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2
2
Chapter 2—Volumetric and Phase Behavior of Oil and Gas Systems . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1
2.2
2.3
2.4
2.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Reservoir-Fluid Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Phase Diagrams for Simple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Retrograde Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Classification of Oilfield Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 3—Gas and Oil Properties and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1
3.2
3.3
3.4
3.5
3.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review of Properties, Nomenclature, and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oil Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IFT and Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K-Value Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
18
22
29
38
40
Chapter 4—Equation-of-State Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cubic EOS’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Phase Flash Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Saturation-Pressure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equilibrium in a Gravity Field: Compositional Gradients . . . . . . . . . . . . . . . . . . . .
Matching an EOS to Measured Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
47
52
55
62
63
65
Chapter 5—Heptanes-Plus Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Molar Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inspection-Properties Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Critical-Properties Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recommended C7) Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grouping and Averaging Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
68
70
77
80
83
83
Chapter 6—Conventional PVT Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1
6.2
6.3
6.4
6.5
6.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wellstream Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multistage-Separator Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constant Composition Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differential Liberation Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constant Volume Depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
88
88
91
93
95
97
Chapter 7—Black-Oil PVT Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1
7.2
7.3
7.4
7.5
7.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traditional Black-Oil Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modified Black-Oil (MBO) Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Applications of MBO Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Partial-Density Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modifications for Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
109
110
116
118
119
Chapter 8—Gas-Injection Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.1
8.2
8.3
8.4
8.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Miscibility and Related Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lean-Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Enriched-Gas Miscible Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
122
128
131
135
Chapter 9—Water/Hydrocarbon Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.1
9.2
9.3
9.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EOS Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
142
142
150
151
Appendix A—Property Tables and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Appendix B—Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Appendix C—Equation-of-State Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Appendix D—Understanding Laboratory Oil PVT Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
vi
Chapter 1
Introduction
1.1 Purpose
This monograph covers a wide range of topics related to phase behavior. Phase behavior is the behavior of vapor, liquid, and solids as a
function of pressure, temperature, and composition. In this monograph, “vapor” is used interchangeably with “gas,” “liquid” refers to
oil and water, and “solids” include hydrates, asphaltenes, and wax.
We are concerned primarily with the volumetric behavior and
composition of phases, including density and isothermal compressibility, and component distribution in each phase. For a mixture with
a known composition, we need to determine the vapor/liquid equilibrium (VLE), including saturation conditions over a wide range of
temperatures and pressures. Transport properties are also needed for
flow calculations (e.g., viscosity in Darcy’s law and molecular diffusion coefficients in Fick’s law).
Phase behavior has many applications in petroleum engineering.
The reservoir engineer relies on pressure/volume/temperature
(PVT) relations to calculate oil and gas reserves, production forecasts, and the efficiency of enhanced oil recovery (EOR) methods.
Most reservoir calculations require PVT properties at reservoir temperature. Production engineers use phase behavior data for surface
separator design and to calculate flow in pipe, where such calculations are made over a range of temperatures from surface to reservoir conditions. Petroleum engineering calculations generally are
made at temperatures from 60 to 350°F and at pressures from about
15 to 15,000 psia.
1.2 Historical Review
Gibbs1,2 and van der Waals3 stated the basic theory of phase behavior in the the late 1800’s and early 1900’s. They formulated the concepts and mathematical relations necessary to describe phase behavior. Katz and Rzasa4 published a comprehensive review of phase
behavior literature from before 1860 to 1945. Muckleroy5 published a bibliography covering 1946 to 1960, and other bibliographies exist for work in phase behavior over the past 30 years.*
Experimental data on reservoir fluids were scarce before the late
1930’s, when Katz et al.4,6-39 at the U. of Michigan, Sage and colleagues40-73 at the California Inst. of Technology, and Eilerts et
al.74-78 at the U.S. Bureau of Mines (USBM) began significant research programs. For 10 years, during the 1950’s, a large amount of
high-quality experimental data was compiled on reservoir fluids.
During the past 40 years, most phase behavior data have been measured by commercial service laboratories and major oil companies.
*SPE Reprint Series No. 15 Phase Behavior gives a recent update of earlier bibliographies.
INTRODUCTION
These data have been used for engineering studies of primary depletion, waterflood evaluation, and gas-injection studies.
Correlation of phase behavior data began in the 1940’s, with notable work by Standing and Katz,17,18 Bicher and Katz,25 Standing,79,80 Eilerts,78 Kennedy and colleagues,81-85 and others. Although equations of state (EOS’s) had been available for more than
50 years (since van der Waals3 published the first cubic EOS in
1873) it was necessary to rely mostly on tables, figures, and chart
correlations, such as nomograms. These correlations provided reliable property estimates for engineering calculations through the
1970’s. Subsequently, empirical equations representing these
graphical correlations were developed and programmed for calculators and computer applications.
With the introduction of electronic computers in the late 1940’s,
application of complicated thermodynamic models became possible. In 1949, Muskat and McDowell86 published one of the earliest
papers in the SPE/AIME Transactions on applications of this new
generation of computers. These authors solved the two-phase flash
calculation with fixed K values for multistage separator design.
Not until Redlich and Kwong87 introduced their classic cubic EOS
in 1949 was it generally accepted that volumetric properties could be
accurately predicted by use of theoretical models. Considerable advances were made in the 1950’s toward correlating volumetric properties of pure components with multiconstant EOS’s.88 By the early
1960’s, there was considerable activity in the application of sophisticated thermodynamic models to multicomponent VLE calculations,
although most of this activity was in process engineering.
In the 1960’s and 1970’s, Starling,89 Soave,90 and Peng and Robinson91 proposed several important modifications of existing
EOS’s. Petroleum engineering EOS applications started seriously in
the late 1970’s and early 1980’s, when EOS-based compositional
reservoir simulators were introduced.92,93 At the same time, several
methods were proposed for EOS fluid characterization of reservoir
fluids, in particular for heptanes and heavier components.94-96 Finally, in the 1980’s, supercomputers appeared and special solution
techniques were developed for compositional simulators,93 thereby
making possible full-field, EOS compositional simulation.
Today’s standard treatment of phase behavior in reservoir simulation is still based on formation volume factors (FVF’s) and surface
gas/oil ratios (GOR’s). This will probably remain true for many
years, in part because many problems can be solved adequately with
a simple PVT formulation and in part because many petroleum engineers are not familiar with more complicated EOS models. This
monograph treats both simple and complicated methods for estimating phase behavior. We suspect that the more complicated PVT
1
models will gradually become the standard, eventually replacing
many of the simpler correlations.
with Chap. 6, Conventional PVT Experiments, and are included as
a supplement to the discussion in that chapter.
1.3 Objectives
This monograph provides the petroleum engineer with a tool to
solve problems that require a description of phase behavior and specific PVT properties. These problems include calculating the FVF
to determine original oil and gas in place and GOR’s, design of “optimal” surface separator conditions, and description of near-critical
phase behavior resulting from the injection of a gas that develops
miscibility with a reservoir oil.
Because of the dramatic evolution in computer technology, petroleum engineers can now study such phenomena as developed miscibility,97 compositional gradients,98 and near-critical phase behavior99 with more sophisticated models. The quality of these models
is sensitive to the EOS fluid characterizations. This monograph
presents phase behavior concepts used in petroleum engineering
and state-of-the-art technology for more complex phase behavior
models, such as cubic EOS’s. We hope the monograph will serve its
purpose for many years to come.
1.5 Nomenclature and Units
SPE-approved symbols are used throughout the monograph. Some
of these symbols will be unfamiliar even to the seasoned SPE reader
(as they are confusing even to the authors!).
One of the most significant changes in nomenclature that we have
introduced is the use of different subscripts for surface and reservoir
phases. Traditionally, o, g, and w are used for oil, gas, and water at
reservoir and at surface conditions, a practice that was difficult to
follow in Chaps. 6 and 7. We have therefore introduced the subscripts o, g, and w for surface phases, retaining o, g, and w for reservoir phases. A better solution to this problem was not apparent,
particularly because some quantities required subscripts for both
reservoir and surface phases—e.g., the gravity of surface gas produced from reservoir oil (written g go in this monograph). To avoid
confusion in the property correlations in Chap. 3, gas and oil specific gravities are still written g g and g o (instead of g g and g o) because
specific gravity is always reported at standard conditions.
We use customary oilfield units (psi, ft3 and bbl, °F and °R, and
lbm). The oilfield unit for mass is pound, written “lbm” to avoid
confusion with pounds force, written “lbf.” Pounds force is never
used explicitly in this monograph. Conversion factors to SI units are
included at the end of each chapter, and Appendix A provides a comprehensive discussion of units and unit conversion tables.
Standard conditions are defined in this monograph as 60°F and
14.7 psia. We recognize that standard pressure varies geographically and the calculation of surface gas volumes in some areas must use
the locally defined value for standard pressure. To accomplish this,
some constants given in the monograph must be recalculated.
1.4 Scope and Organization
The scope of this monograph is limited mostly to two-phase, gas/oil
phase behavior. Multiphase and vapor/solid phase behavior are discussed only briefly. Phase behavior related to chemical (surfactant
and polymer) flooding is not covered because a detailed description
would necessarily reduce coverage of problems more commonly
encountered in petroleum engineering. We also think that this subject should be covered in a separate publication specifically within
the context of chemical flooding technology.
Chaps. 2 and 3 review the “nuts and bolts” of phase behavior principles, relevant PVT properties, and methods to solve most petroleum engineering problems. Useful correlations are presented for
the most common PVT properties.
Chap. 4 discusses cubic EOS’s, including the two-phase flash,
saturation-pressure, and phase-stability calculations and numerical
methods used to solve these VLE calculations. The problem of “tuning” an EOS to match measured PVT data is also addressed.
Chap. 5 describes the characterization of heavy components
(“heptanes plus”) in reservoir fluids for EOS applications. Experimental and mathematical methods describing the heptanes-plus
material are presented, including splitting C7+ into petroleum fractions, estimating critical properties, and grouping an extended fluid
characterization into a reduced number of pseudocomponents.
Chap. 6 covers laboratory measurements of PVT properties and
their application in engineering calculations. The standard PVT
studies include constant composition (mass) expansion, differential
liberation, constant-volume depletion, and the multistage separator
test. Separator and bottomhole sampling methods for establishing
wellstream composition are also discussed.
Chap. 7 describes the black-oil PVT formulation and its extension
to gas condensates, volatile oils, and gas-injection processes. The
black-oil PVT formulation uses FVF’s and solution gas/oil ratios to
relate phase and volumetric properties at reservoir conditions to surface volumes.
Chap. 8 reviews the importance of phase behavior to gas-injection EOR processes. These processes include vaporizing, condensing, and the combined condensing/vaporizing miscible-drive mechanisms. CO2 immiscible and miscible drives and nitrogen injection
are also reviewed.
Chap. 9 covers the behavior of water/hydrocarbon phase and volumetric behavior, including mutual solubilities, water FVF and
compressibility, and the treatment of hydrates.
Appendix A gives tables of component properties, various other
useful tables, and unit conversion factors. Appendix B includes
more than 20 worked examples that range from simple calculations
of ideal gas properties to detailed step-by-step EOS calculations for
a ternary system. Appendix C gives two detailed EOS fluid characterizations, one for a gas condensate and another for a slightly volatile oil. Appendix D is a set of notes by M.B. Standing on understanding laboratory-oil PVT reports. These notes clearly belong
2
References
1. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press,
New Haven, Connecticut (1948) 1.
2. Gibbs, J.W.: On the Equilibrium of Heterogeneous Substances, C.
Works (ed.), Yale U. Press, New Haven, Connecticut (1928) Chap. 1.
3. van der Waals, J.D.: Continuity of the Gaseous and Liquid State of Matter (1873).
4. Katz, D.L. and Rzasa, M.J.: Biblography of Hydrocarbons Under Pressure 1860–1946, University Microfilms Inc. (1946).
5. Muckleroy, J.A.: Biblography on Hydrocarbons, 1946–1960, Gas Processors Assn. (1962).
6. Katz, D.L. and Hachmuth, K.K.: “Vaporization Equilibrium Constants in
a Crude-Oil Natural Gas System,” Ind. & Eng. Chem. (1937) 29, 1072.
7. Katz, D.L.: “Application of Vaporization Equilibrium Constants to Production Engineering Problems,” Trans., AIME (1938) 127, 159.
8. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixture
of Natural Gas and Natural Gasoline Near the Critical Conditions,”
Trans., AIME (1939) 136, 106.
9. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenomena in Oil and Gas Production,” Trans., AIME (1939) 132, 103.
10. Katz, D.L. and Saltman, W.: “Surface Tension of Hydrocarbons,” Ind.
& Eng. Chem. (January 1939) 31, 91.
11. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng.
Chem. (June 1940) 32, No. 6, 817.
12. Wilcox, W.I., Carson, D.B., and Katz, D.L.: “Natural Gas Hydrates,”
Ind. & Eng. Chem. (1941) 33, No. 5, 662.
13. Katz, D.L.: “High Pressure Gas Measurement,” Refiner and Natural
Gasoline Manufacturer (June 1942).
14. Carson, D.B. and Katz, D.L.: “Natural Gas Hydrates,” Trans., AIME
(1942) 146, 150.
15. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile Hydrocarbon
Mixtures,” Trans., AIChE (1942) 38, 995.
16. Katz, D.L.: “Possibilities of Secondary Recovery for the Oklahoma
City Wilcox Sand,” Trans., AIME (1942) 146, 28.
17. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,
AIME (1942) 146, 140.
18. Standing, M.B. and Katz, D.L.: “Density of Crude Oils Saturated with
Natural Gas,” Trans., AIME (1942) 146, 159.
19. Katz, D.L.: “Prediction of the Shrinkage of Crude Oils,” Drill. & Prod.
Prac. (1942) 137.
20. Matthews, T.A., Roland, C.H., and Katz, D.L.: “High Pressure Gas Measurement,” Proc., Natural Gas Assn. of America (NGAA) (1942) 41.
PHASE BEHAVIOR MONOGRAPH
21. Weinaug, C.F. and Katz, D.L.: “Surface Tension of Methane-Propane
Mixtures,” Ind. & Eng. Chem. (1943) 35, No. 2, 239.
22. Bicher, L.B. Jr. and Katz, D.L.: “Viscosities of the Methane-Propane
System,” Ind. & Eng. Chem. (1943) 35, 754.
23. Katz, D.L., Monroe, R.R., and Trainer, R.P.: “Surface Tension of Crude
Oils Containing Dissolved Gases,” Trans., AIME (1943) 155, 624.
24. Standing, M.B. and Katz, D.L.: “Vapor/Liquid Equilibria of Natural
Gas/Crude Oil Systems,” Trans., AIME (1944) 155, 232.
25. Bicher, L.B. Jr. and Katz, D.L.: “Viscosity of Natural Gases,” Trans.,
AIME (1944) 155, 246.
26. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on Sampling
Two-Phase Gas Streams from High Pressure Condensate Wells,” Proc.,
NGAA (September 1945).
27. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. &
Eng. Chem. (February 1945) 37, 195.
28. Katz, D.L.: “Prediction of Conditions for Hydrate Formation in Natural
Gases,” Trans., AIME (1945) 160, 140.
29. Poettman, F.H. and Katz, D.L.: “CO2 in a Natural Gas Condensate System,” Ind. & Eng. Chem. (1946) 38, 530.
30. Brown, G.G. et al..: Natural Gasoline and the Volatile Hydrocarbons,
NGAA, Tulsa, Oklahoma (1948) 24–32.
31. Kobayashi, R. and Katz, D.L.: “Methane-n-Butane-Water System in Twoand Three-Phase Regions,” Ind. & Eng. Chem. (1948) 40, No. 5, 853.
32. Unruh, C.H. and Katz, D.L.: “Gas Hydrates of Carbon Dioxide/Methane Mixtures,” Trans., AIME (1949)186, 83.
33. Rzasa, M.J. and Katz, D.L.: “The Coexistence of Liquid and Vapor
Phases at Pressures Above 10,000 psi,” Trans., AIME (1950) 189, 119.
34. Kobayashi, R. et al.: “Gas Hydrates Formation with Brine and Ethanol
Solutions,” Proc., 30th Annual Convention of NGAA (1951).
35. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,”
Amer. Soc. Petr. Geology Bulletin (February 1952) 36, No. 2, 342.
36. Katz, D.L.: “Possibility of Cycling Deep Depleted Oil Reservoirs After
Compression to a Single Phase,” Trans., AIME (1952) 195, 175.
37. Kobayashi, R. and Katz, D.L.: “Vapor-Liquid Equilibria for Binary Hydrocarbon-Water Systems,” Ind. & Eng. Chem. (1953) 45, No. 2, 440.
38. Donnelly, H.C. and Katz, D.L.: “Phase Equilibria in the Carbon Dioxide-Methane System,” Ind. & Eng. Chem. (1954) 46, 511.
39. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-Hill
Book Co. Inc., New York City (1959).
40. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric Behavior of
Hydrogen Sulfide,” Ind. & Eng. Chem. (1950) 42, 140.
41. Sage, B.H. and Olds, R.H.: “Volumetric Behavior of Oil and Gas from
Several San Joaquin Valley Fields,” Trans., AIME (1947) 170, 156.
42. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Partial Volumetric Behavior
of the Methane-Carbon Dioxide System,” Fundamental Research on
Occurrence and Recovery of Petroleum, API, Dallas (1943) 44.
43. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems—Volumetric Behavior of Ethane-Carbon Dioxide System,” Ind. & Eng.
Chem. (1945) 37, 688.
44. Sage, B.H. and Lacey, W.N.: “Partial Volumetric Behavior of the
Lighter Paraffin Hydrocarbons in the Gas Phase,” Drill. & Prod.
Prac. (1939) 641.
45. Sage, B.H. and Lacey, W.N.: “Thermodynamic Properties of the Light
Paraffin Hydrocarbons and Nitrogen,” API Research Project 37, monograph, API, New York City (1950).
46. Sage, B.H., Hicks, B.L., and Lacey, W.N.: “Partial Volumetric Behavior of the Lighter Hydrocarbons in the Liquid Phase,” Drill. & Prod.
Prac. (1938) 402.
47. Sage, B.H. and Lacey, W.N.: “Apparatus for Determination of Volumetric Behavior of Fluids,” Trans., AIME (1948) 174, 102.
48. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase
Behavior of the Methane-Propane Systems,” Ind. & Eng. Chem.
(1950) 42, 534.
49. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Phase Equilibria in Hydrocarbon Systems: Methane-Propane Systems,” Ind. & Eng. Chem.
(1934) 26, 214.
50. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the Methane-n Butane-Decane System,” Ind. & Eng. Chem.
(1951) 43, 1436.
51. Sage, B.H. and Lacey, W.W.: Volumetric and Phase Behavior of Hydrocarbons, Gulf Publishing Co., Houston (1949).
52. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems,” Ind. & Eng. Chem. (June 1951) 43, 1436.
53. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Behavior in Hydrocarbon System,” Ind. & Eng. Chem. (1951) 43, 2515.
54. Olds, R.H. et al.: “Phase Equilibria in Hydrocarbon Systems. The Butane-Carbon Dioxide System,” Ind. & Eng. Chem. (1949) 41, 475.
INTRODUCTION
55. Reamer, H.H. and Sage, B.H.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the n-Decane-CO2 System,”
J. Chem. Eng. Data (1963) 8, 508.
56. Reamer, H.H., Fiskin, J.M., and Sage, B.H.: “Phase Equilibria in Hydrocarbon Systems: Phase Behavior in the Methane-n-Butane-Decane
System at 160°F,” Ind. & Eng. Chem. (December 1949) 41, 2871.
57. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the Methanen-Heptane System,” Ind. & Eng. Chem. (1956) 1, 29.
58. Sage, B.H., Webster, D.C., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems,” Ind. & Eng. Chem. (1936) 28, 1045.
59. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the MethaneCyclohexane System,” Ind. & Eng. Chem. (1958) 3, 240.
60. Sage, B.H. and Lacey, W.N.: “Effect of Pressure Upon Viscosity of
Methane and Two Natural Gases,” Trans., AIME (1938) 127, 118.
61. Sage, B.H., Yale, W.D., and Lacey, W.N.: “Effect of Pressure on Viscosity of n-Butane and i-Butane,” Ind. & Eng. Chem. (1939) 31, 223.
62. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in
Static Columns of Hydrocarbon Fluids,” Trans., AIME (1939) 132, 120.
63. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the Methane-n-Butane-Decane System,” Ind. & Eng. Chem.
(1947) 39, 77.
64. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the Methane-n-Butane-Decane System,” Ind. & Eng. Chem.
(1952) 44, 1671.
65. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Behavior of Hydrocarbon Mixtures Illustrated by a Simple Case,” API Bulletin (1932) 212, 119.
66. Sage, B.H.: Thermodynamics of Multicomponent Systems, Reinhold
Publishing Co. (1965)
67. Sage, B.H. and Lacey, W.N.: Volumetric and Pha.se Behavior of Hydrocarbons, Stanford Press, Stanford, Connecticut (1939).
68. Sage, B.H. and Reamer, R.H.: “Volumetric Behavior of Oil and Gas
From the Rio Bravo Field,” Trans., AIME (1941) 142, 179.
69. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems. Composition of the Dew-Point Gas of the MethaneWater System,” Ind. & Eng. Chem. (1942) 34, No. 10, 1223.
70. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Composition of the Dew-Point Gas in the Ethane-Water System,” Ind. &
Eng. Chem. (1943) 35, No. 7, 790.
71. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Compositions of the Coexisting Phases of n-Butane-Water System in the
Three-Phase Region,” Ind. & Eng. Chem. (1944) 36, No. 4, 381.
72. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems. n-Butane-Water System in the Two-Phase Region,” Ind. & Eng. Chem. (1952) 44, No. 3, 609.
73. Sage, B.H. and Lacey, W.N.: “Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide,” API Research Project
37, monograph, API, New York City (1955).
74. Eilerts, C.K.: “The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 January 1947) 63.
75. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Reserve
Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 February
1947) 63.
76. Eilerts, C.K., Carlson, H.A., and Mullen, N.B.: “Effect of Added Nitrogen on Compressibility of Natural Gas,” World Oil (June 1948) 129.
77. Eilerts, C.K. et al.: “Phase Relations of a Gas-Condensate Fluid at
Low Temperatures, Including the Critical State,” Pet. Eng. (February
1948) 19, 154.
78. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph
10, USBM, American Gas Assn., New York City (1957) I and II.
79. Standing, M.B.: “Vapor-Liquid Equilibria of Natural Gas-Crude Oil
Systems,” PhD dissertation, U. of Michigan, Ann Arbor, MI (1941).
80. Standing, M.B.: “A Pressure-Volume-Temperature Correlation for Mixtures of California Oils and Gases,” Drill. & Prod. Prac. (1947) 275.
81. Alani, G.H. and Kennedy, H.T.: “Volumes of Liquid Hydrocarbons at
High Temperatures and Pressures,” Trans., AIME (1960) 219, 288.
82. Kennedy, G.C.: “Pressure-Volume-Temperature Relations in CO2 at Elevated Temperatures and Pressures,” Amer. J. Sci. (April 1954) 252, 225.
83. Kennedy, H.T. and Bhagia, N.S.: “An EOS for Condensate Fluids,” JPT
(September 1969) 379.
84. Little, J.E. and Kennedy, H.T.: “A Correlation of the Viscosity of Hydrocarbon Systems with Pressure, Temperature and Composition,”
SPEJ (June 1968) 157; Trans., AIME, 243.
85. Nemeth, L.K. and Kennedy, H.T.: “A Correlation of Dewpoint Pressure
With Fluid Composition and Temperature,” SPEJ (June 1967) 99;
Trans., AIME (1967) 240.
3
86. Muskat, M. and McDowell, J.M.: “An Electrical Computer for Solving
Phase Equilibrium Problems,” Trans., AIME (1949) 186, 291.
87. Redlich, O. and Kwong, J.N.S.: “On the Thermodynamics of Solutions,
V: An Equation of State. Fugacities of Gaseous Solutions,” Chem. Rev.
(1949) 44, 233.
88. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equation
for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures, I. Methane, Ethane, Propane, and n-Butane,” J. Chem. Phy.
(1940) 8, 334.
89. Starling, K.E.: “A New Approach for Determining Equation-of-State
Parameters Using Phase Equilibria Data,” SPEJ (December 1966) 363;
Trans., AIME, 237.
90. Soave, G.: “Equilibrium Constants from a Modified Redlich-Kwong
EOS,” Chem. Eng. Sci. (1972) 27, No. 6, 1197.
91. Peng, D.Y. and Robinson, D.B.: “A New-Constant EOS,” Ind. & Eng.
Chem. Fund. (1976) 15, No. 1, 59.
92. Coats, K.H.: “An EOS Compositional Model,” SPEJ (October 1980)
363; Trans., AIME, 269.
93. Young, L.C. and Stephenson, R.E.: “A Generalized Compositional Approach for Reservoir Simulation,” SPEJ (October 1983) 727; Trans.,
AIME, 275.
94. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of State in Engineering and Re-
4
search, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry
Series, American Chemical Soc. (1978) 182, 386–439.
95. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ
(August 1983) 683; Trans., AIME, 275.
96. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization
of Gas Condensate Mixtures,” C7 Fraction Characterization, L.G.
Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1.
97. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the
Displacement of Oil by Enriched Gases,” paper SPE 15493 presented
at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October.
98. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Column Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, 21–24 September.
99. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,”
JPT (October 1985) 1870.
SI Metric Conversion Factors
°F (°F*32)/1.8
+°C
psi 6.894 757
E)00 +kPa
PHASE BEHAVIOR MONOGRAPH
Chapter 2
Volumetric and Phase Behavior
of Oil and Gas Systems
2.1 Introduction
Petroleum reservoir fluids are naturally occurring mixtures of natural gas and crude oil that exist in the reservoir at elevated temperatures and pressures. Reservoir-fluid compositions typically include
hundreds or thousands of hydrocarbons and a few nonhydrocarbons, such as nitrogen, CO2, and hydrogen sulfide. The physical
properties of these mixtures depend primarily on composition and
temperature and pressure conditions. Reservoir temperature can
usually be assumed to be constant in a given reservoir or to be a weak
function of depth. As oil and gas are produced, reservoir pressure
decreases and the remaining hydrocarbon mixtures change in composition, volumetric properties, and phase behavior. Gas injection
also may change reservoir-fluid composition and properties. Katz
and Williams1 give an excellent review of reservoir fluids and their
general behavior under different operating conditions.
The hydrocarbon phases and connate water sharing the pore volume (PV) in a reservoir are in thermodynamic equilibrium. Strictly
speaking, hydrocarbons and water should be treated simultaneously
in phase-behavior calculations. At typical reservoir conditions, the effect of connate water on hydrocarbon phase behavior can usually be
neglected. Water can, however, affect the total-system phase behavior
(for example, when hydrates form from natural-gas/water mixtures).
This chapter covers only two-phase, vapor/liquid phase behavior.
Chap. 8 briefly covers three- and four-phase systems (vapor/liquid/
liquid and vapor/liquid/liquid/solids) for low-temperature CO2/oil
and rich-gas/oil mixtures, and Chap. 9 gives the behavior of vapor
and solids related to hydrates.
Sec. 2.1 introduces the composition of petroleum reservoir fluids
and emphasizes their chemical complexity. Because reservoir fluids
are made up of many components, a detailed quantitative analysis
is difficult to perform. Organic compounds found in reservoir fluids
are expressed by a general formula that classifies even high-molecular-weight compounds containing sulfur, nitrogen, and oxygen.
This chapter also gives a historical review of the American Petroleum Inst. (API) -supported projects that defined many of the compounds known today.
Simple one- and two-component phase behavior can be helpful in
describing the effects of pressure, temperature, and composition on
the reservoir-fluid phase behavior. Sec. 2.2 presents pressure/temperature ( p-T), pressure/volume ( p-V), and pressure/composition ( p-x)
phase diagrams of simple systems. The behavior of these idealized
systems is qualitatively similar to the behavior of complex reservoir
fluids, as Sec. 2.3 shows.
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Retrograde condensation is perhaps the most unusual phase behavior that petroleum reservoir fluids exhibit.* Sec. 2.4 discusses
the definition of retrograde condensation and the effect of retrograde condensation on the behavior of gas-condensate reservoirs.
Petroleum reservoir fluids can be divided into five general categories, in increasing order of chemical complexity: dry gas, wet gas,
gas condensate, volatile oil, and black oil. However, the phase behaviors of gas condensates and volatile oils are considerably more
complex than those of black oils. The component distribution in a
reservoir fluid, not simply the number of components, determines
how close a fluid is to a critical state. Complex phase behavior is typically associated with systems that are “near critical”: systems that
usually contain 10 to 15 mol% of components that are heptanes and
heavier (C7+).
Since the early 1930’s, experimental data have been measured onfluids of each type listed above. Sec. 2.5 defines each fluid type by
its p-T diagram. Also, general characteristics of reservoir fluids are
summarized in terms of composition and surface properties, such as
GOR and stock-tank-oil gravity.
2.2 ReservoirĆFluid Composition
The nature and composition of a reservoir fluid depends somewhat
on the depositional environment of the formation from which the
fluid is produced. Geologic maturation also influences reservoir-fluid
composition. Several theories offer explanations for the origin and
formation of petroleum over geologic time; no single theory suffices
to explain how oil and gas were formed in all reservoirs. One theory
portrays reservoirs as giant high-temperature/high-pressure reactors
with catalytic rock surfaces that slowly convert deposited organic
matter into oil and gas. Other theories hypothesize that oil and gas
were formed from bacterial action on deposited organic matter. Other
investigators maintain that oil and gas may be formed in the same
geologic formation but that each fluid migrates to “traps” at different
elevations because of fluid-density differences and gravity forces.
Crude oil and natural gas are composed of many chemical compounds with a wide range of molecular weights. Some estimates2-4
suggest that perhaps 3,000 organic compounds can exist in a single
*Historically, retrograde condensation has been considered the most complex phase-behavior phenomenon observed by reservoir fluids. Perhaps equally intriguing are the phenomena
of strong compositional gradients, the condensing/vaporizing miscible mechanism (Chap. 8),
asphaltene precipitation, and low-temperature, multiphase CO2 behavior.
1
TABLE 2.1—COMPOSITION AND PROPERTIES OF SEVERAL RESERVOIR FLUIDS
Composition (mol%)
Gas
Near-Critical
Component
Dry Gas
Wet Gas
Condensate
Oil
Volatile Oil
Black Oil
CO2
0.10
1.41
2.37
1.30
0.93
0.02
N2
2.07
0.25
0.31
0.56
0.21
0.34
C1
86.12
92.46
73.19
69.44
58.77
34.62
C2
5.91
3.18
7.80
7.88
7.57
4.11
C3
3.58
1.01
3.55
4.26
4.09
1.01
i-C4
1.72
n-C4
i-C5
0.50
0.28
0.71
0.89
0.91
0.76
0.24
1.45
2.14
2.09
0.49
0.13
0.64
0.90
0.77
0.43
n-C5
0.08
0.68
1.13
1.15
0.21
C6(s)
0.14
1.09
1.46
1.75
1.61
C7 +
0.82
8.21
10.04
21.76
56.40
Properties
MC
g
7)
C 7)
K wC
7
130
184
219
228
274
0.763
0.816
0.839
0.858
0.920
12.00
11.95
11.98
11.83
11.47
1,490
300
38
24
GOR, scf/STB
∞
105,000
5,450
3,650
OGR, STB/MMscf
0
10
180
275
57
49
45
gAPI
gg
0.61
0.70
0.71
0.70
0.63
psat, psia
3,430
6.560
7,015
5,420
2,810
0.0051
0.0039
2.78
1.73
1.16
9.61
26.7
30.7
38.2
51.4
Bsat, ft3/scf or bbl/STB
ò sat, lbmńft
3
reservoir fluid. The lighter and simpler compounds are produced as
natural gas after surface separation, whereas the heavier and more
complex compounds form crude oil at stock-tank conditions. Table
2.1 gives typical oilfield molar compositions for reservoir mixtures.
The heavier components are usually lumped into a “plus” fraction
instead of being identified individually. Chap. 5 discusses methods
of quantifying and characterizing the components that make up the
plus fraction—usually heptanes-plus.
Natural gas is composed mainly of low-molecular-weight alkanes (methane through butanes), CO2, hydrogen sulfide, nitrogen,
and, in some cases, lesser quantities of helium, hydrogen, CO, and
carbonyl sulfide.5 Most crude oils are composed mainly of hydrocarbons (hydrogen and carbon compounds). The broad spectrum of
organic compounds found in petroleum during the formation of
crude oil also includes sulfur, nitrogen, oxygen, and trace metals.
Tars and asphalts are solid or semisolid mixtures that include bitumen, pitch, waxes, and resins. These high-molecular-weight complex colloidal suspensions exhibit non-Newtonian rheology.
The temperature and pressure gradients in a formation may cause
reservoir-fluid properties to vary as a function of depth. “Compositional grading” is the continual change of composition as a function
of depth.6-8 In compositional grading, reservoir temperature may be
near the critical temperature of reservoir fluid(s) at certain depths in
the reservoir. Physically, the thermodynamic forces of individual
components in a near-critical mixture are of the same order of magnitude as gravity forces that tend to separate the lighter from the
heavier components. The result can be a transition from an undersaturated gas condensate at the highest elevation to an undersaturated
oil at the lowest elevation, with or without a visible phase transition
from gas to oil (gas/oil contact).
In petroleum refining, crude oil is often categorized according to
its base and the hydrocarbon series (paraffin, naphthene, or aromatic) it contains in the highest concentration. Figs. 2.1 and 2.29 illustrate the types and relative amounts of hydrocarbon series that can
be found in typical petroleum-refinery products. Nelson3 gives a
full account of basic hydrocarbon chemistry and test methods that
2
have been used for many years to determine petroleum composition
and inspection properties for refining purposes. The more common
test methods include paraffin, naphthene, and aromatic; saturates,
aromatics, resins, and asphaltenes; and Strieter (asphaltenes, resins,
and oils) analyses; oil gravity in °API; Reid vapor pressure; trueboiling-point distillation; flash, fire, cloud, and pour points; color;
and Saybolt and Furol viscosities. Chap. 5 discusses some of these
methods that are used in petroleum engineering.
The empirical formula Cn H2n)h Sa Nb Oc can be used to classify
nearly all compounds found in crude oil. The largest portion of
crude oil is composed of hydrocarbons with carbon number, n, ranging from 1 to about 60, and h numbers ranging from h+)2 for lowmolecular-weight paraffin hydrocarbons to h+*20 for high-molecular-weight organic compounds (e.g., polycyclic aromatic
hydrocarbons). Occasionally, sulfur, nitrogen, and oxygen substitutions occur in high-molecular-weight organic compounds, with a,
b, and c usually ranging from 1 to 3.2,10
Over the past 60 years, petroleum chemists have identified
hundreds of the complex organic compounds found in petroleum.
Beginning in 1927, Rossini and others11,12 conducted a lengthy investigation of the composition of petroleum [API Research Project
6 (API 6)] to develop and improve petroleum-refining processes. It
took API 6 investigators almost 40 years to elucidate the composition of a single midcontinent crude oil from Well No. 6 in South Ponca City, Oklahoma.
Because compounds with carbon numbers u12 could not be isolated from crude oils, during 1940–66, API Research Project 42 focused on synthesizing and characterizing model hydrocarbons with
high molecular weights. These model compounds were used for
identifying compounds that could not be isolated from crude oil. A
crude oil compound with analytical responses that matched those of
a synthesized model compound was inferred to have a similar chemical structure.
Other API projects13 followed API 6, and increasingly more complex petroleum compounds were identified. API 48 focused on sulfur compounds, API 52 on nitrogen compounds, and API 56 on orPHASE BEHAVIOR
Fig. 2.1—Petroleum products identified according to carbon number.
ganometallic compounds. API 60 extended the work of API 6 to
include petroleum heavy ends.
In 1975, API stopped sponsoring basic research into the composition of petroleum. From 1975 to 1982, the petroleum engineering
industry made additional advances in analytical techniques mainly
because of the synfuels effort. The most sophisticated analytical
techniques now in use include highly selective solvent extraction14-16; simulated distillation; gel permeation, high-performance
liquid,17 and supercritical chromatography18; and mass infrared,
13C nuclear magnetic resonance,19 and Fourier-transform infrared
spectroscopy. The American Chemical Soc. Div. of Petroleum
Chemistry provides a comprehensive review of this area of research
every 2 to 3 years.
Table 2.220 shows an example of a crude-oil distillate classified
by h number (in the general formula Cn H2n)h Sa Nb Oc ) and probable structural type, which determines the range of possible n numbers. Within and across each hydrocarbon class, many isomers share
h and n numbers. The alkane (paraffin) series (h+2) has completely
saturated hydrocarbon chains that are chemically very stable. The
alkene (olefin) and alkyne (acetylene) series (h+0 and h+*2) are
composed of unsaturated, straight-chain hydrocarbons. Because alkenes and alkynes are reactive, they are not usually found in naturally occurring petroleum deposits.
The naphthene series (h+0), saturated-ring or cyclic compounds,
are found in nearly all crudes. The aromatic or “benzene” series
(h+*6) are unsaturated cyclic compounds. Low-boiling-point aromatics, which are also reactive, are found in relatively low concentrations in crude oil. Heavier crude oils are characterized by unsaturated
polycyclic aromatic hydrocarbons with increasingly negative h numVOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
bers. As molecular weight increases, these compounds assume varying degrees of fused-ring saturation, with occasional hydrocarbon
side chains. Sulfur, nitrogen, and oxygen can be substituted in the
fused hydrocarbon rings to form heterocyclics or can occupy various
positions on side chains.21 Metals, such as nickel and vanadium, can
form organometallic compounds (porphyrins) in crude oil.2,10
Asphalts, bitumens, and tars are complex colloidal mixtures of carboids, carbenes, asphaltenes, and maltenes (resins and oils). Micellar
structures of carboids, carbenes, and asphaltenes are formed by aromatic polycondensation reactions and are maintained in colloidal suspension by the maltenes. These fractions are separated according to
their solubility or lack of solubility in certain low-molecular-weight
solvents, such as propane, pentane, n-hexane, and carbon disulfide.
Fig. 2.316 shows a hypothetical chemical structure of an asphaltene. The bracket around the structure implies that the structure is
repeated three times. Although asphalt mixtures are complex in
composition and rheology, they follow certain molecular-weight
distributions that can be characterized as discussed in Chap. 5.
Understanding the nature of asphaltenes is important in petroleum
engineering because, even in low concentrations, asphaltenes can
markedly affect reservoir-fluid phase behavior.22 Because asphaltenes are polar and hydrogen bonding, they alter reservoir wettability
by adsorbing onto the rock surface.23 This alteration of reservoir
wettability may affect capillary pressure, relative-permeability relations, residual oil saturations, waterflood behavior, dispersion, and
electrical properties. Figs. 2.2 and 2.3 vividly show that the composition of crude oil is considerably more complex than the Cn H2n)2
straight-chain models commonly thought of as “oil.” This complexity
3
Fig. 2.2—Summary of hydrocarbons to be expected in crude-oil fractions (from Neumann et al.9).
should be borne in mind when modeling the phase behavior of complex reservoir fluids, particularly in gas-injection projects.23,24
2.3 Phase Diagrams for Simple Systems
The dependence of volumetric and phase behavior on temperature,
pressure, and composition is similar for simple (two- and threecomponent) and complex (multicomponent) systems. Traditionally,
the introduction to phase behavior of complex reservoir fluids starts
with a description of the vapor-pressure and volumetric behavior of
single components. The introduction then proceeds to the behavior
of two- and three-component systems, and finally to the behavior of
complex multicomponent systems. Part of the rationale for this procession lies in the Gibbs phase rule.25,26
The Gibbs phase rule states that the number of intensive variables
(i.e., degrees of freedom), F, that must be specified to determine the
thermodynamic state of equilibrium for a mixture containing n components distributed in P phases (gas, liquid, and/or solid), is
F + n * P ) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.1)
Intensive (thermodynamic) variables, such as temperature, pressure, and density, do not depend on the amount of material in the system. Extensive variables, such as flow rate, total mass, or liquid volume, depend on the extent of the system.
To attain equilibrium requires that no net interphase mass transfer
can occur. Thus, the temperatures and pressures of the coexisting
4
phases must be the same and the chemical potentials of each component in each phase must be equal. A more stringent definition of
phase equilibrium includes other forces in addition to chemical potential (e.g., gravity and capillarity).
On the basis of Eq. 2.1, for a two-phase, single-component system,
F+1 and only temperature or pressure needs to be specified to determine the thermodynamic state of the system. For a two-phase, twocomponent system, F+2 and both temperature and pressure need to
be specified to define the thermodynamic state of the mixture. Twophase binary systems allow one to focus on the effect of temperature
and pressure on the composition and the relative amounts of each of
the two phases, regardless of the composition of the overall mixture.
The Gibbs phase rule implies that as the number of components
increases to n in a two-phase mixture, n*2 composition variables
must be specified in addition to temperature and pressure. If more
than two phases are present, then n*P variables must be specified
in addition to temperature and pressure. Because reservoir fluids
comprise many components, the number of variables that must be
defined to determine the state of a reservoir fluid is conceptually unmanageable. Therefore, simple systems are often used to model the
basic volumetric and phase behavior of crude oil mixtures.
Note that the phase rule must be modified if other potential fields
are considered. For example, if the force of gravity is considered, as
PHASE BEHAVIOR
TABLE 2.2—DISTRIBUTION OF h SERIES FROM 698 TO
995°F DISTILLATE OF SWAN HILLS CRUDE OIL (Ref. 20)
Mass h Series
Probable Type
*12
Naphthalenes
*14
Naphthenonaphthalenes and/or biphenyls
*16
Dinaphthenaphthalenes and/or
*18
Trinaphthenaphthalenes and/or
*20
Tetranaphthenaphthalenes and/or
*22
Pentanaphthenaphthalenes and/or
*24
Hexanaphthenaphthalenes and/or
naphthenobiphenyls
dinaphthenobiphenyls
trinaphthenobiphenyls
tetranaphthenobiphenyls
pentanaphthenobiphenyls
*26
Heptanaphthenaphthalenes and/or
*28
Octanaphthenaphthalenes and/or
*4S
Tricyclic sulfides
*6S
Tetracyclic sulfides
*8S
Pentacyclic sulfides
*10S
Hexacyclic sulfides
hexanaphthenobiphenyls
heptanaphthenobiphenyls
*8S
Thiaindanes/thiatetralins
*10S
Naphthenothiaindanes/thiatetralins
*12S
Dinaphthenothiaindanes/thiatetralins
*14S
Trinaphthenothiaindanes/thiatetralins
*10S
Benzothiophenes
*12S
Naphthenobenzothiophenes
is done when calculating compositional variation with depth, the
phase rule is F+n*P)3.7
2.3.1 Single-Component Systems. The p-T curve shown in Fig. 2.4
is a portion of the vapor-pressure curve for a typical hydrocarbon
compound. Above and to the left of the curve, the hydrocarbon behaves as a liquid; below and to the right, the hydrocarbon behaves
as a vapor. Saturated liquid and vapor coexist at every point along
the vapor-pressure curve. The curve ends at the critical temperature
and critical pressure of the hydrocarbon (the “critical point”). Fig.
2.5 shows a 3D PVT diagram of a pure compound.
The critical temperature of a single component defines the temperature above which any gas/liquid mixture cannot coexist, regardless of pressure. Similarly, the critical pressure defines the pressure
above which liquid and vapor cannot coexist, regardless of temperature. Along the vapor-pressure curve, two phases coexist in equilibrium. At the critical point, the vapor and liquid phases can no longer
be distinguished, and their intensive properties are identical.
For a multicomponent system, the definition of the critical point
is also based on a temperature and pressure at which the vapor and
liquid phases are indistinguishable. However, for a single-component system, the two-phase region terminates at the critical point. In
a multicomponent system, the two-phase region can extend beyond
the system’s critical point (i.e., at temperatures greater than the critical temperature and pressures greater than the critical pressure).
Fig. 2.627 illustrates the continuity of gas and liquid phases for pure
components. In this figure, the darker shading corresponds to higher
density. A sharp contrast in phase densities is readily apparent along
the vapor-pressure curve. As temperature increases along the vaporpressure curve, the discontinuity becomes harder to discern, until finally, at the critical point, the contrast in shading is hardly noticeable.
Qualitatively, the behavior described by the shading in Fig. 2.6 is the
same for multicomponent mixtures in the undersaturated region.
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Fig. 2.3—Hypothetical structure of a petroleum asphaltene (after
Speight and Moschopedis14).
pc
Fig. 2.4—p-T diagram for a single component in the region of vapor/liquid behavior near the critical point ( pc +critical pressure
and Tc +critical temperature).
Phase changes do not have to take place abruptly if certain temperature and pressure paths are followed. A process can start as a
saturated liquid and end as a saturated vapor, with no abrupt change
in phase. The path D–A–E–F–G–B–D in Fig. 2.4 is an example of
a process that changes phases without crossing the vapor-pressure
curve. Pure components actually exist as a saturated “liquid” and
“vapor” only along the vapor-pressure curve. At other pressures and
temperatures, the component only behaves “liquid-like” or “vaporlike,” depending on the location of the system temperature and pressure relative to the system’s critical point. Katz28 suggested calling
a pure substance “single-phase fluid” at pressures greater than the
critical pressure. Strictly speaking, the terms liquid-like and vaporlike should be used to describe undersaturated fluids.
5
Fig. 2.5—Three-dimensional schematic of the PVT surface of a
pure compound (source unknown).
ponent is a saturated liquid. Similarly, the saturation curve to the
right of the critical point (Point B to Point C) defines the dewpoint
curve, along which the component is a saturated vapor.
For any temperature less than the critical temperature, successive
decreases in volume will elevate the pressure of the vapor until the
“dewpoint” (vapor pressure) is reached (Point B on Fig. 2.7). At
these conditions, the component is a saturated vapor in equilibrium
with an infinitesimal amount of saturated liquid. Further decreases
in the volume at constant temperature will result in proportionate increases in the amount of saturated liquid condensed, but the pressure
does not change (i.e., the system pressure remains equal to the vapor
pressure). While more liquid is being formed, the total volume (at
Point D) is being reduced. However, the densities and other intensive properties of the saturated vapor and saturated liquid remain
constant as a consequence of the Gibbs phase rule.
A simple mass balance further shows that the ratio of liquid to vapor equals the ratio of Curve B–D to Curve D–A. Further decreases
in volume will condense more liquid until the bubblepoint is
reached. At the bubblepoint, the system is 100% saturated liquid in
equilibrium with an infinitesimal amount of saturated vapor. Further
decreases in volume beyond the bubblepoint are accompanied by a
large increase in pressure because the liquid is only slightly compressible. This is indicated by the nearly vertical isotherms on the
left side of Fig. 2.7. In the undersaturated vapor region on the right
side of the diagram, a large change in volume reduces pressure only
slightly because the vapor is highly compressible.
Fig. 2.726 shows a p-V diagram for ethane. The area enclosed by
the saturation envelope represents the two-phase region. The area
to the left of the envelope is the liquid-like region, and the area to
the right is the vapor-like region. Point C represents the critical
point. The saturation curve to the left of the critical point (from Point
A to Point C) defines the bubblepoint curve, along which the com-
2.3.2 Two-Component Systems. Two-component systems are
slightly more complex than single-component systems because
both temperature and pressure affect phase behavior in the saturated
region. Two important differences between single- and two-component systems exist. The saturated p-T projection is represented by a
phase envelope rather than by a vapor-pressure curve, and the criti-
3,000
2,000
1,000
0
0
100
200
300
400
500
600
Temperature, °F
Fig. 2.6—Continuity of vapor and liquid states for a single component along the vapor-pressure curve
and at supercritical conditions (after Katz and Kurata27).
6
PHASE BEHAVIOR
Specific volume, ft3/lbm
Fig. 2.7—p-V diagram for ethane at three temperatures (from
Standing26).
cal temperature and critical pressure no longer define the extent of
the two-phase, vapor/liquid region. Fig. 2.829 compares the p-T and
p-V behavior of pure compounds and mixtures.
Fig. 2.926 is a p-T projection of the ethane/n-heptane system for
a fixed composition. For a single-component system, the dew- and
bubblepoint curves are one in the same; i.e., they coincide with the
vapor-pressure curve. In a binary (or other multicomponent) system, the dew- and bubblepoint curves no longer coincide, and a
phase envelope results instead of a vapor-pressure curve. To the left
of the phase envelope, the mixture behaves liquid-like, and to the
right it behaves vapor-like.
For binary or other multicomponent systems, the critical temperature and pressure are defined as the point where the dew- and bubblepoint curves intersect. At this point, the equilibrium phases are
physically indistinguishable. Also, in contrast to the single-component system, two phases can exist at temperatures and pressures
greater than the critical temperature and pressure. The highest temperature at which two phases can coexist in equilibrium is defined
as the cricondentherm (Tangent b–b in Fig. 2.9). Similarly, the highest pressure at which two phases can coexist is defined as the cricondenbar (Tangent a–a).
In the single-phase region, vapor and liquid are distinguished
only by their densities and other physical properties. The region just
beyond the critical point of a mixture has often been called the “supercritical” or “dense-fluid” region. Here, the fluid is considered to
be neither gas nor liquid because the fluid properties are not strictly
liquid-like or vapor-like.
Kay30 measured the phase behavior of the binary ethane/n-heptane system for several compositions, as Fig. 2.10 shows. On the left
side of this figure, the curve terminating at Point C is the vapor pressure of pure ethane; the curve on the right, terminating at Point C7,
is the vapor pressure of pure n-heptane. Points C1 through C3 are
the critical points of ethane/n-heptane mixtures at different compositions. The dashed line represents the locus of critical points for
the infinite number of possible ethane/n-heptane mixtures. Each
mixture composition has its own p-T phase envelope.
The three compositions shown, which are 90.22, 50.25, and 9.78
wt% ethane, represent a system that is mainly ethane, a system that
is one-half ethane and one-half n-heptane (by weight), and a system
that is mainly n-heptane, respectively. Several interesting features
of binary and multicomponent systems can be studied from these
three mixtures. As composition changes, the location of the critical
point and the shape of the p-T phase diagram also change.
Note that the critical pressures of many (but not all) mixtures are
higher than the critical pressures of the components composing the
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
mixture. With a mixture composed mainly of ethane, the critical
point lies to the left of the cricondentherm. Such a system is analogous to a reservoir gas-condensate system. As the percentage of
ethane in the mixture increases further, the critical point of the system approaches that of pure ethane.
The critical point for the mixture composed mostly of n-heptane
lies below the cricondenbar. This system is analogous to a reservoir
black-oil system. As the percentage of n-heptane increases, the critical point of the mixture approaches that of pure n-heptane. With
equal percentages of ethane and n-heptane, the critical pressure is
close to the cricondenbar of ethane and n-heptane. As the concentration of each component becomes similar, the two-phase region becomes larger.
Other binaries provide additional insight into the effect of widely
differing boiling points of the components making up the system.
Fig. 2.1131 shows the vapor pressure of several hydrocarbons and
the critical loci of their binary mixtures with methane. As the boiling
points of the methane/hydrocarbon binary become more dissimilar,
the two-phase region becomes larger and the critical pressure increases. For binaries with components that have similar molecular
structures, the loci of critical points are relatively flat.
2.3.3 Multicomponent Systems. Phase diagrams for naturally occurring reservoir fluids are similar to those for binary mixtures. Fig.
2.125 is the first p-T phase diagram measured for a complex gascondensate system. This p-T diagram is particularly useful because
it exhibits oil-like to gas-like behavior over a range of typical reservoir temperatures, from 80 to 240°F. Katz and coworkers32 used a
glass-windowed cell to measure the distribution of gas and liquid
phases throughout the two-phase region and near the mixture’s critical point. Fig. 2.135 shows isotherms of volume percent vs. pressure
that were measured to determine the two-phase boundary and the
volume-percent quality lines in the p-T diagram in Fig. 2.12.
2.4 Retrograde Condensation
Kurata and Katz33 give the most concise and relevant discussion of
retrograde phenomena related to petroleum engineering. In 1892,
Kuenen34 used the term “retrograde condensation” to describe the
anomalous behavior of a mixture that forms a liquid by an isothermal decrease in pressure or by an isobaric increase in temperature.
Conversely, “retrograde vaporization” can be used to describe the
formation of vapor by an isothermal increase in pressure or by an
isobaric decrease in temperature. Neither form of retrograde behavior occurs in single-component systems.
Fig. 2.14 is a constant-composition p-T projection of a multicomponent system. The diagram shows lines of constant liquid volume
percent (quality). Although total composition is fixed, the respective compositions of saturated vapor and liquid phases change along
the quality lines. The bubblepoint curve represents the locus of
100% liquid, and the dewpoint curve represents the locus of 0% liquid. The bubble- and dewpoint curves meet at the mixture critical
point. Lines of constant quality also converge at the mixture critical
point. The regions of retrograde behavior are defined by the lines of
constant quality that exhibit a maximum with respect to temperature
or pressure. Fig. 2.14 shows that for retrograde phenomena to occur,
the temperature must be between the critical temperature and the
cricondentherm. Fig. 2.1535 illustrates the liquid volumetric behavior of a lean gas-condensate system measured by Eilerts et al.35-37
Fig. 2.12 shows the p-T diagram of a reservoir mixture that would
be considered a gas condensate if it had been discovered at a reservoir
temperature of, for example, 200°F and an initial pressure of 2,700
psia. For these initial conditions, if reservoir pressure drops below
2,560 psia from depletion, the dewpoint will be passed and a liquid
phase will develop in the reservoir. Liquid dropout will continue to
increase until the pressure reaches 2,300 psia, when a maximum of
25 vol% liquid will have accumulated. According to Fig. 2.12, further
pressure reduction will revaporize most of the condensed liquid.
These comments assume that the overall composition of the reservoir mixture remains constant during depletion, a reasonable assumption in the context of this general discussion. In reality, howev7
Fig. 2.8—Qualitative p-T and p-V plots for pure fluids and mixtures; Vc +critical volume (after Edmister and Lee29).
er, the behavior of liquid dropout and revaporization differs from
that suggested by constant-composition analysis. The retrograde
liquid saturation is usually less than the saturation needed to mobilize the liquid phase. Because the heavier components in the original
mixture constitute most of the (immobile) condensate saturation,
the overall molecular weight of the remaining reservoir fluid increases during depletion. The phase envelope for this heavier reservoir mixture is pushed down and to the right of the original phase
diagram (Fig. 2.16); the critical point is shifted to the right toward
a higher temperature. It is not unusual that a retrograde-condensate
mixture under depletion will reach a condition where the overall
composition would exhibit a bubblepoint pressure if the reservoir
were repressured (i.e., the overall mixture critical temperature becomes greater than the reservoir temperature). This change in overall reservoir composition results in less revaporization at lower pressures. Fig. 2.17 shows the difference between constant-composition
and “depletion” liquid-dropout curves.
8
2.5 Classification of Oilfield Systems
One might assume that the name used to identify a reservoir fluid
should not influence how the fluid is treated as long as its physical
properties are correctly defined. In theory this is true, but in practice
we are usually required to define petroleum reservoir fluids as either
“oil” or “gas.” For example, regulatory bodies require the definition
of reservoir fluid for well spacing and determining allowable production rates and field-development strategy (e.g., unitization).
The classification of a reservoir fluid as dry gas, wet gas, gas condensate, volatile oil, or black oil is determined (1) by the location of
the reservoir temperature with respect to the critical temperature and
the cricondentherm and (2) by the location of the first-stage separator temperature and pressure with respect to the phase diagram of
the reservoir fluid. Fig. 2.18 illustrates how four types of depletion
reservoirs for the same hydrocarbon system are defined by the location of the initial reservoir temperature and pressure.
PHASE BEHAVIOR
Fig. 2.10—p-T diagram for the C2/n-C7 system at various concentrations of C2 (after Kay30).
Fig. 2.9—p-T diagram for a C2/n-C7 mixture with 96.83 mol%
ethane (from Standing26).
Fig. 2.11—p-T diagram for various hydrocarbon binaries illustrating the effects of molecular-weight differences on criticalpoint loci (after Brown et al.31).
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
A reservoir fluid is classified as dry gas when the reservoir temperature is greater than the cricondentherm and surface/transport conditions are outside the two-phase envelope; as wet gas when the reservoir temperature is greater than the cricondentherm but the surface
conditions are in the two-phase region; as gas condensate when the
reservoir temperature is less than the cricondentherm and greater than
the critical temperature; and as an oil (volatile or black oil) when the
reservoir temperature is less than the mixture critical temperature.
For a given reservoir temperature and pressure, Fig. 2.1938 shows
the spectrum of reservoir fluids from wet gas to black oil expressed
in terms of surface GOR’s and oil/gas ratios (OGR’s). A more quantitative classification is also given in Fig. 2.19 in terms of molar
composition, by use of a ternary diagram. In the near-critical region,
gas condensates have a C7+ concentration less than [12.5 mol%
and volatile oils fall between 12.5 to 17.5 mol% C7+.
Retrograde gas-condensate reservoirs26,39 typically exhibit
GOR’s between 3,000 and 150,000 scf/STB (OGR’s from about 350
to 5 STB/MMscf) and liquid gravities between 40 and 60°API. The
color of stock-tank liquid is expected to lighten from volatile-oil to
gas-condensate systems, although light volatile oils may be yellowish or water-white and some condensate liquids can be dark brown
Fig. 2.12—p-T diagram for a gas-condensate system (after Katz
et al.5).
9
Fig. 2.14—Hypothetical p-T diagram for a gas condensate showing the isothermal retrograde region.
Fig. 2.13—Volume isotherms for the gas-condensate p-T diagram in Fig. 2.12 (after Katz et al.5)
or even black. 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.
In some cases, for condensate recovery from a surface process facility, the reservoir fluid is mistakenly interpreted to be a gas condensate. Strictly speaking, the definition of a gas condensate depends only on reservoir temperature. The definition of a reservoir
fluid as wet or dry gas depends on conditions at the surface. This
makes differentiation between dry and wet gas difficult because any
gas can conceivably be cooled enough to condense a liquid phase.
The classification of a fluid as an oil is unambiguous because the
only requirement is that the reservoir temperature be less than the
BUBBLEPOINT
Fig. 2.15—Liquid volume (expressed as a liquid/gas ratio) behavior for a lean-gas-condensate
system (from Eilerts et al.35).
10
PHASE BEHAVIOR
Fig. 2.16—Change in phase envelope during the depletion of a gas condensate.
critical temperature. However, the distinction between a black oil
and a volatile oil is more arbitrary. Generally speaking, a volatile oil
is a mixture with a relatively high solution gas/oil ratio. Volatile oils
exhibit large changes in properties when pressure is reduced only
somewhat below the bubblepoint. In an extreme case, the oil volume
may shrink from 100 to 50% with a reduction in pressure of only 100
psi below the bubblepoint. Black-oil properties, on the other hand,
exhibit gradual changes, with nearly linear pressure dependence below the bubblepoint.
Volatile oils typically yield stock-tank-oil gravities greater than
35°API, surface GOR’s between 1,000 and 3,000 scf/STB, and
FVF’s (see Formation Volume Factors in Chap. 6) greater than [1.5
RB/STB. Solution gas released from a volatile oil contains significant quantities of stock-tank liquid (condensate) when this gas is
produced to the surface. Solution gas from black oils is usually considered “dry,” yielding insignificant stock-tank liquids when produced to surface conditions.
For engineering calculations, the liquid content of released solution gas is perhaps the most important distinction between volatile
oils and black oils. This difference is also the basis for the modification of standard black-oil PVT properties discussed in Chap. 7. A
reasonable engineering distinction between black oils and volatile
oils can be made on the basis of simple reservoir material-balance
calculations. If the total surface oil and gas recoveries calculated by
a reservoir material balance with the standard black-oil PVT formulation are sufficiently close to the recoveries calculated by a
compositional material balance, the oil can probably be considered
a black oil (see Chap. 7). If calculated oil recoveries are significantly
different, the reservoir mixture should be treated as a volatile oil by
use of a compositional approach or the modified black-oil PVT
properties outlined in Chap. 7. Several researchers40,41 have shown
that a compositional material balance for depletion of volatile-oil
reservoirs may predict from two to four times the surface liquid reBubblepoint
or
dissolved gas
reservoirs
Dewpoint
or
gas condensate
reservoirs
SingleĆphase
gas reservoirs
CVD lower because of loss
of C7+ in early depletion stages
CCE has stronger revaporization at low
pressures because of greater (initial)
mass of gas remaining in cell
Fig. 2.17—Retrograde volumes for constant-composition and
constant-volume depletion experiments.
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Fig. 2.18—p-T diagram of a reservoir fluid illustrating different
types of depletion reservoirs.
11
ple, the gas is probably saturated at initial reservoir conditions, and
an equilibrium oil could exist at some lower elevation.
Discovery of a saturated reservoir fluid will usually require further field delineation to substantiate the presence of a second equilibrium phase above or below the tested interval. This may entail
running a repeat-formation-tester tool to determine the fluid-pressure gradient as a function of depth, or a new well may be required
updip or downdip to the discovery well. Representative samples of
saturated fluids may be difficult to obtain during a production test.42
Standing26 discusses the situation of an undersaturated gas condensate sampled during a test where bottomhole flowing pressure drops
below the dewpoint pressure. The produced fluid, which is not representative of the original reservoir fluid, may have a dewpoint
equal to initial reservoir pressure. This situation would incorrectly
imply that the reservoir is saturated at initial conditions and that an
underlying oil rim may exist.
References
OGR (STB/MMscf)
GOR (scf/STB)
Fig. 2.19—Spectrum of reservoir fluids in order of increasing
chemical complexity from wet gas to black oil (from Cronquist38).
covery predicted by conventional material balances that are based
on the standard black-oil PVT formulation.
Fluid samples obtained from a new field discovery may be instrumental in defining the existence of an overlying gas cap or an underlying oil rim. Referring to Fig. 2.20, if the initial reservoir pressure
equals the measured bubblepoint pressure of a bottomhole or recombined sample, the oil is probably saturated at initial reservoir
conditions. This implies that an equilibrium gas cap could exist at
some higher elevation. Likewise, if the initial reservoir pressure is
the same as the measured dewpoint pressure of a reservoir gas sam-
}
Retrograde dewpoint
= Resevoir pressure
Bubblepoint
Fig. 2.20—p-T phase diagram of a gas-cap fluid in equilibrium
with an underlying saturated oil.
12
1. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,”
Amer. Soc. Pet. Geologists Bulletin (February 1952) 36, No. 2, 342.
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Dallas (1959) 433.
3. Nelson, W.L.: Petroleum Refinery Engineering, fourth edition,
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4. Nelson, W.L.: “Does Crude Boil at 1400°F?,” Oil & Gas J. (1968) 125.
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6. Muskat, M.: “Distribution of Non-Reacting Fluids in the Gravitational
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7. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in
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9. Neumann, H.J., Paczynska-Lahme, B., and Severin, D.: Composition
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11. Rossini, F.D.: “The Chemical Constitution of the Gasoline Fraction of
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12. Rossini, F.D. and Mair, B.J.: “The Work of the API Research Project on
the Composition of Petroleum,” Proc., Fifth World Pet. Cong. (1954) 223.
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Pet. Cong. (1955) 27.
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Chemical Soc. (1981) 195, 1.
15. Speight, J.G., Long, R.B., and Trowbridge, T.D.: “Factors Influencing
the Separation of Asphaltenes from Heavy Petroleum Feedstocks,”
Fuel (1984) 63, 616.
16. Speight, J.G. and Pancirov, R.J.: “Structural Types in Petroleum Asphaltenes as Deduced from Pyrolysis/Gas Chromatography/Mass
Spectrometry,” Liquid Fuels Technology (1984) 2, No. 3, 287.
17. Such, C., Brulé, B., and Baluja-Santos, C.: “Characterization of a Road
Asphalt by Chromatographic Techniques (GPC and HPLC),” J. Liquid
Chrom. (1979) 2, No. 3, 437.
18. Fetzer, J.C. et al.: “Characterization of Carbonaceous Materials Using
Extraction with Supercritical Pentane,” report, Contract No. DOE/
ER/00854-29, U.S. DOE (1980).
19. Helm, R.V. and Petersen, J.C.: “Compositional Studies of an Asphalt and
Its Molecular Distillation Fractions by Nuclear Magnetic Resonance and
Infrared Spectrometry,” Analytical Chemistry (1968) 40, No. 7, 1100.
20. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Swan Hills,” Hydro. Proc. (April 1974) 53, 93.
21. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Comparisons,”
Hydro. Proc. (Nov. 1974) 53, 187.
22. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. &
Eng. Chem. (February 1945) 37, 195.
23. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 and
Rich-Gas Flooding,” SPERE (February 1991) 17.
24. Bossler, R.B. and Crawford, P.B.: “Miscible-Phase Floods May Precipitate Asphalt,” Oil & Gas J. (23 February 1959) 57, 137.
PHASE BEHAVIOR
25. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press,
New Haven, Connecticut (1948).
26. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, SPE, Richardson, Texas (1977).
27. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng.
Chem. (June 1940) 32, No. 6, 817.
28. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenomena in Oil and Gas Production,” Trans., AIME (1939) 132, 103.
29. Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics,
second edition, Gulf Publishing Co., Houston (1984) I.
30. Kay, W.B.: “The Ethane-Heptane System,” Ind. & Eng. Chem. (1938)
30, 459.
31. Brown, G.G. et al.: Natural Gasoline and the Volatile Hydrocarbons,
NGAA, Tulsa, Oklahoma (1948) 24–32.
32. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixture
of Natural Gas and Natural Gasoline Near the Critical Conditions,”
Trans., AIME (1939) 136, 165.
33. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile Hydrocarbon
Mixtures,” Trans., AIChE (1942) 38, 995.
34. Kuenen, J.P.: “On Retrograde Condensation and the Critical Phenomena of Two Substances,” Commun. Phys. Lab. U. Leiden (1892) 4, 7.
35. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph
10, USBM, American Gas Assn., New York City (1957) I and II.
36. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1
February 1947) 63.
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
37. Eilerts, C.K. et al.: “Phase Relations of a Gas-Condensate Fluid at
Low Tempertures, Including the Critical State,” Pet. Eng. (February
1948) 19, 154.
38. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast Reservoir
Oils,” JPT (May 1973) 538.
39. Moses, P.L.: “Engineering Applications of Phase Behavior of Crude Oil
and Condensate Systems,” JPT (July 1986) 715.
40. Lohrenz, J., Clark, G.C., and Francis, R.J.: “A Compositional Material
Balance for Combination Drive Reservoirs with Gas and Water Injection,” JPT (November 1963) 1233; Trans., AIME, 228.
41. Reudelhuber, F.O. and Hinds, R.F.: “Compositional Material-Balance
Method for Prediction of Recovery From Volatile Oil Depletion Drive
Reservoirs,” JPT (1957) 19; Trans., AIME, 210.
42. Fevang, Ø. and Whitson, C.H.: “Accurate In-Situ Compositions in Petroleum Reservoirs,” paper SPE 28829 presented at the 1994 European
Petroleum Conference, London, 25–27 October.
SI Metric Conversion Factors
°API 141.5/(131.5)°API)
+g/cm3
bbl 1.589 873
E*01 +m3
ft3 2.831 685
E*02 +m3
°F (°F*32)/1.8
+°C
gal 3.785 412
E*03 +m3
lbm 4.535 924
E*01 +kg
psi 6.894 757
E)00 +kPa
13
Chapter 3
Gas and Oil Properties and Correlations
3.1 Introduction
Chap. 3 covers the properties of oil and gas systems, their nomenclature and units, and correlations used for their prediction. Sec. 3.2
covers the fundamental engineering quantities used to describe
phase behavior, including molecular quantities, critical and reduced
properties, component fractions, mixing rules, volumetric properties, transport properties, and interfacial tension (IFT).
Sec. 3.3 discusses the properties of gas mixtures, including correlations for Z factor, pseudocritical properties and wellstream gravity, gas viscosity, dewpoint pressure, and total formation volume
factor (FVF). Sec. 3.4 covers oil properties, including correlations
for bubblepoint pressure, compressibility, FVF, density, and viscosity. Sec. 3.5 gives correlations for IFT and diffusion coefficients.
Sec. 3.6 reviews the estimation of K values for low-pressure applications, such as surface separator design, and convergence-pressure methods used for reservoir calculations.
3.2 Review of Properties, Nomenclature, and Units
3.2.1 Molecular Quantities. All matter is composed of elements that
cannot be decomposed by ordinary chemical reactions. Carbon (C),
hydrogen (H), sulfur (S), nitrogen (N), and oxygen (O) are examples
of the elements found in naturally occurring petroleum systems.
The physical unit of the element is the atom. Two or more elements may combine to form a chemical compound. Carbon dioxide
(CO2), methane (CH4), and hydrogen sulfide (H2S) are examples of
compounds found in naturally occurring petroleum systems. When
two atoms of the same element combine, they form diatomic compounds, such as nitrogen (N2) and oxygen (O2). The physical unit
of the compound is the molecule.
Mass is the basic quantity for measuring the amount of a substance.
Because chemical compounds always combine in a definite proportion (i.e., as a simple ratio of whole numbers), the mass of the atoms
of different elements can be conveniently compared by relating them
with a standard. The current standard is carbon-12, where the element
carbon has been assigned a relative atomic mass of 12.011.
The relative atomic mass of all other elements have been determined relative to the carbon-12 standard. The smallest element is
hydrogen, which has a relative atomic mass of 1.0079. The relative
atomic mass of one element contains the same number of atoms as
the relative atomic mass of any other element. This is true regardless
of the units used to measure mass.
According to the SI standard, the definition of the mole reads “the
mole is the amount of substance of a system which contains as many
elementary entities as there are atoms in 0.012 kilograms of car18
bon-12.” The SI symbol for mole is mol, which is numerically identical to the traditional g mol.
The SPE SI standard1 uses kmol as the unit for a mole where kmol
designates “an amount of substance which contains as many kilograms (groups of molecules) as there are atoms in 12.0 kg (incorrectly written as 0.012 kg in the original SPE publication) of carbon-12 multiplied by the relative molecular mass of the substance
involved.”
A practical way to interpret kmol is “kg mol” where kmol is numerically equivalent to 1,000 g mol (i.e., 1,000 mol). Otherwise, the
following conversions apply.
1 kmol
+ 1,000 mol
+ 1,000 g mol
+ 2.2046 lbm mol
1 lbm mol + 0.45359 kmol
+ 453.59 mol
+ 453.59 g mol
1 mol
+ 1 g mol
+ 0.001 kmol
+ 0.0022046 lbm mol
The term molecular weight has been replaced in the SI system by
molar mass. Molar mass, M, is defined as the mass per mole
(M+m/n) of a given substance where the unit mole must be consistent with the unit of mass. The numerical value of molecular weight
is independent of the units used for mass and moles, as long as the
units are consistent. For example, the molar mass of methane is
16.04, which for various units can be written
M+
+
+
+
16.04 kg/kmol
16.04 lbm/lbm mol
16.04 g/g mol
16.04 g/mol
3.2.2 Critical and Reduced Properties. Most equations of state
(EOS’s) do not use pressure and temperature explicitly to define the
state of a system, but instead they generalize according to corresponding-states theory by use of two or more reduced properties,
which are dimensionless.2
T r + TńT c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1a)
PHASE BEHAVIOR
the following relation for volume fractions x vi, based on component
densities at standard conditions ò i or specific gravities g i.
x vi +
m ińò i
+
j
j
j+1
x i M ińg i
ȍ x M ńg
j
j
j
x i M ińò i
ȍ x M ńò
N
j
j
j
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.4)
N
j
+
N
j
j+1
+
n i M i ńò i
ȍ m ńò ȍ n M ńò
N
j
j+1
Fig. 3.1—Reservoir densities as functions of pressure and temperature.
where the sum of x vi is unity.
Having defined component fractions, we can introduce some
common mixing rules for averaging the properties of mixtures.
Kay’s5 mixing rule, the simplest and most widely used, is given by
a mole-fraction average,
ȍz q .
N
p r + pńp c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1b)
V r + VńV c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1c)
and ò r + òńò c, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1d)
where ò r + 1ńV r . Absolute units must be used when calculating
reduced pressure and temperature. p c, T c, V c, and ò c are the true
critical properties of a pure component, or some average for a mixture. In most petroleum engineering applications, the range of reduced pressure is from 0.02 to 30 for gases and 0.03 to 40 for oils;
reduced temperature ranges from t1 to 2.5 for gases and from 0.4
to 1.1 for oils. Reduced density can vary from 0 at low pressures to
about 3.5 at high pressures.
Average mixture, or pseudocritical, properties are calculated
from simple mixing rules or mixture specific gravity.3,4 Denoting
a mixture pseudocritical property by q pc, the pseudoreduced property is defined q pr + qń q pc. Pseudocritical properties are not approximations of the true critical properties, but are chosen instead so that
mixture properties will be estimated correctly with correspondingstates correlations.
3.2.3 Component Fractions and Mixing Rules. Petroleum reservoir mixtures contain hundreds of well-defined and “undefined”
components. These components are quantified on the basis of mole,
weight, and volume fractions. For a mixture having N components,
i + 1, . . . , N, the overall mole fractions are given by
zi +
ni
mi ń Mi
+
ȍ n ȍ m ńM
N
N
j
j
j+1
,
. . . . . . . . . . . . . . . . . . . . . . . (3.2)
j
j+1
where n+moles, m+mass, M+molecular weight, and the sum of
z i is 1.0. In general, oil composition is denoted by x i and gas composition by y i.
Weight or mass fractions, wi , are given by
wi +
mi
+
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . (3.3)
N
j
j
i i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.5)
i+1
This mixing rule is usually adequate for molecular weight, pseudocritical temperature, and acentric factor.6 We can write a generalized
linear mixing rule as
ȍf q
N
i i
q+
i+1
N
ȍf
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.6)
i
i+1
where f i is usually one of the following weighting factors: f i + z i,
mole fraction (Kay’s rule); f i + w i , weight fraction; or f i + x vi,
volume fraction. Depending on the quantity being averaged, other
mixing rules and definitions of f i may be appropriate.7,8 For example, the mixing rules used for constants in an EOS (Chap. 4) can be
chosen on the basis of statistical thermodynamics.
3.2.4 Volumetric Properties. Density, ò, is defined as the ratio of
mass to volume,
ò + mńV, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.7)
expressed in such units as lbm/ft3, kg/m3, and g/cm3. Fig. 3.1 shows
the magnitudes of density for reservoir mixtures. Molar density,
ò M , gives the volume per mole:
ò M + nńV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.8)
Specific volume, v^, is defined as the ratio of volume to mass and is
equal to the reciprocal of density.
v^ + Vńm + 1ńò. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.9)
Molar volume, v, defines the ratio of volume per mole,
v + Vńn + Mńò + 1ńò M , . . . . . . . . . . . . . . . . . . . . . (3.10)
and is typically used in cubic EOS’s. Molar density, ò M , is given by
ò M + 1ńv + òńM, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.11)
n i Mi
ȍm ȍn M
N
q+
j
j+1
where the sum of w i + 1.0. Although the composition of a mixture
is usually expressed in terms of mole fraction, the measurement of
composition is usually based on mass, which is converted to mole
fraction with component molecular weights.
For oil mixtures at standard conditions (14.7 psia and 60°F), the
total volume can be approximated by the sum of the volumes of individual components, assuming ideal-solution mixing. This results in
GAS AND OIL PROPERTIES AND CORRELATIONS
and is used in the formulas of some EOS’s.
According to the SI standard, relative density replaces specific
gravity as the term used to define the ratio of the density of a mixture
to the density of a reference material. The conditions of pressure and
temperature must be specified for both materials, and the densities
of both materials are generally measured at standard conditions
(standard conditions are usually 14.7 psia and 60°F).
g+
ò ǒ p sc, T scǓ
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12a)
ò ref ǒ p sc, T scǓ
19
Fig. 3.2—Reservoir compressibilities as functions of pressure.
Fig. 3.3—Reservoir FVF’s as functions of pressure.
ǒò oǓ
sc
go +
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12b)
ǒò wǓ
sc
and g g +
ǒò gǓ
sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12c)
ǒò airǓ
sc
Air is used as the reference material for gases, and water is used as
the reference material for liquids. Specific gravity is dimensionless,
although it is customary and useful to specify the material used as a
reference (air+1 or water+1). In older references, liquid specific
gravities are sometimes followed by the temperatures of both the liquid and water, respectively; for example, g o + 0.823 60ń60 , where the
temperature units here are understood to be in degrees Fahrenheit.
The oil gravity, g API, in degrees API is used to classify crude oils
on the basis of the following relation,
B+
V mixture ǒ p, T Ǔ
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.16)
V product ǒ p sc, T scǓ
The units of B are bbl/STB for oil and water, and ft3/scf or bbl/Mscf
for gas. The surface product phase may consist of all or only part of
the original mixture.
Primarily, four volume factors are used in petroleum engineering.
They are oil FVF, B o; water FVF, B w; gas FVF, B g; and total FVF
of a gas/oil system, B t, where
g API + 141.5
g * 131.5 . . . . . . . . . . . . . . . . . . . . . . . . . . (3.13a)
Bo +
Vo
V
+ o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17a)
Vo
(V o) sc
141.5
, . . . . . . . . . . . . . . . . . . . . . . . . (3.13b)
g API ) 131.5
Bw +
Vw
V
+ w , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17b)
Vw
(V w) sc
o
and g o +
where g o +oil specific gravity (water+1). Officially, the SPE does
not recognize g API in its SI standard, but because oil gravity (in degrees API) is so widely used (and understood) and because it is
found in many property correlations, its continued use is justified
for qualitative description of stock-tank oils.
Isothermal compressibility, c, of a fixed mass of material is defined as
ǒ Ǔ
c + * 1 ēV
V ēp
ǒ Ǔ
+ * 1^ ēv
v ēp
T
^
T
ǒ Ǔ,
+ * 1v ēv
ēp
. . . . . (3.14)
T
where the units are psi*1 or kPa*1. In terms of density, ò, and FVF,
B, isothermal compressibility is given by
ǒ Ǔ
1 ēò
c+ò
ēp
T
ǒ Ǔ,
+ 1 ēB
B ēp
. . . . . . . . . . . . . . . . . . . . . (3.15)
T
where B is defined in the next section. Fig. 3.2 shows the variation
in compressibility with pressure for typical reservoir mixtures. A
discontinuity in oil compressibility occurs at the bubblepoint because gas comes out of solution. When two or more phases are present, a total compressibility is useful.8,9
3.2.5 Black-Oil Pressure/Volume/Temperature (PVT) Properties. The FVF, B; solution gas/oil ratio, R s ; and solution oil/gas ratio,
r s, are volumetric ratios used to simplify engineering calculations.
Specifically, they allow for the introduction of surface volumes of
gas, oil, and water into material-balance equations. These are not
standard engineering quantities, and they must be defined precisely.
These properties constitute the black-oil or “beta” PVT formula
used in petroleum engineering. Chap. 7 gives a detailed discussion
of black-oil properties.
20
FVF, or simply volume factor, is used to convert a volume at elevated pressure and temperature to surface volume, and vice versa.
More specifically, FVF is defined as the volume of a mixture at specified pressure and temperature divided by the volume of a product
phase measured at standard conditions,
Bg +
Vg
ǒV gǓ
and B t +
+
sc
Vg
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17c)
Vg
Vo ) Vg
Vo ) Vg
Vt
+
+
; . . . . . . . . . . . (3.17d)
Vo
(V o) sc
(V o) sc
and the total FVF of a gas/water system is
B tw +
Vg ) Vw
Vt
+
. . . . . . . . . . . . . . . . . . . . . . . (3.17e)
Vw
(V w) sc
In Eq. 3.17, V o +oil volume at p and T ; V g +gas volume at p and
T ; V w +water/brine volume at p and T ; V o +(V o) sc +stock-tankoil volume at standard conditions; V w + (V w) sc +stock-tank-water volume at standard conditions; and V g + ǒV gǓ sc+surface-gas
volume at standard conditions.
Because gas FVF is inversely proportional to pressure, a reciprocal gas volume factor, b g (equal to 1/ B g), is sometimes used, where
the units of b g may be scf/ft3 or Mscf/bbl. Fig. 3.3 shows FVF’s of
typical reservoir systems. Inverse oil FVF, b o (equal to 1/ B o) is also
used in reservoir simulation.
Wet gas and gas-condensate reservoir fluids produce liquids at the
surface, and for these gases the surface product (separator gas) consists of only part of the original reservoir gas mixture. Two gas
FVF’s are used for these systems: the “dry” FVF, B gd, and the “wet”
FVF, B gw (or just B g). B gd gives the ratio of reservoir gas volume to
the actual surface separator gas. B gw gives the ratio of reservoir gas
volume to a hypothetical “wet” surface-gas volume (the actual separator-gas volume plus the stock-tank condensate converted to an
equivalent surface-gas volume). Chap. 7 describes when B gd and
B gw are used. The standard definition of B g + (p scńT sc)(ZTńp) (see
Eq. 3.38) represents the wet-gas FVF.
PHASE BEHAVIOR
Fig. 3.4—Solution gas/oil ratios for brine, Rsw , and reservoir oils,
Rs , and inverse solution oil/gas ratio for reservoir gases, 1/rs , as
functions of pressure.
When a reservoir mixture produces both surface gas and oil, the
GOR, R go, defines the ratio of standard gas volume to a reference
oil volume (stock-tank- or separator-oil volume),
ǒV gǓ
V
+ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.18a)
R go +
Vo
(V o) sc
and R sp +
sc
ǒV gǓ
sc
(V o) sp
+
Vg
(V o) sp
. . . . . . . . . . . . . . . . . . . . . . (3.18b)
in units of scf/STB and scf/bbl, respectively. The separator conditions should be reported when separator GOR is used.
Solution gas/oil ratio, R s , is the volume of gas (at standard conditions) liberated from a single-phase oil at elevated pressure and temperature divided by the resulting stock-tank-oil volume, with units
scf/STB. R s is constant at pressures greater than the bubblepoint and
decreases as gas is liberated at pressures below the bubblepoint.
The producing GOR, R p, defines the instantaneous ratio of the total surface-gas volume produced divided by the total stock-tank-oil
volume. At pressures greater than bubblepoint, R p is constant and
equal to R s at bubblepoint. At pressures less than the bubblepoint,
R p may be equal to, less than, or greater than the R s of the flowing
reservoir oil. Typically, R p will increase 10 to 20 times the initial R s
because of increasing gas mobility and decreasing oil mobility during pressure depletion.
The surface volume ratio for gas condensates is usually expressed
as an oil/gas ratio (OGR), r og.
r og +
(V o) sc
ǒV gǓ
sc
+
Vo
+ 1 . . . . . . . . . . . . . . . . . . . . . . . (3.19)
Vg
R go
The unit for r og is STB/scf or, more commonly, “barrels per million”
(STB/MMscf). To avoid misinterpretation, it should be clearly specified whether the OGR includes natural gas liquids (NGL’s) in
addition to stock-tank condensate. In most petroleum engineering
calculations, NGL’s are not included in the OGR.
The ratio of surface oil to surface gas produced from a singlephase reservoir gas is referred to as the solution oil/gas ratio, r s. At
pressures above the dewpoint, the producing OGR, r p is constant
and equal to r s at the dewpoint. At pressures below the dewpoint, r p
is typically equal to or just slightly greater than r s; the contribution
of flowing reservoir oil to surface-oil production is negligible in
most gas-condensate reservoirs.
In the definitions of R p and r p, the total producing surface-gas
volume equals the surface gas from the reservoir gas plus the solution gas from the reservoir oil; likewise, the total producing surface
oil equals the stock-tank oil from the reservoir oil plus the condensate from the reservoir gas. Fig. 3.4 shows the behavior of R p, R s ,
and 1ńr s as a function of pressure.
GAS AND OIL PROPERTIES AND CORRELATIONS
Fig. 3.5—Reservoir viscosities as functions of pressure.
3.2.6 Viscosity. Two types of viscosity are used in engineering calculations: dynamic viscosity, m, and kinematic viscosity, n. The definition of m for Newtonian flow (which most petroleum mixtures
follow) is
m+
tg c
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.20)
duńdy
where t+shear stress per unit area in the shear plane parallel to the
direction of flow, du/dy+velocity gradient perpendicular to the
plane of shear, and g c +a units conversion from mass to force. The
two viscosities are related by density, where m+n ò.
Most petroleum engineering applications use dynamic viscosity,
which is the property reported in commercial laboratory studies.
The unit of dynamic viscosity is centipoise (cp), or in SI units,
mPa@s, where 1 cp+1 mPa@s. Kinematic viscosity is usually reported in centistoke (cSt), which is obtained by dividing m in cp by
ò in g/cm3; the SI unit for n is mm2/s, which is numerically equivalent to centistoke. Fig. 3.5 shows oil, gas, and water viscosities for
typical reservoir systems.
3.2.7 Diffusion Coefficients. In the absence of bulk flow, components in a single-phase mixture are transported according to gradients in concentration (i.e., chemical potential). Fick’s10 law for 1D
molecular diffusion in a binary system is given by
u i + * D i j ǒdC ińd xǓ , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.21)
where u i +molar velocity of Component i; D ij +binary diffusion
coefficient; and C i +molar concentration of Component i + y iò M,
where y i +mole fraction; and x+distance.
Eq. 3.21 clearly shows that mass transfer by molecular diffusion
can be significant for three reasons: (1) large diffusion coefficients,
(2) large concentration differences, and (3) short distances. A combination of moderate diffusion coefficients, concentration gradients, and distance may also result in significant diffusive flow. Molecular diffusion is particularly important in naturally fractured
reservoirs11,12 because of relatively short distances (e.g., small matrix block sizes).
Low-pressure binary diffusion coefficients for gases, D oij , are independent of composition and can be calculated accurately from
fundamental gas theory (Chapman and Enskog6), which are basically the same relations used to estimate low-pressure gas viscosity. No
well-accepted method is available to correct D oij for mixtures at high
pressure, but two types of corresponding-states correlations have
been proposed: D ij + D oij f(T r, p r) and D ij + D oij f(ò r).
At low pressures, diffusion coefficients are several orders of magnitude smaller in liquids than in gases. At reservoir conditions, the
difference between gas and liquid diffusion coefficients may be less
than one order of magnitude.
3.2.8 IFT. Interfacial forces act between equilibrium gas, oil, and
water phases coexisting in the pores of a reservoir rock. These forces
21
are generally quantified in terms of IFT, s; units of s are dynes/cm
(or equivalently, mN/m). The magnitude of IFT varies from [50
dynes/cm for crude-oil/gas systems at standard conditions to t0.1
dyne/cm for high-pressure gas/oil mixtures. Gas/oil capillary pressure, P c, is usually considered proportional to IFT according to the
Young-Laplace equation P c + 2sńr, where r is an average pore radius.13-15 Recovery mechanisms that are influenced by capillary
pressure (e.g., gas injection in naturally fractured reservoirs) will
necessarily be sensitive to IFT.
3.3 Gas Mixtures
This section gives correlations for PVT properties of natural gases,
including the following.
1. Review of gas volumetric properties.
2. Z-factor correlations.
3. Gas pseudocritical properties.
4. Wellstream gravity of wet gases and gas condensates.
5. Gas viscosity.
6. Dewpoint pressure.
7. Total volume factor.
3.3.1 Review of Gas Volumetric Properties. The properties of gas
mixtures are well understood and have been accurately correlated
for many years with graphical charts and EOS’s based on extensive
experimental data.16-19 The behavior of gases at low pressures was
originally quantified on the basis of experimental work by Charles
and Boyle, which resulted in the ideal-gas law,3
pV + nRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.22)
where R is the universal gas constant given in Appendix A for various units (Table A-2). In customary units,
R + 10.73146
psia
ft 3
, . . . . . . . . . . . . . . . . . . (3.23)
° R lbm mol
while for other units, R can be calculated from the relation
ǒ ǓǒT°R Ǔǒ Ǔǒmlbm Ǔ .
p
R + 10.73146 unit
psia
unit
V unit
ft 3
unit
. . . . . . . . (3.24)
For example, the gas constant for SPE-preferred SI units is given by
ǒ
kPa
6.894757
psia
R + 10.73146
ǒ
3
0.02831685 m3
ft
Ǔ ǒ
Ǔ
ǒ1.8 °R
Ǔ
K
2.204623 lbm
kg
Ǔ
kPa @ m 3
+ 8.3143
. . . . . . . . . . . . . . . . . . . . . . . . . (3.25)
K @ kmol
The gas constant can also be expressed in terms of energy units (e.g.,
R+8.3143 J/mol@K); note that J+N@m+(N/m2) m3+Pa@m3. In
this case, the conversion from one unit system to another is given by
R + 8.3143
ǒEJ ǓǒTK Ǔǒmg Ǔ .
unit
unit
unit
+
10.73146(60 ) 459.67)
14.7
+ 379.4 scfńlbm mol
+ 23.69 std m 3ńkmol .
. . . . . . . . . . . . . . . . . . . (3.27)
Second, the specific gravity of a gas directly reflects the gas molecular weight at standard conditions,
gg +
ǒò gǓ
Mg
Mg
sc
+
+
ǒò airǓ
M air
28.97
sc
and M g + 28.97 g g .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
For gas mixtures at moderate to high pressure or at low temperature the ideal-gas law does not hold because the volume of the constituent molecules and their intermolecular forces strongly affect the
volumetric behavior of the gas. Comparison of experimental data
for real gases with the behavior predicted by the ideal-gas law shows
significant deviations. The deviation from ideal behavior can be expressed as a factor, Z, defined as the ratio of the actual volume of one
mole of a real-gas mixture to the volume of one mole of an ideal gas,
Z+
volume of 1 mole of real gas at p and T
,
volume of 1 mole of ideal gas at p and T
. . . . . . . . . . . . . . . . . . . . (3.29)
where Z is a dimensionless quantity. Terms used for Z include deviation factor, compressibility factor, and Z factor. Z factor is used in
this monograph, as will the SPE reserve symbol Z (instead of the recommended SPE symbol z) to avoid confusion with the symbol z
used for feed composition.
From Eqs. 3.22 and 3.29, we can write the real-gas law including
the Z factor as
pV + nZRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.30)
which is the standard equation for describing the volumetric behavior of reservoir gases. Another form of the real-gas law written in
terms of specific volume ( v^ + 1ńò) is
pv^ + ZRTńM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.31)
or, in terms of molar volume (v + Mńò),
pv + ZRT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.32)
Z factor, defined by Eq. 3.30,
Z + pVńnRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.33)
is used for both phases in EOS applications (see Chap. 4). In this
monograph we use both Z and Z g for gases and Z o for oils; Z without
a subscript always implies the Z factor of a “gas-like” phase.
All volumetric properties of gases can be derived from the realgas law. Gas density is given by
ò g + pM gńZRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
. . . . . . . . . . . . . . . . (3.26)
An ideal gas is a hypothetical mixture with molecules that are
negligible in size and have no intermolecular forces. Real gases
mimic the behavior of an ideal gas at low pressures and high temperatures because the mixture volume is much larger than the volume
of the molecules making up the mixture. That is, the mean free path
between molecules that are moving randomly within the total volume is very large and intermolecular forces are thus very small.
Most gases at low pressure follow the ideal-gas law. Application
of the ideal-gas law results in two useful engineering approximations. First, the standard molar volume representing the volume occupied by one mole of gas at standard conditions is independent of
the gas composition.
22
ǒV Ǔ
sc
ǒv gǓ + v g + ng sc + RT
p sc
sc
or, in terms of gas specific gravity, by
p gg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.35)
ZRT
For wet-gas and gas-condensate mixtures, wellstream gravity, g w,
must be used instead of g g in Eq. 3.35.3 Gas density may range
from 0.05 lbm/ft3 at standard conditions to 30 lbm/ft3 for highpressure gases.
Gas molar volume, v g , is given by
ò g + 28.97
v g + ZRTńp,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.36)
where typical values of v g at reservoir conditions range from 1 to 1.5
ft3/lbm mol compared with 379 ft3/lbm mol for gases at standard
conditions. In Eqs. 3.30 through 3.36, R+universal gas constant.
PHASE BEHAVIOR
Pseudoreduced
Temperature
1 January 1941
Fig. 3.6—Standing-Katz4 Z-factor chart.
Gas compressibility, c g , is given by
ǒ Ǔ
ēV g
cg + * 1
V g ēp
ǒ Ǔ
+ 1p * 1 ēZ
Z ēp
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.37)
T
For sweet natural gas (i.e., not containing H2S) at pressures less than
[1,000 psia, the second term in Eq. 3.37 is negligible and c g + 1ńp
is a reasonable approximation.
Gas volume factor, B g, is defined as the ratio of gas volume at specified p and T to the ideal-gas volume at standard conditions,
Bg +
ǒTp Ǔ ZTp .
sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
For customary units ( psc +14.7 psia and Tsc +520°R), this is
B g + 0.02827 ZT
p , . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.39)
with temperature in °R and pressure in psia. This definition of B g
assumes that the gas volume at p and T remains as a gas at standard
conditions. For wet gases and gas condensates, the surface gas will
not contain all the original gas mixture because liquid is produced
GAS AND OIL PROPERTIES AND CORRELATIONS
after separation. For these mixtures, the traditional definition of B g
may still be useful; however, we refer to this quantity as a hypothetical wet-gas volume factor, B gw, which is calculated from Eq. 3.38.
Because B g is inversely proportional to pressure, the inverse volume factor, b g + 1ńB g , is commonly used. For field units,
p
. . . . . . . . . . . . . . . . . . . . . . (3.40a)
ZT
p
. . . . . . . . . . . . . . . (3.40b)
and b g in Mscfńbbl + 0.1985
ZT
b g in scfńft 3 + 35.37
If the reservoir gas yields condensate at the surface, the dry-gas
volume factor, B gd, is sometimes used.20
B gd +
ǒTp ǓǒZTpǓǒF1 Ǔ,
sc
sc
. . . . . . . . . . . . . . . . . . . . . . . (3.41)
gg
where F gg+ratio of moles of surface gas, n g , to moles of wellstream
mixture (i.e., reservoir gas, n g); see Eqs. 7.10 and 7.11 of Chap. 7.
3.3.2 Z-Factor Correlations. Standing and Katz4 present a generalized Z-factor chart (Fig. 3.6), which has become an industry standard for predicting the volumetric behavior of natural gases. Many
empirical equations and EOS’s have been fit to the original Standing-Katz chart. For example, Hall and Yarborough21,22 present an
23
accurate representation of the Standing-Katz chart using a Carnahan-Starling hard-sphere EOS,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
where a + 0.06125 t exp[* 1.2(1 * t) 2], where t + 1ńT pr.
The reduced-density parameter, y (the product of a van der Waals
covolume and density), is obtained by solving
f( y) + 0 + * ap pr )
y ) y2 ) y3 * y4
(1 * y) 3
* (14.76t * 9.76t 2 ) 4.58t 3)y 2
) (90.7t–242.2t 2 ) 42.4t 3)y 2.18)2.82 t,
with
*
1 ) 4y )
df(y)
+
dy
(1 * y) 4
4y 2
4y 3
)
. . . . . . . . . (3.43)
y4
* (29.52t * 19.52t 2) 9.16t 3)y
) (2.18 ) 2.82t)(90.7t * 242.2t 2 ) 42.4t 3)
y 1.18)2.82 t .
. . . . . . . . . . . . . . . . . . . . . . . . . (3.44)
The derivative ēZ/ēp used in the definition of c g is given by
ǒēZēpǓ
+ pa
T
pc
ap ńy
ƪ1y * df(y)ńdy
ƫ . . . . . . . . . . . . . . . . . . . . (3.45)
pr
2
An initial value of y+0.001 can be used with a Newton-Raphson
procedure, where convergence should be obtained in 3 to 10 iterations for Ťf( y)Ť + 1 10 *8.
On the basis of Takacs’23 comparison of eight correlations representing the Standing-Katz4 chart, the Hall and Yarborough21 and the
Dranchuk and Abou-Kassem24 equations give the most accurate
representation for a broad range of temperatures and pressures. Both
equations are valid for 1 x T r x 3 and 0.2 x p r x 25 to 30.
For many petroleum engineering applications, the Brill and
Beggs25 equation gives a satisfactory representation ("1 to 2%) of
the original Standing-Katz Z-factor chart for 1.2 t T r t 2. Also,
this equation can be solved explicitly for Z. The main limitations are
that reduced temperature must be u1.2 ([80°F) and t2.0
([340°F) and reduced pressure should be t15 ([10,000 psia).
The Standing and Katz Z-factor correlation may require special
treatment for wet gas and gas-condensate fluids containing significant amounts of heptanes-plus material and for gas mixtures with
significant amounts of nonhydrocarbons. An apparent discrepancy
in the Standing-Katz Z-factor chart for 1.05 t T r t 1.15 has been
“smoothed” in the Hall-Yarborough21 correlations. The Hall and
Yarborough (or Dranchuk and Abou-Kassem24) equation is recommended for most natural gases. With today’s computing capabilities, choosing simple, less-reliable equations, such as the Brill and
Beggs25 equation, is normally unnecessary.
The Lee-Kesler,26,27 AGA-8,28 and DDMIX29 correlations for Z
factor were developed with multiconstant EOS’s to give accurate
volumetric predictions for both pure components and mixtures.
They require more computation but are very accurate. These equations are particularly useful in custody-transfer calculations. They
also are required for gases containing water and concentrations of
nonhydrocarbons that exceed the limits of the Wichert and Aziz
method.30,31
3.3.3 Gas Pseudocritical Properties. Z factor, viscosity, and other
gas properties have been correlated accurately with correspondingstates principles, where the property is correlated as a function of reduced pressure and temperature.
Z + fǒ p r , T rǓ
and m g ńm gsc + fǒ p r , T rǓ, . . . . . . . . . . . . . . . . . . . . . . . . . (3.46)
24
Fig. 3.7—Gas pseudocritical properties as functions of specific
gravity.
where p r + pńp c and T r + TńT c. Such corresponding-states relations should be valid for most pure compounds when component
critical properties p c and T c are used. The same relations can be
used for gas mixtures if the mixture pseudocritical properties p pc
and T pc are used. Pseudocritical properties of gases can be estimated
with gas composition and mixing rules or from correlations based
on gas specific gravity.
Sutton7 suggests the following correlations for hydrocarbon gas
mixtures.
T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC . . . . . . . . . . . (3.47a)
and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC . . . . . . . . . . . (3.47b)
He claims that Eqs. 3.47a and 3.47b are the most reliable correlations for calculating pseudocritical properties with the Standing-Katz Z-factor chart. He even claims that this method is superior
to the use of composition and mixing rules.
Standing3 gives two sets of correlations: one for dry hydrocarbon
gases ( g gHC t 0.75),
T pcHC + 168 ) 325g gHC * 12.5g 2gHC . . . . . . . . . . . . . . (3.48a)
and p pcHC + 667 ) 15.0 g gHC * 37.5g 2gHC , . . . . . . . . . . (3.48b)
and one for wet-gas mixtures ( g gHC y 0.75),
T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a)
and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC . . . . . . . . . . . (3.49b)
The Standing correlations are used extensively in the industry; Fig.
3.7 compares them with the Sutton correlations. The Sutton and the
Standing wet-gas correlations for T pc give basically the same results,
whereas the three p pc correlations are quite different at g g u 0.85.
Kay’s5 mixing rule is typically used when gas composition is
available.
ȍy M ,
N
M+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
i+1
PHASE BEHAVIOR
ƪǒ
and å + 120 y CO ) y H
ǒ
2
4
) 15 y 0.5
H S * yH
2
2S
2S
Ǔ
Ǔ,
0.9
ǒ
* y CO ) y H
2
2S
Ǔ ƫ
1.6
. . . . . . . . . . . . . . . . . . . . . . . (3.52c)
* and p * are mixture pseudocriticals based on Kay’s mixwhere T pc
pc
ing rule. This method was developed from extensive data from natural gases containing nonhydrocarbons, with CO2 molar concentration ranging from 0 to 55% and H2S molar concentrations ranging
from 0 to 74%.
If only gas gravity and nonhydrocarbon content are known, the
hydrocarbon specific gravity is first calculated from
ǒ
Ǔ
g g * y N M N ) y CO M CO )y H S M H S ńM air
2
2
2
2
2
2
.
g gHC +
1 * y N * y CO * y H S
2
2
2
. . . . . . . . . . . . . . . . . . . . (3.53)
Hydrocarbon pseudocriticals are then calculated from Eqs. 3.47a
and 3.47b, and these values are adjusted for nonhydrocarbon content on the basis of Kay’s5 mixing rule.
ǒ
p *pc + 1 * y N * y CO * y H
2
2
2S
Ǔp
pcHC
) y N p cN ) y CO p c CO ) y H S p cH
2
Fig. 3.8—Heptanes-plus (pseudo)critical properties recommended for reservoir gases (from Standing,33 after Matthews et
al.32).
2
ȍy T
N
i
ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
i+1
ȍy p
N
and p pc +
i
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
i+1
where the pseudocritical properties of the C7+ fraction can be estimated from the Matthews et al.32 correlations (Fig. 3.8),3
Tc C
7)
+ 608 ) 364 logǒ M C
) ǒ2, 450 log M C
and p c C
7)
7)
7)
* 71.2 Ǔ
* 3, 800Ǔ log g C
+ 1, 188 * 431 logǒ M C
ƪ
7)
7)
. . . . . . (3.51a)
* 61.1 Ǔ
) 2, 319 * 852 logǒ M C * 53.7 Ǔ
7)
ƫǒg
C 7)*
0.8 Ǔ.
. . . . . . . . . . . . . . . . . . . (3.51b)
Kay’s mixing rule is usually adequate for lean natural gases that
contain no nonhydrocarbons. Sutton suggests that pseudocriticals
calculated with Kay’s mixing rule are adequate up to g g [ 0.85, but
that errors in calculated Z factors increase linearly at higher specific
gravities, reaching 10 to 15% for g g u 1.5. This bias may be a result
of the C7+ critical-property correlations used by Sutton (not Eqs.
3.51a and 3.51b).
When significant quantities of CO2 and H2S nonhydrocarbons
are present, Wichert and Aziz33,31 suggest corrections to arrive at
pseudocritical properties that will yield reliable Z factors from the
Standing-Katz chart. The Wichert and Aziz corrections are given by
*
* å , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.52a)
T pc + T pc
p pc +
p *pcǒ Tpc* * å Ǔ
*
T pc
) yH
2S
ǒ1 * y Ǔå
, . . . . . . . . . . . . . . . . . (3.52b)
H 2S
GAS AND OIL PROPERTIES AND CORRELATIONS
2
2
2S
. . . . . . . . . . (3.54a)
and T *pc + (1 * y N * y CO * y H S)T pcHC
2
2
2
) y N T cN ) y CO T c CO ) y H S T cH S .
2
T pc +
2
2
2
2
2
2
. . . . (3.54b)
T c* and p *c are used in the Wichert-Aziz equations with CO2 and H2S
mole fractions to obtain mixture T pc and p pc.
The Sutton7 correlations (Eqs. 3.47a and 3.47b) are recommended for hydrocarbon pseudocritical properties. If composition
is available, Kay’s mixing rule should be used with the Matthews et
al.32 pseudocriticals for C7+. Gases containing significant amounts
of CO2 and H2S nonhydrocarbons should always be corrected with
the Wichert-Aziz equations. Finally, for gas-condensate fluids the
wellstream specific gravity, g w (discussed in the next section),
should replace g g in the equations above.
3.3.4 Wellstream Specific Gravity. Gas mixtures that produce condensate at surface conditions may exist as a single-phase gas in the
reservoir and production tubing. This can be verified by determining the dewpoint pressure at the prevailing temperature. If wellstream properties are desired at conditions where the mixture is
single-phase, surface-gas and -oil properties must be converted to
a wellstream specific gravity, g w. This gravity should be used
instead of g g to estimate pseudocritical properties.
Wellstream gravity r p represents the average molecular weight of
the produced mixture (relative to air) and is readily calculated from
the producing-oil (condensate)/gas ratio, r p; average surface-gas
gravity g g ; surface-condensate gravity, g o ; and surface-condensate
molecular weight M o .
gw +
g g ) 4, 580 r p g o
, . . . . . . . . . . . . . . . . . . . (3.55)
1 ) 133, 000 r p ǒ gńM Ǔ o
with r p in STB/scf. Average surface-gas gravity is given by
N sp
ȍR
gg +
pi g gi
i+1
N sp
ȍR
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.56)
pi
i+1
where R pi +GOR of Separator Stage i. Standing33 presents Eq.
3.55 graphically in Fig. 3.9.
When M o is not available, Standing gives the following correlation.
25
Solution Gas/Oil Ratio, scf/STB
Oil/Gas Ratio, STB/MMscf
Fig. 3.9—Wellstream gravity relative to surface average gas gravity as a function of solution oil/gas ratio and surface gravities.
M o + 240 * 2.22 g API . . . . . . . . . . . . . . . . . . . . . . . . . (3.57)
This relation should not be extrapolated outside the range
45 t g API t 60. Eilerts34 gives a relation for ( gńM) o ,
ǒ gńMǓ + ǒ1.892
o
* ǒ4.52
10 *3Ǔ ) ǒ7.35
10 *5Ǔg API
2
10 *8Ǔg API
, . . . . . . . . . . . . . . . . . . . (3.58)
which should be reliable for most condensates. When condensate
molecular weight is not available, the recommended correlation for
M o is the Cragoe35 correlation,
Mo +
6, 084
,
g API * 5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
which gives reasonable values for all surface condensates and
stock-tank oils.
A typical problem that often arises in the engineering of gas-condensate reservoirs is that all the data required to calculate wellstream gas volumes and wellstream specific gravity are not available and must be estimated.36-38 In practice, we often report only the
first-stage-separator GOR (relative to stock-tank-oil volume) and
gas specific gravity, R s1 and g g1, respectively; the stock-tank-oil
gravity, g o ; and the primary-separator conditions, p sp1 and T sp1.
To calculate g w from Eq. 3.55 we need total producing OGR, r p,
which equals the inverse of R s1 plus the additional gas that will be
released from the first-stage separator oil, R s),
rp +
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.60)
ǒR s1 ) R s)Ǔ
R s) can be estimated from several correlations.37,39 Whitson38 proposes use of a bubblepoint pressure correlation (e.g., the Standing40
correlation),
R s) + A 1g g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.61a)
and A 1 +
p
ƪǒ18.2
) 1.4 Ǔ10 ǒ
sp1
0.0125g API*0.00091T sp1
Ǔ
ƫ
,
with p sp1 in psia, T sp1 in °F, and R s) in scf/STB. g g) is the gas gravity of the additional solution gas released from the separator oil. The
Katz41 correlation (Fig. 3.10) can be used to estimate g g), where a
best-fit representation of his graphical correlation is
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.62)
where A 2 + 0.25 ) 0.02g API and A 3 + * (3.57
26
Solving Eqs. 3.61 and 3.62 for R s) gives
R s) +
A1 A2
ǒ1 * A 1 A 3Ǔ
10 *6)g API .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.63)
Average surface separator gas gravity, g g, is given by
gg +
g g1 R s1 ) g g) R s)
.
R s1 ) R s)
. . . . . . . . . . . . . . . . . . . . . . (3.64)
Although the Katz correlation is only approximate, the impact of a
few percent error in g g) is not of practical consequence to the calculation of g w because R s) is usually much less than R s1 .
3.3.5 Gas Viscosity. Viscosity of reservoir gases generally ranges
from 0.01 to 0.03 cp at standard and reservoir conditions, reaching
up to 0.1 cp for near-critical gas condensates. Estimation of gas viscosities at elevated pressure and temperature is typically a two-step
procedure: (1) calculating mixture low-pressure viscosity m gsc at
p sc and T from Chapman-Enskog theory3,6 and (2) correcting this
value for the effect of pressure and temperature with a corresponding-states or dense-gas correlation. These correlations relate the actual viscosity m g at p and T to low-pressure viscosity by use of the
ratio m gńm gsc or difference ( m g * m gsc) as a function of pseudoreduced properties p pr and T pr or as a function of pseudoreduced density ò pr.
Gas viscosities are rarely measured because most laboratories do
not have the required equipment; thus, the prediction of gas viscosity is particularly important. Gas viscosity of reservoir systems is
often estimated from the graphical correlation m gńm gsc + f(T r, p r)
proposed by Carr et al.42 (Fig. 3.11). Dempsey43 gives a polynomial
approximation of the Carr et al. correlation. With these correlations,
gas viscosities can be estimated with an accuracy of about "3% for
most applications. The Dempsey correlation is valid in the range
1.2 x T r x 3 and 1 x p r x 20.
The Lee-Gonzalez gas viscosity correlation (used by most PVT
laboratories when reporting gas viscosities) is given by44
1.205
. . . . . . . . . . . . . . . . . . . (3.61b)
g g) + A 2 ) A 3 R s) ,
Fig. 3.10—Correlation for separator-oil dissolved gas gravity as
a function of stock-tank-oil gravity and separator-oil GOR (from
Ref. 41).
mg + A1
where A 1 +
10 *4 expǒA 2 ò g 3Ǔ ,
A
. . . . . . . . . . . . . . . . . . (3.65a)
ǒ9.379 ) 0.01607M gǓT 1.5
209.2 ) 19.26M g ) T
,
A 2 + 3.448 ) ǒ986.4ńTǓ ) 0.01009M g ,
and A 3 + 2.447 * 0.2224A 2 , . . . . . . . . . . . . . . . . . . . . (3.65b)
with m g in cp, ò g in g/cm3, and T in °R. McCain19 indicates the accuracy of this correlation is 2 to 4% for gg t1.0, with errors up to
20% for rich gas condensates with g g u 1.5.
PHASE BEHAVIOR
Gas Gravity (air+1)
N2, mol%
CO2, mol%
H2S, mol%
g
o
Molecular Weight
Pseudoreduced Temperature, Tr
Fig. 3.11—Carr et al.42 gas-viscosity correlation.
Lucas45 proposes the following gas viscosity correlation, which
is valid in the range 1 t T r t 40 and 0 t p r t 100 (Fig. 3.12)6:
m gńm gsc + 1 )
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ *1
(1.245
where A 1 +
A 1 p 1.3088
pr
A
10 *3)
A
,
expǒ
T pr
. . . . . . . (3.66a)
5.1726T *0.3286
pr
Ǔ
0.4489 expǒ3.0578T *37.7332
pr
,
T pr
A4 +
Ǔ
1.7368 expǒ2.2310T *7.6351
pr
,
T pr
Ǔ
,
where m gsc c + ƪ0.807T pr0.618 * 0.357 expǒ* 0.449T prǓ
) 0.340 expǒ* 4.058T prǓ ) 0.018ƫ ,
,
N
i
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ , . . . . . . . . . . . (3.66b)
GAS AND OIL PROPERTIES AND CORRELATIONS
ȍy Z
and p pc + RT pc
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ ,
A3 +
ǒ Ǔ
1ń6
T pc
c + 9, 490
M 3p 4pc
ci
i+1
N
ȍy v
,
. . . . . . . . . . . . . . . . . . . . . . . . (3.67)
i ci
i+1
with c in cp*1, T and T c in °R, and p c in psia. Special corrections
should be applied to the Lucas correlation when polar compounds,
such as H2S and water, are present in a gas mixture. The effect of
H2S is always t1% and can be neglected, and appropriate corrections can be made to treat water if necessary.
Given its wide range of applicability, the Lucas method is recommended for general use. When compositions are not available, correlations for pseudocritical properties in terms of specific gravity
can be used instead. Standing2 gives equations for m gsc in terms of
g g, temperature, and nonhydrocarbon content,
m gsc + ǒ m gscǓ uncorrected ) Dm N ) Dm CO ) Dm H S ,
2
2
2
. . . . . . . . . . . . . . . . . . . . (3.68a)
27
) A 7ǒ z C
pr +p/pc ; T/Tc ; h in mp
4
c+0.176(Tc /M3/pc )1/6; Tc in K, pc in bar
ȳ
C 7)
Ǔ 3) A ȱ
8
ȧ
ȧ
ȲǒgC7) ) 0.0001 Ǔȴ
M
7)
ȱ
) Aȧ
Ȳǒ g
MC
7)
ȳ )A ȱ
ȧ ȧǒg
) 0.0001 Ǔȴ
Ȳ
2
MC
9
7)
3
10
C 7)
ȳ
ȧ
) 0.0001 Ǔȴ
MC
C 7)
7)
) A 11 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.69)
where A1+*2.0623054, A2+6.6259728, A3+*4.4670559
10*3, A4+1.0448346 10*4, A5+3.2673714 10*2, A6+
*3.6453277 10*3, A7+7.4299951 10*5, A8+*1.1381195
10*1, A9+6.2476497 10*4, A10+*1.0716866 10*6, and
A11+1.0746622 101.
The range of properties used to develop this correlation includes
dewpoints from 1,000 to 10,000 psia, temperatures from 40 to 320°F,
and a wide range of reservoir compositions. The correlation usually
can be expected to predict dewpoints with an accuracy of "10% for
condensates that do not contain large amounts of nonhydrocarbons.
This is acceptable in light of the fact that experimental dewpoint pressures are probably determined with an accuracy of only "5%. The
correlation is generally used only for preliminary reservoir studies
conducted before an experimental dewpoint is available.
Organick and Golding50 and Kurata and Katz51 present graphical
correlations for dewpoint pressure.
pr +0
pr +0
Fig. 3.12—Lucas45 corresponding-states generalized viscosity
correlation (Ref. 6); h+dynamic viscosity and mp+micropoise+10*6 poise+10*4 cp.
3.3.7 Total FVF. Total FVF,3,17,46 B t, is defined as the volume of
a two-phase, gas-oil mixture (or sometimes a single-phase mixture)
at elevated pressure and temperature divided by the stock-tank-oil
volume resulting when the two-phase mixture is brought to surface
conditions,
Bt +
10 *3Ǔ ) ƪǒ1.709
where ǒ m gscǓ uncorrected + ǒ8.188
10 *6Ǔg gƫT * ǒ6.15
* ǒ2.062
Dm N + y N ƪǒ8.48
2
Dm CO + y CO ƪǒ9.08
2
10 *3Ǔ log g g ,
10 *3Ǔƫ,
10 *3Ǔ log g g ) ǒ9.59
2
10 *3Ǔƫ ,
10 *3Ǔ log g g ) ǒ6.24
2
and Dm H2S + y H 2Sƪǒ8.49
10 *5Ǔ
10 Ǔƫ.
10 Ǔ log g g ) ǒ3.73
*3
*3
. . . . . . . . . . . . . . . . . . . (3.68b)
Reid et al.6 review other gas viscosity correlations with accuracy
similar to that of the Lucas correlation.
3.3.6 Dewpoint Pressure. The prediction of retrograde dewpoint
pressure is not widely practiced. It is generally recognized that the
complexity of retrograde phase behavior necessitates experimental
determination of the dewpoint condition. Sage and Olds’46 data are
perhaps the most extensive tabular correlation of dewpoint pressures. Eilerts et al.47,48 also present dewpoint pressures for several
light-condensate systems.
Nemeth and Kennedy49 have proposed a dewpoint correlation
based on composition and C7+ properties.
ƪ
ln p d+A 1 z C ) z CO ) z H S )z C )2(z C )z C ) ) z C
2
2
2
ƫ
) 0.4z C )0.2z N ) A 2 g C
1
) A 4T )ǒA 5z C
28
2
7)
MC
7)
3
6
ȱ
) Aȧ
Ȳǒz
7)
4
ȳ
ȧ
) 0.002 Ǔȴ
zC
3
C 7)
5
Ǔ) A6ǒzC7)MC7)Ǔ2
1
Vo ) Vg
Vo ) Vg
+
. . . . . . . . . . . . . . . . . . . . . . (3.70)
Vo
(V o) sc
B t is used for calculating the oil in place for gas-condensate reservoirs, where V o + 0 in Eq. 3.70. Assuming 1 res bbl of hydrocarbon PV, the initial condensate in place is given by N + 1ńB t (in
STB) and the initial “dry” separator gas in place is G + Nńr p ,
where r p +initial producing (solution) OGR.
For gas-condensate systems, Sage and Olds46 give a tabulated
correlation for B t.
R pT
B t + p Z *,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.71)
where R p +producing GOR in scf/STB, B t is in bbl/STB, T is in °R,
and p is in psia. Z* varies with pressure and temperature, where the
tabulated correlation for Z* is well represented by
p
p 1.5
, . . . . . . . . (3.72)
Z * + A 0 ) A 1p ) A 2p 1.5 ) A 3 ) A 4
T
T
*
*
3
6
where A0+5.050 10 , A1+*2.740 10 , A2+3.331 10*8,
A3+2.198 10*3, and A4+*2.675 10*5 with p in psia and T
in °R. Although the Sage and Olds data only cover the range
600tpt3,000 psia and 100tTt250°F, Eq. 3.72 gives acceptable
results up to 10,000 psia and 350°F (when gas volume is much larger
than oil volume).
When reservoir hydrocarbon volume consists only of gas, the following relations apply for total FVF.
B t + B gd R p + B gw ǒR p ) C og Ǔ ,
. . . . . . . . . . . . . . . . . (3.73a)
C og + 133, 000 ǒg ońM oǓ , . . . . . . . . . . . . . . . . . . . . . . (3.73b)
M o [ 6, 084ńǒ g API * 5.9Ǔ ,
. . . . . . . . . . . . . . . . . . . . . (3.73c)
and g API + 141.5ń(131.5 ) g o) ,
. . . . . . . . . . . . . . . . . (3.73d)
B gw +wet-gas FVF in ft3/scf
where B gd +dry gas FVF in
(given by Eq. 3.38), C og +gas equivalent conversion factor in scf/
STB (see Chap. 7), and R p +producing GOR in scf/STB.
ft3/scf,
PHASE BEHAVIOR
PR EOS
Glasø Uncorrected
Glasø Corrected
C7+ Watson Characterization Factor, KwC7+
C7+ Watson Characterization Factor, KwC7+
Fig. 3.13—Effect of paraffinicity, Kw , on bubblepoint pressure.
Standing3 gives a graphical correlation for B t using a correlation
parameter A defined as
0.5
A + R p T 0.3 g ao , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.74)
gg
where a + 2.9 10 *0.00027 Rp. Standing’s correlation is valid for
both oil and gas-condensate systems and can be represented with
log B t + * 5.262 *
47.4
, . . . . . . . . . (3.75a)
* 12.22 ) log A *
matically.8 An accurate method is needed to correlate the
bubblepoint pressure, temperature, and solution gas/oil ratio.
Oil properties can be grouped into two categories: saturated and
undersaturated properties. Saturated properties apply at pressures at
or below the bubblepoint, and undersaturated properties apply at
pressures greater than the bubblepoint. For oils with initial GOR’s
less than [500 scf/STB, assuming linear variation of undersaturated-oil properties with pressure is usually acceptable.
. . (3.75b)
3.4.1 Bubblepoint Pressure. The correlation of bubblepoint pressure has received more attention than any other oil-property correlation. Standing3,17,40 developed the first accurate bubblepoint correlation, which was based on California crude oils.
and A is given by Eq. 3.74. On the basis of data from North Sea oils,
Glasø52 gives a correlation for B t using the Standing correlation parameter A (Eq. 3.74):
p b + 18.2ǒ A * 1.4 Ǔ, . . . . . . . . . . . . . . . . . . . . . . . . . . (3.78)
where A + ǒR ńg Ǔ 0.83 10 ǒ0.00091T*0.0125g APIǓ, with R in scf/STB, T
ǒ
where log A * + log A * 10.1 *
log B t + ǒ8.0135
96.8
6.604 ) log p
Ǔ
10 *2Ǔ ) 0.47257 log A *
) 0.17351ǒlog A * Ǔ 2 , . . . . . . . . . . . . . . . . . . . . . (3.76)
where A*+A p*1.1089.
Either the Standing or the Glasø correlations for B t can be used
with approximately the same accuracy. However, neither correlation is consistent with the limiting conditions
B t + B o for V g + 0 . . . . . . . . . . . . . . . . . . . . . . . . . . (3.77a)
s
s
g
in °F, and p b in psia.
Lasater53 used a somewhat different approach to correlate bubblepoint pressure, where mole fraction y g of solution gas in the reservoir oil is used as the main correlating parameter17:
p b + A gT , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.79)
g
with T in °R and p b in psia. The function A( y g ) is given graphically
by Lasater, and his correlation can be accurately described by
A + 0.83918
; y g x 0.6 . . . . . . . . (3.80a)
10 1.17664yg y 0.57246
g
and B t + B gd R p for V o + 0. . . . . . . . . . . . . . . . . . . . (3.77b)
and A + 0.83918
B t correlations evaluated at a bubblepoint usually will underpredict
the actual B ob by [0.2.
133, 000 ǒ gńMǓ o
where y g + 1 )
Rs
3.4 Oil Mixtures
This section gives correlations for PVT properties of reservoir oils,
including bubblepoint pressure and oil density, compressibility,
FVF, and viscosity. With only a few exceptions, oil properties have
been correlated in terms of surface-oil and -gas properties, including
solution gas/oil ratio, oil gravity, average surface-gas gravity, and
temperature. A few correlations are also given in terms of composition and component properties.
Reservoir oils typically contain dissolved gas consisting mainly
of methane and ethane, some intermediates (C3 through C6), and
lesser quantities of nonhydrocarbons. The amount of dissolved gas
has an important effect on oil properties. At the bubblepoint a discontinuity in the system volumetric behavior is caused by gas coming out of solution, with the system compressibility changing draGAS AND OIL PROPERTIES AND CORRELATIONS
10 1.08000yg y 0.31109
; y g u 0.6, . . . . . (3.80b)
g
ƪ
ƫ
*1
, . . . . . . . . . . . . (3.81)
with R s in scf/STB. In this correlation, the gas mole fraction is dependent mainly on solution gas/oil ratio, but also on the properties
of the stock-tank oil. The Cragoe35 correlation given by Eq. 3.59 is
recommended for estimating M o when stock-tank-oil molecular
weight is not known.
Standing’s approach was used by Glasø52 for North Sea oils, resulting in the correlation
log p b + 1.7669 ) 1.7447 log A * 0.30218(log A) 2 ,
. . . . . . . . . . . . . . . . . . . . (3.82)
ǒT
Ǔ with p b in psia, T in °F and
where A + ǒR sńg gǓ
R s in scf/STB. Glasø’s corrections for nonhydrocarbon content
and stock-tank-oil paraffinicity are not widely used, primarily be0.816
0.172
ńg 0.989
API
29
cause the necessary data are not available. Sutton and Farshad54
mention that the API correction for paraffinicity worsened bubblepoint predictions for gulf coast fluids. Fig. 3.13 gives an explanation for this observation.
Fig. 3.13 shows the effect of paraffinicity (which is quantified by
the Watson characterization factor, K w) on bubblepoint pressure;
the figure is based on calculations with a tuned EOS for an Asian oil
(solid circles). The oil composition is constant in the example calculation. The 12 C7+ fractions are each split into a paraffinic pseudocomponent and an aromatic pseudocomponent (i.e., 24 C7+ pseudocomponents). The paraffinic fraction was varied, and bubblepoint
calculations were made. The variation in paraffinicity is expressed
in terms of the overall C7+ Watson characterization factor. Also
shown in the figure are the variation in solution gas/oil ratio and the
oil specific gravity with K wC .
7)
The actual reservoir oil has a K wC + 11.55, where the EOS
7)
bubblepoint is close to the uncorrected Glasø bubblepoint prediction. When the correction for paraffinicity is applied, the correction
gives a poorer bubblepoint prediction (even though the overall trend
in bubblepoints is improved by the Glasø paraffinicity corrections).
A quantitatively similar correction to the Glasø correction (but
easier to use) is based on the estimate for Whitson’s55,56 Watson
characterization factor, K w, and yields
ǒ g oǓ
corrected+
ǒ g oǓ measuredǒ K wń11.9 Ǔ 1.1824.
. . . . . . . . . . . (3.83)
The corrected specific gravity correlation is used in the Glasø bubblepoint correlation instead of the measured specific gravity. An estimate
of Kw for the stock-tank oil must be available to use this correction.
Vazquez and Beggs57 give the following correlations. For
g API x 30,
pb
ȱ R ǒ
+ȧ27.64ǒg Ǔ10
Ȳ
*11.172 g API
s
gc
T)460
Ǔȳ
ȧ
ȴ
0.9143
,
In summary, significant differences in predicted bubblepoint pressures should not be expected for most reservoir oils with most of the
previous correlations. The Lasater and Standing equations are recommended for general use and as a starting point for developing reservoir-specific correlations. Correlations developed for a specific region, such as Glasø’s correlation for the North Sea, should probably
be used in that region and, in the case of Glasø’s correlation, may be
extended to other regions by use of the paraffinicity correction.
3.4.2 Oil Density. Density of reservoir oil varies from 30 lbm/ft3 for
light volatile oils to 60 lbm/ft3 for heavy crudes with little or no solution gas. Oil compressibility may range from 3 10*6 psi*1 for
heavy crude oils to 50 10*6 psi*1 for light oils. The variation of
oil compressibility with pressure is usually small, although for volatile oils the effect can be significant, particularly for material-balance
and reservoir-simulation calculations of highly undersaturated volatile oils. Several methods have been used successfully to correlate oil
volumetric properties, including extensions of ideal-solution mixing,
EOS’s, corresponding-states correlations, and empirical correlations.
Oil density based on black-oil properties is given by
òo +
62.4g o ) 0.0136g g R s
, . . . . . . . . . . . . . . . . . . . . (3.88)
Bo
with ò o in lbm/ft3, B o in bbl/STB, and R s in scf/STB. Correlations
can be used to estimate R s and B o from g o, g g, p, and T.
Standing-Katz Method. Standing3,17 and Standing and Katz58
give an accurate method for estimating oil densities that uses an extension of ideal-solution mixing.
ò o + ò po ) D ò p * D ò T , . . . . . . . . . . . . . . . . . . . . . . (3.89)
where ò po is the pseudoliquid density at standard conditions and the
terms Dò T and Dò p give corrections for temperature and pressure,
respectively. Pseudoliquid density is calculated with ideal-solution
mixing and correlations for the apparent liquid densities of ethane
. . . . . . . . . . (3.84)
and, for g API u 30,
pb +
ƪ
ǒ Ǔ ǒ
R
56.06 g s 10
gc
T)460
ƫ
Ǔ
*10.393g API
0.8425
, . . . . . . . . . (3.85)
with p b in psia, T in °F and R s in scf/STB. These equations are based
on a large number of data from commercial laboratories. Vazquez
and Beggs correct for the effect of separator conditions using a modified gas specific gravity, g gc , which is correlated with first-stageseparator pressure and temperature, and stock-tank-oil gravity.
ƪ
g gc + g g 1 ) ǒ0.5912
10 *4Ǔ g APIT sp log
p
ǒ114.7
Ǔƫ,
sp
. . . . . . . . . . . . . . . . . . . . (3.86)
with T sp in °F and p sp in psia.
Standing’s correlation can be used to develop field- or reservoirspecific bubblepoint correlations. A linear relation is usually assumed between bubblepoint pressure and the Standing correlating
coefficient. This is a standard approach used in the industry, and the
Standing bubblepoint correlating parameter is well suited for developing field-specific correlations.
Sometimes the solution gas/oil ratio is needed at a given pressure,
and this is readily calculated by solving the bubblepoint correlation
for R s . For the Standing correlation,
ƪ(0.055p )10 1.4)10
ƫ
0.0125g API
1.205
. . . . . . . . . (3.87)
System Density at 60°F and 14.7 psia, g/cm3
similar relations can be derived for the other bubblepoint correlations.
Fig. 3.14—Apparent liquid densities of methane and ethane
(from Standing33).
Rs + gg
30
0.00091T
;
PHASE BEHAVIOR
g
Fig. 3.15—Chart for calculating pseudoliquid density of reservoir oil (from Standing33).
and methane at standard conditions. Given oil composition x i, ò po
is calculated from
ȍx M
N
i
ò po +
i
i+1
ȍǒx M ńò Ǔ
N
i
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.90)
i
i+1
where Standing and Katz show that apparent liquid densities ò i of
C2 and C1 are functions of the densities ò 2) and ò po , respectively
(Fig. 3.14).
ò C + 15.3 ) 0.3167 ò C
2
2)
ò C + 0.312 ) 0.45 ò po , . . . . . . . . . . . . . . . . . . . . . . . (3.91)
1
C 7)
ȍxM
i
where ò C
2)
+
i
i+C 2
C 7)
ȍ ǒx M ńò Ǔ
i
i
,
. . . . . . . . . . . . . . . . . . . . (3.92)
i
i+C 2
GAS AND OIL PROPERTIES AND CORRELATIONS
with ò in lbm/ft3. Application of these correlations results in an apparent trial-and-error calculation for ò po . Standing33 presents a
graphical correlation (Fig. 3.15) based on these relations, where ò po
is found from ò C3) and weight fractions of C2 and C1 (w C2 and w C1,
respectively).
Figs. 3.16 and 3.17 show the pressure and temperature corrections, D ò p and D ò T , graphically. D ò p is a function of ò po, and D ò T
is a function of ( ò po ) D ò p ). Madrazo59 introduced modified
curves for D ò p and D ò T that improve predictions at higher pressures and temperatures. Standing3 gives best-fit equations for his original graphical correlations of D ò p and D ò T (Eqs. 3.98 and 3.99),
which are not recommended at temperatures u240°F; instead, Madrazo’s graphical correlation can be used. The correction factors can
also be used to determine isothermal compressibility and oil FVF at
undersaturated conditions.
The treatment of nonhydrocarbons in the Standing-Katz method
has not received much attention, and the method is not recommended when concentrations of nonhydrocarbons exceed 10 mol%.
Standing3 suggests that an apparent liquid density of 29.9 lbm/ft3
can be used for nitrogen but does not address how the nonhydrocarbons should be considered in the calculation procedure (i.e., as part
of the C3+ material or following the calculation of ò C and ò C ).
2
1
Madrazo indicates that the volume contribution of nonhydrocar31
Density of System at 60°F and 14.7 psia, lbm/ft3
Fig. 3.16—Pressure correction to the pseudoliquid density at 14.7 psia and 60°F (from Ref. 59).
bons can be neglected completely if the total content is t6 mol%.
Vogel and Yarborough60 suggest that the weight fraction of nitrogen
should be added to the weight fraction of ethane.
Using additive volumes and Eqs. 3.91 and 3.92, we can show that
ò C and ò po can be calculated explicitly. Thus, the following is the
2)
most direct procedure for calculating ò o from the Standing-Katz
method.
1. Calculate the mass of each component.
m i + x i M i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.93)
2. Calculate V C
C 7)
VC
3)
+
ȍ
i+C 3
3)
.
mi
ò i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.94)
where ò i are component densities at standard conditions (Appendix A).
3. Calculate ò C .
2)
* b ) Ǹb 2 * 4ac
, . . . . . . . . . . . . . . . . . . . . (3.95)
òC +
2)
2a
32
where a + 0.3167V C , b + m C * 0.3167 m C ) 15.3 V C ,
3)
2
2)
3)
and c + * 15.3m C .
2)
4. Calculate V C .
2)
VC
2)
+ VC
mC
)ò 2
3)
C
2
+ VC
3)
)
mC
2
15.3 ) 0.3167ò C
. . . . . . . . . . . . . (3.96)
2)
5. Calculate ò po.
ò po +
*b ) Ǹb 2 * 4ac
, . . . . . . . . . . . . . . . . . . . . . (3.97)
2a
where a + 0.45V C , b + m C * 0.45m C ) 0.312V C , and
2)
1
1)
2)
c + * 0.312m C .
1)
6. Calculate the pressure effect on density.
D ò p + 10 *3 ƪ0.167 ) ǒ16.181
* 10 *8 ƪ0.299 ) ǒ263
10 *0.0425òpoǓƫ p
10 *0.0603òpoǓƫ p 2. . . . . . (3.98)
PHASE BEHAVIOR
Density of System at Pressure and 60°F, lbm/ft3
Fig. 3.17—Temperature correction to the pseudoliquid density at pressure and 60°F (from Ref. 59).
ò ga + 38.52
7. Calculate the temperature effect on density.
ƪ
D ò T + (T * 60) 0.0133 ) 152.4ǒò po ) D ò pǓ
NJ
* (T * 60) ǒ8.1
2
* ƪ0.0622
*2.45
ƫ
10 Ǔ
*6
Nj
10 *0.0764(òpo)D òp)ƫ .
. . . . . . . . . . . (3.99)
8. Calculate mixture density from Eq. 3.89.
In the absence of oil composition, Katz41 suggests calculating the
pseudoliquid density from stock-tank-oil gravity, g o, solution gas/
oil ratio, R s , and apparent liquid density of the surface gas, ò ga, taken from a graphical correlation (Fig. 3.18),
ò po +
62.4g o ) 0.0136 R s g g
1 ) 0.0136ǒR s g gńò gaǓ
. . . . . . . . . . . . . . . . . . . (3.100)
Standing gives an equation for ò ga.
GAS AND OIL PROPERTIES AND CORRELATIONS
10 *0.00326 g API
) (94.75 * 33.93 log g API) log g g , . . . . . . . . . . . (3.101)
with ò ga in lbm/ft3 and Rs in scf/STB.
Alani-Kennedy 61 Method. The Alani-Kennedy method for calculating oil density is a modification of the original van der Waals
EOS, with constants a and b given as functions of temperature for
normal paraffins C1 to C10 and iso-butane (Table 3.1); two sets of
coefficients are reported for methane (for temperatures from 70 to
300°F and from 301 to 460°F) and two sets for ethane (for temperatures from 100 to 249°F and from 250 to 460°F). Lohrenz et al.62
give Alani-Kennedy temperature-dependent coefficients for nonhydrocarbons N2, CO2, and H2S (Table 3.1). The Alani-Kennedy
equations are summarized next. Eqs. 3.102b and 3.102c are in the
original van der Waals EOS but are not used.
p + RT * a2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102a)
v*b v
33
where log a C
+ ǒ3.8405985
10 *3Ǔ M C
* ǒ9.5638281
MC
10 *4Ǔ g 7) ) 261.80818
T
C
7)
7)
7)
) ǒ7.3104464
10
*6
Ǔ
M 2C
7)
) 10.753517 . . . . . . . . . . . . . . . . . . . . . (3.103a)
and b C
7)
+ ǒ3.499274
10 *2Ǔ M C
) ǒ2.232395
10 *4ǓT * ǒ1.6322572
7)
* 7.2725403 g C
7)
MC
10 *2Ǔ g 7)
C
7)
) 6.2256545, . . . . . . . . . . . . . . . . . . . . . . . (3.103b)
with ò in lbm/ft3, v in ft3/lbm mol, T in °R, p in psia, and R+universal gas constant+10.73.
Solution of the cubic equation for volume is presented in Chap.
4. Density is given by ò+M/v, where M is the mixture molecular
weight and v is the molar volume given by the solution to the cubic
equation. The Alani-Kennedy method can also be used to estimate
oil compressibilities.
Rackett,63 Hankinson and Thomson,64 and Hankinson et al.65
give accurate correlations for pure-component saturated-liquid densities, and although these correlations can be extended to mixtures,
they have not been tested extensively for reservoir systems. Cullick
et al.66 give a modified corresponding-states method for predicting
density of reservoir fluids, The method has a better foundation and
extrapolating capability than the methods discussed previously
(particularly for systems with nonhydrocarbons); however, space
does not allow presentation of the method in its entirety.
Either the Standing-Katz or Alani-Kennedy method should estimate the densities of most reservoir oils with an accuracy of "2%.
The Alani-Kennedy method is suggested for systems at temperatures u250°F and for systems containing appreciable amounts of
nonhydrocarbons (u5 mol%). Cubic EOS’s (e.g., Peng-Robinson
or Soave-Redlich-Kwong) that use volume translation also estimate
liquid densities with an accuracy of a few percent (e.g., the recommended characterization procedures in Chap. 5 or other proposed
characterizations 67,68).
Fig. 3.18—Apparent pseudoliquid density of separator gas (from
Standing,33 after Katz41).
R 2T 2
a i + 27 p ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102b)
ci
64
RT
b i + 1 p ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102c)
8 ci
ȍx a
N
a+
i
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102d)
i+1
ȍx b ,
N
b+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102e)
i+1
ai +
a 1i
) log a 2i; i 0 C 7) ,
T
. . . . . . . . . . . . . . . . . . (3.102f)
and b i + b 1iT ) b 2i ; i 0 C 7), . . . . . . . . . . . . . . . . . (3.102g)
TABLE 3.1—CONSTANTS FOR ALANI-KENNEDY61 OIL DENSITY CORRELATION
Component
a1
a2
N2
4,300
CO2
8,166
126.0
2.293
H2 S
13,200
0.0
b1
104
b2
4.49
0.3853
0.1818
0.3872
17.9
0.3945
C1
At 70 to 300°F
At 300 to 460°F
9,160.6413
147.47333
61.893223
3,247.4533
*3.3162472
*14.072637
0.50874303
1.8326695
C2
34
At 100 to 250°F
46,709.573
At 250 to 460°F
17,495.343
*404.48844
34.163551
5.1520981
0.52239654
2.8201736
0.62309877
C3
20,247.757
190.24420
2.1586448
0.90832519
i-C4
32,204.420
131.63171
3.3862284
1.1013834
n-C4
33,016.212
146.15445
2.902157
1.1168144
i-C5
37,046.234
299.62630
2.1954785
1.4364289
n-C5
37,046.234
299.62630
2.1954785
1.4364289
n-C6
52,093.006
254.56097
3.6961858
1.5929406
n-C7
82,295.457
5.2577968
1.7299902
n-C8
89,185.432
n-C9
124,062.650
n-C10
146,643.830
64.380112
149.39026
5.9897530
1.9310993
37.917238
6.7299934
2.1519973
26.524103
7.8561789
2.3329874
PHASE BEHAVIOR
3.4.3 Undersaturated-Oil Compressibility. With measured data
or an appropriate correlation for B o or ò o , Eq. 3.14 readily defines
the isothermal compressibility of an oil at pressures greater than the
bubblepoint. “Instantaneous” undersaturated-oil compressibility,
defined by Eq. 3.15 with the pressure derivative evaluated at a specific pressure, is used in reservoir simulation and well-test interpretation. Another definition of oil compressibility may be used in
material-balance calculations (e.g., Craft and Hawkins69)—the
“cumulative” or “average” compressibility defines the cumulative
volumetric change of oil from the initial reservoir pressure to current reservoir pressure.
Compressibility at
Bubblepoint +
pi
V oi
c oǒ p Ǔ +
ŕ c ǒ p Ǔ dp
o
Bubblepoint
Plus 1,000 psia
p
V oi ǒ p i * pǓ
+*
ǒV1 ǓƪV p**Vpǒ pǓƫ. . . . . . . . . . . . . . . . .
o
oi
i
oi
(3.104)
Bubblepoint
Plus 2,000 psia
The cumulative compressibility is readily identified because it is
multiplied by the cumulative reservoir pressure drop, p i * p R. Usually c o is assumed constant; however, this assumption may not be
justified for high-pressure volatile oils.
Oil compressibility is used to calculate the variation in undersaturated density and FVF with pressure.
Fig. 3.19—Undersaturated-oil-compressibility correlation (from
Standing33).
ò o + ò ob expƪc oǒ p * p bǓƫ
[ ò ob ƪ1 * c oǒ p b * pǓƫ . . . . . . . . . . . . . . . . . . . . . (3.105a)
and B o + B ob expƪc oǒ p b * pǓƫ
[ B ob ƪ1 * c oǒ p * p bǓƫ ,
ǒV1 ǓƪV ǒpp*Ǔ *p V
o
ob
b
ob
ƫ
. . . . . . . . . . . . . . . . (3.106)
Strictly speaking, the compressibility of an oil mixture is defined
only for pressures greater than the bubblepoint pressure. If an oil is
at its bubblepoint, the compressibility can be determined and defined only for a positive change in pressure. A reduction in pressure
from the bubblepoint results in gas coming out of solution and, subsequently, a change in the mass of the original system for which
compressibility is to be determined. Implicit in the definition of
compressibility is that the system mass remains constant.
Vazquez and Beggs57 propose the following correlation for
instantaneous undersaturated-oil compressibility.
c o + Ańp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)
where
A+ 10 *5(5R sb ) 17.2T * 1, 180g gc ) 12.61g API* 1, 433), with
c o in psi*1, R sb in scf/STB, T in °F, and p in psia. With this correlation for oil compressibility, undersaturated-oil FVF can be calculated analytically from
B o + B ob( p bńp) A.
Constant A determined in this way is a useful correlating parameter,
one that helps to identify erroneous undersaturated p- V o data.
Standing33 gives a graphical correlation for undersaturated c o
(Fig. 3.19) that can be represented by
. . . . . . . . . . . . . . . . . (3.105b)
where consistent units must be used. These equations are derived
from the definition of isothermal compressibility assuming that co is
constant. When oil compressibility varies significantly with pressure,
Eqs. 3.105a and 3.105b are not really valid. The approximations
ò o [ ò ob [1 * c o( p b * p)] and B o [ B ob [1 * c o( p * p b)] are
used in many applications, and to predict volumetric behavior correctly with these relations requires that co be defined by
c o( p) + *
Bubblepoint Oil Density, lbm/ft3
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.108)
If measured pressure/volume data are available (see Sec. 6.4 in
Chap. 6), these data can be used to determine A (e.g., by plotting
V ońV ob vs. pńp b on log-log paper). Constant A can then be used to
compute compressibilities from the simple relation c o + Ańp.
GAS AND OIL PROPERTIES AND CORRELATIONS
c o + 10 *6 exp
ƪ
ƫ
ò ob ) 0.004347 ǒ p * p bǓ * 79.1
,
(7.141 10 *4)ǒ p * p bǓ * 12.938
. . . . . . . . . . . . . . . . . . . (3.109)
psi*1,
ò ob in lbm/ft3, and p in psia.
with c o in
The Alani-Kennedy EOS can also be solved analytically for oil
compressibility, and Trube70 gives a corresponding-states method
for determining oil compressibility with charts.
Any of the correlations mentioned here should yield reasonable
estimates of c o. However, we recommend that experimental data be
used for volatile oils when c o is greater than about 20 10*6 psi*1.
A simple polynomial fit of the relative volume data, V ro + V ońV ob ,
from a PVT report allows an accurate and explicit equation for undersaturated-oil compressibility.
V ro + A 0 ) A 1 p ) A 2 p 2 . . . . . . . . . . . . . . . . . . . . . . . (3.110a)
ǒ Ǔ
ēV ro
and c o + * 1
V ro ēp
+
T
* ǒ A 1 ) 2A 2 p Ǔ
A0 ) A1 p ) A2 p2
, . . . . . . . . . . . . . . . . . . . . (3.110b)
where A0, A1, and A2 are determined from experimental data. Alternatively, measured data can be fit by use of Eq. 3.108.
3.4.4 Bubblepoint-Oil FVF. Oil FVF ranges from 1 bbl/STB for
oils containing little solution gas to about 2.5 bbl/STB for volatile
oils. B ob increases more or less linearly with the amount of gas in
solution, a fact which explains why B ob correlations are similar to
bubblepoint pressure correlations. For example, Standing’s3,17,40
correlation for California crude oils is
B ob + 0.9759 ) ǒ12
where A + R sǒg gńg oǓ
0.5
10 *5Ǔ A 1.2,
. . . . . . . . . . . . . . . (3.111)
) 1.25T.
35
Glasø’s52 correlation for North Sea crude oils is
logǒ B ob * 1 Ǔ + * 6.585 ) 2.9133 log A * 0.2768ǒlog AǓ 2 ,
. . . . . . . . . . . . . . . . . . . . (3.112)
where A + R s ǒg gńg oǓ 0.526) 0.968T.
The Vazquez and Beggs57 correlation, based on data from commercial laboratories, is
B ob + 1 ) ǒ4.677
10 *4ǓR s ) ǒ0.1751
ǒgg Ǔ * ǒ1.8106
API
gc
10 *4Ǔ(T * 60)
ǒ Ǔ
g
10 Ǔ R s(T * 60) gAPI
gc
*8
4,000
2,000
1,000
800
600
400
200
100°F
100
80
60
120°F
140°F
160°F
40
. . . . . . . . . . . . . . . . . . (3.113a)
20
180°F
for g API x 30 and
B ob + 1 ) ǒ4.67
* ǒ0.1337
10 ǓR s ) ǒ0.11
*4
ǒ Ǔ
g
10 Ǔ(T * 60) gAPI
gc
*4
ǒ Ǔ
g
10 *8ǓR s(T * 60) gAPI . . . . . . . . . (3.113b)
gc
10
8
6
200°F
220°F
4
240°F
2
1
0.8
0.6
Sources of Data
Beal (1946)
Frick (1962)
0.4
for g API u 30, where the effect of separator conditions is included
by use of a corrected gas gravity g gc (Eq. 3.86).
The Standing and the Vazquez-Beggs correlations indicate that a
plot of B o vs. R s should correlate almost linearly. This plot is useful
for checking the consistency of reported PVT data from a differential liberation plot. Eq. 3.114,71 which performs considerably better
for Middle Eastern oils, also suggests a linear relationship between
B ob and R s.
B ob+ 1.0 ) ǒ0.177342
10 *3Ǔ R s ) ǒ0.220163
R sǒ g gńg oǓ )ǒ4.292580
) ǒ0.528707
10 *3Ǔ
10 *6Ǔ R s(T * 60)(1 * g o)
10 *3Ǔ(T * 60). . . . . . . . . . . . . . . . (3.114)
All three B ob correlations (Eqs. 3.113a, 3.113b, and 3.114)
should give approximately the same accuracy. Sutton and Farshad’s54 comparative study of gulf coast oils indicates that Standing’s correlation is slightly better for B ob t 1.4 and that Glasø’s
correlation is best for B ob u 1.4.
3.4.5 Saturated-Oil Compressibility. Perrine8 introduces a definition for the compressibility of a saturated oil that includes the
shrinkage effect of saturated-oil FVF, ēB ońēp, and the expansion
effect of gas coming out of solution, B g(ēR sńēp),
ǒ Ǔ
ēB o
co + * 1
B o ēp
)
T
ǒ Ǔ.
1 B g ēR s
5.615 B o ēp
. . . . . . . . . (3.115)
T
c o is used in the definition of total system compressibility, c t .
c t + c f ) c w S w ) c o S o ) c g S g , . . . . . . . . . . . . . . (3.116)
where c f +rock compressibility. B g has units ft3/scf. R s is in scf/
STB, and B o in bbl/STB+saturated-oil FVF at the pressure of interest, at or below the original oil’s bubblepoint pressure (where both
gas and oil are present).
3.4.6 Oil Viscosity. Typical oil viscosities range from 0.1 cp for
near-critical oils to u100 cp for heavy crudes. Temperature, stocktank-oil density, and dissolved gas are the key parameters determining oil viscosity, where viscosity decreases with decreasing stocktank-oil density (increasing oil gravity), increasing temperature,
and increasing solution gas.
Oil viscosity is one of the most difficult properties to estimate,
and most methods offer an accuracy of only about 10 to 20%. Two
approaches are used to estimate oil viscosity: empirical and corresponding-states correlations. The empirical methods correlate
gas-saturated-oil viscosity in terms of dead-oil (residual or stocktank-oil) viscosity and solution gas/oil ratio. Undersaturated-oil
viscosity is related to bubblepoint viscosity and the ratio or differ36
0.2
0.1
0
10
20
30
40
50
60
Fig. 3.20—Beal dead-oil (stock-tank-oil) viscosity correlation including data in Frick (from Standing33).
ence in actual and bubblepoint pressures. Corresponding-states
methods use reduced density or reduced pressure and temperature
as correlating parameters.
3.4.7 Dead-Oil (Residual- or Stock-Tank-Oil) Viscosity. Several
correlations for dead-oil viscosity are given in terms of oil gravity
and temperature. Standing,3 for example, gives best-fit equations
for the original Beal72 graphical correlation,
m oD +
ǒ
7
0.32 ) 1.8 4.5310
g API
Ǔǒ
Ǔ
360
A
, . . . . . . . . . (3.117)
T ) 200
where A + 10 ƪ0.43)ǒ8.33ńg APIǓƫ .
A somewhat modified version of the original correlation is given
in Fig. 3.20 by Standing.33 Beggs18 and Beggs and Robinson73 give
the following equation for the original Beal correlation,
m oD + * 1 ) 10 ƪT
*1.163 expǒ6.9824*0.04658g
Ǔƫ .
API
. . . . . . (3.118)
Bergman* claims that the temperature dependence of the Beggs and
Robinson correlation is not valid at lower temperatures (t70°F)
and suggests the following correlation, based on viscosity data, for
pure compounds and reservoir oils.
ln ln( m oD ) 1) + A 0 ) A 1 ln(T ) 310), . . . . . . . . . (3.119)
where A 0 + 22.33* 0.194gAPI ) 0.00033 g 2API and A 1 + * 3.20
) 0.0185 g API .
Glasø52 gives a relation (used in the paraffinicity correction of his
bubblepoint pressure correlation) for oils with K w + 11.9.
m oD + (3.141
10 10)T *3.444(log g API) [10.313(log T )*36.447].
. . . . . . . . . . . . . . . . . . . (3.120)
Al-Khafaji et
m oD +
al.74
give the correlation
10 4.9563*0.00488T
,
ǒ g API ) T ń 30 * 14.29 Ǔ 2.709
. . . . . . . . . . . (3.121)
with T in °F and m oD in cp for Eqs. 3.117 through 3.121.
*Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
PHASE BEHAVIOR
Solution gas/oil ratio, scf/STB
Fig. 3.22—Live-oil (saturated) viscosity as a function of dead-oil
viscosity and solution gas/oil ratio (from Standing,33 after Beal72
correlation).
Fig. 3.21 shows dead-oil viscosities calculated at 100°F for a range
of paraffinicities expressed in terms of K w, together with the Bergman* and Glasø48 correlations.
Fig. 3.21—Dead-oil (stock-tank-oil) viscosities at 100°F for varying paraffinicity (from Ref. 33).
3.4.8 Bubblepoint-Oil Viscosity. The original approach by Chew
and Connally76 for correlating saturated-oil viscosity in terms of
dead-oil viscosity and solution gas/oil ratio is still widely used.
Standing75 gives a relation for dead-oil viscosity in terms of deadoil density, temperature, and the Watson characterization factor.
m ob + A 1 ǒm oDǓ A2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.123)
log( m oD ńò o) +
1
*2.17,
A 3ƪK w*ǒ8.24ńg oǓƫ ) 1.639A 2*1.059
. . . . . . . . . . . . . . . . . . . (3.122a)
where A 1 + 1 ) 8.69 log T ) 460,
560
A 2 + 1 ) 0.554 log T ) 460 ,
560
. . . . . . . . . . . . . (3.122b)
A 1 + 10.715(R s ) 100) *0.515 . . . . . . . . . . . . . . . . . . (3.124a)
. . . . . . . . . . . . . . . . . (3.122c)
and A 2 + 5.44(R s ) 150) *0.338 . . . . . . . . . . . . . . . . . . (3.124b)
ǒ2.87A 1 * 1Ǔg o
A 3 + * 0.1285
, . . . . . . . . . . . . . . . (3.122d)
2.87A 1 * g o
and ò o +
Fig. 3.22 shows the variation in m ob with m oD as a function of R s.
The functional relations for A1 and A2 reported by various authors
differ somewhat, but most are best-fit equations of Chew and Connally’s tabulated results.
Beggs and Robinson. 73
go
,
1 ) 0.000321(T * 60)10 0.00462gAPI
. . . . . . (3.122e)
with m in cp, T in °F, and ò in g/cm3 for Eqs. 3.117 through 3.122.
Eqs. 3.122a through 3.122e represent a best fit of the nomograph for
viscosity in terms of temperature, gravity, and characterization factor. Eq. 3.122e (at standard pressure and temperature) is a best fit of
thermal expansion data for crude oils.
Dead-oil viscosity is one of the most unreliable properties to predict with correlations primarily because of the large effect that oil
type (paraffinicity, aromaticity, and asphaltene content) has on viscosity. For example, the oil viscosity of a crude oil with K w + 12
may be 3 to 100 times the viscosity of a less paraffinic crude oil having the same gravity and K w + 11. For this reason the Standing correlation based on the Watson characterization factor is recommended when K w is known. Using an incorrectly estimated K w,
however, may lead to a potentially large error in dead-oil viscosity.
GAS AND OIL PROPERTIES AND CORRELATIONS
Bergman. *
ln A 1 + 4.768 * 0.8359 ln(R s ) 300) . . . . . . . . . . (3.125a)
and A 2 + 0.555 )
133.5 . . . . . . . . . . . . . . . . . . (3.125b)
R s ) 300
Standing. 3
A 1 + 10 *ǒ7.4
and A 2 +
10 *4ǓR s)ǒ2.2
10 *7ǓR 2s
. . . . . . . . . . . . . . . (3.126a)
0.68
0.25
0.062
)
)
.
10 ǒ8.62 10 *5ǓRs 10 ǒ1.1 10*3ǓRs 10 ǒ3.74 10*3ǓRs
. . . . . . . . . . . . . . . . . . (3.126b)
Aziz et
al. 77
A 1 + 0.20 ) ǒ0.80
and A 2 + 0.43 ) ǒ0.57
10 –0.00081 RsǓ . . . . . . . . . . . . . . (3.127a)
10 –0.00072 RsǓ . . . . . . . . . . . (3.127b)
*Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
37
Al-Khafaji et al.74 extend the Chew-Connally76 correlation to higher GOR’s (up to 2,000 scf/STB).
A 1 + 0.247)0.2824 A 0) 0.5657 A 20
* 0.4065 A 30 ) 0.0631 A 40 . . . . . . . . . . . . . . . . . . . (3.128a)
ƪǒ m * m oǓc T ) 10 *4ƫ
and A 2 + 0.894 ) 0.0546 A 0 ) 0.07667A 20
* 0.0736 A 30 ) 0.01008 A 40 ,
is therefore desired. Several corresponding-states viscosity correlations can be used for both oil and high-pressure gas mixtures; the
Lohrenz et al.62 correlation has become a standard in compositional
reservoir simulation. Lohrenz et al. use the Jossi et al.82 correlation
for dense-gas mixtures ( ò pru0.1),6
. . . . . . . . . . . . . . (3.128b)
where A 0 + log(R s) and R s + 0.1 yields A 1 + A 2 + 1. R s is
given in scf/STB for Eqs. 3.124 through 3.128. Chew and Connally
indicate that their correlation is based primarily on data with GOR’s
of t1,000 scf/STB and that the scatter in A 1 at higher GOR’s is
probably the result of insufficient data. Eqs. 3.128a and 3.128b are
based on additional data at higher GOR’s. Eqs. 3.127a and 3.128b
appear to be the most well behaved.
An interesting observation by Abu-Khamsin and Al-Marhoun78
is that saturated-oil viscosity, m ob, correlates very well with saturated-oil density, ò ob .
ln m ob + * 2.652294 ) 8.484462
ò 4ob ,
. . . . . . . . . . (3.129)
This behavior is expected from the Lohrenz et
with ò ob in
al.62 correlation discussed later. Although Abu-Khamsin and AlMarhoun do not comment on the applicability of Eq. 3.129 to undersaturated oils, it would seem reasonable that their correlation should
apply to both saturated and undersaturated oils. In fact, the correlation even appears to predict accurately dead-oil viscosities, m oD, except at low temperatures for heavy crudes. Simon and Graue give
graphical correlations for the viscosity of saturated CO2/oil systems
(see Chap. 8).79
m o * m ob
0.56
+ 0.024m 1.6
ob ) 0.038m ob .
0.001( p * p b)
The Vazquez and
Beggs57
m o + m ob ǒ pńp bǓ A ,
. . . . . . . . (3.130)
) 0.0093324ò 4pr , . . . . . . . . . . . . . . . . . . . . . . (3.133a)
ǒ Ǔ
ò
ò pr + ò
pc
where A + 2.6 p 1.187 expƪ* 11.513 * ǒ8.98 10 *5Ǔpƫ.
A more recent correlation by Abdul-Majeed et al.80 is
ƫ,
. . . . . . . . (3.132a)
where A + 1.9311 * 0.89941 ǒln R sǓ * 0.001194 g 2API
) 0.0092545 g API ǒln R sǓ.
. . . . . . . . . . . . . . . . . . (3.133b)
ò
v , . . . . . . . . . . . . . . . . . . . . . . . . (3.133c)
M pc
+
ȍ z m ǸM
N
i
and m +
o
i
i
i+1
N
ȍ z ǸM
i
.
. . . . . . . . . . . . . . . . . . . . . . . (3.133d)
i
i+1
Pseudocritical properties T pc, p pc, and v pc are calculated with
Kay’s mixing rule.
Component viscosities, m i , can be calculated from the Lucas45
low-pressure correlation Eq. 3.67 or from the Stiel and Thodos83
correlation (as suggested by Lohrenz et al.62).
m i c Ti + ǒ34
10 *5ǓT ri0.94 . . . . . . . . . . . . . . . . . . . . . . (3.134a)
for Tri x1.5, and
m i c Ti + ǒ17.78
10 *5Ǔ(4.58T ri * 1.67) 5ń8 . . . . . . (3.134b)
for T ri u 1.5, where c Ti + 5.35ǒT ci M 3ińp 4ciǓ .
Lohrenz et al.62 give a special relation for v c C of the C7+ fraction.
1ń6
7)
+ 21.573 ) 0.015122M C
) 0.070615M C
g
,
7) C 7)
7)
* 27.656g C
7)
. . . . . . . . . . . . . . . . (3.135)
with m in cp, c in cp*1, ò in lbm/ft3, v in ft3/lbm mol, T in °R, p
in psia, and M in lbm/lbm mol. The Lohrenz et al. method is very
sensitive to mixture density and to the critical volumes of heavy
components. Adjustment of the critical volumes of heavy (and
sometimes light) components to match experimental oil viscosities is usually necessary.
. . . . . . . . . . . . . . . (3.132b)
Eq. 3.132 is based on the observation that a plot of log(m o * m ob)
vs. log(p * p b) plots as a straight line with slope of [1.11. Because this observation appears to be fairly general, it can be used
to check reported undersaturated-oil viscosities and to develop
field-specific correlations.
Sutton and Farshad54 and Khan et al.81 present results that indicate
that the Standing equation gives good results and that the Vazquez-Beggs correlation tends to overpredict viscosities somewhat.
Abdul-Majeed et al.80 indicate that both the Standing and Vazquez-Beggs correlations overpredict viscosities of North African
and Middle Eastern oils (253 data), and that their own correlation performed best for these data and for the data used by Vazquez
and Beggs.
3.4.10 Compositional Correlation. In compositional reservoir
simulation of miscible-gas-injection processes and the depletion of
near-critical reservoir fluids, the oil and gas compositions may be
very similar. A single viscosity relation consistent for both phases
38
,
7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.131)
logǒ p*p bǓ
1ń6
T pc
where c T + 5.35
M 3p 4pc
v cC
correlation is
mo + m ob ) 10 ƪA* 5.2106 ) 1.11
+ 0.10230 ) 0.023364ò pr
) 0.058533ò 2pr * 0.040758ò 3pr
g/cm3.
3.4.9 Undersaturated-Oil Viscosities. Beal72 gives the variation of
undersaturated-oil viscosity with pressure graphically where it has
been curve fit by Standing.2
1ń4
3.5 IFT and Diffusion Coefficients
3.5.1 IFT. Weinaug and Katz84 propose an extension of the Macleod85 relationship for multicomponent mixtures.
ȍ P ǒx Mò
N
s 1ń4
go +
o
i
i+1
i
o
* yi
Ǔ
òg
, . . . . . . . . . . . . . . . . . (3.136)
Mg
with s in dynes/cm (mN/m) and ò in g/cm3. P i is the parachor of
Component i, which can be calculated by group contributions, as
shown in Table 3.2. For n-alkanes, the parachors can be expressed by
P i + 25.2 ) 2.86M i .
. . . . . . . . . . . . . . . . . . . . . . . . (3.137)
Several authors propose parachors for pure hydrocarbons that deviate from the group-contribution values. For example, P C +77 is
1
often cited for methane instead of the group-contribution value of
P C +71. Likewise, P N +41 is often used for nitrogen instead of
2
1
the group-contribution value of P N +35. Fig. 3.23 plots parachors
2
vs. molecular weight for pure components and petroleum fractions.
PHASE BEHAVIOR
TABLE 3.2—PARACHORS FOR PURE COMPONENTS AND
COMPOUND GROUPS
n-paraffins
Heptanes plus of Ref. 4
Gasolines
Crude oil
Pure Component
C1
71
C2
111
C3
151
C4 (also i-C4)
191
C5 (also i-C5)
231
C6
271
C7
311
C8
351
C9
391
C10
431
N2
35
CO2
49
H2 S
80
Group
C
9.0
H
15.5
CH3
55.5
CH2 [in (CH2)n ]
40.0
N
17.5
O
20.0
S
49.1
Example: For methane, CH4. PC1=PC+4(PH)=9+4(15.5)=71.
Fig. 3.23—Hydrocarbon parachors.
Nokay86 gives a simple relation for parachors of pure hydrocarbons
(paraffins, olefins, naphthenes, and aromatics) with a normal boiling
point between 400 and 1,400°R and specific gravity t1,
log P i + * 4.20895 ) 2.29319 log
ǒ Ǔ
T bi
, . . . . . (3.138)
g 0.5937
i
with T b in °R.
Katz and Saltman87 and Katz et al.88 give parachor data for C7+
fractions measured by Standing and Katz,58,89 which are approximately correlated by
P i + 35 ) 2.40M i . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.139)
The API recommended procedure for estimating petroleum fraction
IFT’s is based on an unpublished correlation.27 The graphical correlation can be expressed by
602(1 * T ri) 1.194
,
si +
K wi
ǒ Ǔ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.141)
where ò sL ơ ò sv is assumed. The saturated-liquid density can be
estimated, for example, with the Rackett63 equation.
ò sLi +
P i + 11.4 ) 3.23 M i * 0.0022 M 2i .
M i p ci *ƪ1)ǒ1*TriǓ2ń7ƫ
Z
,
RT ci Ri
. . . . . . . . . . . . . . . . . (3.142)
where Z Ri [ Z ci [ 0.291 * 0.08 w i
. . . . . . . . . . . . . . (3.143)
and R+universal gas constant. The parachors predicted from Eqs.
3.140 through 3.143 are practically constant for a given petroleum
fraction (i.e., the temperature effect cancels out).
GAS AND OIL PROPERTIES AND CORRELATIONS
. . . . . . . . . . . (3.144)
They also discuss the qualitative effect of asphaltenes on IFT and
suggest that the parachor of asphaltic substances generally will not
follow the relations of lighter C7+ fractions.
Ramey91 gives a method for estimating gas/oil IFT with black-oil
PVT properties. We extend the method here to include the effect of
solution oil/gas ratio, r s. Considering stock-tank oil and separator
gas as the “components” ( o and g) making up reservoir oil and gas,
respectively, the Weinaug-Katz84 relation can be written
ƪ ǒMò Ǔ * y ǒMò Ǔƫ ) P ƪx ǒMò Ǔ * y ǒMò Ǔƫ,
s¼
go + P o x o
. . . . . . . . . . . . . . . . . . . . . . . (3.140)
where K w + T 1ń3
ńg, with T b in °R. The parachor can be estimated
b
with the Macleod relation,
Mi
P i + s 1ń4
ò sLi ,
i
Firoozabadi et al.90 give an equation that can be used to approximate the parachor of pure hydrocarbons from C1 through C6 and for
C7+ fractions,
g
o
o
o
o
g
o
g
g
o
g
g
. . . . . . . . . . . . . . . . . . . (3.145a)
where x o +
1 ) (7.52
1
, . . . . . . . . (3.145b)
10 *6)R sǒM ońg oǓ
x g + 1 * x o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145c)
yo +
1 ) (7.52
1
,
10 *6)ǒ M ońg o Ǔr s
. . . . . . . . . . . (3.145d)
y g + 1 * y o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145e)
òo +
62.4g o ) 0.0136g g R s
,
62.4 B o
. . . . . . . . . . . . . . . . . . (3.145f)
ò g + 0.0932ǒ pM gńZT Ǔ , . . . . . . . . . . . . . . . . . . . . . . (3.145g)
M o + x o M o ) x g M g , . . . . . . . . . . . . . . . . . . . . . . . . (3.145h)
39
M g + y o M o ) y g M g ,ĂĂ . . . . . . . . . . . . . . . . . . . . . . . . (3.145i)
)
M o + 6, 084ńg API * 5.9 , . . . . . . . . . . . . . . . . . . . . . (3.145j)
M g + 28.97g g ,
. . . . . . . . . . . . . . . . . . (3.149b)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145k)
P o + ǒ2.376 ) 0.0102g APIǓńM o ,
. . . . . . . . . . . . . . . (3.145l)
1.03587
1.76474
)
,
expǒ1.52996T ijǓ expǒ3.89411T ijǓ
T ij +
T , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.149c)
(åńk) ij
and P g + 25.2 ) 2.86M g , . . . . . . . . . . . . . . . . . . . . . (3.145m)
ǒ åńk Ǔ ij+ ƪǒ åńk Ǔ i ǒåńkǓ jƫ 1ń2 , . . . . . . . . . . . . . . . . . . . . (3.149d)
with ò in g/cm3, R s in scf/STB, B o in bbl/STB, T in °R, and p in psia
and where x o and x g+mole fractions of the surface-oil and -gas
components, respectively, in the oil phase, and y o and y g+mole
fractions of the surface-oil and -gas components, respectively, in the
gas phase. In the traditional black-oil approach r s + 0, simplifying
the relations to those originally suggested by Ramey.91
Eq. 3.145 is useful in black-oil reservoir simulation and when
compositional data are not available. The black-oil approach generally is not recommended for predicting gas/oil IFT’s unless the surface-oil parachor has been fit to experimental IFT data (or to IFT’s
calculated with compositions and densities from an EOS characterization by use of Eq. 3.136).
ǒ åńk Ǔ i + 65.3T ci Z 18ń5 ,
3.5.2 Diffusion Coefficients. Molecular diffusion in multicomponent mixtures is a complex problem. The standard engineering approach uses an effective diffusion coefficient for Component i in a
mixture, D im, where D im can be calculated in one of two ways: (1)
from binary diffusion coefficients and mixture composition or (2)
from Component i properties and mixture viscosity. The first approach uses the Wilke92 formula to calculate D im.
D im +
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.146)
N
j
ij
j+1
j0i
where y i +mixture mole fraction and D ij + D ji is the binary diffusivity at the pressure and temperature of the mixture.
Sigmund93 correlates the effect of pressure and temperature on
diffusion coefficients using a corresponding-states approach with
reduced density.
ò M D ij
+ 0.99589 ) 0.096016ò pr * 0.22035ò 2pr
ò oM D oij
) 0.032874ò 3pr , . . . . . . . . . . . . . . . . . . . . . . . (3.147)
where D ij +diffusion coefficient at pressure and temperature,
ò pr+pseudoreduced density+ ò Mńò Mpc + ǒ òńM Ǔv pc , ò M +mixture molar density, ò oM D oij +low-pressure density-diffusivity product, and v pc +pseudocritical molar volume calculated with Kay’s5
mixing rule. Note that the ratio ò M D ijńò oD oij is the same for all
binary pairs in a mixture because ò pr is a function of only mixture
density and composition.
da Silva and Belery12 note that the Sigmund correlation does not
work well for liquid systems and propose the following extrapolation for ò pru3.
ò MD ij
+ 0.18839 exp(3 * ò pr) , . . . . . . . . . . . . . . . . (3.148)
ò oM D oij
which avoids negative D ij for oils at ò pru3.7 as estimated by the
Sigmund correlation.
Low-pressure binary gas diffusion coefficients,6 D oij , can be estimated from
D oij + 0.001883
T 3ń2ƪǒ1ńM iǓ ) ǒ1ńM jǓƫ
p os 2ijW ij
0.193
)
where W ij + 1.06036
T ij0.1561
expǒ0.47635T ijǓ
40
. . . . . . . . . . . . . . . . . . . . . . . . (3.149e)
s ij + 0.5ǒs i ) s jǓ , . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.149f)
and s i + 0.1866
v 1ń3
ci
Z 6ń5
ci
,
. . . . . . . . . . . . . . . . . . . . . . . . (3.149g)
with the diffusion coefficient, D oij , in cm2/s; molecular weight, M,
in kg/kmol; temperature, T, in K, pressure; p, in bar; Lennard Jones
12-6 potential parameter, s, in Å; Lennard-Jones 12-6 potential
parameter, e/k, in K; and critical volume, vc , in m3/kmol and where
Z c +critical compressibility factor and i and j+diffusing and concentrated species, respectively.
To obtain the low-pressure density-diffusivity product, we use the
ideal-gas law, ò oM + p ońRT, to get
D oij ò oM
+ ǒ2.2648
10
*5
Ǔ
T 1ń2ƪǒ1ńM iǓ ) ǒ1ńM jǓƫ
1ń2
s 2ij W ij
,
. . . . . . . . . . . . . . . . . . (3.150)
1 * yi
ȍ y ńD
ci
0.5
, . . . . . . . . (3.149a)
where ò and ò M have units g mol/cm3.
The accuracy of the Sigmund correlation for liquids is not known,
but the extension proposed by da Silva and Belery (Eq. 148) for
large reduced densities does avoid negative diffusivities calculated
by the Sigmund equation.94 Renner95 proposes a generalized correlation for effective diffusion coefficients of light hydrocarbons
and CO2 in reservoir liquids that can be used as an alternative to the
Sigmund-type correlation.
*1.831 4.524
M *0.6898
ò 1.706
T
,
D im + 10 *9 m *0.4562
o
Mi p
i
. . . . . . . . . . . . . . . . . . . (3.151)
with D in cm2/s and where m o +oil viscosity in cp, M i +molecular
weight, ò Mi +molar density of Component i at p and T in g mol/cm3,
p+pressure in psia, and T+temperature in K. This correlation is
based on 141 experimental data with the following property ranges:
0.2t m ot134 cp; 16tM it44; 0.04t ò Mit7 kmol/m3; 14.7tp
t2,560 psia; and 273tTt333 K, where i+CO2, C1, C2, and C3.
Renner also gives a correlation for diffusivity of CO2 in water/
brine systems.
D CO
2*w
+ ǒ6.392
6.911
10 3Ǔ m CO
m w*0.1584, . . . . . . . . . (3.152)
2
with D in cm2/s and m in cp.
3.6 KĆValue Correlations
This section covers the estimation of equilibrium K values by correlations and the calculation of two-phase equilibrium when K values are known. The K value is defined as the ratio of equilibrium gas
composition yi to the equilibrium liquid composition x i,
K i 5 y ińx i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.153)
K i is a function of pressure, temperature, and overall composition.
K values can be estimated with empirical correlations or by satisfying the equal-fugacity constraint with an EOS (see Chap. 4).
Although the increasing use of EOS’s has tended to lessen interest
in empirical K-value correlations, empirical methods are still useful
for such engineering calculations as (1) multistage surface separation, (2) compositional reservoir material balance, and (3) checking
the consistency of separator-oil and gas compositions.
PHASE BEHAVIOR
Fig. 3.24—General behavior of a K value vs. pressure plot on log-log scale.
Several methods for correlating K values have appeared in the
past 50 years. Most rely on two limiting conditions for describing
the pressure dependence of K values. First, at low pressures,
Raoult’s and Dalton’s laws3 can be used to show that
K i [ p vi ǒ T Ǔńp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.154)
where p v +component vapor pressure at the system temperature.
The limitations of this equation are that temperature must be less than
the component critical temperature (because vapor pressure is not defined at supercritical temperatures) and that the component behaves
as an ideal gas. Also, the equation implies that the K value is independent of overall composition. In fact, the pressure dependence of lowpressure K values is closely approximated by Eq. 3.154.
The second observation is that, at high pressures, the K values of
all components in a mixture tend to converge to unity at the same
pressure. This pressure is called the convergence pressure96 and, for
binaries, represents the actual mixture critical pressure. For multicomponent mixtures, the convergence pressure is a nonphysical
condition unless the system temperature equals the mixture critical
temperature.97,98 This is because a mixture becomes single phase at
the bubblepoint or dewpoint pressure before reaching the convergence pressure.
The log-log plot of K i vs. pressure in Fig. 3.24 shows how the
ideal-gas and convergence-pressure conditions define the K-value
behavior at limiting conditions. For light components (where
T u T ci ), K values decrease monotonically toward the convergence pressure. For heavier components (where T t T ci ), K values
initially decrease as a function of pressure at low pressures, passing
through unity when system pressure equals the vapor pressure of a
particular component, reaching a minimum, and finally increasing
toward unity at the convergence pressure.
GAS AND OIL PROPERTIES AND CORRELATIONS
For reservoir fluids, the pressure where K values reach a minimum is usually u1,000 psia (Fig. 3.25), implying that K values are
more or less independent of convergence pressure (i.e., composition) at pressures t1,000 psia. This observation has been used to
develop general “low-pressure” K-value correlations for surfaceseparator calculations.
3.6.1 Hoffman et al. Method. Hoffman et al.99 propose a method
for correlating K values that has received widespread application.
Ki +
10
ǒA0 ) A1 Fi Ǔ
p
or log K i p + A 0 ) A 1 F i , . . . . . . . . . . . . . . . . . . . . . . (3.155)
where F i +
1ńT bi * 1ńT
logǒ p cińp scǓ ; . . . . . . . . . . . . (3.156)
1ńT bi * 1ńT ci
T c +critical temperature; p c +pressure; T b +normal boiling point;
p sc +pressure at standard conditions; and A 1 and A 0 +slope and intercept, respectively, of the plot log(K i p) vs. F i.
Hoffman et al. show that measured K values for a reservoir gas
condensate correlate well with the proposed equation. They found
that trend of log(K i p) vs. F i is linear for components C1 through C6
at all pressures, while the function turns downward for heavier components at low pressures. Interestingly, the trend becomes more linear for all components at higher pressures.
As Fig. 3.26 shows, Slope A 1 and Intercept A 0 vary with pressure. For low pressures, K i [ p vńp. With the Clapeyron vapor pressure relation,5 log(p v) + a * bńT results in A 0 + log(p sc) and
A 1 + 1. These limiting values of A 0 and A 1 are close to the values
found when A 0( p) and A 1( p) are extrapolated to p + p sc. Because
41
Fig. 3.25—K values at 120°F for binary- and reservoir-fluid systems with convergence pressures
ranging from 800 to 10,000 psia (from Standing3).
K values tend toward unity as pressure approaches the convergence
pressure, p K , it is necessary that A 0 + log(p K) and A 1 ³ 0. Several
authors have noted that plots of log(K i p) vs. F i tend to converge at
a common point. Brinkman and Sicking101 suggest that this “pivot”
point represents the convergence pressure where K i + 1 and
p + p K. The value of F i at the pivot point, F K, is easily shown to
equal log(p Kńp sc).
It is interesting to note that the well-known Wilson102,103 equation,
Ki +
Ǔ
exp 5.37(1 ) w i)ǒ1 * T *1
ri
, . . . . . . . . . . . . . . (3.157)
p ri
is identical to the Hoffman et al.99 relation for A 0 + log(p sc) and
A 1 + 1 when the Edmister104 correlation for acentric factor equation,
42
T bińT ci
wi + 3
logǒ p cińp scǓ * 1 , . . . . . . . . . . . . (3.158)
7 1 * T bińT ci
is used in the Wilson equation. Note that 5.37+(7/3) ln (10).
Whitson and Torp100 suggest a generalized form of the Hoffman
et al.99 equation in terms of convergence pressure and acentric factor.
ǒ Ǔ
p
K i + pci
K
A 1*1
Ǔƫ
expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1
ri
,
p ri
. . . . . . . . . . . . . . . . . . . (3.159)
where A 1 +a function of pressure, with A 1 + 1 at p + p sc and
A 1 + 0 at p + p K. The key characteristics of K values vs. pressure
PHASE BEHAVIOR
log pK
TABLE 3.3—VALUES OF b AND Tb FOR USE IN
STANDING LOW-PRESSURE K-VALUE CORRELATION
–
Component, i
bi
(cycle-°R)
Tbi
°R
470
109
N2
Intercept A0
Slope A1
Pressure, psia
Fig. 3.26—Pressure dependence of slope, A1, and intercept, A0,
in Hoffman et al. Kp-F relationship (Eq. 3.155) for a North Sea
gas condesate NS-1 (from Whitson and Torp100).
CO2
652
194
H2 S
1,136
331
C1
300
94
C2
1,145
303
C3
1,799
416
i-C4
2,037
471
n-C4
2,153
491
i-C5
2,368
542
n-C5
2,480
557
C6 (lumped)
2,738
610
n-C6
2,780
616
n-C7
3,068
669
n-C8
3,335
718
n-C9
3,590
763
n-C10
3,828
805
For C7+ fractions, see Eqs. 3.161f through 3.161h
and temperature are correctly predicted by Eq. 3.159, where the following pressure dependence for A 1 is suggested.
A 1 + 1 * (pńp K) A 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.160)
where A 2 ranges from 0.5 to 0.8 and pressures p and p K are given
in psig. Canfield105 also suggests a simple K-value correlation
based on convergence pressure.
3.6.2 Standing Low-Pressure K Values. Standing106 uses the Hoffman et al.99 method to generate a low-pressure K-value equation
for surface-separator calculations ( p sp t 1, 000 psia and
T sp t 200°F). Standing fits A 1 and A 0 in Eq. 3.155 as a function
of pressure using K-value data from an Oklahoma City crude oil. He
treats the C 7) by correlating the behavior of K C as a function of
7)
“effective” carbon number n C . The Standing equations are
7)
ǒA0 ) A1 Fi Ǔ
, . . . . . . . . . . . . . . . . . . . . . . . . (3.161a)
F i + b iǒ1ńT bi * 1ńTǓ,
. . . . . . . . . . . . . . . . . . . . . . . (3.161b)
K i + p1 10
sp
b i + logǒ p cińp scǓńǒ1ńT bi * 1ńT ciǓ , . . . . . . . . . . . . . . . (3.161c)
A 0ǒ pǓ + 1.2 ) ǒ4.5
10
*4
Ǔp ) ǒ15
10
*8
Ǔp ,
10 *4Ǔp * ǒ3.5
10 *8Ǔp 2,
. . . . . . . . . . . . . . . . . . . (3.161e)
nC
7)
+ 7.3 ) 0.0075T ) 0.0016p,
bC
7)
+ 1, 013 ) 324n C
and T bC
7)
7)
+ 301 ) 59.85n C
. . . . . . . . . . . . . (3.161f)
* 4.256n 2C
7)
7)
,
* 0.971n 2C
. . . . . . . (3.161g)
,
7)
. . . . . (3.161h)
with T in °R except when calculating n C (for n C , T is in °F) and
7)
7)
p in psia. Standing suggests modified values of b i and T bi for nonhydrocarbons, methane, and ethane (Table 3.3). Glasø and Whitson107
show that these equations are accurate for separator flash calculations of crude oils with GOR’s ranging from 300 to 1,500 scf/STB
and oil gravity ranging from 26 to 48°API. Experience shows, however, that significant errors in calculated GOR may result for lean
gas condensates, probably because of inaccurate C 1 and
GAS AND OIL PROPERTIES AND CORRELATIONS
3.6.3 Galimberti-Campbell Method. Galimberti and Campbell108,109 suggested another useful approach for correlating K values where
log K i + A 0 ) A 1T ci2
. . . . . . . . . . . . . . . . . . . . . . . . . (3.162)
is shown to correlate K values for several simple mixtures containing hydrocarbons C 1 through C 10 at pressures up to 3,000 psia and
temperatures from *60 to 300°F.
Whitson developed a low-pressure K-value correlation, based on
data from Roland,110 at pressures t1,000 psia and temperatures
from 40 to 200°F, for separator calculations of gas condensates.
A 0 + 4.276 * ǒ7.6
10 *4ǓT
) ƪ* 1.18 ) ǒ5.675
10 *4ǓTƫ log p , . . . . . . . . (3.163a)
NJ
A 1 + 10 *6 ǒ* 4.9563 ) 0.00955T Ǔ ) ƪǒ1.9094
2
. . . . . . . . . . . . . . . . . . (3.161d)
A 1ǒ pǓ + 0.890 * ǒ1.7
C 7) K values. The Hoffman et al. method with Standing’s low-pressure correlations are particularly useful for checking the consistency of separator-gas and -oil compositions.
* ǒ1.235
10 *5ǓT ) ǒ3.34
10 *3Ǔ
Nj
10 *8ǓT 2ƫ p , . . . (3.163b)
T cC1 + 343 * 0.04p, . . . . . . . . . . . . . . . . . . . . . . . . . (3.163c)
and T c C7) + 1, 052.5 * 0.5125T ) 0.00375T 2 , . . . . (3.163d)
with p in psia, T in °F, and T c in °R.
3.6.4 Nonhydrocarbon K Values. Lohrenz et al.111 reported nonhydrocarbon K values as a function of pressure, temperature, and
convergence pressure.
ln K H
2S
ǒ
p
+ 1*p
K
Ǔ ƪ6.3992127 ) 1, 399.2204
T
0.8
* 0.76885112 ln p *
*
ƫ
1, 112, 446.2
,
T2
18.215052 ln p
T
. . . . . . . . . . . . . . . . . . . . . (3.164a)
43
p
1, 184.2409
+ ǒ1 * p Ǔ ǒ11.294748 *
T
0.4
ln K N
2
K
Ǔ
* 0.90459907 ln p , . . . . . . . . . . . . . . . . . . . (3.164b)
ǒ
p
ln K CO + 1 * p
2
0.6
K
ln p )
Ǔ ǒ7.0201913 * 152.7291
* 1.8896974
T
Ǔ
1, 719.2956 ln p 644, 740.69 ln p
*
,
T
T2
. . . . . . . . . . . . . . . . . . . (3.164c)
with p in psia and T in °R. For low-pressure K-value estimation, the
first term in Eq. 3.164 simplifies to unity (assuming that
1 * pńp K [ 1) and the K values become functions of pressure and
temperature only. However, these equations do not give the correct
low-pressure value of ē(ln K i)ńē(ln p) + * 1
3.6.5 Convergence-Pressure Estimation. For correlation purposes, convergence pressure is used as a variable to define the composition dependence of K values. Convergence pressure is a function of overall composition and temperature. Whitson and
Michelsen112 show that convergence pressure is a thermodynamic
phenomenon, with the characteristics of a true mixture critical point,
that can be predicted with EOS’s.
Rzasa et al.113 give an empirical correlation for convergence
pressure as a function of temperature and the product (Mg) C .
7)
Standing2 suggests that convergence pressure of reservoir fluids varies almost linearly with C 7) molecular weight.
Convergence pressure can also be calculated with a trial-and-error procedure suggested by Rowe.97,98,114 This procedure involves
the use of several empirical correlations for estimating mixture critical pressure and temperature, pseudocomponent critical properties,
and the K values of methane and octane. The Galimberti and Campbell108,109 K-value method is used to estimate K values of other
components by interpolation and extrapolation of the C 1 and C 8 K
values. This approach to convergence pressure is necessary if the K
values are used for processes that approach critical conditions or
where K values change significantly because of overall composition
effects. The method cannot, of course, be more accurate than the
correlations it uses and therefore is expected to yield only qualitatively correct results.
For reservoir calculations where convergence pressure can be assumed constant (e.g., pressure depletion), a more direct approach to
determining convergence pressure is suggested. With a K-value correlation of the form K i + K( p K, p, T ) as in Eq. 3.159, the convergence pressure can be estimated from a single experimental saturation pressure. For a bubblepoint and a dewpoint, Eqs. 3.165 and
3.166, respectively, must be satisfied.
F ǒ p KǓ + 1 *
ȍ z K ǒp , p , TǓ + 0
N
i
i
K
b
. . . . . . . . . . . . . (3.165)
i+1
and Fǒ p KǓ + 1 *
ȍ
N
zi
i+1 K i ǒ p K, p d , T Ǔ
+ 0, . . . . . . . . . . . (3.166)
where z i, p b, or p d and T are specified and p K is determined.
The two-phase flash calculation, with K values given, is discussed in Chap. 4 in the Phase-Split Calculation section.
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45
100. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion
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46
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SI Metric Conversion Factors
Å 1.0*
E*01 +nm
°API 141.5/(131.5)°API)
+g/cm3
bar 1.0*
E)05 +Pa
bbl 1.589 873
E*01 +m3
Btu/lbm mol 2.236
E)03 +J/mol
cp 1.0*
E*03 +Pa@s
cSt 1.0*
E*06 +m2/s
dyne/cm 1.0*
E)00 +mN/m
ft 3.048*
E*01 +m
E*02 +m2
ft2 9.290 304*
ft3 2.831 685
E*02 +m3
ft3/lbm mol 6.242 796
E*02 +m3/kmol
°F (°F*32)/1.8
+°C
°F (°F)459.67)/1.8
+K
E)00 +cm2
in.2 6.451 6*
lbm 4.535 924
E*01 +kg
lbm mol 4.535 924
E*01 +kmol
psi 6.894 757
E)00 +kPa
E*01 +kPa*1
psi*1 1.450 377
°R 5/9
+K
*Conversion factor is exact.
PHASE BEHAVIOR
Chapter 4
EquationĆofĆState Calculations
4.1 Introduction
Cubic equations of state (EOS’s) are simple equations relating pressure, volume, and temperature (PVT). They accurately describe the
volumetric and phase behavior of pure compounds and mixtures, requiring only critical properties and acentric factor of each component. The same equation is used to calculate the properties of all
phases, thereby ensuring consistency in reservoir processes that approach critical conditions (e.g., miscible-gas injection and depletion
of volatile-oil/gas-condensate reservoirs). Problems involving multiphase behavior, such as low-temperature CO2 flooding, can be
treated with an EOS, and even water-/hydrocarbon-phase behavior
can be predicted accurately with a cubic EOS.
Volumetric behavior is calculated by solving a simple cubic equation, usually expressed in terms of Z+pv/RT,
Z 3 ) A 2 Z 2 ) A 1 Z ) A 0 + 0, . . . . . . . . . . . . . . . . . . . . (4.1)
where constants A0, A1, and A2 are functions of pressure, temperature, and phase composition.
Phase equilibria are calculated with an EOS by satisfying the
condition of chemical equilibrium. For a two-phase system, the
chemical potential of each component in the liquid phase mi (x) must
equal the chemical potential of each component in the vapor phase
mi ( y), mi ( x)+mi ( y). Chemical potential is usually expressed in
terms of fugacity, fi , where mi +RT ln fi )li (T ) and li (T ) are
constant terms that drop out in most problems.1-3 It is readily shown
that the condition mi (x)+mi ( y) is satisfied by the equal-fugacity
constraint, fLi +fvi , where fugacity is given by
R
f
ln f i + ln i + 1
RT
yi p
ŕ ǒēnēp * RTVǓ dV * ln Z.
. . . . . . (4.2)
i
V
Other thermodynamic properties, such as Helmholz energy, enthalpy, and entropy, can be readily defined in terms of the fugacity coefficient. Michelsen4 gives a particularly compact and useful discussion of the relation between thermodynamic properties aimed at
making efficient EOS calculations.
A component material balance is also required to solve vapor/liquid equilibrium problems: zi +Fv yi )(1*Fv )xi , where Fv +mole
fraction of the vapor phase+nv /(nv )nL ). Integrating the component
balance in the two-phase flash calculation is discussed in Sec. 4.3.1.
Solving phase equilibria with an EOS is a trial-and-error procedure, requiring considerable computations. With today’s computers, however, the task is fast and reliable. The accuracy of EOS predictions has also improved considerably during the past 15 years,
EQUATION-OF-STATE CALCULATIONS
during which emphasis has been on improved liquid volumetric predictions and treating the heptanes-plus fraction (Chap. 5).
This chapter provides the equations and algorithms necessary for
calculating phase and volumetric behavior of reservoir fluids with
a cubic EOS. Sec. 4.2 reviews the most important cubic equations,
starting with van der Waals’5 EOS from 1873 and concluding with
the method of volume translation, which has greatly improved the
volumetric capabilities of cubic EOS’s.
In Secs. 4.3 through 4.5, we present algorithms for solving vapor/
liquid equilibrium (VLE) problems, including the two-phase flash,
phase-stability-test, and saturation-pressure calculations. Reference is also made to methods for solving three-phase and criticalpoint calculations. Sec. 4.6 deals specifically with compositional
gradients with depth caused by gravity and thermal diffusion. Finally, Sec. 4.7 covers how to “tune” an EOS to match experimental
PVT data (see also Appendix C).
4.2 Cubic EOS's
Since the introduction of the van der Waals EOS, many cubic EOS’s
have been proposed—e.g., the Redlich and Kwong6 EOS (RK EOS)
in 1949, the Peng and Robinson7 EOS (PR EOS) in 1976, and the
Martin8 EOS in 1979, to name only a few.9-15 Most of these equations retain the original van der Waals repulsive term RT/(v*b),
modifying only the denominator in the attractive term. The RedlichKwong equation has been the most popular basis for developing
new EOS’s. Another trend has been to propose generalized three-,
four-, and five-constant cubic equations that can be simplified to the
PR EOS, RK EOS, or other familiar forms. Kumar and Starling16,17
use the most general five-constant cubic EOS to fit volumetric and
phase behavior of nonpolar compounds, although they do not apply
the equation to mixtures.
Most petroleum engineering applications rely on the PR EOS or a
modification of the RK EOS. Several modified Redlich-Kwong
equations have found acceptance, with Soave’s18 modification (SRK
EOS) being the simplest and most widely used. Unfortunately the
SRK EOS yields poor liquid densities. Zudkevitch and Joffe19 proposed a modified RK EOS, the ZJRK EOS, where both EOS
constants are corrected by temperature-dependent functions, resulting in improved volumetric predictions. Yarborough11 proposed a
generalized form of the ZJRK EOS for petroleum reservoir mixtures.
The PR EOS is comparable with the SRK EOS in simplicity and
form. Peng and Robinson7 report that their equation predicts liquid
densities better than the SRK EOS, although PR EOS densities are
usually inferior to those calculated by the ZJRK EOS. A distinct advantage of the Peng-Robinson and Soave-Redlich-Kwong equa1
van der Waals also stated the critical criteria that are used to define
the two EOS constants a and b—namely, that the first and second
derivatives of pressure with respect to volume equal zero at the critical point of a pure component.
ǒēpēvǓ
p c,T c,v c
+
ǒēēvpǓ
2
2 p ,T ,v
c c c
+ 0. . . . . . . . . . . . . . . . . . . (4.5)
Martin and Hou21 show that this constraint is equivalent to the condition (Z*Zc )3+0 at the critical point. Fig. 4.1 shows the p-v relation
of a pure compound for TtTc , T+Tc , and TuTc , indicating the inflection point on the critical isotherm that represents the van der
Waals critical criteria. Imposing Eq. 4.5 on Eq. 4.3 and specifying pc
and Tc (as opposed to specifying two of the other critical properties),
the constants a and b in the van der Waals equation are given by
a + 27
64
R 2 T c2
pc
R Tc
and b + 1 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.6)
8 c
The critical volume is given by vc +(3/8)( RTc /pc ), resulting in a
constant critical compressibility factor.
Zc +
pc vc
+ 3 + 0.375.
8
R Tc
. . . . . . . . . . . . . . . . . . . . . . . (4.7)
The van der Waals equation also can be written in terms of the Z factor (Z+pv/RT ).
Z 3 * (B ) 1) Z 2 ) A Z * AB + 0 ,
Fig. 4.1—p-V relation of a pure component at subcritical, critical,
and supercritical temperatures.
tions, where a simple temperature-dependent correction is used for
EOS constant A, is reproducibility. The ZJRK EOS’s rely on tables
or complex functions to represent the highly nonlinear correction
terms for EOS constants A and B.
Peneloux et al.’s20 volume-translation method modifies a twoconstant cubic equation by introducing a third EOS constant, c,
without changing the equilibrium calculations of the original twoconstant equation. The volume-translation constant c eliminates the
inherent volumetric deficiency suffered by all two-constant equations, and, for practical purposes, volume translation makes any
two-constant EOS as accurate as any three-constant equation.12-15
4.2.1 van der Waals5 Equation. van der Waals proposed the first cubic EOS in 1873. The van der Waals EOS gives a simple, qualitatively
accurate relation between pressure, temperature, and molar volume.
p + RT * a2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.3)
v*b v
where a+“attraction” parameter, b+“repulsion” parameter, and
R+universal gas constant. Comparing this equation with the ideal
gas law, p+RT/v, we see that the van der Waals equation offers two
important improvements. First, the prediction of liquid behavior is
more accurate because volume approaches a limiting value, b, at
high pressures,
lim v ǒ p Ǔ + b , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.4)
p ³ R
where b is sometimes referred to as the “covolume” (effective molecular volume). The term RT/(v*b) dictates liquid behavior and
physically represents the repulsive component of pressure on a molecular scale.
The van der Waals equation also improves the description of nonideal gas behavior, where the term RT/(v*b) approximates ideal
gas behavior ( p[RT/v) and the term a/v2 accounts for nonideal behavior. The a/v2 term reduces system pressure and traditionally is
interpreted as the attractive component of pressure.
2
. . . . . . . . . . . . . . . (4.8)
p
pr
where A + a
+ 27 2
64 T r
(RT) 2
and B + b
pr
p
+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.9)
RT
8 Tr
Abbott22 gives an interesting historical review of the van der
Waals EOS, its strengths and weaknesses, and its analogy to other
cubic EOS’s.
4.2.2 Redlich-Kwong6 Equations. The RK EOS is
a
p + RT *
. . . . . . . . . . . . . . . . . . . . . . . . (4.10)
v * b v (v ) b)
or, in terms of Z factor,
Z 3 * Z 2 ) ǒ A * B * B 2 Ǔ Z * AB + 0
and Z c + 1ń3 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.11)
with EOS constants defined as
R 2T 2
a + W oa p c a(T r),
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.12a)
where W oa + 0.42748;
RT
b + W ob p c ,
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.12b)
+ 0.08664;
p
pr
A+a
+ W oa 2 a(T r), . . . . . . . . . . . . . . . . . . . . (4.12c)
(RT) 2
Tr
where
W ob
where a(T r) + T *0.5
;
r
p
pr
and B + b
+ W ob . . . . . . . . . . . . . . . . . . . . . . . . . (4.12d)
RT
Tr
The fugacity expression for pure components is
ǒ
Ǔ
f
ln p + ln f + Z * 1 * ln(Z * B) * A ln 1 ) B .
B
Z
. . . . . . . . . . . . . . . . . . . . (4.13)
PHASE BEHAVIOR
TABLE 4.1—BIP’s FOR THE PR EOS AND SRK EOS
PR EOS*
SRK EOS**
N2
CO2
H2 S
N2
CO2
H2 S
N2
—
—
—
—
—
—
CO2
0.000
—
—
0.000
—
—
0.120
—
H2 S
0.130
0.135
—
0.120†
C1
0.025
0.105
0.070
0.020
0.120
0.080
C2
0.010
0.130
0.085
0.060
0.150
0.070
C3
0.090
0.125
0.080
0.080
0.150
0.070
i-C4
0.095
0.120
0.075
0.080
0.150
0.060
C4
0.095
0.115
0.075
0.080
0.150
0.060
i-C5
0.100
0.115
0.070
0.080
0.150
0.060
C5
0.110
0.115
0.070
0.080
0.150
0.060
C6
0.110
0.115
0.055
0.080
0.150
0.050
C7 +
0.110
0.115
0.050‡
0.080
0.150
0.030‡
*Nonhydrocarbon
BIP’s from Nagy and
**Nonhydrocarbon
Shirkovskiy.24
Use for both the original PR EOS (Ref. 7) and modified PR EOS (Ref. 25).
BIP’s from Reid et al.3
†Not reported by Reid et al.3
‡Should decrease gradually with increasing carbon number.
The cubic Z-factor equation can readily be solved with an analytical or a trial-and-error approach.1,2 One or three real roots may exist, where the smallest root (assuming that it is greater than B) is typically chosen for liquids and the largest root is chosen for vapors. The
middle root is always discarded as a nonphysical value. For mixtures, the choice of lower or upper root is not known a priori and the
correct root is chosen as the one with the lowest normalized Gibbs
energy, g *,23
ȍy
N
g *y +
ln f i ǒ yǓ
i
i+1
ȍx
N
and g *x +
ln f i ǒ x Ǔ , . . . . . . . . . . . . . . . . . . . . . . . . . (4.14)
i
i+1
where yi and xi +mole fractions of vapor and liquid, respectively,
and fi +multicomponent fugacity given (for a vapor phase) by
ln
B
fi
+ ln f i + i (Z * 1) * ln(Z * B)
B
yi p
)A
B
ǒ
Bi 2
*
B
A
ȍy A
N
j
j+1
Ǔ
ij
ǒ
Ǔ
ln 1 ) B .
Z
. . . . . . . (4.15)
The traditional quadratic mixing rule is used for A, and a linear mixing rule is used for B. For a vapor phase with composition yi , these
are given by
ȍȍ y y A
N
A+
N
i j
ij ,
i+1 j+1
ȍy B ,
N
B+
i
i
i+1
and A ij + ǒ1 * k ijǓ ǸA i A j , . . . . . . . . . . . . . . . . . . . . . . . (4.16)
where kij +binary-interaction parameters (BIP’s), where kii +0 and
kij +kji . Usually, kij +0 for most hydrocarbon/hydrocarbon (HC/
HC) pairs, except perhaps C1/C7) pairs. Nonhydrocarbon/HC kij
are usually nonzero, where kij [0.1 to 0.15 for N2/HC and CO2/HC
pairs (Table 4.1).3,24,25
EQUATION-OF-STATE CALCULATIONS
Many students of the RK EOS have been intrigued by its simplicity, accuracy, and the pleasure of deriving its thermodynamic properties. This has led to innumerable attempts to improve and extend
the original Redlich-Kwong equation. Certainly hundreds, if not
thousands, of technical papers and theses have been written about
the RK EOS. With the advent of digital computers, this “craze” developed into what Abbott10 called the Redlich-Kwong decade
(1967–77). Abbott claims that the remarkable success of the RK
EOS results from its excellent prediction of the second virial coefficient (securing good performance at low densities) and reliable predictions at high densities in the supercritical region. This latter observation results from the compromise fit of densities in the
near-critical region; all components have a critical compressibility
factor of Z c +1/3, where, in fact, Z c ranges from 0.29 for methane
to t0.2 for heavy C7) fractions. The Redlich-Kwong value of
Z c +1/3 is reasonable for lighter hydrocarbons but is unsatisfactory
for heavier components.
4.2.3 Soave-Redlich-Kwong. Several attempts have been made to
improve VLE predictions of the RK EOS by introducing a component-dependent correction term a for EOS constant A. Soave18 used
vapor pressures to determine the functional relation for the correction factor used in Eq. 4.12,
a + ƪ1 ) mǒ1 * T r 0.5 Ǔƫ
2
and m + 0.480 ) 1.574 w * 0.176 w 2. . . . . . . . . . . . . . (4.17)
Acentric factor w is defined in Chap. 5, and values for pure components can be found in Appendix A. Table 4.1 gives nonhydrocarbon
BIP’s for the SRK EOS as recommended by Reid et al.3; kij +0 is
generally recommended for HC/HC pairs.
The Soave-Redlich-Kwong equation is the most widely used RK
EOS proposed to date even though it grossly overestimates liquid
volumes (and underestimates liquid densities) of petroleum mixtures. The present use of the SRK EOS results from historical and
practical reasons. It offers an excellent predictive tool for systems
requiring accurate predictions of VLE and vapor properties. Volume
translation (discussed in Sec. 4.2.6) is highly recommended, if not
mandatory, when liquid densities are needed from the EOS. The
Pedersen et al.26,27 C7) characterization method is recommended
when the SRK EOS is used.
3
plicity and overall accuracy (particularly when used with volume
translation). The ZJRK EOS is surprisingly accurate for both liquid
and vapor property estimations, where its main disadvantage is the
complexity of functions used to represent temperature-dependent
corrections for the EOS constants A and B.
0.09
0.08
4.2.5 Peng-Robinson.7 In 1976, Peng and Robinson proposed a
two-constant equation that created great expectations for improved
EOS predictions and improved liquid-density predictions in particular. The PR EOS is given by
0.07
0.06
0.05
a
. . . . . . . . . . . . . . (4.19)
p + RT *
v * b v(v ) b) ) b(v * b)
0.04
0.03
or, in terms of Z factor,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1.0
Reduced Temperature
Z 3 * (1 * B)Z 2 ) ǒ A * 3B 2 * 2B ǓZ
* ǒAB * B 2 * B 3Ǔ + 0
and Z c + 0.3074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20)
0.45
The EOS constants are given by
0.40
R 2T 2
a + W oa p c a,
c
0.35
0.30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.21a)
where W oa + 0.45724;
0.25
RT
b + W ob p c ,
c
0.20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.21b)
where W ob + 0.07780;
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1.0
Reduced Temperature
Fig. 4.2—Temperature and component-dependent EOS terms
W oaa(T r, w) and W oaa(T r, w) for the ZJRK EOS (from Yarborough11).
a + ƪ1 ) mǒ1 * ǸT rǓƫ ; . . . . . . . . . . . . . . . . . . . . . . (4.21c)
2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . (4.21d)
al.25
4.2.4 Zudkevitch-Joffe-Redlich-Kwong. Zudkevitch and Joffe19
proposed a novel procedure for improving the volumetric predictions of the RK EOS without sacrificing VLE capabilities of the
original equation. They suggest that the EOS constants A and B
should be corrected as functions of temperature to match saturated
liquid densities and liquid fugacities. They show that vapor fugacities and fugacity ratios (K values) remain essentially unaffected and
that their procedure does not greatly affect vapor densities. Shortly
after the original modification appeared, Joffe et al.28 suggested that
vapor pressures should be used instead of liquid fugacities. This is
the approach used today in what is still referred to as the Zudkevitch-Joffe modification, the ZJRK EOS.
Haman et al.29 proposed the correction terms a and b for EOS
constants A and B in equation form for pure paraffins. Yarborough11
proposed generalized a and b charts for petroleum reservoir fluids
that include heavy petroleum fractions.
R 2T 2
a + W oa p c T r*0.5 aǒ T r , w Ǔ
c
RT
and b + W ob p c bǒ T r , w Ǔ .
c
and Robinson and Peng30 proposed a
In 1979, Robinson et
modified expression for m that is recommended for heavier components (wu0.49).
m + 0.3796 ) 1.485w * 0.1644w 2 ) 0.01667w 3 .
. . . . . . . . . . . . . . . . . . . . (4.22)
Fugacity expressions are given by
f
ln p +ln f + Z * 1 * ln(Z* B)
*
and ln
A ln
2 Ǹ2 B
ƪ
Z ) ǒ1 ) Ǹ2ǓB
Z * ǒ1 * Ǹ2ǓB
ƫ
fi
B
+ ln f i + i (Z * 1) * ln(Z * B)
B
yi p
)
A
2 Ǹ2 B
ǒ
Bi 2
*
B
A
ȍyA
N
j
j+1
Ǔƪ
ij
ln
Z ) ǒ1 ) Ǹ2ǓB
Z * ǒ1 * Ǹ2ǓB
ƫ
,
. . . . . . . . . . . . . . . . . . . . . . . (4.18)
. . . . . . . . . . . . . . . . . . . . (4.23)
Unfortunately, the temperature-dependent functions are complex
because they are represented by higher-order polynomials or cubic
splines (see Fig. 4.2). The behavior of these functions is highly nonlinear near Tr +1, and a discontinuity is introduced by setting the
correction factors a+b+1 at Tr y1. A single set of a and b corrections is not used in the industry, making reproducing results from
one version to another difficult. Preferably, a table of a and b correction factors should be provided when reporting a fluid characterization based on a ZJRK EOS.
Two Redlich-Kwong modifications, the SRK EOS and ZJRK
EOS, have found widespread application to petroleum reservoir
fluids. The Soave equation is sometimes preferred because of its sim-
where traditional mixing rules (Eq. 4.16) are used in the derivation
of the multicomponent fugacity expression.
The PR EOS does not calculate inferior VLE’s compared with the
RK EOS equations, and the temperature-dependent correction term
for EOS constant A is very similar to the Soave correction. The largest improvement offered by the PR EOS is a universal critical compressibility factor of 0.307, which is somewhat lower than the Redlich-Kwong value of one-third and closer to experimental values for
heavier hydrocarbons. The difference between PR EOS and SRK
EOS liquid volumetric predictions can be substantial, although, in
many cases, the error in oil densities is unacceptable from both
equations. Some evidence exists that the PR EOS underpredicts sat-
4
PHASE BEHAVIOR
TABLE 4.2—JHAVERI-YOUNGREN31
VOLUME-TRANSLATION CORRELATION FOR
C7) FRACTIONS WITH THE PR EOS
s i + 1 * A 0ńM
A1
i
A0
A1
Paraffins
2.258
0.1823
Naphthenes
3.004
0.2324
Aromatics
2.516
0.2008
Hydrocarbon Family
TABLE 4.3—VOLUME-TRANSLATION COEFFICIENTS
(si +ci /bi ) FOR PURE COMPOUNDS FOR THE
PR EOS AND SRK EOS
Component
PR EOS
SRK EOS
N2
*0.1927
*0.0079
CO2
*0.0817
0.0833
Fig. 4.3—p-V diagram of a pure component as calculated by a cubic EOS illustrating the van der Waals’s “loop” defining vapor
pressure by the equal-area rule.
H2 S
*0.1288
0.0466
C1
*0.1595
0.0234
uration pressure of reservoir fluids compared with the SRK EOS,
thereby requiring somewhat larger HC/HC (C1/C7)) BIP’s for the
PR EOS.
In review, the Peng-Robinson and Soave-Redlich-Kwong equations
are the two most widely used cubic EOS’s. They provide the same accuracy for VLE predictions and satisfactory volumetric predictions for
vapor and liquid phases when used with volume translation.
C2
*0.1134
0.0605
C3
*0.0863
0.0825
i-C4
*0.0844
0.0830
n-C4
*0.0675
0.0975
i-C5
*0.0608
0.1022
n-C5
*0.0390
0.1209
n-C6
*0.0080
0.1467
n-C7
0.0033
0.1554
n-C8
0.0314
0.1794
n-C9
0.0408
0.1868
n-C10
0.0655
0.2080
4.2.6 Volume Translation. In 1979, Martin8 proposed a new concept in cubic EOS’s, volume translation. His application was to ease
the comparison of his proposed generalized EOS with previously
published equations. In an independent study, Peneloux et al.20 used
volume translation to improve volumetric capabilities of the SRK
EOS. Peneloux et al.’s key contribution was to show that the volume
shift does not affect equilibrium calculations for pure components
or mixtures and therefore does not affect the original VLE capabilities of the SRK EOS. Volume translation works equally well with
any two-constant EOS, as Jhaveri and Youngren31 show for the
Peng-Robinson equation.
Volume translation solves the main problem with two-constant
EOS’s, poor liquid volumetric predictions. A simple correction term
is applied to the EOS-calculated molar volume.
v + v EOS * c,
ǒ f viǓ
modified+
ǒ
ȍx c
N
i i
i+1
ȍy c ,
N
and v v + v vEOS *
i i
. . . . . . . . . . . . . . . . . . . . . . . (4.25)
i+1
and v EOS
where v EOS
v +EOS-calculated liquid and vapor molar volL
umes, respectively; xi and yi +liquid and vapor compositions, respectively; and ci +component-dependent volume-shift parameEQUATION-OF-STATE CALCULATIONS
and ǒ f LiǓ
Ǔ
ǒf viǓ original exp * c i p
RT
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.24)
where v+corrected molar volume, vEOS+EOS-calculated volume,
and c+component-specific constant. The shift in volume is actually
equivalent to adding a third constant to the EOS but is special because equilibrium conditions are unaltered. This is readily seen for
a pure component, where the van der Waals “loop” (Fig. 4.3) defines
vapor pressure by making the areas above and below the p+pv line
on a p-v plot equal. Shifting the p-v plot to the left or right along the
volume axis does not change the equal-area (fugacity) balance, and
it can be readily seen that vapor-pressure predictions are unaltered
by introducing the volume-shift term c.
Peneloux et al.20 also show that multicomponent VLE is unaltered by introducing the correction term as a mole-fraction average.
v L + v LEOS *
ters. When the volume shift is introduced to the EOS for mixtures,
the resulting expressions for fugacity are
modified+
ǒ
(f Li) original exp * c i
Ǔ
p
.
RT
. . . . . . . . . . (4.26)
This implies that fugacity ratios are unaltered by the volume shift,
ǒ f Li ń f viǓ modified+ ǒf Li ń f viǓ original .
. . . . . . . . . . . . . . . . . . (4.27)
Applications that require direct use of fugacity (e.g., compositionalgradient calculation and semisolid phase equilibrium) must include
the volume-translation coefficient in the fugacity expression. Also,
the constant c can be temperature dependent but cannot include
pressure or composition dependence without derivation of new fugacity expressions.
Peneloux et al. propose that ci be determined for each component
separately by matching the saturated-liquid density at Tr +0.7. ci
can actually be determined by matching the EOS to any density value at a specified pressure and temperature. Jhaveri and Youngren31
write ci as a ratio, si +ci /bi , suggesting the following equation for
C7) fractions,
A
si + ci ń bi + 1 * A0 ń Mi 1 .
. . . . . . . . . . . . . . . . . . . (4.28)
Table 4.2 gives A0 and A1 values, and Table 4.3 gives si values for
selected pure components that have been determined by matching
5
PR EOS
S C1 through C10
paraffins fit at T+0.7
— Jhaveri-Youngren
for paraffins
Methane Concentration, mol%
Fig. 4.4—Variation of volume-translation parameter si +ci /bi vs.
molecular weight.
the saturated liquid density at Tr +0.7. Fig. 4.4 shows the variation
of si with M.
Volume translation can be applied to any two-constant cubic
equation, thereby eliminating the volumetric deficiency suffered by
all two-constant equations. For practical purposes volume translation makes any two-constant EOS as accurate as any three-constant
equation12-15 (see Fig. 4.5).
4.3 TwoĆPhase Flash Calculation
The isothermal two-phase flash calculation is the workhorse of most
EOS applications. The problem consists of defining the amounts and
compositions of equilibrium phases, usually liquid and vapor, given
the pressure, temperature, and overall composition. An inherent obstacle to solving this problem is not knowing whether two equilibrium
phases form at the specified pressure and temperature. The mixture
may exist as a single phase or may split into two or more phases.
The algorithms presented in this section assume that a mathematical solution to the two-phase problem exists: either a solution yielding equilibrium phase compositions or a “trivial” solution. Even
when the results appear physically consistent, a rigorous check of
the solution with the phase-stability test (discussed in Sec. 4.4) may
be required. Alternatively, defining the phase stability before a twophase flash calculation is made improves the reliability of the flash
results but adds computations. Mathematically, the two-phase flash
calculation can be solved by either (1) satisfying the equal-fugacity
and material-balance constraints with a successive-substitution or
Newton-Raphson algorithm32,33 or (2) minimizing the mixture
Gibbs free energy function.34
The first approach is used almost exclusively because it is readily
implemented with one of several iterative algorithms. Gibbs energy
minimization has received less attention, and it is unclear whether
it has any fundamental advantages over the simpler and more direct
equal-fugacity approach, at least for two-phase problems.
The usual constraint equations for solving the two-phase flash
problem are equal fugacities and a component/phase material balance. Assuming that all other forces are negligible (e.g., gravity),
the criterion of thermodynamic equilibrium is that the chemical potential of Component i in Phase 1 equals the chemical potential of
Component i in Phase 2; this is true for all Components i+ 1, . . . , N
(and all phases). Fugacity, fi , is a useful expression for the chemical
potential, mi , where mi +RT ln fi )li (T), and the equal-chemical-potential constraint can be written as
f Li + f vi ,
i + 1, . . . , N.
. . . . . . . . . . . . . . . . . . . . . . . (4.29)
This constraint can be solved numerically by use of some measure
of convergence, such as
6
Fig. 4.5—Comparison of measured and EOS-calculated saturated-liquid densities of the binary system C1/C10 systems at
100°F; SW+Schmidt-Wenzel.14.
ȍǒff
Li
i+1
Ǔ t e,
2
N
vi
*1
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.30)
where e is a convergence tolerance (e.g., 1
10*13).
4.3.1 Two-Phase Split Calculation (Rachford-Rice35 Procedure).
The component and phase material-balance constraints state that n total moles of feed with Composition zi distribute into nv moles of vapor
with Composition yi and nL moles of liquid with Composition xi without loss of matter or chemical alteration of the component species.
The material-balance constraints can be written as
n + nv ) nL
and n z i + n v y i ) n L x i , i + 1, . . . , N. . . . . . . . . . . . . . . . (4.31)
Introducing the vapor mole fraction Fv +nv /(nL )nv ), Eq. 4.31 can
be written as
z i + F v y i ) (1 * F v) x i . . . . . . . . . . . . . . . . . . . . . . . . (4.32)
Additionally, the mole fractions of equilibrium phases and the overall mixture must sum to unity.
ȍ y + ȍ x + ȍ z + 1.
N
N
i
N
i
i+1
i
i+1
. . . . . . . . . . . . . . . . . . . . (4.33)
i+1
This constraint can be expressed as
ȍǒ y * x Ǔ + 0.
N
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.34)
i+1
Introducing the equilibrium ratio Ki ,
K i + y ińx i ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.35)
the number of unknowns can be reduced from 2 N)1 ( yi , xi , and Fv )
to N)1 (Ki and Fv ). By use of the component material balance (Eq.
4.31) and by replacing yi by Ki xi , Eq. 4.34 can be solved in terms
of a single variable Fv .
* 1)
ȍǒy * x Ǔ + ȍ 1 )z (K
+ 0.
F (K * 1)
N
h(F v) +
N
i
i
i+1
i
i+1
i
v
. . . (4.36)
i
Eq. 4.36 is usually referred to as the Rachford-Rice35 equation. Fig.
4.6 shows the function h(Fv ) for a five-component mixture.
With feed composition and K values known, the only remaining
unknown is Fv. h(Fv ) has asymptotes at Fv +1/(1*Ki ), where each
K value gives an asymptote.36,37 Mathematically, it can be shown
that the only physically meaningful solution of h(Fv )—i.e., where
PHASE BEHAVIOR
Phase compositions are calculated from the material-balance
equations
zi
xi +
F v (K i * 1) ) 1
zi Ki
+ Ki xi .
F v (K i * 1) ) 1
and y i +
. . . . . . . . . . . . . . . . (4.41)
4.3.2 EOS Two-Phase Flash Algorithm. The flash calculation is
initialized by estimating a set of K values; the Wilson39 equation is
commonly used.
Ki +
Fig. 4.6—Rachford-Rice35 function h(FV ) for a five-component
mixture (from Ref. 37).
all Compositions xi and yi are positive—lies in the region
FvmintFv tFvmax, where
F v min +
1
1 * K max
and F v max +
1
.
1 * K min
. . . . . . . . . . . . . . . . . . . . . . . . . (4.37)
It can be shown that Fvmint0 and Fvmaxu1 if at least one K value
is t1 and one K value is u1. This implies that the solution for
h(Fv )+0 should always be limited to the region FvmintFv tFvmax.
Because h(Fv ) is monotonic and the derivative hȀ(Fv )+dh/dFv
can be expressed analytically, the Newton-Raphson algorithm is
commonly used to solve for Fv .
+ F nv *
F n)1
v
hǒF nvǓ
hȀǒF nvǓ
hȀ(F v) + dh + *
dF v
N
v
i+1
where dh + *
d Fv
ȍ
Ǔƫ
. . . . . . . . . . . . . . (4.42)
pr i
K values from this equation are not accurate at high pressures, which
potentially cause the two-phase flash to converge incorrectly to a
trivial solution. Results from a phase-stability test provide the most
reliable K-value estimates for initializing the two-phase flash but are
relatively expensive to obtain. Reliable K-value estimates can be
taken from a converged flash of the same mixture or a “related” mixture at a pressure and temperature not too far removed from the
conditions of the present flash calculation. For example, in simulating a depletion experiment with an EOS, the K values at the saturation pressure can be used as initial estimates for the flash at the first
depletion stage, the converged K values from this flash can be used
for the flash at the second stage, and so on at lower pressures.
With estimated K values, the Rachford-Rice35 equation is solved
for Fv, with the search for Fv always bounded by Fvmin and Fvmax.
F v min +
1
t0
1 * K max
and F v max +
1
u 1. . . . . . . . . . . . . . . . . . . . . . . (4.43)
1 * K min
Phase compositions are calculated from the material-balance
equations. Having calculated xi and yi , phase Z factors ZL and Zv and
component fugacities fLi and fvi are calculated with the EOS.
and Z v + F EOSǒ y, p, T Ǔ . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.44)
ȍ
z i (K i * 1)
N
i+1
zi
+ 0,
v ) ci
N
ǒ
Z L + F EOSǒ x, p, T Ǔ
2
ƪF v (Ki * 1) ) 1ƫ
2
,
. . . . (4.38)
where n+iteration counter. The first guess for Fv can be chosen arbitrarily as 0.5.
In 1949, Muskat and McDowell38 proposed a solution to the twophase split calculation that is basically the same as the one proposed
by Rachford and Rice35 but numerically more efficient. Introducing
the quantity ci +1/(Ki *1), where ci +R for Ki +1, Muskat and
McDowell proposed the following form of the function h(Fv ).
h(F ) + ȍ
F
ƪ
exp 5.37ǒ1 ) w i Ǔ 1 * T *1
ri
. . . . . . . . . . . . . . . . . . . . . . (4.39)
zi
ǒF v ) c i Ǔ
and f Li + F EOSǒ x, Z L, p, T Ǔ
and f vi + F EOSǒ y, Z v, p, T Ǔ.
. . . . . . . . . . . . . . . . . . . . . . . (4.45)
The “normalized” Gibbs energy function, g *, of each phase is calculated from
ȍ x ln f
N
g *L +
i
Li
i+1
ȍ y ln f
N
and g *v +
i
vi ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.46)
i+1
and the normalized mixture Gibbs energy is given by
.
2
. . . . . . . . . . . . . . . . . (4.40)
g *mix + F v g *v ) (1 * F v)g *L .
. . . . . . . . . . . . . . . . . . . . (4.47)
If a Newton estimate from Eq. 4.38 with either the MuskatMcDowell or Rachford-Rice equations for h gives an estimate of Fv
outside the range FvmintFv tFvmax, the Newton method should be
replaced by interval bisection or modified regula falsi until convergence is achieved. Severe round-off errors may cause any solution
technique to fail when both K and z of one component are very small
(e.g., KN +1 10*12 and zN +1 10*20).*
If multiple Z-factor roots are found for either phase, the root with the
lowest Gibbs energy should be chosen.23 For example, if three liquid Z-factor roots were calculated ( Z L1, Z L2, and Z L3), the middle
root, Z L2, is automatically discarded and the two Gibbs energy functions, g *L1 and g *L3 , are calculated; f L1i are calculated with Z L1, and
f L3i are calculated with Z L3. If g *L3 t g *L1, Z L3 should be chosen;
otherwise, choose Z L1 for g *L1 t g *L3 .
Zick* suggests that this method of choosing the Z-factor root is not
fail-safe because, at early iterations in the flash calculation, the incor-
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1991).
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
i+1
EQUATION-OF-STATE CALCULATIONS
7
TABLE 4.4—SEQUENCE OF FLASH CALCULATIONS TO
ENSURE CORRECT SOLUTION WITH MULTIPLE ROOTS
Liquid ZL
Root Chosen
Possible Order
of Multiple Flash
Calculations
Smallest
1
n
Largest
n
3
n
n
Smallest
Largest
n
2
4
Vapor, Zv
Root Chosen
n
n
n
rect root may have a lower Gibbs energy than the correct root. He proposes that the flash calculation be converged completely with a consistent choice of roots (e.g., the smallest root always chosen for the
liquid phase and the largest root always chosen for the vapor phase).
If multiple roots in either phase are detected during this flash calculation, a second, third, and potentially fourth flash calculation must be
completed, as summarized in Table 4.4. The two-phase solution with
the lowest mixture Gibbs energy is chosen as the correct solution.
With fugacities calculated for each phase, the fugacity constraint
(Eq. 4.30) is checked. The recommended convergence tolerance is
1 10*13, although a less stringent value can be used in some applications. If convergence is not achieved, the K values can be modified with successive substitution.
+ K (n)
K (n)1)
i
i
f Li(n)
f (n)
vi
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.48)
where the superscripts (n) and (n)1) indicate the iteration level.
With new K values, the Rachford-Rice35 equation is solved again
(with new values of Fvmin and Fvmax), phase compositions are calculated with the converged Fv value, phase Z factors and component
fugacities are calculated from the EOS, and the fugacity constraint
is rechecked. This iterative procedure is repeated until convergence
is achieved. Three types of converged solutions can be obtained.
1. A physically acceptable solution is found with 0xFv x1,
where Fv +0 corresponds to a bubblepoint condition, Fv +1 corresponds to a dewpoint condition, and 0tFv t1 indicates a twophase condition.
2. A physically unacceptable solution is found with Fv t0 or
Fv u1,37 where the calculated equilibrium compositions satisfy the
equal-fugacity constraint and the mathematical material-balance
equation. This solution indicates that the mixture is thermodynamically stable as a single phase and will not split into two phases. For
this solution, the calculated equilibrium compositions would coexist in thermodynamic equilibrium at the given pressure and temperature if they were mixed together in a physically meaningful proportion (creating, of course, a different mixture composition).
3. A so-called trivial solution is found where the calculated phase
compositions are identical to the mixture composition and K values
equal one (xi +yi +zi and Ki +1).
The first solution is usually a “correct” solution. However, if a potential three-phase solution exists, the two-phase solution may represent
only a local minimum in the mixture Gibbs energy surface and the mixture Gibbs energy may be reduced further by locating the three-phase
solution or another two-phase solution. Michelsen32 suggests that this
problem is best dealt with by use of phase-stability analysis.
Whitson and Michelsen37 refer to the second solution to the flash
as a “negative” flash because one of the phase mole fractions is negative (and the other phase fraction is u1). Although this condition
is physically unacceptable, the solution still has practical application. For example, phase properties and compositions are continuous across the phase boundaries. Also, a nontrivial negative flash
solution indicates phase stability with the same certainty as the
phase-stability test, although the negative flash calculation requires
better initial K-value estimates than does the phase-stability test.
A trivial solution to the flash calculation should always be
checked with the phase-stability test to verify that the mixture is in
fact single phase. Trivial solutions arise for several reasons, the most
8
Fig. 4.7—p-T phase envelope and envelopes indicating the limit
of a nontrivial negative flash and a nontrival stability test for the
binary C2/n-C4 system (from Ref. 37).
common being poor initial K-value estimates (e.g., from the Wilson39 equation). A “valid” trivial solution occurs when two-phase
solutions do not exist. This occurs outside the p-T envelope that
Whitson and Michelsen define as the convergence-pressure envelope, where Fv !"R in the negative flash (Fig. 4.7). Along the
phase boundary and near a critical point, the Newton-Raphson flash
technique tends to converge to a trivial solution more readily than
do successive-substitution methods. Finally, as Michelsen23 shows,
the two-phase flash never converges to a trivial solution with
successive substitution under the following conditions.
1. The phase-stability test indicates that the mixture is unstable.
2. The K values resulting from the stability test are used to initialize the flash calculation.
3. The mixture Gibbs energy g *1
mix at the first iteration is less than
the mixture Gibbs energy g *z .
The flash calculation initialized by a successful phase-stability
test is the safest solution method available, albeit more expensive
than a direct two-phase flash calculation.
Successive substitution is the safest solution technique for the
two-phase flash problem, but it becomes slow when fugacity coefficients are strongly composition dependent. The method is particularly slow near phase boundaries and critical points, where many
thousands of iterations may be required to reduce the convergence
criterion to an acceptable value. Successive substitution can be accelerated with one of several methods as described in Refs. 33 and
40 through 43 among others. Michelsen32 recommends the general
dominant eigenvalue method44 (GDEM); he shows that this method
is particularly well suited for the two-phase flash problem because
two dominant eigenvalues are found near phase boundaries and the
critical point. He recommends preceding each GDEM promotion
(acceleration) with five successive-substitution iterations, where
the GDEM K-value correction is given by
+ ln K (n)
)
ln K (n)1)
i
i
Du (n)
* m 2 Du (n*1)
i
i
1 ) m 1 ) m 2 , . . . . . . . . . (4.49)
where Du i 5 ln ǒ f Lińf viǓ
and m 1 + ǒb 02 b 12 * b 01 b 22Ǔńǒb 11 b 22 * b 12 b 12Ǔ , . . . . . . . (4.50a)
m 2 + ǒb 01 b 12 * b 02 b 11Ǔńǒb 11 b 22 * b 12 b 12Ǔ ,
ȍ Du
N
and b jk +
ǒ n*j Ǔ
Du (n*k)
.
i
i
. . . . . (4.50b)
. . . . . . . . . . . . . . . . . . . . (4.50c)
i+1
PHASE BEHAVIOR
m1 and m2 are coefficients reflecting the relative magnitudes of
dominant eigenvalues l1 and l2. Michelsen suggests that promotions be rejected (or reduced) if the mixture Gibbs energy increases
after a promotion.
Zick* shows that the coefficients m1 and m2 calculated with Eqs.
4.50a and 4.50b can be seriously affected by round-off error. He
suggests that the substitution ejk 5(bjk *b12)/b12 eliminates the
round-off problem and that this transformation of variables results
in promotion coefficients m1 and m2 that can be used even near a critical point. Also, the Michelsen32 suggestion to switch to a NewtonRaphson method after two GDEM iterations is unnecessary with the
modified GDEM coefficients. For most practical reservoir applications, GDEM will converge in two to three promotions (11 to 16 iterations), with near-critical problems requiring up to six promotions
(31 iterations).
In summary, the two-phase flash calculation can be outlined with
the following step-by-step procedure.
1. Estimate K values.
2. Calculate Kmin and Kmax.
3. Solve the Rachford-Rice phase-split calculation (Eq. 4.36)
for Fv, limited between Fvmin and Fvmax (Eq. 4.43).
4. Calculate phase compositions x and y (Eq. 4.41).
5. Calculate phase Z factors ZL and Zv from the EOS.
6. Calculate component fugacities fLi and fvi from the EOS.
7. Calculate phase Gibbs energy functions g *L and g *v (Eq. 4.46),
determine the correct Z-factor roots of each phase (if multiple roots
exist), and calculate the mixture Gibbs energy (Eq. 4.47).
8. Check the equal-fugacity constraint (Eq. 4.30).
9. (a) If convergence is reached, stop. (b) If convergence is not
reached, update the K values with the fugacity ratios (Eq. 4.48) or
a GDEM promotion (Eq. 4.49); alternatively, use another acceleration technique or a Newton-Raphson K-value update.
10. Check for convergence at a trivial solution (Ki !1) with the
condition
ȍǒln K Ǔ
N
i
2
t 10 *4.
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.51)
i+1
11. If a trivial solution is not detected, return to Step 2. Otherwise,
confirm the trivial solution with a stability test.
For reservoir simulation, a Newton-Raphson solution to the flash
problem can be used because initial K-value estimates (from earlier
timesteps and neighboring gridblocks) should be reliable, and the reduced computation time of a Newton method compared with an accelerated successive-substitution method can be significant.45 Michelsen’s23 implementation of the Newton-Raphson method is considered
a very efficient algorithm and is cited here directly from his original
publication (with the exception of some Nomenclature changes).
“The set of equations to be solved is
e i(K) + ƪlnǒn vińn vǓ ) ln f viƫ * ƪlnǒn Lińn LǓ ) ln f Liƫ + 0,
. . . . . . . . . . . . . . . . . . . . (4.52)
where nvi and nLi +number of moles of Component i in the vapor
and liquid phases, respectively.
“The Jacobian matrix is given by
J ij +
ēe i
,
ē ln K j
yielding
J + B A *1 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.56)
z
n vn L
with B ij + x yi d ij * 1 )
nv ) nL
i i
ƪǒ Ǔ ǒ Ǔ ƫ
ē ln f i
ēn j
)
v
ē ln f i
ēn j
L
. . . . . . . . . . . . . . . . . . . . (4.57)
and A ij +
zi
d * 1. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.58)
x i y i ij
“Since B is symmetric, we can use the decomposition
B + LDL T, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.59)
where L is unit lower triangular and D is diagonal with positive elements for a positive definite B.
“Then,
b + * AL *TD *1L *1e, . . . . . . . . . . . . . . . . . . . . . . . . (4.60)
where the cost of the decomposition and the subsequent backsubstitution is only about half of that required for conventional solution
of Eq. 4.54 by Gaussian elimination.” Application of the Michaelsen Newton-Raphson algorithm, as proposed here and without
proper precautions, will lead to convergence problems near phase
boundaries because both matrices become singular at phase boundaries and the solution will be severely affected by round-off errors.
4.4 Phase Stability
One of the most difficult aspects of making VLE calculations with
an EOS is knowing whether a mixture will actually split into two (or
more) phases at the specified pressure and temperature. Traditionally, this problem has been solved either by conducting a two-phase
flash or by making a saturation-pressure calculation; both methods
are expensive and not entirely reliable.
In 1982, two papers32,46 showed how the Gibbs tangent-plane criterion could be used to establish the thermodynamic stability of a
phase [i.e., whether a given composition has a lower energy remaining as a single phase (stable) or whether the mixture Gibbs energy will
decrease by splitting the mixture into two or more phases (unstable)].
Ref. 46 shows graphically how the Gibbs tangent-plane criterion is
used to establish phase stability of simple binary systems, and Ref. 32
gives an algorithm to establish phase stability numerically. This section on phase stability follows these references closely.
Phase stability deals with the question of whether a mixture can
attain a lower energy by splitting into two or more phases. The Gibbs
energy for n moles of mixture Composition z i considered as a homogeneous phase is given by
ȍǒn m Ǔ
N
Gz +
i z
i
i+1
ȍz m .
N
+n
i
zi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.61)
i+1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.53)
The mixture will split into two phases y and x if the mixture Gibbs
energy, Gmix, is less than Gz , where Gmix is given by
and the correction b with bi +Dln Ki is found from
Jb + * e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.54)
ȍǒn m Ǔ
N
G mix +
i
“The Jacobian matrix is calculated from
ȍǒn
N
N
ēe i ēn L k
ēe i
+
, . . . . . . . . . . . . . . . . . (4.55)
J ij +
ēn L k ē ln K j
ē ln K j
k+1
ȍ
+
EQUATION-OF-STATE CALCULATIONS
) ǒn i m i Ǔ L ; m Li + m vi + m i
vi
) n LiǓm i
i+1
ȍ nƪF y ) (1 * F ) x ƫ m .
N
+
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
i v
i+1
v i
v
i
i
. . . . . . . . . . . . . . (4.62)
i+1
9
Mole Fraction Component 1
Fig. 4.9—p-x plot of a two-component mixture exhibiting various
two- and three-phase equilibrium conditions (Ref. 46).
Fig. 4.8—Gibbs energy surface for a binary system.
The Gibbs tangent-plane criterion considers the energy surface
for a homogenous phase. In terms of overall mole fractions zi +ni /n
with fugacities evaluated for z, the normalized Gibbs energy function, g * + GńRT, is given by
ȍz
N
g *z +
i
ln f i (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.63)
i+1
g *z is a normalized Gibbs energy for the mixture composition. For
a binary mixture, the energy surface g * represents a curve that can
be plotted vs. one of the mole fractions (Fig. 4.8). For a ternary system, the energy surface can be plotted in three dimensions ( g * vs.
two of the mole fractions), but a graphical representation is not possible for systems with more than three components.
Graphically, the condition of equilibrium for a binary system is established on a g * plot by drawing a straight line that is tangent to the
TABLE 4.5—PHASES IN EACH PRESSURE INTERVAL
Region
Phases Present
I
Only single-phase vapor, V
II
Single-phase liquid, L1
Two-phase vapor/liquid, V/L1
Single-phase vapor, V
III
Single-phase liquid, L1
Three-phase vapor/liquid/liquid, V/L1/L2
Single-phase vapor, V
IV
Single-phase liquid, L1
Liquid/liquid, L1/L2
Single-phase liquid, L2
Vapor/liquid, V/L2
Single-phase vapor, V
V
Single-phase liquid, L1
Liquid/liquid, L1/L2
Single-phase liquid, L2
10
curve at two (or more) compositions. A valid tangent plane cannot intersect the Gibbs energy surface anywhere except at the points of tangency. For example, the vapor/liquid tangent passes through the two
points ǒx, g *LǓ and ǒy, g *v Ǔ in Fig. 4.8. The compositions through which
the tangent passes are equilibrium phases that satisfy the equal-fugacity condition. A physically acceptable two-phase solution requires
that the mixture composition lie between the two equilibrium compositions, xtzty. If z lies outside the compositions bounded by x
and y (ztx or zuy), the material-balance constraint is violated and
the mixture is stable. Likewise, z+y and z+x indicate stable conditions for a mixture at its dewpoint and bubblepoint, respectively.
When the overall composition z lies between the equilibrium compositions (xtzty), the mixture is unstable and will split into the two
equilibrium phases with compositions y and x, having a mixture
Gibbs energy given by g *mix + F vg *v ) (1 * F v)g *L. with g *mix
t g *z . The value of g *mix is read directly from the tangent line at the
mixture composition, and the vapor mole fraction F v is given by the
distance from z to y, relative to the total distance between x and y
ƪF v + (z * y)ń(x * y)ƫ.
Baker et al.46 discuss the mathematical conditions associated
with the Gibbs tangent-plane criterion and illustrate the technique
for a binary system that exhibits two- and three-phase behavior at
various pressures and a fixed temperature. Fig. 4.9, a p-x diagram
divided into five pressure intervals, is adapted from their example.
Depending on the mixture composition, various combinations of
the three potential phases [vapor (V), lower liquid (L1), and upper
liquid (L2)] can form in each pressure interval. Table 4.5 shows the
phases for each interval.
Figs. 4.10A through 4.10G and 4.11A through 4.11F present
Gibbs energy plots for Regions II, III, and IV together with the p-x
diagram (Fig. 4.9). Fig. 4.10A shows the g * curve for a low pressure
in Region II where only two “valleys” exist, and thereby only one tangent can be drawn. Equilibrium compositions are located at the two
points where the tangent touches the g * curve, y and xL1, each of
which is near the bottom of a valley. Figs. 4.10B through 4.10D show
the g * curve for a higher pressure in Region II, where a middle valley
develops between the two valleys exhibited in Fig. 4.10A. Only one
valid tangent can be drawn, between the L1 and V valleys. This tangent is valid because it does not pass through the g * curve at compositions other than the points of tangency, xL1 and y. Two other tangents
can be drawn, one yielding a liquid/liquid (L1/L2) solution between
the left and middle valleys and the other yielding a liquid/vapor
(L2/V) solution between the middle and right valleys. These two tangents are, however, invalid because they lie above the g * curve in
violation of the tangent-plane criterion. Such tangents represent false
two-phase solutions that satisfy the equal-fugacity constraint but
PHASE BEHAVIOR
Developing Second Liquid
Phase “Valley”
Fig. 4.10A—Gibbs energy plot for the Baker et al.46 binary example, Region II.
yield only a local minimum in the mixture Gibbs energy. False twophase solutions are difficult to detect unless one has a priori knowledge of the actual equilibrium condition. Low-temperatures and highCO2 concentrations are conditions associated with three-phase
behavior that may be susceptible to false two-phase solutions.
Fig. 4.10E shows the g * curve for the three-phase pressure (Region
III). A single line can be drawn that is tangent to three compositions
( y, xL1, and xL2). The three-phase solution is physically valid for any
composition lying between the lower liquid (xL1) and the vapor (y)
compositions, with the relative amounts of each phase in a two-phase
mixture being determined by the overall composition. For ztxL1 and
zuy, the mixture is stable and remains as a single phase.
Fig. 4.10F shows the g * curve for a pressure in Region IV where
the middle valley decreases relative to the left and right valleys. This
creates a curve that has two valid tangents, one representing a L1/L2
solution and the other representing a LȀ2ńV solution. Valid two-phase
solutions are found for mixture compositions in either the L1/L2 interval, xL1tztxL2, or the LȀ2ńV interval xȀL2tzty. Mixture compositions outside these two intervals will remain as a stable single phase.
The tangent that can be drawn between a lower liquid and vapor phase
(dashed line) is not a valid two-phase solution because the tangent lies
above the g * curve in the middle region of compositions (Fig. 4.10G).
However, this is a potential two-phase solution that could readily be
calculated and mistaken for a valid solution.
In Figs. 4.10A through 4.10G, the tangent-plane solutions that
pass through compositions where g * is convex have been ignored.
This follows from the observation that any mixture composition
with the condition ǒē 2g *ńēz 2Ǔ t 0 is intrinsically unstable,37 and
any search for a solution to the tangent-plane criterion will move
away from such “convex” solutions. Also, these tangents violate the
tangent-plane criterion because they lie above the energy surface
(see Fig. 4.11A).
Baker et al.’s46 graphical interpretation of stability analysis is
particularly useful for describing the Gibbs tangent-plane criterion
but does not lend itself to being implemented as a numerical algorithm that can be used to calculate phase stability. Michelsen32 proposes an algorithm that determines whether a mixture will remain
EQUATION-OF-STATE CALCULATIONS
Fig. 4.10B—Gibbs energy plot for the Baker et al.46 binary example: Region II, correct two-phase V/L1 solution.
single phase or split into multiple phases. Michelsen’s algorithm is
similar to a flash calculation but is faster and safer (accurate K-value
estimates are not needed for the stability test).
The Michelsen stability test is based on locating “second-phase”
compositions that have tangent planes parallel to the tangent plane
of the mixture composition. If any of the parallel tangent planes lie
below the tangent plane of the mixture composition, the mixture is
unstable and will split into at least two phases. If all compositions
having parallel tangent planes lie above the mixture tangent plane
or no composition has a parallel tangent plane, the mixture is stable
as a single phase. In addition, if a composition (not equal to the mixture composition) lies on the same tangent plane as the mixture, the
mixture is at a bubble- or a dewpoint and the second phase is an incipient equilibrium phase. Figs. 4.11B through 4.11F graphically illustrate the Michelsen stability-test criteria for stable and unstable
mixture compositions.
The mathematical description of Michelsen’s stability test is not
within the scope of this monograph, but his stability-test algorithm
follows. Actually, two tests are usually required; one test assumes that
the second phase is vapor-like, and the other assumes that the second
phase is liquid-like. This corresponds to initializing the search for a
second phase with two compositions where each search is conducted
separately. The compositions used to initialize each search should
represent “poor” guesses (i.e., very vapor-like and very liquid-like
compositions) to expand the composition space being searched. One
could conceivably use N stability tests (N+number of components),
each starting with a pure component as the initial composition estimate, but this would be unnecessarily time consuming.
Michelsen shows that locating a second-phase composition with
a tangent plane parallel to the tangent plane of the mixture composition is equivalent to locating a composition y with component fugacities f yi equal to mixture component fugacities f zi times a constant,
f zi
+ S + I, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.64)
f yi
11
False V/L2 Two-Phase
Equilibrium Condition
False L1/L
Two-Phase
Equilibrium
Condition
Fig. 4.10C—Gibbs energy plot for the Baker et al.46 binary example: Region II, false two-phase V/L2 solution.
Fig. 4.10D—Gibbs energy plot for the Baker et al.46 binary example: Region II, false two-phase L1/L2 solution.
where I+constant. A successive-substitution algorithm, summarized in the following procedure, can readily be used to solve the Michelsen stability test. Note that each test is conducted separately
(e.g., converging the vapor-like search first, then converging the liquid-like search).
1. Calculate the mixture fugacities, f zi ; with multiple Z-factor
roots, choose the root with the lowest g *.
2. Use the Wilson equation (Eq. 4.42) to estimate initial K values.
Ǔƫ
expƪ5.37(1 ) w i)ǒ1 * T *1
ri
.
p ri
K 1i +
. . . . . . . . . . . . . (4.42)
3. Calculate second-phase mole numbers, Yi , using the mixture
composition z i and the present K-value estimates.
(Y i) v + z i (K i) v
or (Y i) L + z i ń(K i) L .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.65)
4. Sum the mole numbers.
ȍǒY Ǔ
N
Sv +
j v
j+1
ȍǒY Ǔ .
N
or S L +
j L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.66)
j+1
5. Normalize the second-phase mole numbers to get mole fractions, yi .
(y i) v +
(Y i) v
ȍǒY Ǔ
N
j v
j+1
12
+
(Y i) v
Sv
Fig. 4.10E—Gibbs energy plot for the Baker et al.46 binary example: Region III, correct three-phase solution.
PHASE BEHAVIOR
False V-L1 Two-Phase
Equilibrium Condition
Fig. 4.10F—Gibbs energy plot for the Baker et al.46 binary example: Region IV, two possible correct two-phase solutions (L1/L2
or V/L2).
(Y i) L
or (y i) L+
ȍǒY Ǔ
N
+
(Y i) L
.
SL
. . . . . . . . . . . . . . . . . . . . . (4.67)
j L
Fig. 4.10G—Gibbs energy plot for the Baker et al.46 binary example: Region IV, false two-phase V/L1 solution.
Michelsen suggests that Step 9 of the successive substitution can
be accelerated with the GDEM approach with one eigenvalue (only
one eigenvalue approaches 1 near the critical point in a stability
test). He recommends that four successive-substitution iterations
precede each promotion. The GDEM update is given by
j+1
l
6. Calculate the second-phase fugacities ( fyi )v or ( fyi )L from the
EOS; with multiple Z-factor roots (for a given phase), choose the
root with the lowest Gibbs energy g *.
7. Calculate the fugacity-ratio corrections for successive-substitution update of the K values.
(R i) v +
f zi
1
ǒf yiǓ L
f zi
2
SL .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.68)
10*12).
t e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.69)
9. If convergence is not obtained, update the K values.
K (n)1)
+ K (n)
R (n)
.
i
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.70)
10. Check whether a trivial solution is being approached using
the criterion
N
i
2
ȍ ln R
N
b 01 +
(n)
ln
i
R (n*1)
,
i
ȍ ln R
(n*1)
i
ln R (n*1)
,
i
. . . . . . . . . . . . . . . . . (4.72)
i+1
i+1
ȍǒln K Ǔ
Ť
b 11
,
11 * b 01
N
N
i
Ťb
and b 11 +
8. Check whether convergence is achieved (e.g., et1
ȍ(R * 1)
l+
i+1
ǒf yiǓ v S v
or (R i) L +
ƪR (n)
ƫ,
+ K (n)
K (n)1)
i
i
i
t1
10 *4.
. . . . . . . . . . . . . . . . . . . . . . (4.71)
i+1
11. If a trivial solution is not indicated, go to Step 3 for another
iteration.
EQUATION-OF-STATE CALCULATIONS
where the superscript (n) is the iteration counter.
Table 4.6 summarizes the interpretation of the two-part stability
test. The mixture (very likely) is stable if both tests yield Sx1, if
both tests converge to a trivial solution, or if one test yields Sx1 and
the other converges to a trivial solution. Theoretically, it is impossible to establish without a doubt that a mixture is stable until all compositions have been tested. However, both solutions indicating stability from the two-part Michelsen test usually ensures that a
mixture is in fact single phase.
On the other hand, only one test indicating Su1 is sufficient to
determine that a mixture is definitely unstable. For an unstable solution, the resulting K values from the stability test can be used to initialize the two-phase flash. Potentially both SL and Sv are u1, in
which case the best initial K values for the flash are given by
Ki +( yi )v /( yi )L +(Ki )v (Ki )L , requiring that both tests be completed
(even though the first test positively indicates an unstable mixture).
Fig. 4.12 shows Nghiem and Li’s47 EOS calculations identifying
the phase boundary of a reservoir oil. Also shown is the envelope
within the phase boundary (dashed line) where one of the stability
13
Region of Compositions Where
Stability Test Converges Nontrivial
Fig. 4.11A—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for region of compositions where stability test converges nontrivial.
Fig. 4.11B—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for liquid-like feed, z, with one unstable condition, y,
located.
tests converges to a trivial solution. The lower dashed line (starting
from the critical point) shows where the liquid-like stability test
converges to a trivial solution, and the upper dashed curve shows
where the vapor-like stability test converges to a trivial solution. Inside the dashed-curve envelope, both the liquid- and vapor-like stability tests converge to a nontrivial unstable solution (both SL and
Sv are u0). Fig. 4.13 illustrates the behavior of SL and Sv vs. pressure at a fixed temperature for this system.
Michelsen’s phase-stability test has many applications; the following summarizes the most important ones.
1. Determining whether a mixture composition is thermodynamically stable as a single phase. If the test indicates stability (assuming
Fig. 4.11C—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for vapor-like feed, z, with one stable condition, y, located.
Fig. 4.11D—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for liquid-like feed, z, with two unstable conditions, yL
and yv, located.
14
PHASE BEHAVIOR
Fig. 4.11E—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for vapor-like feed, z, with one unstable condition, yL ,
located.
that both liquid- and vapor-like second phases have been tested), it
is very likely that a two-phase solution does not exist.
2. With at least one unstable solution, initializing the two-phase
flash calculation with K values determined from the unstable solution(s) of the stability test. This is particularly useful if K values
from a converged flash at nearby conditions are not available.
3. Initializing and limiting the pressure range in a saturation-pressure calculation (see Sec. 4.5).
4. Checking the stability of a converged two-phase flash when
three-phase behavior is suspected (e.g., for low-temperature and
high-CO2 systems). This requires, however, two modifications of the
stability test: (a) choice of appropriate initial K-value estimates for the
“third”-phase search and (b) use of the converged two-phase fugaci-
Fig. 4.11F—Gibbs energy plot for a hypothetical binary system
showing a graphical interpretation of Michelsen’s32 phase-stability test for liquid-like feed, z, with one unstable condition, yv ,
located.
ties, feqi +fvi +fLi , instead of fzi in the new search (i.e., locate a third
composition y so that feqi /fyi equals a constant S, with Sx1 indicating
stability; Su1 would indicate an unstable condition for the two-phase
solution, thereby guaranteeing a multiphase solution).
TABLE 4.6—SUMMARY OF POSSIBLE
PHASE-STABILITY-TEST RESULTS
Second Phase
Vapor-Like
ǒKiǓ
Stable
Unstable
ǒyiǓ
+ z
v
i
v
Liquid-Like
ǒKiǓ
l
+
zi
ǒyiǓ
l
Probable
Number of
Valleys on g*
TS
TS
1
SL x1
TS
2
TS
SL x1
2
Sv x1
SL x1
3
Sv u1
TS
2
TS
SL u1
2
Sv u1
SL u1
2
Sv u1
SL x1
3
Sv x1
SL u1
3
TS+trivial solution.
EQUATION-OF-STATE CALCULATIONS
Fig. 4.12—Phase and stability-limit envelopes for a reservoir oil;
stability limit represents the condition when one of the stability
tests first converges to a trivial solution (from Nghiem and Li47).
15
LIQUID-lIKE SECOND PHASE
VAPOR-LIKE SECOND PHASE
Nghiem et al.48 use the condition of zero tangent-plane distance,
d TP +0, to solve for saturation pressure, psat, and incipient-phase
composition y.
d TPǒ p sat , y Ǔ + ln
ǒȍ Ǔ
N
Yi
+ 0.
. . . . . . . . . . . . . . . . . (4.77)
i+1
The recommended approach for determining saturation pressure
is based on a slightly different approach proposed by Michelsen49;
he uses the condition
TS
TS
Qǒ p sat , y Ǔ + 1 *
ȍ z ƪf (z)ńf ǒ y Ǔƫ + 0
N
i
i
i
i+1
ȍ y ǒf
N
+1*
i
Ǔ
zi ń f yi
i+1
ȍY ,
N
+1*
i
. . . . . . . . . . . . . . . . . . . . . . . (4.78)
i+1
where incipient-phase mole fractions are defined by
Fig. 4.13—Behavior of mole number sums from stability test, SL
and Sv, vs. pressure for a fixed temperature; TS+trivial solution.
4.5 SaturationĆPressure Calculation
For a mixture composition z at fixed temperature T, the saturationpressure calculation involves finding the pressure(s) where the mixture is in equilibrium with an infinitesimal amount of an incipient
phase. In terms of a two-phase flash, the saturation pressure defines
a pressure where the vapor mole fraction, Fv, equals zero or one
(Fv +0 at bubblepoint and Fv +1 at dewpoint).
One way to locate the saturation pressure of a mixture would be
to make a 1D search in pressure for Fv +0 or 1, where the two-phase
flash is converged at each pressure estimate during the search. Although this approach would be safe, it also would be very slow. Several alternative saturation-pressure algorithms are available that are
both efficient and reliable when used with stability analysis.
The two conditions defining a saturation pressure are that the fugacities of all components are equal in both phases,
f zi + f yi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.73)
and that the mole fractions of the incipient phase, y, equal unity,
ȍ y + 1.
N
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.74)
i
Expressing the incipient-phase mole fractions in terms of K values
( yi +zi Ki for a bubblepoint and yi +zi /Ki for a dewpoint), the traditional equations used to solve bubble- and dewpoint calculations,
respectively, are
ȍz K + 0
N
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.75a)
i+1
ȍ z ńK + 0.
N
and 1 *
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . (4.75b)
i+1
In terms of stability analysis, the saturation-pressure condition
corresponds to finding a second phase with a tangent plane that is
parallel to the mixture composition’s tangent plane, with zero distance between the two tangent planes. This is equivalent to the sum
of incipient-phase mole numbers equaling unity.
ȍ Y + 1.
N
i
i+1
16
Yi
ȍY
.
N
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.79)
j
j+1
An efficient method to solve this equation uses a Newton-Raphson
update for pressure and accelerated successive substitution (GDEM)
for composition. The following procedure outlines this approach.
1. Guess a saturation type: bubble- or dewpoint. An incorrect
guess will not affect convergence, but the final K values may be “upside down.”
2. Guess a pressure p *.
3. Perform Michelsen’s stability test at p *.
4. (a) If the mixture is stable for the current value of p *, this pressure represents p * the upper bound of the search for a saturation
pressure on the upper curve of the phase envelope. Return to Step
1 and try a lower pressure to look for an unstable condition. (b)
With an unstable condition at p *, this pressure represents the lower
bound in the search for a saturation pressure on the upper curve of
the phase envelope.
5. Having found an unstable solution, use the K values from the
stability test to calculate incipient-phase mole numbers at bubbleand dewpoint with Eqs. 4.80a and 4.80b, respectively.
Y i + z i K i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.80a)
and Y i + z ińK i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.80b)
i+1
1*
yi 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.76)
If two unstable solutions were found in the stability test, use the K
values for the test with the largest mole number sum S.
At this point, the initialization is complete and the iteration sequence begins.
6. Calculate the normalized incipient-phase compositions.
yi +
Yi
ȍY
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.81)
N
j
j+1
7. Calculate phase Z factors, Zz and Zy, and component fugacities,
fzi and fyi , from the EOS at the present saturation-pressure estimate.
When multiple Z-factor roots are found for a given phase, the root
giving the lowest Gibbs energy should be chosen.
8. Calculate fugacity-ratio corrections.
f zi
Ri +
f yi
ǒȍ Ǔ
N
Yj
*1
.
. . . . . . . . . . . . . . . . . . . . . . . . . (4.82)
j+1
PHASE BEHAVIOR
9. Update incipient-phase mole numbers with the fugacity-ratio
corrections,
ƫ l,
Y i(n)1) + Y i(n) ƪR (n)
i
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.83)
where four iterations use successive substitution (l+1) followed by
a GDEM promotion with l given by
l+
Ťb
11
Ť
b 11
,
* b 01
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.84)
ȍ ln R
N
where b 01 +
(n)
ln R (n*1)
i
i
4.6 Equilibrium in a Gravity Field:
Compositional Gradients
Gibbs53 was the first to give the formula for calculating compositional variation under the force of gravity for an isothermal system.
The condition of equilibrium is satisfied by the constraint
m i ǒ p ref , z ref , T Ǔ + m iǒ p, z, TǓ ) M i gǒh * h refǓ ,
i+1
ȍ ln R
Michelsen52 also has proposed a critical-point calculation algorithm that is, surprisingly, as fast or faster than a two-phase flash calculation. The critical point is determined by a simple 2D search (in
temperature and volume) with a function that requires only evaluation of the mixture fugacities.
N
and b 11 +
(n*1)
ln R (n*1)
.
i
i
i + 1, 2, . . . , N, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.89)
i+1
10. Calculate a new estimate of saturation pressure using a Newton-Raphson update.
+ p (n)
p (n)1)
sat
sat *
where
Q (n)
ǒēQ
Ǔ
ēp
(n)
,
. . . . . . . . . . . . . . . . . . . . . . (4.85)
ȍ Y R ǒēfēp f1 * ēfēp f1 Ǔ
N
ēQ
+
ēp
yi
i
i
i+1
zi
yi
. . . . . . . . . . . . . (4.86)
zi
is evaluated at Iteration (n).
If searching for an upper saturation pressure, the new pressure estimate must be higher than p *. If the new estimate is lower than p *,
go to Step 1 and use a new initial-pressure estimate higher than the
present p * value.
11. Check for convergence. Zick* suggests the following two
criteria.
Ť
and
ȍY
N
1*
i
i+1
ƪ
Ť
t 10 *13
ƫ
)
ȍ lnln(R
ǒY ńz Ǔ
N
i
i
i+1
i
2
t 10 *8. . . . . . . . . . . . . . . . . . . . . (4.87)
In addition, check for a trivial solution using the criterion
ȍǒln Yz Ǔ
N
i
i+1
i
2
t 10 *4 .
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.88)
12. (a) If convergence is not achieved, return to Step 6. (b) If convergence is achieved, determine the saturation type by comparing
the mole fraction of the heaviest component in the mixture with that
in the incipient phase, where yN tzN indicates a bubblepoint with
Ki +yi /zi and yN uzN indicates a dewpoint where Ki +zi /yi , or by
comparing the density of the incipient phase with that of the feed.
This algorithm can be modified to search for both lower and upper saturation pressures as well as saturation temperature at a specified pressure.
Michelsen50 also gives an efficient procedure for calculating the
entire phase envelope, including calculations through the critical
point. More recently, he presented an approximate phase-envelope algorithm51 that is up to 10 times faster than his original algorithm using a trial-and-error solution directly for pressure and temperature
(component fugacities do not need to be converged at each point on
the phase envelope). Surprisingly, the results are extremely close to
fully converged saturation conditions and provide excellent starting
estimates for a rigorous saturation-point calculation. He also shows
that the approximate solution is always inside the phase envelope,
thus representing an unstable thermodynamic condition.
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
EQUATION-OF-STATE CALCULATIONS
where mi +chemical potential of Component i, zref+homogeneous
(single-phase) mixture at pressure pref at a reference depth href, and
p+pressure and z+mixture composition at depth h. The entire system is at constant temperature (dT/dh+0).
In 1930, Muskat54 provided exact solutions to Eq. 4.89 for a simplified EOS (ideal mixing). Numerical examples based on this oversimplified EOS led to the misleading conclusion that gravity has
negligible effect on compositional variation in reservoir systems. In
1938, Sage and Lacey55 evaluated Eq. 4.89 using a more realistic
EOS model. They provide examples showing significant variations
of composition with depth for reservoir mixtures. Furthermore, they
made the key observation that significant compositional variations
should be expected in systems in the vicinity of a critical condition.
From 1938 to 1980, the petroleum literature is apparently void of
publications regarding calculation of compositional gradients. Several references during this period do, however, mention reservoirs exhibiting compositional variation. Schulte56 cites most of these references. He appears to be the first to solve Eq. 4.89 with a cubic EOS.
His classic paper illustrates that significant compositional variation
can result from gravity segregation in petroleum reservoirs. Schulte
gives examples showing the effect of oil type (aromatic content) and
interaction coefficients (used in the mixing rules of a cubic EOS) on
compositional gradients. He also compares gradients calculated with
the Peng-Robinson and Soave-Redlich-Kwong equations.
In 1980, significant compositional gradients were reported in the
Brent field, North Sea.56-58 In the Brent formation of the Brent field,
a significant composition gradient was observed, with the transition
from gas to oil occurring at a saturated gas/oil contact (GOC). These
papers also describe the unusual transition from gas to oil in the absence of a saturated GOC. This transition occurs at a depth where the
reservoir fluid is a critical mixture, with a critical temperature equal
to the reservoir temperature and a critical pressure less than the reservoir pressure. Apparently, the Statfjord formation in the Brent field
is an example of a reservoir with such an “undersaturated GOC.”
In 1983, Holt et al.59 presented a formulation of the compositionalgradient problem that includes thermal diffusion. Example calculations in this paper were, unfortunately, limited to binary systems. Numerous publications on the subject of compositional gradient were
presented in 1984 and 1985.60,61 Most of these were field case histories; in fact, a special session of the 1985 SPE Annual Technical Conference and Exhibition was dedicated to this subject.62-64
Hirschberg60 discusses the influence of asphaltenes on compositional grading. He uses a simplified two-component model with one
component representing asphaltenes and the other representing the
remaining deasphalted oil. He makes the observation that compositional grading in heavier oils ( go u0.85 or gAPIt35°API) can be
strongly influenced by both the amount and the properties of asphaltenes, which implies that quantitatively accurate estimates of compositional grading resulting from asphaltenes are extremely difficult because of the strong dependence of calculated results on
physical properties of the oil and asphaltene(s). Finally, Hirschberg
discusses two mechanisms for the development of a tar mat.
Riemens et al.61 present an interesting evaluation of the compositional grading in the Birba field, Oman. On the basis of isothermal
gravity/chemical equilibrium (GCE) calculations and field measure17
ments of PVT data, they show that a significant compositional gradient
exists. The authors also evaluate the possibility of injecting gas into the
undersaturated oil zone where multicontact miscibility can develop.
Montel and Gouel65 suggest an algorithm for solving the isothermal
GCE problem. The procedure is only approximate because it calculates
pressure with an incremental hydrostatic term instead of solving directly for pressure. They discuss the effect of fluid characterization on compositional grading and the effect of reservoir temperature and pressure.
Finally, the authors suggest that including thermal diffusion may improve the reliability of calculated compositional gradients (although
they do not include this effect in their study).
Metcalfe et al.63 report measured variation of composition and
physical properties of reservoir fluids in the Anschutz Ranch East
field in the U.S. Overthrust Belt. These authors use an EOS to characterize the PVT behavior of the entire range of fluids sampled from
the reservoir. However, instead of calculating the compositional
variation using gravity/chemical equilibrium and the developed
EOS characterization, they correlate compositional variation graphically on the basis of measured data.
Creek and Schrader62 report compositional grading data for another Overthrust Belt reservoir, the East Painter field. Considerable data
are presented together with comparison of measured compositional
gradients and those calculated with the isothermal GCE model. They
report difficulty in matching observed saturation-pressure and GOR
gradients. Finally, the authors indicate that most reservoirs along the
Overthrust Belt have varying degrees of compositional grading.
Belery and da Silva66 present a formulation describing the combined effects of gravity and thermal diffusion for a system of zero net
mass flux. After assessing various approaches for treating thermal
diffusion, they selected the Dougherty and Drickamer67 method.
Belery and da Silva extend this formulation (originally valid only for
binary systems) to multicomponent systems. They use a field example with EOS characterization and measured gradient data from the
North Sea Ekofisk field to illustrate the gravity/thermal model. Because measured PVT gradients were very scattered (probably because of sampling problems), the comparison is not quantitatively accurate (with or without thermal diffusion). However, the calculations
show qualitatively the effect of thermal diffusion and are the first such
calculations reported for multicomponent systems.
Wheaton68 discusses an isothermal GCE model that includes the
influence of capillary pressure. The addition of capillary forces was
apparently justified in an effort to assist in the initialization of reservoir simulators. Simulators use capillary pressure curves to initialize
saturation and pressure distributions discretely in the vertical direction. Results of the calculated examples in Wheaton’s paper suggest
that neglecting compositional variations in a gas-condensate reservoir may result in large errors in the initial hydrocarbons in place. Obviously, these results are primarily a consequence of neglecting the
compositional variation resulting from gravity/chemical equilibrium.
Quantitatively similar results would have been obtained with or without the inclusion of capillary pressures. Finally, his observation that
neglecting compositional gradients will lead to incorrect specification of initial oil and gas in place is equally applicable to gas-condensate and oil reservoirs (i.e., practically any petroleum reservoir).
In his discussion of Wheaton’s paper, Chaback69 makes the observation that nonisothermal effects can be on the same order of
magnitude as gravity effects. More importantly, he notes that a nonisothermal system will never reach equilibrium (zero energy flux)
even though a stationary (steady-state) condition of zero net mass
flux is reached.
Montel70 discusses compositional grading, including comments on
treating thermal diffusion. He provides an equation for calculating the
Rayleigh-Darcy number that is used to indicate whether a fluid/rock
system will experience convection (mechanical instability).
Bedrikovetsky 71 gives an extensive discussion and formal mathematical treatment of compositional grading, including gravity,
thermal, and capillary forces. The treatment yields complicated expressions, which, in a few cases, are solved for simple conditions
(idealized EOS and binary systems). Many of the results are similar
to those given by Muskat.54 No examples are given for multicomponent mixtures with a realistic thermodynamic model.
18
Recently, Faissat et al.72 gave a theoretical review of equilibrium
formulations that include gravity and thermal diffusion. Belery and
da Silva66 mention most of the formulations, but Faissat et al. formalize the thermal-diffusion term in a generic way. Unfortunately, calculations are not provided for comparing the different formulations.
4.6.1 Isothermal GCE. Eq. 4.89 gives the condition for isothermal
GCE, which is sometimes written in differential form as
dm i ) M i gdh + 0, i + 1, 2, . . . , N . This condition represents N
equations. Together with the constraint that the sum of mole fractions
z(h) must add to one,
ȍ z (h) + 1,
N
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.90)
i+1
it is possible to solve for composition z(h) and pressure p(h) at a specified depth h.
Because chemical potential can be expressed as mi +RT
ln fi )l(T ), Eq. 4.89 can be expressed in terms of fugacity.
ln f i ǒ p ref , z ref , TǓ + ln f i ǒ p, z, T Ǔ ) 1 M i gǒh * h refǓ ,
RT
i + 1, 2, . . . , N. . . . . . . . . . . . . . . . . . . . . . . . . . (4.91)
For convenience, we define fi (h)+fi [ p(h),z(h),T ] and fi (href)+fi
( pref,zref,T ), yielding
ƪ
f i (h) + f iǒh refǓ exp *
ƫ
M i gǒh * h refǓ
,
RT
i + 1, 2, . . . , N. . . . . . . . . . . . . . . . . . . . . . . . . . (4.92)
The volume-translation method is widely used for correcting volumetric deficiencies of the original Soave-Redlich-Kwong and PengRobinson equations. The method involves calculating a linearly translated volume, vȀ, by adding a constant c to the molar volume, v,
calculated from the original EOS, vȀ+v)c. Peneloux et al.20 show
that the volume shift modifies the component fugacity as
fi exp[ci ( p/RT)] (see Eqs. 4.26 and 4.96). This correction must be included in the fugacity expressions used for gradient calculations and
also must be included in the pressure derivative of fugacity used in the
recommended algorithm for solving the isothermal GCE problem.
On the basis of the Gibbs-Duhem equation,53 combining the mechanical-equilibrium condition, dpńdh +* òg , with the GCE condition, Eq. 4.89, guarantees automatic satisfaction of the condition
ŕ ò(h)gdh .
h
p(h) + pǒh refǓ )
. . . . . . . . . . . . . . . . . . (4.93)
h ref
Interestingly, the isothermal GCE equations are still valid and satisfy
this condition when a saturated GOC is located between href and h [i.e.,
even when ò(h) is not a continuous function].
4.6.2 Isothermal GCE Algorithm. Eqs. 4.89 and 4.90 represent
equations similar to those used to calculate saturation pressure.
Michelsen51 gives an efficient method for solving the saturationpressure calculation, which has been modified here to solve the
GCE problem,
Qǒ p, z Ǔ + 1 *
ȍ z ƪf ǒp
N
~
i
i
Ǔ ń f i ǒ p, z Ǔƫ
ref, z ref
i+1
ȍY ,
N
+1*
i
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.94)
i+1
where Y i + z i ƪ f i ǒ p ref, z refǓńf i ǒ p, z Ǔƫ . . . . . . . . . . . . . . . . . . (4.95)
~
ƪ
and f i ǒ p ref, z refǓ + f i ǒ p ref, z refǓ exp *
~
ƫ
M i gǒh * h refǓ
.
RT
. . . . . . . . . . . . . . . . . . . . (4.96)
PHASE BEHAVIOR
An efficient algorithm for solving Eq. 4.94 uses a Newton-Raphson
update for pressure and accelerated successive substitution
this approach.
(GDEM44) for composition. The following outlines
~
1. Calculate fugacities of ~the reference feed f i ( y ref, z ref) and the
gravity-corrected fugacity f i ( p ref, z ref) from Eqs. 4.26 and 4.96.
This calculation needs to be made only once. Initial estimates of
composition and pressure at h are simply values at the reference
depth, z 1(h) + z ref and p 1(h) + p ref .
2. Calculate fugacities of the composition estimate z at the pressure estimate p. Calculate mole numbers from Eq. 4.95. Calculate
fugacity-ratio corrections with
f i ǒp ref , z refǓ
~
Ri +
f i ǒ p, zǓ
ǒȍ Ǔ
N
*1
Yj
.
. . . . . . . . . . . . . . . . . . (4.97)
j+1
3. Update mole numbers using Eqs. 4.83 and 4.84.
from Y (n)1)
using
4. Calculate z (n)1)
i
i
zi + Yi ń
ǒȍ Ǔ
N
Yj .
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.98)
j+1
5. Update the pressure estimate using a Newton-Raphson estimate.
p (n)1) + p (n) *
where
ēQ
+
ēp
Q (n)
ǒēQńēpǓ
ȍY R
N
i
i+1
i
(n)
,
ǒēf ińēpǓ
.
f i ǒ p, z Ǔ
. . . . . . . . . . . . . . . . . . . . (4.99)
. . . . . . . . . . . . . . . . . . (4.100)
6. Check for convergence using Eq. 4.87.
7. Iterate until convergence is achieved.
After finding the composition z(h) and pressure p(h) that satisfy
Eqs. 4.89 and 4.90, a phase-stability test32 must be made to establish
whether the solution is valid. A valid solution is single phase (thermodynamically stable). An unstable solution indicates that the calculated z and p will split into two (or more) phases, thereby making
the solution invalid.
If the gradient solution is unstable, then the stability-test composition y should be used to reinitialize the gradient calculation. The
starting pressure for the new gradient calculation can be pref or, preferably, the converged pressure from the gradient calculation that
led to the unstable solution. Note that unstable gradient solutions
usually occur only a short distance beyond a saturated GOC.
Locating a potential GOC requires a trial-and-error search. For a
saturated GOC, three approaches might be considered: (1) stability
tests, (2) negative flash calculations,37 or (3) saturation-pressure
calculations. The first and second methods should be the fastest,
with the negative flash probably being faster because information
from previous flash calculations can be used for initialization of
subsequent flash calculations.
Unfortunately, an algorithm based on either the stability test or
negative flash results may suffer from the fact that only trivial solutions exist over a large part of the reservoir thickness. On the other
hand, either method can be used efficiently to determine the saturated GOC once a nontrivial stability condition is found.
If an undersaturated GOC exists (i.e., a transition from gas to oil
through a critical mixture), only a search based on saturation-pressure calculations can be used. The following algorithm is recommended for locating both saturated and undersaturated GOC’s.
First, calculate the composition and pressure at the top (zT and pT )
and the bottom (z B and pB ) of the reservoir; then, calculate saturation pressures psT and psB . If the saturation types (bubblepoint/dewpoint) are the same at the top and bottom, then no GOC exists.
Otherwise, a search for the GOC, hGOC, is made.
A straightforward search algorithm would be interval halving
based on the saturation type. At Iteration n, a solution with a dew+ h (n) for the
point at depth h (n) would replace the top depth h (n)1)
T
next iteration, and a solution with a bubblepoint at a given depth
+ h (n) . The depth estimate
would replace the bottom depth h (n)1)
B
ƫ. The
) h (n)
for a given iteration is calculated from h (n) + 0.5ƪh (n)
B
T
EQUATION-OF-STATE CALCULATIONS
number of iterations required to meet a tolerance dh would be
1.5 lnƪ(h T * h B)ńdhƫ . For example, only 13 gradient and saturation-pressure calculations would be needed to achieve dh+0.33 ft
for a total thickness (h T * h B)+1,640 ft.
More efficient algorithms for locating the GOC can probably be
developed, particularly if a nontrivial stability solution can be located. Alternatively, Michelsen’s52 critical-point algorithm or his
new method for calculating accurate approximations for saturation
pressure and temperature51 may provide a good starting point for
developing an improved algorithm.
Whitson and Belery73 give a detailed discussion of compositional-gradient calculations, including the application of isothermal and
nonisothermal compositional-gradient algorithms to reservoir fluid
systems ranging from a saturated low-GOR black-oil/dry-gas system to a near-critical system.
4.7 Matching an EOS to Measured Data
Most EOS characterizations (see Chap. 5) are not truly predictive74,75
because errors in saturation pressure are commonly "10%, those in
densities are "5%, and compositions may be off by several mole percent for key components. Also, the EOS may predict a dewpoint incorrectly when the measured saturation condition is a bubblepoint, or
vice versa. This lack of predictive capability by the EOS can be because of insufficient compositional data for the C7) fractions, inaccurate properties for the C7) fractions, inadequate BIP’s, or incorrect
overall composition.
The EOS characterization can be improved in a number of ways.
First, however, the experimental data and fluid compositions should
be checked for consistency (see Chap. 6). If the PVT data appear
consistent and the fluid compositions are considered representative
of the material that was analyzed in the PVT laboratory, modifying
the parameters in the EOS to improve the fluid characterization will
be necessary. Refs. 26 and 74 through 79 present methods for modifying the cubic EOS to fit experimental PVT data. Most of these
methods modify the properties of fractions making up the C7) (Tc ,
pc , w, or direct multipliers on the EOS constants Wa and Wb ) and
BIP’s kij between methane and C7) fractions. When an injection
gas containing significant amounts of nonhydrocarbons is being
studied, the kij between nonhydrocarbon and C7) fractions may
also be modified.
Some methods use nonlinear regression to modify the EOS parameters automatically.74,78,79 Others have tried simply to make
manual adjustments to the EOS parameters through a trial-and-error
approach.75,77,80 The trend is now to automate the EOS modification procedure with nonlinear regression, including large amounts
of measured PVT and compositional data.81
Coats and Smart74 recommend five standard EOS modifications:
Wa and Wb of methane; Wa and Wb of the heaviest C7) fraction; and
kij between methane and the heaviest C7) fraction. Additional parameters (nonhydrocarbon Wa and Wb and kij ) are used for systems
with significant amounts of nonhydrocarbon components. Their approach differs from other methods in that they do not use volume
translation. As a result, significant methane corrections had to be applied to EOS constants Wa and Wb . Using the Coats and Smart approach with the PR EOS typically results in multipliers of the EOS
constants Wa and Wb ranging from 1.2 to 1.5 for methane and from
0.6 to 0.8 for the heaviest C7) fraction; kij of the methane/C7)
heavy fraction varies from 0 to 0.3. The W corrections can be interpreted as modifications of the critical properties.75
With a somewhat untraditional regression approach, Coats and
Smart minimize a sum of weighted absolute deviations using linear
programming. They suggest weighting factors of 40 for saturation
pressures, 10 for saturation densities, and 1 for most other data.
Their results are impressive, showing excellent matches of nearcritical fluids, hydrocarbon and nonhydrocarbon gas injection in
oils and retrograde condensate systems, and simple depletion data.
With a two-constant cubic EOS with volume translation, the modifications of EOS parameters (or critical properties) is typically only
5 to 10% compared with the "30 to 40% modifications required
with the Coats and Smart approach without volume translation. This
is explained by the initial predictions being much better with vol19
ume translation, thereby requiring fewer modifications to achieve
the same quality fit of measured data.
Interestingly, the same five standard regression parameters originally suggested by Coats and Smart can be used with an EOS that
uses volume translation. However, the result is usually that methane
corrections to Wa and Wb remain close to 1.0 and corrections to Wa
and Wb for the heaviest C7) fraction range from 0.9 to 1.1. Therefore, it may be better to drop methane corrections to Wa and Wb and
use instead one set of corrections to the Wa and Wb for the heaviest
C7) fraction, and another set of corrections to the Wa and Wb for the
next-to-heaviest C7) fraction. This approach is particularly helpful
when matching liquid-dropout curves with a “tail” (see Appendix
C) or in multicontact vaporization experiments.
Finally, an alternative to use of corrections to Wa and Wb directly
would be to modify Tc and pc instead (modification of w is not recommended). Be aware, however, that the sensitivity of the minimization problem to Tc and pc is probably less than to Wa and Wb ,
thereby making the mathematical search for a minimum more difficult. Appendix C gives a thorough discussion of how nonlinear regression can be used to adjust EOS parameters systematically to fit
measured PVT data.
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EQUATION-OF-STATE CALCULATIONS
67. Dougherty, E.L. Jr. and Drickamer, H.G.: “Thermal Diffusion and Molecular Motion in Liquids,” J. Phys. Chem. (1955) 59, 443.
68. Wheaton, R.J.: “Treatment of Variation of Composition With Depth in
Gas-Condensate Reservoirs,” SPERE (May 1991) 239.
69. Chaback, J.J.: “Discussion of Treatment of Variations of Composition With Depth in Gas-Condensate Reservoirs,” SPERE (February
1992) 157.
70. Montel, F.: “Phase Equilibria Needs for Petroleum Exploration and
Production Industry,” Fluid Phase Equilibria (1993) 84, 343.
71. Bedrikovetsky, P.G.: Mathematical Theory of Oil and Gas Recovery,
Petroleum Engineering & Development Studies, Cluwer Academic,
Horthreht, Russia (1993) 4.
72. Faissat, B. et al.: “Fundamental Statements about Thermal Diffusion
for a Multicomponent Mixture in a Porous Medium,” Fluid Phase Equilibria (1995) 100, 1.
73. Whitson, C.H. and Belery, P.: “Compositional Gradients in Petroleum
Reservoirs,” paper SPE 28000 presented at the 1994 U. of Tulsa/SPE
Centennial Petroleum Engineering Symposium, Tulsa, Oklahoma,
29–31 August.
74. Coats, K.H. and Smart, G.T.: “Application of a Regression-Based EOS
PVT Program to Laboratory Data,” SPERE (May 1986) 277.
75. Whitson, C.H.: “Effect of C7) Properties on Equation-of-State Predictions,” SPEJ (December 1984) 685; Trans., AIME, 277.
76. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,”
JPT (October 1985) 1870.
77. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “On the Dangers
of Tuning Equation of State Parameters,” paper SPE 14487 available
from SPE, Richardson, Texas (1985).
78. Agarwal, R., Li, Y.K., and Nghiem, L.X.: “A Regression Technique
With Dynamic-Parameter Selection for Phase Behavior Matching,”
SPERE (February 1990) 115.
79. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State,” Dr.Ing. dissertion, Norwegian Inst. of Technology, Trondheim, Norway (1989).
80. Turek, E.A. et al.: “Phase Equilibria in CO2-Multicomponent Hydrocarbon Systems: Experimental Data and an Improved Prediction Technique,” SPEJ (June 1984) 308.
81. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the
Displacement of Oil by Enriched Gases,” paper SPE 15493 presented
at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October.
SI Metric Conversion Factors
°API 141.5/(131.5)°API)
+g/cm3
bar 1.0*
E)05 +Pa
ft 3.048*
E*01 +m
°F (°F*32)/1.8
+°C
°F (°F)459.67)/1.8
+K
psi 6.894 757
E)00 +kPa
*Conversion factor is exact.
21
Chapter 5
HeptanesĆPlus Characterization
5.1 Introduction
Some phase-behavior applications require the use of an equation of
state (EOS) to predict properties of petroleum reservoir fluids. The
critical properties, acentric factor, molecular weight, and binary-interaction parameters (BIP’s) of components in a mixture are required
for EOS calculations. With existing chemical-separation techniques,
we usually cannot identify the many hundreds and thousands of components found in reservoir fluids. Even if accurate separation were
possible, the critical properties and other EOS parameters of compounds heavier than approximately C20 would not be known accurately. Practically speaking, we resolve this problem by making an
approximate characterization of the heavier compounds with experimental and mathematical methods. The characterization of heptanesplus (C7)) fractions can be grouped into three main tasks.1–3
1. Dividing the C7) fraction into a number of fractions with
known molar compositions.
2. Defining the molecular weight, specific gravity, and boiling
point of each C7) fraction.
3. Estimating the critical properties and acentric factor of each C7)
fraction and the key BIP’s for the specific EOS being used.
This chapter presents methods for performing these tasks and
gives guidelines on when each method can be used. A unique characterization does not exist for a given reservoir fluid. For example,
different component properties are required for different EOS’s;
therefore, the engineer must determine the quality of a given characterization by testing the predictions of reservoir-fluid behavior
against measured pressure/volume/temperature (PVT) data.
The amount of C7) typically found in reservoir fluids varies from
u50 mol% for heavy oils to t1 mol% for light reservoir fluids.4
Average C7) properties also vary widely. For example, C7) molecular weight can vary from 110 to u300 and specific gravity from
0.7 to 1.0. Because the C7) fraction is a mixture of many hundreds
of paraffinic, naphthenic, aromatic, and other organic compounds,5
the C7) fraction cannot be resolved into its individual components
with any precision. We must therefore resort to approximate descriptions of the C7) fraction.
Sec. 5.2 discusses experimental methods available for quantifying C7) into discrete fractions. True-boiling-point (TBP) distillation provides the necessary data for complete C7) characterization,
including mass and molar quantities, and the key inspection data for
each fraction (specific gravity, molecular weight, and boiling point).
Gas chromatography (GC) is a less-expensive, time-saving alternative to TBP distillation. However, GC analysis quantifies only the
mass of C7) fractions; such properties as specific gravity and boiling point are not provided by GC analysis.
HEPTANES-PLUS CHARACTERIZATION
Typically, the practicing engineer is faced with how to characterize a C7) fraction when only z C7) the mole fraction, ; molecular
weight, M C7); and specific gravity, g C7) , are known. Sec. 5.3 reviews methods for splitting C7) into an arbitrary number of subfractions. Most methods assume that mole fraction decreases exponentially as a function of molecular weight or carbon number. A
more general model based on the gamma distribution has been successfully applied to many oil and gas-condensate systems. Other
splitting schemes can also be found in the literature; we summarize
the available methods.
Sec. 5.4 discusses how to estimate inspection properties g and Tb
for C7) fractions determined by GC analysis or calculated from a
mathematical split. Katz and Firoozabadi’s6 generalized single carbon number (SCN) properties are widely used. Other methods for
estimating specific gravities of C7) subfractions are based on forcing the calculated g C7) to match the measured value.
Many empirical correlations are available for estimating critical
properties of pure compounds and C7) fractions. Critical properties
can also be estimated by forcing the EOS to match the boiling point and
specific gravity of each C7) fraction separately. In Sec. 5.5, we review
the most commonly used methods for estimating critical properties.
Finally, Sec. 5.6 discusses methods for reducing the number of
components describing a reservoir mixture and, in particular, the
C7) fraction. Splitting the C7) into pseudocomponents is particularly important for EOS-based compositional reservoir simulation.
A large part of the computing time during a compositional reservoir
simulation is used to solve the flash calculations; accordingly, minimizing the number of components without jeopardizing the quality
of the fluid characterization is necessary.
5.2 Experimental Analyses
The most reliable basis for C7) characterization is experimental
data obtained from high-temperature distillation or GC. Many experimental procedures are available for performing these analyses;
in the following discussion, we review the most commonly used
methods. TBP distillation provides the key data for C7) characterization, including mass and molar quantities, specific gravity, molecular weight, and boiling point of each distillation cut. Other such
inspection data as kinematic viscosity and refractive index also may
be measured on distillation cuts.
Simulated distillation by GC requires smaller samples and less
time than TBP distillation.7-9 However, GC analysis measures only
the mass of carbon-number fractions. Simulated distillation results
can be calibrated against TBP data, thus providing physical properties for the individual fractions. For many oils, simulated distillation
1
Cutoff (n-paraffin) boiling point
Midvolume (“normal”) boiling point
Dp
Dp
N2
N2
Fig. 5.2—TBP curve for a North Sea gas-condensate sample illustrating the midvolume-point method for calculating average
boiling point (after Austad et al.7).
Fig. 5.1—Standard apparatus for conducting TBP analysis of
crude-oil and condensate samples at atmospheric and subatmospheric pressures (after Ref. 11).
provides the necessary information for C7) characterization in far
less the time and at far less cost than that required for a complete
TBP analysis. We recommend, however, that at least one complete
TBP analysis be measured for (1) oil reservoirs that may be candidates for gas injection and (2) most gas-condensate reservoirs.
5.2.1 TBP Distillation. In TBP distillation, a stock-tank liquid (oil
or condensate) is separated into fractions or “cuts” by boiling-point
range. TBP distillation differs from the Hempel and American Soc.
for Testing Materials (ASTM) D-158 distillations10 because TBP
analysis requires a high degree of separation, which is usually controlled by the number of theoretical trays in the apparatus and the
reflux ratio. TBP fractions are often treated as components having
unique boiling points, critical temperatures, critical pressures, and
other properties identified for pure compounds. This treatment is
obviously more valid for a cut with a narrow boiling-point range.
The ASTM D-289211 procedure is a useful standard for TBP
analysis of stock-tank liquids. ASTM D-2892 specifies the general
procedure for TBP distillation, including equipment specifications
(see Fig. 5.1), reflux ratio, sample size, and calculations necessary
to arrive at a plot of cumulative volume percent vs. normal boiling
point. Normal boiling point implies that boiling point is measured
at normal or atmospheric pressure. In practice, to avoid thermal decomposition (cracking), distillation starts at atmospheric pressure
and is changed to subatmospheric distillation after reaching a limiting temperature. Subatmospheric boiling-point temperatures are
converted to normal boiling-point temperatures by use of a vaporpressure correlation that corrects for the amount of vacuum and the
fraction’s chemical composition. The boiling-point range for fractions is not specified in the ASTM standard. Katz and Firoozabadi6
recommend use of paraffin normal boiling points (plus 0.5°C) as
boundaries, a practice that has been widely accepted.
2
Fig. 5.27 shows a plot of typical TBP data for a North Sea sample.
Normal boiling point is plotted vs. cumulative volume percent.
Table 5.1 gives the data, including measured specific gravities and
molecular weights. Average boiling point is usually taken as the value found at the midvolume percent of a cut. For example, the third
cut in Table 5.1 boils from 258.8 to 303.8°F, with an initial 27.49
vol% and a final 37.56 vol%. The midvolume percent is
(27.49)37.56)/2+32.5 vol%; from Fig. 5.2, the boiling point at
this volume is [282°F. For normal-paraffin boiling-point intervals,
Katz and Firoozabadi’s6 average boiling points of SCN fractions
can be used (see Table 5.2).
The mass, m i, of each distillation cut is measured directly during
a TBP analysis. The cut is quantified in moles n i with molecular
weight, M i, and the measured mass m i, where n i + m ińM i. Volume
of the fraction is calculated from the mass and the density, ò i (or specific gravity, g i), where V i + m ińò i . M i is measured by a cryoscopic method based on freezing-point depression, and ò i is measured
by a pycnometer or electronic densitometer. Table 5.1 gives cumulative weight, mole, and volume percents for the North Sea sample.
Average C7) properties are given by
ȍm
N
MC
7)
+
i
i+1
N
ȍn
i
i+1
ȍm
N
and ò C
7)
+
i
i+1
N
ȍV
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.1)
i
i+1
where ò C7) + g C7)ò w with ò w +pure water density at standard
conditions. These calculated averages are compared with measured values of the C7) sample, and discrepancies are reported as
“lost” material.
Refs. 7 and 15 through 20 give procedures for calculating properties from TBP analyses. Also, the ASTM D-289211 procedure gives
details on experimental equipment and the procedure for conducting
TBP analysis at atmospheric and subatmospheric conditions. Table
5.3 gives an example TBP analysis from a commercial laboratory.
PHASE BEHAVIOR
TABLE 5.1—EXPERIMENTAL TBP RESULTS FOR A NORTH SEA CONDENSATE
Upper
Tbi
(°F)
Average
Tbi *
(°F)
C7
208.4
194.0
90.2
0.7283
96
123.9
0.940
C8
258.8
235.4
214.6
0.7459
110
287.7
C9
303.8
282.2
225.3
0.7658
122
C10
347.0
325.4
199.3
0.7711
C11
381.2
363.2
128.8
C12
420.8
401.1
C13
455.0
C14
4.35
4.80
7.80
4.35
4.80
11.92
1.951
10.35
11.15
16.19
14.70
15.95
11.88
294.2
1.847
10.87
11.40
15.33
25.57
27.35
11.82
137
258.5
1.455
9.61
10.02
12.07
35.18
37.37
11.96
0.7830
151
164.5
0.853
6.21
6.37
7.08
41.40
43.74
11.97
136.8
0.7909
161
173.0
0.850
6.60
6.70
7.05
48.00
50.44
12.03
438.8
123.8
0.8047
181
153.8
0.684
5.97
5.96
5.68
53.97
56.41
11.99
492.8
474.8
120.5
0.8221
193
146.6
0.624
5.81
5.68
5.18
59.78
62.09
11.89
C15
523.4
509.0
101.6
0.8236
212
123.4
0.479
4.90
4.78
3.98
64.68
66.87
12.01
C16
550.4
537.8
74.1
0.8278
230
89.5
0.322
3.57
3.47
2.67
68.26
70.33
12.07
C17
579.2
564.8
76.8
0.8290
245
92.6
0.313
3.70
3.59
2.60
71.96
73.92
12.16
C18
604.4
591.8
58.2
0.8378
259
69.5
0.225
2.81
2.69
1.87
74.77
76.62
12.14
C19
629.6
617.0
50.2
0.8466
266
59.3
0.189
2.42
2.30
1.57
77.19
78.91
12.11
C20
653.0
642.2
45.3
0.8536
280
53.1
0.162
2.19
2.06
1.34
79.37
80.97
12.10
427.6
0.8708
370
491.1
1.156
20.63
19.03
9.59
100.00
100.00
2,580.5
12.049
100.00
100.00
100.00
Sum
Mi
(g/mol)
gi **
2,073.1
Average
0.8034
Vi
(cm3)
ni
(mol)
wi
(%)
SxVi
%
xi
%
C21)
mi
(g)
Swi
%
xVi
%
Fraction
172
Kw
11.98
Reflux ratio+1 : 5; reflux cycle+18 seconds; distillation at atmospheric pressure+201.2 to 347°F; distillation at 100 mm Hg+347 to 471.2°F; and distillation at 10 mm Hg+471.2
to 653°F.
Vi +mi /gi /0.9991; ni +mi /Mi ; wi +100
*Average taken at midvolume point.
**Water+1.
mi /2073.1; xVi +100
Vi/2580.5; xi +100
ni /12.049; Swi +Swi ; SxVi +SxVi ; and Kw +(Tbi +460)1/3/gi .
Boiling points are not reported because normal-paraffin boiling-point
intervals are used as a standard; accordingly, the average boiling
points suggested by Katz and Firoozabadi6 (Table 5.2) can be used.
5.2.2 Chromatography. GC and, to a lesser extent, liquid chromatography are used to quantify the relative amount of compounds
found in oil and gas systems. The most important application of
chromatography to C7) characterization is simulated distillation by
GC techniques.
Fig. 5.3 shows an example gas chromatogram for the North Sea
sample considered earlier. The dominant peaks are for normal paraffins, which are identified up to n-C22. As a good approximation for
a paraffinic sample, the GC response for carbon number Ci starts at
the bottom response of n-Ci*1 and extends to the bottom response
of n-Ci . The mass of carbon number Ci is calculated as the area under
the curve from the baseline to the GC response in the n-Ci*1 to n-Ci
interval (see the shaded area for fraction C9 in Fig. 5.3). As Fig. 5.47
shows schematically, the baseline should be determined before running the actual chromatogram.
Because stock-tank samples cannot be separated completely by
standard GC analysis, an internal standard must be used to relate GC
area to mass fraction. Normal hexane was used as an internal standard for the sample in Fig. 5.3. The internal standard’s response factor may need to be adjusted to achieve consistency between simulated and TBP distillation results. This factor will probably be
constant for a given oil, and the factor should be determined on the
basis of TBP analysis of at least one sample from a given field. Fig.
5.5 shows the simulated vs. TBP distillation curves for the Austad
et al.7 sample. A 15% correction to the internal-standard response
factor was used to match the two distillation curves.
As an alternative to correcting the internal standard, Maddox and
Erbar15 suggest that the reported chromatographic boiling points be
adjusted by a correction factor that depends on the reported boiling
HEPTANES-PLUS CHARACTERIZATION
point and the “paraffinicity” of the composite sample. This correction factor varies from 1 to 1.15 and is slightly larger for aromatic
than paraffinic samples.
Several laboratories have calibrated GC analysis to provide simulated-distillation results up to C40. However, checking the accuracy
of simulated distillation for SCN fractions greater than approximately C25 is difficult because C25 is usually the upper limit for reliable TBP distillation. The main disadvantage of simulated distillation is that inspection data are not determined directly for each
fraction and must therefore either be correlated from TBP data or estimated from correlations (see Sec. 5.4).
Sophisticated analytical methods, such as mass spectroscopy,
may provide detailed information on the compounds separated by
GC. For example, mass spectroscopy GC can establish the relative
amounts of paraffins, naphthenes, and aromatics (PNA’s) for carbon-number fractions distilled by TBP analysis. Detailed PNA information should provide more accurate estimation of the critical
properties of petroleum fractions, but the analysis is relatively costly and time-consuming from a practical point of view. Recent work
has shown that PNA analysis3,19-23 may improve C7) characterization for modeling phase behavior with EOS’s. Our experience, however, is that PNA data have limited usefulness for improving EOS
fluid characterizations.
5.3 Molar Distribution
Molar distribution is usually thought of as the relation between mole
fraction and molecular weight. In fact, this concept is misleading because a unique relation does not exist between molecular weight and
mole fraction unless the fractions are separated in a consistent manner. Consider for example a C7) sample distilled into 10 cuts separated by normal-paraffin boiling points. If the same C7) sample is
distilled with constant 10-vol% cuts, the two sets of data will not
3
TABLE 5.2—SINGLE CARBON NUMBER PROPERTIES FOR HEPTANES-PLUS (after Katz and Firoozabadi6)
Katz-Firoozabadi Generalized Properties
Lee-Kesler12/Kesler-Lee13
Correlations
Tb Interval*
Fraction
Number
Defined
Kw
Tc
(°R)
pc
(psia)
ą
ąw
Riazi14
Defined
Vc
(ft3/lbm mol)
Zc
Lower
(°F)
Upper
(°F)
Average Tb
(°F)
(°R)
ăăg*ă
M
6
97.7
156.7
147.0
606.7
0.690
84
12.27
914
476
0.271
5.6
0.273
7
156.7
210.0
197.4
657.1
0.727
96
11.96
976
457
0.310
6.2
0.272
8
210.0
259.0
242.1
701.7
0.749
107
11.86
1,027
428
0.349
6.9
0.269
9
259.0
304.3
288.0
747.6
0.768
121
11.82
1,077
397
0.392
7.7
0.266
10
304.3
346.3
330.4
790.1
0.782
134
11.82
1,120
367
0.437
8.6
0.262
11
346.3
385.5
369.0
828.6
0.793
147
11.84
1,158
341
0.479
9.4
0.257
12
385.5
422.2
406.9
866.6
0.804
161
11.86
1,195
318
0.523
10.2
0.253
13
422.2
456.6
441.0
900.6
0.815
175
11.85
1,228
301
0.561
10.9
0.249
14
456.6
489.0
475.5
935.2
0.826
190
11.84
1,261
284
0.601
11.7
0.245
15
489.0
520.0
510.8
970.5
0.836
206
11.84
1,294
268
0.644
12.5
0.241
16
520.0
548.6
541.4
1,001.1
0.843
222
11.87
1,321
253
0.684
13.3
0.236
17
548.6
577.4
572.0
1,031.7
0.851
237
11.87
1,349
240
0.723
14.0
0.232
18
577.4
602.6
595.4
1,055.1
0.856
251
11.89
1,369
230
0.754
14.6
0.229
19
602.6
627.8
617.0
1,076.7
0.861
263
11.90
1,388
221
0.784
15.2
0.226
20
627.8
651.2
640.4
1,100.1
0.866
275
11.92
1,408
212
0.816
15.9
0.222
21
651.2
674.6
663.8
1,123.5
0.871
291
11.94
1,428
203
0.849
16.5
0.219
22
674.6
692.6
685.4
1,145.1
0.876
305
11.94
1,447
195
0.879
17.1
0.215
23
692.6
717.8
707.0
1,166.7
0.881
318
11.95
1,466
188
0.909
17.7
0.212
24
717.8
737.6
726.8
1,186.5
0.885
331
11.96
1,482
182
0.936
18.3
0.209
25
737.6
755.6
746.6
1,206.3
0.888
345
11.99
1,498
175
0.965
18.9
0.206
26
755.6
775.4
766.4
1,226.1
0.892
359
12.00
1,515
168
0.992
19.5
0.203
27
775.4
793.4
786.2
1,245.9
0.896
374
12.01
1,531
163
1.019
20.1
0.199
28
793.4
809.6
804.2
1,263.9
0.899
388
12.03
1,545
157
1.044
20.7
0.196
29
809.6
825.8
820.4
1,280.1
0.902
402
12.04
1,559
152
1.065
21.3
0.194
30
825.8
842.0
834.8
1,294.5
0.905
416
12.04
1,571
149
1.084
21.7
0.191
31
842.0
858.2
851.0
1,310.7
0.909
430
12.04
1,584
145
1.104
22.2
0.189
32
858.2
874.4
865.4
1,325.1
0.912
444
12.04
1,596
141
1.122
22.7
0.187
33
874.4
888.8
879.8
1,339.5
0.915
458
12.05
1,608
138
1.141
23.1
0.185
34
888.8
901.4
892.4
1,352.1
0.917
472
12.06
1,618
135
1.157
23.5
0.183
35
901.4
915.8
906.8
1,366.5
0.920
486
12.06
1,630
131
1.175
24.0
0.180
36
919.4
1,379.1
0.922
500
12.07
1,640
128
1.192
24.5
0.178
37
932.0
1,391.7
0.925
514
12.07
1,650
126
1.207
24.9
0.176
38
946.4
1,406.1
0.927
528
12.09
1,661
122
1.226
25.4
0.174
39
959.0
1,418.7
0.929
542
12.10
1,671
119
1.242
25.8
0.172
40
971.6
1,431.3
0.931
556
12.10
1,681
116
1.258
26.3
0.170
41
982.4
1,442.1
0.933
570
12.11
1,690
114
1.272
26.7
0.168
42
993.2
1,452.9
0.934
584
12.13
1,697
112
1.287
27.1
0.166
43
1,004.0
1,463.7
0.936
598
12.13
1,706
109
1.300
27.5
0.164
44
1,016.6
1,476.3
0.938
612
12.14
1,716
107
1.316
27.9
0.162
45
1,027.4
1,487.1
0.940
626
12.14
1,724
105
1.328
28.3
0.160
*At 1 atmosphere.
**Water+1.
produce the same plot of mole fraction vs. molecular weight. However, a plot of cumulative mole fraction,
ȍz
vs. cumulative average molecular weight,
ȍz M
i
Q zi +
j+1
ȍz
j
,ĂĂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.2)
N
j+1
4
i
j
j
Q Mi +
j+1
ȍz
j
,
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.3)
j
j+1
PHASE BEHAVIOR
TABLE 5.3—STANDARD TBP RESULTS FROM COMMERCIAL PVT LABORATORY
Component
mol%
wt%
Density
(g/cm3)
Gravity
ągAPI
Molecular
Weight
Heptanes
Octanes
Nonanes
Decanes
Undecanes
Dodecanes
Tridecanes
Tetradecanes
Pentadecanes plus
1.12
1.30
1.18
0.98
0.62
0.57
0.74
0.53
4.10
2.52
3.08
3.15
2.96
2.10
2.18
3.05
2.39
31.61
0.7258
0.7470
0.7654
0.7751
0.7808
0.7971
0.8105
0.8235
0.8736
63.2
57.7
53.1
50.9
49.5
45.8
42.9
40.1
30.3
96
101
114
129
144
163
177
192
330
*At 60°F.
Note: Katz and Firoozabadi6 average boiling points (Table 5.2) can be used when normal paraffin boiling-point intervals are used.
should produce a single curve. Strictly speaking, therefore, molar
distribution is the relation between cumulative molar quantity and
some expression for cumulative molecular weight.
In this section, we review methods commonly used to describe
molar distribution. Some methods use a consistent separation of
fractions (e.g., by SCN) so the molar distribution can be expressed
directly as a relationship between mole fraction and molecular
weight of individual cuts. Most methods in this category assume that
C7) mole fractions decrease exponentially. A more general approach uses the continuous three-parameter gamma probability
function to describe molar distribution.
5.3.1 Exponential Distributions. The Lohrenz-Bray-Clark24
(LBC) viscosity correlation is one of the earliest attempts to use an
exponential-type distribution for splitting C7). The LBC method
splits C7) into normal paraffins C7 though C40 with the relation
z i + z C exp[A 1(i * 6) ) A 2(i * 6) 2], . . . . . . . . . . . . . (5.4)
6
where i+carbon number and z C6 +measured C6 mole fraction.
Constants A1 and A2 are determined by trial and error so that
ȍz
7)
7)
+
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.5)
MC
7)
+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . (5.6)
i+7
are satisfied. Paraffin molecular weights (Mi +14i)2) are used in
Eq. 5.6. A Newton-Raphson algorithm can be used to solve Eqs. 5.5
and 5.6. Note that the LBC model cannot be used when z C7) t z C6
and M C7) u M C40. The LBC form of the exponential distribution
has not found widespread application.
More commonly, a linear form of the exponential distribution is
used to split the C7) fraction. Writing the exponential distribution
in a general form for any Cn) fraction (n+7 being a special case),
z i + z Cn exp A[(i * n)],
. . . . . . . . . . . . . . . . . . . . . . . . (5.7)
where i+carbon number, z Cn +mole fraction of Cn , and
A+constant indicating the slope on a plot of ln z i vs. i. The constants
z Cn and A can be determined explicitly. With the general expression
M i + 14 i ) h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.8)
for molecular weight of Ci and the assumption that the distribution
is infinite, constants z Cn and A are given by
40
zC
ȍz M
40
and z C
z Cn +
i+7
14
M Cn) * 14(n * 1) * h
and A + lnǒ1 * z CnǓ
. . . . . . . . . . . . . . . . . . (5.9)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.10)
R
so that
ȍz + 1
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.11)
i+n
(a)
(b)
(c)
Fig. 5.3—Simulated distillation by GC of the North Sea gas-condensate sample in Fig. 5.2 (after Austad et al.7).
HEPTANES-PLUS CHARACTERIZATION
Fig. 5.4—GC simulated distillation chromatograms (a) without
any sample (used to determine the baseline), (b) for a crude oil,
and (c) for a crude oil with internal standard (after MacAllister
and DeRuiter9).
5
700 to 1,000°F Distillate
1,000 to 1,250°F Distillate
1,200°F Residue
Fig. 5.5—Comparison of TBP and GC-simulated distillation for
a North Sea gas-condensate sample (after Austad et al.7).
R
and
ȍz M + M
i
i
C n)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.12)
i+n
are satisfied.
Eqs. 5.9 and 5.10 imply that once a molecular weight relation is chosen (i.e., h is fixed), the distribution is uniquely defined by C7) molecular weight. Realistically, all reservoir fluids having a given C7) molecular weight will not have the same molar distribution, which is one
reason why more complicated models have been proposed.
5.3.2 Gamma-Distribution Model. The three-parameter gamma
distribution is a more general model for describing molar distribution. Whitson2,25,26 and Whitson et al.27 discuss the gamma distribution and its application to molar distribution. They give results
for 44 oil and condensate C7) samples that were fit by the gamma
distribution with data from complete TBP analyses. The absolute
average deviation in estimated cut molecular weight was 2.5 amu
(molecular weight units) for the 44 samples.
The gamma probability density function is
p(M) +
(M * h) a*1 exp NJ* ƪǒ M * h ǓńbƫNj
b a G(a)
, . . . . . . . . (5.13)
where G+gamma function and b is given by
b+
MC
7)
a
*h
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.14)
The three parameters in the gamma distribution are a, h, and
M C7) The key parameter a defines the form of the distribution, and
its value usually ranges from 0.5 to 2.5 for reservoir fluids; a+1
gives an exponential distribution. Application of the gamma distribution to heavy oils, bitumen, and petroleum residues indicates
that the upper limit for a is 25 to 30, which statistically is approaching a log-normal distribution (see Fig. 5.628).
The parameter h can be physically interpreted as the minimum
molecular weight found in the C7) fraction. An approximate relation between a and h is
110
h[
. . . . . . . . . . . . . . . . . . . . . . . . . (5.15)
1 * ǒ1 ) 4 ńa 0.7Ǔ
6
Fig. 5.6—Gamma distributions for petroleum residue (after
Brulé et al.28).
for reservoir-fluid C7) fractions. Practically, h should be considered
as a mathematical constant more than as a physical property, either
calculated from Eq. 5.15 or determined by fitting measured TBP data.
Fig. 5.7 shows the function p(M) for the Hoffman et al.29 oil and
a North Sea oil. Parameters for these two oils were determined by fitting experimental TBP data. The Hoffman et al. oil has a relatively
large a of 2.27, a relatively small h of 75.7, with M C7)+198; the
North Sea oil is described by a+0.82, h+93.2, and M C7)+227.
The continuous distribution p(M ) is applied to petroleum fractions by dividing the area under the p(M ) curve into sections (shown
schematically in Fig. 5.8). By definition, total area under the p(M )
curve from h to R is unity. The area of a section is defined as
normalized mole fraction z ińz C 7) for the range of molecular
weights Mbi*1 to Mbi . If the area from h to molecular-weight
boundary Mb is defined as P0(Mb ), then the area of Section i is
P0(Mbi )*P0(Mbi*1), also shown schematically in Fig. 5.8. Mole
fraction zi can be written
zi + zC
7)
ƪP ǒM Ǔ * P ǒM Ǔƫ .
0
bi
0
b i*1
. . . . . . . . . . . . . . . (5.16)
Average molecular weight in the same interval is given by
Mi + h ) a b
P 1ǒM b iǓ * P 1ǒM b i*1Ǔ
P 0ǒM b iǓ * P 0ǒM b i*1Ǔ
,
. . . . . . . . . . . (5.17)
where P 0 + Q S, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.18)
ǒ
Ǔ
1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.19)
and P 1 + Q S * a
PHASE BEHAVIOR
The gamma distribution can be fit to experimental molar-distribution data by use of a nonlinear least-squares algorithm to determine
a, h, and b. Experimental TBP data are required, including weight
fraction and molecular weight for at least five C7) fractions (use of
more than 10 fractions is recommended to ensure a unique fit of model parameters). The sum-of-squares function can be defined as
a + 2.273
h + 75.7
M C + 198.4
7)
a + 0.817
h + 93.2
M C + 227
Fǒ a, h , b Ǔ +
7)
ȍ (D
N*1
Mi)
2
,
. . . . . . . . . . . . . . . . . . . . . . . (5.24)
i+1
where D Mi +
Fig. 5.7—Gamma density function for the Hoffman et al.29 oil
(dashed line) and a North Sea volatile oil (solid line). After Whitson et al.27
where Q + e *y y a G(a), . . . . . . . . . . . . . . . . . . . . . . . . . (5.20)
R
S+
ƪ
ȍ y Ȋ(a ) k)
j
j
j+0
and y +
k+0
ƫ
,
. . . . . . . . . . . . . . . . . . . (5.21)
i
. . . . . . . . . . . . . . . . . . (5.25)
ȍw ,
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.26)
j+1
vs. the cumulative dimensionless molecular-weight variable,
8
i
.
Subscripts mod and exp+model and experimental, respectively. This
sum-of-squares function weights the lower molecular weights more
than higher molecular weights, in accordance with the expected accuracy for measurement of molecular weight. Also, the sum-of-squares
function does not include the last molecular weight because this molecular weight may be inaccurate or backcalculated to match the measured average C7) molecular weight. If the last fraction is not included, the model average molecular weight, (M C7)) mod + h ) ab,
can be compared with the experimental value as an independent
check of the fit.
A simple graphical procedure can be used to fit parameters a and
h if experimental M C7) is fixed and used to define b. Fig. 5.10
shows a plot of cumulative weight fraction,
Q wi +
Note that P0(Mb0+h)+P1(Mb0+h)+0.
The summation in Eq. 5.21 should be performed until the last
term is t1 10*8. The gamma function can be estimated by30
ȍA x
(M i) exp
i
*1
Mb * h
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.22)
b
Gǒ x ) 1Ǔ + 1 )
(M i) mod * (M i) exp
, . . . . . . . . . . . . . . . . . . . . . (5.23)
i+1
where A1+*0.577191652, A2+0.988205891, A3+*0.897056937,
A4+0.918206857, A5+*0.756704078, A6+0.482199394, A7+
*0.193527818, and A8+0.035868343 for 0xxx1. The recurrence
formula, G(x)1)+xG(x), is used for xu1 and xt1; furthermore,
G(1)+1.
The equations for calculating zi and Mi are summarized in a short
FORTRAN program GAMSPL found in Appendix A. In this simple
program, the boundary molecular weights are chosen arbitrarily at
increments of 14 for the first 19 fractions, starting with h as the first
lower boundary. The last fraction is calculated by setting the upper
molecular-weight boundary equal to 10,000. Table 5.4 gives three
sample outputs from GAMSPL for a+0.5, 1, and 2 with h+90 and
M C7)+200 held constant. Fig. 5.9 plots the results as log zi vs. Mi .
The amount and molecular weight of the C26) fraction varies for
each value of a.
Q *M i +
QM i * h
.
MC * h
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.27)
7)
Table 5.5 and the following outline describe the procedure for determining model parameters with Fig. 5.10 and TBP data.
1. Tabulate measured mole fractions zi and molecular weights Mi
for each fraction.
2. Calculate experimental weight fractions, w i + (z i M i)
B (z C 7)M C7)), if they are not reported.
3. Normalize weight fractions and calculate cumulative normalized weight fraction Q w i .
4. Calculate cumulative molecular weight Q M i from Eq. 5.3.
5. Assume several values of h (e.g., from 75 to 100) and calculate
Q *M i for each value of the estimated h.
6. For each estimate of h, plot Q *M i vs. Q wi on a copy of Fig. 5.10
and choose the curve that fits one of the model curves best. Read the
value of a from Fig. 5.10.
7. Calculate molecular weights and mole fractions of Fractions i
using the best-fit curve in Fig. 5.10. Enter the curve at measured values of Q wi , read Q *M i , and calculate Mi from
Mi + h ) ǒMC
7)
* hǓ
ƪǒ
Q wi * Q wi*1
Ǔ * ǒQ wi*1ńQ *M i*1Ǔƫ
Q wińQ *M i
.
. . . . . . . . . . . . . . . . . . . (5.28)
p(M)
h
A + z ińz C 7)
+ P 0ǒM biǓ * P 0ǒM bi*1Ǔ
h M bi
A + P 0ǒM biǓ
h M bi*1
A + P 0ǒM bi*1Ǔ
Fig. 5.8—Schematic showing the graphical interpretation of areas under the gamma density
function p(M) that are proportional to normalized mole fraction; A+area.
HEPTANES-PLUS CHARACTERIZATION
7
TABLE 5.4—RESULTS OF GAMSPL PROGRAM FOR THREE DATA SETS WITH DIFFERENT
GAMMA-DISTRIBUTION PARAMETER a
ăąąąăa+0.5
ăąąąăa+1.0
Mole
Molecular
Mole
Molecular
Mole
Molecular
Number
Fraction
Weight
Fraction
Weight
Fraction
Weight
1
0.2787233
94.588
0.1195065
96.852
0.0273900
99.132
2
0.1073842
110.525
0.1052247
110.852
0.0655834
111.490
3
0.0772607
124.690
0.0926497
124.852
0.0852269
125.172
4
0.0610991
138.758
0.0815774
138.852
0.0927292
139.038
5
0.0505020
152.796
0.0718284
152.852
0.0925552
152.963
6
0.0428377
166.819
0.0632444
166.852
0.0877762
166.916
7
0.0369618
180.836
0.0556863
180.852
0.0804707
180.883
8
0.0322804
194.848
0.0490314
194.852
0.0720157
194.859
9
0.0284480
208.857
0.0431719
208.852
0.0632969
208.841
10
0.0252470
222.864
0.0380125
222.852
0.0548597
222.826
11
0.0225321
236.870
0.0334698
236.852
0.0470180
236.814
12
0.0202013
250.875
0.0294699
250.852
0.0399302
250.805
13
0.0181808
264.879
0.0259481
264.852
0.0336535
264.797
14
0.0164152
278.883
0.0228471
278.852
0.0281813
278.790
15
0.0148619
292.886
0.0201167
292.852
0.0234690
292.784
16
0.0134879
306.888
0.0177127
306.852
0.0194514
306.778
17
0.0122665
320.890
0.0155959
320.852
0.0160543
320.774
18
0.0111762
334.892
0.0137321
334.852
0.0132017
334.770
19
0.0101996
348.894
0.0120910
348.852
0.0108204
348.766
0.1199341
539.651
0.0890834
466.000
0.0463166
420.424
20
Total
1.0000000
1.0000000
Average
200
For all three cases h + 90 and M C
+ 200.
7)
ǒ
Qw i
Q *M i
*
Q w i*1
Q *M i*1
Ǔ
.
. . . . . . . . . . . . . . . . . . . (5.29)
+ 200
(
}
7)
h + 90
DM b + 14
0.8
(
V a + 2.0
MC
1.0
)
a + 0.5
a + 1.0
200
Fig. 5.11 shows a Q *M i * Q wi match for the Hoffman et al.29 oil
with h+70, 72.5, 75, and 80 and indicates that a best fit is achieved
for h+72.5 and a+2.5 (see Fig. 5.12).
Although the gamma-distribution model has the flexibility of
treating reservoir fluids from light condensates to bitumen, most
reservoir fluids can be characterized with an exponential molar distribution (a+1) without adversely affecting the quality of EOS pre)
For computer applications, Q wi and Q *M i can be calculated exactly
from Eqs. 5.16 through 5.23 with little extra effort.
f
1.0000000
200
Mole fractions zi are given by
z i + z C 7)
ăąąąăa+2.0
Fraction
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Normalized Mole Fraction, Qzi
Fig. 5.9—Three example molar distributions for an oil sample
with M C 7)= 200 and h = 90, calculated with the GAMSPL program
(Table A-4) in Table 5.4.
8
Fig. 5.10—Cumulative-distribution type curve for fitting experimental TBP data to the gamma-distribution model. Parameters
a and h are determined with M C held constant.
7)
PHASE BEHAVIOR
TABLE 5.5—CALCULATION OF CUMULATIVE WEIGHT FRACTION AND
CUMULATIVE MOLECULAR WEIGHT VARIABLE FOR HOFFMAN et al.29 OIL
Q *Mi
Component
i
zi
ąSzi ą
Mi
zi Mi
ăSzi Mi ă
Qwi
QMi
h+70
h+72.5
h+75
h+80
h+85
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0.0263
0.0234
0.0235
0.0224
0.0241
0.0246
0.0266
0.0326
0.0363
0.0229
0.0171
0.0143
0.0130
0.0108
0.0087
0.0072
0.0058
0.0048
0.0039
0.0034
0.0028
0.0025
0.0023
0.0091
0.0263
0.0497
0.0732
0.0956
0.1197
0.1443
0.1709
0.2035
0.2398
0.2627
0.2799
0.2941
0.3072
0.3180
0.3267
0.3338
0.3396
0.3444
0.3483
0.3517
0.3545
0.3570
0.3593
0.3684
99
110
121
132
145
158
172
186
203
222
238
252
266
279
290
301
315
329
343
357
371
385
399
444
2.604
2.574
2.844
2.957
3.497
3.882
4.570
6.067
7.371
5.093
4.079
3.596
3.466
3.008
2.526
2.152
1.811
1.582
1.351
1.196
1.039
0.963
0.926
4.049
2.604
5.178
8.021
10.978
14.475
18.357
22.928
28.995
36.366
41.458
45.538
49.134
52.600
55.607
58.133
60.285
62.097
63.679
65.031
66.227
67.265
68.228
69.154
73.203
0.036
0.071
0.110
0.150
0.198
0.251
0.313
0.396
0.497
0.566
0.622
0.671
0.719
0.760
0.794
0.824
0.848
0.870
0.888
0.905
0.919
0.932
0.945
1.000
99.0
104.2
109.6
114.8
120.9
127.2
134.2
142.5
151.7
157.8
162.7
167.0
171.2
174.9
178.0
180.6
182.9
184.9
186.7
188.3
189.8
191.1
192.5
198.7
0.225
0.266
0.308
0.348
0.396
0.445
0.499
0.563
0.634
0.682
0.720
0.754
0.787
0.815
0.839
0.859
0.877
0.893
0.907
0.919
0.931
0.941
0.952
1.000
0.210
0.251
0.294
0.335
0.384
0.434
0.489
0.555
0.627
0.676
0.715
0.749
0.782
0.811
0.836
0.857
0.875
0.891
0.905
0.918
0.929
0.940
0.951
1.000
0.194
0.236
0.280
0.322
0.371
0.422
0.478
0.546
0.620
0.669
0.709
0.744
0.778
0.808
0.832
0.854
0.872
0.889
0.903
0.916
0.928
0.939
0.950
1.000
0.160
0.204
0.249
0.293
0.345
0.398
0.457
0.526
0.604
0.655
0.697
0.733
0.769
0.799
0.825
0.847
0.867
0.884
0.899
0.913
0.925
0.936
0.948
1.000
0.123
0.169
0.216
0.262
0.316
0.371
0.433
0.506
0.586
0.640
0.683
0.722
0.758
0.791
0.818
0.841
0.861
0.879
0.894
0.909
0.921
0.933
0.945
1.000
Total
0.3684
198.7
73.203
dictions. Whitson et al.27 proposed perhaps the most useful application of the gamma-distribution model. With Gaussian quadrature,
their method allows multiple reservoir-fluid samples from a common reservoir to be treated simultaneously with a single fluid characterization. Each fluid sample can have different C7) properties
when the split is made so that each split fraction has the same molecular weight (and other properties, such as g, Tb , Tc , pc , and w), while
1.0
Y h + 65
0.8
J h + 70
F h + 75
X h + 80
the mole fractions are different for each fluid sample. Example applications include the characterization of a gas cap and underlying
reservoir oil and a reservoir with compositional gradient.
The following outlines the procedure for applying Gaussian
quadrature to the gamma-distribution function.
1. Determine the number of C7) fractions, N, and obtain the
quadrature values Xi and Wi from Table 5.6 (values are given for
N+3 and N+5).
2. Specify h and a. When TBP data are not available to determine
these parameters, recommended values are h+90 and a+1.
3. Specify the heaviest molecular weight of fraction N (recommended value is M N + 2.5M C7)). Calculate a modified b* term,
b * + ǒ M N * h ǓńX N .
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Normalized Mole Fraction, Qzi
Fig. 5.11—Graphical fit of the Hoffman et al.29 oil molar distribution by use of the cumulative-distribution type curve. Best-fit
model parameters are a = 2.5 and h = 72.5.
HEPTANES-PLUS CHARACTERIZATION
Fig. 5.12—Calculated normalized mole fraction vs. molecular
weight of fractions for the Hoffman et al.29 oil based on the best
fit in Fig. 5.11 with a = 2.5 and h = 72.5.
9
TABLE 5.6—GAUSSIAN QUADRATURE FUNCTION
VARIABLES, X, AND WEIGHT FACTORS, W
X
W
Three Quadrature Points (plus fractions)
1
2
3
7.110 930 099 29 10*1
0.415 774 556 783
2.294 280 360 279
6.289 945 082 937
2.785 177 335 69 10*1
1.038 925 650 16 10*2
Five Quadrature Points (plus fractions)
5.217 556 105 83 10*1
1
2
3
4
5
3.986 668 110 83 10*1
0.263 560 319 718
1.413 403 059 107
3.596 425 771 041
7.085 810 005 859
12.640 800 844 276
7.594 244 968 17 10*2
3.611 758 679 92 10*3
2.336 997 238 58 10*5
Quadrature function values and weight factors can be found for other quadrature numbers
in mathematical handbooks.30
4. Calculate the parameter d.
d + exp
ǒ
a b*
*1
MC * h
7)
Ǔ
.
. . . . . . . . . . . . . . . . . . . (5.30)
5. Calculate the C7) mole fraction zi and Mi for each fraction.
zi + zC
7)
ƪ W i f (X i)ƫ,
Mi + h ) b* Xi ,
and f(X) +
(X) a*1 ǒ1 ) ln dǓ a
.
G(a)
dX
. . . . . . . . . . . . . . . . . . (5.31)
6. Check whether the calculated M C7) from Eq. 5.12 equals the
measured value used in Step 4 to define d. Because Gaussian quadrature is only approximate, the calculated M C7) may be slightly in
error. This can be corrected by (slightly) modifying the value of d,
and repeating Steps 5 and 6 until a satisfactory match is achieved.
When characterizing multiple samples simultaneously, the values
of MN , h, and b* must be the same for all samples. Individual sample
values of M C7) and a can, however, be different. The result of this
characterization is one set of molecular weights for the C7) fractions, while each sample has different mole fractions zi (so that their
average molecular weights M C7) are honored).
Specific gravities for the C7) fractions can be calculated with
one of the correlations given in Sec. 5.4 (e.g., Eq. 5.44), where the
characterization factor (e.g., Fc ) must be the same for all mixtures.
The specific gravities, g C7) , of each sample will not be exactly reproduced with this procedure (calculated with Eq. 5.37), but the average characterization factor can be chosen so that the differences
are very small ( g"0.0005). Having defined Mi and gi for the C7)
fractions, a complete fluid characterization can be determined with
correlations in Sec. 5.5.
5.4 InspectionĆProperties Estimation
5.4.1 Generalized Properties. The molecular weight, specific gravity, and boiling point of C7) fractions must be estimated in the absence of experimental TBP data. This situation arises when simulated
distillation is used or when no experimental analysis of C7) is available and a synthetic split must be made by use of a molar-distribution
model. For either situation, inspection data from TBP analysis of a
sample from the same field would be the most reliable source of M,
g, and Tb for each C7) fraction. The next-best source would be measured TBP data from a field producing similar oil or condensate from
the same geological formation. Generalized properties from a producing region, such as the North Sea, have been proposed.31
Katz and Firoozabadi6 suggest a generalized set of SCN properties for petroleum fractions C6 through C45. Table 5.2 gives an extended version of the Katz-Firoozabadi property table. Molecular
10
weights can be used to convert weight fractions, wi , from simulated
distillation to mole fractions,
zi +
w i ńM i
ȍ w ńM
.
N
j
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.32)
j
j+7
However, the molecular weight of the heaviest fraction, C N, is not
known. From a mass balance, M N is given by
wN
, . . . . . . . . . . . . (5.33)
MN +
N*1
ǒwC ńMC Ǔ * ǒwińMiǓ
7)
7)
ȍ
i+7
where Mi for i+7,…, N*1 are taken from Table 5.2. Unfortunately,
the calculated molecular weight M N is often unrealistic because of
measurement errors in M C7) or in the chromatographic analysis and
because generalized molecular weights are only approximate. Both
w N and M C7) can be adjusted to give a “reasonable” M N, but caution
is required to avoid nonphysical adjustments. The same problem is
inherent with backcalculating M N with any set of generalized molecular weights used for SCN Fractions 7 to N*1 (e.g., paraffin values).
During the remainder of this section, molecular weights and mole
fractions are assumed to be known for C7) fractions, either from
chromatographic analysis or from a synthetic split. The generalized
properties for specific gravity and boiling point can be assigned to
SCN fractions, but the heaviest specific gravity must be backcalculated to match the measured C7) specific gravity. The calculated gN
also may be unrealistic, requiring some adjustment to generalized
specific gravities. Finally, the boiling point of the heaviest fraction
must be estimated. TbN can be estimated from a correlation relating
boiling point to specific gravity and molecular weight.
5.4.2 Characterization Factors. Inspection properties M, g, and Tb
reflect the chemical makeup of petroleum fractions. Some methods
for estimating specific gravity and boiling point assume that a particular characterization factor is constant for all C7) fractions.
These methods are only approximate but are widely used.
Watson or Universal Oil Products (UOP) Characterization Factor. The Watson or UOP factor, Kw, is based on normal boiling point,
Tb , in °R and specific gravity, g.32,33
T 1ń3
K w 5 gb .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.34)
Kw varies roughly from 8.5 to 13.5. For paraffinic compounds,
Kw +12.5 to 13.5; for naphthenic compounds, Kw +11.0 to 12.5;
and for aromatic compounds, Kw +8.5 to 11.0. Some overlap in Kw
exists among these three families of hydrocarbons, and a combination of paraffins and aromatics will obviously “appear” naphthenic.
However, the utility of this and other characterization factors is that
they give a qualitative measure of the composition of a petroleum
fraction. The Watson characterization factor has been found to be
useful for approximate characterization and is widely used as a parameter for correlating petroleum-fraction properties, such as molecular weight, viscosity, vapor pressure, and critical properties.
An approximate relation2 for the Watson factor, based on molecular weight and specific gravity, is
K w [ 4.5579 M 0.15178 g *0.84573 . . . . . . . . . . . . . . . . . . . (5.35)
This relation is derived from the Riazi-Daubert14 correlation for
molecular weight and is generally valid for petroleum fractions with
normal boiling points ranging from 560 to 1,310°R (C7 through
C30). Experience has shown, however, that Eq. 5.35 is not very accurate for fractions heavier than C20.
Kw calculated with M C7) and g C7) in Eq. 5.35 is often constant
for a given field. Figs. 5.13A and 5.13B7 plot molecular weight vs.
specific gravity for C7) fractions from two North Sea fields. Data
for the gas condensate in Fig. 5.13A indicate an average
K wC7)+11.99"0.01 for a range of molecular weights from 135 to
150. The volatile oil shown in Fig. 5.13B has an average
K wC7)+11.90"0.01 for a range of molecular weights from 220 to
PHASE BEHAVIOR
Molecular Weight, MC 7+
Fig. 5.13A—Specific gravity vs. molecular weight for C7) fractions for a North Sea Gas-Condensate Field 2 (after Austad et al.7).
255. The high degree of correlation for these two fields suggests accurate molecular-weight measurements by the laboratory. In general, the spread in K wC7) values will exceed "0.01 when measurements are performed by a commercial laboratory.
When the characterization factor for a field can be determined,
Eq. 5.35 is useful for checking the consistency of C7) molecularweight and specific-gravity measurements. Significant deviation in
K wC7) , such as "0.03 for the North Sea fields above, indicates possible error in the measured data. Because molecular weight is more
prone to error than determination of specific gravity, an anomalous
K wC7) usually indicates an erroneous molecular-weight measurement. For the gas condensate in Fig. 5.13A, a C7) sample with specific gravity of 0.775 would be expected to have a molecular weight
of [141 (for K wC7)+ 11.99). If the measured value was 135, the
Watson characterization factor would be 11.90, which is significantly lower than the field average of 11.99. In this case, the C7) molecular weight should be redetermined.
Eq. 5.35 can also be used to calculate specific gravity of C7) fractions determined by simulated distillation or a synthetic split (i.e.,
when only mole fractions and molecular weights are known). Assuming a constant Kw for each fraction, specific gravity, gi , can be
calculated from
g i + 6.0108 M i0.17947 K w*1.18241 .
. . . . . . . . . . . . . . . . . (5.36)
Kw must be chosen so that experimentally measured C7) specific
gravity, (g C7)) exp, is calculated correctly.
ǒgC7)Ǔ
exp
+
zC
7)
MC
7)
ȍǒz M ńg Ǔ
N
i
i
.
. . . . . . . . . . . . . . . . . . . . . (5.37)
i
i+1
The Watson factor satisfying Eq. 5.37 is given by
Kw +
ƪ
0.16637 g C
zC
7)
MC
ȍz M
A
7) 0
7)
ƫ
*0.84573
,
. . . . . . . . . . . . . . . (5.38)
Molecular Weight, MC 7+
Fig. 5.13B—Specific gravity vs. molecular weight for C7) fractions for a North Sea Volatile-Oil Field 3B(after Austad et al.7).
Jacoby Correlation
(Aromaticity Factor, Ja )
Present Correlation
(Watson Factor, Kw )
Ja
Fig. 5.14—Specific gravity vs. molecular weight for constant values of the Jacoby aromaticity factor (solid lines) and the Watson
characterization factor (dashed lines). After Whitson.25
Boiling points, Tbi , can be estimated from Eq. 5.36.
3
T bi + (K wg i) .
i
0.82053
.
i
. . . . . . . . . . . . . . . . . . . . . . (5.39)
i+1
HEPTANES-PLUS CHARACTERIZATION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.40)
Unfortunately, Eqs. 5.36 through 5.40 overpredict g and Tb at molecular weights greater than [250 (an original limitation of the
Riazi-Daubert14 molecular-weight correlation).
Jacoby Aromaticity Factor. The Jacoby aromaticity factor, Ja , is
an alternative characterization factor for describing the relative
composition of petroleum fractions.34 Fig. 5.142 shows the original
Jacoby relation between specific gravity and molecular weight for
several values of Ja . The behavior of specific gravity as a function
of molecular weight is similar for the Jacoby factor and the relation
for a constant Kw. However, specific gravity calculated with the
Jacoby method increases more rapidly at low molecular weights,
flattening at high molecular weights (a more physically consistent
behavior). A relation for the Jacoby factor is
N
where A 0 +
Kw
Ja +
g * 0.8468 ) ǒ15.8ńMǓ
. . . . . . . . . . . . . . . . . . (5.41)
0.2456 * ǒ1.77ńMǓ
11
Cf typically has a value between 0.27 and 0.31 and is determined for
a specific C7) sample by satisfying Eq. 5.37.
5.4.3 Boiling-Point Estimation. Boiling point can be estimated
from molecular weight and specific gravity with one of several correlations. Søreide also developed a boiling-point correlation based
on 843 TBP fractions from 68 reservoir C7) samples,
T b + 1928.3 * ǒ1.695
exp ƪ * ǒ4.922
) ǒ3.462
Fig. 5.15—Specific gravity vs. carbon number for constant values of the Yarborough aromaticity factor (after Yarborough1).
or, in terms of specific gravity,
ǒ
Ǔ
g + 0.8468 * 15.8 ) J a 0.2456 * 1.77 . . . . . . . (5.42)
M
M
The first two terms in Eq. 5.42 (i.e., when Ja +0) express the relation
between specific gravity and molecular weight for normal paraffins.
The Jacoby factor can also be used to estimate fraction specific
gravities when mole fractions and molecular weights are available
from simulated distillation or a synthetic split. The Jacoby factor
satisfying measured C7) specific gravity (Eq. 5.37) must be calculated by trial and error. We have found that this relation is particularly accurate for gas-condensate systems.27
Yarborough Aromaticity Factor. Yarborough1 modified the
Jacoby aromaticity factor specifically for estimating specific gravities when mole fractions and molecular weights are known. Yarborough tries to improve the original Jacoby relation by reflecting the
changing character of fractions up to C13 better and by representing
the larger naphthenic content of heavier fractions better. Fig. 5.15
shows how the Yarborough aromaticity factor, Ya , is related to specific gravity and carbon number. A simple relation representing Ya
is not available; however, Whitson26 has fit the seven aromaticity
curves originally presented by Yarborough using the equation
g i + expƪA 0 ) A 1 i *1 ) A 2 i ) A 3 ln(i)ƫ , . . . . . . . . . . (5.43)
where i+carbon number. Table 5.7 gives the constants for Eq. 5.43.
The aromaticity factor required to satisfy measured C7) specific
gravity (Eq. 5.37) is determined by trial and error. Linear interpolation of specific gravity should be used to calculate specific gravity
for a Ya value falling between two values of Ya in Table 5.7.
Søreide 35 Correlations. Søreide developed an accurate specificgravity correlation based on the analysis of 843 TBP fractions from
68 reservoir C7) samples.
g i + 0.2855 ) C f (M i * 66) 0.13 .
. . . . . . . . . . . . . . . (5.44)
10 5Ǔ M *0.03522 g 3.266
10 *3Ǔ M * 4.7685 g
10 *3Ǔ Mgƫ ,
. . . . . . . . . . . . . . . . . . . . . (5.45)
with Tb in °R.
Table 5.8 gives estimated specific gravities determined with the
methods just described for a C7) sample with the exponential split
given in Table 5.4 (a+1, h+90, M C7)+200) and g C7)+0.832.
The following equations also relate molecular weight to boiling
point and specific gravity; any of these correlations can be solved
for boiling point in terms of M and g. We recommend, however, the
Søreide correlation for estimating Tb from M and g.
Kesler and Lee. 12
M + ƪ* 12, 272.6 ) 9, 486.4g ) (4.6523 * 3.3287g)T bƫ
) NJ ǒ1 * 0.77084g * 0.02058g 2Ǔ
Ǔ
ƪǒ1.3437 * 720.79T *1
b
Nj
10 7ƫ T *1
b
) NJǒ1 * 0.80882g ) 0.02226g 2Ǔ
Ǔ
ƪǒ1.8828 * 181.98T –1
b
Nj.
10 12ƫ T *3
b
. . . . . . . . (5.46)
Riazi and Daubert. 14
M + (4.5673
10 *5)T b2.1962 g *1.0164 .
. . . . . . . . . . . . (5.47)
American Petroleum Inst. (API). 36
M + ǒ2.0438
10 2Ǔ T b0.118 g 1.88 expǒ0.00218T b * 3.07gǓ .
. . . . . . . . . . . . . . . . . . . . (5.48)
Rao and
Bardon. 37
ln M + (1.27 ) 0.071K w) ln
ǒ22.311.8T
Ǔ.
) 1.68K
b
w
. . . . . . . . . . . . . . . . . . . . (5.49)
Riazi and Daubert. 18
M + 581.96 T b0.97476 g 6.51274 expƪǒ5.43076
* 9.53384 g ) ǒ1.11056
10 *3ǓT b
10 *3ǓT bgƫ . . . . . . . . . . (5.50)
TABLE 5.7—COEFFICIENTS FOR YARBOROUGH AROMATICITY FACTOR CORRELATION1,26
12
Ya
A0
A1
0.0
*7.43855 10*2
*1.72341
0.1
*4.25800 10*1
0.2
A2
A2
1.38058 10*3
*3.34169 10*2
*7.00017 10*1
*3.30947 10*5
8.65465 10*2
*4.47553 10*1
*7.65111 10*1
1.77982 10*4
1.07746 10*1
0.3
*4.39105 10*1
*9.44068 10*1
4.93708 10*4
1.19267 10*1
0.4
*2.73719 10*1
*1.39960
3.80564 10*3
5.92005 10*2
0.6
*7.39412 10*3
*1.97063
5.87273 10*3
*1.67141 10*2
0.8
*3.17618 10*1
*7.78432 10*1
2.58616 10*3
1.08382 10*3
PHASE BEHAVIOR
TABLE 5.8—COMPARISON OF SPECIFIC GRAVITIES WITH CORRELATIONS BY USE OF
DIFFERENT CHARACTERIZATION FACTORS
gi for Different Correlations With Constant Characterization
Factor Chosen To Match g C + 0.832
7)
Kw +12.080
Ja +0.2395
Ya +0.2794
Cf +0.2864
96.8
0.7177
0.7472
0.7051
0.7327
0.1052
110.8
0.7353
0.7684
0.7286
0.7550
0.0926
124.8
0.7511
0.7849
0.7486
0.7719
4
0.0816
138.8
0.7656
0.7981
0.7660
0.7856
5
0.0718
152.8
0.7789
0.8088
0.7813
0.7972
6
0.0632
166.8
0.7913
0.8178
0.7951
0.8072
7
0.0557
180.8
0.8028
0.8253
0.8075
0.8161
8
0.0490
194.8
0.8136
0.8318
0.8189
0.8241
9
0.0432
208.8
0.8238
0.8374
0.8294
0.8314
10
0.0380
222.8
0.8335
0.8423
0.8391
0.8380
Fraction
zi
1
0.1195
2
3
Mi
11
0.0335
236.8
0.8426
0.8466
0.8482
0.8442
12
0.0295
250.8
0.8514
0.8505
0.8567
0.8500
13
0.0259
264.8
0.8597
0.8539
0.8646
0.8554
14
0.0228
278.8
0.8677
0.8570
0.8722
0.8604
15
0.0201
292.8
0.8753
0.8598
0.8793
0.8652
16
0.0177
306.8
0.8827
0.8623
0.8861
0.8697
17
0.0156
320.8
0.8898
0.8646
0.8926
0.8740
18
0.0137
334.8
0.8966
0.8668
0.8988
0.8782
19
0.0121
348.8
0.9033
0.8687
0.9048
0.8821
20
0.0891
466.0
0.9514
0.8805
0.9468
0.9096
1.0000
200.0
0.8320
0.8320
0.8320
0.8320
Total
5.5 CriticalĆProperties Estimation
Kesler-Lee. 12
Thus far, we have discussed how to split the C7) fraction into
pseudocomponents described by mole fraction, molecular weight,
specific gravity, and boiling point. Now we must consider the problem of assigning critical properties to each pseudocomponent. Critical temperature, Tc ; critical pressure, pc ; and acentric factor, w, of
each component in a mixture are required by most cubic EOS’s.
Critical volume, vc , is used instead of critical pressure in the Benedict-Webb-Rubin38 (BWR) EOS, and critical molar volume is
used with the LBC viscosity correlation.24 Critical compressibility
factor has been introduced as a parameter in three- and four-constant
cubic EOS’s.
Critical-property estimation of petroleum fractions has a long history beginning as early as the 1930’s; several reviews22,25,26,39,40
are available. We present the most commonly used correlations and
a graphical comparison (Figs. 5.16 through 5.18) that is intended
to highlight differences between the correlations. Finally, correlations based on perturbation expansion (a concept borrowed from
statistical mechanics) are discussed separately.
The units for the remaining equations in this section are Tb in °R,
TbF in °F+Tb *459.67, Tc in °R, pc in psia, and vc in ft3/lbm mol.
Oil gravity is denoted gAPI and is related to specific gravity by
gAPI+141.5/g*131.5.
T c + 341.7 ) 811g ) (0.4244 ) 0.1174g)T b
5.5.1 Critical Temperature. Tc is perhaps the most reliably correlated critical property for petroleum fractions. The following critical-temperature correlations can be used for petroleum fractions.
Roess. 41 (modified by API36).
T c + 645.83 ) 1.6667ƪgǒ T bF ) 100 Ǔƫ
* ǒ0.7127
2
10 *3Ǔƪgǒ T bF ) 100 Ǔƫ .
HEPTANES-PLUS CHARACTERIZATION
) (0.4669 * 3.2623g)
. . . . . . . . . . . . (5.52)
Cavett. 42
T c + 768.07121 ) 1.7133693T bF
* ǒ0.10834003
2
10 *2ǓT bF
* ǒ0.89212579
10 *2Ǔ g APIT bF
) ǒ0.38890584
3
10 *6ǓT bF
) ǒ0.5309492
) ǒ0.327116
2
10 *5Ǔ g APIT bF
2
.
10 *7Ǔ g 2APIT bF
. . . . . . . . . . . . . . . (5.53)
Riazi-Daubert. 14
T c + 24.27871T b0.58848 g 0.3596 .
. . . . . . . . . . . . . . . . . . (5.54)
Nokay. 43
T c + 19.078 T b0.62164 g 0.2985 .
. . . . . . . . . . . . . . . . . . . . (5.55)
5.5.2 Critical Pressure. pc correlations are less reliable than Tc correlations. The following are pc correlations that can be used for petroleum fractions.
Kesler-Lee. 12
ln p c + 8.3634 * 0.0566
g
*
. . . . . . . . . . . (5.51)
10 5T *1
b .
ƪǒ
0.11857
0.24244 ) 2.2898
g ) g2
Ǔ
ƫ
10 *3 T b
13
Fig. 5.16—Comparison of critical-temperature correlations for
boiling points from 600 to 1,500°R assuming a constant Watson
characterization factor of 12.
)
*
ƪǒ
ƪǒ
0.47227
1.4685 ) 3.648
g ) g2
0.42019 ) 1.6977
g2
ƫ
Ǔ
10 *7 T 2b
ƫ
Ǔ
Fig. 5.17—Comparison of critical-pressure correlations for boiling points from 600 to 1,500°R assuming a constant Watson
characterization factor of 12.
10 *10 T 3b .
. . . . . (5.56)
Cavett. 42
log p c + 2.8290406 ) ǒ0.94120109
* ǒ0.30474749
* ǒ0.2087611
10 *3ǓT bF
2
10 *5ǓT bF
10 *4Ǔ g APIT bF
) ǒ0.15184103
3
10 *8ǓT bF
) ǒ0.11047899
2
10 *7Ǔ g APIT bF
* ǒ0.48271599
10 *7Ǔ g 2APIT bF
) ǒ0.13949619
2
. . . . . . . . . . . . (5.57)
10 *9Ǔ g 2APIT bF
Riazi-Daubert. 14
p c + ǒ3.12281
10 9ǓT *2.3125
g 2.3201 . . . . . . . . . . . . . . (5.58)
b
5.5.3 Acentric Factor. Pitzer et
ǒǓ
p*
w 5 * log pv * 1,
c
al.44
Lee-Kesler. 13 (Tbr +Tb /Tc t0.8).
defined acentric factor as
w+
– lnǒ p cń14.7Ǔ ) A 1 ) A 2 T *1
) A 3 ln T br ) A 4 T br6
br
. . . . . . . . . . . . . . . . . . . . . . . . (5.59)
where p *v+vapor pressure at temperature T+0.7Tc (Tr +0.7).
Practically, acentric factor gives a measure of the steepness of the
vapor-pressure curve from Tr +0.7 to Tr +1, where p *v /pc +0.1 for
w+0 and p *v /pc +0.01 for w+1. Numerically, w[0.01 for methane, [0.25 for C5, and [0.5 for C8 (see Table A.1 for literature values of acentric factor for pure compounds). w increases to u1.0 for
petroleum fractions heavier than approximately C25 (see Table 5.2).
The Kesler-Lee12 acentric factor correlation (for Tb /Tc u0.8) is
developed specifically for petroleum fractions, whereas the correlation for Tb /Tc t0.8 is based on an accurate vapor-pressure correlation for pure compounds. The Edmister45 correlation is limited to
pure hydrocarbons and should not be used for C7) fractions. The
three correlations follow.
14
Fig. 5.18—Comparison of acentric factor correlations for boiling
points from 600 to 1500°R assuming a constant Watson characterization factor of 12.
A 5 ) A 6 T *1
) A 7 ln T br ) A 8 T br6
br
,
. . . . . . . . . . . . . . . . . . . . (5.60)
where A1+*5.92714, A2+ 6.09648, A3+ 1.28862, A4+
*0.169347, A5+ 15.2518, A6+*15.6875, A7+*13.4721,
and A8+ 0.43577.
Kesler-Lee. 12 (Tbr +Tb /Tc u0.8).
w + * 7.904 ) 0.1352K w * 0.007465K 2w
) 8.359T br ) (1.408 * 0.01063K w)T *1
br .
. . . . . . . (5.61)
Edmister. 45
logǒ p cń14.7Ǔ
w+3
* 1.
7 ƪǒT cńT bǓ * 1ƫ
. . . . . . . . . . . . . . . . . . . . . (5.62)
PHASE BEHAVIOR
5.5.4 Critical Volume. The Hall-Yarborough46 critical-volume
correlation is given in terms of molecular weight and specific gravity, whereas the Riazi-Daubert14 correlation uses normal boiling
point and specific gravity.
Hall-Yarborough. 46
v c + 0.025 M 1.15 g *0.7935 .
. . . . . . . . . . . . . . . . . . . . . . (5.63)
Riazi-Daubert. 14
v c + ǒ7.0434
10 *7Ǔ T b2.3829 g *1.683.
. . . . . . . . . . . . . (5.64)
Critical compressibility factor, Zc , is defined as
Zc +
p cv c
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.65)
RT c
where R+universal gas constant. Thus, Zc can be calculated directly
from critical pressure, critical volume, and critical temperature. Reid
et al.40 and Pitzer et al.44 give an approximate relation for Zc .
Z c [ 0.291 * 0.08w.
Paraffin molecular weight, MP, is not explicitly a function of Tb , and
Eqs. 5.67 through 5.73 must be solved iteratively; an initial guess
is given by
MP [
Tb
.
10.44 * 0.0052T b
. . . . . . . . . . . . . . . . . . . . . (5.74)
Twu claims that the normal-paraffin correlations are valid for C1
through C100, although the properties at higher carbon numbers are
only approximate because experimental data for paraffins heavier
than approximately C20 do not exist. The following relations are
used to calculate petroleum-fraction properties.
Critical Temperature.
ǒ11 )* 2f2f Ǔ ,
2
T c + T cP
T
T
f T + Dg T
ƪ
* 0.362456 )
T b0.5
Ǔ ƫ
ǒ
0.0398285 * 0.948125
Dg T ,
T b0.5
. . . . . . . . . . . . . . . . . . . . . . . . . (5.66)
Eq. 5.66 is not particularly accurate (grossly overestimating Zc for
heavier compounds) and is used only for approximate calculations.
and Dg T + exp[5(g P * g)] * 1.
. . . . . . . . . . . . . . . . . . (5.75)
Critical Volume.
ǒ11 )* 2f2f Ǔ ,
2
5.5.5 Correlations Based on Perturbation Expansions. Correlations for critical temperature, critical pressure, critical volume, and
molecular weight have been developed for petroleum fractions with
a perturbation-expansion model with normal paraffins as the reference system. To calculate critical pressure, for example, critical
temperature, critical volume, and specific gravity of a paraffin with
the same boiling point as the petroleum fraction must be calculated
first. Kesler et al.47 first used the perturbation expansion (with n-alkanes as the reference fluid) to develop a suite of critical-property
and acentric-factor correlations.
Twu48 uses the same approach to develop a suite of critical-property correlations. We give his normal-paraffin correlations first,
then the correlations for petroleum fractions.
Normal Paraffins (Alkanes).
ƪ
T cP + T b 0.533272 ) ǒ0.191017
) ǒ0.779681
10
*3
(0.959468 10 2)
)
ǒ0.01T bǓ 13
ƫ
10 *10ǓT b3
*1
,
. . . . . . . . . . . . . . . . . . (5.67)
2
. . . . . . . . . . . . . . . (5.68)
v cP + [ 1 * (0.419869 * 0.505839a * 1.56436a 3
,
. . . . . . . . . . . . . . . . . . . . . . . . . (5.69)
g P + 0.843593 * 0.128624a * 3.36159a 3
* 13749.5a 12 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.70)
and T b + exp(5.71419 ) 2.71579q * 0.28659q 2
* 39.8544q *1 * 0.122488q *2)
* 24.7522q ) 35.3155q 2 ,
T
where a + 1 * b
T cP
and q + ln M P .
0.466590 )
T b0 .5
ǒ
Ǔ ƫ
* 0.182421 ) 3.01721
Dg v ,
T b0.5
and Dg v + expƪ4ǒg 2P * g 2Ǔƫ * 1.
. . . . . . . . . . . . . . . . . . (5.76)
Critical Pressure.
ǒTT ǓǒVV Ǔǒ11 )* 2f2f Ǔ ,
2
p c + p cP
)
c
cP
p
cP
c
p
ƪǒ
2.53262 * 46.1955
* 0.00127885T b
T b0.5
Ǔ
Ǔ ƫ
ǒ
* 11.4277 ) 252.14
) 0.00230535T b Dg p ,
T b0.5
and Dg p + exp[0.5(g P * g)] * 1. . . . . . . . . . . . . . . . . . (5.77)
1 ) 2f
Ǔ,
ln M + ln M ǒ
1 * 2f
2
M
) 36.1952a 2 ) 104.193a 4) ,
*8
ƪ
f v + Dg v
v
Molecular Weight.
p cP + (3.83354 ) 1.19629a 0.5 ) 34.8888a
* 9481.7a 14)]
v
f p + Dg p
ǓT b
10 *7ǓT b2 * ǒ0.284376
v c + v cP
. . . . . . . . . . . . . . . . . (5.71)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.72)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.73)
HEPTANES-PLUS CHARACTERIZATION
P
ƪ ǒ
f M + Dg M |x| )
M
Ǔ ƫ
* 0.0175691 ) 0.193168
Dg M ,
T b0.5
x + 0.012342 * 0.328086
,
T b0.5
and Dg M + exp[5(g P * g)].
. . . . . . . . . . . . . . . . . . . . . (5.78)
Figs. 5.16 through 5.18 compare the various critical-property correlations for a range of boiling points from 600 to 1,500°R.
5.5.6 Methods Based on an EOS. Fig. 5.1928 illustrates the important influence that critical properties have on EOS-calculated properties of pure components. Vapor pressure is particularly sensitive to
critical temperature. For example, the Riazi-Daubert19 critical-temperature correlation for toluene overpredicts the experimental value
15
a+1, h+90) with Gaussian-quadrature or equal-mass fractions or
(2) the exponential distribution (Eq. 5.7). Specific gravities should
be estimated with the Søreide35 correlation (Eq. 5.44), choosing Cf
to match measured C7) specific gravity (Eq. 5.37). Boiling points
should be estimated from the Søreide correlation (Eq. 5.45).
For the PR EOS, we recommend the nonhydrocarbon BIP’s given
in Chap. 4 and the modified Chueh-Prausnitz54 equation for C1
through C7) pairs,
k ij
ȱ
+ Aȧ1 *
Ȳ
ǒ
2v 1ń6
v 1ń6
ci
cj
v 1ń3
) v 1ń3
ci
cj
Ǔ ȳȴȧ
B
,
. . . . . . . . . . . . . . . . (5.79)
with A+0.18 and B+6.
Tc underpredicted←
→Tc overpredicted
Deviation From Experimental Value, %
Fig. 5.19—Effect of critical temperature on vapor-pressure prediction of toluene with the PR EOS; AAD+absolute average deviation (after Brulé et al.28).
by only 1.7%. Even with this slight error in Tc , the average error in
vapor pressures predicted by the Peng-Robinson49 (PR) EOS is 16%.
The effect of critical properties and acentric factor on EOS calculations for reservoir-fluid mixtures is summarized by Whitson.26
In principle, the EOS used for mixtures should also predict the behavior of individual components found in the mixture. For pure
compounds, the vapor pressure is accurately predicted because all
EOS’s force fit vapor-pressure data. Some EOS’s are also fit to saturated-liquid densities at subcritical temperatures. The measured
properties of petroleum fractions, boiling point, and specific gravity
can also be fit by the EOS, as discussed later.
For each petroleum fraction separately, two of the EOS parameters (Tc ; pc ; w; volume-shift factor, s; or multipliers of EOS constants
A and B) can be chosen so that the EOS exactly reproduces experimental boiling point and specific gravity. Because only two inspection properties are available (Tb and g), only two of the EOS parameters can be determined. Whitson50 suggests fixing the value of w
with an empirical correlation and adjusting Tc and pc to match normal boiling point and molar volume (M/g) at standard conditions.
Critical properties satisfying these criteria are given for a wide range
of petroleum fractions by the PR EOS and the Soave-RedlichKwong (SRK) EOS.22,23 A better (and recommended) approach for
cubic EOS’s is to use the volume-shift factor s (see Chap. 4) to match
specific gravity or a saturated liquid density and acentric factor to
match normal boiling point.
Other methods for forcing the EOS to match boiling point and
specific gravity have also been devised. Brulé and Starling51 proposed a method that uses viscosity as an additional inspection property of the fraction for determining critical properties. This approach proved particularly successful when applied to the BWR
EOS for residual-oil supercritical extraction (ROSE).28
5.6 Recommended C7) Characterizations
We recommend the following C7) characterization procedure for
cubic EOS’s.
1. Use the Twu48 (or Lee-Kesler12) critical property correlation
for Tc and pc .
2. Choose the acentric factor to match Tb ; alternatively, use the
Lee-Kesler12/Kesler-Lee13 correlations.
3. Determine volume-translation coefficients, si , to match specific
gravities; alternatively, use Peneloux et al.’s52 correlation for the SRK
EOS22,23 or Jhaveri and Youngren’s53 correlation for the PR EOS.49
When measured TBP data are not available, a mathematical split
should be made with either (1) the gamma distribution (default
16
5.6.1 SRK-Recommended Characterization. Alternatively, the
Pedersen et al.55 characterization procedure can be used with the
SRK EOS.
1. Split the plus fraction Cn) (preferably nu10) into SCN fractions up to C80 using Eqs. 5.7 through 5.11 and h+*4.
2. Calculate SCN densities ò i (gi + ò i /0.999) using the equation
ò i+A0)A1 ln(i), where A0 and A1 are determined by satisfying the
experimental-plus density, ò n), and measured (or assumed) density, ò n *1 ( ò6+0.690 can be used for C7)).
3. Calculate critical properties of all C7) fractions (distillation
cuts from C7 to Cn*1 and split SCN fractions from Cn through C80)
using the correlations
T c + 163.12 ò ) 86.052 ln M ) 0.43475 M * 1877.4 ,
M
,
ln p c + * 0.13408 ) 2.5019 ò ) 208.46 * 3987.2
M
M2
and m SRK + 0.48 ) 1.574 w * 0.176 w 2
+ 0.7431 ) 0.0048122 M ) 0.0096707 ò
* ǒ3.7184
10 *6ǓM 2. . . . . . . . . . . . . . . . . . . (5.80)
Note that the use of acentric factor is circumvented by directly calculating the term m used in the a correction term to EOS Constant A.
4. Group C7) into 3 to 12 fractions using equal-weight fractions
in each group; use weight-average mixing rules.
5. Calculate volume-translation parameters for C7) fractions to
match specific gravities; pure component c values are taken from
Peneloux et al.52
6. All hydrocarbon/hydrocarbon BIP’s are set to zero. SRK BIP’s
given in Chap. 4 are used for nonhydrocarbon/hydrocarbon pairs.
The two recommended C7) characterization procedures outlined previously for the PR EOS and SRK EOS are probably the best
currently available (other EOS characterizations, such as the Redlich-Kwong EOS modified by Zudkevitch and Joffe,56 and some
three-constant characterizations should provide similar accuracy
but are not significantly better). Practically, the two characterization
procedures give the same results for almost all PVT properties (usually within 1 to 2%). With these EOS-characterization procedures,
we can expect reasonable predictions of densities and Z factors ("1
to 5%), saturation pressures ("5 to 15%), gas/oil ratios and formation volume factors ("2 to 5%), and condensate-liquid dropout
("5 to 10% for maximum dropout, with poorer prediction of taillike behavior just below the dewpoint).
The recommended EOS methods are less reliable for prediction
of minimum miscibility conditions, near-critical saturation pressure
and saturation type (bubblepoint or dewpoint), and both retrograde
and near-critical liquid volumes. Improved predictions can be obtained only by tuning EOS parameters to accurate PVT data covering a relatively wide range of pressures, temperatures, and compositions (see Sec. 4.7 and Appendix C).
5.7 Grouping and Averaging Properties
The cost and computer resources required for compositional reservoir simulation increase substantially with the number of compoPHASE BEHAVIOR
nents used to describe the reservoir fluid. A compromise between
accuracy and the number of components must be made according
to the process being simulated (i.e., according to the expected effect
that phase behavior will have on simulated results). For example, a
detailed fluid description with 12 to 15 components may be needed
to simulate developed miscibility in a slim-tube experiment. With
current computer technology, however, a full-field simulation with
fluids exhibiting near-critical phase behavior is not feasible for a
15-component mixture. The following are the main questions regarding component grouping.
1. How many components should be used?
2. How should the components be chosen from the original fluid
description?
3. How should the properties of pseudocomponents be calculated?
the method is general and can be applied to any molar-distribution
model and for any number of C7) groups.
In general, most authors have found that broader grouping of C7)
as C7 through C10, C11 through C15, C16 through C20, and C21) is
substantially better than splitting only the first few carbon-number
fractions (e.g., C7, C8, C9, and C10)). Gaussian quadrature is recommended for choosing the pseudocomponents in a C7) fraction;
equal-mass fractions or the Li et al.59 approach are valid alternatives.
5.7.2 Mixing Rules. Several methods have been proposed for calculating critical properties of pseudocomponents. The simplest and
most common mixing rule is
ȍz q
q +
ȍz ,
i i
iŮI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.83)
I
5.7.1 How Many and Which Components To Group. The number
of components used to describe a reservoir fluid depends mainly on
the process being simulated. However, the following rule of thumb
reduces the number of components for most systems: group N2 with
methane, CO2 with ethane, iso-butane with n-butane, and iso-pentane with n-pentane. Nonhydrocarbon content should be less than
a few percent in both the reservoir fluid and the injection gas if a
nonhydrocarbon is to be grouped with a hydrocarbon.
Five- to eight-component fluid characterizations should be sufficient to simulate practically any reservoir process, including (1) reservoir depletion of volatile-oil and gas-condensate reservoirs, (2) gas
cycling above and below the dewpoint of a gas-condensate reservoir,
(3) retrograde condensation near the wellbore of a producing well,
and (4) immiscible and miscible gas-injection. Coats57 discusses a
method for combining a modified black-oil formula with a simplified
EOS representation of separator oil and gas streams. The “oil” and
“gas” pseudocomponents in this model contain all the original fluid
components in contrast to the typical method of grouping where each
pseudocomponent is made up of only selected original components.
Lee et al.58 suggest that C7) fractions can be grouped into two
pseudocomponents according to a characterization factor determined by averaging the tangents of fraction properties M, g, and Ja
plotted vs. boiling point.
Whitson2 suggests that the C7) fraction can be grouped into NH
pseudocomponents given by
i
iŮI
where qi +any property (Tc , pc , w, or M) and zi +original mole fraction for components (i+1,..., I) making up Pseudocomponent I. Average specific gravity should always be calculated with the assumption of ideal solution mixing.
ȍz M
.
g +
ȍǒz M ńg Ǔ
i
i
iŮI
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.84)
I
i
i
i
iŮI
Pedersen et al.55 and others suggest use of weight fraction instead
of mole fraction. Wu and Batycky’s63 empirical mixing-rule approach uses both the molar- and weight-average mixing rules and
a proportioning factor, F, to calculate pcI , TcI , and wI .
qI +
ȍf q ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.85)
i i
iŮI
where qI represents pcI , TcI , and wI and fi +average of the molar and
weight fractions,
f i + F q iz i ) (1 * F) q i w i
and w i +
N H + 1 ) 3.3 log(N * 7), . . . . . . . . . . . . . . . . . . . . . (5.81)
zi Mi
ȍz M
,
N
j
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.86)
j
j+1
where N+carbon number of the heaviest fraction in the original
fluid description. The groups are separated by molecular weights MI
given by
MI + MC
7
ǒM
N ń MC
7
Ǔ
1ńN H
,
. . . . . . . . . . . . . . . . . . . (5.82)
where I+1,..., NH . Molecular weights, Mi , from the original fluid
description (i+7,..., N) falling within boundaries MI*1 to MI are included in Group I. This method should only be used when C7) fractions are originally separated on a carbon-number basis and for N
greater than [20.
Li et al.59 suggest a method for grouping components of an original fluid description that uses K values from a flash at reservoir temperature and the “average” operating pressure. The original mixture
is divided arbitrarily into “light” components (H2S, N2, CO2, and C1
through C6) and “heavy” components (C7)). Different criteria are
used to determine the number of light and heavy pseudocomponents. Li et al. also suggest use of phase diagrams and compositional
simulation to verify the grouped fluid description (a practice that we
highly recommend).
Still other pseudoization methods have been proposed60,61; Schlijper’s61 method also treats the problem of retrieving detailed compositional information from pseudoized (grouped) components. Behrens
and Sandler62 suggest a grouping method for C7) fractions based
on application of the Gaussian-quadrature method to continuous
thermodynamics. Although a simple exponential distribution is
used with only two quadrature points (i.e., the C7) fractions are
grouped into two pseudocomponents), Whitson et al.27 show that
HEPTANES-PLUS CHARACTERIZATION
with 0xFx1.
A generalized mixing rule for BIP’s can be written
k IJ +
ȍȍf f k
j ij ,
i
. . . . . . . . . . . . . . . . . . . . . . . . . (5.87)
iŮI jŮJ
where fi is also given by Eq. 5.86.
On the basis of Chueh and Prausnitz’s54 arguments, Lee-Kesler13
proposed the mixing rules in Eqs. 5.88 through 5.92.
v cI +
ƪ ȍȍ ǒ
ƪ ȍȍ
1
8
T cI +
)
v 1ń3
cj
iŮI jŮJ
1
8v cI
2
i
,
. . . (5.88)
iŮI
ǒ
z i z j ǒT ci T cjǓ 1ń2 v 1ń3
) v 1ń3
ci
cj
iŮI jŮJ
ǒ Ǔ
ǒȍ Ǔń ǒȍ Ǔ
B
ȍz
ƫǒ Ǔ
Ǔ ń ȍz
3
z i z j v 1ń3
ci
Ǔ
3
ƫ
2
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.89)
iŮI
wI +
zi wi
iŮI
zi ,
Z cI + 0.2905 * 0.085w I ,
and p cI +
Z cI R T cI
v cI .
. . . . . . . . . . . . . . . . . . . (5.90)
iŮI
. . . . . . . . . . . . . . . . . . . . . . (5.91)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.92)
17
TABLE 5.9—EXAMPLE STEPWISE-REGRESSION PROCEDURE FOR PSEUDOIZATION
TO FEWER COMPONENTS FOR A GAS CONDENSATE FLUID UNDERGOING DEPLETION
Original
Component
Original
Number
Component
Step 1
Step 2
Step 3
Step 4
Step 5
1
N2
N2)C1*
N2)C1
N2)C1
N2)C1)CO2)C2*
N2)C1)CO2)C2
2
CO2
CO2)C2*
CO2)C2
CO2)C2
C3)i-C4)n-C4
C3)i-C4)n-C4
)i-C5)n-C5)C6*
)i-C5)n-C5)C6
3
C1
C3
C3
C3)i-C4)n-C4*
F1
F1
4
C2
i-C4
i-C4)n-C4*
i-C5)n-C5)C6*
F2
F2)F3*
5
C3
n-C4
i-C5)n-C5*
F1
F3
6
i-C4
i-C5
C6
F2
7
n-C4
n-C5
F1
F3
8
i-C5
C6
F2
F3
9
n-C5
F1
10
C6
F2
F3
11
F1
12
F2
13
F3
Regression Parameters
kij
1, 9, 10, and 11
1, 7, 8, and 9
1, 5, 6, and 7
1, 3, 4, and 5
1, 3, and 4
Wa
1
4
3
1
3
Wb
1
4
3
1
3
Wa
2
5
4
2
4
Wb
2
5
4
2
4
*Indicates the grouped pseudocomponents being regressed in a particular step.
Lee et al.58 and Whitson2 consider an alternative method for calculating C7) critical properties based on the specific gravities and
boiling points of grouped pseudocomponents.
Coats57 presents a method of pseudoization that basically eliminates the effect of mixing rules on pseudocomponent properties.
The approach is simple and accurate. Coats requires the pseudoized
characterization to reproduce exactly the volumetric behavior of the
original reservoir fluid at undersaturated conditions. This is
achieved by ensuring that the mixture EOS constants A and B are
identical for the original and the pseudoized characterizations. First,
pseudocritical properties ( pcI , TcI , and wI ) are estimated with any
mixing rule (e.g., Kay’s64 mixing rule). Then W aI and W bI are determined to satisfy the following equations.
ƪȍ
W aI +
iŮI
ȍ zi zj aiaj ǒ1 * kijǓ
jŮJ
zi
iŮI
ǒR 2TcI2 ńpcIǓa I(TrI, w I)
ǒȍ Ǔń ǒȍ Ǔ
zi bi
and W bI +
ƫńǒȍ Ǔ
2
iŮI
zi
iŮI
ǒRT cIńp cIǓb I(T rI, w I)
,
. . . . . . . . . . . . . . . . (5.93)
R 2T 2
where a i + W ai p ci a i (T ri, w i)
ci
RT
and b i + W bi p ci b i(T ri, w i) .
ci
. . . . . . . . . . . . . . . . . . . . . (5.94)
W ai and W bi may include previously determined corrections to the
numerical constants W oa and W ob. This approach to determining
pseudocomponent properties, together with Eq. 5.87 for k I J , is surprisingly accurate even for VLE calculations. Coats also gives an
18
analogous procedure for determining pseudocomponent vcI for the
LBC24 viscosity correlation.
Coats’ approach is preferred to all the other proposed methods. It
ensures accurate volumetric calculations that are consistent with the
original EOS characterization, and the method is easy to implement.
5.7.3 Stepwise Regression. A reduced-component characterization
should strive to reproduce the original complete characterization
that has been used to match measured PVT data. One approach to
achieve this goal is stepwise regression, summarized in the following procedure.
1. Complete a comprehensive match of all existing PVT data with
a characterization containing light and intermediate pure components and at least three to five C7) fractions.
2. Simulate a suite of depletion and multicontact gas-injection
PVT experiments that cover the expected range of compositions in
the particular application.
3. Use the simulated PVT data as “real” data for pseudoization
based on regression.
4. Create two new pseudocomponents from the existing set of
components. Use the pseudoization procedure of Coats to obtain
WaI and WbI values, and use Eq. 5.87 for k I J .
5. Use regression to fine tune the W aI and W bI values estimated
in Step 4; also regress on key BIP’s, such as (N2)C1)*C7),
(CO2)C2)*C7), and other nonzero BIP’s involving pseudocomponents from Step 4.
6. Repeat Steps 4 and 5 until the quality of the characterization
deteriorates beyond an acceptable fluid description. Table 5.9
shows an example five-step pseudoization procedure.
In summary, any grouping of a complete EOS characterization
into a limited number of pseudocomponents should be checked to
ensure that predicted phase behavior (e.g., multicontact gas injection data, saturation pressures, and densities) are reasonably close
to the predictions for the original (complete) characterization. Stepwise regression is the best approach to determine the number and
PHASE BEHAVIOR
properties of pseudocomponents that can accurately describe a reservoir fluid’s phase behavior. If stepwise regression is not possible,
standard grouping of the light and intermediates (N2)C1,
CO2)C2, i-C4)n-C4, and i-C5)n-C5) and Gaussian quadrature
for C7) (or equal-mass fractions) is recommended; a valid alternative is the Li et al.59 method. The Coats57 method (Eqs. 5.93
and 5.94) is always recommended for calculating pseudocomponent properties.
References
1. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry
Series, American Chemical Soc., Washington, DC (1978) 182, 386.
2. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ
(August 1983) 683; Trans., AIME, 275.
3. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “SRK-EOS Calculation for Crude Oils,” Fluid Phase Equilibria (1983) 14, 209.
4. Craft, B.C., Hawkins, M., and Terry, R.E.: Applied Petroleum Reservoir
Engineering, second edition, Prentice-Hall Inc., Englewood Cliffs,
New Jersey (1991).
5. McCain, W.D. Jr.: The Properties of Petroleum Fluids, second edition,
PennWell Publishing Co., Tulsa, Oklahoma (1990).
6. Katz, D.L. and Firoozabadi, A.: “Predicting Phase Behavior of Condensate/Crude-Oil Systems Using Methane Interaction Coefficients,” JPT
(November 1978) 1649; Trans., AIME, 265.
7. Austad, T. et al.: “Practical Aspects of Characterizing Petroleum
Fluids,” paper presented at the 1983 North Sea Condensate Reservoirs
and Their Development Conference, London, 24–25 May.
8. Chorn, L.G.: “Simulated Distillation of Petroleum Crude Oil by Gas
Chromatography—Characterizing the Heptanes-Plus Fraction,” J.
Chrom. Sci. (January 1984) 17.
9. MacAllister, D.J. and DeRuiter, R.A.: “Further Development and Application of Simulated Distillation for Enhanced Oil Recovery,” paper
SPE 14335 presented at the 1985 SPE Annual Technical Conference
and Exhibition, Las Vegas, Nevada, 22–25 September.
10. Designation D158, Saybolt Distillation of Crude Petroleum, Annual
Book of ASTM Standards, ASTM, Philadelphia, Pennsylvania (1984).
11. Designation D2892-84, Distillation of Crude Petroleum (15.Theoretical Plate Column), Annual Book of ASTM Standards, ASTM, Philadelphia, Pennsylvania (1984) 8210.
12. Kesler, M.G. and Lee, B.I.: “Improve Predictions of Enthalpy of Fractions,” Hydro. Proc. (March 1976) 55, 153.
13. Lee, B.I. and Kesler, M.G.: “A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States,” AIChE J.
(1975) 21, 510.
14. Riazi, M.R. and Daubert, T.E.: “Simplify Property Predictions,” Hydro.
Proc. (March 1980) 115.
15. Maddox, R.N. and Erbar, J.H.: Gas Conditioning and Processing—Advanced Techniques and Applications, Campbell Petroleum Series, Norman, Oklahoma (1982) 3.
16. Organick, E.I. and Golding, B.H.: “Prediction of Saturation Pressures
for Condensate-Gas and Volatile-Oil Mixtures,” Trans., AIME
(1952) 195, 135.
17. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-Hill
Book Co. Inc., New York City (1959).
18. Riazi, M.R. and Daubert, T.E.: “Analytical Correlations Interconvert
Distillation-Curve Types,” Oil & Gas J. (August 1986) 50.
19. Riazi, M.R. and Daubert, T.E.: “Characterization Parameters for Petroleum Fractions,” Ind. Eng. Chem. Res. (1987) 26, 755.
20. Robinson, D.B. and Peng, D.Y.: “The Characterization of the Heptanes
and Heavier Fractions,” Research Report 28, Gas Producers Assn., Tulsa, Oklahoma (1978).
21. Riazi, M.R. and Daubert, T.E.: “Prediction of the Composition of Petroleum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1980) 19, 289.
22. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 1. Phase
Envelope Calculations by Use of the Soave-Redlich-Kwong Equation
of State,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, 163.
23. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 2. Flash
and PVT Calculations with the SRK Equation of State,” Ind. Eng.
Chem. Proc. Des. Dev. (1984) 23, 566.
24. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of
Reservoir Fluids From Their Compositions,” JPT (October 1964)
1171; Trans., AIME, 231.
HEPTANES-PLUS CHARACTERIZATION
25. Whitson, C.H.: “Effect of C7) Properties on Equation-of-State Predictions,” paper SPE 11200 presented at the 1982 SPE Annual Technical
Conference and Exhibition, New Orleans, 26–29 September.
26. Whitson, C.H.: “Effect of C7) Properties on Equation-of-State Predictions,” SPEJ (December 1984) 685; Trans., AIME, 277.
27. Whitson, C.H., Andersen, T.F., and Søreide, I.: “C7) Characterization of Related Equilibrium Fluids Using the Gamma Distribution,”
C7 ) Fraction Characterization, L.G. Chorn and G.A. Mansoori
(eds.), Advances in Thermodynamics, Taylor & Francis, New York
City (1989) 1, 35–56.
28. Brulé, M.R., Kumar, K.H., and Watansiri, S.: “Characterization
Methods Improve Phase-Behavior Predictions,” Oil & Gas J. (11
February 1985) 87.
29. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium
Constants for a Gas-Condensate System,” Trans., AIME (1953) 198, 1.
30. Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Functions, Dover Publications Inc., New York City (1970) 923.
31. Haaland, S.: “Characterization of North Sea Crude Oils and Petroleum
Fractions,” MS thesis, Norwegian Inst. of Technology, Trondheim,
Norway (1981).
32. Watson, K.M., Nelson, E.F., and Murphy, G.B.: “Characterization of
Petroleum Fractions,” Ind. Eng. Chem. (1935) 27, 1460.
33. Watson, K.M. and Nelson, E.F.: “Improved Methods for Approximating Critical and Thermal Properties of Petroleum,” Ind. Eng. Chem.
(1933) 25, No. 8, 880.
34. Jacoby, R.H. and Rzasa, M.J.: “Equilibrium Vaporization Ratios for Nitrogen, Methane, Carbon Dioxide, Ethane, and Hydrogen Sulfide in
Absorber Oil/Natural Gas and Crude Oil/Natural Gas Systems,” Trans.,
AIME (1952) 195, 99.
35. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State,” Dr.Ing. dissertation, Norwegian Inst. of Technology, Trondheim, Norway (1989).
36. Technical Data Book—Petroleum Refining, third edition, API, New
York City (1977).
37. Rao, V.K. and Bardon, M.F.: “Estimating the Molecular Weight of Petroleum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1985) 24, 498.
38. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equation
for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures, I. Methane, Ethane, Propane, and n-Butane,” J. Chem. Phys.
(1940) 8, 334.
39. Reid, R.C.: “Present, Past, and Future Property Estimation Techniques,” Chem. Eng. Prog. (1968) 64, No. 5, 1.
40. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases
and Liquids, fourth edition, McGraw-Hill Book Co. Inc., New York
City (1987) 12–24.
41. Roess, L.C.: “Determination of Critical Temperature and Pressure of
Petroleum Fractions,” J. Inst. Pet. Tech. (October 1936) 22, 1270.
42. Cavett, R.H.: “Physical Data for Distillation Calculations-Vapor-Liquid Equilibria,” Proc., 27th API Meeting, San Francisco (1962) 351.
43. Nokay, R.: “Estimate Petrochemical Properties,” Chem. Eng. (23 February 1959) 147.
44. Pitzer, K.S. et al.: “The Volumetric and Thermodynamic Properties of
Fluids, II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization,” J. Amer. Chem. Soc. (1955) 77, No. 13, 3433.
45. Edmister, W.C.: “Applied Hydrocarbon Thermodynamics, Part 4:
Compressibility Factors and Equations of State,” Pet. Ref. (April
1958) 37, 173.
46. Hall, K.R. and Yarborough, L.: “New, Simple Correlation for Predicting Critical Volume,” Chem. Eng. (November 1971) 76.
47. Kesler, M.G., Lee, B.I., and Sandler, S.I.: “A Third Parameter for Use
in Generalized Thermodynamic Correlations,” Ind. Eng. Chem. Fund.
(1979) 18, No. 1, 49.
48. Twu, C.H.: “An Internally Consistent Correlation for Predicting the
Critical Properties and Molecular Weights of Petroleum and Coal-Tar
Liquids,” Fluid Phase Equilibria (1984) No. 16, 137.
49. Peng, D.Y. and Robinson, D.B.: “A New-Constant Equation of State,”
Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59.
50. Whitson, C.H.: “Critical Properties Estimation From an Equation of
State,” paper SPE 12634 presented at the 1984 SPE/DOE Symposium
on Enhanced Oil Recovery, Tulsa, Oklahoma, 15–18 April.
51. Brulé, M.R. and Starling, K.E.: “Thermophysical Properties of Complex Systems: Applications of Multiproperty Analysis,” Ind. Eng.
Chem. Proc. Des. Dev. (1984) 23, 833.
19
52. Peneloux, A., Rauzy, E., and Freze, R.: “A Consistent Correction for
Redlich-Kwong-Soave Volumes,” Fluid Phase Equilibria (1982) 8, 7.
53. Jhaveri, B.S. and Youngren, G.K.: “Three-Parameter Modification of
the Peng-Robinson Equation of State To Improve Volumetric Predictions,” SPERE (August 1988) 1033; Trans., AIME, 285.
54. Chueh, P.L. and Prausnitz, J.M.: “Calculation of High-Pressure Vapor–
Liquid Equilibria,” Ind. Eng. Chem. (1968) 60, No. 13.
55. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization
of Gas Condensate Mixtures,” C7) Fraction Characterization, L.G.
Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1.
56. Zudkevitch, D. and Joffe, J: “Correlation and Prediction of Vapor-Liquid Equilibrium with the Redlich-Kwong Equation of State,” AIChE J.
(1970) 16, 112.
57. Coats, K.H.: “Simulation of Gas-Condensate-Reservoir Performance,”
JPT (October 1985) 1870.
58. Lee, S.T. et al.: “Experiments and Theoretical Simulation on the Fluid
Properties Required for Simulation of Thermal Processes,” SPEJ (October 1982) 535.
59. Li, Y.-K., Nghiem, L.X., and Siu, A.: “Phase Behavior Computation for
Reservoir Fluid: Effects of Pseudo Component on Phase Diagrams and
Simulations Results,” paper CIM 84-35-19 presented at the 1984 Petroleum Soc. of CIM Annual Meeting, Calgary, 10–13 June.
20
60. Newley, T.M.J. and Merrill, R.C. Jr.: “Pseudocomponent Selection
for Compositional Simulation,” SPERE (November 1991) 490;
Trans., AIME, 291.
61. Schlijper, A.G.: “Simulation of Compositional Processes: The Use of
Pseudocomponents in Equation-of-State Calculations,” SPERE (September 1986) 441; Trans., AIME, 282.
62. Behrens, R.A. and Sandler, S.I.: “The Use of Semicontinuous Description To Model the C7) Fraction in Equation of State Calculations,” paper SPE 14925 presented at the 1986 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 23–23 April.
63. Wu, R.S. and Batycky, J.P.: “Pseudocomponent Characterization for
Hydrocarbon Miscible Displacement,” paper SPE 15404 presented at
the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–6 October.
64. Kay, W.B.: “The Ethane-Heptane System,” Ind. & Eng. Chem. (1938)
30, 459.
SI Metric Conversion Factors
ft3/lbm mol 6.242 796
°F (°F*32)/1.8
°F (°F)459.67)/1.8
psi 6.894 757
°R 5/9
E*02 +m3/kmol
+°C
+K
E)00 +kPa
+K
PHASE BEHAVIOR
Chapter 6
Conventional PVT Measurements
6.1 Introduction
This chapter reviews the standard experiments performed by pressure/volume/temperature (PVT) laboratories on reservoir fluid
samples: compositional analysis, multistage surface separation,
constant composition expansion (CCE), differential liberation expansion (DLE), and constant volume depletion (CVD). We present
data from actual laboratory reports and give methods for checking
the consistency of reported data for each experiment. Chaps. 5 and
8 discuss special laboratory studies, including true-boiling-point
(TBP) distillation and multicontact gas-injection tests, respectively.
Table 6.1 summarizes experiments typically performed on oils
and gas condensates. From this table, we see that the DLE experiment is the only test never performed on gas-condensate systems.
We begin by discussing standard analyses performed on oil and gascondensate samples.
6.1.1 General Information Sheet. Most commercial laboratories
report general information on a cover sheet of the laboratory report,
including formation and well characteristics and sampling conditions. Tables 6.2 and 6.31,2 show this information, which may be
important for correct application and interpretation of the fluid analyses. This is particularly true for wells where separator samples
must be recombined to give a representative wellstream composition. Most of these data are supplied by the contractor of the fluid
study and are recorded during sampling. Therefore, the representative for the company contracting the fluid study is responsible for
the correctness and completeness of reported data.
We strongly recommend that the following data always be reported
in a general information sheet: (1) separator gas/oil ratio (GOR) in
standard cubic feet/separator barrel, (2) separator conditions at sampling, (3) field shrinkage factor used ( + B osp), (4) flowing bottomhole pressure (FBHP) at sampling, (5) static reservoir pressure, (6)
minimum FBHP before and during sampling, (7) time and date of
sampling, (8) production rates during sampling, (9) dimensions of
sample container, (10) total number and types of samples collected
during the drillstem test, and (11) perforation intervals.
6.1.2 Oil PVT Analyses. Standard PVT analyses performed on reservoir oils usually include (1) bottomhole wellstream compositional
analysis through C 7), (2) CCE, (3) DLE, and (4) multistage-separator tests. The CCE experiment determines the bubblepoint pressure
and volumetric properties of the undersaturated oil. It also gives
two-phase volumetric behavior below the bubblepoint; however,
these data are rarely used. The DLE experiment and separator test
are used together to calculate traditional black-oil properties, B o
and R s, for reservoir-engineering calculations. Occasionally,
88
instead of a DLE study, a CVD experiment is run on a volatile oil.
Also, the C 7) fraction may be separated into single-carbon-number
cuts from C 7 through approximately C 20) by TBP analysis or simulated distillation (see Chap. 5).
6.1.3 Gas-Condensate PVT Analyses. The standard experimental
program for a gas-condensate fluid includes (1) recombined wellstream compositional analysis through C 7), (2) CCE, and (3) CVD.
The CCE and CVD data are measured in a high-pressure visual cell
where the dewpoint pressure is determined visually. Total volume/
pressure and liquid-dropout behavior is measured in the CCE experiment. Phase volumes defining retrograde behavior are measured in the CVD experiment together with Z factors and
produced-gas compositions through C 7). Optionally, a multistageseparator test can be performed as well as TBP analysis or simulated
distillation of the C 7) into single-carbon-number cuts from C 7 to
about C 20) (see Chap. 5).
6.2 Wellstream Compositions
PVT studies usually are based on one or more samples taken during
a production test. Bottomhole samples can be obtained by wireline
with a high-pressure container during either production testing or a
shut-in period. Alternatively, separator samples can be taken during
a production test. Bottomhole sampling is the preferred method for
most oil reservoirs, while recombined samples are traditionally used
for gas-condensate reservoirs.3-8 Taking both bottomhole and separator samples in oil wells is not uncommon. The advantage of separator samples is that they can be recombined in varying proportions
to achieve a desired bubblepoint pressure (e.g., initial reservoir
pressure); these larger samples are needed for special PVT tests
(e.g., TBP and slim tube among others).
6.2.1 Bottomhole Sample. Table 6.4 shows the reported wellstream
composition of a reservoir oil where C 7) specific gravity and molecular weight are also reported. In the example report, composition is
given both as mole and weight percent although many laboratories report only molar composition. Experimentally, the composition of a
bottomhole sample is determined by the following (Fig. 6.1).
1. Flashing the sample to atmospheric conditions.
2. Measuring the volumes of surface gas, V g , and surface oil, V o .
3. Determining the normalized weight fractions, w gi and w oi, of
surface samples by gas chromatography.
4. Measuring surface-oil molecular weight, M o , and specific
gravity, g o .
PHASE BEHAVIOR
TABLE 6.1—LABORATORY ANALYSES PERFORMED ON
RESERVOIR-OIL AND GAS-CONDENSATE SYSTEMS
TABLE 6.2—EXAMPLE GENERAL INFORMATION SHEET
FOR GOOD OIL CO. WELL 4 OIL SAMPLE
Laboratory Analysis
Oils
Gas Condensates
Bottomhole sample
D
d
Recombined composition
d
D
First well completed
C7+ TBP distillation
d
d
Original reservoir pressure at 8,692 ft, psig
C7+ simulated distillation
d
d
Original produced GOR, scf/bbl
Constant-composition expansion
D
D
Production rate, B/D
Multistage surface separation
D
d
Differential liberation
D
Separator temperature, °F
N
Separator pressure, psig
CVD
d
D
Multicontact gas injection
d
d
D+standard, d+can be performed, and N+not performed.
z i + F g y i ) (1 * F g) x i ; . . . . . . . . . . . . . . . . . . . . . . . . (6.1)
1
, . . . . . . . . . . . . . . . . . . (6.2)
1 ) ƪ133, 300ǒ gńM Ǔ ońR sƫ
where R s +GOR V gńV o in scf/STB from the single-stage flash;
yi +
ȍ ǒw
ǒ
ńM jǓ ) w g C
j0C 7)
xi +
7)
7)
w o ińM i
ǒ
ńM jǓ ) w o C
oj
j0C 7)
+
7)
ńM g C
Ǔ
; . . . . . . . . . (6.3)
wo C
ǒ1ńM Ǔ *
o
7)
7)
ȍ ǒw
ńM o C
7)
Ǔ
;
600
300
75
200
Oil gravity at 60°F, °API
Datum
8,000
No
Well Characteristics
Elevation, ft
610
Total depth, ft
8,943
Producing interval, ft
8,684 to 8,700
Tubing size, in.
27/8
Tubing depth, ft
8,600
PI at 300 B/D, B-D/psi
1.1
Last reservoir pressure at 8,500 ft, psig
3,954*
/ /19
Reservoir temperature at 8,500 ft, °F
Shut in 72 hours
Amerada
Normal production rate, B/D
300
GOR, scf/bbl
600
Separator pressure, psig
200
Separator temperature, °F
Ǔ
ojńM j
. . . . . . . . . . . . . . (6.5)
j0C 7)
Surface gas usually contains less than 1 mol% C 7) material consisting mainly of heptanes and octanes; M g C + 105 is usually a
7)
good assumption. Surface oil contains less than
1 mol% of the light
constituents C 1, C 2, and nonhydrocarbons. Low-temperature distillation can be used to improve the accuracy of reported weight
fractions for intermediate components in the surface oil ( C 3 through
C 6); however, gas chromatography is more widely used.
The most probable source of error in wellstream composition of a
bottomhole sample is the surface-oil molecular weight, M o , which
appears in Eq. 6.2 for F g and Eq. 6.4 for x i . M o is usually accurate
within "4 to 10%. In Chap. 5, we showed that the Watson characterization factor, K w, of surface oil (Eq. 5.35) should be constant (to
within "0.03 of the determined value) for a given reservoir. Once an
average has been established for a reservoir (usually requiring three
separate measurements), potential errors in M o can be checked. A
calculated K w that deviates from the field-average K w by more than
"0.03 may indicate an erroneous molecular-weight measurement.
Eqs. 6.1 through 6.4 show that all component compositions are
affected by M o C , which is backcalculated from M o with Eq.
7)
6.5. Fortunately, the
amount of lighter components (particularly C 1)
in the surface oil are small, so the real effect on conversion from
weight to mole fractions of the surface oil usually is not significant.
6.2.2 Recombined Samples. Tables 6.5 and 6.6 present the separator-oil and -gas compositional analyses of a gas-condensate fluid
and recombined wellstream composition. The separator-oil composition is obtained by use of the same procedure as that used for
bottomhole oil samples (Eqs. 6.1 through 6.5). This involves bringing the separator oil to standard conditions, measuring properties
CONVENTIONAL PVT MEASUREMENTS
(m/d/y)
217*
Well status
. . . . . . . . . . (6.4)
(m/d/y)
4,100
Pressure gauge
ȍ ǒw
and M o C
/ /19
Date
wg i ń Mi
gj
Cretaceous
Original gas cap
5. Converting w gi weight fractions to normalized mole fractions
y i and x i .
6. Recombining mathematically to the wellstream composition, z i.
Eqs. 6.1 through 6.5 give Steps 1 through 6 mathematically.
Fg +
Formation Characteristics
Name
75
Base pressure, psia
14.65
Well making water, % water cut
0
Sampling Conditions
Sample depth, ft
8,500
Well status
Shut in 72 hours
GOR
Separator pressure, psig
Separator temperature, °F
Tubing pressure, psig
1,400
Casing pressure, psig
Sampled by
Sampler type
Wofford
*Pressure and temperature extrapolated to the midpoint of the producing
interval+4,010 psig and 220°F.
and compositions of the resulting surface oil and gas, and recombining these compositions to give the separator-oil composition; Tables
6.5 and 6.6 report the results.
Separator gas is introduced directly into a gas chromatograph,
which yields weight fractions, w g . These weight fractions are converted to mole fractions, y i , by use of appropriate molecular
weights; Tables 6.5 and 6.6 show the results. C 7) molecular weight
is backcalculated with measured separator-gas specific gravity, g g .
Mg C
7)
+ w gC
7)
ǒ
1
*
28.97g g
Ǔ
*1
ȍ
wg i
Mi
i0C 7)
. . . . . . . . (6.6)
89
TABLE 6.3—EXAMPLE GENERAL INFORMATION SHEET
FOR GOOD OIL CO. WELL 7 GAS CONDENSATE
Formation Characteristics
Formation name
Pay sand
Date first well completed
/ /19
Original reservoir pressure at 10,148 ft, psig
(m/d/y)
5,713
Original produced-gas/liquid ratio, scf/bbl
Production rate, B/D
Separator pressure, psig
Separator temperature, °F
Liquid gravity at 60°F, °API
Datum, ft subsea
8,000
Well Characteristics
Elevation, ft KB
2,214
Total depth, ft
10,348
Producing interval, ft
10,124 to 10,176
Tubing size, in.
2
Tubing depth, ft
10,100
Open-flow potential, MMscf/D
Last reservoir pressure at 10,148 ft, psig
5,713
Date
/ /19
Reservoir temperature at 10,148 ft, °F
(m/d/y)
186
Status of well status
Shut in
Pressure gauge
Amerada
Sampling Conditions
Flowing tubing pressure, psig
3,375
FBHP, psig
5,500
Primary-separator pressure, psig
300
Primary-separator temperature, °F
62
Secondary-separator pressure, psig
20
Secondary-separator temperature, °F
60
Field stock-tank-liquid gravity at 60°F, °API
58.5
Primary-separator-gas production rate, Mscf/D
762.14
Pressure base, psia
14.696
Temperature base, °F
60
Compressibility factor, Fpv
1.043
Gas gravity (laboratory)
0.737
Gas-gravity factor, Fg
0.902
Stock-tank-liquid production rate at 60°F, B/D
127.3
Primary-separator-gas/stock-tank-liquid ratio
In scf/bbl
In bbl/MMscf
5,987
167.0
Sampled by
For the example PVT report (Tables 6.5 and 6.6), the separator
gas/oil ratio, R sp, during sampling is reported as standard gas volume per separator-oil volume (4,428 scf/bbl). In this report, the units
are incorrectly labeled scf/bbl at 60°F, where in fact the separator-oil
volume is measured at separator pressure (300 psig) and temperature (62°F). The separator-oil formation volume factor (FVF), B osp,
is 1.352 bbl/STB and represents the volume of separator oil required
to yield 1 STB of oil (i.e., condensate).
The equation used to calculate wellstream composition, z i, is
z i + F gsp y i ) (1 * F gsp) x i ,
ǒ
F gsp + 1 )
2, 130ò osp
M osp R sp
Ǔ
*1
, . . . . . . . . . . . . . . . . . . . . . (6.8)
ȍx M . . . . . . . . . . . . . . . . . . . . . . . . . . . .
N
where M osp +
i
i
(6.9)
i+1
. . . . . . . . . . . . . . . . . . . . . (6.7)
where F gsp +mole fraction of wellstream mixture that becomes
separator gas. In the laboratory report, F gsp is reported as “primary90
separator gas/wellstream ratio” (801.66 Mscf/MMscf), which is
equivalent to mole per mole ( F gsp +0.80166 mol/mol). The reported value of F gsp can be checked with
ò osp in lbm/ft3 is calculated with a correlation (e.g., Standing-Katz9)
or with the relation (62.4g o ) 0.0136g g R s)ńB o , where R s and
B o +separator-oil values in scf/STB and bbl/STB, respectively;
PHASE BEHAVIOR
TABLE 6.4—WELLSTREAM (RESERVOIR-FLUID)
COMPOSITION FOR GOOD OIL CO. WELL 4
BOTTOMHOLE OIL SAMPLE
Component
H2 S
CO2
N2
Methane
Ethane
Propane
i-butane
n-butane
i-pentane
n-pentane
Hexanes
Heptanes plus
Total
mol%
Nil
0.91
0.16
36.47
9.67
6.95
1.44
3.93
1.44
1.41
4.33
33.29
100.00
wt%
Nil
0.43
0.05
6.24
3.10
3.27
0.89
2.44
1.11
1.09
3.97
77.41
100.00
Density*
(g/cm3)
°API*
Molecular
Weight
DV g
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.11)
V osp
D R sp +
0.8515
34.5
218
and D R s +
DV g
.
Vo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.12)
Total GOR is calculated by adding the stock-tank-oil-based GOR’s
from each separator stage.
*At 60°F.
g o +stock-tank-oil density; and g g +gravity of gas released from
the separator oil.
Finally, the value of stock-tank-liquid/wellstream ratio in bbl/MMscf
represents the separator barrels produced per 1 MMscf of wellstream.
In terms of F gsp and separator properties, this value equals
ǒ
removed gas, n g ; and specific gravity of removed gas, g g. If requested, the gas samples can be analyzed chromatographically to
give molar composition, y.
The oil remaining after gas removal is brought to the conditions
of the next separator stage. The gas is removed again and quantified
by moles and specific gravity. Oil volume is noted, and the process
is repeated until stock-tank conditions are reached. Final oil volume,
V o , and specific gravity, g o , are measured at 60°F.
Table 6.7 gives results from four separator tests, each consisting
of two stages of separation. The first-stage-separator pressure is varied from 50 to 300 psig, and stock-tank conditions are held constant
at 0 psig and 75°F. GOR’s are reported as standard gas volume per
separator-oil volume, R sp, and as standard gas volume per stocktank-oil volume, R s, respectively.
Ǔ
bbl + 470(1*F gsp) M ospńò osp , . . . . . . . . . . . . . . (6.10)
B osp
MMscf
where 470+(1 million scf/MMscf)/[(379 scf/lbm mol)(5.615 ft3/bbl)].
The separator-oil and -gas compositions can be checked for consistency with the Hoffman et al.10 K-value method and Standing’s11
low-pressure K-value equations.
6.3 MultistageĆSeparator Test
The multistage-separator test is performed on an oil sample primarily to provide a basis for converting differential-liberation data from
a residual-oil to a stock-tank-oil basis. Occasionally, several separator tests are run to help choose separator conditions that maximize
stock-tank-oil production. Usually, two or three stages of separation
are used, with the last stage at atmospheric pressure and near-ambient temperature (60 to 80°F). The multistage-separator test can also
be conducted for high-liquid-yield gas-condensate fluids.
Fig. 6.2 illustrates schematically how the separator test is performed. Initially, the reservoir sample is at saturation conditions and
the volume is measured ( V ob or V gd ). The sample is then brought to
the pressure and temperature of the first-stage separator. All the gas
is removed, and the oil volume at the separator stage, V osp, is noted
together with the volume of removed gas, DV g ; number of moles of
N sp
Rs +
ȍǒD R Ǔ
s k
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.13)
k+1
Separator-oil FVF’s, B osp, are reported as the ratio of separator-oil
volume to stock-tank-oil volume.
B osp +
V osp
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.14)
Vo
Accordingly, the relation between separator gas/oil ratio and stocktank gas/oil ratio at a given stage is
D Rs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.15)
B osp
D R sp +
Because B osp u 1, it follows that R sp t R s.
Bubblepoint-oil FVF, B ob , is the ratio of bubblepoint-oil volume
to stock-tank-oil volume.
B ob +
V ob
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.16)
Vo
The average gas gravity, g g , is used in oil PVT correlations and
to calculate reservoir densities on the basis of black-oil properties.
The average gas gravity is calculated from
N sp
ȍǒ g Ǔ ǒD R Ǔ
g k
gg +
s k
k+1
N sp
ȍǒD R Ǔ
, . . . . . . . . . . . . . . . . . . . . . . . . . . (6.17)
s k
k+1
Fig. 6.1—Procedure for recombining single-stage separator samples to obtain wellstream
composition of a bottomhole sample; BHS + bottomhole sampler, GC + gas chromatograph,
FDP + freezing-point depression, and DM + densitometer.
CONVENTIONAL PVT MEASUREMENTS
91
TABLE 6.5—SEPARATOR AND RECOMBINED WELLSTREAM COMPOSITIONS
FOR GOOD OIL CO. WELL 7 GAS CONDENSATE
Separator Products Hydrocarbon Analysis
Separator Liquid
Component
(mol%)
CO2
Trace
N2
Separator Gas
(mol%)
(gal/Mscf)
0.22
Wellstream
(mol%)
(gal/Mscf)
0.18
Trace
0.16
0.13
Methane
7.78
75.31
61.92
Ethane
10.02
15.08
Propane
15.08
6.68
1.832
14.08
8.35
2.290
iso-butane
2.77
0.52
0.170
0.97
0.317
n-butane
11.39
1.44
0.453
3.41
1.073
iso-pentane
3.52
0.18
0.066
0.84
0.306
n-pentane
6.50
0.24
0.087
1.48
0.535
Hexanes
8.61
0.11
0.045
1.79
0.734
34.33
0.06
0.028
6.85
3.904
100.00
100.00
2.681
100.00
9.159
Heptanes plus
Total
Heptanes-Plus Properties
Oil gravity, °API
46.6
Specific gravity
at 60/60°F
0.7946
Molecular weight
0.795
143
103
143
Parameters
Calculated separator gas gravity (air+1.000)
0.735
Calculated gross heating value for separator gas at 14.696 psia and
60°F, BTU/ft3 dry gas
1,295
Primary-separator-gas*/-separator-liquid* ratio, scf/bbl at 60°F
4,428
Primary-separator-gas/stock-tank-liquid ratio at 60°F, bbl at 60°F/bbl
1.352
Primary-separator-gas/wellstream ratio, Mscf/MMscf
801.66
Stock-tank-liquid/wellstream ratio, bbl/MMscf
133.9
*Primary separator gas and liquid collected at 300 psig and 62°F.
TABLE 6.6—MATERIAL-BALANCE CALCULATIONS FOR
GOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE
Liquid Composition at Specified Pressures
(mol%)
Component
At 3,500 psig
At 2,900 psig
At 2,100 psig
At 1,300 psig
At 605 psig
CO2
0.18
0.18
0.18
0.15
0.08
N2
0.13
0.08
0.06
0.03
0.01
C1
13.18
45.04
32.22
19.69
11.77
C2
8.12
14.05
13.99
12.32
7.44
C3
12.59
9.67
11.25
11.66
9.31
i-C4
3.44
1.14
1.59
1.85
1.64
n-C4
5.21
4.82
6.12
7.35
7.17
i-C5
2.67
1.25
1.77
2.43
2.79
n-C5
5.74
2.16
3.48
4.62
5.50
C6
8.47
3.11
4.55
6.40
8.37
C7+
Total
M o, gńmol
M oC 7), gńmol
ò o, gńcm 3
92
40.27
18.51
24.79
33.50
45.91
100.00
100.00
100.00
100.00
100.00
96.6
54.1
64.3
78.2
95.6
168.8
160.1
152.1
149.9
150.3
0.3235
0.2642
0.1625
0.0892
0.0398
PHASE BEHAVIOR
where M i +molecular weight and ò i + component liquid density
in lbm/ft3 at standard conditions (Table A-1). The C 7) material in
separator gases is usually treated as normal heptane.
6.4 Constant Composition Expansion
pst+14.7 psia
Tst+60°F
Fig. 6.2—Schematic of a multistage-separator test.
where ǒg gǓ k +separator-gas gravity at Stage k. This relation is based
on the ideal gas law at standard conditions, where moles of gas are directly proportional with standard gas volume ( v g +379 scf/lbm mol).
Table 6.8 gives the composition of the first-stage-separator gas
at 50 psig and 75°F. The gross heating value, H g , of this gas is calculated by Kay’s12 mixing rule and component heating values, H i,
given in Table A-1.
ȍy H .
N
Hg +
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.18)
i+1
Component liquid yields, L i , represent the liquid volumes of a
component or group of components that can theoretically be processed from 1 Mscf of separator gas (gallons per million standard
cubic feet). Li can be calculated from
ǒ Ǔ
M
L i + 19.73y i ò i , . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.19)
i
6.4.1 Oil Samples. For an oil sample, the CCE experiment is used
to determine bubblepoint pressure, undersaturated-oil density, isothermal oil compressibility, and two-phase volumetric behavior at
pressures below the bubblepoint. Table 6.9 presents data from an
example CCE experiment for a reservoir oil.
Fig. 6.3 illustrates the procedure for the CCE experiment. A blind
cell (i.e., a cell without a window) is filled with a known mass of reservoir fluid. Reservoir temperature is held constant during the experiment. The sample initially is brought to a condition somewhat above
initial reservoir pressure, ensuring that the fluid is single phase. As the
pressure is lowered, oil volume expands and is recorded.
The fluid is agitated at each pressure by rotating the cell. This
avoids the phenomenon of supersaturation, or metastable equilibrium, where a mixture remains as a single phase even though it should
exist as two phases.13-15 Sometimes supersaturation occurs 50 to
100 psi below actual bubblepoint pressure. By agitating the mixture
at each new pressure, the condition of supersaturation is avoided, allowing more accurate determination of the bubblepoint.
Just below the bubblepoint, the measured volume will increase
more rapidly because gas evolves from the oil, yielding a higher system compressibility. The total volume, V t, is recorded after the twophase mixture is brought to equilibrium. Pressure is lowered in steps
of 5 to 200 psi, where equilibrium is obtained at each pressure.
When the lowest pressure is reached, total volume is three to five
times larger than the original bubblepoint volume.
The recorded cell volumes are plotted vs. pressure, and the resulting curve should be similar to one of the curves in Fig. 6.4.16 For a
black oil (far from its critical temperature), the discontinuity in volume at the bubblepoint is sharp and the bubblepoint pressure and
volume are easily read from the intersection of the p-V trends in the
single- and two-phase regions.
Volatile oils do not exhibit the same clear discontinuity in volumetric behavior at the bubblepoint pressure. Instead, the p-V curve
is practically continuous in the region of the bubblepoint because
the undersaturated-oil compressibility is similar to the effective
two-phase compressibility. This makes determining the bubblepoint of volatile oils in a blind cell difficult. Instead, a windowed cell
TABLE 6.7—SEPARATOR TESTS (RESERVOIR-FLUID) OF
GOOD OIL CO. WELL 4 OIL SAMPLE
Separator
Pressure
(psia)
Separator
Temperature
(°F)
GORb
(ft3/bbl)
GORc
(ft3/bbl)
50
to
0
75
715
737
75
41
41
100
to
0
75
637
676
75
91
92
200
to
0
75
542
602
75
177
178
300
to
0
75
478
549
75
245
246
Stock-Tank
Gravity
(°API)
40.5
40.7
40.4
40.1
FVFd
(bbl/bbl)
1.481
1.474
1.483
1.495
aGauge.
bIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of oil at indicated pressure and
cIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of stock-tank oil at 60°F.
dIn barrels of saturated oil at 2,620 psi gauge and 220°F per barrel of stock-tank oil at 60°F.
eIn barrels of oil at indicated pressure and temperature per barrel of stock-tank oil at 60°F.
CONVENTIONAL PVT MEASUREMENTS
Separator
Volume
Factore
(bbl/bbl)
Flashed-Gas
Specific
Gravity
1.031
0.840
1.007
1.338
1.062
0.786
1.007
1.363
1.112
0.732
1.007
1.329
1.148
0.704
1.007
1.286
temperature.
93
TABLE 6.8—FIRST-STAGE SEPARATOR-GAS
COMPOSITION AND GROSS HEATING VALUE FOR
GOOD OIL CO. WELL 4 OIL SAMPLE*
Component
mol%
gal/Mscf
H2 S
Nil
CO2
1.62
N2
0.30
C1
67.00
C2
16.04
4.265
C3
8.95
2.449
i-C4
1.29
0.420
n-C4
2.91
0.912
i-C5
0.53
0.193
n-C5
0.41
0.155
C6
0.44
0.178
C7+
0.49
0.221
Total
100.00
TABLE 6.9—CCE DATA (RESERVOIR-FLUID)
FOR GOOD OIL CO. WELL 4 OIL SAMPLE
Saturation (bubblepoint) pressure*, psig
0.02441
Thermal expansion of undersaturated
oil at 5,000 psi+V at 220°F/V at 76°F
1.08790
Compressibility of saturated oil at
reservoir temperature
From 5,000 to 4,000 psi, vol/vol-psi
From 4,000 to 3,000 psi, vol/vol-psi
From 3,000 to 2,620 psi, vol/vol-psi
Calculated gross heating value,
BTU/ft3
8.793
0.840
1,405
dry gas at 14.65 psia and 60°F
*Collected at 50 psig and 75°F in the laboratory.
is used to observe visually the first bubble of gas and the liquid volumes below the bubblepoint.
Reported data from commercial laboratories usually include bubblepoint pressure, p b ; bubblepoint density, ò ob, or specific volume,
v ob(v + 1ńò); and isothermal compressibility of the undersaturated
oil, co , at pressures above the bubblepoint (Table 6.9). The table also
shows the oil’s thermal expansion, indicated by a ratio of undersaturated-oil volume at a specific pressure and reservoir temperature to
the oil volume at the same pressure and a lower temperature.
Total volumes are reported relative to the bubblepoint volume.
V rt +
Vt
.
V ob
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.20)
Traditionally, isothermal compressibility data are reported for pressure intervals above the bubblepoint. In fact, the undersaturated-oil
compressibility varies continuously with pressure, and, because
V t + V o (V rt + V ro) for p u p b, oil compressibility can be expressed as
ǒ Ǔ
ēV rt
c+ 1
V rt ēp
ǒ Ǔ;
ēV ro
+ 1
ēp
V
ro
T
p u p b . . . . . . . . . . (6.21)
T
13.48 x 10 – 6
15.88 x 10 – 6
18.75 x 10 – 6
Pressure/Volume Relations*
Heating Value
Calculated gas gravity (air+1.000)
2,620
Specific volume at saturation
pressure*, ft3/lbm
Pressure
(psig)
Relative volume
(L)†
5,000
0.9639
4,500
0.9703
4,000
0.9771
3,500
0.9846
3,000
0.9929
2,900
0.9946
2,800
0.9964
2,700
0.9983
2,620**
1.0000
2,605
1.0022
2.574
2,591
1.0041
2.688
2,516
1.0154
2.673
2,401
1.0350
2.593
2,253
1.0645
2.510
2,090
1.1040
2.422
1,897
1.1633
2.316
1,698
1.2426
2.219
1,477
1.3618
2.118
1,292
1.5012
2.028
1,040
1.7802
1.920
830
2.1623
1.823
640
2.7513
1.727
472
3.7226
1.621
Y function‡
*
**
1
At 220°F.
Saturation pressure.
Relative volume+V/Vsat in barrels at indicated pressure per barrel at saturation
pressure.
‡ Y function+( p *p)/(p
sat
abs)(V/Vsat*1).
The V rt function at undersaturated conditions may be fit with a secondĆdegree polynomial, resulting in an explicit relation for undersaturated-oil compressibility (see Chap. 3).
Total volumes below the bubblepoint can be correlated by the Y
function,16,17 defined as
Y+
pb * p
pb * p
+
,
p(V rt * 1)
pƪǒV tńV bǓ * 1ƫ
. . . . . . . . . . . . . . (6.22)
where p and p b are given in absolute pressure units. As Fig. 6.5
shows, Y vs. pressure should plot as a straight line and the linear
trend can be used to smooth V rt data at pressures below the bubblepoint. Standing16 and Clark17 discuss other smoothing techniques
and corrections that may be necessary when reservoir conditions
and laboratory PVT conditions are not the same.
Fig. 6.3—Schematic of a CCE experiment for an oil and a gas
condensate.
94
6.4.2 Gas-Condensate Samples. The CCE data for a gas condensate
usually include total relative volume, V rt , defined as the volume of
gas or of gas-plus-oil mixture divided by the dewpoint volume. Z facPHASE BEHAVIOR
at 290 psia
Fig. 6.4—Volume vs. pressure for an oil during a DLE test (after Standing16).
tors are reported at pressures greater than and equal to the dewpoint
pressure. Table 6.10 gives these data for a gas-condensate example.
Reciprocal wet-gas FVF, b gw, is reported at dewpoint and initial
reservoir pressures, where these values represent the gas equivalent
or wet-gas volume at standard conditions produced from 1 bbl of
reservoir gas volume.
b gw + ǒ5.615
p
T p
10 *3Ǔ p sc
+ 0.198
,
sc ZT
ZT
. . . . . . . . (6.23)
with b gw in Mscf/bbl, p in psia, and T in °R.
Most CCE experiments are conducted in a visual cell for gas condensates, and relative oil (condensate) volumes, V ro, are reported at
pressures below the dewpoint. V ro normally is defined as the oil volume divided by the total volume of gas and oil, although some reports define it as the oil volume divided by the dewpoint volume.
6.5 Differential Liberation Expansion
The DLE experiment is designed to approximate the depletion process of an oil reservoir18 and thereby provide suitable PVT data to
CONVENTIONAL PVT MEASUREMENTS
calculate reservoir performance.16,19-21 Fig. 6.6 illustrates the laboratory procedure of a DLE experiment. Figs. 6.7A through 6.7C
and Table 6.11 give DLE data for an oil sample.
A blind cell is filled with an oil sample, which is brought to a
single phase at reservoir temperature. Pressure is decreased until the
fluid reaches its bubblepoint, where the oil volume, V ob , is recorded.
Because the initial mass of the sample is known, bubblepoint density, ò ob, can be calculated.
The pressure is decreased below the bubblepoint, and the cell is
agitated until equilibrium is reached. All gas is removed at constant
pressure. Then, the volume, DV g; moles, Dn g; and specific gravity,
g g, of the removed gas are measured. The remaining oil volume, V o ,
is also recorded. This procedure is repeated 10 to 15 times at decreasing pressures and finally at atmospheric pressure. Residual-oil
volume, V or , and specific gravity, g or , are measured at 60°F.
Other properties are calculated on the basis of measured data
( DV g , V o , Dn g , g g , V or , and g or), including differential solution
gas/oil ratio, R sd ; differential oil FVF, B od ; oil density, ò o ; and gas
Z factor, Z. For Stage k, these properties can be determined from
95
Bubblepoint
Temperature
°5F
80
163
185
205
Pressure
psia
1,970
2,437
2,520
2,615
Volume
cm3
82.30
86.88
87.92
89.05
Fig. 6.5—PVT relation and plot of Y function for an oil sample at pressures below the bubblepoint.
ȍ 379ǒDn Ǔ
calculations, volume factors, R s and B o , are used to relate reservoiroil volumes, V o, to produced surface volumes, V g and V o; i.e.,
k
g j
ǒR sdǓ +
k
ǒB odǓ k +
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . (6.24)
V or
ǒ V oǓ k
V or
,
Rs +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.25)
ȍǒ28.97ń5.615ǓǒDn Ǔ ǒg Ǔ
Vg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.28)
Vo
and B o +
k
V or(62.4g or) )
g j
ǒò oǓ +
k
ǒ V oǓ
350g or )
k
ȍ 0.0764ǒDR
k
+
g j
j+1
j+1
5.615ǒB odǓ k
Ǔ ǒg gǓ
j
and (Z) k + ǒ1ńRTǓǒ pDV gńDn gǓ k ,
and B od +
. . . . . . . . . . . . . . . . . . (6.27)
with V or and V o in bbl, R sd in scf/bbl, B od in bbl/bbl, DV g in ft3, p
in psia, Dn g in lbm mol, ò o in lbm/ft3, and T in °R. Note that the subscript j+1 indicates the final DLE stage at atmospheric pressure and
reservoir temperature. Reported oil densities are actually calculated
by material balance, not measured directly.
6.5.1 Converting From Differential to Stock-Tank Basis. Perhaps
the most important step in the application of oil PVT data for reservoir
calculations is conversion of the differential solution gas/oil ratio,
R sd, and oil FVF, B od , to a stock-tank-oil basis.16,20 For engineering
96
Vg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.30)
V or
,
. . . . . . . . . . . . . . . . . . (6.26)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.29)
Differential properties R sd and B od reported in the DLE report are
relative to residual-oil volume (i.e., the oil volume at the end of the
DLE experiment, corrected from reservoir to standard temperature).
R sd +
sd j
Vo
.
Vo
Vo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.31)
V or
The equations commonly used to convert differential volume factors to a stock-tank basis are
ǒBB Ǔ
R s + R sb * ǒR sdb * R sdǓ
and B o + B od
ǒBB Ǔ ,
ob
ob
. . . . . . . . . . . . . . . . . . (6.32)
odb
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.33)
odb
where B ob +bubblepoint-oil FVF, R sb +solution gas/oil ratio
from a multistage-separator flash, and R sdb and B odb +differential
volume factors at the bubblepoint pressure. The term ( B obńB odb),
PHASE BEHAVIOR
TABLE 6.10—CCE DATA FOR GOOD OIL CO.
WELL 7 GAS-CONDENSATE SAMPLE
Pressure
(psig)
Relative volume
Deviation Factor
Z
6,000
0.8808
1.144
5,713*
0.8948
1.107**
5,300
0.9158
1.051
5,000
0.9317
1.009
4,800
0.9434
0.981
4,600
0.9559
0.953
4,400
0.9690
0.924
4,300
0.9758
0.909
4,200
0.9832
0.895
4,100
0.9914
0.881
4,000†
1.0000
0.867‡
3,905
1.0089
3,800
1.0194
3,710
1.0299
3,500
1.0559
3,300
1.0878
3,000
1.1496
2,705
1.2430
2,205
1.5246
1,605
2.1035
1,010
3.5665
Pressure/volume relations of reservoir fluid at 186°F.
* Reservoir pressure.
** Gas FVF+1.591 Mscf/bbl.
† Dewpoint pressure.
‡ Gas FVF+1.424 Mscf/bbl.
representing the volume ratio, V orńV o , is used to eliminate the residual-oil volume, V or , from the Rsd and Bod data. Note that the conversion from differential to “flash” data depends on the separator
conditions because B ob and R sb depend on separator conditions.
Although, the conversions given by Eqs. 6.32 and 6.33 typically
are used, they are only approximate. The preferred method, as originally suggested by Dodson et al.,22 requires that some equilibrium
oil be taken at each stage of the DLE experiment and flashed through
a multistage separator to give the volume ratios, R s and B o . This laboratory procedure is costly and time-consuming and is seldom used.
However, the method is readily incorporated into an equation-ofstate (EOS) -based PVT program.
6.6 Constant Volume Depletion
The CVD experiment is designed to provide volumetric and compositional data for gas-condensate and volatile-oil reservoirs producing by pressure depletion. Fig. 6.8 shows the stepwise procedure
of a CVD experiment schematically, and Figs. 6.9A through 6.9D
and Table 6.12 give CVD data for an example gas-condensate fluid.
The CVD experiment provides data that can be used directly by
the reservoir engineer, including (1) a reservoir material balance
that gives average reservoir pressure vs. recovery of total wellstream (wet-gas recovery), sales gas, condensate, and natural gas
liquids; (2) produced-wellstream composition and surface products
vs. reservoir pressure; and (3) average oil saturation in the reservoir
(liquid dropout and revaporization) that occurs during pressure
depletion. For many gas-condensate reservoirs, the recoveries and
oil saturation vs. pressure data from the CVD analysis closely
approximate actual field performance for reservoirs producing by
pressure depletion. When other recovery mechanisms, such as waterdrive and gas cycling, are considered, the basic data required for
reservoir engineering are still taken mainly from a CVD report. This
section provides a description of the data provided in a standard
CONVENTIONAL PVT MEASUREMENTS
Fig. 6.6—Schematic of DLE experiment.
CVD analysis, ways to check the data for consistency,23-25 and how
to extract reservoir-engineering quantities from the data.23,26
Initially, the dewpoint, p d , or bubblepoint pressure, p b , of the reservoir sample is established visually and the cell volume, V cell, at
saturated conditions is recorded. The pressure is then reduced by
300 to 800 psi and usually by smaller amounts (50 to 250 psi) just
below the saturation pressure of more-volatile systems. The cell is
agitated until equilibrium is achieved, and volumes V o and V g are
measured. At constant pressure, sufficient gas, DV g, is removed to
return the cell volume to the original saturated volume.
In the laboratory, the removed gas (wellstream) is brought to atmospheric conditions, where the amount of surface gas and condensate are measured. Surface compositions y g and x o of the produced
surface volumes from the reservoir gas are measured, as are the volumes DV o and DV g , densities ò o and ò g and oil molecular weight
M o . From these quantities, we can calculate the moles of gas removed, Dn g.
D ng +
DV o ò o DV g
)
. . . . . . . . . . . . . . . . . . . . . . . . . (6.34)
Mo
379
These data are reported as cumulative wellstream produced, n p , relative to the initial moles n.
ǒnn Ǔ
p
k
+ 1n
ȍ(Dn ) ,
k
g j
. . . . . . . . . . . . . . . . . . . . . . . . . (6.35)
j+1
where j+1 corresponds to saturation pressure and (Dn g) 1 + 0. The
initial amount (in moles) of the saturated fluid is known when the cell
is charged. The quantity n pńn is usually reported as cumulative wet
gas produced in MMscf/MMscf, which is equivalent to mol/mol.
Surface compositions y g and x o of the removed reservoir gas and
properties of the removed gas are not reported directly in the laboratory report but are recombined to yield the equilibrium gas (wellstream) composition, y i , which also represents the equilibrium gas
remaining in the cell. The C 7) molecular weight of the wellstream,
M gC7), is backcalculated from measured specific gravity
( g w + g g ) and reservoir-gas composition, y. C 7) specific gravity of
the produced gas, g gC7) , is also reported, but this value is calculated
from a correlation.
Knowing the cumulative moles removed and its volume occupied
as a single-phase gas at the removal pressure, we can calculate the
equilibrium gas Z factor from
Z+
pDV g
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.36)
D n g RT
A “two-phase” Z factor is also reported that is calculated assuming that the gas-condensate reservoir depletes according to the material balance for a dry gas and that the initial condition of the reservoir is at dewpoint pressure.
97
Fig. 6.7A—DLE data for an oil sample from Good Oil Co. Well 4; differential solution gas/oil
ratio, Rsd .
ǒ Ǔǒ1 * GG Ǔ,
pd
p
+
Zd
Z2
pw
. . . . . . . . . . . . . . . . . . . . . . . . (6.37)
w
where G pw +cumulative wellstream (wet gas) produced and
G w +initial wet gas in place. As defined in Eq. 6.37, the term G pwńG w
equals n pńn reported in the CVD report. From Eq. 6.37, the only unknown at a given pressure is Z 2 , and the two-phase Z factor is then given by
Z2 +
p
. . . . . . . . . . . . . . . . . . . . . . (6.38)
ǒ p dńZ dǓƪ1 * ǒ n pńn Ǔƫ
Theoretical liquid yields, L i , are also reported for C 3) through
C 5) groups in the produced wellstreams at each pressure-depletion
stage. These values are calculated with
ǒ Ǔ
M
L i + 19.73y i ò i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.39)
i
and by summing the yields of components in the particular “plus”
group. For example, the liquid yield of C 5) material at CVD Stage
k is given by
ǒL Ǔ
C 5)
98
C 7)
k
+
ȍ ǒL Ǔ
j k
j+i C 5
M
+ 19.73 ȍ ǒy Ǔ ǒ ò Ǔ .
C 7)
j
j k
j+i C 5
j
. . . . . (6.40)
Table 6.13 gives various calculated cumulative recoveries based
on the reservoir initially being at its dewpoint. The basis for the calculations is 1 MMscf of dewpoint wet gas in place, G w ; the corresponding initial moles in place at dewpoint pressure is given by
G
n + vw
g
+
1 10 6 scf + 2, 638 lbm mol. . . . . . . . . . (6.41)
379 scfńlbm mol
The first row of recoveries (wellstream) simply represents the
cumulative moles produced, n pńn, expressed as wet-gas volumes,
G pw, in Mscf.
ǒ Ǔ
np
G pw + nv g n
+ (2, 638 lbm mol)ǒ379 scfńlbm molǓ
ǒ1
+ 1
ǒ Ǔ
np
10 3 MscfńscfǓ n
10 3
ǒnn Ǔ.
p
. . . . . . . . . . . . . . . . . . . . . . . . . (6.42)
Recoveries in Rows 2 through 4 (Normal Temperature Separation, Total Plant Products in Primary-Separator Gas, and Total Plant
Products in Second-Stage-Separator Gas) refer to production when
the reservoir is produced through a three-stage separator. Fig. 6.10
PHASE BEHAVIOR
Fig. 6.7B—DLE data for an oil sample from Good Oil Co. Well 4; differential oil FVF (relative
volume), Bod .
illustrates the process schematically. The calculated recoveries are
based on multistage-separator calculations that use low-pressure K
values and a set of separator conditions chosen arbitrarily or specified when the PVT study is requested.
6.6.1 Recoveries: “Normal Temperature Separation.” Column
1: Initial in Place. In Column 1, Row 2a the stock-tank oil in solution in the initial dewpoint fluid (N+135.7 STB) is calculated by
flashing 1 MMscf of the original dewpoint fluid, G w , through a
multistage separator.
Rows 2b through 2d give the volumes of separator gas at each
stage of a three-stage flash of the initial dewpoint fluid: 757.87,
96.68, and 24.23 Mscf, respectively. The mole fraction of wellstream resulting as a surface gas F gg is given by
G
F gg + d + ǒ757.87 ) 96.68 ) 24.23 Mscfńlbm molǓ
Gw
ǒ1
10 3 scfńMscfǓńǒ379 scfńlbm molǓ
+ 0.8788 lbm molńlbm mol,
. . . . . . . . . . . . . . . (6.43)
where G d +total separator “dry” gas and the corresponding mole
fraction of stock-tank oil is 0.1212 mol/mol. F gg is used to calculate
dry-gas FVF (see Eq. 3.41). For the dewpoint pressure, this gives
CONVENTIONAL PVT MEASUREMENTS
B gd +
+
ǒ p scńT scǓǒ ZTńp Ǔ
B gw
+
F gg
F gg
ǒ14.7ń520ǓNJ[0.867(186 ) 460)]ń4015Nj
0.8788
+ 4.487
10 *3 ft 3ńscf. . . . . . . . . . . . . . . . . . . . (6.44)
The producing GOR of the dewpoint mixture for the specified
separator conditions can be calculated as
R p + G + ƪǒ757.87 ) 96.68 ) 24.23 Mscfńlbm molǓ
N
ǒ1
10 3 scfńMscfǓƫń135.7 STBńlbm mol
+ 6, 476 scfńSTB .
. . . . . . . . . . . . . . . . . . . . . . . . (6.45)
The dewpoint solution oil/gas ratio, r sd, is simply the inverse of R p .
r sd + r p + 1
Rp
+ 1.544
10 *4 STBńscf + 154.4 STBńMMscf.
. . . . . . . . . . . . . . . (6.46)
Note that specific gravities of stock-tank oil and separator gases are
not reported for the separator calculations.
99
Fig. 6.7C—DLE data for an oil sample from Good Oil Co. Well 4; oil viscosity, mo .
Column 2 and Higher. On the basis of 1 MMscf of initial dewpoint fluid, Rows 2a through 2d give cumulative volumes of separator products at each depletion pressure ( N p, G p1, G p2, and G p3 ).
The producing GOR of the wellstream produced during a depletion
stage is given by
ǒR pǓ +
k
ǒG p1 ) G p2 ) G p3Ǔ
k
* ǒG p1 ) G p2 ) G p3Ǔ
ǒN pǓ * ǒN pǓ
k
k*1
k*1
R p + NJ[(301.57 ) 20.75 ) 5.61) * (124.78 ) 12.09 ) 3.16)]
. . . . . . . . . . . . . . . . . . . . . . . . (6.48)
10 *5 STBńscf
+ 45.8 STBńscf . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.49)
100
* ǒG p1 ) G p2 ) G p3Ǔ
G w ƪǒ n pńn Ǔ k * ǒ n pńn Ǔ k*1ƫ
k*1
.
For p+2,100 psig, this gives
F gg + [(301.57 ) 20.75 ) 5.61)*(124.78 ) 12.09
10 3Ǔńǒ1
10 6Ǔ(0.35096 * 0.15438)
+ 0.9558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.51)
B gd +
ǒ14.7ń520Ǔƪ0.762(186 ) 460)ń2, 115ƫ
0.9558
+ 6.884
rs + rp + 1
Rp
1
+ 4.58
21, 580 scfńSTB
k
The dry-gas FVF at 2,100 psig is
10 3ǓNjń(24.0 * 15.4)
In terms of the solution oil/gas ratio,
+
ǒG p1 ) G p2 ) G p3Ǔ
) 3.16)]ǒ1
For 2,100 psig, this gives
+ 21, 850 scfńSTB.
ǒF ggǓ +
k
. . . . . . . . . . . . . . . . . . (6.50)
.
. . . . . . . . . . . . . . . . . . (6.47)
ǒ1
At a given pressure, the mole fraction of the removed CVD gas
wellstream that becomes dry separator gas is given by
10 *3 ft 3ńscf . . . . . . . . . . . . . . . . . . . . (6.52)
In summary, the information provided in the rows labeled Normal
Temperature Separation gives estimates of the condensate and
sales-gas recoveries assuming a multistage surface separation. For
example, at an abandonment pressure of 605 psig, the condensate
recovery is 35.1 STB of the 135.7 STB initially in place (in solution
in the dewpoint mixture), or 26% condensate recovery. Dry-gas recovery is (685.02)37.79)10.40)+733.21 Mscf of the 878.78
PHASE BEHAVIOR
TABLE 6.11—DLE DATA FOR GOOD OIL CO. WELL 4 OIL SAMPLE
Pressure
(psig)
Solution
GOR
(scf/bbl*)
Relative
Oil Volume
(RB/bbl*)
2,620
2,350
2,100
1.850
1,600
1,350
1,110
850
600
350
159
0
854
763
684
612
544
479
416
354
292
223
157
0
1.600
1.554
1.515
1.479
1.445
1.412
1.382
1.351
1.320
1.283
1.244
1.075
1.000**
Differential Vaporization
Relative
Oil
Deviation
Total Volume Density
Factor
Z
(RB/bbl*)
(g/cm3)
1.600
1.665
1.748
1.859
2.016
2.244
2.593
3.169
4.254
6.975
14.693
0.6562
0.6655
0.6731
0.6808
0.6889
0.6969
0.7044
0.7121
0.7198
0.7291
0.7382
0.7892
0.846
0.851
0.859
0.872
0.887
0.903
0.922
0.941
0.965
0.984
Gas FVF
(RB/bbl*)
Incremental
Gas Gravity
0.00685
0.00771
0.00882
0.01034
0.01245
0.01552
0.02042
0.02931
0.05065
0.10834
0.825
0.818
0.797
0.791
0.794
0.809
0.831
0.881
0.988
1.213
2.039
DLE Viscosity Data at 220°F
Pressure
(psig)
Oil Viscosity
(cp)
5,000
4,500
4,000
3,500
3,000
2,800
2,620
2,350
2,100
1,850
1,600
1,350
1,100
850
600
350
159
0
0.450
0.434
0.418
0.401
0.385
0.379
0.373
0.396
0.417
0.442
0.469
0.502
0.542
0.592
0.654
0.783
0.855
1.286
Calculated Gas
Viscosity
(cp)
0.0191
0.0180
0.0169
0.0160
0.0151
0.0143
0.0135
0.0126
0.0121
0.0114
0.0093
Gravity of residual oil+35.1°API at 60°F.
*Barrels
**At
of residual oil.
60°F.
Mscf dry gas originally in place, or 83.4%. These recoveries can be
compared with the reported wet-gas (or molar) recovery of 76.787%
at 605 psig. In addition to recoveries, the calculated results in this
section can be used to calculate solution oil/gas ratio, r s, and dry-gas
FVF, B gd , for modified black-oil applications.
6.6.2 Recovery: Plant Products. Rows 3 through 5 consider
theoretical liquid recoveries for propane, butanes, and pentanesplus assuming 100% plant efficiency. Recoveries in Rows 3 and 4
are for the calculated separator gases from Stages 1 and 2 of the
three-stage surface separation. Recoveries in Row 5 are for the produced wellstreams from the CVD experiment and represent the absolute maximum liquid recoveries that can be expected if the reservoir is produced by pressure depletion. Fig. 6.10 illustrates the
recovery calculations schematically. Liquid volumes (in gal/MMscf
of initial dewpoint fluid) at CVD Stage k are calculated from
ǒ ǓƪȍǒDnn Ǔ ǒ y Ǔ ƫ,
M
(L i) k + 19, 730 ò i
i
k
g
i j
j+1
. . . . . . . . . (6.53)
Fig. 6.8—Schematic of CVD experiment.
j
CONVENTIONAL PVT MEASUREMENTS
101
Fig. 6.9A—CVD data for gas-condensate sample from Good Oil Co. Well 7; liquid-dropout curve, Vro .
where j + 1 represents the dewpoint, y i +compositions of wellstream entering the gas plant at various stages of depletion,
M i +component molecular weights, and ò i + liquid component
densities in lbm/ft3 at standard conditions (Table A-1).
Calculated liquid recoveries below the dewpoint use the moles of
wellstream produced ( Dn gńn) and the compositions yi from the separator gas (Rows 3 and 4) or wellstream (Row 5) entering the plant.
Column 1 (Initial in Place) gives the total recoveries assuming that
the entire initial dewpoint fluid is taken to the surface and processed
[i.e., k + 1 and (Dn gńn) 1 + 1 in Eq. 6.53].
Note that cumulative recovery of propanes from the first-stage
separator during depletion (1,276 gal) is larger than the liquid propane produced in the first-stage-separator gas of the original dewpoint mixture (1,198 gal). This means that the stock-tank oil from
the separation of original dewpoint mixture contains more propane
than the cumulative stock-tank-oil volumes produced by depletion
and three-stage separation.
The results given in Rows 3 and 4 cannot be calculated from reported data because surface separator compositions from the threestage separation are not provided in the report. The results in Row
5 can be checked. As an example, consider the C 3 recoveries for the
initial-in-place fluid at 2,100 psig.
ǒL Ǔ
C3
pd
+ 19, 730 ǒ44.09ń31.66Ǔƪ (1)(0.0837)ƫ
+ 2, 299 galńMMscf . . . . . . . . . . . . . . . . . . . . (6.54a)
102
ǒ Ǔ
and L C
3 2100
+ 19, 730 ǒ44.09ń31.66Ǔ [0.0825(0.05374)
) 0.0810(0.15438 * 0.05374)
) 0.0757(0.35096 * 0.15438)]
+ 754 galńMMscf. . . . . . . . . . . . . . . . . (6.54b)
For the C 5) recoveries at the dewpoint,
ǒL Ǔ
C 5)
pd
+ 19, 730 [(72.15ń38.96) (0.0091)
) (72.15ń39.36) (0.0152)
) (86.17ń41.43) (0.0179) ) (143ń49.6) (0.0685)]
+ 5, 513 galńMMscf . . . . . . . . . . . . . . . . . . (6.55)
6.6.3 Correcting Recoveries for Initial Pressure Greater Than
Dewpoint Pressure. All recoveries given in Table 6.13 assume that
the reservoir pressure is initially at dewpoint. This assumption is
made because initial reservoir pressure is not always known with
certainty when PVT calculations are made. However, adjusting reported recoveries is straightforward when initial pressure is greater
than dewpoint pressure. With Q Table as recoveries given in Columns
2 and higher in Table 6.13, Q d as hydrocarbons in place in Column
PHASE BEHAVIOR
Dewpoint
Pressure
Fig. 6.9B—CVD data for gas-condensate sample from Good Oil Co. Well 7; equilibrium gas
compositions, yi .
1 at dewpoint pressure, and Q as actual cumulative recoveries based
on hydrocarbons in place at the initial pressure,
Q + Qd
ƪ
ƫ
ǒ pńZǓ
ǒ pńZǓ
i
*
; p y p d , . . . . . . . . . . . . (6.56)
ǒ pńZǓ
ǒ pńZǓ
d
d
Q + Q Table ) DQ d ; p t p d , . . . . . . . . . . . . . . . . . . . (6.57)
and DQ d + Q d
pńZ)
ƪ((pńZ)
* 1ƫ ,
i
. . . . . . . . . . . . . . . . . . . . (6.58)
d
where DQ d +additional recovery from initial to dewpoint pressure.
For the example report,
DQ d +
ƪ
ƫ
ǒ5, 728ń1.107Ǔ
* 1 Qd
ǒ4, 015ń0.867Ǔ
+ 0.1173 Q d ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.59)
recalling that moles of material at dewpoint is 2,638 lbm mol, moles
of material at initial pressure of 5,728 psig is n +2, 638(1 ) 0.1173)
+ 2, 947 lbm mol, and the basis of calculations is G w + 1.173
MMscf of wet gas in place at initial pressure of 5,728 psia.
The cumulative wellstream produced at the dewpoint pressure of
4,000 psig is 0.1173(1, 000) + 117.3 Mscf. Recovery at 3,500 psig
is 117.3 ) 53.74 + 171.0 Mscf. Likewise, wet-gas recovery
CONVENTIONAL PVT MEASUREMENTS
should be increased by 117.3 Mscf for all depletion pressures in the
CVD table.
For stock-tank-oil recovery, Q d + 135.7 STB, so DQ d + 15.9
STB. Stock-tank-oil recovery at 4,000 psig is 15.9 ) 0 + 15.9
STB; at 3,500 psig the recovery should be 15.9 ) 6.4 + 22.3 STB,
and so on.
On the basis of 1 MMscf wet gas at the dewpoint or 1.1173 MMscf
at initial reservoir pressure, the laboratory hydrocarbon pore volume (HCPV), V pHClab, is the same.
V pHClab + ǒG wB gwǓ d
+ ǒ1
10 6Ǔ
NJǒ Ǔƪ
14.7
520
0.867(186 ) 460)
4, 015
ƫNj
+ 3, 943 ft 3
+ ǒG w B gwǓ i
+ 1.1173
10 6
+ 3, 943 ft 3 .
NJǒ Ǔƪ
14.7
520
1.107(186 ) 460)
5728
ƫNj
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.60)
The actual HCPV of a reservoir is much larger than V pHClab, and the
conversion to obtain recoveries for the actual HCPV is simply
103
Fig. 6.9C—CVD data for gas-condensate sample from Good Oil Co. Well 7; equilibrium gas Z
factor, Zg .
Q actual + Q lab
V pHCactual
,
V pHClab
. . . . . . . . . . . . . . . . . . . . . . . (6.61)
where Q lab +laboratory value given by Eqs. 6.55 and 6.57. As an example, suppose geological data indicate a HCPV of 625,000 bbl
(82.45 acre-ft), or 3.509 106 ft3. Then, original wet gas in place is
G w + 1.1173
6
10 6 3.509 10
3, 943
+ 994.3 MMscf . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.62)
and condensate in solution at initial pressure is given by
6
N + 135.7(1.1173) 3.509 10
3, 943
+ 134, 900 STB .
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.63)
6.6.4 Liquid-Dropout Curve. Table 6.11 and Figs. 6.9A through
6.9D show relative oil volumes, V ro, measured in the example CVD
experiment. V ro is defined as the volume of oil, V o , at a given pressure divided by the original saturation volume, V s. This relative volume is an excellent measure of the average reservoir-oil saturation
(normalized) that will develop during depletion of a gas-condensate
104
reservoir. Correcting for water saturation, S w , the reservoir-oil saturation can be calculated from V ro with
S o + (1 * S w)V ro .
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.64)
For most gas condensates, V ro shows a maximum near 2,000 to
2,500 psia. Cho et al.27 give a correlation for maximum liquid dropout as a function of temperature and C 7) mole percent in the dewpoint mixture.
ǒ V roǓ
max
+ 93.404 ) 4.799 z C
7)
* 19.73 ln T , . . . . . . (6.65)
with z C7) in mole percent and T in °F. The correlation predicts
(V ro) max +23.2% for the example condensate fluid compared with
24% measured experimentally (at 2,100 psig). Fig. 6.11 shows values of (V ro) max vs. T and z C7)from Eq. 6.65.
Considerable attention usually is given to matching the liquiddropout curve when an EOS is used. Some gas condensates havewhat is referred to as a “tail,” where liquid drops out very slowly
(sometimes for several thousand psi below the dewpoint) before finally increasing toward a maximum. Matching this behavior with
an EOS can prove difficult, and the question is whether matching the
tail is really necessary (see Appendix C).
What really matters for reservoir calculations of a gas-condensate
fluid is how much original stock-tank condensate is “lost” because
of retrograde condensation in the reservoir. The shape and magniPHASE BEHAVIOR
Fig. 6.9D—CVD data for gas-condensate sample from Good Oil Co. Well 7; wet-gas material
balance.
tude of liquid dropout reflects the change in producing oil/gas ratio,
r p [ r s . A tail on a liquid-dropout curve implies that the producing
wellstream is becoming only slightly leaner (i.e., r s is decreasing
only slightly). The cumulative condensate recovery is given by
Gp
Np +
ŕ r dG ,
s
p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.66)
0
where G p +cumulative dry gas produced. Cumulative condensate
production is readily evaluated from a plot of r s vs. G p .
One of the most important checks of an EOS characterization for
any gas condensate, particularly one with a tail, is N p calculated
from CVD data vs. N p calculated from the EOS characterization. It
is alarming how much the surface condensate recovery can be underestimated if the tail is not matched properly. We do not recommend matching the dewpoint exactly with a liquid-dropout curve
that is severely overpredicted in the region where measured results
indicate little dropout. If the EOS characterization cannot be modified to honor the tail of liquid-dropout curve, it is preferable to
underpredict the measured dewpoint pressure and match only the
higher liquid-dropout volumes.
In summary, oil relative volume, V ro, is not important per se; however, the effect of liquid dropout on surface condensate production
CONVENTIONAL PVT MEASUREMENTS
should be emphasized. In fact, the effect of shape and magnitude of
liquid dropout on fluid flow in the reservoir is negligible, and any
EOS match will probably have the same effect on fluid flow from the
reservoir into the wellbore (i.e., inflow performance).
6.6.5 Consistency Check of CVD Data. Reudelhuber and Hinds24
give a detailed procedure for checking CVD data consistency that
involves a material-balance check on components and phases and
yields oil compositions, density, molecular weight, and M C7). Together with reported data, these calculated properties allow K values
to be calculated and checked for consistency with the Hoffman et
al.10 method.11,28 Whitson and Torp’s23 material-balance equations
are summarized later. Similar equations can also be derived for a
DLE experiment when equilibrium gas compositions and oil relative volumes are reported. Reported CVD data include temperature,
T ; dewpoint pressure, p d , or bubblepoint pressure, p b ; dewpoint Z
factor, Z d, or bubblepoint-oil density, ò ob . Additional data at each
Depletion Stage k include oil relative volume, V ro; initial fraction
of cumulative moles produced, n pńn; gas Z factor (not the twophase Z factor), Z; equilibrium gas composition, yi ; and equilibrium
gas (wellstream) C 7) molecular weight, M g C7).
The equilibrium gas density, ò g ; molecular weight, M g ; and wellstream gravity, g w + M gńM air , are readily calculated at each
105
TABLE 6.12—CVD DATA FOR GOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE 2*
Reservoir Pressure, psig
5,713**
4,000†
CO2
0.18
0.18
0.18
0.18
0.18
0.19
N2
0.13
0.13
0.13
0.14
0.15
0.15
0.14
C1
61.72
61.72
63.10
65.21
69.79
70.77
66.59
C2
14.10
14.10
14.27
14.10
14.12
14.63
16.06
C3
8.37
8.37
8.26
8.10
7.57
7.73
9.11
i-C4
0.98
0.98
0.91
0.95
0.81
0.79
1.01
n-C4
3.45
3.45
3.40
3.16
2.71
2.59
3.31
i-C5
0.91
0.91
0.86
0.84
0.67
0.55
0.68
n-C5
1.52
1.52
1.40
1.39
0.97
0.81
1.02
C7
1.79
1.79
1.60
1.52
1.03
0.73
0.80
C7+
6.85
6.85
5.90
4.41
2.00
1.06
1.07
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Component, mol%
Total
3,500
2,900
2,100
1,300
605
0‡
0.21
Properties
C7+ molecular weight
143
143
138
128
116
111
110
C7+ specific gravity
0.795
0.795
0.790
0.780
0.767
0.762
0.761
Equilibrium gas deviation factor, Z
1.107
0.867
0.799
0.748
0.762
0.819
0.902
Two-phase deviation factor, Z
1.107
0.867
0.802
0.744
0.704
0.671
0.576
0.000
5.374
15.438
35.096
57.695
76.787
Wellstream produced, cumulative
% of initial
93.515
From smooth compositions
C3+, gal/Mscf
9.218
9.218
8.476
7.174
5.171
4.490
5.307
C4+, gal/Mscf
6.922
6.922
6.224
4.980
3.095
2.370
2.808
C5+, gal/Mscf
5.519
5.519
4.876
3.692
1.978
1.294
1.437
23.9
22.5
18.1
Retrograde Condensation During Gas Depletion
Retrograde liquid volume,
0.0
3.3
19.4
12.6
% hydrocarbon pore space
*Study conducted at 186°F.
** Original reservoir pressure.
† Dewpoint pressure.
‡ 0-psig residual-liquid properties: 47.5°API oil gravity at 60°; 0.7897 specific gravity at 60/60°F; and molecular weight of 140.
Depletion Stage k [and at the dewpoint ( k + 1) for a gas-condensate sample] from
ǒM gǓ +
k
ȍ(y )
N
i k
Mi ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.67)
i+1
ǒò gǓ +
k
ǒ Ǔ
np
(n t) k + 1 * n ,
k
p ǒM gǓ k
(Z) k RT
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.68)
and ǒg gǓ k + ǒ g wǓ k +
ǒM gǓ
k
28.97
.
. . . . . . . . . . . . . . . . . . . . (6.69)
On a basis of 1 mol initial dewpoint fluid ( n + 1), the cell volume is
ǒn gǓ +
k
ǒ p Ǔ ǒV gǓ k
k
(Z) k RT
,
and (n o) k + (n t) k * ǒn gǓ k , . . . . . . . . . . . . . . . . . . . . . . . . (6.73)
and moles and mass of the individual phases remaining in the cell at
Stage k are given by
ȍǒDnn Ǔ ǒM Ǔ ,
k
(m t) k + M s *
g
g j
j +2
Z RT
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.70)
V cell + dp
d
for a gas condensate and
M
V cell + ò ob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.71)
ob
for a volatile oil. Oil and gas volumes, respectively, at Stage k are
(V o) k+ V cell (V ro) k
and ǒV gǓ k + V cellƪ1 * (V ro) kƫ . . . . . . . . . . . . . . . . . . . . . (6.72)
Moles and mass of the total material remaining in the cell at Stage k
are given by
106
j
ǒm gǓ + ǒn gǓ ǒM gǓ ,
k
k
k
and (m o) k + (m t) k * ǒm gǓ k .
. . . . . . . . . . . . . . . . . . . . . . (6.74)
In Eqs. 6.73 and 6.74,
ǒDnn Ǔ + ǒnn Ǔ * ǒnn Ǔ
g
p
j
p
j
, . . . . . . . . . . . . . . . . . . . . (6.75)
j*1
M s +saturated-fluid molecular weight, and (n pńn) 1 + 0.
Densities and molecular weights of the oil phase are calculated from
PHASE BEHAVIOR
TABLE 6.13—CALCULATED RECOVERIES* FROM CVD REPORT
FOR GOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE
Reservoir Pressure (psig)
Initial in Place
4,000**
3,500
2,900
2,100
1,300
605
0
1,000
0
53.74
154.38
350.96
576.95
767.87
935.15
Stock-tank liquid, bbl
135.7
0
6.4
15.4
24.0
29.7
35.1
Primary-separator gas, Mscf
757.87
0
41.95
124.78
301.57
512.32
658.02
Second-stage gas, Mscf
96.68
0
4.74
12.09
20.75
27.95
37.79
Stock-tank gas, Mscf
24.23
0
1.21
3.16
5.61
7.71
10.4
Propane, gal
1,198
0
67
204
513
910
1,276
Butanes, gal
410
0
23
72
190
346
491
Pentanes, gal
180
0
10
31
81
144
192
Propane, gal
669
0
33
86
149
205
286
Butanes, gal
308
0
15
41
76
108
159
Pentanes, gal
138
0
7
19
35
49
69
Propane, gal
2,296
0
121
342
750
1,229
1,706
Butanes, gal
1,403
0
73
202
422
665
927
Pentanes, gal
5,519
0
262
634
1,022
1,315
1,589
Wellstream, Mscf
Normal temperature separation†
Total plant products in primary separator‡
Total plant products in second-stage
separator‡
Total plant products in wellstream‡
* Cumulative
recovery per MMscf of original fluid calculated during depletion.
**Dewpoint pressure.
† Recovery basis: primary separation at 500 psia and 70°F, second-stage separation at 50 psia and 70°F, and stock tank at 14.7 psia and
‡ Recovery assumes 100% plant efficiency.
ǒò oǓ +
k
(m o) k
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.76)
(V o) k
and (M o) k +
(m o) k
(n o) k
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.77)
ƪ
(z i) k + 1
(z i) 1 *
(n t) k
K values can be calculated from K i + y ińx i , and z i +overall composition of the mixture remaining in the cell at Stage k .
ȍǒDnn Ǔ ǒ y Ǔ
k
g
ƫ
i j
j+2
j
. . . . . . . . . . . . (6.79)
C 7) molecular weight of the oil phase can be calculated from
and the oil composition is given by
(n t) k(z i) k * ǒn gǓ k ǒ y iǓ k
(x i) k +
. . . . . . . . . . . . . . . . . . . . (6.78)
(n t) k * ǒn gǓ k
70°F.
ǒM
o C 7)
Ǔ
(M o) k *
k
+
ȍ (x )
i k
i0C 7)
ǒxC7)Ǔk
Mi
. . . . . . . . . . . . . . . (6.80)
Table 6.6 summarizes these calculations for the sample gas-condensate mixture.
(Separator Gas 1)
(Separator Gas 2)
Fig. 6.10—Schematic of method of calculating plant recoveries in a CVD report for a gas
condensate.
CONVENTIONAL PVT MEASUREMENTS
107
Nonphysical
Heptanes Plus, mol%
Fig. 6.11—Calculated maximum retrograde oil relative volumes from the Cho et al.27 correlation.
The oil composition at the last depletion state (605 psig for the example condensate) can be measured, but it must be requested specifically. Also, the residual-oil molecular weight, M or , and specific
gravity, g or, remaining after depletion at atmospheric pressure are
typically measured and reported as shown in Table 6.12. These values can be compared with calculated values by use of the materialbalance equations shown earlier.
The material-balance calculations are more accurate for rich gas
condensates and volatile oils. In fact, obtaining reasonable materialbalance oil properties for lean gas condensates is difficult. Sometimes it is useful to modify the reported oil relative volumes (particularly those close to the dewpoint) to monitor the effect on
calculated oil properties.
An alternative material-balance check that may be even more
useful for determining data consistency (particularly for leaner gas
condensates) involves starting with reported final-stage condensate
composition, (x i) k+N, and adding back the removed gases, (y i) k , for
each stage from k + N to k + 1. This results in the original gas
composition, (z i) k+1 , which can be compared quantitatively with
the laboratory-reported composition.
References
1. “Core Laboratories Good Oil Company Oil Well No. 4 PVT Study,” Core
Laboratories, Houston.
2. “Core Laboratories Good Oil Company Condensate Well No. 7 PVT
Study,” Core Laboratories, Houston.
3. Flaitz, J.M. and Parks, A.S.: “Sampling Gas-Condensate Wells,” Trans.,
AIME (1942) 146, 13.
4. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on Sampling
Two-Phase Gas Streams from High Pressure Condensate Wells,” (September 1945).
5. Reudelhuber, F.O.: “Sampling Procedures for Oil Reservoir Fluids,” JPT
(December 1957) 15.
6. Clark, N.J.: “Sampling and Testing Oil Reservoir Samples,” JPT (Jan.
1962) 12.
7. Clark, N.J.: “Sampling and Testing Gas Reservoir Samples,” JPT
(March 1962) 266.
8. Recommended Practice for Sampling Petroleum Reservoir Fluids, API,
Dallas (1966) 44.
9. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,
AIME, (1942) 146, 140.
10. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants
for a Gas-Condensate System,” Trans., AIME (1953) 198, 1.
11. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratios
of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT
(September 1979) 1193.
12. Kay, W.B.: “The Ethane-Heptane System,” Ind. & Eng. Chem. (1938)
30, 459.
13. Kennedy, H.T. and Olson, C.R.: “Bubble Formation in Supersaturated
Hydrocarbon Mixtures,” Oil & Gas J. (October 1952) 271.
108
14. Silvey, F.C., Reamer, H.H., and Sage, B.H.: “Supersaturation in Hydrocarbon Systems: Methane-n-Decane,” Ind. Eng. Chem. (1958) 3, No. 2, 181.
15. Tindy, R. and Raynal, M.: “Are Test-Cell Saturation Pressures Accurate
Enough?,” Oil & Gas J. (December 1966) 126.
16. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, eighth edition, SPE, Richardson, Texas (1977).
17. Clark, N.J.: “Adjusting Oil Sample Data for Reservoir Studies,” JPT
(February 1962) 143.
18. Moses, P.L.: “Engineering Applications of Phase Behavior of Crude-Oil
and Condensate Systems,” JPT (July 1986) 715.
19. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reservoir Engineering, McGraw-Hill Book Co. Inc., New York City (1960).
20. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,
first edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).
21. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Scientific
Publishing Co., Amsterdam (1978).
22. Dodson, C.R., Goodwill, D., and Mayer, E.H.: “Application of Laboratory PVT Data to Reservoir Engineering Problems,” Trans., AIME
(1953) 198, 287.
23. Whitson, C.H. and Torp, S.B.: “Evaluating Constant-Volume-Depletion
Data,” JPT (March 1983) 610; Trans., AIME, 275.
24. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing the Evaluation of PVT Data,” paper OSEA 88174 presented at the 1988 Offshore
Southeast Asia Conference, Singapore, 2–5 February.
25. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality of
Data From Standard Gas-Condensate PVT Experiments,” paper SPE
17768 presented at the 1988 Gas Technology Symposium, Dallas,
13–15 June.
26. Reudelhuber, F.O. and Hinds, R.F.: “Compositional Material Balance
Method for Prediction of Recovery From Volatile-Oil Depletion-Drive
Reservoirs,” JPT (January 1957) 19; Trans., AIME, 210.
27. Cho, S.J., Civan, F., and Starling, K.E.: “A Correlation To Predict Maximum Condensation for Retrograde Condensation Fluids and Its Use in
Pressure-Depletion Calculations,” paper SPE 14268 presented at the
1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September.
28. Clark, N.J.: “Theoretical Aspects of Oil and Gas Equilibrium Calculations,” JPT (April 1962) 373.
SI Metric Conversion Factors
°API 141.5/(131.5)°API)
+g/cm3
bbl 1.589 873
E*01 +m3
Btu 1.055 056
E)00 +kJ
cp 1.0*
E*03 +Pa@s
ft 3.048*
E*01 +m
E*02 +m3
ft3 2.831 685
°F (°F*32)/1.8
+°C
gal 3.785 412
E*03 +m3
in. 2.54*
E)00 +cm
lbm mol 4.535 924
E*01 +kmol
psi 6.894 757
E)00 +kPa
*Conversion factor is exact.
PHASE BEHAVIOR
Chapter 7
BlackĆOil PVT Formulations
7.1 Introduction
This chapter reviews black-oil pressure/volume/temperature (PVT)
formulations, gives examples of their application, and provides
guidelines for choosing the proper PVT formulation for a given reservoir. Sec. 7.2 reviews the traditional black-oil PVT formulation. The
three basic PVT properties are introduced: solution gas/oil ratio, R s ;
oil formation volume factor (FVF), B o; and gas FVF, B g. These properties define the PVT behavior of reservoir-oil and -gas mixtures by
quantifying the volumetric behavior and the distribution of surfacegas and surface-oil “components” as functions of pressure.
Many reservoirs being discovered today are at great depths, with
a higher percentage of these deep reservoirs containing gas-condensate and volatile-oil fluids. Treatment of these reservoirs requires
modification of the standard PVT formulation, as Sec. 7.3 discusses.
In particular, the additional property solution oil/gas ratio, r s, is
introduced, together with a modified gas FVF.
Sec. 7.4 covers the application of black-oil PVT properties to
well-rate deliverability and material-balance calculations. Sec. 7.5
discusses alternative black-oil PVT formulations, including the partial-density approach. And finally, Sec. 7.6 briefly reviews some
limited compositional formulations that are used in the simulation
of gas-injection processes.
7.2 Traditional BlackĆOil Formulation
It was already clear in the 1920’s that the engineering of oil reservoirs
required knowledge of how much gas was dissolved in the oil at reservoir conditions and how much the oil would shrink when it was
brought to the surface. It was also recognized that free gas at reservoir
conditions would expand up to several hundred times when brought
to surface conditions. Engineering quantities were needed to relate
surface volumes to reservoir volumes and vice versa. Three properties evolved to serve this purpose: solution gas/oil ratio, R s ; oil FVF,
B o; and gas FVF, B g. These properties are defined, respectively, by
Rs +
volume of surface gas dissolved in reservoir oil
,
volume of stock-tank oil from reservoir oil
. . . . . . . . . . . . . . . . . . . . . (7.1a)
Bo +
volume of reservoir oil
,
volume of stock-tank oil from reservoir oil
. . . . . . . . . . . . . . . . . . . . (7.1b)
BLACK-OIL PVT FORMULATIONS
and B g +
volume of reservoir gas
.
volume of surface gas from reservoir gas
. . . . . . . . . . . . . . . . . . . . . (7.1c)
These three properties constitute the traditional black-oil PVT
formulation, which has the following assumptions.
1. Reservoir oil consists of two surface “components,” stock-tank
oil and surface (total separator) gas.
2. Reservoir gas does not yield liquids when brought to the surface.
3. Surface gas released from the reservoir oil has the same properties as the reservoir gas.
4. Properties of stock-tank oil and surface gas do not change during depletion of a reservoir.
Fig. 7.11 illustrates schematically the relation between reservoir
phases and surface components. This simplified PVT formulation
is still the standard for most petroleum engineering applications.
The traditional black-oil quantities, R s, B o, and B g, can be estimated with the correlations in Chap. 3 or can be calculated from differential-liberation and multistage-separator data (Chap. 6).
The validity of the traditional black-oil PVT formulation depends
primarily on the reservoir-oil volatility. Any reservoir oil with less
than [750 scf/STB initial solution gas/oil ratio can probably be
treated with the traditional PVT formulation. Also, any oil reservoir
that produces at higher than its bubblepoint pressure during most of
the reservoir’s productive life can be treated with this formulation
(e.g., strong waterdrive, gas-cap-drive, or waterflooded reservoirs).
Volatile oils usually have an initial gas/oil ratio (GOR) greater
than [1,000 scf/STB and an initial stock-tank-oil gravity
u35°API. The following are the two main depletion characteristics
of a volatile-oil reservoir: (1) varying surface gravity of produced
stock-tank oil and (2) the percentage of produced stock-tank oil
coming from the flowing reservoir gas increases from zero initially
to a significant percentage at depletion (potentially u90%).
For most petroleum engineering calculations, the variation in
stock-tank-oil gravity can be neglected. However, neglecting the
surface oil that is produced from flowing reservoir gas may cause
gross underestimation of the ultimate stock-tank-oil recovery. Fig.
7.2 shows the actual depletion characteristics of a volatile-oil reservoir, where reservoir pressure decreases from 5,000 to 1,800 psia,
produced surface-oil gravity increases from 44 to 62°API, and producing GOR increases from 3,800 to 22,000 scf/STB.
A good check of the traditional black-oil formulation is to
compare reservoir material-balance performance determined on the
basis of standard black-oil PVT properties (e.g., a material bal1
Fig. 7.1—Schematic of traditional black-oil formulation relating
reservoir phases to surface components.
ance2) with depletion characteristics calculated from a compositional material balance. The traditional black-oil formulation
should not be used if the stock-tank-oil recoveries differ significantly (see Figs. 7.3 and 7.4).
Fig. 7.5 is another plot that indicates the relative volatility of an
oil. Differential-liberation relative oil volumes are plotted as shrinkage ( 1 * B odńB odb) vs. normalized pressure ( pńp b ), which indicates whether the shrinkage is rapid or slow. A curve that drops rapidly indicates a highly volatile oil. A “black” oil will tend to plot
above the solid “unit-slope” line shown in Fig. 7.5.
7.3 Modified BlackĆOil (MBO) Formulation
Several modifications of the traditional black-oil formulation have
been introduced to account for the surface-liquid content in reservoir gases. Most formulations introduce an additional PVT property, the solution oil/gas ratio, r s, and a modified definition of the gas
FVF. Fig. 7.6 shows schematically the relation between reservoir
phases and surface components in the MBO formulation.
Because this chapter gives a detailed description of the MBO
PVT formulation, we have introduced a more concise nomenclature
that distinguishes between reservoir and surface phases. Traditionally, we use the subscript o to represent both reservoir oil and stocktank oil and g to represent both reservoir gas and surface separator
Cumulative Surface Oil Produced, fraction
Fig. 7.3—Average reservoir pressure and producing GOR vs. cumulative oil for near-critical oil Reservoir NS-2; comparison of
traditional and MBO formulations.
2
Fig. 7.2—Depletion characteristics of a volatile-oil reservoir
(adapted from Ref. 1).
Pressure, psia
Fig. 7.4—GOR’s vs. pressure for near-critical Reservoir NS-2
and volatile-oil Reservoir NS-3.
PHASE BEHAVIOR
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 7.6—Schematic showing relation between reservoir phases
and surface phases (components) for MBO formulation.
0.9
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 7.5—Oil shrinkage plot used to evaluate volatility of a reservoir oil (from Ref. 3).
gas. In this chapter, we use the following subscripts to distinguish
between reservoir and surface phases: o+reservoir-oil phase at p
and T, g+reservoir gas phase at p and T, oo+stock-tank oil from
reservoir oil, go+surface gas from reservoir oil (“solution” gas),
og+stock-tank oil (condensate) from reservoir gas, gg+surface
gas from reservoir gas, o+total stock-tank oil, and g+total surface
gas, where the overbar indicates a surface-phase (component). To
avoid confusion, the standard term g w is used to represent the wellstream gravity of a reservoir gas (instead of g g).
The four MBO PVT parameters, oil FVF, solution gas/oil ratio,
dry-gas FVF, and solution oil/gas ratio are defined respectively as
Bo +
Vo
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2a)
V oo
Rs +
V go
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2b)
V oo
B gd +
where Vo +reservoir-oil volume, V oo +volume of stock-tank oil
produced from the reservoir oil, V go +volume of surface gas produced from the reservoir oil, V g +reservoir gas volume, V gg +volume of surface gas produced from the reservoir gas, and
V og +stock-tank oil (condensate) produced from the reservoir gas.
Fig. 7.7 outlines one procedure for determining MBO properties.
The equilibrium-gas and -oil phases from a depletion experiment
[constant composition expansion, constant volume depletion
(CVD), or differential liberation] are passed separately through a
multistage separator. The MBO properties are calculated according
to the definitions given in Eq. 7.2. Figs. 7.8 through 7.11 show
MBO properties calculated with the Whitson-Torp4 method for the
gas condensate, near-critical oil, and volatile oils in Table 7.1. Refs.
5 through 11 provide alternative methods.
7.3.1 Surface Gravities. When a well produces both reservoir oil
and gas, the composite surface gravities, g o and g g, will be an average of the surface gravities of the two reservoir phases, g oo and g go
for the reservoir oil and g og and g gg for the reservoir gas. The average gas gravity is given by
g g + F gg g gg ) ǒ1 * F ggǓ g go ,
. . . . . . . . . . . . . . . . . . . . (7.3)
Vg
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2c)
V gg
and r s +
V og
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2d)
V gg
Fig. 7.7—Schematic of the Whitson-Torp4 method for calculating MBO properties on the basis of depletion experiments and
multistage separation.
BLACK-OIL PVT FORMULATIONS
Fig. 7.8—Solution GOR, Rs , vs. pressure for volatile reservoir
Fluids NS-1, NS-2, and NS-3 calculated with the Whitson-Torp4
method.
3
Fig. 7.9—Oil FVF, Bo , vs. pressure for volatile reservoir Fluids
NS-1, NS-2, and NS-3 calculated with the Whitson-Torp4 method.
Fig. 7.10—Solution OGR, rs , vs. pressure for volatile reservoir
Fluids NS-1 and NS-3 calculated with the Whitson-Torp4 method.
where F gg +fraction of total surface gas produced from the reservoir gas.
Clearly, the assumption that g oo + g og + g o makes predicting the
variation in overall stock-tank-oil gravity during depletion impossible. As Fig. 7.2 shows, this variation can be significant.
F gg +
V gg
1 * Rs ń Rp
V gg
+
+
. . . . . . . . . . . . (7.4)
V gg ) V go
Vg
1 * R sr s
The average stock-tank-oil gravity is given by
g o + F oo g oo ) (1 * F oo ) g og , . . . . . . . . . . . . . . . . . . . (7.5)
where F oo +fraction of total stock-tank oil that comes from the reservoir oil.
F oo +
1 * rs Rp
V oo
V oo
+
+
,
V oo ) V og
Vo
1 * R sr s
. . . . . . . . . . . . (7.6)
with R p and R s in scf/STB and r s in STB/scf in Eqs. 7.4 and 7.6.
Surface gravities g oo, g og, g go, and g gg are determined separately for the reservoir-oil and reservoir-gas phases from multistage-separator calculations. Because the compositions of reservoir oil and gas
change during pressure depletion, the surface gravities also vary with
pressure. The variation in g og and g go in Figs. 7.12 and 7.13 is typical
of volatile-oil and gas-condensate mixtures. On the other hand, g oo
and g gg usually do not vary significantly with pressure.
Although the variation in surface gravities should be considered
in engineering calculations, most MBO formulations assume that
g oo + g og + g o + constant
and g go + g gg + g g + constant.
. . . . . . . . . . . . . . . . . . . (7.7)
Fig. 7.11—Inverse dry-gas FVF, bgd (+1/Bgd ), vs. pressure for
Gas-Condensate NS-1 calculated with the Whitson-Torp4 method.
TABLE 7.1—SOLUTION OIL/GAS RATIO CALCULATED FROM FIELD STOCK-TANK-OIL
GRAVITY COMPARED WITH EOS-CALCULATED VALUES
rs (STB/MMscf)
Test Date
pR
(psia)
Rp
(scf/STB)
Rs
( scf/STB)
Bubblepoint
5,555
1,500
January 1979
4,455
June 1980
go
EOS
g oo
EOS
g og
1,500
0.8430
0.843
0.7595
2,215
1,006
0.8353
0.843
0.7467
62
61
3,685
3,840
768
0.8289
0.843
0.7401
43
44
November 1983
3,105
7,480
615
0.8189
0.843
0.7356
32
34
May 1987
2,683
9,480
514
0.8146
0.843
0.7325
28
29
From g o
EOS
100
Note: Whitson-Torp4 method used to calculate R s, g oo, g og, and r s in last column. g oo does not change appreciably with pressure and is therefore assumed constant.
4
PHASE BEHAVIOR
Fig. 7.12—Surface-gas gravities vs. pressure during depletion.
Fig. 7.13—Surface-oil gravities vs. pressure during depletion.
Because the constant-gravity assumption is widely used, it should
be considered when determining the MBO properties R s, B o, B gd,
and r s. For example, Coats8 gives a procedure for determining
MBO properties of a gas condensate where the original mixture is
first passed through a separator to determine the surface gravities;
these gravities are assumed to be constant. A depletion experiment
is then simulated with an equation of state (EOS), and the equilibrium gas from each depletion stage is passed through a separator to
determine r s at the particular pressure. With constant surface gravities and r s as a function of pressure, B gd, B o, and R s, are determined so that reservoir-oil and -gas densities and the oil relative volumes from the depletion experiment are honored.
Surface-oil and -gas gravities are used in reservoir simulators to
convert B o, R s, B gd, and r s to reservoir-oil and -gas densities.
A dry-gas FVF, B gd (defined as the volume of reservoir gas divided by the volume of surface gas resulting after separation of the
reservoir gas), is used for the MBO formulation.
òo +
and ò g
62.4g oo ) 0.0136g go R s
Bo
0.0764g gg ) 350g og r s
+
. . . . . . . . . . . . . . . . (7.8)
B gd
Accurate phase densities can be important for reservoir processes
where gravity affects the recovery mechanism (e.g., gravity drainage in naturally fractured reservoirs). Therefore, manual checking
of MBO properties and surface gravities used as input for reservoir
simulation is recommended to ensure that the reservoir-oil and -gas
densities are calculated accurately.
7.3.2 Gas FVF. The traditional definition of gas FVF assumes that
the number of moles of gas at the surface equals the number of moles
of gas at reservoir conditions. This obviously is not correct if the reservoir gas yields condensate at the surface. The definition is still
used, however, for liquid-yielding reservoir gases and is called the
“wet”-gas FVF, B gw. The surface volume is a hypothetical wet-gas
volume consisting of the “dry”-surface-gas volume and the surface
condensate converted to an equivalent surface-gas volume.
Vg
.
B gw +
V gw
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.9)
With V g + n g ZRTńp and V gw + n g RT scńp sc, B gw is simply given
by the traditional equation for gas FVF.
B gw +
p sc ZT
+ 0.02827 ZT
p ,
T sc p
. . . . . . . . . . . . . . . . . . (7.10)
where B gw is in ft3/scf, T is in °R, and p is in psia.
BLACK-OIL PVT FORMULATIONS
B gd +
Vg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.11)
V gg
With V g + n g ZRTńp and V gg + n gg RT scńp sc, the dry-gas FVF
can be written
B gd +
ZT
p sc ZT
(1 ) C og r s) + 0.02827 p (1 ) C og r s)
T sc p
+ B gw(1 ) C og r s),
. . . . . . . . . . . . . . . . . . . . . . . (7.12)
where r s is in STB/scf, B gd and B gw are in ft3/scf, T is in °R, and p
is in psia. C og is a conversion from surface-oil volume in STB to an
“equivalent” surface gas in scf.
C og + 379
ǒlbmscfmol Ǔ
+ 133, 000
ǒ
5.615
ft Ǔ
ǒSTB
3
62.4
ǒ
Ǔ
g og lbm mol
M og
ft 3
Ǔ
g og scf
. . . . . . . . . . . . . . . . . . . . . . . (7.13)
M og STB
If condensate molecular weight, M og, is not measured, it can be estimated with the Cragoe12 correlation,
Mo +
6, 084
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.14)
g API * 5.9
The term (1 ) C og r s) *1 represents the mole fraction of reservoir
gas that becomes dry surface gas after separation and usually ranges
from 0.85 for rich gases to 1.0 for dry gases. Fig. 7.14 shows the behavior of the ratio as a function of pressure during depletion of a gas
condensate and a volatile oil.
7.3.3 Solution Oil/Gas Ratio. The following simplified relation can
be used to calculate r s in terms of reservoir-gas specific gravity, g w.
rs +
g w * g gg
.
4, 600 g og * C og g w
. . . . . . . . . . . . . . . . . . . . . . (7.15)
gw is reported as a function of pressure in the differential-liberation
experiment and is readily calculated from reported compositions in
a CVD experiment. Assuming that g gg + g g and g og + g o , surface
gravities usually are taken from a multistage separation of the original reservoir mixture and assumed constant throughout depletion.
On the basis of field production data, r s can be calculated in terms
of the actual measured stock-tank-oil gravity, g o.
5
Moles of reservoir oil and gas, respectively, in lbm mol are
n o + n oo ) n go
and n g + n og ) n gg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.21)
V o F ooC oo
,
379
where n oo +
V o F oo R s
n go +
379
n og +
,
V o ǒR p * R s F oo Ǔr sC og
379
V o ǒR p * R s F oo Ǔ
and n gg +
379
.
,
. . . . . . . . . . . . . . . . . . . . (7.22)
This yields
no +
Fig. 7.14—Fraction of reservoir gas that becomes “dry” surface gas vs. pressure during depletion of a gas condensate and
a volatile oil.
rs +
g o * g oo
. . . . . . . . . . . . . . . (7.16)
R sǒ g o * g ogǓ * R p ǒg oo * g ogǓ
Table 7.1 compares r s values from this relation (determined with field
data from a volatile-oil reservoir) with r s from EOS calculations.
7.3.4 Compositional Information. Engineering calculations based
on black-oil properties actually contain more compositional information than is commonly used. Given the compositions of stocktank oil and separator gases, we can calculate oil and gas compositions (and K values) at reservoir conditions using black-oil PVT
properties. Also, wellstream composition can be calculated from the
producing GOR.
To develop the compositional relations, we use a basis of V o
stock-tank barrels of total stock-tank oil. Volume of reservoir-oil
and -gas phases, respectively, is
V o + 5.615 V o F oo B o
and V g + V o B gd ǒR p * R s F ooǓ ,
. . . . . . . . . . . . . . . . . . . (7.17)
with V o and V g in ft3, R p and R s in scf/STB, B o in bbl/STB, and B gd
in ft3/scf. F oo is the fraction of total stock-tank oil that comes from
the reservoir oil (Eq. 7.4).
Mass of reservoir-oil and -gas phases, respectively, in lbm is
m o + m oo ) m go
and m g + m og ) m gg , . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.18)
where m oo + 350 V o F oog oo ,
m og + 350V o ǒR p * R s F oo Ǔ g og ,
and m gg + 0.076V o ǒR p * R s F oo Ǔ g gg . . . . . . . . . . . . . . . (7.19)
379
,
. . . . . . . . . . . . (7.23)
with C oo and C og given by
g oo
M oo
g og
and C og + 133, 000
.
M og
C oo + 133, 000
. . . . . . . . . . . . . . . . . . . . . . . . (7.24)
On the basis of these relations, the mole fractions of surface components in the reservoir oil are
xo +
n oo
1
+
ǒ1 ) R s ń C oo Ǔ
no
n go
and x g +
no
+ 1 * x o, . . . . . . . . . . . . . . . . . . . . . . . . . (7.25)
and the mole fractions of surface components in the reservoir gas are
yo +
n og
ng
+
n gg
and y g +
ng
1
1 ) ǒr s C og Ǔ
*1
+ 1 * y o, . . . . . . . . . . . . . . . . . . . . . . . . . (7.26)
with K values K o + y ońx o and K g + y gńx g . Strictly speaking,
Components o and g are not the same “species” and K values cannot be interpreted physically unless (1) the properties of surface oils
from reservoir gas and oil are equal and constant and (2) the surface
gases from reservoir gas and oil are equal and constant.
The mole fraction of the wellstream that comes from the reservoir
gas is F g + n gń(n g ) n o); therefore,
ƪ
F oo(C oo ) R s)
Fg + 1 )
Ǔ
(1 * F oo)ǒC og ) r *1
s
yi +
ƫ
*1
,
. . . . . . . . . . . (7.27)
y ggi ) ǒC og r sǓ x ogi
1 ) C og r s
and x i +
This yields
m o + V o F oo ǒ350 g oo ) 0.076 R s g goǓ
6
V o ǒR p * R s F oo Ǔǒ1 ) r s C ogǓ
and n g +
with C oo, C og, and R s in scf/STB and r s in STB/scf.
Compositions of reservoir oil, x i, and reservoir gas, y i, can be calculated from black-oil properties R s, r s, and surface properties by
m go + 0.076 V o F oo R sg go ,
and m g + V o ǒR p * R s F ooǓǒ350 g og r s ) 0.076 g gg Ǔ .
V o F ooǒ C oo ) R sǓ
379
. . . (7.20)
y goi ) ǒC oo ńR sǓ x ooi
1 ) C oo ńR s
, . . . . . . . . . . . . . . . . . . . . (7.28)
where y ggi +average composition of surface gases produced from
the reservoir gas; x ogi +composition of surface oil produced from
the reservoir gas; y goi +average composition of surface gases proPHASE BEHAVIOR
TABLE 7.2—EOS-CALCULATED SEPARATOR-GAS AND –OIL COMPOSITIONS FROM THREE-STAGE
SEPARATION OF ORIGINAL DEWPOINT GAS AND EOS-CALCULATED EQUILIBRIUM OIL
Reservoir Gas
Reservoir Oil
Component
y sp1
y sp2
y sp3
y gg
x og
y go
x oo
CO2
0.026092
0.030059
0.036539
0.026388
0.000588
0.027475
0.000627
0.003265
6.12 10*7
10*7
N2
0.003552
0.002154
0.000362
0.003460
C1
0.827710
0.809814
0.389891
0.816791
0.002079
0.809754
0.002103
C2
0.083029
0.099069
0.209316
0.086288
0.006739
0.090387
0.006730
C3
0.033261
0.036388
0.183444
0.036976
0.022803
0.039307
0.022010
5.94
i-C4
0.005535
0.005410
0.039898
0.006376
0.013315
0.006609
0.012297
n-C4
0.010249
0.009582
0.077103
0.011882
0.036997
0.012158
0.033498
i-C5
0.003145
0.002559
0.022571
0.003616
0.030334
0.003486
0.026016
n-C5
0.002939
0.002287
0.020158
0.003355
0.036413
0.003183
0.030772
C6
0.002425
0.001577
0.012855
0.002673
0.081629
0.002398
0.066496
F1
0.001671
0.000953
0.007116
0.001798
0.135151
0.001612
0.111360
F2
0.000380
0.000141
0.000739
3.87 10*4
0.252945
0.000353
0.221341
F3
6.34 10*6
1.03 10*6
2.63 10*6
6.20 10*6
0.223155
6.30 10*6
0.230727
F4
7.62 10*9
3.40 10*10
2.76 10*10
7.37 10*9
0.120536
8.91 10*9
0.162246
10*15
10*16
4.10
F5
10*13
2.90
4.50
3.93
10*13
0.037312
5.90
10*13
0.073772
TABLE 7.3—RESERVOIR EQUILIBRIUM COMPOSITIONS CALCULATED FROM EOS
AND FROM MBO PVT PROPERTIES WITH SURFACE-GAS AND -OIL COMPOSITIONS
Dewpoint*
y
Reservoir Pressure**
x
y
x
Component
Feed
EOS
EOS
MBO
EOS
MBO
CO2
0.0237
0.0237
0.0245
0.0251
0.0206
0.0189
N2
0.0031
0.0031
0.0034
0.0033
0.0018
0.0022
C1
0.7319
0.7319
0.7817
0.7774
0.5316
0.5517
C2
0.0780
0.0780
0.0791
0.0824
0.0737
0.0637
C3
0.0355
0.0355
0.0344
0.0363
0.0401
0.0338
i-C4
0.0071
0.0071
0.0066
0.0067
0.0090
0.0084
n-C4
0.0145
0.0145
0.0133
0.0131
0.0194
0.0190
i-C5
0.0064
0.0064
0.0056
0.0049
0.0097
0.0107
n-C5
0.0068
0.0068
0.0058
0.0050
0.0106
0.0120
C6
0.0109
0.0109
0.0088
0.0065
0.0194
0.0229
0.0367
F1
0.0157
0.0157
0.0115
0.0082
0.0325
F2
0.0267
0.0267
0.0158
0.0126
0.0704
0.0709
F3
0.0233
0.0233
0.0081
0.0108
0.0841
0.0737
F4
0.0126
0.0126
0.0015
0.0058
0.0573
0.0518
F5
0.0039
0.0039
0.0001
0.0018
0.0196
0.0236
C7+
0.0821
0.1302
0.0369
0.0393
0.2639
0.2567
*6,750 psia.
**4,315 psia.
duced from the reservoir oil; x ooi +composition of surface oil produced from the reservoir oil; and C oo, C og, and R s are in scf/STB
and r s is in STB/scf.
Average surface-gas compositions y ggi and y goi are calculated
separately with the relations
N sp
ȍǒy
y ggi +
ggi
Ǔ ńǒ r sǓ j
j
j+1
N sp
ȍǒ1ń r Ǔ
,
. . . . . . . . . . . . . . . . . . . . . . . . . . (7.29)
s j
j+1
N sp
ȍǒy
y goi +
goi
Ǔ ǒ R sǓ j
j
j+1
N sp
ȍǒ R Ǔ
s j
j+1
BLACK-OIL PVT FORMULATIONS
where the subscript j indicates the separator stage. Stage GOR’s
and OGR’s, (R s) j and (r s) j , respectively, are based on stock-tank
volumes.
The four surface compositions (and gravities) are, in principle,
functions of pressure. However, the average separator-gas compositions from reservoir oil and from reservoir gas may be similar, and
7
Fig. 7.15—Calculated compositions for reservoir gas based on
MBO properties and surface-component compositions; comparison with EOS compositions.
y ggi + y goi + constant is a reasonable assumption (as is
g gg + g go + constant). These compositions are readily determined
from a multistage flash of the original reservoir mixture (see Table
7.2). Table 7.3 and Figs. 7.15 through 7.17 show calculated reservoir-phase compositions based on Eq. 7.26 for a gas-condensate
mixture. K values are also calculated (K i + y ińx i) and compared
with EOS results for a simulated CVD experiment (Fig. 7.18).
Wellstream composition, z i, can be calculated from reservoir
phase compositions y i and x i.
z i + y i F g ) x i ǒ1 * F gǓ ,
. . . . . . . . . . . . . . . . . . . . . . (7.30)
where F g is given by Eq. 7.25 in terms of producing GOR, R p
(through the quantity F oo). Note that values of R s and r s used to calculate F g , y i , and x i must be evaluated at the same pressure.
7.4 Applications of MBO Formulation
The MBO PVT approach has been limited mainly to reservoir simulation, although some applications have been reported in well-test
Fig. 7.16—Calculated methane mole fractions for reservoir oil
and gas based on MBO properties and surface-component compositions; comparison with EOS compositions.
analysis, inflow performance, and reservoir material balance.
Multiphase flow in pipe is another obvious application. To aid in the
use of MBO properties for volatile reservoir fluids, we present several engineering equations in terms of MBO properties.
7.4.1 Rate Equations (IPR)—Traditional Black-Oil PVT. Inflow-performance relations (IPR’s) give the relation between total
surface rates, q o and q g ; wellbore flowing pressure, p wf ; and average reservoir pressure, p R. For example, consider the radial-flow
equation for an undersaturated oil well.13
q o + q oo +
khǒp R * p wfǓ
141.2 m o B oƪlnǒr eńr wǓ * 0.75 ) sƫ
, . . . . . (7.31)
where q o is in STB/D, k +absolute permeability at irreducible water saturation, md; h +total reservoir thickness, ft; m o +oil viscosity, cp; B o +oil FVF, bbl/STB; r e +outer drainage radius, ft;
r w +actual wellbore radius, ft; and s +total skin factor.
Modified Black-Oil Properties
EOS
EOS
EOS
Fig. 7.17—Calculated compositions for reservoir oil based on
MBO properties and surface-component compositions; comparison with EOS compositions.
8
Fig. 7.18—Calculated K values for reservoir oil and gas based on
MBO properties and surface component compositions; comparison with EOS compositions.
PHASE BEHAVIOR
The appropriate equations to calculate rates in the production system are14,15
q o + q oo ) q og +
kh
141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ ǒmk B ) 5.615 mk Br Ǔdp
pR
rg s
ro
o
o
gd
g
p wf
and q g + q go ) q gg +
kh
141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ ǒkm BR ) 5.615 m kB Ǔdp ,
pR
ro
s
o
o
rg
g
. . . . . . . . . . . . (7.36)
gd
p wf
with q o in STB/D and q g in scf/D.
The liquid and vapor rates in the tubing or reservoir are given by
q o + q o F oo B o
and q g + q o ǒR p * R s F oo Ǔ B gd ,
. . . . . . . . . . . . . . . . . . . (7.37)
where F oo +fraction of total surface oil coming from the flowing
liquid (Eq. 7.6).
F oo +
Fig. 7.19—Fraction of wellbore rate from reservoir oil, fraction of
surface oil from reservoir oil, GOR, and pwf during depletion of
a volatile-oil reservoir.
For saturated-oil wells producing both reservoir oil and gas, the
oil-rate equation can be written terms of traditional black-oil PVT
properties (r s + 0) as
pR
q o + q oo +
kh
141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ mk B dp.
ro
o
o
1 * Rp rs
q oo
+
.
qo
1 * Rs rs
. . . . . . . . . . . . . . . . . . . . . . . (7.38)
PVT properties used to calculate q o and q g are evaluated at the pressure and temperature in the reservoir or the production tubing.
Evaluation of the integrals in Eq. 7.36 is not straightforward. In fact,
using only one of the two rate equations would be logical, depending
on which phase was dominant. For a predominantly oil system, the oil
rate in Eq. 7.36 should be used for q o and the gas rate could be calculated from the total producing GOR. Likewise, for a predominantly gas
system, the gas rate in Eq. 7.36 should be used for q g and the oil rate
can be calculated from the total producing GOR. Producing GOR
would be available from material-balance calculations.
The volumetric fraction of reservoir fluids flowing as an oil phase
at wellbore conditions is
p wf
. . . . . . . . . . . . . . . . . . . . (7.32)
qo Bo
qo
+
+
qo ) qg
q o B o ) q g B gd
Total gas rate from a saturated-oil well is the product of the oil rate
and total producing GOR.
qg + qo Rp ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.33)
where q g is in scf/D and R p usually is available from material-balance calculations.
The rate of the oil phase flowing anywhere in the tubing or reservoir can be calculated as
qo + qo Bo ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.34)
with q o in B/D and B o evaluated at a specific pressure and temperature.
The flow rate of free gas at the same conditions is calculated from
q g + q oǒR p * R sǓ
B gd
,
5.615
. . . . . . . . . . . . . . . . . . . . . . (7.35)
with q g in B/D, q o in STB/D, R s and R p in scf/STB, and B gd in ft3/scf.
R s and B gd are evaluated at the same pressure and temperature.
7.4.2 IPR—MBO PVT. Eqs. 7.32 and 7.33 are based on the traditional black-oil PVT formulation where reservoir gas is assumed to
have no liquid content. For volatile reservoir fluids, the surface oil
consists of surface oil from the flowing liquid and condensed from
the flowing vapor. Likewise, the surface-gas rate consists of surface
gas from the flowing vapor and released from the flowing liquid.
BLACK-OIL PVT FORMULATIONS
ƪ
1)
ǒR p * Rs F ooǓ Bgd
5.615 F ooB o
ƫ
*1
,
. . . . . . . . . . . . . . . . . . . . (7.39)
where B o, R s, B gd, and r s are evaluated at the wellbore flowing pressure, p wf . For a volatile-oil reservoir, the oil fraction will drop to less
than 50% during depletion (see Fig. 7.19), marking the point when
the gas phase becomes the dominant flowing phase. The relative
amounts of reservoir oil and gas flowing at the wellbore should be
considered in the interpretation of well tests and application of IPR’s.
7.4.3 Reservoir Material Balance—MBO PVT. Reservoir material-balance relations for solution-gas-drive and dry-gas reservoirs
are well known and widely used. Borthne16 presents a reservoir material balance based on MBO properties that can be used for black
oils, volatile oils, and gas condensates. Modifications to the material
balance that account for connate water with dissolved gas, water influx, and other such factors can be included readily.
The basis of calculation is 1 bbl reservoir bulk volume. The conservation-of-mass equations for a single-cell material balance
yields the following difference equations for reservoir-oil and -gas
phases during a timestep Dt k + t k * t k*1 with a change in average
pressure from ( p R) k*1 to ( p R) k .
ǒ A oǓ k * ǒ A oǓ k*1) D N p + 0
and ǒ A gǓ k * ǒ A gǓ k*1 ) DG p + 0 ,
. . . . . . . . . . . . . . . . (7.40)
9
TABLE 7.4—MBO PROPERTIES FOR GAS CONDENSATE NS-1
Pressure
(psia)
Bo
(bbl/STB)
Rs
(scf/STB)
g oo
Bgd
(ft3/scf)
rs
(STB/MMscf)
g og
g gg
6,748.2
2.6490
3,005
0.7837
6,514.7
2.4693
2,662
0.7849
0.7155
0.004244
181.0
0.7689
0.7114
0.8958
0.7171
0.004205
158.2
0.7647
0.7110
6,014.7
2.2241
2,180
0.9051
0.7859
0.7208
0.004226
125.7
0.7575
0.7107
5,514.7
2.0495
0.9194
1,829
0.7859
0.7251
0.004333
102.4
0.7516
0.7106
0.9306
4,314.7
3,114.7
1.7427
1,211
0.7845
0.7397
0.004940
64.0
0.7399
0.7114
0.9516
1.5116
757
0.7832
0.7629
0.006371
39.3
0.7298
0.7139
0.9677
2,114.7
1.3525
456
0.7829
0.7927
0.009179
26.2
0.7224
0.7181
0.9772
1,214.7
1.2277
232
0.7843
0.8324
0.016214
21.2
0.7151
0.7268
0.9808
714.7
1.1651
124
0.7864
0.8663
0.028276
24.7
0.7088
0.7386
0.9771
g go
F gg
Water g+1.
where D N p and DG p +incremental quantities of total surface oil
and total surface gas, respectively, produced during the timestep;
Ao + f
ƪBS ) 5.615(1 *BS * S )r g ƫ
o
w
o
gd
o
*
s o
ƪ
e + ǒA gǓ k * ǒA gǓ k*1) DG p. . . . . . . . . . . . . . . . . . . . . (7.45)
ƫ
S o R s g *g 5.615(1 * S w * S o)
and A g + f
;
)
B gd
Bo
. . . . . (7.41)
g og
and g *o + g
oo
g go
and g *g + g .
gg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.42)
In Eqs. 7.40 through 7.42, D N p and A o are in STB/bbl,
DG p and A g are in scf/bbl, R s is in scf/STB, B o is in bbl/STB, r s is
in STB/scf, and B gd is in ft3/scf. Other quantities used in the material-balance procedure are
E o + 1 ) 5.615r s g *o
E g + R s g *g ) 5.615
Rp +
and
k rg m o B o
k ro m g B gd
,
k rg m o B o
,
k ro m g B gd
ǒ S oǓ k +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.43)
ǒ A oǓ k*1 * ǒDN pǓ k * ƪf(1 * S wi)r s g *oń B gdƫ
ƪfǒ1ńB o * r s g *o ń B gdǓƫ
k
.
^
ƪfǒR s g*gń Bo * 1ńBgdǓƫ
k
. . . . . (7.46)
k
4. Calculate (k rgńk ro) k from (S o) k .
5. Calculate (A o) k , (A g) k , (E o) k , and (E g) k .
6. Calculate DN po , incremental surface oil produced from reservoir oil, where D N po + D G pńE g and E g + 0.5[(E g) k ) (E g) k*1].
7. Calculate D N p, incremental total surface oil produced, where
D N po + D N pńE o and E o + 0.5[(E o) k ) (E o) k*1].
8. Calculate the material-balance error,
. . . . . . . . . . . . . . . . . . . (7.47)
k
7.5 PartialĆDensity Formulation
In 1965, Kniazeff and Naville7 presented the first approach to modeling gas-condensate and volatile-oil systems with a simplified
compositional PVT formulation. They introduced four “partial densities” as PVT parameters in a radial, 1D numerical model to study
the inflow performance of a Middle East gas–condensate field. The
flow and conservation equations were written in terms of mass,
where surface volumes were not considered directly.
Partial densities, ò p , are defined as
ò pij +
4. Calculate (k rgńk ro) k from (S o) k .
5. Calculate (A o) k , (A g) k , (E o) k , and (E g) k .
6. Calculate D N po , incremental surface oil produced from reservoir oil, where D N po + D N pńE o and E o + 0.5[(E o) k ) (E o) k*1].
m ij
,
Vj
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.48)
where m ij +surface mass of Component i in Phase j; V j +reservoir
volume of Phase j; i+g and o+surface gas and oil, respectively; and
j+g and o+reservoir gas and oil, respectively. The four partial densities, ò p , can be expressed as composite terms of MBO properties.
. . . . . . . . . . . . . . . . . . . . (7.44)
10
ǒA gǓ k*1 * ǒDG pǓ k * ƪfǒ1 * S wiǓ ń B gdƫ
9. If e is not sufficiently small, assume a new pressure ( p R) k and
redo Steps 2 through 8.
with m o and m g in cp; R s, R p, and E g in scf/STB; r s in STB/scf;
E o in STB/STB; B o in bbl/STB; and B gd in ft3/scf. PVT properties
and porosity are (g *g ) k functions of pressure only. Application of
these relations is outlined for an oil and a gas-condensate reservoir.
Oil Reservoir.
1. Specify (D N p) k , the total surface oil produced in STB/bbl of
bulk volume.
2. Assume ( p R) k and calculate PVT properties and porosity:
(B o) k , (R s) k , ( m o) k , (g *o ) k , (B gd) k , (r s) k , ( m g) k , (g *g ) k, and (f) k .
3. Calculate oil saturation ( S o) k from Eqs. 7.39 through 7.41.
ǒ S oǓ k +
9. If e is not sufficiently small, assume a new pressure ( p R) k and
redo Steps 2 through 8.
Gas-Condensate Reservoir.
1. Specify (DG p) k, total surface gas produced in scf/bbl of bulk
volume.
2. Assume ( p R) k and calculate PVT properties and porosity:
(B o) k , (R s) k , ( m o) k , (g *o ) k, (B gd) k , (r s) k , ( m g) k , (g *g ) k, and (f) k .
3. Calculate oil saturation ( S o) k from Eqs. 7.39 through 7.41.
e + (A o) k * (A o) k*1 ) DN p .
DG p
,
DN p
k rg
+ fǒ S oǓ ,
k ro
7. Calculate DG p, incremental total surface gas produced, where
DG p + D N po E g and E g + 0.5[(E g) k ) (E g) k*1].
8. Calculate the material-balance error,
òpgg+
0.0763 g gg
,
B gd
ò pog+
350 g og r s
,
B gd
PHASE BEHAVIOR
Gas Injection Parameter, Gi,
Mscf/bbl Oil in Cell
Fig. 7.20—Partial densities vs. pressure for Gas-Condensate
NS-1.
òpgo+
0.0136 g go R s
,
Bo
and ò poo+
62.4 g oo
,
Bo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.49)
with ò p in lbm/ft3, B o in bbl/STB, R s in scf/STB, B gd in ft3/scf, and
r s in STB/scf. Table 7.4 and Fig. 7.20 show the behavior of partial
densities and their relation to MBO properties.
From Eq. 7.47, we see that the variation in surface gravities with
pressure is included directly in the definitions of the PVT properties.
In fact, this is necessary to maintain an exact mass balance. Drohm
and Goldthorpe9 and Drohm et al.10,11 indicate that a similar approach can be used for reservoir simulators on the basis of the MBO
approach. They correct the MBO parameters with surface densities,
which indicates that an exact mass balance can be maintained if the
corrected properties ( B *o, R *s , B *gd, and r *s ) are used instead of the
original parameters ( B o, R s, B gd, and r s ).
B *o +
Bo
,
62.4 g oo
R *s + R s
ǒgg Ǔ ,
go
oo
B *gd +
B gd
,
62.4 g gg
and r *s + r s
ǒgg Ǔ ,
og
Gas Injection Parameter, Gi,
Mscf/bbl Oil in Cell
Fig. 7.21—Variation in black-oil PVT properties with gas-injection parameter Gi (adapted from Ref. 5).
The complexity of some formulations is disturbing because so many
nonphysical quantities are used to correlate compositional effects.
With the increasing speed of compositional simulators and the increase in available computing power, it is difficult to justify the effort
to develop these highly empirical, pseudo-PVT formulations for gasinjection projects where compositional effects are important. If a simplified formulation is used, it should be checked with a compositional
formulation. Tang and Zick19 recently proposed and interesting and
accurate pseudocompositional model with the computational speed
of a black-oil model and the accuracy of an EOS model that is of particular interest in miscible-gas-injection simulations.
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.50)
gg
with densities in lbm/ft3, B o in bbl/STB, R s in scf/STB, B gd in
ft3/scf, and r s in STB/scf. Reservoir models based on the DrohmGoldthorpe or the partial-density approach still do not yield a consistent surface-volume material balance unless surface gravities are
considered pressure dependent.
7.6 Modifications for Gas Injection
Cook et al.5 extend the MBO formulation for vaporizing-gas-injection processes, where a gas-injection parameter, G i, is defined as the
cumulative volume of injection gas entering a grid cell, divided by the
grid-cell volume. PVT properties B o, R s, B gd, and r s are correlated
in tabular form vs. G i (see Fig. 7.21). Lo and Youngren,17 Whitson
et al.,18 and others propose other extensions to the MBO formulation.
BLACK-OIL PVT FORMULATIONS
1. Woods, R.W.: “Case History of Reservoir Performance of a Highly Volatile Type Oil Reservoir,” JPT (October 1955) 156; Trans., AIME, 204.
2. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Scientific Publishing Co., Amsterdam (1978).
3. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast Reservoir
Oils,” JPT (May 1973) 538.
4. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion
Data,” JPT (March 1983) 610; Trans., AIME, 275.
5. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A Beta-Type Reservoir
Simulator for Approximating Compositional Effects During Gas Injection,” SPEJ (October 1974) 471.
6. Spivak, A. and Dixon, T.N.: “Simulation of Gas-Condensate Reservoirs,” paper SPE 4271 presented at the 1973 SPE Annual Meeting,
Houston, 10–12 January.
7. Kniazeff, V.J. and Naville, S.A.: “Two-Phase Flow of Volatile Hydrocarbons,” SPEJ (March 1965) 37; Trans., AIME, 234.
8. Coats, K.H.: “Simulation of Gas-Condensate Reservoir Performance,”
JPT (October 1985) 1870.
11
9. Drohm, J.K. and Goldthorpe, W.H.: “Black Oil PVT Revisited—Use of
Pseudocomponent Mass for an Exact Material Balance,” paper SPE
17081 available from SPE, Richardson, Texas (1987).
10. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing the
Evaluation of PVT Data,” paper SPE 17685 presented at the 1988 SPE
Offshore Southeast Asia Conference, Singapore, 2–5 February.
11. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality of
Data From Standard Gas-Condensate PVT Experiments,” paper SPE
17768 presented at the 1988 SPE Gas Technology Symposium, Dallas,
13–15 June.
12. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,”
U.S. Dept. Commerce, Washington, DC (1929) 97.
13. Golan, M. and Whitson, C.H.: Well Performance, second edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1986).
14. Fetkovich, M.D. et al.: “Oil and Gas Relative Permeabilities Determined From Rate/Time Performance Data,” paper SPE 15431 presented at the 1986 SPE Annual Technical Conference and Exhibition,
New Orleans, 5–8 October.
15. Boe, A., Skjaeveland, S., and Whitson, C.H.: “Two-Phase Pressure Test
Analysis,” SPEFE (December 1989) 604; Trans., AIME, 287.
16. Borthne, G.: “Development of a Material Balance and Inflow Performance for Oil and Gas-Condensate Reservoirs,” MS thesis, U. Trondheim, Norwegian Inst. Technology, Trondheim, Norway (1986).
12
17. Lo, T.S. and Youngren, G.K.: “A New Approach to Limited Compositional Simulation: Direct Solution of the Phase Equilibrium Equations,” SPERE (November 1987) 703; Trans., AIME, 283.
18. Whitson, C.H., da Silva, F.V., and Søreide, I.: “Simplified Compositional Formulation for Modified Black-Oil Simulators,” paper SPE
18315 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, 2–5 October.
19. Tang, D.E. and Zick, A.A.: “A New Limited Compositional Reservoir
Simulator,” paper SPE 25255 presented at the 1993 SPE Symposium on
Reservoir Simulation, New Orleans, 28 February–3 March.
SI Metric Conversion Factors
°API
bbl
ft3
°F
lbm
lbm mol
psi
141.5/(131.5)°API)
+g/cm3
1.589 873
E*01 +m3
2.831 685
E*02 +m3
(°F*32)/1.8
+°C
4.535 924
E*01 +kg
4.535 924
E*01 +kmol
6.894 757
E)00 +kPa
PHASE BEHAVIOR
Chapter 8
GasĆInjection Processes
8.1 Introduction
For the past 50 years, gas injection has been used successfully in
both oil and gas-condensate reservoirs. Hydrocarbon recoveries
have been increased over what could be obtained by primary drive
mechanisms and waterflooding. It was recognized early that the
phase and volumetric behavior of gas/oil systems during gas injection had an important effect on ultimate recovery efficiency. Recovery efficiency is defined as the product of areal and vertical sweep
efficiencies and the microscopic displacement efficiency of the contacted reservoir volume. Fluid properties influence all three components of overall recovery efficiency.
1. Viscosities are found in the definition of mobility ratio, which affects areal and vertical sweep efficiency, including viscous fingering.
2. Phase densities define the degree of gravity segregation, which
in turn affects vertical sweep efficiency by gravity bypassing
(tonguing) in gravity-dominated processes.
3. Interfacial tensions, viscosities, interphase mass transfer (i.e.,
vaporization and condensation), and miscibility affect the residual
oil saturation (ROS) defining microscopic displacement efficiency.
Gas-injection processes are designed to enhance the recovery of
oil. The first application of gas injection was intended simply to
maintain reservoir pressure at a level that would sustain existing
production rates. Another purpose for pressure maintenance in gascondensate reservoirs was to avoid low liquids recovery resulting
from retrograde condensation.
Injection of lean gas consisting mainly of methane or nitrogen
can, by vaporization, recover significant quantities of light and intermediate hydrocarbons (C5 through C12) from reservoir oil. Nitrogen-rich-gas injection can theoretically recover most of the hydrocarbons making up solution gas (C1 through C7). In gas-condensate
reservoirs, lean-gas injection can be miscible if reservoir pressure
is above the dewpoint; otherwise, lean gas can revaporize liquids
that drop out by retrograde condensation, which occurs when reservoir pressure drops below the dewpoint. In oil reservoirs, vaporization at sufficiently high pressure may develop an in-situ gas that becomes sufficiently enriched in intermediate components to displace
the reservoir-oil miscibility; this process is called the vaporizinggas miscible drive.1
Miscibility also can be attained by injecting a gas that is enriched
with liquefied petroleum gases (LPG’s)—mainly propane. Through
phase equilibrium, the injected gas transfers the LPG’s to the reservoir oil, which is typically deficient in these intermediate components. Repeated contacts with enriched gas develops an oil that may
become miscible with the injection gas; this process is traditionally
called the enriched-gas, or condensing-gas, miscible drive.1
GAS-INJECTION PROCESSES
Zick2 shows that another mechanism may develop from injection
of enriched gas that results in miscible-like recoveries (u95%)
without necessarily achieving a miscible condition. The combined
condensing/vaporizing mechanism he describes is a process that exhibits a sharp near-miscible “front.” A condensing mechanism occurs just ahead of the front, and a vaporizing mechanism trails the
front. A practical consequence of this mechanism is that a lower enrichment level can be used for the injection gas than would be estimated from the traditional interpretation of the enriched-gas miscible drive process.
Miscible displacement also can be achieved by a miscible-slug
drive process, where a slug of propane-rich mixture is injected and
mixes miscibly with the reservoir oil on first contact. After a sufficient volume of slug has been injected [5 to 20% of reservoir pore
volume (PV)], a dry gas is injected to drive the slug. The dry gas may
be followed by continuous water injection or by a water-alternatinggas (WAG) injection sequence.
Since the 1970’s, CO2 flooding has been considered one of the
most promising gas-injection processes in the U.S.3-5 Major investments have been made to transport large quantities of CO2 in pipelines from CO2 reservoirs in Colorado and New Mexico to oil reservoirs in west Texas and Oklahoma and from Mississippi to
Louisiana. CO2 flooding has been used successfully in a wide variety of oil reservoirs, with stock-tank-oil gravities ranging from 15
to 45°API, reservoir temperatures from 80 to 300°F, reservoir pressures from less than 1,000 to more than 4,000 psia, and in both sandstone and carbonate formations that vary in thickness from less than
ten to more than several hundred feet.
Recovery mechanisms involved with CO2 flooding include oil
swelling, oil-viscosity reduction, vaporization of intermediate to heavy
hydrocarbons (C5 through C30), and development of multicontact miscibility. Other phase behavior exhibited by CO2/oil systems includes
asphaltene deposition and three-phase [vapor/liquid/liquid (VLL)] behavior in low-temperature systems. Phase and volumetric behavior are
important in both miscible and immiscible CO2 processes.
All the gas-injection methods mentioned can be initiated as secondary or tertiary projects (i.e., following, in conjunction with, or
as a replacement for a waterflood). The occurrence of large water
saturations in tertiary and WAG processes does not appear to influence the role of phase and volumetric behavior on these EOR processes. However, CO2 solubility in water may affect oil recovery if
the loss of CO2 to connate and injected water is significant.
1
Fig. 8.2—Phase behavior of the methane/butane/decane ternary,
including critical-pressure curves for mixtures of fixed composition as functions of temperature (from Refs. 9 and 10).
Fig. 8.1—Phase behavior of ethane/heptane system, including
critical locus defining MMP conditions (from Ref. 8).
8.2 Miscibility and Related Phase Behavior
Miscible gas displacement typically is characterized by high recoveries in slim-tube displacement experiments. These recoveries are usually greater than 90% and somewhat less than the 100% theoretical
recovery expected for “first-contact”-miscible displacement. The
small ROS (2 to 10% of PV) is an immobile, highly viscous oil consisting mainly of heavy, nonvolatile hydrocarbons. Miscible gas displacement may also cause deposition of a solid asphaltene precipitate
that can alter wettability and water injectivity.6,7 As a thermodynamic
condition, miscibility is defined as the condition when two fluids are
mixed in any proportion and the resulting mixture is a single phase.
For example, gasoline and kerosene are miscible at room conditions,
whereas stock-tank oil and water are clearly immiscible.
rameter C + z C4ń(z C4 ) z C10), the locus of critical pressures
indirectly defines the condition of miscibility as a function of temperature. For a specific temperature, Fig. 8.2 gives the composition
dependence of critical pressure. Choosing, for example, 280°F and
2,000 psia, the composition corresponding to this critical condition
is z C1 + 0.5 and C + 0.85 ( z C4 + 0.42, z C10 + 0.08). At 280°F
and 3,000 psia, the critical composition is z C1 + 0.68 and
C + 0.65 ( z C4 + 0.21, z C10 + 0.11).
Knowing only the critical composition of a ternary system at a specific temperature and pressure does not directly determine whether
two mixtures of the three components will be miscible. Graphically,
a ternary composition diagram can be used to determine whether two
mixtures are first-contact miscible or whether the two mixtures can
develop miscibility. Fig. 8.3 shows the ternary diagram for the meth-
8.2.1 Binary Systems. For a binary system, the condition of miscibility is readily defined on a pressure/temperature ( p-T) diagram.
The dashed line in Fig. 8.1 represents the locus of critical points for
all mixtures of ethane and heptane. The critical-locus curve for a
binary system will always enclose the two-phase region for all possible mixtures of the two components. Thus, for a binary mixture at
a specific temperature, the pressure on the critical-locus curve represents the minimum pressure where miscibility can occur independently of overall composition. At all pressures greater than this
minimum miscibility pressure (MMP), any mixture of the binary
will form a single phase. Fig. 8.1 also shows that temperature increases MMP for a binary mixture at lower temperatures, but the effect reverses at higher temperatures, where MMP decreases with increasing the temperature.
8.2.2 Ternary Systems. The condition of miscibility for a ternary
system can also be depicted on a p-T diagram. Fig. 8.2 shows the
curves defining critical pressure vs. temperature for the ternary system methane/butane/decane (C1/n-C4/n-C10). For a specific composition defined in terms of mole percent methane z C1 and the pa2
Fig. 8.3—Ternary composition diagram for C1/n-C4/n-C10 system
at 280°F and 2,000 psia (data from Ref. 10).
PHASE BEHAVIOR
Fig. 8.4A—Path of developed miscibility by the vaporizing-gas
miscible drive process for C1/n-C4/n-C10 system.
Fig. 8.4B—EOS calculated slim-tube profiles for the vaporizing-gas miscible drive process for C1/n-C4/n-C10 system
(adapted from Ref. 2).
ane/butane/decane system at 280°F and 2,000 psia. The critical composition determined from Fig. 8.2 is shown as the critical point, C.
Other compositional data for equilibrium systems at the same condition are plotted on the ternary diagram. These data define the phase
envelope enclosing all compositions that will split into two phases
when brought to this specific pressure and temperature. The upper
curve of the phase diagram defines the dewpoint curve, while the lower curve defines the bubblepoint curve. The dewpoint and bubblepoint curves join at the critical composition, C.
A tie-line is a straight line on a ternary diagram joining an equilibrium-vapor composition with its equilibrium-liquid composition
Fig. 8.5A—EOS calculated slim-tube profiles for the enrichedgas miscible drive process for C1/n-C4/n-C10 system (adapted
from Ref. 2).
Fig. 8.5B—Path of developed miscibility by the enriched-gas
miscible drive process for C1/n-C4/n-C10 system.
GAS-INJECTION PROCESSES
3
(e.g., Line XY). Any system with an overall composition lying on
this tie-line will split into the same equilibrium-liquid and -vapor
compositions defined by X and Y (e.g., overall compositions ZA and
ZB ). Fig. 8.3 shows three tie-lines inside the phase envelope. Every
point on the two-phase envelope is connected to another point on the
envelope by a tie-line. A limiting tie-line can be drawn through the
critical composition (dashed line in Fig. 8.3). This critical tie-line
determines whether two mixtures in a ternary system can develop
miscibility by a multiple-contact process.
Strictly speaking, two fluids are first-contact miscible if the line
connecting the two compositions does not pass through the two-phase
envelope on a ternary diagram. In Fig. 8.3, the G1/G2, G2/O2, and
O1/O2 mixtures are first-contact miscible and the G1/O1, G1/O2, and
G2/O1 mixtures are not. Some systems can develop miscibility by a
multiple-contact process. The criterion for developed multicontact
miscibility in a ternary system is that the two original mixtures lie on
opposite sides of the critical tie-line. The following paragraphs describe two methods of developing miscibility in a ternary system.
G1 and O2 can develop miscibility by the vaporizing-gas miscible
drive process. Here, intermediate and heavy components (C4 and C10)
are vaporized from the original oil, O2, into the lean gas, G1, making
a richer gas that contacts O2 and develops an even richer gas that again
contacts O2. Finally, the gas composition approaches Critical Composition C, which is miscible with O2. Fig. 8.4A shows the path of developed miscibility for this process on a ternary diagram. Fig. 8.4B
shows simulated slim-tube profiles of oil saturation, phase densities,
and K values for the vaporizing-gas miscible drive of the methane/butane/decane system determined with the Peng-Robinson11 EOS.
G2 and O1 can develop miscibility by the traditional enriched-gas
miscible drive process. Here the intermediate component (C4) in the
original gas, G2, transfers to oil, O1. This enriched oil is made even
richer by new contacts with G2 until the oil is modified so that its
composition approaches Critical Composition C. This developed
critical “oil” is miscible with G2. Fig. 8.5A shows the path of developed miscibility for this process on a ternary diagram. Fig. 8.5B
shows simulated slim-tube profiles of oil saturation, phase densities, and K values for enriched-gas miscible drive of the methane/
butane/decane system determined with the Peng-Robinson EOS.
8.2.3 Pseudoternary Diagrams for Multicomponent Systems.
For a true three-component system, first-contact and developed
miscibility can be determined uniquely from a ternary diagram at a
specific pressure and temperature. Pseudoternary diagrams are also
used for multicomponent mixtures, where several components are
grouped together and represented at each apex on the ternary diagram. This method is used despite the inherent limitation that multicomponent phase behavior cannot be represented uniquely with a
ternary diagram. Strictly speaking, a ternary representation of a
multicomponent system is valid only if the relative amounts of all
components defining each pseudocomponent remain constant. This
condition cannot be satisfied for oil systems, but the graphical representation is still used.
Methane, N2, and CO2 are usually treated as the light pseudocomponent in a pseudoternary diagram, with ethane through hexanes
treated as the intermediate pseudocomponent and heptanes-plus as
the heavy pseudocomponent. Sometimes CO2 is included with the
intermediate components. The general characteristic of pseudoternary phase behavior described earlier (namely, that developed miscibility can be achieved if the injection gas and reservoir oil lie on
opposite sides of the critical tie-line) are applied directly to multicomponent systems.
Unlike the pseudoternary phase envelope for a three-component
system, the pseudoternary phase envelope for a multicomponent
system is not unique. It must be developed from a sequence of multiple contacts, where the multicontact procedure starts with the original injection gas and the reservoir oil. Thereafter, the procedures for
vaporizing- and condensing-gas drives are different. A forwardcontact procedure is used for the vaporizing-gas drive, while a backward-contact procedure is used for the enriched-gas drive.
The forward-contact procedure starts by mixing the injection gas
with the reservoir oil to obtain a two-phase mixture. The equilibrium-gas and -oil compositions provide two points and a tie-line on
4
the pseudoternary diagram. The gas from the two-phase mixture is
then removed and put into contact with original reservoir oil to form
a new two-phase mixture, providing two more points and another
tie-line on the pseudoternary diagram. The process of removing
equilibrium gas and mixing it with original reservoir oil is repeated
until either (1) the enriched gas becomes miscible with the original
reservoir oil or (2) the compositions of the equilibrium gas and equilibrium oil no longer change. If Condition 1 is achieved, the process
is multicontact miscible and most of the phase envelope is established up to the critical point. If Condition 2 is achieved, the process
is not multicontact miscible and only part of the phase diagram is
established. When miscibility is not achieved, the reservoir oil is located on an extension of a tie-line with the equilibrium mixtures and
no further component exchange is achieved by mixing the equilibrium gas with the original reservoir oil.
The pseudoternary diagram for the traditional enriched-gas drive
process is developed by a backward-contact procedure. This starts by
mixing the injection gas with reservoir oil to obtain a two-phase mixture. The equilibrium-gas and -oil compositions determine a point
and a tie-line on the pseudoternary diagram. The equilibrium oil is
then put into contact with the original injection gas to form a new twophase mixture, yielding another point and tie-line on the pseudoternary diagram. The process of mixing altered equilibrium oil with original injection gas is repeated until either Condition 1 or 2 (described
in the preceding paragraph) is achieved. Interpretation of the miscibility condition is the same as that for the vaporizing-gas drive process.
Zick2 claims that the pseudoternary representation of enrichedgas injection may lead to erroneous interpretation of the actual recovery mechanism. He further claims that the traditional enrichedgas miscible drive (developed by the multicontact process just
described) may rarely, if ever, occur in reservoir systems. His observations are covered in more detail in the Sec. 8.4. On the other
hand, pseudoternary representation of the vaporizing-gas miscible
drive process probably gives a reasonable description of the actual
displacement mechanism.
Quaternary diagrams have also been used to describe multicontact displacement in multicomponent systems; however, the additional dimension added by the fourth component makes this graphical representation more difficult to understand. Also, the
“uniqueness” (i.e., oversimplification) of a single critical tie-line on
a ternary diagram is no longer valid with a quaternary representation. In their discussion of N2 in a vaporizing-gas miscible drive process (Fig. 8.6), Koch and Hutchinson12 give perhaps the most illustrative use of a quaternary diagram.
8.2.4 Slim-Tube Displacements. A single definition of multicontact miscibility has not been accepted for multicomponent systems.
Most definitions relate to recovery performance curves from labora-
Fig. 8.6—Illustration of phase relations for vaporizing-gas miscible drive process with N2 as injection gas (from Ref. 12).
PHASE BEHAVIOR
H2O From
PositiveDisplacement
Pump
CO2 Supply Cylinder
Sand-Packed Coil
CO2
Test OIl
Solvent
Backpressure
Regulator
Capillary-Tube
Sight Glass
Well Test Meter
100-cm3 Burette
Fig. 8.8A—Experimental slim-tube for CO2 displacement of a
west Texas 30°API oil showing effect of temperature on recovery-pressure behavior (adapted from Ref. 13).
100
Well-Designed Slim Tubes: Miscible Recoveriesu95%
90
Fig. 8.7—Schematic of a slim-tube displacement apparatus;
sand-packed coil consists of 40-ft-long, 1/4-in.-OD stainlesssteel tubing packed with 160/200-mesh Ottawa sand (adapted
from Ref. 13).
80
tory displacement tests. A slim-tube apparatus is used in the displacement experiments. Most slim tubes consist of 0.25-in.-outerdiameter coiled tubing, from 25 to 75 ft in length, packed with
uniform sand or beads and housed in a constant-temperature container. Fig. 8.7 is a schematic of a slim tube. Orr et al.14 summarize
slim-tube characteristics described in various miscible studies.
Slim-tube results are interpreted by plotting cumulative oil recovery vs. PV of gas injected. Two recoveries are usually reported, at
breakthrough and after injection of 1.2 PV of gas. To determine the
MMP, several slim-tube experiments are conducted at varying displacement pressures. Recovery at 1.2 PV of gas injected is then
plotted vs. displacement pressure. For immiscible displacements,
where relative permeabilities and viscosities influence the recovery
process, recovery increases with pressure.
The recovery-pressure curve starts to flatten when the displacement becomes near miscible. Depending on the type of displacement process, temperature, injection gas, and other factors, the transition from immiscible to miscible may be abrupt or gradual (Figs.
8.8A and 8.8B). Table 8.1 gives reservoir-oil and -gas compositions
for Fig. 8.8B. Choice of the “break point” defining MMP is somewhat arbitrary. Some investigators use a specific recovery factor,
such as 90% at 1.2 PV of gas injected, to define MMP. For CO2/oil
systems, Holm and Josendal16-19 use a definition of MMP that requires 80% recovery at breakthrough and 94% recovery at a producing gas/oil ratio (GOR) of 40,000 scf/bbl (occurring at approximately 1.1 to 1.3 PV injected gas).
Color change and lack of multiphase production from the slim-tube
apparatus have also been used to define the MMP. Yellig and Metcalfe13 give a particularly good discussion of criteria for defining
MMP on the basis of slim-tube data for CO2 systems. Fig. 8.9 shows
recovery vs. PV of gas injected for a CO2 miscible displacement;
changing colors of the produced oil are noted on the curve (from dark
to red to orange to yellow to clear). Fig. 8.10 shows the qualitative
character of produced fluids from a series of slim-tube tests at immiscible and miscible conditions for an enriched-gas displacement. The
solid line indicates fluid density based on photoelectric-cell output,
and shading represents two-phase production.
It is generally accepted that slim-tube displacements yield the most
reliable information for defining true multicontact miscibility. Although the slim-tube-determined miscibility condition is affected almost exclusively by the phase behavior of the fluids being studied,
this miscibility condition is not the same as “thermodynamic misci-
50
GAS-INJECTION PROCESSES
Low Miscible Recoveries
Indicate Slim-Tube
Equipment Design Problems
70
60
40
21
23
25
27
29
31
33
35
37
39
41
43
Displacement Pressure, MPa
Fig. 8.8B—Experimental slim-tube results for high-pressure displacement of a reservoir oil showing effect of injected lean-gas
composition on recovery-pressure behavior (from Ref. 15).
TABLE 8.1—RESERVOIR-OIL AND INJECTION-GAS
COMPOSITIONS FOR FIG. 8.8B
Reservoir
Component
Oil
Gas 1
Gas 2
Gas 3
H2 S
0.00
0.0
0.00
0.00
N2
0.06
1.2
0.35
0.31
CO2
2.71
0.0
0.00
0.00
C1
34.66
93.3
81.71
69.61
C2
6.96
3.0
9.03
12.18
C3
6.46
1.1
4.31
8.83
i-C4
1.54
0.0
0.84
1.63
n-C4
4.09
0.7
1.63
3.42
i-C5
1.87
0.0
0.54
1.07
n-C5
2.57
0.7
0.59
1.22
C6
3.58
0.55
0.00
C7
3.66
0.00
1.73
C8
3.46
0.43
C9
3.13
C10
2.61
C11+
22.64
5
Clear and Yellow
Yellow
Orange
Red
Dark
Fig. 8.9—Experimental slim-tube recovery vs. PV injected CO2
curve indicating change in color (as viewed in sight glass) of
produced fluid after breakthrough (adapted from Ref. 13).
bility.” Slim-tube experiments are relatively fast and simple to conduct, they do not require expensive equipment, and the experimental
procedure can be automated readily with standard data-acquisition
tools. The rising-bubble apparatus has also been suggested as a method to arrive at an indication of true miscibility21-24; however, we are
skeptical of this claim for the condensing/vaporizing drive mechanism. Zhou and Orr25 appear to share this skepticism.
8.2.5 Multiple-Contact Pressure/Volume/Temperature (PVT)
Experiments. Although the slim-tube displacement experiment is
the preferred method for determining the MMP of an injection gas,
it does not provide controlled measurements of the system phase
and volumetric behavior. Various PVT experiments can be used to
supplement slim-tube measurements for miscible displacement
projects. Also, PVT experiments provide the only means of obtaining important data, such as viscosities, densities, compositions, and
K values. Multicontact PVT data are particularly useful for tuning
an equation of state (EOS) or any other PVT model that may be used
in reservoir simulation.
PVT experiments designed for gas-injection processes involve
multiple contacts of injection or equilibrium gas with original reservoir oil or previously contacted equilibrium oil. The swelling test
(Fig. 8.11) is the most common multicontact PVT experiment. In
this experiment, injection gas is mixed with original reservoir oil in
varying proportions, with each mixture quantified in terms of a molar percentage of injection gas (e.g., 20 mol% CO2 indicates that 0.2
moles of CO2 has been mixed with 0.8 moles of reservoir oil). The
saturation pressure and phase volumes at more than and less than the
saturation pressure are measured for each mixture. The data are
presented in a pressure/composition ( p-x) diagram, as in Figs. 8.12
and 8.13. Pressure/volume plots are also used, usually as crossplots
to determine quality lines on a p-x diagram. Occasionally, compositions of equilibrium-oil and -gas phases are determined for some
mixtures in a swelling test (usually those at pressures close to the expected operating pressure of the injection project and those close to
the critical point on the p-x diagram).
The forward- and backward-contact PVT experiments (Fig. 8.14)
also provide useful phase and volumetric data for gas-injection studies. The forward-contact experiment follows the procedure described
earlier for the vaporizing-gas miscible displacement process. That is,
the equilibrium gas from each contact is removed and mixed with
more of the original reservoir oil. The amount of gas mixed with original oil at each contact may vary, but the amount should not affect the
results significantly. The developed gas should eventually reach miscibility with the original reservoir oil if the experiment is conducted
at a pressure greater than the MMP. Otherwise, the forward-contact
6
Fig. 8.10—Produced wellstream character indicated by solid line
representing density from photoelectric-cell output; shading indicates two-phase production (from Ref. 20).
experiment gives information about how efficiently the developed
gas vaporizes the original oil without achieving miscibility.
The backward-contact experiment follows the procedure described for the enriched-gas miscible drive process. Here the equilibrium oil resulting from a given contact is mixed with more of the
original injection gas. According to the traditional interpretation of
the enriched-gas miscible drive process, miscibility should develop
between the original injection gas and the altered reservoir oil. Benham et al.27 present backward-contact PVT results that indicate
miscibility can be achieved by this process (Fig. 8.15). On the other
hand, Zick2 presents backward-contact PVT experiments and EOS
simulations that convincingly show that miscibility is not achieved
by this process even at pressures considerably higher than the MMP
determined by slim-tube experiments (Figs. 8.16 and 8.17).
The backward-contact experiment also can be used to investigate
revaporization of retrograde condensate by lean injection gas. Figs.
8.18 and 8.19 show swelling and backward-contact experimental
data reported by Vogel and Yarborough.28 Here, N2 was used in the
backward-contact experiment to revaporize retrograde liquid that
formed when a lean-gas condensate was brought into contact with
50% N2 in a swelling test. Vogel and Yarborough also give experimental results for the effect of N2 on a reservoir oil. Nitrogen was
mixed with the original reservoir oil in varying proportions (0, 0.144,
0.5, and 1.5 PV of N2 per PV of original reservoir oil). For a given
N2/oil mixture, the system was brought to equilibrium at a specified
pressure. The equilibrium gas was completely removed and discarded. The equilibrium oil was analyzed chromatographically, and
a differential liberation experiment was conducted on the oil to determine solution gas/oil ratio and oil volumetric properties. Figs. 8.20
through 8.24 present some of the results from these experiments.
PHASE BEHAVIOR
Fig. 8.11—Schematic of swelling test.
8.2.6 Calculation Algorithms. Several methods have been proposed for calculating MMP by multicontact calculations with an
EOS or K-value model.27,29,30 These methods typically involve either a forward- or backward-contact mixing procedure, with the
intention of simulating either a vaporizing- or condensing-gas drive
process, respectively.
Metcalfe et al.31 proposed a more rigorous calculation approach
based on Cook et al.’s32,33 multicell vaporization model. With this
approach, fluid mixing along a series of connected “cells” is used
to simulate the development of miscible conditions with time (Fig.
8.25). Initially, all cells are filled with reservoir-fluid composition.
Then, a volume of injection gas (approximately 20% of a cell volume) is mixed with the contents of the first cell and brought to equilibrium. Part of the resulting equilibrium gas and oil is mixed with
the contents of Cell 2 and brought to equilibrium, part of the resulting equilibrium gas and oil from Cell 2 is mixed with Cell 3, and so
on. Finally, production is recorded from the last cell; typically,
approximately 50 cells are used.
This series of calculations constitutes one “batch” or “timestep.”
The calculations are repeated with a new volume of injected gas, and
the compositional changes in one or more cells are monitored with
time. Metcalfe et al. plot the results on a ternary diagram to apply
the critical tie-line approach for establishing the condition of miscibility. Injection-gas composition can change at each timestep, there-
Bubblepoint
curve
,
vol%
liquid
BUBBLEPOINTS
DEWPOINTS
N2 in Painter Reservoir, mol%
Fig. 8.12—Experimental p-x diagram for mixtures of lean natural
gas with a Block 31 Devonian reservoir oil (adapted from Ref. 1).
GAS-INJECTION PROCESSES
Fig. 8.13—Experimental p-x diagram for mixtures of N2 with a
Painter reservoir oil (adapted from Ref. 26).
7
INJECTION
GAS
CRUDE
INJECTION
GAS
CRUDE
CRUDE
INJECTION
GAS
INJECTION
GAS
CRUDE
Fig. 8.14—Schematic of forward- and backward-contact experiments.
by allowing the study of miscible-slug displacement and the effect
of driving an enriched gas with a cheaper lean gas.
Metcalfe et al. propose three methods for determining which
phases are passed from one cell to the next and the amount of each
phase passed. The first method, originally proposed by Cook et
al.,32,33 passes all equilibrium gas from cell to cell, simulating the
vaporizing-gas (forward-contact) process. The second method
passes only enough gas and oil to the next cell to ensure that the remaining mixture in the current cell fills the cell volume. The third
passes equilibrium gas and oil according to the mobility ratio
(krg /kro ) (mo /mg ), where the relative permeability ratio krg /kro is
entered as a function of saturation. For any miscible displacement
process, the second and third methods should give the same conditions of miscibility in the limit of small injection volumes and a large
number of cells. The first method is valid only for a vaporizing-gas
drive mechanism.
Short of simulating a slim-tube displacement, the multicell calculation approach is probably the best generalized scheme for determining the conditions required to develop miscibility. It should give
the same conditions of developed miscibility as slim-tube results if
the multicell calculations are interpreted correctly. Calculation
methods based strictly on forward- or backward-contact procedures
are not recommended.27,29,30
Johns et al.34,35 present analytical simulation results based on the
method of characteristics for three- and four-component systems
that verify the mechanisms of the condensing/vaporizing drive
mechanism originally described by Zick.2 Practically, their approach is limited to four-component systems and is more difficult
to program than a one-dimensional (1D) slim-tube or batch-type experiment. Wang and Orr36 recently proposed a generalized algorithm for computing MMP on the basis of complex tie-line analysis
Fig. 8.16—Experimental slim-tube recoveries at 1.2 PV injected
gas vs. injection pressure for Reservoir Oil A displaced by Solvent A (adapted from Ref. 2).
8
Intermediates, mol%
Fig. 8.15—Experimental backward-contact PVT data for enriched-gas/reservoir-oil system (adapted from Ref. 27).
founded in the theory fo the method of characterisitics; the CN
mechanism can be the method of developed miscibility.
8.3 LeanĆGas Injection
Lean-gas injection with methane- and N2-rich gases has been used
for reservoir management during primary production, as an alternative to waterflooding for secondary recovery, and in gravity-stable
tertiary projects. Successful projects include (1) pressure maintenance in oil reservoirs to maintain productivity, sometimes by developing an artificial gas cap; (2) gravity-stable displacement in dipping, high-permeability oil reservoirs; (3) reservoir-voidage
replacement to maintain the oil/water contact in a strong-waterdrive
reservoir; (4) recovery of upstructure “attic” oil and gas in strongwaterdrive reservoirs; (5) high-pressure multicontact miscibility in
oil reservoirs; and (6) partial and full pressure maintenance in gascondensate reservoirs.
Some of the more important justifications for lean-gas injection
include gas availability, better injectivity in low-permeability reservoirs, conservation or environmental constraints, and superior oil
recoveries compared with alternative EOR methods. Not all applications of lean-gas injection require special treatment of phase
Fig. 8.17—EOS-calculated multiple backward-contact PVT experiment for Oil A and Solvent A at 900 psi higher than MMP pressure indicated from experimental and simulated slim-tube results (adapted from Ref. 2).
PHASE BEHAVIOR
0% (gas only, no added N2)
10% cumulative added N2
30% cumulative added N2
50% cumulative added N2
Liquid, vol%
0% (gas only, no added N2)
10% cumulative added N2
30% cumulative added N2
50% cumulative added N2
Liquid, vol%
Fig. 8.18—Effect of N2 on the phase behavior of two gas-condensate reservoir fluids: equilibrium flash volumetric expansion tests
run at 381K and no material removed from p-V cell (from Ref. 28).
and volumetric behavior. However, high-pressure injection in oil
reservoirs and gas cycling in partially depleted condensate reservoirs do require detailed knowledge of how the injected gas behaves
with the reservoir fluids.
8.3.1 Vaporizing-Gas Miscible Drive. Several high-pressure, leangas miscible projects have been reported for light oils with stocktank-oil gravities greater than 35°API and with operational flooding
pressures greater than approximately 3,500 psia.1 Lean gases contain
mostly methane or N2, with methane-rich gases also containing
smaller quantities of ethane and C3+ components. Nitrogen-rich
gases include flue gas, consisting of approximately 88% N2 and 12%
CO2, and pure N2 generated from cryogenic air separation.
Lean gases tend to vaporize intermediate hydrocarbons in the C5
to C12 range, depending on the pressure and injection-gas composition. Nitrogen also tends to “trade places” with the solution gas in
an oil, thereby improving natural gas recovery. At sufficiently high
pressure, lean gas can develop an in-situ gas that is sufficiently rich
in intermediate components (C2 through C4) to develop miscibility
with the reservoir oil. Another condition for developed miscibility
by the vaporizing-gas drive process is that the reservoir should not
have an initial free-gas saturation. That is, in a vaporizing-gas drive
mechanism the gas saturation must always be zero ahead of the miscible displacement front.
Pressure is usually the primary design parameter in a vaporizinggas miscible drive project. Other considerations include slug size,
WAG ratio, and producer/injector pattern. Methane-rich injection
gases tend to develop miscibility at slightly lower pressures than N2rich gases, depending mainly on the methane content in the reservoir oil. Also, the price differential between N2 and lean natural gas
can be significant, so the methane/N2 ratio may be a valid design parameter in some lean-gas miscible projects.
Experimental
Calculated
Cumulative PV
N2 Contacted
0.0 PV N2
0.144 PV N2
0.50 PV N2
1.50 PV N2
N2 Injective, cumulative PV
Fig. 8.19—Revaporization of retrograde condensate by multiple
contacts with N2 (adapted from Ref. 28).
GAS-INJECTION PROCESSES
Fig. 8.20—Experimental oil formation volume factors (FVF’s) for
mixtures of N2 with a reservoir oil (adapted from Ref. 28).
9
Experimental
Experimental
Calculated
Calculated
Cumulative PV
N2 Contacted
0.0 PV N2
0.144 PV N2
0.50 PV N2
Cumulative PV
N2 Contacted
1.50 PV N2
0.50 PV N2
1.50 PV N2
0.144 PV N2
0.0 PV N2
Fig. 8.21—Experimental solution GOR data for mixtures of N2
with reservoir oil (adapted from Ref. 28).
Stalkup1 reports only one MMP correlation for the vaporizinggas drive miscible process. This correlation gives MMP as a function of reservoir-oil bubblepoint pressure; reduced temperature of
the reservoir oil; and mass fraction of three groups in the reservoir
oil: (1) C2 through C6 plus CO2 and H2S, (2) C1 plus N2, and (3) C7+.
8.3.2 Gas Cycling. Gas cycling in condensate reservoirs has been
used for the past 50 years to minimize lossses in liquid recovery.
When reservoir pressure drops below the dewpoint pressure in a
gas-condensate reservoir, liquids condense and remain primarily as
an immobile phase. The produced wellstream becomes leaner (as
reflected by a decreasing condensate yield), and overall condensate
recovery may be as low as 15 to 20%. Further depletion at pressures
less than approximately 2,000 psia may revaporize some of the
“lost” retrograde condensate, but this additional recovery is usually
not significant.
To maximize liquid recovery, reservoir pressure should be kept
higher than the dewpoint pressure to avoid retrograde condensation.
Typically, this is achieved by reinjecting produced gas that has been
separated and processed for condensate and natural gas liquids
Fig. 8.22—Experimental oil density data for mixtures of N2 with
reservoir oil (adapted from Ref. 28).
(NGL’s). Because the produced gas is not sufficient to replace the
reservoir voidage caused by production, makeup gas must be obtained to achieve full pressure maintenance. If the reservoir is initially undersaturated (i.e., the initial pressure is greater than the dewpoint pressure), reinjecting only the produced gas is acceptable until
reservoir pressure approaches the dewpoint.
The economics of delaying gas sales to increase condensate recovery may be prohibitive. Alternatives to delayed gas sales include
reinjection of only part of the produced gas, purchasing cheaper
makeup gas for reinjection, and replacing produced-gas reinjection
Experimental Data
Before Cycling With N2
After Cycling With N2
Experimental
Calculated
20 PV
Cumulative PV
N2 Contacted
1.50 PV N2
0.50 PV N2
0.144 PV N2
0.0 PV N2
Fig. 8.23—Experimental oil viscosity data for mixtures of N2 with
reservoir oil (adapted from Ref. 28).
10
Fig. 8.24—Change in heptanes-plus distribution for oil that has
been in contact with 20 PV of N2: effect of cycling, simulated true
boiling point analysis by temperature-programmed gas chromatography (adapted from Ref. 28).
PHASE BEHAVIOR
(a)
Gas to Next Cell
Gas
Gas
Oil
Oil
Oil
Oil
Original
Cell
Condition
(b)
Final
Cell
Condition
Gas (and Oil) to Next Cell
Gas (and Oil)
Gas
Oil
Oil
Oil
Oil
Original
Cell
Condition
Final
Cell
Condition
Gas (and Oil) to Next Cell
(c)
Gas (and Oil)
Gas
Oil
Gas
Oil
Original
Cell
Condition
Cell 1
Cell 2
Gas (and Oil)
Gas (and Oil)
Injection
Gas Original
Original
Oil
Oil
Batch 1
Injection
Gas
Batch 2
Injection
Gas
Batch N
Gas (and Oil)
Oil
Gas (and Oil)
Oil
Fig. 8.26—Effect of N2 and lean natural gas on the dewpoint
pressure of a gas-condensate reservoir fluid (from Ref. 37).
Cell NN
ters used to optimize recovery and other factors affecting an enriched-gas drive project.
Gas (and Oil)
Original
Oil
Gas (and Oil)
Oil
Oil
Cumulative Gas Injected, scf/RB
Gas
Oil
Final
Cell
Condition
Oil
Gas (and Oil)
Oil
Gas (and Oil)
Gas (and Oil)
Oil
Fig. 8.25—Schematic of multicell calculation method: (a) stagnant
oil, (b) moving excess oil, and (c) oil and gas moved by phase mobilities (adapted from Ref. 31).
8.4.1 Traditional Mechanism. Some difference of opinion exists
concerning the actual displacement mechanism responsible for high
recoveries reported in slim-tube experiments with enriched gases.
The traditional enriched-gas displacement mechanism is based on
an interpretation of a pseudoternary diagram, where miscibility is
developed by repeated contacts of the injection gas with the oil
found at the point of injection. The following is Benham et al.’s27
description (based on their Fig. 3) of this traditional interpretation
of the enriched-gas miscible displacement process (Fig. 8.27).
“Assuming that a phase diagram of this general shape is an appropriate representation, the mechanism for obtaining miscibility
may be illustrated by reference to Fig. 3. This figure has been pre-
with injection of flue gas or N2. Generating large quantities of N2
cryogenically on location has been demonstrated in several large
gas-cycling projects.1 Nitrogen has also been used as makeup gas
to ensure full pressure maintenance.
In the early 1980’s, studies27,32 showed that N2 caused substantial condensation of liquids when mixed with a gas-condensate mixture (Figs. 8.18 and 8.26). This behavior caused concern that N2
might worsen the problem of retrograde condensation if used to
maintain pressure in condensate reservoirs. Subsequent displacement and multicontact tests showed that practically all the liquid
condensed by initial contact with N2 was revaporized by later contacts with the N2 (Fig. 8.19).28,37-39 Slim-tube recoveries with N2
displacing a gas condensate showed behavior similar to that of
methane-rich gas displacements, with both gases yielding practically 100% total hydrocarbon recovery.
8.4 EnrichedĆGas Miscible Drive
Miscible displacement projects with enriched injection gas are reported in the literature for reservoir oils with stock-tank-oil gravities
ranging from 30 to 45°API.1 Typical flooding pressures range from
1,500 to 4,000 psia. Enriched gases usually contain methane,
ethane, and varying quantities of LPG components C3 through C4.
CO2 also may be found in the injection gas without significantly affecting the miscibility condition. Reservoir displacement pressure
and the degree of LPG enrichment are the two main design parameGAS-INJECTION PROCESSES
Fig. 8.27—Pseudoternary representation of the traditional enriched-gas-miscible drive process (adapted from Ref. 27).
11
Fig. 8.28—Benham et al.26 chart for determining maximum methane content in an injection gas for miscibility to develop according to traditional enriched-gas miscible oil withM C +240 at 3,000
5)
psia (adapted from Ref. 27).
pared to demonstrate the mechanism involved in obtaining miscibility between reservoir fluid represented by Point R and an enriched gas
represented by Point RG. The reservoir fluid is in the two-phase region and has a liquid phase of composition (m) and a vapor phase of
composition (a). As gas is first injected, it will tend to move both liquid and vapor until eventually the gas velocity is greater than the liquid velocity. The first mixing will be between liquid (m) and rich gas
(RG). The over-all composition of this mixture could be Point a. This
mixture separates into two phases represented by Points n and b. As
more rich gas is injected, it displaces the gas (b) and mixes with the
liquid (n). These may mix to an overall composition (b), which separates into liquid (o) and vapor (c). Injection of more rich gas will result
in displacement of the vapor (c) and mixing of the liquid (o) with the
injection fluid (RG) to form the mix (g). This continues until injection
fluid (RG) mixes with the liquid (t), at which time a miscible displacement begins. Injection fluid (RG) miscibly displaces the liquid (t),
which miscibly displaces the liquid (s), which miscibly displaces r,
etc. The gases will also be miscibly displaced by the rich gas; therefore, a completely miscible displacement has been achieved. The liquids will gradually build up in saturation with displacement until a
completely single-phase miscible displacement is achieved.
“It may be shown that the leanest mixture that will give a miscible
displacement is represented by a point on the extension of the limiting tie-line (A-B), which passes through the critical point (C).”
Benham et al. use this interpretation of the displacement mechanism to develop a series of working curves for estimating the degree
of enrichment required to attain MMP for a given reservoir oil. Their
graphical correlations require (1) average molecular weight of the
reservoir-oil C5+ mixture, (2) average molecular weight of the C2+
components that will be used to enrich the injection gas, and (3) reservoir temperature. With these three data, the appropriate charts are
entered to obtain the allowable methane concentration in the injection gas. Each chart represents an MMP; charts are provided for
MMP’s of 1,500, 2,000, 2,500, and 3,000 psia (Fig. 8.28). A plot of
LPG enrichment vs. MMP can then be made for design calculations.
Zick2 reports an MMP of 3,100 psia at 185°F for his Reservoir
Fluid A with an injection gas consisting of 39 mol% methane (20%
methane mixed with 80% Solvent A containing 23.5 mol% C1). Fig.
8.29 plots slim-tube recovery at 1.2 PV gas injected vs. dilution of the
solvent with pure methane. With M C2)+40 for the reported solvent
and M C5)+260 for the reservoir oil, the Benham et al. charts give
a maximum methane content for the injection gas somewhat greater
than 50 mol%. That is, the Benham et al. charts indicate that MMP
can be achieved at 3,000 psia with the solvent diluted 35% with methane. Fig. 8.29 indicates that the experimental slim-tube recovery is
only 65% for this injection-gas composition. The Kuo16 MMP correlation predicts a similar overestimation of methane dilution.
8.4.2 Combined Condensing/Vaporizing Mechanism. Zick proposes an alternative mechanism to explain the miscible-like recoveries
that can be achieved by displacing a reservoir oil with enriched gas. The
12
Fig. 8.29—Experimental slim-tube recoveries at 3,100 psig as
functions of solvent dilution with methane for Reservoir Oil A (depleted to 3,000 psig) and Solvent A at 185°F (adapted from Ref. 2).
mechanism is a combination of (1) a leading front that enriches original
oil with light intermediates found in the original injection gas and
middle intermediates (C5 through C30) that have been vaporized from
the reservoir oil behind the front and (2) a trailing front of injection gas
that vaporizes middle-intermediate components. A sharp transition
zone separates condensing and vaporizing fronts. This transition zone
is near miscible, or perhaps miscible in the absence of dispersion, recovering practically all the reservoir oil with only a small ROS.
Fig. 8.30—EOS-calculated slim-tube profiles for condensing-/
vaporizing-gas drive of Reservoir Oil A by Solvent A (adapted
from Ref. 2).
PHASE BEHAVIOR
-
Four-Phase
Injection Gas, mol%
Fig. 8.31—Multiphase behavior for mixture of 81.72 mol% (67.99
vol%) enriched driving gas (32% C1, 37% C2, and 30% C3) and a
reservoir oil at pt2,000 psia and 105°F (adapted from Ref. 40).
Fig. 8.30 shows the profile of oil saturation, phase densities, and
K values for an enriched-gas displacement calculated by an EOS
slim-tube simulator. Five regions are readily identified in this figure.
On the basis of the proposed condensing/vaporizing mechanism,
these five regions can be summarized as follows.
1. Original oil zone.
2. A leading two-phase front with net condensation of intermediate components. The gas contains light-intermediate components
found in the original injection gas and middle-intermediate components that have been vaporized from the reservoir oil.
3. A sharp transition zone with near-miscible behavior. The front
side of the transition zone (toward Zone B) shows dramatic condensation of intermediate and heavy components. The back side of
the transition zone (toward Zone D) shows highly efficient vaporization of intermediate and heavy components. Only a small ROS
is left behind the transition zone.
4. A trailing front of enriched gas, which vaporizes middle-intermediate components found in the remaining residual oil.
5. A stripped ROS, in equilibrium with the injection gas, remains
behind. Little if any mass transfer occurs here. The residual oil consists of a heavy, nonvolatile material and the components making up
the injection gas.
The net mass transfer of components between the gas and oil
phases is reflected by the slope of the K values plotted vs. distance.
Net condensation from the gas phase into the oil phase occurs where
the slope dKi /dx is negative for middle-intermediate and heavy
components (Zone B). Net vaporization from the oil phase into the
gas phase occurs where the slope dKi /dx is positive for the middleintermediate and heavy components (Zone D).
Zick2 gives a fairly detailed summary of the condensing/vaporizing
mechanism. With experimental and simulation results, he shows that
the traditional enriched-gas miscible drive mechanism cannot explain
miscible-like recoveries for three different reservoir-oil/enriched-gas
systems. His arguments basically hinge on the observation that the oil
that should first become miscible with an enriched gas (i.e., the oil nearest the point of injection) does not become miscible in multicontact
PVT experiments or in simulations of slim-tube displacements.
He writes, “When the enriched gas comes into contact with the
oil, the light intermediates condense from the gas into the oil, making the oil lighter. The equilibrium gas is more mobile than the oil,
so it moves on ahead and is replaced by fresh injection gas, from
which more light intermediates condense, making the oil even lightGAS-INJECTION PROCESSES
Injection Gas, mol%
Fig. 8.32—p-x diagram showing multiphase behavior for an enriched gas (32% C1, 37% C2, and 30% C3) mixed with a reservoir
oil at 105°F (adapted from Ref. 40).
er. If this kept occurring until the oil was light enough to be miscible
with the injection gas, it would constitute the condensing-gas drive
mechanism. However, this is unlikely to occur with a real reservoir
oil. As the light intermediates are condensing from the injection gas
into the oil, the middle intermediates are being stripped from the oil
into the gas. Since the injection gas contains none of these middle
intermediates, they cannot be replenished in the oil. After a few contacts between the oil and the injection gas, the oil becomes essentially saturated in the light intermediates, but it continues to lose middle
intermediates, which are stripped out and carried on ahead by the
mobile gas phase. The light intermediates of the injection gas cannot
substitute for the middle intermediates the oil is losing. So after the
first few contacts make the oil lighter by net condensation of [light]
intermediates, subsequent contacts make the oil heavier by net vaporization of [middle] intermediates. Once this begins to occur, the
oil no longer has a chance of becoming miscible with the gas. Ultimately, all the middle intermediates are removed and the residual oil
will be very heavy, containing only the heaviest, nonvolatile fraction and the components present in the injection gas.”
Zick goes on to explain how high recoveries can be obtained with
enriched-gas displacement without necessarily achieving true miscibility. Regardless of whether true miscibility develops, he insists that
the miscibility (or near miscibility) that does occur is not developed
according to the traditional enriched-gas drive mechanism (i.e., between the injection gas and the oil at the point of injection). Instead,
he proposes the combined condensing/vaporizing mechanism. He
claims that reaching miscibility by the traditional enriched-gas process requires higher displacement pressures (or higher enrichment
levels) than the MMP (or minimum miscibility enrichment) determined by slim-tube experiments (Figs. 8.17 and 8.20). A characteristic of the combined condensing/vaporizing mechanism is that a
free-gas saturation always exists ahead of the front and that some
ROS is found behind the front.
Novosad and Costain21 and Novosad et al.22 describe a displacement mechanism for enriched-gas drive that differs from both the
13
TABLE 8.2—CO2 PHYSICAL PROPERTIES
M (g+1.52)
44.01
Tc , °F
88
pc , psia
1,070
ò c, gmńcm (lbmńft )
3
3
0.469 (29.2)
Zc
0.274
w (Pitzer acentric factor)
0.239
Tb ,°F, “dry ice” at 1 atm
*110
CO2 equivalent
1 ton, Mscf
1 lbm, scf
17.2
8.6
condensing-gas and the combined condensing-/vaporizing-gas drive
mechanisms. On the basis of their interpretation, they propose a simple rising-bubble apparatus to determine the enrichment level required to develop miscibility for a given oil. This experimental technique implies, however, a type of vaporizing-gas drive mechanism
that would not seem to apply for most enriched-gas displacements.
Even so, the experimental results they provide seem to give reasonable conditions of developed miscibility for the highly undersaturated
oils used in their studies.
8.4.3 Multiphase Behavior. Enriched-gas injection at low temperatures may yield complex multiphase VLL/solid (VLLS) behavior.
Shelton and Yarborough40 present a thorough study of multiphase
behavior for a reservoir oil in contact with a rich gas consisting of
32% methane, 37% ethane, and 30% propane at 105°F. Figs. 8.31
and 8.32 show some of their study results. The multiphase VLL behavior and asphaltene/wax precipitation are strikingly similar to
those of CO2/oil systems at the same temperature.40,41
Although experimental evidence is lacking, multiphase behavior
probably can be anticipated when the system temperature is not
Fig. 8.34—CO2 Z factor (from Refs. 42 and 43).
14
Fig. 8.33—CO2 density (from Refs. 42 and 43).
Fig. 8.35—CO2 viscosity (from Refs. 42 and 43).
PHASE BEHAVIOR
TABLE 8.3—CO2 DENSITY* (from Ref. 42)
Pressure (bar)
Temperature
(°F)
25
50
75
100
150
200
250
300
68
0.0527
0.1423
0.8100
0.8550
0.9010
0.9335
0.9600
0.9832
86
0.0499
0.1251
0.6550
0.7820
0.8500
0.8887
0.9190
0.9460
104
0.0476
0.1135
0.2305
0.6380
0.7850
0.8415
0.8771
0.9077
122
0.0456
0.1052
0.1932
0.3901
0.7050
0.7855
0.8347
0.8687
140
0.0437
0.0984
0.1726
0.2868
0.6040
0.7240
0.7889
0.8292
158
0.0421
0.0930
0.1584
0.2478
0.5040
0.6605
0.7379
0.7882
176
0.0406
0.0883
0.1469
0.2215
0.4300
0.5935
0.6872
0.7466
194
0.0391
0.0845
0.1381
0.2019
0.3730
0.5325
0.6359
0.7040
212
0.0378
0.0810
0.1305
0.1877
0.3330
0.4815
0.5880
0.6630
230
0.0366
0.0778
0.1239
0.1765
0.3040
0.4378
0.5443
0.6230
248
0.0354
0.0749
0.1187
0.1673
0.2800
0.4015
0.5053
0.5855
266
0.0344
0.0722
0.1141
0.1595
0.2620
0.3718
0.4718
0.5517
284
0.0334
0.0697
0.1094
0.1525
0.2465
0.3470
0.4419
0.5200
302
0.0325
0.0674
0.1054
0.1461
0.2337
0.3267
0.4151
0.4925
320
0.0316
0.0653
0.1018
0.1403
0.2229
0.3089
0.3918
0.4680
*In gm/cm3.
more than approximately 50°F higher than the critical temperature
of the injection gas. For example, the pseudocritical temperature of
the enriched gas used by Shelton and Yarborough was 60°F and significant multiphase behavior was observed at 105°F. CO2 systems
exhibit multiphase behavior up to approximately 130°F, about 40°F
higher than the critical temperature of CO2. Accordingly, an injection gas rich in NGL’s probably experiences multiphase behavior
Fig. 8.36—CO2 phase diagram (from Refs. 42 and 43).
GAS-INJECTION PROCESSES
and asphaltene precipitation at higher reservoir temperatures than
a less enriched gas does.
8.5 CO2 Injection
8.5.1 CO2 Physical Properties. CO2 is a stable, nontoxic compound found in a gaseous state at standard conditions. For petroleum applications, CO2 exists either as a gas or as a liquid-like su-
Fig. 8.37—CO2 FVF (from Refs. 42 and 43).
15
Stock-Tank Oil Molecular Weight
Stock-Tank Oil Specific Gravity
1.06
1.04
1.02
1.00
0.98
0.96
0.94
0.92
0.90
0.88
0.86
0.84
0.82
0.80
0.78
Fig. 8.39—Correlation for swelling of a dead stock-tank oil when
saturated with CO2 (adapted from Ref. 44).
cant. Corrosion in CO2 floods, particularly in WAG projects, requires special attention.
0
0.1
XCO2
0.2
0.3
0.4
0.5
0.6
in Oil With UOP K=11.7
Fig. 8.38—Correlation for solubility of CO2 in dead stock-tank
oils (adapted from Ref. 44).
percritical fluid. Table 8.2 gives the key physical properties of CO2.
Figs. 8.33 through 8.35, respectively, show density, Z factor, and
viscosity of CO2 as functions of pressure and temperature. Table 8.3
gives tabular data for the density of pure CO2.
Fig. 8.36 shows the phase diagram of CO2 with an extrapolation
of the critical isochor. The critical isochor defines supercritical
conditions where phase density equals the critical density of 0.47 g/
cm3. Later, we show that CO2 density at reservoir conditions is the
main parameter that determines MMP of CO2 with reservoir oils. In
fact, the critical isochor drawn in Fig. 8.36 gives a close approximation of the Yellig-Metcalfe13 correlation for CO2 MMP.
Fig. 8.37 gives the reservoir barrels occupied by 1 Mscf of CO2
as a function of pressure and temperature. For most CO2 projects,
approximately 2 Mscf of CO2 is required to fill 1 res bbl PV. Typically, approximately 5 to 10 Mscf of CO2 is the “gross utilization”
required to recover an additional 1 bbl of stock-tank oil by the
CO2-miscible flooding process; gross utilization is driven strongly by economics and may differ from these typical values. As much
as half of the injected CO2 may remain in the reservoir at the end
of a CO2 flood.
CO2, when mixed with water, forms carbonic acid. This acidic
byproduct may affect injectivity in carbonate reservoirs, but the corrosive effect on steel tubulars and surface equipment may be signifi16
8.5.2 Immiscible CO2/Oil Behavior. CO2 flooding has been applied successfully in viscous, heavy-oil reservoirs. Oil swelling and
oil-viscosity reduction are the two primary mechanisms in immiscible CO2 displacement. Low-pressure reservoirs and reservoirs
with stock-tank-oil gravities less than approximately 30°API are
typical candidates for immiscible CO2 displacement. Gravity-stable
displacement with CO2 also may be an efficient immiscible process.
Simon and Graue44 give generalized graphical correlations for solubility (Fig.8.38), swelling (Fig. 8.39), and viscosity reduction for
“dead” stock-tank oils saturated with CO2 (Fig. 8.40). Reported accuracies for the solubility and swelling correlations are 2 and 0.5%, respectively, and 12% deviation is reported for the viscosity correlation.
Fig. 8.38 shows that CO2 solubility in crude oils increases with
decreasing temperature. Solution gas/oil ratio in scf/STB can be calculated from CO2 mole fraction, x CO2 , in a CO2/oil mixture from
R s + 133, 000
x CO2
go
.
M o 1 * x CO
. . . . . . . . . . . . . . . . . . . (8.1)
2
At temperatures less than approximately 200°F, the correction to solubility based on the universal oil products (Watson) characterization
factor is less than 2% for most reservoir oils (11.4 t K w t 12.4).
Fig. 8.39 shows the swelling factor, expressed as the ratio of
CO2-saturated stock-tank-oil volume divided by original stocktank-oil volume. Swelling increases with increasing CO2 solubility
and with decreasing stock-tank-oil molar volume (M ońg o).
Oil-viscosity reduction (Fig. 8.40) is substantial for all API-gravity stock-tank oils at pressures up to approximately 750 psia; the effect diminishes at higher pressures because of reduced CO2 solubility. High-viscosity oils are affected the most by CO2 solubility; oil
viscosity may be reduced by as much as two orders of magnitude.
Practically, Simon and Graue’s correlations are valid only for
heavier oils ( g API t 25 and m o t 5 cp) without solution gas and
at temperatures greater than approximately 120°F. Fig. 8.39 shows
the effect of solution gas on oil swelling. The Simon-Graue correlations cannot be used to calculate solubility and swelling in reservoir oils containing solution gas and also do not predict the draPHASE BEHAVIOR
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Fig. 8.41—Effect of solution gas on swelling of a reservoir oil by
CO2 (adapted from Refs. 16 through 19).
0
Fig. 8.40—Effect of CO2 on oil viscosity (adapted from Ref. 44).
matic change in solubility and swelling behavior exhibited by
some oils at lower temperatures.
8.5.3 Miscible CO2/Oil Behavior. Fig. 8.41 shows the swelling behavior of a stock-tank oil reported by Holm,5,45 Holm and Josendal,16-19 and Holm and O’Brien.46 The experiment starts with a
constant-volume visual cell initially filled approximately one-third
with stock-tank oil. CO2 is added in increments, and the cell is rocked
for each mixture until equilibrium is reached. The final pressure and
oil volume are noted, and the oil volume, relative to the initial oil volume, is plotted vs. pressure. When a certain pressure is reached, the
oil phase, which was being swollen by increasing amounts of dis-
Fig. 8.42—Volumetric behavior of Cabin Creek stock-tank oil as
CO2 is added to a constant-volume visual cell (adapted from
Refs. 16 through 19).
GAS-INJECTION PROCESSES
solved CO2, suddenly decreases in volume. Significant extraction of
intermediate and heavy components (C5 through C30) from the oil
phase into the upper CO2-rich phase causes this dramatic change in
oil volumetric behavior. At sufficiently high pressures, the CO2-rich
phase may even become heavier than the oil (hydrocarbon-rich liquid) phase, resulting in phase inversion (Fig. 8.42).
Holm and Josendal note that the observed discontinuity in swelling behavior is caused by a change in the behavior of CO2-rich
phase from vapor-like to liquid-like that is almost coincident with
the pressure required to develop miscibility in slim-tube measurements. Qualitatively, the change in behavior of the CO2-rich phase
is analogous to the volumetric change that occurs for a pure component when pressure passes through the vapor pressure. That is, the
CO2-rich phase behaves like a vapor at pressures below the “vapor
pressure” and like a liquid at higher pressures. Once the CO2-rich
phase attains liquid-like behavior, it extracts intermediate and heavy
components from the oil, as would be expected with a liquid solvent.
The temperature required for the CO2-rich phase to exhibit sharp,
discontinuous volumetric behavior depends on the oil but is usually
Fig. 8.43—Volumetric behavior of Mead-Strawn stock-tank oil as
CO2 is added to a constant-volume visual cell at different temperatures (adapted from Refs. 16 through 19).
17
Fig. 8.44—Volumetric behavior and slim-tube results for the
Mead-Strawn and Fansworth stock-tank oils at 135°F (adapted
from Refs. 16 through 19).
less than 150°F. Fig. 8.43 shows swelling/extraction experiments for
the 41°API Mead-Strawn crude oil at different temperatures. Holm
and Josendal note that miscibility develops at a pressure only slightly
higher than the pressure where the character of swelling changes (i.e.,
where significant hydrocarbon extraction starts). They also point out
that the sharpness of change in the swelling behavior is more pronounced at lower temperatures and is coincident with the sharpness
in change from immiscible to miscible displacement indicated on a
recovery-pressure curve from slim-tube experiments (Fig. 8.44).
Holm and Josendal correlate miscibility development with the density of pure CO2. They indicate that light oils can develop miscibility
at conditions where CO2 has a density as low as 0.4 g/cm3 (critical
CO2 density is 0.47 g/cm3) and that most oils will develop miscibility
at conditions where CO2 density ranges from 0.5 to 0.7 g/cm3 (Fig.
8.45). Their correlation for MMP shows that the CO2 density required
to develop miscibility depends primarily on the amount of gasoline
and gas/oil components (C5 through C30) found in the stock-tank oil.
Their correlation uses weight percent (w C5 through w C30)ńw C5)as
a correlating parameter, with typical values ranging from 70 to 80%
requiring CO2 densities ranging from 0.65 to 0.55 g/cm3 to develop
Fig. 8.45—Density of CO2 required to develop miscibility for various oils at temperatures from 90°F to 190°F (adapted from Refs.
16 through 19).
miscibility. Fig. 8.46 shows the Holm-Josendal correlation for MMP.
Stalkup1 covers other correlations for CO2 MMP.
The distribution of components in the C5 through C30 cut of an
oil also affects the MMP, but Holm and Josendal do not include this
effect directly in their correlation. They do show, however, that the
fraction of gasolines (C5 through C12) in the C5 through C30 cut has
a measurable effect. Typically, gasolines make up 40 to 50 wt% of
the C5 through C30 cut. Higher gasoline content will decrease the
MMP, and lower gasoline content will increase the MMP. The type
of hydrocarbons (paraffinic vs. aromatic) making up the C5 through
C30 material in a crude oil has negligible effect on MMP. Aromatic
oils appear to have slightly lower MMP’s than paraffinic oils, all
other conditions being the same.
Nitrogen and light C1 through C4 hydrocarbons in the reservoir
oil generally have a negligible effect on CO2 MMP if the MMP is
less than the reservoir-oil bubblepoint pressure. The light components in the reservoir oil are extracted ahead of the miscible front in
a CO2 process (Fig. 8.47). The bank of light components does not
affect the extraction process or developed miscibility. Yellig and
Metcalfe13 point out that the MMP of a reservoir oil equals the bubblepoint of that oil if the bubblepoint pressure is greater than the
MMP determined for a low-GOR sample of the same stock-tank oil.
However, this is true only when considering the traditional vaporizing-gas drive mechanism. With the condensing/vaporizing mechanism, the MMP can be lower than the bubblepoint pressure.
Methane, N2 , and C2 through C4 hydrocarbons mixed with the
CO2 injection gas affect MMP significantly. Methane and N2 tend
135°F and 1,800 psi—near miscible
0.33
PV
0.17
PV
2,500 psi—multicontact miscible
0.15
PV
0.15
PV
2,500 psi—first contact miscible
C5-C30 Content of Oil, (C5*C30)/C5+, wt%
0.24
PV
Fig. 8.46—CO2 MMP correlation equals the pressure corresponding to the CO2 density from the chart at reservoir temperature; MMP may be less than the oil bubblepoint for a C/V miscible
mechanism.
18
Fig. 8.47—Schematic of distribution of components in CO2 displacement at miscible and near-miscible conditions based on
slim-tube simulation results (adapted from Refs. 16 through 19).
PHASE BEHAVIOR
Fig. 8.48—Experimental recoveries from slim-tube displacements for a Wasson stock-tank crude oil displaced by a CO2 slug
pushed by N2 at 1,250 psig and 107°F with no gas in solution and
100-ft coil (adapted from Ref. 47).
to increase MMP, while NGL’s tend to decrease MMP. However, a
sufficiently large PV of injected CO2 can be followed by N2 or lean
gas without affecting MMP (Fig. 8.48).
8.5.4 Multiphase Behavior. CO2/oil systems exhibit multiphase
VLLS behavior similar to that described earlier for enriched-gas/oil
systems.48 Three-phase VLL behavior is limited to reservoir temperatures less than approximately 130°F, pressures from 1,000 to
1,500 psia (somewhat less than the MMP), and CO2 concentrations
greater than approximately 50 mol%. At other conditions, vapor/
liquid or liquid/liquid behavior is expected, with the upper phase
containing mainly CO2 and the lower phase containing mostly hydrocarbons and some dissolved CO2. Asphaltene precipitation can
occur over a relatively large range of pressures and CO2 concentrations (Fig. 8.49), usually including the VLL region. The three
phases in a CO2/oil VLL system include a CO2-rich vapor (the upper phase), a CO2-rich liquid (the middle phase) containing some
hydrocarbons, and a hydrocarbon-rich liquid (the lower phase) containing C5+ with some dissolved natural gas and CO2.
Consider a CO2/oil mixture in the three-phase region in Fig.
8.50. Moving up in pressure through the lower two-phase region,
the hydrocarbon-rich liquid is in equilibrium with a CO2-rich vapor phase. Near 1,000 psia (the dewpoint of the CO2-rich vapor
phase), a CO2-rich liquid phase appears. As pressure increases
through the three-phase region, the volume of the CO2-rich liquid
phase increases, mostly at the expense of the CO2-rich vapor
phase. At few hundred psi higher than the onset of three-phase behavior (the bubblepoint of the CO2-rich liquid phase), the
CO2-rich vapor phase disappears.
The onset of three-phase behavior in a CO2/oil system is related
to the upper CO2-rich phase behaving like a component at its vapor
pressure (see the earlier discussion on miscibility). Because the
CO2-rich phase is actually a mixture, the transition from vapor-like
to liquid-like behavior occurs over a narrow range of pressures
compared with the abrupt change experienced at the vapor pressure
of a pure component.
The volume of hydrocarbon-rich liquid increases because of
swelling in the low-pressure, vapor/liquid region and through the
three-phase region. When the CO2-rich phase completes its transition from vapor-like to liquid-like behavior at the top of the threephase region, the oil phase stops swelling and starts shrinking as a
result of strong extraction of C5 through C30 components by the
CO2-rich liquid phase.
Practically, the effect of three-phase behavior on the CO2 displacement process is small and can be ignored when modeling field
performance. The three-phase region usually is located in geological layers that have experienced CO2 breakthrough, some distance
GAS-INJECTION PROCESSES
CO2 in Mixture, mol%
Fig. 8.49—Experimental p-x diagram for west Texas reservoir oil
and up to 95% CO2 injection gas showing large region of asphaltene precipitation (from Ref. 4).
away from the producing wells, where reservoir pressure is between
1,000 and 1,500 psia. The three-phase region may, however, cause
serious problems for compositional simulators based on a twophase vapor/liquid equilibrium (VLE) algorithm.50 The problem
LOWER LIQUID PHASE, vol%
CO2 in Mixture, mol%
Fig. 8.50—Experimental p-x diagram for Wasson crude oil and
CO2 injection gas at 105°F (from Ref. 49).
19
8.5.5 CO2/Water Behavior. Chap. 9 covers methods for estimating
CO2/water behavior. The two primary design considerations in a
CO2-injection project related to CO2/water phase behavior are the
treatment of corrosion resulting from the formation of carbonic acid
when CO2 mixes with water and the loss of injected CO2 resulting
from the saturation of connate and injected water with CO2.
Fig. 8.51 shows CO2 solubility in water and brines at 100°F. In fact,
CO2 solubility in water is not very sensitive to temperature at temperatures greater than 100°F. Also, solubility increases only slightly at
pressures greater than approximately 3,000 psia. Salinity has a significant effect on CO2 solubility, reducing the solubility in brine by
approximately 30% for every 100,000 ppm of total dissolved solids.
Water density and viscosity change only slightly when saturated with
CO2. Enick and Klara53 reported on the effect of CO2 solubility in
brine on compositional simulation of CO2 flooding.
References
Fig. 8.51—Solubility of CO2 in pure water and NaCl brines at
100°F (adapted from Ref. 4; data from Ref. 52).
arises because, thermodynamically, the flash algorithm is searching
for an equilibrium condition with only two phases. If the mixture being flashed actually exhibits three-phase behavior according to the
thermodynamic model being used, the VLE algorithm must choose
one of several valid two-phase solutions. Any of these two-phase
solutions satisfies the equilibrium constraints, but the two-phase
solutions merely represent local minimums in the Gibbs free energy,
while the three-phase solution represents a global minimum (see
Chap. 4). Numerical instabilities arise when a gridblock oscillates
between one two-phase solution and another.
In CO2 flooding, asphaltene precipitation could be a more serious
multiphase problem than three-phase VLL behavior. First, asphaltene precipitation occurs over a wider range of pressures and CO2
compositions, potentially causing reduced injectivity and productivity. Several authors40,48 have provided laboratory measurements
showing that asphaltene precipitation occurs over a wide range of
conditions. Unfortunately, few investigators have documented the
quantitative effect of asphaltenes on reservoir performance. Christman and Gorell6 give results that indicate that reduced injectivities
experienced in many tertiary CO2 projects can be modeled without
accounting for reduced permeability and altered wettability caused
by asphaltene precipitation. Still, serious operational problems
associated with asphaltenes have been reported in field operations.
Monger and Trujillo41 report on a comprehensive study of the deposition of organic solids during CO2 and rich-gas flooding.
Few thermodynamic models have been suggested for predicting
asphaltene precipitation. Kawanaka et al.51 propose a technique for
predicting organic deposition of asphaltene, wax, and other solidlike materials that may precipitate from reservoir oils. The model
uses a continuous distribution for the solid phase, and the authors
provide results that give reasonable predictions for miscible-solvent
processes. Finally, they give a comprehensive review of literature
on asphaltene precipitation, measurements, and thermodynamic
models for prediction of VLS phase behavior.
20
1. Stalkup, F.I. Jr.: Miscible Displacement, Monograph Series, SPE, Richardson, Texas (1984) 8.
2. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the
Displacement of Oil by Enriched Gases,” paper SPE 15493 presented
at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October.
3. Klins, M.A.: CO2 Flooding, Basic Mechanisms, and Project Design,
Intl. Human Resources Development Corp., Boston (1984).
4. Goodrich, J.H.: “ Target Reservoirs for CO2 Miscible Flooding,” Report DOE/MC/08341-17, U.S. DOE, Washington, DC (1980).
5. Holm, L.W.: “Status of CO2 and Hydrocarbon Miscible Oil Recovery
Methods,” JPT (January 1976) 76.
6. Christman, P.G. and Gorell, S.B.: “Comparison of Laboratory-Observed and Field-Observed CO2 Tertiary Injectivity,” JPT (February
1990) 226; Trans., AIME, 289.
7. Harvey, M.T., Shelton, J.L., and Kelm, C.H.: “Field Injectivity Experiences With Miscible Recovery Projects Using Alternate Rich-Gas and
Water Injection,” JPT (September 1977) 1051.
8. Kay, W.B.: “ The Ethane-Heptane System,” Ind. Eng. Chem. (1938)
30, 459.
9. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-Hill
Book Co. Inc., New York City (1959).
10. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Behavior of Hydrocarbon Mixtures Illustrated by a Simple Case,” API Bulletin (1932) 212, 119.
11. Peng, D.Y. and Robinson, D.B.: “A New-Constant Equation-of-State,”
Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59.
12. Koch, H.A. Jr. and Hutchinson, C.A. Jr.: “Miscible Displacements of
Reservoir Oil Using Flue Gas,” Trans., AIME (1958) 213, 7.
13. Yellig, W.F. and Metcalfe, R.S.: “Determination and Prediction of
CO2 Minimum Miscibility Pressures,” JPT (January 1980) 160;
Trans., AIME, 269.
14. Orr, F.M. Jr. et al.: “Laboratory Experiments To Evaluate Field Prospects for CO2 Flooding,” JPT (April 1982) 888.
15. Auxiette, G. and Chaperon, I.: “Linear Gas Drives in High-Pressure Oil
Reservoirs Compositional Simulation and Experimental Analysis,” paper SPE 10271 presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 4–7 October.
16. Holm, L.W. and Josendal, V.A.: “Mechanisms of Oil Displacements of
CO2,” JPT (December 1974) 1427; Trans., AIME, 257.
17. Holm, L.W. and Josendal, V.A.: “Effect of Oil Composition on Miscible-Type Displacement by CO2,” paper SPE 8814 presented at the
1980 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 20–23 April.
18. Holm, L.W. and Josendal, V.A.: “Discussion of Determination and Prediction of CO2 Minimum Miscibility Pressures,” JPT (May 1980) 870.
19. Holm, L.W. and Josendal, V.A.: “Effect of Oil Composition on MiscibleType Displacement by Carbon Dioxide,” SPEJ (February 1982) 87.
20. Kuo, S.S.: “Prediction of Miscibility for the Enriched-Gas Drive Process,” paper SPE 14152 presented at the 1985 SPE Annual Technical
Conference and Exhibition, Las Vegas, Nevada, 22–25 September.
21. Novosad, Z. and Costain, T.G.: “New Interpretation of Recovery Mechanisms in Enriched Gas Drives,” J. Cdn. Pet. Tech. (March–April 1988)
21, No. 2, 54.
22. Novosad, Z., Sibbald, L.R., and Costain, T.G.: “Design of Miscible Solvents for a Rich Gas Drive—Comparison of Slim Tube and Rising
Bubble Tests,” J. Cdn. Pet. Tech. (January–February 1990) 29, No. 1, 37.
PHASE BEHAVIOR
23. Poettmann, F.H., Christiansen, R.L., and Mihcakan, I.M.: “Discussion
of Methodology for the Specification of Solvent Blends for Miscible
Enriched-Gas Drives,” SPERE (February 1992) 154.
24. Sibbald, L.R., Novosad, Z., and Costain, T.G.: “Authors’ Reply to Discussion of Methodology for the Specification of Solvent Blends for
Miscible Enriched-Gas Drives,” SPERE (February 1992) 156.
25. Zhou, D. and Orr, F.M. Jr.: “Analysis of Rising-Bubble Experiments To
Determine Minimum Miscibility Pressures,” SPE Journal (March
1998) 19.
26. Peterson, A.V.: “Optimal Recovery Experiments With N2 and CO2,”
Pet. Eng. Intl. (November 1978) 40.
27. Benham, A.L., Dowden, W.E., and Kunzman, W.J.: “Miscible Fluid Displacement—Prediction of Miscibility,” Trans., AIME (1960) 219, 229.
28. Vogel, J.L. and Yarborough, L.: “ The Effect of Nitrogen on the Phase
Behavior and Physical Properties of Reservoir Fluids,” paper SPE 8815
presented at the 1980 SPE Annual Technical Conference and Exhibition, Tulsa, Oklahoma, 20–23 April.
29. Jensen, F. and Michelsen, M.L.: “Calculation of First Contact and Multiple Contact Miscibility Pressures,” In Situ (1990) 14, 1.
30. Luks, K.D., Turek, E.A., and Baker, L.E.: “Calculation of Minimum Miscibility Pressure,” SPERE (November 1987) 501; Trans., AIME, 283.
31. Metcalfe, R.S., Fussell, D.D., and Shelton, J.L.: “A Multicell Equilibrium Separation Model for the Study of Multiple-Contact Miscibility in
Rich-Gas Drives,” SPEJ (June 1973) 147; Trans., AIME, 255.
32. Cook, A.B. et al.: “Effects of Pressure, Temperature, and Type of Oil
on Vaporization of Oil During Gas Cycling,” Report RI 7278, U.S. Bureau of Mines, Washington, DC (1969).
33. Cook, A.B., Walter, C.J., and Spencer, G.C.: “Realistic K Values of C7+
Hydrocarbons for Calculating Oil Vaporization During Gas Cycling at
High Pressure,” JPT (July 1969) 901; Trans., AIME, 246.
34. Johns, R.T., Orr, F.M. Jr., and Dindoruk, B.: “Analytical Theory of
Combined Condensing/Vaporizing Gas Drives,” paper SPE 24112
presented at the 1992 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 22–24 April.
35. Johns, R.T., Fayers, J.F., and Orr, F.M. Jr.: “Effect of Gas Enrichment
and Dispersion on Nearly Miscible Displacement in Condensing/Vaporizing Drives,” paper SPE 24938 presented at the 1992 SPE Annual
Technical Conference and Exhibition, Washington, DC, 4–7 October.
36. Wang, Y. and Orr, F.M. Jr.: “Analytical Calculation of Minimum Miscibility Pressure,” Fluid Phase Equilibria (1997) 139, 101.
37. Moses, P.L. and Wilson, K.: “Phase Equilibrium Considerations in Using Nitrogen for Improved Recovery From Retrograde Condensate
Reservoirs,” JPT (February 1981) 256; Trans., AIME, 271.
38. Donohoe, C.W. and Buchanan, R.D. Jr.: “Economic Evaluation of
Cycling Gas-Condensate Reservoirs With Nitrogen,” JPT (February
1981) 263; Trans., AIME, 271.
39. Renner, T.A. et al.: “Displacement of a Rich-Gas Condensate by Nitrogen: Laboratory Corefloods and Numerical Simulations,” SPERE
(February 1989) 52; Trans., AIME, 287.
GAS-INJECTION PROCESSES
40. Shelton, J.L. and Yarborough, L.: “Multiple-Phase Behavior in Porous Media During CO2 or Rich-Gas Flooding,” JPT (September
1977) 1171.
41. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 and
Rich-Gas Flooding,” SPERE (February 1991) 17; Trans., AIME, 291.
42. Kennedy, G.C.: “Pressure-Volume-Temperature Relations in CO2 at
Elevated Temperatures and Pressures,” Amer. J. Sci. (April 1954)
252, 225.
43. Kennedy, J.T. and Thodos, G.: “ The Transport Properties of CO2,”
AIChE J. (1961) 7, 625.
44. Simon, R. and Graue, D.J.: “Generalized Correlations for Predicting
Solubility, Swelling, and Viscosity Behavior of CO2/Crude Oil Systems,” JPT (January 1965) 102; Trans., AIME, 234.
45. Holm, L.W.: “CO2 Requirements in CO2 Slug and Carbonated Water
Oil Recovery Processes,” Prod. Monthly (September 1963).
46. Holm, L.W. and O’Brien, L.J.: “CO2 Test at the Mead-Strawn Field,”
JPT (April 1971) 431.
47. O’Leary et al.: Nitrogen-Driven CO2 Slugs Reduced Costs,” Pet. Eng.
Intl. (May 1979) 130.
48. Orr, F.M. Jr., Yu, A.D., and Lein, C.L.: “Phase Behavior of CO2 and
Crude Oil in Low-Temperature Reservoirs,” SPEJ (August 1981) 480.
49. Gardner, J.W., Orr, F.M. Jr., and Patel, P.D.: “ The Effect of Phase Behavior on CO2-Flood Displacement Efficiency,” JPT (November
1981) 2067.
50. Perschke, D.R., Pope, G.A., and Sepehrnoori, K.: “Phase Identification
During Compositional Simulation,” paper SPE 19442 available from
SPE, Richardson, Texas (1989).
51. Kawanaka, S., Park, S.J., and Mansoori, G.A.: “Organic Deposition
From Reservoir Fluids: A Thermodynamic Predictive Technique,”
SPERE (May 1991)185.
52. McRee, B.C.: “How It Works, Where It Works,” Pet. Eng. Intl. (November 1977) 52.
53. Enick, R.M. and Klara, S.M.: “Effects of CO2 Solubility in Brine on the
Compositional Simulation of CO2 Floods,” SPERE (May 1992) 253.
SI Metric Conversion Factors
atm 1.013 250
E)05 +Pa
°API 141.5/(131.5)°API)
+g/cm3
bar 1.0*
E)05 +Pa
bbl 1.589 873
E*01 +m3
cp 1.0*
E*03 +Pa@s
ft 3.048*
E*01 +m
E*02 +m3
ft3 2.831 685
°F (°F*32)/1.8
+°C
in. 2.54*
E)00 +cm
lbm 4.535 924
E*01 +kg
psi 6.894 757
E)00 +kPa
ton 9.071 847
E*01 +Mg
*Conversion factor is exact.
21
Chapter 9
Water/Hydrocarbon Systems
9.1 Introduction
The connate or “original” water found in petroleum reservoirs usually contains both dissolved salts (consisting mainly of NaCl) and
solution gas (consisting mainly of methane and ethane). Initial water saturation can range from 5 to 50% of the pore volume (PV) in
the net-pay intervals of a reservoir (where production is primarily
oil and gas). Higher water saturations are found in the aquifer and
where water has swept oil or gas during a waterflood.
From a reservoir-depletion point of view, the amount of water
connected with a reservoir is as important as the properties of the
water, particularly in material-balance calculations where water expansion (compressibility times water volume) may contribute significantly to pressure support.1,2 From a production point of view,
water mobility is important, requiring determination of water saturations, water viscosity, and formation volume factor (FVF). For
surface-processing calculations, water composition, water content
in the produced wellstream, and conditions where water and hydrocarbons coexist must be defined.
The three most important aspects of phase behavior involving water/hydrocarbon systems are mutual solubilities of gas and water, volumetric behavior of reservoir brines, and hydrate formation and treatment. Sec. 9.2 presents pressure/volume/temperature (PVT)
correlations for water/hydrocarbon systems. Standard PVT properties—solution gas/water ratio, Rsw ; isothermal water compressibility,
cw ; water FVF, Bw ; water viscosity, mw ; and water content in gas,
rsw —are correlated in terms of pressure, temperature, and salinity by
use of graphical charts and empirical equations. Correlations for water/hydrocarbon interfacial tension (IFT), s wh, are also presented.
At very high temperatures and pressures, some correlations and
the existing water-property data base are not adequate. Equations of
state (EOS’s) have been used with reasonable success in predicting
mutual solubilities and phase properties of hydrocarbon/water systems up to 400°F and greater than 10,000 psia,3-8 as discussed in
Sec. 9.3. The effect of salinity on gas/water phase behavior has also
been treated to some extent by the EOS methods.9
Sec. 9.4 covers the physical structure of hydrates and how to calculate conditions under which hydrates form. Hydrate formation
can have a significant effect on production and surface-facilities
equipment and even on deep drilling. Water/hydrocarbon phase diagrams give the conditions of initial hydrate formation. These diagrams are particularly useful for designing a production system to
avoid hydrate formation. The formation of hydrates can also be estimated with vapor/solid equilibrium ratios.
WATER/HYDROCARBON SYSTEMS
9.2 Properties and Correlations
Like all reservoir fluids, formation-water properties depend on
composition, temperature, and pressure. Reservoir water is seldom
pure and usually contains dissolved gases and salts. Total dissolved
solids (TDS), usually consisting mainly of NaCl, ranges from
10,000 to [300,000 ppm; seawater salinity is [ 30,000 ppm.
Water is limited as to how much salt it can keep in solution. The
limiting concentration for NaCl brine is10
C *sw + 262, 180 ) 72T ) 1.06T 2 ,
. . . . . . . . . . . . . . . . (9.1)
with T in °C and C w in ppm. If reservoir temperature is known but
a water sample cannot be obtained, this relation gives the limiting
salinity of the reservoir brine. Salinity of a brine usually is less than
80% of the value given by Eq. 9.1. Otherwise, the best estimate of
brine salinity can be taken from a neighboring reservoir in the same
geological formation.
Scale buildup in tubing and surface equipment is caused by the
precipitation of salts in produced brine,11 usually calcium carbonate, calcium sulfate (e.g., gypsum), barium or strontium sulfates,
and iron compounds. Temperature, pressure, total salinity, and salt
composition are the primary variables determining the severity of
scaling. Note that Eq. 9.1 should not be used to detect conditions that
result in scale buildup.
Dissolved gas in water is usually less than 30 scf/STB (approximately 0.4 mol%) at normal reservoir conditions. The effect of salt
and gas content on water properties can be important, and the following discussion gives methods to estimate fluid properties in terms of
temperature, pressure, dissolved gas, and salinity. Methods for estimating PVT properties of formation water usually are based on initial
estimates of the pure-water properties at reservoir temperature and
pressure that are then corrected for salinity and dissolved gas.
9.2.1 Salinity. The cations dissolved in formation waters usually include Na+, K+, Ca++, and Mg++, and the anions include Cl *, SO**
4 ,
. Most formation waters contain primarily NaCl. Susand HCO **
3
pended salts, entrained solids, and corrosion-causing bacteria may
also be present in reservoir waters, but these constituents usually do
not affect formation-water PVT properties. The geochemistry of
formation waters can be useful in detecting foreign-water encroachment and in determining its source. Table 9.1 gives example compositions of reservoir brines.
Salinity defines the concentration of salts in a saline solution
(brine) and may be specified as one of several quantities: weight
fraction, w s; mole fraction, x s; molality, c sw; molarity, c sv; parts per
million by weight, C sw; and parts per million by volume, C sv. Table
1
TABLE 9.1—EXAMPLE COMPOSITIONS OF FORMATION BRINES
Dodson-Standing13
Component
Seawater
(ppm)
Brine A
(ppm)
Brine B
(ppm)
Arun Field
(mg/L)
Gulf Coast Frioa
(mg/L)
Kansas Wilcoxb
(mg/L)
Kansas Wilcoxa
(mg/L)
Sodium (Na)
10,560
3,160
12,100
5,212
40,600
10,800
142,500
Calcium (Ca)
Magnesium (Mg)
400
58
520
80
5,100
790
14,400
1,270
40
380
5
1,000
5,560
68,500
Sulfate (SO4)
2,650
0
5
262
110
80
300
Chloride (Cl)
18,980
4,680
20,000
7,090
69,100
10,870
142,600
140
696
980
1,536
990
20
530
0
0
130
0
0
0
3
65
0
0
0
0
80
350
515
0
0
0
0
0
0
Bicarbonate (HCO3)
Iodide (I)
Bromide (Br)
Others
Total
34,580
8,630
34,110
14,190
116,900
28,200
369,180
Specific gravity
1.0243c
1.006d,e
1.024d,e
1.014d
1.086d,e
1.015d
1.140d
aMaximum salt-containing composition reported for field/formation.
bMinimum salt-containing composition reported for field/formation.
cAt 20°F.
dAt 60°F.
eEstimated with Eq. 9.3.
TABLE 9.2—DEFINITIONS OF SALT CONCENTRATIONS
Symbol
Unit
Definition
Weight fraction
ws
g/g
m sńǒm s ) m owǓ
Mole fraction
xs
g mol/g mol
n sńǒn s ) n owǓ
Molality
csw
g mol/kg
10 3n sńm ow
Molarity
csv
g mol/L
10 3n sńV w
Quantity
ppm, weight basis
Csw
mg/kg
10 m sńǒm s ) m owǓ
ppm, volume basis
Csv
mg/L
10 6n sńV w
6
o
m s + mass salt, m o
w + mass pure water, n s + moles salt, n w + moles pure
water, and V w + volume brine mixture.
9.2 formally defines these quantities; in the table, m s +mass of salt
in grams, m ow+mass of pure water in grams, n s +moles of salt in
gram moles, n ow+moles of pure water in gram moles, and V w+volume of the brine mixture in cubic centimeters.
Some common conversions for the various concentrations are
C sv + ò wC sw ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2a)
C
C sw + ò sv + 10 6 w s ,
w
c sw +
17.1
,
10 6 C *1
sw * 1
and C sw +
. . . . . . . . . . . . . . . . . . . . . . . . . (9.2b)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2c)
10 6
,
17.1 c *1
sw ) 1
. . . . . . . . . . . . . . . . . . . . . . . (9.2d)
where the Eqs. 9.2c and 9.2d apply for NaCl brines. If brine density,
ò w, at standard conditions (14.7 psia and 60°F) is not reported, it can
be estimated from the Rowe-Chou12 density correlation for NaCl.
ò wǒ p sc, T scǓ + ǒ1.0009 * 0.7114w s )
0.26055 ws2
Ǔ
*1
,
with ò w in g/cm3 and w s in weight fraction TDS. For many engineering applications, ò w+1 g/cm3 is assumed and the mass of salt
is considered negligible compared with the mass of pure water, resulting in the approximate relations
c sv [ c sw + c s ,
2
10 *6ǓC s ,
ln x i + ln f i * ln H i *
. . . . . . . . . . . . . . . . . . . . . . . (9.4)
v~ i ǒ p * p vwǓ
,
RT
. . . . . . . . . . . . . (9.5)
where x i+solubility of gas Component i in water, f i +partial fugacity, H i+Henry’s constant, and v~ i +modified molar volume. H i and
v~ i are nonlinear functions of temperature. Cramer18 uses a similar
approach to correlate gas solubilities for methane/water and methane/NaCl-brine systems over a wide range of pressures, temperatures, and salinities.
At reservoir conditions, the solubility of methane in water and the
effect of salinity are the most important variables affecting water
properties. The following empirical equation gives a reasonable fit
of the Culberson and McKetta14,19 solubility data for methane in
pure water at conditions 100tTt350°F and 0tpt10,000 psia,
ƪȍǒȍ Ǔ ƫ
3
x C + 10 *3
3
A i jT j p i ,
i+0
. . . . . . . . . . . . . . . (9.6)
j+0
where A00+0.299, A01+*1.273 10*3, A02+0.000, A03+0.000,
A10+2.283 10*3, A11+*1.870 10*5, A12+7.494 10*8,
A13+*7.881 10*11, A20+*2.850 10*7, A21+2.720 10*9,
A22+*1.123 10*11, A23+1.361 10*14, A30+1.181 10*11,
A31+*1.082 10*13, A32+4.275 10*16, and A33+*4.846
10*19, with T in °F and p in psia. Gas solubility expressed as a solution gas/water ratio, R sw at standard conditions is
R sw + 7, 370
C sw [ C sv + C s ,
10*6 applies for NaCl brines.
9.2.2 Gas Solubilities in Water/Brine.
The solubility of natural gases in water is rather complicated to estimate from empirical correlations. However, the effect of gas solubility usually is minor except at high temperatures. At temperatures less
than approximately 300°F and pressures less than 5,000 psia, solubility usually is less than 0.4 mol%, or approximately 30 scf/STB. According to Dodson and Standing’s13 results, this amount of dissolved
gas causes an increase of approximately 25% in water compressibility
(e.g., from 3.8 10*6 to 4.8 10*6 psi*1). Experimental gas solubilities for C1 through C4 hydrocarbons, nonhydrocarbons, natural
gas, and a few binaries and ternaries are available in the literature.
Figs. 9.1 through 9.3 present some of these data.
Kobayashi and Katz17 give a method for estimating gas solubilities in pure water based on Henry’s law for dilute solutions.
1
. . . . . . . . . . . . . . . . . . . . . (9.3)
and c s [ ǒ17.1
where the constant 17.1
xg
[ 7, 370 x g , . . . . . . . . . . . . . . . . . (9.7)
1 * xg
with R sw in scf/STB.
PHASE BEHAVIOR
ks +
lim
cs ³ 0
ƪ
c *1
log
s
ƫ
ǒf R
Ǔ
i w
,
ǒf R
Ǔo
i w
. . . . . . . . . . . . . . . . . (9.8)
where k s+Setchenow constant, c s+salt concentration, and (f R
i )w
o
and (f R
i ) w+fugacity coefficients of Component i at infinite dilution in the salt solution and in pure water, respectively. Both molality and molarity have been used in the literature for defining Setchenow constants; however, molality, c sw, is now considered to be the
preferred concentration. The unit for the Setchenow constant is
M*1 (i.e., kg/g mol), where M+molarity.
The ratio of infinite-dilution fugacity coefficients is traditionally
assumed to give an accurate estimate of the ratio of solubilities,
yielding the relation
xg
R sw
ǒ
*k c
[ x o + 10 s s [ 10 * 17.1
R osw
g
Fig. 9.1—Gas-solubility data for methane in pure water (adapted
from Ref. 14).
10 *6Ǔk s C s
,
. . . . . . . . . (9.9)
where R osw +solubility of gas in pure water and R sw +solubility of gas
in brine. For k su0, the gas solubility is less in brines than in pure water, a fact that has led to the term “salting-out coefficient” for k s.
The Setchenow constant is more or less independent of pressure
but is a strong function of temperature. Cramer18 gives a detailed
treatment of Setchenow (and Henry’s) constants for the C1/NaCl system using data at temperatures up to 570°F and pressures up to 2,000
psia. He proposes the temperature dependence of k s shown in Fig.
9.4. This figure also shows values of k s reported elsewhere for the
C1/NaCl system, illustrating the relatively large uncertainty in salting-out coefficients, even for such a well-defined system. Søreide and
Whitson9 give a best-fit relation for the Cramer correlation.
Amirijafari and Campbell20 give experimental component solubilities and an empirical method for calculating the total gas solubility of the C1/C2/C3 ternary mixture. However, for most applications
gas solubility can be estimated by assuming that the gas consists
only of methane. A standard two-phase flash calculation with a cubic EOS gives a surprisingly accurate prediction of gas solubilities,
as discussed in Sec. 9.3. This approach is the recommended procedure for estimating gas solubilities of hydrocarbon/water/brine mixtures at high pressures and temperatures.
9.2.3 Salinity Correction for Solubilities. Refs. 9 and 21 give the
Setchenow (sometimes written Secenov) relation for correcting hydrocarbon solubility in pure water for salt content.
Pressure,
psia
Fig. 9.2—Gas-solubility data for natural gas in pure water
(adapted from Ref. 13).
WATER/HYDROCARBON SYSTEMS
Pressure, psia
1,000
Fig. 9.3—Gas-solubility data for CO2 in pure water (adapted from
Refs. 15 and 16).
3
f f f Methane
V V V Ethane
Propane
n-butane
Fig. 9.4—Temperature dependence of the Setchenow (saltingout) coefficient for light hydrocarbons (Ref. 9).
(k s) C
1*NaCl
+ 0.1813 * ǒ7.692
) ǒ2.6614
10 *4ǓT
10 *6ǓT 2 * ǒ2.612
10 *9ǓT 3,
. . . . . . . . . . . . . . . . . . . . (9.10)
M*1
and T in °F. Using relations suggested by Pawliwith k s in
kowski and Prausnitz21 relating k s of methane to k s of other hydrocarbons, Søreide and Whitson9 propose the following relation for
Hydrocarbon i.
k si + (k s) C
1*NaCl
) 0.000445ǒ T bi * 111.6 Ǔ ,
. . . . . . . (9.11)
with k s in M*1 and the normal boiling point, T bi, in K. Fig. 9.4
shows the temperature dependence of k s for light hydrocarbons (C2
through C4) based on Eqs. 9.10 and 9.11.
Clever and Holland22 give salting-out correlations for C1/NaCl
and CO2/NaCl systems. The correlation for CO2/NaCl is
(k s) CO
2*NaCl
+ 0.257555 * ǒ0.157492
* ǒ0.253024
10 *3ǓT
10 *5ǓT 2 ) ǒ0.438362
10 *8ǓT 3 ,
. . . . . . . . . . . . . . . . . . (9.12)
M*1.
The temperature range for Eq. 9.12 is
with T in K and k s in
40tTt660°F. The Setchenow coefficient varies somewhat with
pressure for the CO2/NaCl system, thereby making Eq. 9.12 less accurate than hydrocarbon/NaCl correlations. Fig. 9.5 illustrates the
effect of salts other than NaCl on low-pressure solubilities by use of
lines of equal gas solubility vs. molality of the salt, where NaCl is
the reference salt.
9.2.4 Equilibrium Conditions in Oil/Gas/Water Systems. All
phases (oil, gas, and water) in a reservoir are initially in thermodynamic equilibrium. This implies that the water phase contains finite
quantities of all hydrocarbon and nonhydrocarbon components
found in the hydrocarbon phases and that the hydrocarbon phases
contain a finite quantity of water. The amount of lighter compounds
(C1, C2, N2, CO2, and H2S) in the water phase can be significant and
depends mainly on the amount of each component in the hydrocarbon phase(s). The amount of C3+ hydrocarbons found in water is
usually small and can be neglected.
The K value representing the ratio of the mole fraction of Component i in the hydrocarbon phase to the mole fraction of Component
i in the water phase ( K i + z i,HCńx i,aq) is approximately constant at
a given pressure and temperature, independent of overall hydrocar4
Fig. 9.5—Lines of equal gas solubility for various salts with NaCl
as a reference (adapted from Ref. 23).
bon composition and whether the hydrocarbon is single phase or
two phase. For example, the amount of methane dissolved in water
for a methane-rich natural gas will be higher than the amount of
methane dissolved in water for an oil (above its bubblepoint). Furthermore, the amount of methane dissolved in water for a gas/oil
system with overall methane content of 40 mol% will probably be
about the same as for a single-phase oil with 40 mol% methane.
An oil that is undersaturated (with respect to gas) is still in equilibrium with the water phase. When pressure is lowered, a new equilibrium
state is reached between the undersaturated oil and water. The result is
that some of the methane will move from the water to the oil (without
free gas forming); i.e., the solution gas/water ratio decreases. At some
lower pressure, the oil will reach its bubblepoint and further reduction
in pressure will yield two sources of free gas: gas coming out of solution from the oil and gas coming out of solution from the water.
Therefore, for an undersaturated-oil reservoir, the solution gas/
water ratio of reservoir brine will decrease continuously from the
initial reservoir pressure to the reservoir-oil bubblepoint pressure
and even further at lower pressures. Correspondingly, the reservoiroil solution gas/oil ratio will increase (albeit slightly) from initial to
bubblepoint pressure and then decrease below the bubblepoint. An
EOS must be used to quantify the changing solution gas/water and
solution gas/oil ratios in this situation.
Fig. 9.6 shows calculations with an EOS that illustrate the relative
gas solubility in a reservoir oil and a reservoir gas. The oil and gas
compositions are in equilibrium at approximately 3,500 psia. At
higher pressures, the gas solubility in water is higher in the gas/water system than in the oil/water system. At less than 3,500 psia, three
phases will exist in either system and the two-phase flash calculation gives only approximate solubilities on the basis of treating the
hydrocarbon as a single phase.
9.2.5 Water/Brine FVF and Compressibility. The FVF of reservoir water, Bw, depends on pressure, temperature, salinity, and dissolved gas. Fig. 9.7 gives Dodson and Standing’s13 results for pure
water with and without solution gas. Contrary to saturated-oil volumetric behavior, the liquid volume of a gas-saturated water increases with decreasing pressure. That is, the expansion caused by
isothermal compressibility is larger than the shrinkage caused by
gas coming out of solution.
The pressure dependence of Bw that Dodson and Standing give
for gas-saturated water/brine applies to all gas and oil reservoirs that
have appreciable solution gas. Even if the oil is undersaturated, as
discussed earlier, the solution gas/water ratio decreases continuousPHASE BEHAVIOR
T+258°F
Gas/Water
Oil/Water
Gas/Oil Bubblepoint, 3,500 psia
Fig. 9.6—Gas dissolved in water for reservoir-oil/water and reservoir-gas/water systems, EOS two-phase calculations.
ly from the initial pressure to the oil bubblepoint pressure and further thereafter. This precludes the pressure dependence of water
FVF shown in Fig. 9.8, where a discontinuity occurs at some bubblepoint condition. The only way a reservoir brine could have this
behavior is if the hydrocarbons that originally saturated the brine
had migrated away completely and the reservoir pressure subsequently increased with further burial (creating an undersaturated
condition for the brine with respect to hydrocarbon components).
The FVF of brine at atmospheric pressure, reservoir temperature,
and without dissolved gas, B ow, is
ò wǒ p sc, T scǓ
v oǒ p sc, TǓ
B ow + o
+ w
.
v wǒ p sc, T scǓ
ò wǒ p sc, TǓ
. . . . . . . . . . . . . . . . (9.13)
Long and Chierici24,25 give experimental data and correlations for the
density of pure water and NaCl-brine solutions, although the proposed correlations extrapolate poorly at temperatures greater than
approximately 250°F. Kutasov26 gives several accurate correlations
for FVF’s of pure water, but the equation for Bw results in a constant
isothermal compressibility that is independent of pressure.
Rowe and Chou12 give the following correlation for water and
NaCl-brine specific volume at zero pressure (also applicable at atmospheric pressure).
v woǒ p sc, TǓ +
Fig. 9.7—FVF of pure water with and without natural gas
(adapted from Ref. 13).
which, when integrated, gives
B *wǒ p, TǓ
+*
ln o
B wǒ p sc, TǓ
p
ŕ c ǒ p, TǓ dp.
*
w
. . . . . . . . . . . . . . . . (9.16)
0
With the compressibility data reported by Rowe and Chou covering the conditions 70tTt350°F, 150tpt4,500 psia, and
0t w st0.3, a general correlation for the compressibility of a brine
(without solution gas), c *w, is
c *wǒ p, T Ǔ + ǒ A 0 ) A 1 p Ǔ
*1
,
1
+ A 0 ) A 1w s ) A 2w 2s ,
ò woǒ p sc, TǓ
where A 0 + 5.91635 * 0.01035794T
) ǒ0.9270048
10 *5ǓT 2
* 1, 127.522T *1 ) 100, 674.1T *2 ,
A 1 + * 2.5166 ) 0.0111766T * ǒ0.170552
10 *4ǓT 2 ,
and A 2 + 2.84851 * 0.0154305T ) ǒ0.223982
10 *4ǓT 2 ,
. . . . . . . . . . . . . . . . . . . . (9.14)
v wo
in cm3/g, T in K, and w s in weight fraction of NaCl. The efwith
fect of pressure on FVF can be calculated by use of the definition of
water compressibility,
ǒ Ǔ
ēB w
c *w + * 1
B w ēp
C s ,T ,
. . . . . . . . . . . . . . . . . . . . . . . (9.15)
WATER/HYDROCARBON SYSTEMS
Fig. 9.8—Effect of gas solubility on water FVF at saturated and
undersaturated conditions, EOS two-phase calculations.
5
Fig. 9.10—Effect of CO2 solubility (in terms of saturation pressure) on water viscosity.
9.4.6 Water/Brine Viscosity. Fig. 9.9 presents the viscosities of
pure water and NaCl brines as functions of temperature and salinity.
The following equations (except for the pressure correction A0) are
presented by Kestin et al.,29 who report an accuracy of "0.5% in
the range 70tTt300°F, 0tpt5,000 psia, and 0t C swt300,000
ppm (0t c swt5 M).
m w + ǒ1 ) A 0 pǓm *w ,
Fig. 9.9—Water/NaCl-brine viscosity as a function of temperature and salinity.
where A 0 + 10 6ƪ0.314 ) 0.58w s ) ǒ1.9
*ǒ1.45
A1 +
ǒ
A
1) 1 p
A0
Ǔ
ǒ1ńA1Ǔ
,
. . . . . . . . . . (9.18)
. . . . . . . (9.19)
with R sw in scf/STB. This relation fits the Dodson-Standing data
at 150, 200, and 250°F but overpredicts the effect of dissolved gas
at 100°F.
Dodson and Standing also give a correction for the effect of dissolved gas on water/brine compressibility.
c wǒ p, T, R swǓ + c *wǒ p, TǓ ǒ1 ) 0.00877 R swǓ ,
. . . . . . . . (9.20)
with R sw in scf/STB. This relation is valid only for undersaturatedoil/water systems at higher than oil bubblepoint pressure. For gas/
water systems and saturated-oil/water systems, the total compressibility effect is given by the Perrine formula,28
6
T, R sw
A2 +
ȍa
mo
log m ow +
w20
1.5Ǔ
,
B wǒ p, T, R swǓ + B *wǒ p, TǓǒ1 ) 0.0001 R sw
ǒ Ǔ
i
1i c sw
,
i
2 ic sw
,
i+1
where A0 and A1 are given by Eq. 9.17. Eq. 9.18 results in water and
brine densities that are within 0.5% of values given by Rogers and Pitzer’s27 highly accurate correlation for 60tTt400°F, 0tpt15,000
psia, and 0t C st300,000 ppm. For the same range of conditions,
Eq. 9.17 calculates isothermal compressibilities within approximately 5% of Rogers and Pitzer’s values.
With Dodson and Standing’s13 data for pure water saturated with
a natural gas, an approximate correction for dissolved gas on water/
brine FVF at saturated conditions is
ēB w
c tw + * 1
B w ēp
ȍa
[0.8 ) 0.01(T * 90) exp(* 0.25c sw)],
3
. . . . . . . . . . . . . . . . . (9.17)
w s in weight fraction of NaCl.
with
Solving Eq. 9.16 for the FVF of a brine without solution gas, B *w, gives
+
A 0 + 10
10 *6ǓT 2ƫ
B owǒ p sc, TǓ
w20
*3
3
10 *4ǓT
in psi*1, p in psia, T in °F, and
B *wǒ p, TǓ
w
i+1
and A 1 + 8 ) 50w s * 0.125w sT,
c *w
mo
m *w
log m o + A 1 ) A 2 log m ow ,
ǒ Ǔ.
B g ēR sw
) 1
5.615 B w ēp
T
4
i+1
ǒ20 * T Ǔ i
,
3i
96 ) T
and m ow20 + 1.002 cp,
. . . . . . . . . . . . . . . . . . . . . . . . . . (9.22)
where a11+3.324 10*2, a12+3.624 10*3, a13+*1.879 10*4,
a21+*3.96 10*2, a22+1.02 10*2, a23+*7.02 10*4, a31+
1.2378, a32+*1.303 10*3, a33+3.060 10*6, a34+2.550
10*8, with m in cp, T in °C, and p in MPa. Kestin et al.’s pressure
correction A0 contains 13 constants and does not extrapolate well at
high temperatures. The pressure correction for A0 in Eq. 9.22 is
more well-behaved, with only small deviations from the original
Kestin et al. correlation at low temperatures.
The effect of dissolved gas on water viscosity has not been reported. Intuitively, one might suspect that water viscosity decreases
with increasing gas solubility, although Collins30 suggests that dissolved gas may increase brine viscosity. As Fig. 9.10 shows, systems saturated with CO2 show an increase in viscosity with increasing gas solubility.
9.2.7 Solubility of Water in Natural Gas. Fig. 9.11 shows the solubility of pure water in methane. McKetta and Wehe31 give two chart
inserts for correcting pure-water solubilities for salinity and gas
gravity (based mainly on Dodson and Standing’s13 values). A bestfit equation for these charts is
y w + y ow A g A s,
ln y ow +
. . . . (9.21)
ȍa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.23a)
0.05227p ) 142.3 ln p * 9, 625
T ) 460
* 1.117 ln p ) 16.44,
. . . . . . . . . . . . . . . . . (9.23b)
PHASE BEHAVIOR
Dewpoint of Natural Gas
J.J. McKetta and A.H. Wehe,
U. of Texas (1958)
Fig. 9.11—Water solubility in natural gases, including gas-composition and salinity effects
(adapted from Ref. 31).
Ag + 1 )
(1.55
g g * 0.55
10 4)g g T *1.446 * (1.83
10 4) T *1.288
,
. . . . . . . . . . . . . . . . . . (9.23c)
A s + 1 * ǒ2.222
and A s + 1 * ǒ3.92
10 *6ǓC s ,
. . . . . . . . . . . . . . . . . (9.23d)
10 *9 ǓC 1.44
s ,
. . . . . . . . . . . . . . . (9.23e)
with T in °F, p in psia, and C s in ppm or mg/L. Eq. 9.23 yields an
absolute average deviation of 2.5% for y ow, with a maximum error
WATER/HYDROCARBON SYSTEMS
less than 10% for 100tTt460°F and 200tpt10,000 psia. Eq.
9.23d is from the Dodson-Standing correlation and is not recommended. Eq. 9.23e is from the Katz et al.32 correlation and is recommended. Mole fraction of water in gas, y w, can be converted to a water/gas ratio, r sw , with
r sw + 135
yw
[ 135y w ,
1 * yw
. . . . . . . . . . . . . . . . . . . (9.24)
where r sw is in STB/MMscf. Replacing the constant 135 with
47,300 yields r sw in lbm/MMscf.
7
ow
Fig. 9.12—Water/brine/oil IFT data correlated with the McLeod
parameter (adapted from Ref. 33).
Temperature, °F
The correction term for salinity that Dodson and Standing13 proposed is based on limited results for one low-salinity brine. The Katz
et al.32 salinity correction is based on lowering of vapor pressure for
brine solutions at 100°C, where the assumption is made that
p vw(100°C, C s)
yw
y ow [ p ovw(100°C) ,
. . . . . . . . . . . . . . . . . . . . . . . . . (9.25)
where p vw+brine vapor pressure and p w+pure-water vapor pressure, both measured at 100°C. Very little data are available to confirm these two salinity corrections. However, EOS calculations indicate that the Katz et al. correlation is probably valid up to M[3;
at higher molalities, the EOS-calculated ratio y wńy is less than that
predicted by the Katz et al. correlation (see Sec. 9.3).
Finally, water dissolved in reservoir gas and oil mixtures will not
contain salts (i.e., it is fresh water), a fact that can help in identifying
where produced water comes from.
9.2.8 Water/Brine/Hydrocarbon IFT. The IFT of water/hydrocarbon
systems, Ds wh , varies from approximately 72 dynes/cm for water/
brine/gas systems at atmospheric conditions to 20 to 30 dynes/cm for
water/brine/stock-tank-oil systems at atmospheric conditions. The variation in s wh is nearly linear with the density difference between water
and the hydrocarbon phase Ds wh (i.e., Ds wo or Ds wg ), where
s wh+72 dynes/cm at Ds wh + Ds wg +1. This can be expressed in
equation form as
s wh + s o ) (72 * s o)Dò wh , . . . . . . . . . . . . . . . . . . . (9.26)
where s o+intercept at Dò wh +0.
Ramey33 proposes a correlation for s wh based on the Macleod parameter s ¼ńDò. This parameter was plotted vs. Dò (Fig. 9.12) with
data for brines with stock-tank oil, saturated and undersaturated reservoir oils, and natural gases. Eq. 9.26, where s o+15, represents
Ramey’s graphical correlation surprisingly well. A near-exact fit of
his correlation is
s wg + 20 ) 36 Dò wh .
Fig. 9.13—Water/brine/oil IFT data correlation (adapted from
Ref. 34).
ported by various authors show considerable scatter, and it seems
that any correlation will give only approximate IFT values for such
systems until consistent data become available. Mutual-solubility
effects of gas dissolved in water and water dissolved in gas may affect IFT’s, perhaps explaining some of the difference in methane/
brine and methane/water IFT’s in Fig. 9.14. Otherwise, the seemingly erratic behavior of some water/brine/oil IFT data may be
explained by aromatic compounds and asphaltenes. Also, crude-oil
samples exposed to atmospheric conditions for long periods of time
may experience oxidation that can affect IFT measurements.
. . . . . . . . . . . . . . . . . . . . . . . (9.27)
Ramey’s data that do not lie on his general correlation are accurately
represented by Eq. 9.26, with s o ranging from 5 to 30. Fig. 9.13
shows a graphical correlation for s wg given by Standing34 for water/
brine/methane systems (apparently based on Hocott’s35 naturalgas/brine data).
Firoozabadi and Ramey36 consider the IFT of water and hydrocarbons using data for distilled water and pure hydrocarbons. They
arrive at a graphical relation similar to Ramey’s33 original correlation, with the addition of reduced temperature as a correlating parameter. Unfortunately, their correlation does not predict water/
brine/oil IFT’s with more accuracy than the original Ramey
correlation (or Eq. 9.27). As Fig. 9.14 shows, water/gas IFT’s re8
Salt Concentration, ppm
Fig. 9.14—Methane/water and methane/brine IFT’s.
PHASE BEHAVIOR
which can be used for 0.44t T rt0.72 (60tTt400°F). Alternatively, the Søreide-Whitson9 relation for a H O can be used with the
2
Peng-Robinson 38 EOS (PR EOS).
ƪ
1.1Ǔ
ǒ
a 0.5
H O + 1 ) 0.453 1 * T r H O 1 * 0.0103c sw
2
ǒ
2
) 0.0034 T r*3
*1
H O
Fig. 9.15—Pure-water and NaCl-brine vapor-pressure curves.
9.3 EOS Predictions
Mutual solubilities and volumetric properties of water/hydrocarbon
systems can be predicted with reasonable accuracy with one of several modifications to existing cubic EOS’s. Other types of EOS’s also
have been applied to these systems but do not show a clearly superior
predictive capability. Although cubic EOS’s are not widely used for
reservoir water/hydrocarbon systems, this approach eventually is expected to replace the empirical correlations currently being used.
To improve vapor-pressure predictions of water (and solubilities of
water in the nonaqueous phase), Peng and Robinson37 proposed a
modified correction term, a (applied to EOS Constant a), for water.
aH
2O
+ ƪ1.008568 ) 0.8215ǒ1 *
0.5
T rw
Ǔƫ ,
2
. . . . . . . . (9.28)
2
Ǔƫ .
. . . . . . . . . . . . . . . . (9.29)
Eq. 9.29 predicts pure-water vapor pressures within 0.2% of steamtable values for 0.44t T rwt1 (i.e., Tu60°F) and can be used to predict vapor pressures of NaCl solutions with the same accuracy. Fig.
9.15 shows vapor pressures of pure water and NaCl-brine solutions reported by Haas.10 With a correction for salinity in the a term, the predicted water solubilities in nonaqueous phases are expected to improve.
The most important modification of existing cubic EOS’s for water/hydrocarbon systems is the introduction of alternative mixing
rules for EOS Constant A, where different binary-interaction parameters (BIP’s), k ij, are used for the aqueous and nonaqueous (hydrocarbon) phases. Peng and Robinson37 propose a simple EOS modification for hydrocarbon/water systems; namely, they define two
sets of k ij: k ij,HC for the hydrocarbon phase(s) and k ij,aq for the
aqueous phase. EOS Constant A is therefore calculated differently
for the hydrocarbon and aqueous phases,
ȍȍy
N
A HC +
N
i,HC y j,HC
A i A j ǒ1 * k i j,HCǓ
i+1 j+1
ȍȍx
N
and A aq +
N
i,aq x j,aq
A i A j ǒ1 * k i j,aqǓ ,
. . . . . . . . . (9.30)
i+1 j+1
TABLE 9.3—RECOMMENDED BIP’s FOR THE PR EOS TO PREDICT SOLUBILITIES
IN WATER/HYDROCARBON SYSTEMS*
Aqueous Phase
k ij, aq + ǒ1 ) a 0c swǓA 0 ) ǒ1 ) a 1c swǓA 1T ri ) ǒ1 ) a 2c swǓA 2T 2ri ,
Hydrocarbons
where a 0 + 0.017407, a 1 + 0.033516, a 2 + 0.011478
A 0 + 1.112 * 1.7369w *0.1
, A 1 + 1.1001 ) 0.83w i
i
A2+*0.15742*1.0988wi , i+hydrocarbons, and j+water/brine.
0.75Ǔ
Ǔ
ǒ
k ij, aq + * 1.70235ǒ1 ) 0.025587c 0.75
sw ) 0.44338 1 ) 0.08126c sw T ri ,
N2
where i+N2 and j+water/brine.
Ǔ
k ij, aq + * 0.31092ǒ1 ) 0.15587c 0.75
sw
CO2
Ǔ
) 0.2358ǒ1 ) 0.17837c 0.98
sw T ri * 21.2566 exp(* 6.7222T r * c sw),
where i+CO2 and j+water/brine.
k ij, aq + * 0.20441 ) 0.23426T ri , where i + H 2S and j + waterńbrine.
H2 S
Nonaqueous Phase
i
kij ,HC, where j+water
C1
0.4850
C2
0.4920
C3
0.5525
C4
0.5091
C5)
0.5000
N2
0.4778
CO2
0.1896
H2 S
0.19031*0.05965Tri
Acentric factors w used in developing hydrocarbon/water BIP’s are C1+0.0108, C2+0.0998, C3+0.1517, and C4+0.1931.
*Modified Peng-Robinson a term for water/brine, Eq. 9.29.
WATER/HYDROCARBON SYSTEMS
9
Brine Salinity, Csw
fff 0
V V V 0.86
1.71
2.57
+++ 3.42
KKKK 5.13
D D D Katz et al.32 Correlation
V V V Calculated at 100°F
Calculated at 250°F
Fig. 9.16—Predicted gas-phase water solubilities for methane/
NaCl-brine mixtures at 250°F determined with the general aw
term (Eq. 9.31).
respectively, where y i,HC +hydrocarbon composition (gas or oil) and
x i,aq+water-phase composition. Using two sets of k ij has been applied successfully to correlate mutual solubilities of hydrocarbon/
water and nonhydrocarbon/water binary systems. Table 9.3 gives
recommended k ij relations for aqueous and nonaqueous phases for
the PR EOS, where these interaction coefficients must be used with
the general a H O relation (Eq. 9.29). The CO2/water/brine correla2
tion gives the best results at pressures less than approximately 5,000
psia because data in this region have been given more weight in development of the correlation.
Considerable data on solubilities of hydrocarbon and nonhydrocarbon gases in brine solutions were used in making the salinity
corrections for aqueous-phase k ij. Similar data were not available
for solubilities of water in the nonaqueous phase for mixtures containing brines. Until more data become available, it will be necessary to assume that the effect of salinity is adequately treated by the
modified a H O term (Eq. 9.30).
2
Fig. 9.16 shows predicted water solubilities for methane/NaClbrine mixtures with varying salt concentration. The predicted reduction in water solubility for mixtures containing brine, relative to solubility for mixtures containing pure water, is more or less
independent of pressure and temperature. Fig. 9.17 correlates the ratio y wńy ow calculated by the modified PR EOS (with a H O from Eq.
2
9.30) vs. salinity. The effect of salinity is clearly less than that predicted by the Dodson-Standing13 correlation (Eq. 9.23d), whereas
the Katz et al.32 correlation (Eq. 9.23e) appears to be consistent with
the EOS calculations up to M[3.
Simultaneous application of aqueous- and nonaqueous-phase interaction coefficients requires modification of the standard EOS implementation (which uses a single set of k ij). Figs. 9.18 through
9.22 show the accuracy of this approach for mutual-solubility predictions of binaries and natural-gas/water/brine mixtures, suggesting that the required modification is probably warranted.
A standard implementation of the PR EOS can still be used with
the BIP’s in Table 9.3. If only gas solubility in the water phase is
needed, accurate gas solubilities can be predicted with the aqueousphase k ij,aq for both phases; however, calculated hydrocarbon-phase
composition will not be accurate. Likewise, if only water solubility
in the hydrocarbon phase is needed, the hydrocarbon-phase k ij,HC
can be used for both phases, but calculated aqueous-phase compositions will not be accurate in this case. Fig. 9.23 compares experimental solubilities for the methane/water system with results predicted by the modified PR EOS (with two sets of k ij) and by the
original PR EOS with a single set of k ij.
10
Fig. 9.17—Effect of salinity on gas-phase water solubility
for methane/NaCl-brine mixtures determined with the general
aw term (Eq. 9.31).
Composition- and density-dependent mixing rules have also been
proposed for modifying cubic EOS’s for water/hydrocarbon systems. Panagiotopoulos and Reid’s39 linear composition-dependent
mixing rule has received considerable interest. Unfortunately, as
Kistenmacher and Michelsen40 point out, it violates several fundamental thermodynamic conditions. Enick et al.8 propose temperature-dependent correction terms for both EOS Constants A and B of
water, together with a linear composition-dependent mixing rule for
Constant A. With this approach, they successfully describe multiphase equilibria for a multicomponent water/oil/CO2 system.
Several noncubic EOS’s3-5,41,42 have been proposed for water/
hydrocarbon systems, including conventional activity-coefficient
models that are limited to relatively low pressures and more general
electrolyte EOS models. However, these models do not appear to be
better than the simpler modifications of cubic EOS’s.
9.4 Hydrates
Gas hydrates are solutions of gases in crystalline solids called clathrates. Gas molecules occupy the void spaces (cages) in the watercrystal lattice. Hydrates can form at temperatures considerably higher
than the freezing point of pure water. For example, in high-pressure
wells (more than 15,000 psia), hydrates have been observed at temperatures much higher than 100°F. Hydrates resemble wet snow and,
like ice, will float on water. In the oil field, hydrates look like a grayish
snow cone. When hydrate “snow” is tossed on the ground, the hydrocarbons escaping can be heard easily, giving the impression that the
hydrocarbons were physically trapped in the snow. The distinctive
crackling sound is in fact caused by escaping natural-gas molecules
rupturing the crystal lattice of the hydrate molecules.
Hydrates were discovered in 1810 by Davy and were investigated
only as curiosities of physical chemistry for many years thereafter.43 In
1888, Villard became the first to determine the existence of hydrates
with typical components of natural gas, such as methane, ethane, and
propane.43 However, the real push to measure hydrate phase behavior
did not begin until the 1930’s when Hammerschmidt44 pointed out that
hydrates were the culprits that were choking wellhead and production
equipment in gas fields. He also suggested ways to inhibit their formation. Although hydrate inhibition has been practiced for more than 50
years, the severe conditions encountered in arctic and deep drilling
have sparked a new wave of interest in measurement of hydrate formation and inhibition at these conditions.
Although the kinetics and fluid mechanics of hydrate formation
and dissociation are not covered here, they are nonetheless important in deepwater drilling operations. Because vast deposits of natuPHASE BEHAVIOR
Mole Fraction Water in Vapor Phase, yw
Fig. 9.18—EOS predictions of mutual solubilities for methane/
water system determined with different sets of BIP’s for aqueous
and nonaqueous phases.
ral-gas hydrates exist in the Arctic, a great deal of Russian research
has been conducted on both the kinetics and thermodynamics of hydrate formation and dissociation.43 Recovery of natural gas entrapped in these vast hydrate deposits in permafrost regions (by hydrate dissociation) has also been studied recently.45
The three most widely used calculation methods for predicting
hydrate formation are (1) the vapor/solid K-value method of Katz
and his coworkers46-51 and equations fitting the developed K-value
charts; (2) methods of Campbell and his coworkers52-54; and (3)
combined methods based on statistical thermodynamics (van der
Waals and Platteeuw55) for the hydrate phase and EOS’s for the fluid
phases. These methods are discussed later.
9.4.1 Crystallography of Hydrates. In the presence of a free-water
phase, hydrates will form below a certain temperature often referred
to as the “hydrate temperature.” Hydrate crystals generally grow only
in the presence of a free-liquid-water phase at typical oilfield conditions. Hydrates can also form in the presence of a dense-vapor-water
phase at temperatures sufficiently low to ensure hydrogen bonding.
The general conditions under which hydrates form include gas at or
below its water dewpoint (which can yield the free-water phase necessary for hydrate formation in the system) and conditions at moderately low temperature or high pressure. With respect to components
WATER/HYDROCARBON SYSTEMS
Mole Fraction Natural Gas, xng
Fig. 9.19—EOS predictions of mutual solubilities for naturalgas/water system determined with different sets of BIP’s for
aqueous and nonaqueous phases.
normally found in natural gas, hydrate formation has been observed
and measured only for the light constituents found in natural gas: C1
through C4 alkanes (including i-C4), N2, CO2, and H2S.
Fig. 9.24 shows a schematic of the natural-gas hydrate-crystal
lattice. Two common types of hydrate-crystal structures have been
proposed from interpretation of results of von Stackelberg and
Müller’s56 X-ray diffraction studies of hydrates. Structure I is usually a body-centered lattice, and Structure II has a diamond lattice.
Structures I and II have different sized cages (i.e., void spaces). In
Structure I hydrates, methane can fill the smaller cages, while the
larger cages can be filled only by larger hydrocarbon molecules,
such as ethane. The cages in Structure II hydrates are larger, allowing entrapment of propane and i-butane in addition to methane and
ethane. Fig. 9.25 summarizes the components and corresponding
size ranges that fit into Structure I and II cavities. Light components,
such as methane, ethane, and CO2, form Structure I hydrates; nitrogen and the heavier alkanes, such as propane, n-butane, i-butane,
and neopentane, form Structure II hydrates.
Enough cages must be filled with hydrocarbon molecules to stabilize the crystal lattice. Because all the cages do not have to be full,
11
Mole Fraction Natural Gas, xng
Mole Fraction CO2 in Aqueous Phase, xCO 2
Fig. 9.20—EOS prediction of gas solubility for CO2/water/brine
systems at 302°F determined with different sets of BIP’s for
aqueous and nonaqueous phases; symbols+experimental and
lines+calculated.
the molecular weight of a clathrate hydrate is not fixed. The “vacancy” of the hydrate-crystal lattice depends on which “guest” naturalgas molecules happen to be available to occupy the void locations
between the interstices of the host water molecules and on the conditions under which the crystal lattice is formed. Thus, the presence
of methane and ethane leads only to the formation of Structure I hydrates and the presence of methane, ethane, and propane leads to the
formation of a mixture of Structure I and II hydrates.
The general trends of hydrate formation can be qualitatively predicted for a particular natural-gas component. The two important factors in formation of the two different structures of hydrates are size
and solubility of the natural-gas molecules. The rate of clathration is
partially dependent on solubility because the more soluble a gaseous
component is in water, the higher the probability that it will be
“caught” in a cage as the hydrate crystal is being formed. The size of
the guest molecule not only determines the structure type but also the
rate of formation. For example, comparing the rate of clathration of
methane with that of ethane, a higher pressure is required to form pure
methane hydrates than pure ethane hydrates, even though methane is
considerably more soluble in water than ethane. The reason is that
methane is a smaller molecule that is more difficult to entrap as the
cage of the crystal lattice closes. Furthermore, hydrates form more
readily from natural-gas mixtures than from pure components because the range of molecular sizes in natural gas has a higher probability of filling enough cavities to stabilize the hydrate-crystal lattice.
Researchers only recently proved that butanes are hydrate formers. McLeod and Campbell53 and others showed that the butanes
are hydrate formers when methane is present to occupy the smaller
cavities in Structure II hydrates. They found that, like ethane and
propane and in contrast to n-pentane, the butanes lower the hydrateforming pressure. Hydrates with n-butane are very unstable, and, at
pressures higher than 10,000 psia, n-butane behavior reverts to that
of a nonhydrate former. Alkanes with a higher carbon number than
n-butane are not believed to form hydrates.
9.4.2 Phase Diagrams for Hydrates. At cryogenic temperatures
and subatmospheric pressures, phase diagrams show a multitude of
hydrate forms. We cover only the simpler phase diagrams that represent the most common conditions encountered in subsurface engineering and in surface facilities.
The temperature and pressure conditions for hydrate formation in
surface gas-processing facilities are generally much lower than
those considered in production and reservoir engineering. The
12
Mole Fraction Natural Gas, xng
Fig. 9.21—EOS prediction of gas solubility for natural-gas/brine
system determined with different sets of BIP’s for aqueous and
nonaqueous phases.
conditions of initial hydrate formation are often given by simple p-T
phase diagrams for water/hydrocarbon systems. In 1885, Roozeboom defined a lower hydrate quadruple point, Q 1 ( IńL wńHńV),
and an upper quadruple point, Q 2 ( L wńHńVńL HC), as on Fig. 9.26.43
His nomenclature for the phases is I+pure ice, L w+liquid water,
L HC+liquid hydrocarbon, V+vapor, and H+hydrate. The quadruple point defines the condition at which four phases are in equilibrium. Because the Gibbs phase rule leads to zero degrees of freedom
for this system, the values of these quadruple points (Table 9.4) for
the eight natural-gas hydrate formers are unique and invariant and
provide a quantitative basis for classification of hydrate formers.
Each quadruple point is at the intersection of four three-phase
lines. The lower quadruple point, Q 1, represents the transition of L w
to I. As temperature decreases to Point Q 1, hydrates cease forming
from vapor and liquid water and are forming from vapor and ice.
The upper quadruple point, Q 2, is the approximate intersection of
Line L wńHńV with the vapor pressure of the hydrate former and represents the upper temperature limit for hydrate formation for that
component. Some of the lighter natural-gas components, such as
methane and nitrogen, do not have an upper quadruple point, so no
upper temperature limit exists for hydrate formation. This is the reason that hydrates can still form at high temperatures (up to 120°F)
in the surface facilities of high-pressure wells.
PHASE BEHAVIOR
Experimental
kij Different for Each Phase (modified aw),
kij Same for Each Phase (modified aw),
kij Same for Each Phase (original aw)
kij+0.485 (nonaqueous phase)
kij+*0.260 (nonaqueous phase)
Mole Fraction Nitrogen in Aqueous Phase, xN 2
Fig. 9.22—EOS prediction of gas solubility for N2/NaCl-brine
system at 217°F determined with different sets of BIP’s for
aqueous and nonaqueous phases; symbols+experimental and
lines+PR EOS predicted.
Fig. 9.27 shows the main area of hydrate formation in petroleumengineering applications. Line FEG represents the natural-gas-mixture dewpoint curve. The dewpoint line is analogous to the vaporpressure curves of the individual components in Fig. 9.26. Point E
is the maximum hydrate-forming temperature (analogous to the
quadruple points, Q 2, of the individual components in Fig. 9.26).
The hydrate curve is Line BE. At the intersection of the dewpoint
and hydrate curves, the hydrate curve for many natural-gas systems
becomes nearly vertical and establishes the maximum hydrateforming temperature. For a natural-gas system with very high concentrations of methane, such as encountered in the deep natural-gas
plays in the Anadarko basin, the maximum hydrate-forming temperature may be essentially nonexistent (observe that no Q 2 exists
for the methane curve in Fig. 9.26). The general approach to hydrate
prediction in most engineering applications is to determine Hydrate
Line BE and the position of Dewpoint Line FEG on Line BE. Sec.
9.4.3 discusses calculation methods.
Fig. 9.28 shows Deaton and Frost’s58 data for hydrate-formation
conditions for methane/propane mixtures. These data show how hydrate-formation conditions for natural gas are strongly dependent
on the propane concentration. The general effect of increasing propane concentration is to lower the hydrate-forming pressure and to
increase the hydrate-forming temperature. Katz et al.32 and Wilcox
et al.49 developed Fig. 9.29 to determine hydrate-forming conditions for natural gas at different specific gravities. Because Fig. 9.29
is based on gas gravity, it is particularly useful as a quick guide to
estimate the hydrate temperature for a natural gas. Fig. 9.29 should
not be used if CO2 or H2S is present at a combined concentration y1
mol%. At pressures less than 12,000 psia, the Joule-Thompson expansion of a natural gas, for example, across a separator choke, reduces the temperature of the gas. Katz et al.32 present charts (Figs.
9.30 through 9.32) that show the maximum permissible expansion
of natural gases before hydrate formation occurs.
9.4.3 Calculation Method of Katz and Coworkers.32,47,49,51 By
applying the analogy of vapor/liquid equilibrium K values to a solid
solution, Carson and Katz47 and Wilcox et al.49 developed the concept of a vapor/solid K value for predicting the temperature and
pressure conditions under which hydrates form or dissociate.
K i (v*s)
y
+ x i ,
i (s)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.31)
WATER/HYDROCARBON SYSTEMS
Experimental
kij Different for Each Phase (modified aw),
kij Same for Each Phase (modified aw),
kij Same for Each Phase (original aw)
kij+0.485 (nonaqueous phase)
kij+*0.260 (nonaqueous phase)
Fig. 9.23—Predicted mutual solubilities of methane/water system at 100°F determined with the modified PR EOS with one and
two sets of kij ; all kij from Table 9.3.
where K i(v*s)+vapor/solid equilibrium value of Component i,
y i+gas composition, and x i(s)+mole fraction of Component i in the
solid on a water-free basis. Calculation of hydrate-formation temperature is analogous to calculation of a dewpoint temperature (discussed in Chap. 3). A gas in the presence of a free-water phase will
form a hydrate if
ȍK y
N
i
y 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.32)
i (v*s)
i+1
Conversely, hydrate-dissociation temperature can be treated like a
bubblepoint calculation. A hydrate will dissociate if
ȍx K
N
i
i (v*s)
y 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.33)
i+1
Because K i(v*s) is based on the mole fraction of a guest naturalgas component in the solid-phase hydrate mixture on a water-free
basis, the concept of K i(v*s) is only an approximation of the original
13
Hydrate Former
3Å
Cavities
Occupied
No Hydrates
A
Kr
N2
O2
4Å
52/3 H2O
512 + 51264
S-II
53/4 H2O
512 + 51262
S-I
CH4
Xe; H2S
5Å
CO2
C2H6
6Å
(CH2)3O
Fig. 9.24—Schematic of hydrate-crystal lattice; circles represent
water molecules, lines represent hydrogen bonds (from Ref. 52).
definition of vapor/liquid equilibrium ratios, K i. For example, the
concept of the vapor/solid K value cannot be used to calculate hydrate-phase splits or equilibrium-phase compositions. The vapor/
solid K value can be used only to predict the temperature or pressure
where hydrates form or dissociate. However, on the basis of component K i(v*s) values, where the natural-gas components will concentrate can be determined qualitatively. If K i(v*s) for a natural-gas
component is greater than unity (nitrogen is a typical example), the
component will tend to concentrate in the gaseous phase rather than
in the hydrate phase. If K i(v*s) is less than unity (for example, propane), the component will tend to concentrate in the hydrate. Katz
and his coworkers provide K i(v*s) nomograms for several naturalgas components as functions of temperature and pressure.
Sloan57 developed the following polynomial-fit equation of the
Katz-Carson charts, which can be used to estimate K i(v*s).
ln K i(v*s) + A 0 ) A 1 T ) A 2 p ) A 3 T *1 ) A 4 ńp ) A 5 pT
) A 6 T 2 ) A 7 p 2 ) A 8 ǒ pńTǓ ) A 9 ln ǒ pńT Ǔ
) A 10 p *2 ) A 11 ǒ Tńp Ǔ ) A 12 ǒ T 2ńp Ǔ ) A 13 ǒpńT 2Ǔ
) A 14 ǒTńp 3Ǔ ) A 15 T 3 ) A 16 ǒp 3ńT 2Ǔ ) A 17 T 4 .
. . . . . . . . . . . . . . . . . . (9.34)
Table 9.5 gives the values of Constants A0 through A17.
K i(v*s) for nonhydrate formers are assumed to be infinity in the
calculation. The original work assumed that nitrogen and butanes
were not hydrate formers, which was subsequently shown to be incorrect. However, fairly reliable estimates can be obtained by assuming that the K i(v*s) for nitrogen and the butanes are also infinity
as long as the pressure is less than approximately 1,000 psia. This
method becomes less reliable for pressures higher than 1,000 psia.
9.4.4 Calculation Methods of Campbell and Coworkers.52-54 To
address the pressure limitations of the K i(v*s) method of Katz and his
coworkers as well as the hydrate-temperature-depression effects of
molecules too large to fit into the cavities of the hydrate crystal,
Campbell and his coworkers52-54 developed additional empirical
procedures. In general, these methods can be used for quick estimates
of hydrate-formation temperatures when pressures exceed the
1,000-psia limitation of the K i(v*s) method. The Trekell-Campbell54
14
51262
S-I
72/3 H2O
c-C3H6
C3H8
iso-C4H10
7Å
51264
S-II
17 H2O
n-C4H10
No S-I or S-II Hydrates
8Å
Fig. 9.25—Summary of natural-gas components fitting into Structure I and II (S-I and S-II, respectively) cavities (from Ref. 57).
method covers pressures from 1,000 to 6,000 psia, and the
McLeod-Campbell53 method covers pressures from 6,000 to
10,000 psia.
The Trekell-Campbell method calculates additive effects of gas
molecules on the hydrate-forming temperature of methane. They
give eight nomograms, six of which give positive displacements as
functions of pressure for C3, n-C4, and i-C4 and two that give negative corrections (depression) for nonhydrate formers, such as C5+.
This method is strictly empirical and must be used with caution, but
it is useful as a quick estimate at pressures up to 6,000 psia.
McLeod and Campbell developed another method to predict hydrate-formation temperatures at very high pressures encountered in
deep-gaswell drilling. They prepared a very simple correlation
based on a modified Clapeyron equation to describe the energy of
phase transition at pressures 6,000 to 10,000 psia.
T + 3.89 ǸC
ȍy C ,
N
and C +
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.35)
i+1
where T is in °R, y i+gas molar composition, and Table 9.6 gives
the hydrate-former constants C i for C1 through C4 hydrocarbons.
The hydrate-prediction methods of Campbell and his coworkers
are mostly empirical but do provide a reliable answer when computer
programs for the more theoretical models described in the next secPHASE BEHAVIOR
van der Waals and Platteeuw’s statistical-mechanical solid-solution
theory of clathrates. These authors developed an adsorption model
based on statistical mechanics to derive a relation for the chemical
potential of water in the hydrate phase. Their method is based on an
equation that relates the chemical potential of water in the hydrate
structure in much the same way that chemical potential of a component is related to the activity of a component in a mixture.
m i + m oi ) RT ln a i ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (9.36)
where m i+chemical potential of pure Component i (see Chap. 4)
and a i+activity of Component i in the mixture.
van der Waals and Platteeuw propose the following Langmuir adsorption-isotherm analogy that accounts for the microscopic hydrate structure.
m wH + m wMT ) RT
ȍn
i
ci ln
ǒ
1*
ȍy
j
Ǔ
ji
,
. . . . . . . . (9.37)
where m wH+chemical potential of water in the filled hydrate,
m wMT +chemical potential of water in the empty hydrate,
n ci +number of Type i cavities per water molecule in hydrate-crystal lattice, and y ji+fraction (probability) of Type j molecule occupying Type i cavity.
The Langmuir adsorption theory is applicable because “clathration” and “declathration” are analogous to adsorption and desorption,
respectively. The probability term, y ji, depends on the interaction between the guest gas molecule and its “cage” (the “site” by analogy
with the original Langmuir theory). The term y ji also depends on the
fugacities of the components in the gas phase, which can be calculated with an EOS. Parrish and Prausnitz59 were the first to extend the
van der Waals and Platteeuw statistical-mechanics model to multicomponent systems. They used the Kihara potential to calculate the
Langmuir constants. John et al.60 and Schroeter et al.61 also used the
Kihara potential to calculate the Langmuir constants.
Erickson and Sloan62 developed a calculation procedure using the
van der Waals and Platteeuw model. A computer program
(CSMHYD) is included with Ref. 62, and a complete description of
the calculation algorithm and computer-program flow chart are also
provided. Ref. 63 provides a description and algorithm for a similar
approach. These methods are fairly difficult to program from the literature; therefore, Ref. 62 with the program diskettes is recommended.
Other researchers have developed calculation methods based on
the van der Waals and Platteeuw in combination with an EOS. Ng
et al.64 and Robinson and Mehta65 made predictions of hydrateformation conditions using the PR EOS and developed a computerbased method that is available through the Gas Processors Assn.
Schroeter et al.61 used the Benedict et al. EOS66 to model the fluid
phase in hydrate calculations with sour-gas (including H2S) systems. Munck et al.67 used the Soave-Redlich-Kwong EOS with the
van der Waals and Platteeuw adsorption model to calculate fugacities of liquid and gaseous phases in equilibrium with hydrates. They
used the Michelsen68 stability algorithms (see Chap. 4) to develop
a computer program that predicts hydrate-formation conditions
without prior knowledge of the phases. To account for the effects of
nonelectrolyte inhibitors, Munck et al.67 used the UNIQUAC activity-coefficient model. They obtained good agreement for hydrates
in equilibrium with North Sea reservoir fluids.
MT
Fig. 9.26—Hydrate-formation conditions for natural-gas hydrate
formers (from Ref. 57).
tion are not available. They can also be used as a check of the more
sophisticated estimation methods (also described in the next section).
9.4.5 van der Waals and Platteeuw55 Model. Most modern computer-based methods of predicting hydrate formation are based on
WATER/HYDROCARBON SYSTEMS
9.4.6 Water Content of Vapor in Equilibrium With Hydrates. The
concentration of water in the vapor phase in equilibrium with hydrate
is usually very small, on the order of 0.001 mol% or less. Phase diagrams and nomograms for determining the water content of vapor in
equilibrium with hydrates are complicated by metastable equilibrium
in the gas/ice region and are cumbersome to use for the many possible
combinations of compositions. Song and Kobayashi69 present a
mathematical approach for determining the water content of gases in
the vapor/hydrate region. They studied methane-rich and CO2-rich
systems, which are especially important in EOR operations (Fig.
9.33). Sloan55 proposes a slight improvement to the Kobayashi et
al.50 method and provides the necessary equations, along with an extensive table of coefficients and an example of how to use the method.
15
TABLE 9.4—QUADRUPLE POINTS FOR NATURAL-GAS HYDRATE FORMERS (from Ref. 57)
Lower Quadruple, Q1
Upper Quadruple, Q2
Natural-Gas
Temperature
Pressure
Temperature
Component
(°F)
(psi)
(°F)
Pressure
C1
29.9
371.7
C2
31.9
76.9
58.4
491.7
C3
31.9
24.9
42.2
80.6
i-C4
31.9
16.4
35.3
24.2
CO2
31.9
182.2
49.7
N2
29.8
2,079.5
H2 S
31.4
13.5
(psi)
No Q2
652.5
No Q2
85.2
324.7
n-C4 does not form hydrate by itself; it requires the presence of a “help gas.”
Fig. 9.27—Characteristics of hydrate-forming natural-gas mixture at typical production conditions (from Ref. 52).
Algorithms for predicting hydrocarbon concentration in vapor in
equilibrium with hydrate are also discussed.
Methods for predicting hydrate-formation conditions have improved, and the prediction of hydrate-formation conditions with
the van der Waals and Platteeuw49 method can be used reliably. In
extreme operating conditions, such as those encountered in deep
drilling, calculation methods for predicting hydrate formation
may not be reliable. In these situations, laboratory measurements
are recommended.
9.4.7 Hydrate Inhibition. Hammerschmidt 70 presented a relation
for predicting the depression of the hydrate-forming temperature of
natural gases in contact with dilute aqueous solutions of antifreezes,
such as methanol and glycols (e.g., ethylene glycol). Hammerschmidt’s equation originates from the relationship for determining the
colligative properties (in this case, freezing or hydrate-forming
point) of an ideal solution.
DT [ 2, 335
16
w
,
100 M * w M
. . . . . . . . . . . . . . . . . . . (9.38)
Fig. 9.28—Hydrate-formation conditions for methane/propane/
water mixtures (from Ref. 57).
with DT in °F, M+molecular weight of the antifreeze agent (e.g.,
M+32 for methanol), and w+weight percent of the antifreeze
agent in solution. Fig. 9.34 shows how the hydrate-formation temperature is depressed with the addition of methanol to water and a
typical natural-gas component.
Use of the Hammerschmidt equation should be restricted to sweet
natural gases with antifreeze concentrations of less than 0.20 mol%.
Campbell51 suggests that for glycols, the factor 2,335 should be replaced by 4,000. For concentrated methanol solutions, like those
used to free a plugged-up tubing string in a high-pressure well, Nielsen and Bucklin71 propose the following modification of the Hammerschmidt equation.
DT + * 129.6 lnǒ1 * x MeOHǓ ,
. . . . . . . . . . . . . . . . . . (9.39)
PHASE BEHAVIOR
Initial Temperature, °F
Fig. 9.29—Temperature and pressure conditions of hydrate
formation for natural gases (from Ref. 32).
where DT+depression of the hydrate-forming temperature in °F
and x MeOH+mole fraction of methanol inhibitor.
References
1. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,
first edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).
2. Fetkovich, M.J., Reese, D.E., and Whitson, C.H.: “Application of a General Material Balance for High-Pressure Gas Reservoirs,” SPE Journal
(March 1998) 3.
3. Li, Y-.K. and Nghiem, L.X.: “Phase Equilibria of Oil, Gas, and Water/
Brine Mixtures From a Cubic Equation of State and Henry’s Law,” Cdn.
J. Chem. Eng. (June 1986) 64, 486.
4. Carroll, J.J. and Mather, A.E.: “Equilibrium in the System Water-Hydrogen Sulfide: Modelling the Phase Behavior with an Equation of State,”
Cdn. J. Chem. Eng. (1989) 67.
5. Michel, S., Hooper, H.H., and Prausnitz, J.M.: “Mutual Solubilities of
Water and Hydrocarbons From an Equation of State. Need for an Unconventional Mixing Rule,” Fluid Phase Equilibria (1989) 45.
6. Firoozabadi, A. et al.: “EOS Predictions of Compressibility and Phase Behavior in Systems Containing Water, Hydrocarbons, and CO2,” SPERE
(May 1988) 673.
WATER/HYDROCARBON SYSTEMS
Fig. 9.30—Maximum permissible expansion of 0.6-gravity natural gas without hydrate formation (from Ref. 32).
7. Nutakki, R. et al.: “Calculation of Multiphase Equilibria for Water-Hydrocarbon Systems at High Temperature,” paper SPE 17390 presented
at the 1988 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma, 17–20 April.
8. Enick, R.M., Holder, G.D., and Mohamed, R.: “Four-Phase Flash Equilibrium Calculations Using the Peng-Robinson Equation of State and A
Mixing Rule for Asymmetric Systems,” SPERE (November 1987) 687.
9. Søreide, I. and Whitson, C.H.: “Peng-Robinson Predictions for Hydrocarbons, CO2, N2 and H2S With Pure Water and NaCl-Brines,” Fluid Phase
Equilibria (1992).
10. Haas, J.L. Jr.: “Physical Properties of the Coexisting Phases and Thermochemical Properties of the H2O Component in Boiling NaCl Solutions,” Geological Survey Bulletin (1976) 1421-A and -B.
11. Patton, C.C.: Oil Field Water Systems, Campbell Petroleum Series, Norman, Oklahoma (1981).
17
Initial Temperature, °F
Initial Temperature, °F
Fig. 9.31—Maximum permissible expansion of 0.7-gravity natural gas without hydrate formation (from Ref. 32).
12. Rowe, A.M. Jr. and Chou, J.C.S.: “Pressure-Volume-Temperature-Concentration Relation of Aqueous NaCl Solutions,” J. Chem. Eng. Data
(1970) 15, 61.
13. Dodson, C.R. and Standing, M.B.: “Pressure, Volume, Temperature and
Solubility Relations for Natural Gas-Water Mixtures,” Drill. & Prod.
Prac. (1944) 173.
14. Culberson, O.L. and McKetta, J.J. Jr.: “Phase Equilibria in Hydrocarbon/
Water Systems. III The Solubility of Methane in Water at Pressures to
10,000 psi,” Trans., AIME (1951) 192, 223.
18
Fig. 9.32—Maximum permissible expansion of 0.8-gravity natural gas without hydrate formation (from Ref. 32).
15. Wiebe, R. and Gaddy, V.L.: “The Solubility of Carbon Dioxide in Water
at Various Temperatures from 12 to 40°C and at Pressures to 500 Atmospheres,” J. Amer. Chem. Soc. (1940) 62, 815.
16. Wiebe, R. and Gaddy, V.L.: “Vapor Phase Composition of Carbon Dioxide-Water Mixtures at Various Temperatures and at Pressures to 700 Atmospheres,” J. Amer. Chem. Soc. (1941) 63, 475.
17. Kobayashi, R. and Katz, D.L.: “Vapor-Liquid Equilibria for Binary Hydrocarbon-Water Systems,” Ind. Eng. Chem. (1953) 45, No. 2, 440.
18. Cramer, S.D.: “Solubility of Methane in Brines From 0 to 300°C,” Ind.
Eng. Chem. Proc. Des. Dev. (1984) 23, No. 3, 533.
PHASE BEHAVIOR
TABLE 9.5—VALUES OF COEFFICIENTS A0 THROUGH A17 IN EQ. 9.34
Coefficients
Component
A0
A1
A2
A3
A4
A5
CH4
1.63636
0.0
0.0
31.6621
*49.3534
5.31 x 10*6
C2 H6
6.41934
0.0
0.0
*290.283
2,629.10
0.0
C3 H8
*7.8499
0.0
0.0
47.056
0.0
*1.17 x 10*6
i-C4H10
*2.17137
0.0
0.0
0.0
0.0
0.0
n-C4H10
*37.211
0.86564
0.0
732.20
0.0
0.0
N2
1.78857
0.0
*0.001356
*6.187
0.0
0.0
CO2
9.0242
0.0
0.0
*207.033
0.0
4.66 x 10*5
H2 S
*4.7071
0.06192
0.0
82.627
0.0
*7.39 x 10*6
A6
A7
A8
A9
A10
A11
CH4
0.0
0.0
0.128525
*0.78338
0.0
0.0
C2 H6
0.0
9.0 x 10*8
0.129759
*1.19703
*8.46 x 104
*71.0352
C3 H8
7.145 x 10*4
0.0
0.0
0.12348
1.669 x 104
0.0
i-C4H10
1.251 x 10*3
1.0 x 10*8
0.166097
*2.75945
0.0
0.0
n-C4H10
0.0
9.37 x 10*6
*1.07657
0.0
0.0
*66.221
N2
0.0
2.5 x 10*7
0.0
0.0
0.0
0.0
CO2
*6.992x 10*3
2.89 x 10*6
*6.223 x 10*3
0.0
0.0
0.0
H2 S
0.0
0.0
0.240869
*0.64405
0.0
0.0
A12
A13
A14
A15
A16
A17
CH4
0.0
*5.3569
0.0
*2.3 x 10*7
*2.0x 10*8
0.0
C2 H6
0.596404
*4.7437
7.82 x 104
0.0
0.0
0.0
C3 H8
0.23319
0.0
*4.48 x 104
5.5 x 10*6
0.0
0.0
i-C4H10
0.0
0.0
*8.84 x 102
0.0
*5.7 x 10*7
*1.0 x 10*8
0.0
9.17 x 105
0.0
4.98x 10*6
*1.26 x 10*6
0.0
5.87 x 105
0.0
1.0 x 10*8
1.1x 10*7
2.55 x 10*6
0.0
0.0
0.0
n-C4H10
0.0
N2
0.0
CO2
0.27098
0.0
0.0
8.82 x 10*5
H2 S
0.0
*12.704
0.0
*1.3x 10*6
TABLE 9.6—COEFFICIENTS FOR EQ. 9.35
AS FUNCTIONS OF PRESSURE
Hydrate-Former C Values
Pressure
(psia)
C1
C2
C3
i-C4
n-C4
6,000
18,933
20,806
28,382
30,696
17,340
7,000
19,096
20,848
28,709
30,913
17,358
8,000
19,246
20,932
28,764
30,935
17,491
9,000
19,367
21,094
29,182
31,109
17,868
10,000
19,489
21,105
29,200
30,935
17,868
19. Culberson, O.L. and McKetta, J.J. Jr.: “Phase Equilibria in Hydrocarbon/
Water Systems. IV Vapor Liquid Equilibrium Constants in the Methane/
Water and Ethane/Water Systems,” Trans., AIME (1951) 192, 297.
20. Amirijafari, B. and Campbell, J.M.: “Solubility of Gaseous Hydrocarbons Mixtures in Water,” SPEJ (February 1972) 21; Trans., AIME, 253.
21. Pawlikowski, E.M. and Prausnitz, J.M.: “Estimation of Setchenow
Constants for Nonpolar Gases in Common Salts at Moderate Temperatures,” Ind. Eng. Chem. Fund. (1983).
22. Clever, H.L. and Holland, C.J.: “Solubility of Argon Gas in Aqueous Alkali Halide Solutions,” J. Chem. Eng. Data (July 1968) 13, No. 3, 411.
23. Markham, A.E. and Kobe, K.A.: “The Solubility of Carbon Dioxide
and Nitrous Oxide in Aqueous Salt Solutions,” J. Amer. Chem. Soc.
(1941) 63, 449.
24. Long, G. and Chierici, G.L.: “Compressibilité et Masse Specifique des
Eaux de Gisement dans les Conditions des Gisements. Application à
Quelques Problemes de ‘Reservoir Engineering’,” Proc., Fifth World Petroleum Congress (1959) 187.
25. Long, G. and Chierici, G.: “Salt Content Changes Compressibility of
Reservoir Brines,” Pet. Eng. (July 1961) B-25.
WATER/HYDROCARBON SYSTEMS
26. Kutasov, I.M.: “Correlation simplifies obtaining downhole brine density,” Oil & Gas J. (5 August 1991) 48.
27. Rogers, P.S.Z. and Pitzer, K.S.: “Volumetric Properties of Aqueous Sodium Chloride Solutions,” J. Phys. Chem. Ref. Data (1982) 11, No. 1, 15.
28. Sutton, R.P.: “Compressibility Factors for High-Molecular Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September.
29. Kestin, J., Khalifa, H.E., and Correia, R.J.: “Tables of the Dynamic and
Kinematic Viscosity of Aqueous NaCl Solutions in the Temperature
Range 20–150°C and the Pressure Range 0.1–35 MPa,” J. Phys. Chem.
Ref. Data (1981) 10, No. 1, 71.
30. Collins, A.G.: “Properties of Produced Waters,” Petroleum Engineering
Handbook, H.B. Bradley et al. (eds.), SPE, Richardson, Texas (1987)
Chap. 24, 1–23.
31. McKetta, J.J. Jr. and Wehe, A.H.: “Hydrocarbon/Water and Formation
Water Correlations,” Petroleum Production Handbook, T.C. Frick and
R.W. Taylor (eds.), SPE, Richardson, Texas (1962) II, 22.
32. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-Hill
Book Co. Inc., New York City (1959).
33. Ramey, H.J. Jr.: “Correlations of Surface and IFT’s of Reservoir Fluids,”
paper SPE 4429 available from SPE, Richardson, Texas (1973).
34. Standing, M.B.: Petroleum Engineering Data Book, Norwegian Inst. of
Technology, Trondheim, Norway (1974).
35. Hocott, C.R.: “IFT Between Water and Oil Under Reservoir Conditions,” Trans., AIME (1939) 132, 184.
36. Firoozabadi, A. and Ramey, H.J. Jr.: “Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions,” paper CIM 873830, Calgary,
7–10 June 1987.
37. Peng, D.Y. and Robinson, D.B.: “Two and Three Phase Equilibrium Calculations for Coal Gasification and Related Processes,” Thermodynamics of Aqueous Systems with Industrial Applications, ACS Symposium
Series 133 (1980).
38. Peng, D.Y. and Robinson, D.B.: “A New-Constant Equation of State,”
Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59.
19
DTȀ
DT
Fig. 9.33—Hydrate-formation conditions for CO2/water systems
(adapted from Ref. 69).
39. Panagiotopoulos, A.Z. and Reid, R.C.: “New Mixing Rule for Cubic
Equations of State for Highly Polar, Asymmetric Systems,” Equations
of State: Theories and Applications, K.C. Chao and R.L. Robinson
(eds.), ACS Symposium Series (1986) 571.
40. Kistenmacher, H. and Michelsen, M.L.: “On Composition-Dependent
Interaction Coefficients,” Fluid Phase Equilibria (1992).
41. Harvey, A.H. and Prausnitz, J.M.: “Thermodynamics of High-Pressure Aqueous Systems Containing Gases and Salts,” AIChE J. (1989)
35, No. 4, 635.
42. Ludecke, D. and Prausnitz, J.M.: “Phase Equilibria for Strongly Nonideal Mixtures From an Equation of State with Density-Dependent Mixing Rules,” Fluid Phase Equilibria (1985) 22, 1.
43. Makogon, Y.F.: Hydrates of Natural Gas, PennWell Books, Tulsa, Oklahoma (1981).
44. Hammerschmidt, E.G.: “Preventing and Removing Hydrates in Natural
Gas Pipelines,” Oil & Gas J. (1939) 37, No. 8, 66.
45. Holder, G.D., Malone, R.D., and Lawsa, W.F.: “Effects of Gas Composition and Geothermal Properties on the Thickness and Depth of NaturalGas-Hydrate Zones,” JPT (September 1987) 1142.
46. Katz, D.L.: “Prediction of Conditions for Hydrate Formation in Natural
Gases,” Trans., AIME (1945) 160, 141.
47. Carson, D.B. and Katz, D.L.: “Natural Gas Hydrates,” Trans., AIME
(1942) 146, 150.
48. Unruh, C.H. and Katz, D.L.: “Gas Hydrates of Carbon Dioxide/Methane
Mixtures,” Trans., AIME (1949) 83.
49. Wilcox, W.I., Carson, D.B., and Katz, D.L.: “Natural Gas Hydrates,”
Ind. Eng. Chem. (1941) 33, No. 5, 662.
50. Katz, D.L. and Lee, R.L.: Natural Gas Engineering, Chemical Engineering Series, McGraw-Hill Book Co. Inc., New York City (1990).
51. Kobayashi, R. et al.: “Gas Hydrates Formation with Brine and Ethanol
Solutions,” paper presented at the 1951 Natural Gasoline Assn. of America Annual Convention.
52. Campbell, J.M.: Gas Conditioning and Processing, sixth edition, Campbell Petroleum Series, Norman, Oklahoma (1984).
53. McLeod, H.D. Jr. and Campbell, J.M.: “Natural Gas Hydrates at Pressures to 10,000 psia,” JPT (June 1961) 590.
54. Trekell, R.E. and Campbell, J.M.: Petr. Chem. Div. (March 1966) 61.
55. van der Waals, J.H. and Platteeuw, J.C.: “Clathrate Solutions,” Adv.
Chem. Phys. II, I. Prigogine (ed.), Interscience Publishers, New York
City (1959) 1–58.
56. von Stackelberg, M. and Müller, H.G.: “On the Structure of Gas Hydrates,” J. Phys. Chem. (1951) 19, 1319.
57. Sloan, E.D.: “Phase Equilibria of Natural Gas Hydrates,” paper presented at the 1984 Gas Producers Assn. Annual Convention, New Orleans,
19–21 March.
58. Deaton, W.M. and Frost, E.M.: Gas Hydrates and Their Relation to the
Operation of Natural Gas Pipelines, Monograph 8, U.S. Bureau of
Mines, Washington, DC (1946).
20
Fig. 9.34—General effect of methanol added to water/ethane
system (adapted from Ref. 71).
59. Parrish, W.R. and Prausnitz, J.M.: “Dissociation Pressures of Gas Hydrates Formed by Gas Mixtures,” Ind. Eng. Chem. Proc. Des. Dev.
(1972) 11, No. 1, 26.
60. John, V.T., Papadopoulos, K.D., and Holder, G.D.: “A Generalized Model for Predicting Equilibrium Conditions for Gas Hydrates,” AIChE J.
(1985) 31, No. 2, 252.
61. Schroeter, J.P., Kobayashi, R., and Hildebrand, M.A.: “Hydrate Decomposition Conditions in the System H2S-Methane-Propane,” Ind. Eng.
Chem. Fund. (1983) 22, 361.
62. Ericksen and Sloan, E.D.: “Calculation Procedure Using vdW-Platteeuw Model,” Clathrate Hydrates of Natural Gas, Marcel Dekker, New
York City (1990).
63. Technical Data Book—Petroleum Refining, third edition, API, New York
City (1977).
64. Ng, H.-J., Chen, C.-J., and Saeterstad, T.: “Hydrate Formation and Inhibition in Gas Condensate and Hydrocarbon Liquid Systems,” Fluid
Phase Equilibria (1987) 36, 99.
65. Robinson, D.B. and Mehta, B.R.: “Hydrates in the Propane-Carbon
Dioxide-Water System,” J. Cdn. Pet. Tech. (January–March 1971) 33.
66. Starling, K.E. and Powers, J.E.: “Enthalpy of Mixtures by Modified
BWR Equations,” Ind. & Eng. Chem. Fund. (1970) 9, 531.
67. Munck, J., Skjold-J¢rgensen, S., and Rasmussen, P.: “Computations of the
Formation of Gas Hydrates,” Chem. Eng. Sci. (1988) 43, No. 10, 2661.
68. Michelsen, M.L.: “The Isothermal Flash Problem. Part I. Stability,”
Fluid Phase Equilibria (1982) 9, 1.
69. Song, K.Y. and Kobayashi, R.: “Water Content of CO2 in Equilibrium
With Liquid Water and/or Hydrates,” SPEFE (December 1987) 500;
Trans., AIME, 283.
70. Hammerschmidt, E.G.: “Formation of Gas Hydrates in Natural Gas
Transmission Lines,” Ind. & Eng. Chem. (August 1934) 26, No. 8, 851.
71. Nielsen, R.B. and Bucklin, R.W.: “Why Not Use Methanol for Hydrate
Control?” Hydro. Proc. (April 1983) 71.
SI Metric Conversion Factors
bar 1.0*
bbl 1.589 873
cp 1.0*
dyne/cm 1.0*
ft3 2.831 685
°F (°F*32)/1.8
°F (°F)459.67)/1.8
g mol 1.0*
lbm 4.535 924
psi 6.894 757
psi*1 1.450 377
E)05 +Pa
E*01 +m3
E*03 +Pa@s
E)00 +mN/m
E*02 +m3
+°C
+K
E*03 +kmol
E*01 +kg
E)00 +kPa
E*01 +kPa*1
*Conversion factor is exact.
PHASE BEHAVIOR
Appendix A
Property Tables and Units
TABLE A-1A—COMPONENT PROPERTIES FOR CUSTOMARY UNITS
Compound
Nitrogen
Molecular
Weight
Specific
M
(lbm/lbm mol)
Gravity*
ăągąă
Liquid
Density
ò sc
(lbm/ft3)
Critical Constants
pc
(psia)
Tc
(°R)
Acentric
Normal
Boiling
Point
Ideal
Liquid
Yield
Gross
Heating
Value
L
(gal/Mscf)
H
(Btu/scf)
vc
(ft3/lbm mol)
Zc
Factor
ąw
Tb
(°R)
N2
28.02
0.4700
29.31
493.0
227.3
1.443
0.2916
0.0450
139.3
Carbon dioxide
CO2
44.01
0.5000
31.18
1,070.6
547.6
1.505
0.2742
0.2310
350.4
Hydrogen sulfide
H2S
34.08
0.5000
31.18
1,306.0
672.4
1.564
0.2831
0.1000
383.1
672
Methane
C1
16.04
0.3300
20.58
667.8
343.0
1.590
0.2884
0.0115
201.0
1,012
Ethane
C2
30.07
0.4500
28.06
707.8
549.8
2.370
0.2843
0.0908
332.2
Propane
C3
44.09
0.5077
31.66
616.3
665.7
3.250
0.2804
0.1454
416.0
27.4
2,557
iso-butane
i-C4
58.12
0.5613
35.01
529.1
734.7
4.208
0.2824
0.1756
470.6
32.7
3,354
Butane
n-C4
58.12
0.5844
36.45
550.7
765.3
4.080
0.2736
0.1928
490.8
31.4
3,369
iso-pentane
i-C5
72.15
0.6274
39.13
490.4
828.8
4.899
0.2701
0.2273
541.8
36.3
4,001
Pentane
n-C5
72.15
0.6301
39.30
488.6
845.4
4.870
0.2623
0.2510
556.6
36.2
4,009
Hexane
n-C6
86.17
0.6604
41.19
436.9
913.4
5.929
0.2643
0.2957
615.4
41.2
4,756
Heptane
n-C7
100.20
0.6828
42.58
396.8
972.5
6.924
0.2633
0.3506
668.8
46.3
5,503
Octane
n-C8
114.20
0.7086
44.19
360.6
1,023.9
7.882
0.2587
0.3978
717.9
50.9
6,250
Nonane
n-C9
128.30
0.7271
45.35
332.0
1,070.3
8.773
0.2536
0.4437
763.1
55.7
6,996
Decane
n-C10
142.30
0.7324
45.68
304.0
1,111.8
9.661
0.2462
0.4902
805.2
61.4
7,743
28.97
0.4700
29.31
547.0
239.0
1.364
0.2910
0.0400
141.9
Air
Water
Oxygen
H2O
18.02
1.0000
62.37
3,206.0
1,165.0
0.916
0.2350
0.3440
671.6
O2
32.00
0.5000
31.18
732.0
278.0
1.174
0.2880
0.0250
162.2
1,783
*Water+1.
PROPERTY TABLES AND UNITS
1
TABLE A-1B—COMPONENT PROPERTIES IN SI METRIC UNITS
Compound
Nitrogen
Molecular
Weight
Specific
M
(kg/kmol)
Gravity*
ągą
Liquid
Density
ò sc
Critical Constants
Acentric
Normal
Boiling
Point
Ideal
Liquid
Yield
Gross
Heating
Value
Tb
(K)
L
(m3/1000 m3)
H
(MJ/std m3)
(kg/m3)
pc
(kPa)
Tc
(K)
vc
(m3/kmol)
Zc
Factor
ąw
N2
28.02
0.4700
469.5
3 399
126.3
0.0901
0.2916
0.0450
77.39
Carbon dioxide
CO2
44.01
0.5000
499.5
7 382
304.2
0.0940
0.2742
0.2310
194.67
Hydrogen sulfide
H2S
34.08
0.5000
499.5
9 005
373.6
0.0976
0.2831
0.1000
212.83
25.04
Methane
C1
16.04
0.3300
329.7
4 604
190.6
0.0993
0.2884
0.0115
111.67
37.71
Ethane
C2
30.07
0.4500
449.6
4 880
305.4
0.1479
0.2843
0.0908
184.56
Propane
C3
44.09
0.5077
507.2
4 249
369.8
0.2029
0.2804
0.1454
231.11
3.67
95.27
iso-butane
i-C4
58.12
0.5613
560.7
3 648
408.2
0.2627
0.2824
0.1756
261.44
4.37
125.0
Butane
n-C4
58.12
0.5844
583.8
3 797
425.2
0.2547
0.2736
0.1928
272.67
4.20
125.5
iso-pentane
i-C5
72.15
0.6274
626.8
3 381
460.4
0.3058
0.2701
0.2273
301.00
4.86
149.1
Pentane
n-C5
72.15
0.6301
629.5
3 369
469.7
0.3040
0.2623
0.2510
309.22
4.83
149.4
Hexane
n-C6
86.17
0.6604
659.7
3 012
507.4
0.3701
0.2643
0.2957
341.89
5.51
177.2
Heptane
n-C7
100.20
0.6828
682.1
2 736
540.3
0.4322
0.2633
0.3506
371.56
6.20
205.0
Octane
n-C8
114.20
0.7086
707.9
2 486
568.8
0.4920
0.2587
0.3978
398.83
6.80
232.9
Nonane
n-C9
128.30
0.7271
726.4
2 289
594.6
0.5477
0.2536
0.4437
423.94
7.45
260.7
Decane
n-C10
142.30
0.7324
731.7
2 096
617.7
0.6031
0.2462
0.4902
447.33
8.20
288.5
28.97
0.4700
469.5
3 771
132.8
0.0852
0.2910
0.0400
78.83
Air
Water
Oxygen
H2O
18.02
1.0000
999.0
22 105
647.2
0.0572
0.2350
0.3440
373.11
O2
32.00
0.5000
499.5
5 047
154.4
0.0733
0.2880
0.0250
90.11
66.43
*Water+1.
2
PHASE BEHAVIOR
TABLE A-2—UNIVERSAL GAS CONSTANT FOR DIFFERENT UNITS
Pressure
Volume
Temperature
Mass (mole)
Unit
Unit
Unit
Unit
Gas Constant
R
psia
ft3
°R
lbm
10.7315
psia
cm3
°R
lbm
303,880
psia
cm3
°R
g
669.94
bar
ft3
°R
lbm
0.73991
atm
ft3
°R
lbm
0.73023
atm
cm3
°R
g
45.586
Pa
m3
K
kg
8314.3
Pa
m3
K
g
8.3143
kPa
m3
K
kg
8.3143
kPa
cm3
K
g
8314.3
bar
m3
K
kg
0.083143
bar
cm3
K
g
83.143
atm
m3
K
kg
0.082055
atm
cm3
K
g
82.055
Btu
°R
lbm
1.9858
Btu
°R
g
0.0043780
calorie
°R
lbm
500.76
calorie
°R
g
1.1040
kcal
°R
lbm
0.50076
kcal
°R
g
0.0011040
calorie
K
kg
1985.8
calorie
K
g
1.9858
erg
K
kg
8.3143 1010
erg
K
g
8.3143 107
J
K
kg
8314.3
J
K
g
8.3143
Energy Unit
TABLE A-3—RECOMMENDED BIP’s
FOR PR EOS AND SRK EOS FOR
NONHYDROCARBON/HYDROCARBON COMPONENT PAIRS
PR EOS*
N2
CO2
SRK EOS**
H2 S
N2
CO2
H2 S
N2
—
—
—
—
—
—
CO2
0.000
—
—
0.000
—
—
H2 S
0.130
0.135
—
0.120†
0.120
—
C1
0.025
0.105
0.070
0.020
0.120
0.080
C2
0.010
0.130
0.085
0.060
0.150
0.070
0.070
C3
0.090
0.125
0.080
0.080
0.150
i-C4
0.095
0.120
0.075
0.080
0.150
0.060
C4
0.095
0.115
0.075
0.080
0.150
0.060
i-C5
0.100
0.115
0.070
0.080
0.150
0.060
C5
0.110
0.115
0.070
0.080
0.150
0.060
C6
0.110
0.115
0.055
0.080
0.150
0.050
C7+
0.110
0.115
0.050‡
0.080
0.150
0.030‡
*Nonhydrocarbon
BIP’s from Ref. 1.
**Nonhydrocarbon
BIP’s from Ref. 2.
†Not
reported in Ref. 2.
‡Should
decrease gradually with increasing carbon number.
BIP+binary interaction parameter, PR EOS+Peng-Robinson equation of state, and
SRK EOS+Soave-Redlich-Kwong equation of state.
PROPERTY TABLES AND UNITS
3
TABLE A-4—FORTRAN PROGRAM FOR CALCULATING SPLIT OF C7+ WITH GAMMA DISTRIBUTION
C
C––––
C
PROGRAM GAMSPL
IMPLICIT DOUBLE PRECISION (A*H,O*Z)
DOUBLE PRECISION MWBL,MWBU,MWAV,MW7P
OPEN(10,FILE+’GAMSPL.OUT’)
WRITE(*,*) ’Input ALFA, ETA, M7+ >’
READ (*,*) ALFA,ETA,MW7P
BETA+(MW7P*ETA)/ALFA
MWBU+ETA
S1+0.0
S2+0.0
WRITE(10,2000) ALFA,ETA,MW7P
DO 100 I+1,20
MWBL+MWBU
MWBU+MWBL)14.0
IF (I.EQ.20) MWBU+10000.0
CALL P0P1(ALFA,ETA,BETA,MWBL,P0L,P1L)
CALL P0P1(ALFA,ETA,BETA,MWBU,P0U,P1U)
Z+P0U*P0L
S1+S1)Z
MWAV+ETA+ALFA*BETA*(P1U*P1L)/(P0U*P0L)
S2+S2)Z*MWAV
WRITE(10,2100) I,Z,MWAV
100 CONTINUE
WRITE(10,2200) S1,S2/S1
2000 FORMAT (/
. ’ ALFA ........ :’,F10.3/
. ’ ETA ......... :’,F10.3/
. ’ MW7P ........ :’,F10.3/
. ’ –––––––––––––––––––––––––––––’/
. ’ Frac.
Mole
Molecular ’/
. ’ No.
Fraction
Weight ’/
. ’ –––––
–––––––––– –––––––––– ’)
2100 FORMAT (1X,I3,3X,F10.7,2X,F10.3)
2200 FORMAT (’
–––––––––
––––––– ’/7X,F10.7,2X,F10.3)
END
SUBROUTINE P0P1 (ALFA,ETA,BETA,MWB,P0,P1)
IMPLICIT DOUBLE PRECISION (A*H,O*Z)
DOUBLE PRECISION MWB
P0+0.0
P1+0.0
IF (MWB.LE.ETA) RETURN
Y+(MWB*ETA)/BETA
Q+DEXP(*Y)*Y**ALFA/GAMA(ALFA)
TERM+1.0/ALFA
S+TERM
DO 100 J+1,10000
TERM+TERM*Y/(ALFA)DFLOAT(J))
S+S)TERM
IF (DABS(TERM).LE.1.0D *8) GOTO 200
100 CONTINUE
WRITE (*,2000)
200 CONTINUE
P0+Q*S
P1+Q*(S*1.0/ALFA)
2000 FORMAT (1X,’*** PR : SUM DOES NOT CONVERGE’)
RETURN
END
DOUBLE PRECISION FUNCTION GAMA (X)
IMPLICIT DOUBLE PRECISION(A*H,O*Z)
DIMENSION B(8)
DATA B /*0.577191652, 0.988205891,*0.897056937,
.
0.918206857,*0.756704078, 0.482199394,
.
*0.193527818, 0.035868343 /
CONST+1.0
XX+X
IF (X.LT.1.0) XX+X)1.0
100 IF (XX.LE.2.0) GOTO 200
XX+XX*1.0
CONST+XX*CONST
GOTO 100
200 XX+XX*1.0
Y+1.0
DO 300 I+1,8
Y+Y)B(I)*XX**I
300 CONTINUE
GAMA+CONST*Y
IF (X.LT.1.0) GAMA+GAMA/X
RETURN
END
4
PHASE BEHAVIOR
TABLE A-5—GREEK ALPHABET
TABLE A-6—SI SYSTEM UNITS
Upper Case
Lower Case
Name
A
a
Alpha
B
b
Beta
G
g
Gamma
D
d
Delta
E
e
Epsilon
Z
z
Zeta
H
Q
h
q
Eta
Theta
I
i
Iota
K
k
Kappa
L
l
Lambda
M
m
N
C
O
P
Base SI Units Used in Phase Behavior
Quantity
Length
Unit
Symbol
meter
m
Time
second
s
Mass
kilogram
kg
Temperature
kelvin
K
Amount of substance
mole
mol
Quantity
Mass
Volume
Unit
Symbol
tonne
Mg
liter
Definition
1
L
SI Term
Mg + 103 kg
1 L+1
Mg
dm3
dm3
TABLE A-7—SI PREFIXES
Multiplication Factor
Prefix
1012
Tera
Symbol*
T
Mu
109
Giga
G
n
Nu
106
Mega
M
c
Xi
103
Kilo
k
o
Omicron
102
Hecto
h
p
Pi
10
Deka
da
10*1
Deci
d
10*2
R
ò
Rho
S
s
Centi
c
Sigma
10*3
Milli
m
T
t
Tau
10*6
Micro
m
U
u
Upsilon
10*9
Nano
n
Pico
p
F
f
Phi
10*12
X
x
Chi
10*15
Femto
f
10*18
Atto
a
Y
y
Psi
W
w
Omega
PROPERTY TABLES AND UNITS
*Only the symbols T (tera), G (giga), and M (mega) are capital letters. Compound
prefixes are not allowed; e.g., use nm (nanometer) rather than mmm (millimicrometer).
5
TABLE A-8—PHYSICAL CONSTANTS AND VALUES (from Ref. 3)
Triple point of water
273.16 exactly
K*
0.01 exactly
°C
491.688 exactly
°R
32.018 exactly
°F
0.00 exactly
K*
*273.15 exactly
°C
0.00 exactly
°R
Absolute zero
*459.67 exactly
°F
8.3143
J@mol*1@K*1*
10.731 5
psia@ft3@(lbm-mol)*1@°R*1
Density of water at 60°F
999.014
kg@m*3*
[15.56°C, 288.71 K]
0.999 014
g@cm*3
62.366 4
lbm@ft*3
Gas constant, R
Standard atmosphere
1.013 2
bar
14.696 0
psia
1.223 2
kg@m*3*
1.223 2 10*3
g@cm*3
0.076 362
lbm@ft*3
9.806 650
m@s*2*
980.665 0
cm@s–2
32.174 05
ft@s*2
1.000 000
kg@m@N*1@s*2*
1.000 000
g@cm@dyne*1@s*2
32.174 05
lbm@ft@lbf*1@s*2
Earth’s gravitational acceleration, g
gc
p
Pa*
1.013 25
Density of air at 1 atm, 60°F
[15.56°C, 288.71 K]
105
3.141 593 …
gAPI, °API
[141.5/g(60°F)]*131.5
*SI values. All quantities are consistent with conversion factors for the current SI system.
TABLE A-9—TEMPERATURE SCALE CONVERSIONS (from Ref. 3)
To Convert
To
Solve
degree Fahrenheit, TF
kelvin, TK
TK = (TF + 459.67)/1.8
degree Rankine, TR
kelvin, TK
TK = TR /1.8
degree Fahrenheit, TF
degree Rankine, TR
degree Fahrenheit, TF
degree Celsius, TC
TR = TF + 459.67
TC = (TF *32)/1.8
degree Celsius, TC
kelvin, TK
TK = TC + 273.15
The SI standard, the kelvin (K), is defined so that the triple point of water is 273.16 K exactly. The SI temperature symbol is written
K, without a degree symbol. The cgs (and common) temperature unit is degree Celsius, °C; the common oilfield unit is degree
Fahrenheit, °F, or degree Rankine, °R.
6
PHASE BEHAVIOR
TABLE A-10—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)
To Convert From
To
Multiply By
Inverse
Area
acre (acre)
square meter (m2)*
square foot (ft2)
4.046 856
4.356 000**
E + 03
E + 04
2.471 054
2.295 684
E – 04
E – 05
darcy (darcy)
square meter (m2)*
square centimeter (cm2)
square micrometer (mm2)
millidarcy (md)
cm2-cp@sec*1@atm*1
9.869 23
9.869 23
9.869 23
1.000 000**
1.000 000**
E – 13
E – 09
E – 01
E + 03
E + 00
1.013 25
1.013 25
1.013 25
1.000 000**
1.000 000**
E + 12
E + 08
E + 00
E – 03
E + 00
square foot (ft2)
square meter (m2)*
square centimeter (cm2)
square inch (in.2)
9.290 304**
9.290 304**
1.440 000**
E – 02
E + 02
E + 02
1.076 391
1.076 391
6.944 444
E + 01
E – 03
E – 03
hectare (ha)
square meter (m2)*
acre
1.000 000**
2.471 054
E + 04
E + 00
1.000 000**
4.046 856
E – 04
E – 01
square mile (sq mile)
square meter (m2)*
acre
2.589 988
6.400 000**
E + 06
E + 02
3.861 022
1.562 500**
E – 07
E – 03
gram per cubic centimeter (g/cm3)
kilogram/cubic meter (kg/m3)*
pound-mass/cubic foot (lbm/ft3)
pound-mass/gallon (lbm/gal)
pound-mass/barrel (lbm/bbl)
1.000 000**
6.242 797
8.345 405
3.505 070
E + 03
E + 01
E + 00
E + 02
1.000 000**
1.601 846
1.198 264
2.853 010
E – 03
E – 02
E – 01
E – 03
pound-mass per cubic foot (lbm/ft3)
kilogram/cubic meter (kg/m3)*
pound-mass/gallon (lbm/gal)
pound-mass/barrel (lbm/bbl)
1.601 846
1.336 805
5.614 583
E + 01
E – 01
E + 00
6.242 797
7.480 520
1.781 076
E – 02
E + 00
E – 01
pound-mass per gallon (lbm/gal)
kilogram/cubic meter (kg/m3)*
pound-mass/barrel (lbm/bbl)
1.198 264
4.200 000
E + 02
E + 01
8.345 406
2.380 952
E – 03
E – 02
dyne (dyne)
newton (N)*
pound-force (lbf)
1.000 000**
2.248 089
E – 05
E – 06
1.000 000**
4.448 222
E + 05
E + 05
kilogram-force (kgf)
newton (N)*
pound-force (lbf)
9.806 650**
2.204 622
E + 00
E + 00
1.019 716
4.535 924
E – 01
E – 01
pound-force (lbf)
newton (N)*
4.448 222
E + 00
2.248 089
E – 01
angstrom (Å)
meter (m)*
1.000 000**
E – 10
1.000 000**
E + 10
centimeter (cm)
meter (m)*
1.000 000**
E – 02
1.000 000**
E + 02
foot (ft)
meter (m)*
centimeter (cm)
3.048 000**
3.048 000**
E – 01
E + 01
3.280 840
3.280 840
E + 00
E – 02
inch (in.)
meter (m)*
centimeter (cm)
2.540 000**
2.540 000**
E – 02
E + 00
3.937 008
3.937 008
E + 01
E – 01
micron (mm)
meter (m)*
1.000 000**
E – 06
1.000 000**
E + 06
mile (U.S. statute)
meter (m)*
foot
1.609 344**
5.280 000**
E + 03
E + 03
6.213 712
1.893 939
E – 04
E – 04
Density
Force
Length
*SI conversions. All quantities are current to SI standards as of 1974.
**Conversion factor is exact and all following digits are zero. All other factors have been rounded.
The notation E + 03 is used in place of 103, and so on.
PROPERTY TABLES AND UNITS
7
TABLE A-10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)
To Convert From
To
Multiply By
Inverse
Mass
gram-mass
kilogram (kg)*
1.000 000**
E – 03
1.000 000**
E + 03
ounce-mass (avoirdupois)
kilogram (kg)*
gram (g)
2.834 952
2.834 952
E – 02
E + 01
3.527 397
3.527 397
E + 01
E – 02
pound-mass
kilogram (kg)*
ounce-mass
4.535 923 7** E – 01
1.600 000**
E + 01
2.204 623
6.250 000**
E + 00
E – 02
slug
kilogram (kg)*
pound-mass (lbm)
1.459 390
3.217 405
E + 01
E + 01
6.852 178
3.108 095
E – 02
E – 02
ton (U.S. short)
kilogram (kg)*
pound-mass (lbm)
9.071 847
2.000 000**
E + 02
E + 03
1.102 311
5.000 000**
E – 03
E – 04
ton (U.S. long)
kilogram (kg)*
pound-mass (lbm)
1.016 047
2.240 000**
E + 03
E + 03
9.842 064
4.464 286
E – 04
E – 04
ton (metric)
kilogram (kg)*
1.000 000**
E + 03
1.000 000**
E – 03
tonne
kilogram (kg)*
1.000 000**
E + 03
1.000 000**
E – 03
atmosphere (atm)
(Normal is 760 mm Hg)
pascal (Pa)*
mm Hg (0°C)
feet water (4°C)
pound-force/square inch (psi)
bar
1.013 25
7.600 000**
3.389 95
1.469 60
1.013 25
E + 05
E + 02
E + 01
E + 01
E + 00
9.869 23
1.315 789
2.949 90
6.804 60
9.869 23
E – 06
E – 03
E – 02
E – 02
E – 01
bar (bar)
pascal (Pa)*
pound-force/square inch (psi)
1.000 000**
1.450 377
E + 05
E + 01
1.000 000**
6.894 757
E – 05
E – 02
centimeter of Hg (32°F)
pascal (Pa)*
pound-force/square inch (psi)
1.333 22
1.933 67
E + 03
E – 01
7.500 64
5.171 51
E – 04
E + 00
dyne/square centimeter (dyne/cm2)
pascal (Pa)*
pound force/square inch (psi)
1.000 000**
1.450 377
E – 01
E – 05
1.000 000**
6.894 757
E + 01
E + 04
feet of water (39.2°F)
pascal (Pa)*
pound force/square inch (psi)
2.988 98
4.335 15
E + 03
E – 01
3.345 62
2.306 73
E – 04
E + 00
kilogram-force/square centimeter
pascal (Pa)*
bar
pound force/square inch (psi)
9.806 650**
9.806 650**
1.422 334
E + 04
E – 01
E + 01
1.019 716
1.019 716
7.030 695
E – 05
E + 00
E – 02
pound-force/inch2 (psi)
pascal (Pa)*
6.894 757
E + 03
1.450 377
E – 04
day (d)
second (s)*
minute (min)
hour (h)
8.640 000**
1.440 000**
2.400 000**
E + 04
E + 03
E + 01
1.157 407
6.944 444
4.166 667
E – 05
E – 04
E – 02
hour (h)
second (s)*
minute (min)
3.600 000**
6.000 000**
E + 03
E + 01
2.777 778
1.666 667
E – 04
E – 02
minute (min)
second (s)*
6.000 000**
E + 01
1.666 667
E – 02
Pressure
Time
*SI conversions. All quantities are current to SI standards as of 1974.
**Conversion factor is exact and all following digits are zero. All other factors have been rounded.
The notation E + 03 is used in place of 103, and so on.
8
PHASE BEHAVIOR
TABLE A-10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)
To Convert From
To
Multiply By
Inverse
Viscosity
centipoise (cp)
pascal-second (Pa@s)*
dyne-second/square centimeter
(dyne-s/cm2)
pound-mass/foot-second (lbm/ft-sec)
pound-force-second/square foot
(lbf-sec/ft2)
pound-mass/foot-hour (lbm/ft-hr)
1.000 000**
1.000 000**
6.719 689
2.088 543
2.419 088
centistoke (cSt)
square meter/second (m2/s)*
centipoise/gram-cubic centimeter
(cp/g-cm3)
1.000 000** E – 06
1.000 000** E + 00
1.000 000** E + 06
1.000 000** E + 00
poise
pascal-second (Pa@s)*
1.000 000** E – 01
1.000 000** E + 01
pound-mass/foot-second (lbm/ft-sec)
pascal-second (Pa@s)*
1.488 164
E + 00
6.719 689
E – 01
pound-mass/foot-hour (lbm/ft-hr)
pascal-second (Pa@s)*
4.133 789
E – 04
2.419 088
E + 03
pascal-second (Pa@s)*
4.788 026
E + 01
2.088 543
E – 02
acre-foot (acre-ft)
cubic meter (m3)*
cubic foot (ft3)
barrel (bbl)
1.233 482
E + 03
4.356 000** E + 04
7.758 368
E + 03
8.107 131
2.295 684
1.288 931
E – 04
E – 05
E – 04
barrel (bbl)
cubic meter (m3)*
cubic foot (ft3)
gallon (gal)
1.589 873
E – 01
5.614 583
E + 00
4.200 000** E + 01
6.289 811
1.781 076
2.380 952
E + 00
E – 01
E – 02
cubic foot (ft3)
cubic meter (m3)*
cubic inch (in.3)
gallon (gal)
2.831 685
1.728 000
7.480 520
E – 02
E + 03
E + 00
3.531 466
5.787 037
1.336 805
E + 01
E – 04
E – 01
gallon (gal)
cubic meter (m3)*
cubic inch (in.3)
3.785 412
2.310 001
E – 03
E + 02
2.641 720
4.329 003
E + 02
E – 03
liter (L)
cubic meter (m3)*
1.000 000** E – 03
1.000 000** E + 03
barrel/day (B/D)
cubic meter/second (m3/s)*
cubic meter/hour (m3/h)
cubic meter/day (m3/d)
cubic centimeter/second (cm3/s)
cubic foot/minute (ft3/min)
gallon/minute (gal/min)
1.840 131
6.624 472
1.589 873
1.840 131
3.899 016
2.916 667
E – 06
E – 03
E – 01
E + 00
E – 03
E – 02
5.434 396
1.509 554
6.289 810
5.434 396
2.564 750
3.428 571
E + 05
E + 02
E + 00
E – 01
E + 02
E + 01
cubic foot/minute (ft3/min)
cubic meter/second (m3/s)*
4.719 474
E – 04
2.118 880
E + 03
cubic foot/second (ft3/sec)
cubic meter/second (m3/s)*
2.831 685
E – 02
3.531 466
E + 01
(m3/s)*
6.309 020
E – 05
1.585 032
E + 04
pound-force-second/square foot
(lbf-sec/ft2)
E – 03
E – 02
E – 04
E – 05
E + 00
1.000 000**
1.000 000**
1.488 164
4.788 026
4.133 789
E + 03
E + 02
E + 03
E + 04
E – 01
Volume
Volumetric rate
gallon/minute (gal/min)
cubic meter/second
*SI conversions. All quantities are current to SI standards as of 1974.
**Conversion factor is exact and all following digits are zero. All other factors have been rounded.
The notation E + 03 is used in place of 103, and so on.
PROPERTY TABLES AND UNITS
9
TABLE A-11—ADDITIONAL CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR
To Convert From
To
Multiply By
Inverse
Amount of substance
mole (mol)
kilomole (kmol)
pound-mass mole (lbm mol)
2.204 623
E + 03
4.535 923
E – 04
gram mole (gmol)
1.000 000*
E + 00
1.000 000*
E + 00
kilomole (kmol)
1.000 000*
E – 03
1.000 000*
E + 03
mole (gmol)
1.000 000*
E + 03
1.000 000*
E – 03
gram mole (gmol)
1.000 000*
E + 03
1.000 000*
E – 03
pound-mass mole (lbm mol)
4.535 923
E – 01
2.204 623
E + 00
square meter/second (m2/s)
1.000 000*
E – 04
1.000 000*
E + 04
square millimeter/second (mm2/s)
1.000 000*
E + 02
1.000 000*
E – 02
square foot/second (ft2/sec)
1.076 390
E – 03
9.290 304
E + 02
square foot/hour (ft2/hr)
3.875 000
E + 00
2.580 640
E – 01
dyne/centimeter (dyne/cm)
1.000 000*
E + 00
1.000 000*
E + 00
kiloJoule (kJ)
1.055 056
E + 00
9.478 160
E – 01
calorie (cal)
2.521 640
E + 02
3.965 660
E – 03
kilocalorie (kcal)
2.521 640
E – 01
3.965 660
E + 00
erg
1.055 056
E + 10
9.478 160
E – 11
Diffusivity
square centimeter/second (cm2/s)
Surface tension
milliNewton/meter (mN/m)
Energy
British thermal unit (Btu)
*Conversion factor is exact.
References
1. Nagy, Z. and Shirkovskiy, A.I.: “Mathematical Simulation of Natural Gas
Condensation Processes Using the Peng-Robinson Equation of State,”
paper SPE 10982 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, 26–29 September.
10
2. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases
and Liquids, fourth edition, McGraw-Hill Book Co. Inc., New York City
(1987).
3. Earlougher, R.C. Jr.: Advances in Well Test Analysis, SPE Monograph Series, SPE, Richardson, Texas (1977) 5.
PHASE BEHAVIOR
Appendix B
Example Problems
Introduction
Many of the problems presented here were introduced by Standing
during his 2 years as visiting professor at the Norwegian Inst. of
Technology in Trondheim during 1973–74. Some of the problems
have been modified or expanded, and additional problems have
been included to cover subjects presented in the monograph that
were not necessarily covered in Standing’s problems.
Problem 1
Problem. A light-hydrocarbon gas has the compositional analysis
given in Table B-1.
Calculate the following properties.
a. Weight composition.
b. Molecular weight.
c. Specific gravity.
d. Density in lbm/ft3 at 20 psia and 120°F, assuming ideal gas behavior.
e. Density in kg/m3 at 3.1 atm and 50°C, assuming ideal gas behavior.
Solution. The problem is solved by calculating mass, mi +xi Mi , and
mass (weight) fractions, as shown in Table B-2. The following
equations have been used.
wi +
mi
n i Mi
+
ȍm ȍn M
N
j
j
j+1
gg +
;
N
. . . . . . . . . . . . . . . . . . . . . . . (3.3)
j
j+1
ǒò gǓ
sc
ǒò airǓ
sc
+
Mg
Mg
+
M air
28.97
and M g + 28.97 g g ;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
ò g + pM gńZRT;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
ȍy M ,
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
ȍy p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
a. Weight composition is given as wi in Eq. 3.3.
b. The ratio of total mass, S mi , to total moles, Syi , gives the average molecular weight.
M g + (24.97)ń(1.00) + 24.97 lbmńlbm mol.
c. Gas specific gravity is given by
g g + (24.97)ń(28.97) + 0.864 (air + 1).
d. Gas density is calculated with Eq. 3.35.
ò g + (20)(24.97)ń[(1)(10.732)(120 ) 460)]
+ 0.0801 lbmńft 3.
e. Gas density in SI units is also calculated with Eq. 3.35 with the
correct gas constant, R, from Table A-2 in Appendix A.
ò g + (3.1)(24.97)ń[(1)(0.082055)(50 ) 273)] + 2.92 kgńm 3.
Problem 2
Problem. Table B-3 gives the compositional analysis of a relatively-high-sulfur-content Canadian gas. If the gas-processing plant
that treats the gas removes 100% of the H2S and converts it to elemental sulfur, how many long tons (2,200 lbm) of sulfur will result
from processing 1,000 Mscf of field gas?
Solution. Total mass in lbm mol of 1,000 Mscf gas is calculated
from the real gas law,
pV + nZRT.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.30)
N
and M +
i
i+1
ȍy T
N
T pc +
i
Component
i+1
N
and p pc +
TABLE B-1—GAS COMPOSITIONAL ANALYSIS
(PROBLEM 1)
i
ci .
i+1
EXAMPLE PROBLEMS
Mole Fraction
Methane
0.49
Ethane
0.38
Propane
0.13
1
TABLE B-2—MASS AND MASS (WEIGHT) FRACTIONS (PROBLEM 1)
Component i
Molecular Weight
Mi
(lbm/lbm mol)
Mole Fraction
xi
C1
16.04
0.49
7.84
0.314
C2
30.07
0.38
11.40
0.456
C3
44.09
0.13
5.73
0.230
1.00
24.97
1.000
Total
Mass
mi = xi Mi
(lbm)
TABLE B-3—GAS COMPOSITIONAL ANALYSIS
(PROBLEM 2)
Mole Fraction
zi
Component i
0.0112
C3
0.05
0.2609
n-C4
0.10
0.5575
n-C5
0.15
C2
0.0760
n-C6
0.70
C3
0.0433
i-C4
0.0061
n-C4
0.0137
i-C5
0.0033
n-C5
0.0052
C6
0.0053
0.0175
+ 128 and g C
7)
+ 0.780.
Note: Canadian standard pressure base is 14.65 psia. Assume that Z=1 at standard
conditions.
Solving for n,
n + pVńZRT + (14.65)ǒ1
10 6Ǔń[(1.0)(10.73)(60 ) 460)]
+ 2, 625 lbm mol.
Moles of H2S is calculated by multiplying the total moles by the
mole fraction of H2S.
nH
2S
Liquid Volume Fraction
xVi
Component i
H2 S
C7+
7)
TABLE B-4—LIQUID VOLUME COMPOSITION
(PROBLEM 3)
CO2
C1
MC
Weight Fraction
wi = mi /(Smj )
+ (0.2609)(2, 625) + 685 lbm mol.
Problem 3
Problem. At 15.56°C, a storage tank contains 1,000 m3 of gasoline
with the liquid volume composition given in Table B-4.
Calculate the following.
a. Weight (mass) composition.
b. Molar composition.
c. Molecular weight.
d. Specific gravity.
e. Oil gravity (°API).
f. Moles in kilogram moles (kmol) of n C in the tank.
6
g. Gallons of n C in the tank.
5
h. Pounds of n C in the tank.
4
Note: Use component properties from Appendix A and values
from Table B-5.
Solution.
a. Weight composition from Column 5, where wi +mi /(Smj ).
b. Mole composition from Column 8, where xi +ni /(Snj ).
c. Molecular weight from
M + (640.0 kg)ń(8.248 kmol)
+ 77.6 kgńkmol + 77.6 lbmńlbm mol.
There is one mole of sulfur (S) per mole of H2S, so
n S + (0.2609)(2, 625) + 685 lbm mol.
d. Density from
The mass of sulfur equals the moles of sulfur times the molecular
weight of sulfur (MS+32),
m S + (685)(32)ńǒ2, 200 lbmńtonǓ
ò o + m ońV o + (640.0 kg)ńǒ1.0 m 3Ǔ + 640.0 kgńm 3.
Specific gravity is calculated from g o + ò ońò w (Eq. 3.12), where
densities are at standard conditions.
g o + ǒ640.0 kgńm 3Ǔńǒ999.0 kgńm 3Ǔ + 0.640 (water + 1).
+ 9.96 long tonsń1, 000 Mscf produced gas.
TABLE B-5—COMPOSITION CONVERSIONS FOR MIXTURES (PROBLEM 3)
Column
Component i
1
Liquid Volume
Fraction
xVi
2
Liquid Volume*
Vi
(m3)
3
Liquid Density
òi
(kg/m3)
4
Mass
m i + òV i
(kg)
5
Weight
Fraction
wi
6
Molecular Weight
Mi
(kg/kmol)
7
Moles
ni = mi /Mi k
(kmol)
8
Mole
Fraction
xi
C3
0.05
0.05
507.2
25.36
0.040
44.09
0.575
0.070
n-C4
0.10
0.10
583.9
58.39
0.091
58.12
1.005
0.122
n-C5
0.15
0.15
629.5
94.43
0.148
72.15
1.309
0.159
n-C6
0.70
0.70
659.8
461.86
0.722
86.17
5.360
0.650
640.04
1.000
8.248
1.000
Total
1.00
*On the basis of 1 m3.
2
PHASE BEHAVIOR
TABLE B-6—SEPARATOR GAS AND SEPARATOR OIL
COMPOSITIONS FOR WELLSTREAM RECOMBINATION
CALCULATION (PROBLEM 4)
Component i
Gas Mole Fraction
yi
Liquid Volume Fraction
xVi
C1
0.968
0.020
C2
0.010
0.006
C3
0.011
0.011
i-C4
0.003
0.009
n-C4
0.003
0.013
i-C5
0.002
0.016
n-C5
0.001
0.010
C6
0.002
0.038
C7+
0.000
0.877
Problem 5
Problem. A new well was completed with perforations in three separate intervals. Initial pressure at midperforations (4,650 ft subsurface) was 2,000 psig at 150°F. The first 24-hour production test gave
the information in Table B-8.
On the basis of these data, which of the following do you consider
best describes the well effluent.
a. Production of a single phase from a gas-condensate reservoir.
b. Production of separate gas and liquid phases into the well.
c. Production of undersaturated liquid into the well.
Explain the basis for your decision.
M C7) + 144 and g C7) + 0.775.
e. g API + (141.5)ń(0.640) * (131.5) + 89.4°API.
f. Moles of n C + ǒ1000 m 3Ǔǒ5.36 kmolńm 3Ǔ + 5360 kmol.
6
g. Volume of n C + ǒ1000 m 3Ǔǒ0.15 m 3ńm 3Ǔǒ6.289 bblńm 3Ǔ
5
ǒ42 galńbblǓ + 3.962 10 4 gal.
h. Mass of n C + ǒ1000 m 3Ǔǒ58.39 kgńm 3Ǔǒ2.205 lbmńkgǓ
4
+ 1.2875
10 5 lbm.
Problem 4
Problem. During a 24-hour test, a well produced 463 STB oil and
5,783 Mscf of separator gas (these volumes are expressed at 14.4 psia
and 60°F). Table B-6 gives oil and gas compositions. Calculate the
well-effluent composition in mole fraction. Use apparent liquid densities for methane and ethane of 0.30 and 0.45 g/cm3, respectively.
Solution. From Eq. 3.18, the producing gas/oil ratio (GOR) is
R p + q gńq o + ǒ5.783
Table B-7 calculates oil molar composition and recombined wellstream composition with 1 STB oil volume as a basis. Ideal solution
mixing is assumed for the stock-tank oil. Also, note that the component moles in the stock-tank oil are given by n oi + 5.6146 V i ò ińM i
(Eq. 3.4).
10 6Ǔń(463) + 12, 500 scfńSTB,
or in terms of the producing oil/gas ratio (OGR) from Eq. 3.19,
r p + 1ńR p + ǒ10 6 scfńMMscfǓńǒ12, 500 scfńSTBǓ
+ 80 STBńMMscf.
On a basis of 1 STB, the moles of gas produced is given by solving
for ng from the real gas law [ pV+nZRT (Eq. 3.30)], with Z+1,
n g + [(14.4)(12, 500)]ń[(1.0)(10.73)(60 ) 460)]
+ 32.3 lbm mol.
Solution. The GOR of 19,000 might be descriptive of a gas-condensate system (Answer a). However, at the reservoir pressure of 2,000
psi and 150°F, it would be unlikely that a 27°API liquid could dissolve in the gas phase. The reservoir gas probably has been or currently is in contact with a reservoir oil. At 2,000 psia, the K values
(Ki +yi /xi ) of the heavy components that make up a 27°API crude
would be extremely small (mostly t10*3) and the heaviest components would have the lowest K values. Even if the reservoir oil
contacting the reservoir gas is very heavy, the resulting amounts of
heavy components found in the equilibrium gas would be very small
and proportionally more of the lighter fractions would be found in
the reservoir gas. The condensate from such an equilibrium gas
would tend to have a lower gravity (e.g., gAPIu50°API).
Answer c is also wrong because it is not possible to dissolve
19,000 scf of gas in 1 STB of such a heavy crude oil.
Consequently, Answer b is the best answer. Both reservoir oil
(with a gravity somewhat heavier than 27°API) and reservoir gas
(with a much lighter condensate gravity) are both flowing into the
well simultaneously. Coning, leakage behind the casing, or multiple
completion intervals are three situations that might cause the production characteristics seen in this well.
Problem 6
Problem. Table B-9 gives the gas composition of the Sabine field
in Texas. This is a typical composition of field gases produced from
primary separators. Assuming that this gas is to be compressed and
reinjected into a reservoir at 200°F, calculate the compressibility
factor, Z; gas formation volume factor (FVF), Bg ; and gas density,
ò g , at 2,000 psig and 160°F. Make the calculations using pseudocritical properties calculated from the gas composition in Table B-9
and from gas gravity.
TABLE B-7—OIL MOLAR COMPOSITION AND RECOMBINED WELLSTREAM COMPOSITION (PROBLEM 4)
Gas
Component Mole Fraction
i
yi
Gas Moles
ngi +ng yi
(lbm mol)
Oil Volume
Voi
(STB)
Liquid Density
òi
(lbm/ft3)
Molecular Weight
Mi
(lbm/lbm mol)
Oil Moles
noi
(lbm mol)
Total Moles
ni +ngi + noi
(lbm mol)
Wellstream
Mole Fraction
zi
C1
0.968
31.266
0.020
18.73
16.04
0.131
31.398
0.9123
C2
0.010
0.323
0.006
28.09
30.07
0.031
0.354
0.0103
C3
0.011
0.355
0.011
31.66
44.09
0.044
0.400
0.0116
i-C4
0.003
0.097
0.009
35.01
58.12
0.030
0.127
0.0037
C4
0.003
0.097
0.013
36.45
58.12
0.046
0.143
0.0041
i-C5
0.002
0.065
0.016
39.13
72.15
0.049
0.113
0.0033
C5
0.001
0.032
0.010
39.30
72.15
0.031
0.063
0.0018
C6
0.002
0.065
0.038
41.19
86.17
0.102
0.167
0.0048
C7+
0.000
0.0000
0.877
48.33
144.00
1.653
1.653
0.0480
2.117
34.417
1.0000
Total
1.000
1.000
Compostions are calculated on the basis of 1 STB oil volume.
EXAMPLE PROBLEMS
3
TABLE B-8—RESULTS OF FIRST 24-HOUR
PRODUCTION TEST (PROBLEM 5)
TABLE B-9—GAS COMPOSITION (PROBLEM 6)
Mole Fraction
yi
Oil produced, STB
65
Component
Stock-tank-oil gravity, °API
27
C1
0.875
C2
0.083
C3
0.021
i-C4
0.006
n-C4
0.008
i-C5
0.003
n-C5
0.002
C6
0.001
C7+
0.001
Gas produced, MMscf
1.23
Gas/oil ratio, scf/bbl separator oil
19,000
Solution. Properties From Composition.
M g + 28.97 g g ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
ȍy M ,
N
M+
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
i
i+1
ȍy T
and Z + 0.846.
N
T pc +
i
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
i+1
ò g + pM gńZRT ,
ȍy p
N
and p pc +
Gas density is given by
i
ci .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
which yields
ò g + (2, 015)(18.83)ń[(0.846)(10.73)(160 ) 460)]
i+1
With the pseudocritical properties in Table B-10, these equations give,
T pc + 376°R,
+ 6.74 lbmńft 3.
Properties From Specific Gravity Correlations.
The Sutton3 correlations for pseudocritical properties are
p pc + 667 psia,
T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC . . . . . . . . . . (3.47a)
M g + 18.83 (KayȀs mixing rule),
and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC ,
g g + (18.83)ń(28.97) + 0.65 (air + 1),
. . . . . . . . . (3.47b)
which give
T pr + TńT pc + (160 ) 460)ń376 + 1.65,
2
T pc + 169.2 ) 349.5(0.65) * 74.0(0.65) + 365°R,
and p pr + pńp pc + 2, 015ń667 + 3.02.
2
p pc + 756.8 * 131.0(0.65) * 3.6(0.65) + 670 psia ,
Gas Z factor is given by the Hall-Yarborough1,2 correlation.
T pr + TńT pc + (160 ) 460)ń365 + 1.70,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
p pr + pńp pc + 2, 015ń670 + 3.01,
where a + 0.06125 t expƪ* 1.2(1 * t) ƫ, where t + 1ńT pr . This
2
Z + 0.865,
and ò g + 6.59 lbmńft 3.
gives
t + 1ńT pr + 1ń1.65 + 0.606,
a + (0.06125)(0.606) expƪ(* 1.2)(1 * 0.606)
2
ƫ + 0.0308,
y + 0.10996 ǒd Fńdy + 0.79798Ǔ,
Problem 7
Problem. Calculate the viscosity of the Sabine field gas of Problem
6 under reservoir conditions of 2,000 psig and 160°F. Use the Lucas4 and Lohrenz-Bray-Clark5 viscosity correlations based on gas
composition.
TABLE B-10—PSEUDOCRITICAL PROPERTIES (PROBLEM 6)
Component
zi
Mi
pci
(psia)
Tci
°R
zi Mi
zi pci
(psia)
zi Tci
°R
C1
0.8750
16.04
667.8
C2
0.0830
30.07
707.8
343.0
14.04
584.3
300.1
549.8
2.50
58.7
C3
0.0210
44.09
45.6
616.3
665.7
0.93
12.9
14.0
i-C4
0.0060
58.12
529.1
734.7
0.35
3.2
4.4
C4
0.0080
58.12
550.7
765.3
0.46
4.4
6.1
i-C5
0.0030
72.15
490.4
828.8
0.22
1.5
2.5
C5
0.0020
72.15
488.6
845.4
0.14
1.0
1.7
86.17
436.9
913.4
0.09
0.4
0.9
360.6
1,023.9
0.11
0.4
1.0
18.83
666.8
376.4
C6
0.0010
C7+*
0.0010
Total
1.0000
114.0
*Use properties for n-C8.
4
PHASE BEHAVIOR
TABLE B-11—LOHRENZ-BRAY-CLARK5 VISCOSITY CALCULATIONS (PROBLEM 7)
Component
zi
vci
(ft3/lbm mol)
Zci
zi vci
(ft3/lbm mol)
zi Zci
C1
0.8750
1.590
0.2884
1.391
0.2524
C2
0.0830
2.370
0.2843
0.197
0.0236
C3
0.0210
3.250
0.2804
0.068
0.0059
i-C4
0.0060
4.208
0.2824
0.025
0.0017
C4
0.0080
4.080
0.2736
0.033
0.0022
i-C5
0.0030
4.899
0.2701
0.015
0.0008
C5
0.0020
4.870
0.2623
0.010
0.0005
C6
0.0010
5.929
0.2643
0.006
0.0003
C7+
0.0010
7.882
0.2587
Total
1.0000
m gńm gsc + 1 )
where A 1 +
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ
A
A
*1
,
. . . . . . (3.66a)
A 3 + 0.272,
A 4 + 1.105,
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ ,
A3 +
Ǔ
0.4489 expǒ3.0578T *37.7332
pr
,
T pr
A4 +
Ǔ
1.7368 expǒ2.2310T *7.6351
pr
,
T pr
A 1 + 0.0607,
A 2 + 0.0886,
Ǔ
10 *3) expǒ5.1726T *0.3286
pr
,
T pr
(1.245
A 5 + 0.7473,
m gńm gsc + 1.360,
and m g + 0.0167 cp.
Lohrenz-Bray-Clark Correlation. Eqs. 3.133 through 3.135 give
the Lohrenz-Bray-Clark correlation.
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ ,
. . . . . . . . . . . (3.66b)
ƪǒm * m oǓz T ) 10 *4ƫ
where m gsc c + ƪ0.807T pr0.618 * 0.357 expǒ* 0.449T prǓ
) 0.340 expǒ* 4.058T prǓ ) 0.018ƫ ,
ǒ Ǔ
ȍy Z
) 0.058533ò 2pr * 0.040758ò 3pr ) 0.0093324ò 4pr ,
ò
ò pr + ò
pc
N
and p pc +
ci
ȍy v
i
.
. . . . . . . . . . . . . . . . . . . . . . . . (3.67)
and m o +
i ci
,
ȍ z ǸM
i
.
. . . . . . . . . . . . . . . . . . . . . . . . (3.133)
i
i+1
The Lucas correlation gives
m iz Ti + ǒ34
T pc + 376°R,
10 *5ǓT ri0.94
. . . . . . . . . . . . . . . . . . . . . (3.134a)
for Tri x1.5, and
Z pc + 0.2876,
m iz Ti + ǒ17.78
v pc + 1.752 ft 3ńlbm mol,
p pc + 663 psia,
10 *5Ǔ(4.58T ri * 1.67)
for Tri u1.5, where z Ti + 5.35ǒT ci M 3ińp 4ciǓ
M + 18.83 lbmńlbm mol,
v cC
T pr + TńT pc + (160 ) 460)ń376 + 1.65,
1ń6
+ 77.3 cp
7)
+ 21.573 ) 0.015122M C
) 0.070615M C
p pr + pńp pc + 2, 015ń663 + 3.04,
*1
0.618
m gscc + NJ0.807(1.65)
* 0.357 exp[(* 0.449)(1.65)]
) 0.340 exp[(* 4.058)(1.65)] ) 0.018Nj + 0.948,
EXAMPLE PROBLEMS
i
i+1
N
i
i+1
3
4
c + 9, 490NJ(376)ńƪ(18.83) (663) ƫNj
1ń6
ò
v ,
M pc
+
ȍ z m ǸM
N
i
+ 0.10230 ) 0.023364ò pr
ǒ Ǔ
,
i+1
RT pc N
1ń4
T pc
where z T + 5.35
M 3p 4pc
1ń6
T pc
c + 9, 490
M 3p 4pc
0.0003
0.2876
m gsc + ǒ m gsc c Ǔńc + 0.948ń77.3 + 0.0123 cp,
Solution. Lucas Correlation With Composition.
A 1 p 1.3088
pr
0.008
1.752
g
.
7) C 7)
7)
5ń8
1ń6
. . . . . . . (3.134b)
.
* 27.656g C
7)
. . . . . . . . . . . . . . . (3.135)
On the basis of the data in Tables B-11 and B-12, this correlation
yields
T pc + 376°R,
T pr + 1.65,
p pc + 663 psia,
5
TABLE B-12—LOHRENZ-BRAY-CLARK VISCOSITY5 CALCULATIONS (PROBLEM 7)
Tri
ci
mi
(cp)
343.0
1.81
0.0463
707.8
549.8
1.13
44.09
616.3
665.7
0.0060
58.12
529.1
0.0080
58.12
550.7
i-C5
0.0030
72.15
C5
0.0020
C6
C7+
Mi
pci
(psia)
Tci
(°R)
0.8750
16.04
667.8
C2
0.0830
30.07
C3
0.0210
i-C4
C4
zi mi M½
i
z i M½
i
0.0125
0.0438
3.504
0.0352
0.0108
0.0049
0.455
0.93
0.0329
0.0097
0.0013
0.139
734.7
0.84
0.0322
0.0090
0.0004
0.046
765.3
0.81
0.0316
0.0088
0.0005
0.061
490.4
828.8
0.75
0.0310
0.0083
0.0002
0.025
72.15
488.6
845.4
0.73
0.0312
0.0081
0.0001
0.017
0.0010
86.17
436.9
913.4
0.68
0.0312
0.0076
0.0001
0.009
0.0010
114.00
360.6
1,023.9
0.61
0.0314
0.0068
0.0001
0.011
0.0516
4.268
Component
zi
C1
Total
1.0000
TABLE B-13—ANALYSIS OF SOUR CANADIAN GAS
(PROBLEM 8)
p pc +
* * eǓ
p *pcǒ Tpc
*
T pc
) yH
2S
ǒ1 * y Ǔe
Component
i
Mole Fraction
yi
CO2
0.0112
H2 S
0.2609
C1
0.5575
C2
0.0760
C3
0.0433
i-C4
0.0061
e + 29.8,
n-C4
0.0137
T pc + 489.6 * 29.8 + 459.8°R,
i-C5
0.0033
n-C5
0.0052
C6
0.0053
C7+
0.0175
M C7) + 128 and g C7) + 0.780.
and e + 120
ǒ
ƪǒ
y CO ) y H
2
4
) 15 y 0.5
H S * yH
2
2S
2S
Ǔ
Ǔ,
0.9
ǒ
* y CO ) y H
2
2S
Ǔ
1.6
ƫ
. . . . . . . . . . . . . . . . . . . . . . . (3.52c)
which (with the pseudocritical properties in Table B-14) gives
p pc +
(829.5)(489.6 * 29.8)
+ 770 psia,
(489.6) ) (0.2609)(1 * 0.2609)(29.8)
T pr + 696ń459.8 + 1.51,
and p pr + 3, 065ń770 + 3.98.
Where the Standing-Katz8 Z-factor chart is fit by the Hall-Yarborough1,2 correlation,
p pr + 3.04,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
M + 18.83,
v Mpc + 1.752 ft 3ńlbm mol,
where a + 0.06125 t expƪ* 1.2(1 * t) 2ƫ, where t + 1ńT pr ,
ò pr + ǒ6.74ń18.83Ǔ(1.752) + 0.627,
and F( y) + 0 + * ap pr )
NJ
c T + 5.35 (376)ńƪ(18.83) (663)
3
4
ƫNj
1ń6
and m g + 0.0121 ) ƪ(0.131) * 10 *4ƫń(0.0436) + 0.0166 cp.
4
Problem 8
) ǒ90.7t–242.2t 2 ) 42.4t 3Ǔy 2.18)2.82t,
. . . . . . (3.43)
with t + 1ń1.51 + 0.6622,
a + 0.06125(0.6622) expƪ* 1.2(1 * 0.6622)
2
ƫ + 0.03537,
y + 0.18088,
Problem. Table B-13 gives the analysis of the sour Canadian gas of
Problem 2. Use the method developed by Wichert and Aziz6,7 and
calculate adjusted pseudocritical properties for use with the Standing-Katz8 Z-factor chart. Then, calculate the gas FVF, Bg , at reservoir conditions of 3,050 psig and 236°F. Note that Canadian standard conditions are 14.65 psia and 60°F.
and gas FVF given by
Solution. The Wichert-Aziz pseudocritical correction is given by
yields
T pc + T *pc * e, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.52a)
y ) y2 ) y3 * y4
(1 * y) 3
* ǒ14.76t * 9.76t 2 ) 4.58t 3Ǔy 2
+ 0.0436,
m gsc + 0.0516ń4.268 + 0.0121 cp,
6
, . . . . . . . . . . . . . . . . . (3.52b)
H 2S
and Z + 0.778,
Bg +
ǒTp Ǔ ZTp
sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
B g + ǒ14.65ń520Ǔƪ(0.778)(696)ń(3, 065)ƫ + 0.00498 ft 3ńscf.
PHASE BEHAVIOR
TABLE B-14—PSEUDOCRITICAL-PROPERTY CALCULATIONS
FOR A SOUR GAS (PROBLEM 8)
zi
Mi
pci
(psia)
Tci
(°R)
zi Mi ă
CO2
0.0112
44.01
H2 S
0.2609
34.08
C1
0.5575
C2
0.0760
C3
1,070.6
547.6
0.49
12.0
6.1
1,306.0
672.4
8.89
340.7
175.4
16.04
667.8
343.0
8.94
372.3
191.2
30.07
707.8
549.8
2.29
53.8
41.8
0.0433
44.09
616.3
665.7
1.91
26.7
28.8
i-C4
0.0061
58.12
529.1
734.7
0.35
3.2
4.5
C4
0.0137
58.12
550.7
765.3
0.80
7.5
10.5
i-C5
0.0033
72.15
490.4
828.8
0.24
1.6
2.7
C5
0.0052
72.15
488.6
845.4
0.38
2.5
4.4
C6
0.0053
86.17
436.9
913.4
0.46
2.3
4.8
C7+*
0.0175
386.7
1,099.5
2.24
6.8
19.2
26.98
829.5
489.6
Total
128.0
1.0000
zi pci
(psia)
zi Tci
(°R)
*C7+ pseudocriticals from Eq. 3.51.
A 3 + * 3.57
TABLE B-15—SURFACE PRODUCTION DATA
(PROBLEM 9)
Reservoir pressure, psia
5,200
Reservoir temperature °F
250
Separator pressure, psia
950
Separator temperature, °F
160
Primary separator gas rate, Mscf/D
4,265
Primary separator gas gravity (air = 1)
0.70
Tank-oil rate, STB/D
370
Tank-oil gravity, °API
45
R s) +
10 *6(45) + * 1.607
(385)(1.15)
1 * (385)(* 1.607
10 *4)
10 *4,
+ 417 scfńSTB,
10 *4(417) + 1.08 (air + 1).
and g gs1 + 1.15 * 1.607
The total GOR’s and OGR’s are given by
R 1 + ǒ4.265
10 6Ǔń(370) + 11, 527 scfńSTB,
R p + 11, 527 ) 417 + 11, 944 scfńSTB,
and r p + 1ńR p + 8.37
10 *5 STBńscf + 83.7 STBńMMscf.
Total gas specific gravity is given by
Problem 9
Problem. Calculate the reservoir voidage, DVR , expressed as cubic
feet, resulting from 1 day of production from the gas-condensate
reservoir with surface production data given in Table B-15.
DV R + DV g + ǒDV gńD tǓD t B gd + q g(1 day) B gd + q g B gd .
Surface-gas rate is q g + q o R p + q oǒR 1 ) R s)Ǔ , where R 1 is the
separator gas/oil ratio (per stock-tank barrel of condensate) and R s)
is the solution gas/oil ratio of the separator oil.
Estimating the additional gas from the separator oil (Eqs. 3.61
through 3.63),
R s) + A 1g gs1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.61a)
p
ƪǒ18.2
) 1.4 Ǔ10 ǒ
sp1
0.0125g API*0..00091T sp1
Ǔ
ƫ
;
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.62)
10 *6)g API ;
where A 2 + 0.25 ) 0.2g API and A 3 + * (3.57
and R s) +
A1 A2
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.63)
ǒ1 * A 1 A 3Ǔ
gives A 1 +
ƪǒ
Ǔ
ƫ
950 ) 1.4 10 (0.0125)(45)*(0.00091)(160)
18.2
A 2 + 0.25 ) 0.02(45) + 1.15,
EXAMPLE PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . (3.64)
which yields
+ 0.713 (air + 1).
The condensate stock-tank-oil molecular weight is estimated
from the Cragoe9 correlation (Eq. 3.59),
Mo +
6, 084
,
g API * 5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
resulting in
M o + 6, 084ń(45 * 5.9) + 156,
1.205
. . . . . . . . . . . . . . . . . . . (3.61b)
g gs1 + A 2 ) A 3 R s) ,
g g1 R s1 ) g gs1 R s)
,
R s1 ) R s)
g g + [11, 527(0.70) ) 417(1.08)]ń(11, 527 ) 417)
Solution. On the basis of 1 day of production,
and A 1 +
gg +
1.205
+ 385,
which gives the wellstream specific gravity from Eq. 3.55.
gw +
g g ) 4, 580 r p g o
1 ) 133, 000 r p ǒ gńM Ǔ o
.
. . . . . . . . . . . . . . . . . . (3.55)
This yields
gw +
0.713 ) (4, 580)(83.7 10 *6)(0.8017)
1 ) (133, 000)(83.7 10 *6)ǒ0.8017ń156Ǔ
+ 0.963 (air + 1) .
The Sutton3 pseudocritical correlations
T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC
and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC
. . . . . . . . . . (3.47a)
. . . . . . . . . (3.47b)
7
give T pc + 437°R and p pc + 627 psia, and reduced properties are
T pr + TńT pc + 710ń437 + 1.625
VC
and p pr + pńp pc + 5, 200ń627 + 8.293.
The gas volumetric properties are given by Eqs. 3.42 and 3.43,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
where a + 0.06125 t expƪ* (1.21 * t) ƫ, where t + 1ńT pr ,
2
and F(y) + 0 + * ap pr )
3
. . . . . (3.43)
giving Z+1.024. With Eq. 7.12,
sc
sc
and C og given by
C og + 133, 000
Ǔ
og r s
2
and m C
Recalling Eq. 3.95,
2)
og
+
* b ) Ǹb 2 * 4ac
,
2a
. . . . . . . . . . . . . . . . . . . (3.95)
a + 0.3167(1.385) + 0.439;
b + 3.40 * 0.3167(69.97) ) 15.3(1.385) + 2.43;
c + * 15.3(69.97) + * 1, 071;
. . . . . . . . . . . . . . . . . . . . (7.12)
2)
+
+ 46.70 lbmńft 3 ,
. . . . . . . . . . . . . . . . . . . . . . . . . (7.13)
og
* (2.43) ) Ǹ(2.43) * 4(0.439) (* 1, 071)
2(0.439)
2
and ò C
ǒMg Ǔ
+ 69.97 lbm.
2)
2)
) ǒ90.7t * 242.2t 2 ) 42.4t 3Ǔy 2.18)2.82t,
ǒTp Ǔ ZTp ǒ1 ) C
m C + 3.40 lbm,
where a + 0.3167V C , b + m C * 0.3167 m C ) 15.3 V C ,
3)
2
2)
3)
and c + * 15.3m C , we calculate
* ǒ14.76t * 9.76t 2 ) 4.58t 3Ǔy 2
B gd +
+ 1.385 ft 3,
3)
òC
y ) y2 ) y3 * y4
(1 * y)
through 3.97. From Table B-17 and Eqs. 3.93 and 3.94, volumes and
masses needed for the calculations are
the pseudoliquid density of the C2+ mixture at standard conditions.
From Eq. 3.96,
+ 133, 000ǒ0.8017ń156Ǔ
+ 683 scfńSTB.
VC
So with r s + 1ńR p ,
2)
+ VC
B gd + ƪǒ14.7ń520Ǔ(1.024)(160 ) 460)ń(5, 200)ƫ
3)
mC
)ò 2
C
3)
)
+ VC
2
mC
2
15.3 ) 0.3167ò C
ƪ1 ) ǒ683ń11, 944Ǔƫ
,
. . . . . . . . . . . (3.96)
2)
TABLE B-16—OIL COMPOSITION (PROBLEM 10)
+ 0.00395 ft 3ńscf.
Component
The initial daily reservoir voidage is then
DV g + ǒDV gńD tǓǒD tǓǒB gǓ
+ (370)(11, 944)(0.00395) + 17, 470 ft 3 + 3, 110 bbl .
Problem 10
Problem. Table B-16 shows the composition of a reservoir oil in the
Kabob field, Canada. Bubblepoint pressure is 3,100 psia at 236°F
reservoir temperature. Calculate the density in lbm/ft3 of the reservoir oil at bubblepoint conditions using ideal-solution principles according to the Standing-Katz8 method.
Solution. Following the calculation procedure outlined in Chap .3,
pseudoliquid density, ò po, is calculated explicitly with Eqs. 3.94
Mole Fraction
CO2
0.0111
C1
0.3950
C2
0.0969
C3
0.784
i-C4
0.0159
n-C4
0.0372
i-C5
0.0123
n-C5
0.0211
C6
0.0295
C7+
0.3026
M C7) + 182 and g C7) + 0.8275.
TABLE B-17—STANDING-KATZ8 DENSITY CALCULATION (PROBLEM 10)
mi + zi Mi
(lbm)
V i + m ińò i
(ft3)
zi
C1
0.3950
16.04
6.34
C2
0.0969
30.07
2.91
CO2
0.0111
44.01
C3
0.0784
44.09
31.66
3.46
0.109
i-C4
0.0159
58.12
35.01
0.92
0.026
C4
0.0372
58.12
36.45
2.16
0.059
i-C5
0.0123
72.15
39.13
0.89
0.023
C5
0.0211
72.15
39.30
1.52
0.039
C6
0.0295
86.17
41.19
2.54
0.062
C7+
0.3026
182.00
51.61
55.07
1.067
Total
1.0000
Mi
òi
(lbm/ft3)
Component
0.49
76.31
Note: CO2 is treated as C2.
8
PHASE BEHAVIOR
which gives
VC
2)
+ 1.385 ) (3.40)ń[15.3 ) (0.3167)(46.70)]
+ 1.50 ft 3 .
The mass of methane and of the total mixture (C1+) are taken from
Table B-17.
m C 1 + 6.34 lbm and m C
1)
+ 76.31 lbm.
From Eq. 3.97, the pseudoliquid density of the overall mixture is
calculated at standard conditions.
ò po +
*b ) Ǹb 2 * 4ac
,
2a
which results in
* (27.53) ) Ǹ(27.53) * 4(0.674)(* 23.81)
2(0.674)
ò ga + 38.52
2
Pseudoliquid oil density is given by
The pressure correction is calculated with Eq. 3.98 and ò po +
41.69 lbm/ft3.
Dò p + 10 *3 ƪ0.167 ) ǒ16.181
* 10 *8 ƪ0.299 ) ǒ263
10 *0.0425òpoǓƫ p
10 *0.0603òpoǓƫ p 2,
Dò p + 10 *3 NJ0.167 ) ƪ16.181
* 10 *8 NJ0.299 ) ƪ263
. . . . (3.98)
10 *0.0603(41.69)ƫNj (3, 500)
2
10 *8Ǔ(3, 500)
2
+ 1.26 lbmńft 3.
The temperature correction is calculated with Eq. 3.99 and
ò po ) Dò p +41.69)1.26+42.95 lbm/ft3.
ƪ
Dò T + (T * 60) 0.0133 ) 152.4ǒò po ) Dò pǓ
*2.45
ƫ
10 *6Ǔ
Nj
. . . . . . . . . . . (3.99)
giving
Dò T + (238 * 60)ƪ0.0133 ) 152.4(42.95)
NJ
2
* (238 * 60) ǒ8.1
* ƪ0.0622
*2.45
ƫ
10 *6Ǔ
Nj
10 *0.0764(42.95)ƫ + 5.85 lbmńft 3 .
Eq. 3.89 gives the oil density at 3,100 psia and 238°F.
ò o + ò po ) Dò p * Dò T ,
. . . . . . . . . . . . . . . . . . . . . . (3.89)
resulting in
ò o + 41.69 ) 1.26 * 5.85 + 37.10 lbmńft 3 .
EXAMPLE PROBLEMS
ò po +
1 ) 0.0136ǒR s g gńò gaǓ
,
. . . . . . . . . . . . . . . . . (3.100)
62.4(0.845) ) 0.0136(900)(0.85)
+ 45.3 lbmńft 3.
1 ) (0.0136)ƪ(900)(0.85)ń(26.4)ƫ
ò o + ò po ) Dò p * Dò T ,
. . . . . . . . . . . . . . . . . . . . . . (3.89)
as ò o + 45.3 ) 1.1 * 3.5 + 42.9 lbmńft 3.
Pressure gradient with depth (dp/dh) in psi/ft is given by dp/dh
+ ò oǒgńg cǓń144, where ò is in lbm/ft3, g+32 ft/sec2, and gc +32
lbm-ft/(lbf-sec2), giving
dp/dh+42.9(32/32)/144+0.298 psi/ft.
Assuming that this gradient is more or less constant from 7,200
to 6,000 ft subsea, the oil pressure at a depth of 6,000 ft subsea is
ǒ p RǓ
10 *0.0764ǒòpo)DòpǓƫ ,
62.4g o ) 0.0136 R s g g
The pressure correction to density, if given by Eq. 3.98 is
Dò p + 1.1 lbmńft 3 and ò po ) Dò p + 45.3 ) 1.1 lbmńft 3 +
46.4 lbm/ft3. On the basis of ò po ) Dò p , the temperature correction
is given by Eq. 3.99 and is Dò T + 3.5 lbmńft 3, yielding the reservoir oil density from Eq. 3.89,
10 *0.0425(41.69)ƫNj (3, 500)
10 *3Ǔ(3, 500) * ǒ1.104
ò po +
which gives
giving
* ƪ0.0622
10 *(0.00326)(36) ) ƪ94.75 * (33.93) log (36)ƫ
log (0.85) + 26.4 lbmńft 3
+ 41.69 lbmńft 3.
NJ
10 *0.00326g API
) ǒ94.75 * 33.93 log g APIǓ log g g , . . . . . . . . . . . (3.101)
c + * 0.312(76.31) + * 23.81,
* (T * 60) 2 ǒ8.1
Apparent gas pseudoliquid density is given by
ò ga + 38.52
b + 6.34 * 0.45(76.31) ) 0.312(1.50) + * 27.53,
+ ǒ0.441
Solution. From Eq. 3.100, pseudoliquid density, ò po, can be calculated from oil and gas surface gravities, g o and g g, respectively;
solution gas/oil ratio, R s ; and apparent liquid density of separator
gas, ò ga. Stock-tank-oil gravity is
g o + 141.5ń(131.5 ) 36) + 0.845 (water + 1).
. . . . . . . . . . . . . . . . . . . . (3.97)
where a + 0.45(1.50) + 0.674,
and ò po +
Problem 11
Problem. An oil well produces at a total GOR of 900 scf/STB. Total
gas gravity is 0.85 (air+1). Stock-tank-oil gravity is 36°API. Calculate, using ideal-solution principles and apparent liquid density of
the gas, the density of the reservoir oil at 3,300 psia and 190°F. If
reservoir pressure is 3,300 psia at 7,200 ft subsea, what would the
reservoir pressure be at a datum level of 6,000 ft subsea?
6000
+ 3, 300 * 0.298(7, 200 * 6, 000) + 2, 942 psia.
This result assumes that a continuous oil column exists from 6,000
to 7,200 ft subsea.
Problem 12
Problem. For the reservoir considered in Problem 11, use the Standing10 bubblepoint correlation to estimate bubblepoint pressure. On
the basis of this estimate, is it possible that a gas cap might be found
between the test depth of 7,200 ft subsea and the structure top at
6,000 ft subsea? If so, at what depth?
Solution. The Standing bubblepoint-pressure correlation, Eq. 3.78,
p b + 18.2ǒ A * 1.4 Ǔ,
where A + ǒR sńg gǓ 0.83
A + ǒ900ń0.85Ǔ
0.83
. . . . . . . . . . . . . . . . . . . . . . . . . (3.78)
10 ǒ0.00091T*0.0125gAPIǓ, gives
10 [0.00091(190)*0.0125(36)] + 171.2
and p b + 18.2(171.2 * 1.4) + 3, 090 psia.
9
If this bubblepoint-pressure estimate is accurate (even though the
correlation accuracy is probably only "5%), a gas cap may be expected at a subsea depth, calculated from
ǒ p RǓ
GOC
+ p b + 3, 090 + 3, 300 * 0.298ǒ7, 200 * D GOCǓ ,
where 0.298 psi/ft is the oil gradient calculated in Problem 11. Solving this relation for D GOC gives D GOC + 6, 500 ft subsea.
Problem 13
Problem. If the hydrocarbon pore volume (HCPV) of the reservoir
in Problem 11 is approximately 40 106 ft3/ft reservoir thickness,
estimate the original oil in place, N, and original gas in place, G. The
water/oil contact (WOC) is at 7,300 ft subsea, the gas/oil contact
(GOC) depth is given in Problem 12 as 6,500 ft subsea, and the top
of the structure is at 6,000 ft subsea.
Solution. To solve this problem, oil and gas FVF’s must be estimated.
The oil FVF will vary throughout the 800-ft oil column. The oil is saturated at the GOC and undersaturated at depths down to the WOC.
Several assumptions must be made because so little data are available.
1. Constant temperature is assumed throughout the reservoir, although a gradient of 1 to 2°F/100 ft probably exists.
2. Oil composition is assumed to be uniform in the oil column, although it would not be surprising if the GOR decreased somewhat
from the GOC to the WOC (e.g., from 900 to 800 scf/STB).
3. Gas composition is assumed to be uniform in the gas cap (probably a reasonable assumption).
4. A condensate yield must be assumed for the reservoir gas.
From data in the literature (or from a similar reservoir in the same
geographical area), we can find a similar reservoir oil/gas system.
An initial solution OGR, rsi , of 40 STB/MMscf is assumed here.
5. Surface condensate gravity of 60°API ( g og + 0.739) is also
assumed.
6. The surface-gas gravity for the reservoir gas is assumed to be
slightly less than the surface-gas gravity for the reservoir oil,
g gg + 0.80 (see, for example, Fig. 7.12).
The gas and oil column HCPV’s, V HCg and V HCo , respectively,
are given by
V pHCg + ǒ40
10 6Ǔ(6, 500 * 6, 000) + 20
and V pHCo + ǒ40
10 9 ft 3
6, 084
,
g API * 5.9
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
where a + 0.06125 t exp[* 1.2(1 * t) 2], with t + 1ńT pr, gives
Z+0.808.
Gas density is calculated at the GOC to obtain a gas gradient for
estimating the average pressure in the gas cap.
òg +
(3, 090)(28.97)(0.904)
+ 14.36 lbmńft 3 ,
(0.808)(10.73)(190 ) 460)
ǒd pńdhǓ + 14.36ń144 + 0.0997 psińft
g
and ǒ p RǓ g + [3, 090 * (0.0997)(6, 500 * 6, 000)]ń2
+ 3, 065 psia.
Pseudoreduced pressure at ( p R) g is p pr + 3, 065ń650 + 4.715,
and the Z factor is 0.806. The wet-gas FVF at ( p R) g is given by
Bg +
ǒTp Ǔ ZTp,
sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
resulting in
B gw + (0.02827)(0.806)(190 ) 460)ń(3, 065)
+ 0.00483 ft 3ńscf
and b gw + 1ńB gw + 207 scfńft 3.
However, the dry-gas FVF is needed to calculate dry surface gas
for the estimated V pHCg . Eqs. 7.12 and 7.11 are used to calculate B gd .
B gd +
ZT
p sc ZT ǒ
1 ) C og r sǓ + 0.02827 p ǒ1 ) C og r sǓ
T sc p
+ B gwǒ1 ) C og r sǓ , . . . . . . . . . . . . . . . . . . . . . . . . (7.12)
where C og + (133, 000)(0.739)(112) + 876 scfńSTB,
and B gd + V gńV gg ,
10 9 bbl.
Initial gas in place represents the free gas in place plus the gas in
solution in the oil column. To calculate gas FVF, a wellstream gravity, g w, must be calculated first. With g og + 0.739, Eq. 3.59 gives
an estimate of the condensate molecular weight.
Mo +
are Tpc +426°R and ppc +650 psia.
At the GOC, reduced properties are Tpr +(190)460)/426+1.526
and ppr +3,090/650+4.754.
The Standing-Katz8 Z-factor correlation (Eq. 3.42),
10 9 ft 3
10 6Ǔ(7, 300 * 6, 500) + 32
+ 5.700
and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC , . . . . . . . . . . (3.49b)
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
giving
M og + 6, 084ń(60 * 5.9) + 112.
From Eq. 3.55,
gw +
g g ) 4, 580 r p g o
,
1 ) 133, 000 r p ǒ gńM Ǔ o
gw +
0.8 ) ǒ4, 580 Ǔ (40 10 *6)(0.739)
1 ) 133, 000(40 10 *6)ƪ(0.739)ń(112) ƫ
. . . . . . . . . . . . . . . . . . (3.55)
+ 0.904 (air + 1).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.11)
giving
B gd + (0.00483)ƪ1 ) (876)ǒ40
10 *6Ǔƫ + 0.00500 ft 3ńscf
and b gd + 1ńB gd + 200 scfńft 3.
For the oil column, oil FVF must be estimated at an average oil
pressure ( p R) o.
ǒ p RǓ + 3, 090 ) (0.298)(7, 300 * 6, 500)ń2 + 3, 209 psia.
o
Bubblepoint oil FVF is estimated from the Standing correlation
(Eq. 3.111),
B ob + 0.9759 ) ǒ12
10 *5Ǔ A 1.2,
. . . . . . . . . . . . . . . (3.111)
where A + R sǒg gńg oǓ 0.5) 1.25T, giving
0.5
A + 900ǒ0.85ń0.845Ǔ ) 1.25(190) + 1, 140
and B ob + 0.9759 ) ǒ12
10 *5Ǔ(1, 140)
1.2
+ 1.535 bblńSTB.
With g gg + g gHC + 0.80, pseudocritical properties from the Standing10 “wet-gas” correlations (Eq. 3.49),
Undersaturated oil FVF can be calculated with an estimate of the
undersaturated oil compressibility with Eq. 3.107 for c o at ( p R) o ,
and Eq. 3.105 for B o.
T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a)
c o + Ańp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)
10
PHASE BEHAVIOR
Problem 14
Problem. Estimate oil and gas viscosities at 2,500 psia and 190°F
for the reservoir considered in Problems 11 through 13.
giving
c o + 10 *5ƪ(5)(900) ) (17.2)(190) * (1, 180)(0.85)
) (12.61)(36) * 1, 433ƫń(3, 209) + 18.0
10 *6 psi *1.
ò o + ò ob expƪc oǒp * p bǓƫ
[ ò ob ƪ1 * c oǒ p b * p Ǔƫ
. . . . . . . . . . . . . . . . . . . . (3.105a)
Solution. Gas viscosity can be estimated from the Lucas4 correlation (Eq. 3.66).
m gńm gsc + 1 )
and B o + B ob expƪc oǒ p b * p Ǔƫ
[ B ob ƪ1 * c o ǒp * p bǓƫ ,
. . . . . . . . . . . . . . . . . (3.105b)
which give
B o + 1.535 expƪǒ18.3
However, a more exact approach uses Eq. 109, which properly accounts for the pressure dependence of oil compressibility.
ƪ
ƫ
ò ob ) 0.004347 ǒ p * p bǓ * 79.1
,
(7.141 10 *4)ǒ p * p bǓ * 12.938
. . . . . . . . . . . . . . . . . . . (3.109)
resulting in
A + 10 *5[(5)(900) ) (17.2)(190) * (1, 180)(0.85)
) (12.61)(36) * 1, 433] + 0.05776
0.05776
+ 1.532 bblńSTB.
and B o + 1.535ǒ3, 090ń3, 209Ǔ
The two approaches result in almost no difference in B o for this
example of slight undersaturation. However, for higher degrees of
undersaturation, the difference can be significant; therefore, in general, Eq. 3.109 is recommended.
Initial oil in place in the oil column is given by
(N) o + V pHCońB oi + ǒ5.700
+ 3.720
10 9Ǔ(200) + 4.000
10 12 scf.
10 Ǔǒ40
12
10 Ǔ
*6
10 9Ǔ(900) + 3.348
N + (N) o ) (N) g + ǒ3.720
10 12 scf .
10 9Ǔ ) ǒ0.160
10 9Ǔ
10 9 STB
and G + (G) g ) (G) o + ǒ4.000
10 12Ǔ ) ǒ3.348
10 12Ǔ
10 12 scf.
Note that significant gas reserves are found as solution gas in this
oil reservoir. This is not uncommon for volatile and even moderately volatile oil reservoirs (GORu750 scf/STB). In general, in larger
field developments, the economic value of solution gas cannot be
ignored as both production revenue for depletion drive and lost income in waterflooding projects.
EXAMPLE PROBLEMS
Ǔ
0.4489 expǒ3.0578T *37.7332
pr
,
T pr
A4 +
Ǔ
1.7368 expǒ2.2310T *7.6351
pr
,
T pr
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ .
. . . . . . . . . . . (3.66b)
Pseudocritical properties are estimated from reservoir gas (wellstream) gravity, g w. The initial wellstream gravity of 0.904 calculated
in Problem 13 is somewhat higher than would be expected for the
equilibrium gas at 2,500 psia (see, for example, Table 6.11). We therefore assume a current wellstream gravity of g w + g gHC + 0.85.
With the Standing10 wet-gas correlations (Eq. 3.49) for pseudocritical properties,
T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a)
+ 416°R
and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC ,
. . . . . . . . . (3.49b)
+ 654 psia,
The gas molecular weight is
M g + (28.97)(0.85) + 24.62 lbmńlbm mol,
which is used to calculate c.
4
ƫ
1ń6
+ 69.3 cp *1,
giving
Thus, the initial stock-tank oil plus condensate in place, N, and the
initial dry gas plus solution gas in place, G, are, respectively,
+ 7.348
A3 +
3
Initial gas in solution in the oil column is given by
+ 3.880
, . . . . . . . (3.66a)
Ǔ
10 *3) expǒ5.1726T *0.3286
pr
,
T pr
(1.245
c + 9, 490ƪ(416)ń(24.62) ń(654)
10 6 STB.
(G) o + NR si + ǒ3.720
*1
and p pr + pńp pc + 2, 500ń654 + 3.823.
Initial condensate in place in solution in the gas column is given by
+ 160
A
T pr + TńT pc + (190 ) 460)ń416 + 1.562
Initial (dry) gas in place in the gas column is given by
(N) g + G d r si + ǒ4.000
A
giving pseudoreduced properties
10 9Ǔń(1.532)
10 9 STB.
(G) g + V pHCo b gd + ǒ20
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ ,
10 *6Ǔ(3, 090 * 3.209)ƫ
+ 1.532 bblńSTB.
c o + 10 *6 exp
where A 1 +
A 1 p 1.3088
pr
m gsc + ǒ m gsc c Ǔńc + 0.9046ń69.3 + 0.0131 cp,
m gńm gsc + 1.601,
and m g + 0.0210 cp.
Use the Lee-Gonzalez correlation (Eq. 3.65) to calculate gas
viscosity.11
mg + A1
where A 1 +
10 *4 expǒA 2 ò g 3Ǔ ,
A
. . . . . . . . . . . . . . . . . . (3.65a)
ǒ9.379 ) 0.01607M gǓT 1.5
209.2 ) 19.26M g ) T
,
A 2 + 3.448 ) ǒ986.4ńTǓ ) 0.01009M g ,
and A 3 + 2.447 * 0.2224A 2 .
. . . . . . . . . . . . . . . . . . . (3.65b)
Gas density must be calculated first with
ò g + pM gńZRT.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
11
Using the Chew and Connally12 correlation (Eq. 3.123),
TABLE B-18—THREE-COMPONENT-SYSTEM
COMPOSITION (PROBLEM 15)
m ob + A 1 ǒm oDǓ A2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.123)
Component
Mole Fraction
C1
0.20
C3
0.32
n-C5
0.48
and the Bergman equations for constants A 1 and A 2 (Eq. 3.125),
ln A 1 + 4.768 * 0.8359 ln(R s ) 300) . . . . . . . . . . (3.125a)
and A 2 + 0.555 )
TABLE B-19—COMPONENT PROPERTIES (PROBLEM 15)
Component
pci
(psia)
Tci
(°R)
wi
Ki
C1
667.8
343.0
0.0115
9.208
C3
616.3
665.7
0.1454
1.439
n-C5
488.6
845.4
0.2510
0.358
TABLE B-20—CALCULATED RESULTS
FROM ITERATIONS (PROBLEM 15)
Iteration
Fv
h(Fv )
*1.75
dh/dFv
10*2
*0.98880
1
0.5
2
0.48227
1.48 10*4
*1.00606
3
0.48242
1.16
10*8
*1.00590
4
0.48242
7.07 10*17
*1.00590
This gives
òg +
A 2 + 0.555 ) (133.5)ń(700 ) 300) + 0.6885 ,
and m ob + (0.3656)(1.78)
0.6885
+ 0.544 cp.
Problem 15
Problem. Table B-18 gives the composition of a three-component
system of methane, propane, and normal pentane. Use the modified
Wilson13 K-value equation (Eq. 3.159) with a convergence pressure
of 2,000 psia to estimate K values at 500 psia and 160°F. Make a
flash calculation using the Muskat-McDowell14 (or RachfordRice15) algorithm given by Eqs. 4.36 through 4.40.
Solution. Table B-19 gives component properties taken from Appendix A needed to calculate K values from the modified Wilson Kvalue equation. A 0 + 0.7 is used in the modified Wilson K-value
correlation, where A + 1 * ( pńp k ) 0.7 in Eq. 3.159. For example,
the K value for methane is given by
p
K i + pci
k
Ǔƫ
expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1
ri
,
p ri
A 1*1
+ 11.0 lbmńft 3 + 0.176 gńcm 3 ,
. . . . . . . . . . . . . . . . . . (3.159)
where Z+0.803 is estimated from the Standing-Katz8 correlation
(Eqs. 3.42 and 3.43). From Eq. 3.65b, the constants in the gas viscosity correlation are
[9.379 ) 0.01607(24.62)](650)
A1 +
209.2 ) 19.26(24.62) ) 650
1.5
+ 121.5,
resulting in
A+1*
ǒ
and K C + 667.8
1
2, 000
Ǔ expƪ(5.214)(0.176)
+ 0.627,
1.287
ƫ + 0.0212 cp.
The oil viscosity is calculated by first estimating dead-oil viscosity, m oD. With the Bergman* correlation (Eq. 3.119),
ln lnǒ m oD ) 1Ǔ + A 0 ) A 1 ln(T ) 310), . . . . . . . . . (3.119)
Ǔ
0.627*1
expƪ5.37(0.627)(1 ) 0.0115)ǒ1 * 1ń1.807Ǔƫ
+ 9.21,
0.749
giving
10
0.7
1
and A 3 + 2.447 * 0.2224(5.214) + 1.287 ,
m g + ǒ121.5
* 14.7 Ǔ
ǒ2,500
000 * 14.7
(T r) C + (160 ) 460)ń343 + 1.807,
A 2 + 3.448 ) ǒ986.4ń650Ǔ ) 0.01009(24.62) + 5.214,
*4
. . . . . . . . . . . . . . . . . (3.125b)
gives A 1 + expƪ4.768 * 0.8359 ln(700 ) 300)ƫ + 0.3656,
ǒ Ǔ
(2, 500)(24.62)
(0.803)(10.73)(190 ) 460)
133.5 ,
R s ) 300
and the ci value for methane is
c i + 1ń(K i * 1) + 1ń(9.21 * 1) + 0.122.
With these K values, four iterations are used to solve the MuskatMcDowell equation,
2
where A 0 + 22.33* 0.194(36)) 0.00033 (36) + 15.77,
and m oD + * 1 ) expNJ exp[15.77*2.534 ln(190 ) 310)] Nj
+ 1.78 cp.
The viscosity correction for a saturated live oil depends on the
amount of gas in solution, R s . We estimate the solution gas/oil ratio
using Standing’s10 bubblepoint pressure correlation (Eq. 3.78), setting the current pressure of 2,500 psia as the bubblepoint of the saturated oil and solving for R s as given by Eq. 3.87,
R s + (0.85)
NJ[(0.055)(2, 500)10 ) (1.4)]
(0.00091)(190)
Nj
10 (0.0125)(36)
+ 700 scfńSTB.
*Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
12
ȍF
N
h(F v) +
A 1 + * 3.20 ) (0.0185)(36) + * 2.534,
i+1
v
zi
+ 0,
) ci
. . . . . . . . . . . . . . . . . . . . . . (4.39)
where c i + 1ń(K i * 1). Table B-20 summarizes the calculated results from the iterations, and Table B-21 gives the final results for
the flash calculation, including equilibrium vapor and liquid compositions.
Problem 16
Problem. Calculate the bubblepoint pressure for the ternary system
in Problem 15 at 160°F using the modified Wilson K-value equation. Eq. 3.165 is used to solve for bubblepoint pressure given a Kvalue correlation based on convergence pressure.
Fǒ p KǓ + 1 *
ȍ z K ǒ p , p , TǓ + 0.
N
i
i
K
b
. . . . . . . . . . . . (3.165)
i+1
PHASE BEHAVIOR
TABLE B-21—FINAL FLASH-CALCULATION RESULTS (PROBLEM 15)
zi /(Fv + ci )2
zi
Ki
ci
zi /(Fv + ci )
C1
0.20
9.208
0.122
0.331
0.548
0.0403
0.3713
C3
0.32
1.439
2.278
0.116
0.042
0.2641
0.3800
n-C5
0.48
0.358
*1.556
Total
*0.447
7.07 10*17
1.00
h(Fv )+7.07
xi
yi
0.416
0.6956
0.2487
1.00590
1.0000
1.0000
10*17 and hȀ(Fv )+*1.00590.
TABLE B-22—PRESSURE-GUESS CALCULATIONS* (PROBLEM 16)
pK +1,300 psia
pK +1,375 psia (correct)
pK +1,500 psia
Component
zi
Ki
yi +zi Ki
Ki
yi +zi Ki
Ki
yi +zi Ki
C1
0.20
2.181
0.436
1.982
0.396
1.703
0.341
C3
0.32
1.003
0.321
0.996
0.319
0.988
0.316
C5
0.48
0.560
0.269
0.594
0.285
0.657
0.315
Total
1*Syi +*0.02591
1.00
1*Syi +*0.00002[0
1*Syi +0.028024
*At T+160°F.
a. Calculate the convergence pressure, p K, that matches the measured bubblepoint pressure. Use the modified Wilson K-value equation (Eq. 3.159) with A 0 + 0.7.
b. Use the K-value correlation developed in Part a to make a
single-stage separator flash calculation to 14.7 psia and 60°F. Report the stock-tank-gas and -oil compositions, GOR, oil gravity in
°API, and gas specific gravity.
TABLE B-23—OIL COMPOSITION (PROBLEM 17)
Component
Mole Fraction
CO2
0.0111
C1
0.3950
C2
0.0969
C3
0.0784
i-C4
0.0159
n-C4
0.0372
i-C5
0.0123
Solution. Table B-25 gives relevant component properties for this
problem. The K values at reservoir conditions are calculated with
T+236°F. The modified Wilson equation (Eq. 3.159) is
ǒ Ǔ
p
K i + pci
K
A 1*1
Ǔƫ
expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1
ri
,
p ri
n-C5
0.0211
C6
0.0295
. . . . . . . . . . . . . . . . . . (3.159)
C7+
0.3026
where A + 1 * ( pńp K) and A 0 + 0.7 is assumed. By adjusting
convergence pressure, p K, the bubblepoint condition given by Eq.
3.165 is satisfied with p K + 4, 052.8 psia.
Table B-26 gives the K values and incipient-phase gas composition. The K-value correlation is then used to make a flash calculation
at standard conditions p+14.7 psia and T+60°F. With K values at
these conditions, the Rachford-Rice equation, Eq. 4.36, is solved for
gas-phase mole fraction ( F v + F g) where F g + 0.64241; stocktank-oil and separator-gas compositions are given later.
On the basis of the surface-gas composition, specific gravity g g is
MC
gC
0.7
182
7)
0.8275
7)
K wC
11.79
7)
C7+ is split into three fractions; Table B-24 gives mole fractions and properties.
Solution. Although an iterative procedure, such as Newton-Raphson, can be solved analytically with the modified Wilson K-value
equation, it takes only a few guesses to locate the pressure that satisfies Eq. 3.165. Table B-22 summarizes the results of the calculations for three guesses of pressure, where p K + 1, 375 psia gives
a satisfactory result for bubblepoint pressure.
g g + 27.32ń28.97 + 0.943 (air + 1)
and stock-tank oil properties are
M o + ǒSm o iǓńǒSn o iǓ + 167.8ń1.0 + 167.8 lbmńlbm mol,
ò o + ǒSm o iǓńǒSV o iǓ + 167.8ń3.323 + 50.48 lbmńft 3 ,
Problem 17
Problem. Tables B-23 and B-24 show the composition of a reservoir oil in the Kabob field, Canada. Bubblepoint pressure is 3,100
psia at 236°F reservoir temperature.
g o + ò ońò w + 0.8094 (water + 1),
and g API + 141.5ńg o * 131.5 + 43.3°API,
TABLE B-24—MOLE FRACTIONS AND PROPERTIES OF C7+ COMPONENT (PROBLEM 17)
C7+ Fraction
zi
Mi
Tci
(°R)
pci
(psia)
ąăwi ąă
F1
0.1578
114.1
1065.5
409.6
0.3255
0.7674
727.0
F2
0.1243
223.1
1356.0
235.1
0.6538
0.8403
1,029.5
F3
0.0205
455.0
1689.1
134.6
1.1489
0.9254
1,410.1
0.3026
182.0
Total
ąăgi *
Tbi
(°R)
0.8275
*Water+1.
EXAMPLE PROBLEMS
13
TABLE B-25—COMPONENT PROPERTIES (PROBLEM 17)
Component
zi
Mi
wi
0.0111
44.01
31.18
1,070.6
547.6
0.2310
0.3950
16.04
20.58
667.8
343.0
0.0115
C2
0.0969
30.07
28.06
707.8
549.8
0.0908
C3
0.0784
44.09
31.66
616.3
665.7
0.1454
i-C4
0.0159
58.12
35.01
529.1
734.7
0.1756
C4
0.0372
58.12
36.45
550.7
765.3
0.1928
i-C5
0.0123
72.15
39.13
490.4
828.8
0.2273
C5
0.0211
72.15
39.30
488.6
845.4
0.2510
C6
0.0295
86.17
41.19
436.9
913.4
0.2957
F1
0.1578
114.10
47.86
409.6
1,065.5
0.3255
F2
0.1243
233.10
52.41
235.1
1,356.0
0.6538
F3
0.0205
455.00
57.72
134.6
1,689.1
1.1489
1.0000
pri
Modified
Wilson
Ki
1.270
2.90
1.324
0.0147
C1
2.028
4.64
1.539
0.6079
C2
1.265
4.38
1.196
0.1159
CO2
Tci
(°R)
C1
TABLE B-26—K VALUES AND INCIPIENT-PHASE
GAS COMPOSITION (PROBLEM 17)
Tri
pci
(psia)
CO2
Total
Component
òi
(lbm/ft3)
Incipient
Phase
yi +zi Ki
where n o i + x i, m o i + x i M i, and V o i + x i M ińò i .
On the basis of 1 mole of feed, the surface volumes are given by
V g + 379F g + 379(0.64241) + 243.5 scf
and V o + ǒ1 * F gǓǒM ońò oǓ + (1 * 0.64241)ǒ167.8ń50.48Ǔ
+ 1.188 ft 3 + 0.2117 STB
and the GOR is
C3
1.045
5.03
0.990
0.0776
i-C4
0.947
5.86
0.867
0.0138
C4
0.909
5.63
0.831
0.0309
i-C5
0.839
6.32
0.733
0.0090
C5
0.823
6.34
0.709
0.0150
Problem 18
C6
0.762
7.10
0.614
0.0181
F1
0.653
7.57
0.461
0.0727
F2
0.513
13.19
0.189
0.0234
F3
0.412
23.03
0.04302
Problem. Make equation-of-state (EOS) calculations using the
Peng-Robinson 16 EOS (PR EOS) for the ternary system described
in Table B-28. Use the cubic m term (Eq. 4.22) for wu0.4 (C10).
a. Make a two-phase flash calculation at 500 psia and 280°F.
b. Make a Michelsen phase-stability test followed by a two-phase
flash calculation at 1,500 psia and 280°F.
Total
0.0009
1.0000
R go + V gńV o + 243.5ń0.2117 + 1, 150 scfńSTB.
Table B-27 summarizes the results.
TABLE B-27—SEPARATOR FLASH CALCULATION (PROBLEM 17)
Modified
Wilson
Ki
MuskatMcDowell
zi /(F g + ci )
Stock-Tank
Oil
xi
Separator
Gas
yi
V oi + x i M ińò i
(ft3)
m gi + y i M i
0.01
0.000
0.75
0.03
0.002
9.84
0.1484
0.13
0.005
4.46
0.1136
0.67
0.021
5.01
0.0204
0.45
0.013
1.19
Component
Tri
pri
CO2
0.949
0.014
51.05
0.0168
0.0003
0.0171
C1
1.515
0.022
287.94
06116
0.0021
0.6137
C2
0.945
0.021
34.28
0.1441
0.0043
C3
0.781
0.024
7.44
0.0983
0.0153
i-C4
0.707
0.028
2.64
0.0127
0.0077
(lbm)
C4
0.679
0.027
1.81
0.0199
0.0244
0.0443
1.42
0.039
2.58
i-C5
0.627
0.030
0.662
*0.0053
0.0157
0.0104
1.13
0.029
0.75
C5
0.615
0.030
0.493
*0.0159
0.0313
0.0154
2.26
0.057
1.11
C6
0.569
0.034
0.153
*0.0549
0.0647
0.0099
5.58
0.135
0.85
F1
0.488
0.036
1.58 x 10*2
*0.4223
0.4291
0.0068
48.96
1.023
0.77
*0.3476
0.3476
0.0000
81.02
1.546
0.00
*0.0573
0.0573
0.0000
26.08
0.452
0.00
0.0000
1.0000
1.0000
167.76
3.323
27.32
F2
0.383
0.063
9.93x 10*6
F3
0.308
0.109
4.83x 10*11
Total
14
m oi + x i M i
(lbm)
PHASE BEHAVIOR
TABLE B-28—TERNARY SYSTEM (PROBLEM 18)
Component
i
zi
Mi
Tci
(°R)
pci
(psia)
wi
si +ci /bi
C1
0.50
16.04
343.0
667.8
0.0115
*0.1595
C4
0.42
58.12
765.3
550.7
0.1928
*0.0675
C10
0.08
142.29
1,111.8
304.0
0.4902
0.0655
TABLE B-29—CHANGES DURING ITERATIONS
(PROBLEM 18)
Convergence
Tolerance
log[S(1–fLi /fvi )2]
Iteration
Trivial Solution
Indicator
S(ln Ki )2
Vapor-Phase
Mole Fraction
Fv
1
0.708
30.73
0.852187
2
*2.230
15.24
0.853914
3
*4.380
14.66
0.853528
4
*6.454
14.61
0.853423
5
*8.457
14.61
0.853405
6
*15.236
14.61
0.853401
Solution.
a. The flash calculation is made with five successive-substitution iterations followed by a general dominant eigenvalue method (GDEM)
promotion. Tables B-29 and B-30 give the results of the calculations
for the six iterations required to solve the flash problem. Table B-31
shows the change in convergence tolerance, the trivial-solution indicator, and vapor-phase mole fraction during each iteration. The convergence tolerance indicates how close the phase fugacities of each
component have come to one another. Convergence was specified as
10 *12 in this example. The trivial-solution indicator stabilizes after
three iterations. Convergence toward a trivial solution is usually indicated for values S(ln K i) 2 t 10 *4. Details of the EOS calculations for the first iteration are summarized later, step by step.
K-Value Estimate. The Wilson13 equation is used to estimate K
values.
Ki +
ƪ
ǒ
exp 5.37ǒ1 ) w i Ǔ 1 * T *1
ri
Ǔƫ
.
pr i
. . . . . . . . . . . . . (4.42)
This gives
(m) C + 0.3796 ) 1.485(0.4902) * 0.1644(0.4902)
expƪ5.37(1 ) 0.4902)ǒ1 * 1ń0.666Ǔƫ
+ 0.0108,
ǒ500ń304Ǔ
and for the other components, K C1 + 24.58 and K C4 + 0.8820.
Phase Split. With K-value estimates and the feed composition
known, a phase split is made with either the Rachford-Rice15 or Muskat-McDowell14 algorithms. This results in vapor-phase mole fraction, F v + 0.852187, and the compositions given in Table B-30.
EOS Constants for Each Phase Separately. EOS Constants A
and B must now be calculated separately for the vapor and liquid
phases on the basis Compositions y i and x i. For decane,
3
) 0.01667(0.4902) + 1.070,
(T r) C + (280 ) 460)(1, 111.8) + 0.666,
10
ƪ
(a) C + 1 ) (1.070)ǒ1 * Ǹ0.666Ǔ
10
(a) C +
10
2
ƫ + 1.432,
2
2
W a R 2 T c2
(10.73) (1, 111.8)
(1.432)
p c a + 0.45724
304.0
+ 306, 500 psia-ft 3ńlbm mol *1,
RT
(10.73)(1, 111.8)
and (b) C + W b p c + 0.07780
c
10
304.0
+ 3.053 ft 3ńlbm mol.
From Eq. 4.9,
A+a
p
(RT)
and B + b
2
pr
+ 27 2
64 T r
pr
p
+1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.9)
RT
8 Tr
at 500 psia and 280°F. EOS Constants A and B for decane are
and (B) C + (3.053)
10
500
+ 2.435
2
2
(10.73) (280 ) 460)
500
+ 0.1922,
(10.73)(280 ) 460)
and for other components A C + 0.04906, A C + 0.4544, B C
1
+ 0.02701, and B C + 0. 07308.
4
1
4
The A i and B i constants are the same for both phases. To calculate A and B constants for the vapor phase ( A v and B v) and the liquid
phase ( A L and B L ), traditional mixing rules are used (Eq. 4.16).
ȍȍy y A
N
Av +
N
i j
ij,
i+1 j+1
ȍȍx x A
N
N
R 2T 2
a + W oa p c a , where W oa + 0.45724;
c
AL +
RT
b + W ob p c , where W ob + 0.07780;
c
A i j + ǒ1 * k i jǓ ǸA i A j
a + ƪ1 ) mǒ1 * ǸT rǓƫ ;
Bv +
i j
ij ,
i+1 j+1
2
ȍy B ,
N
and m + 0.37464 ) 1.54226 w * 0.26992 w .
2
. . . . . . . (4.21)
The modified relation for m (Eq. 4.22) is used for decane because
its acentric factor is greater than 0.4,
EXAMPLE PROBLEMS
2
10
10
T r + TńT c + (280 ) 640)ń(1, 111.8) + 0.666,
10
. . . . . . . . . . . . . . . . . . . . (4.22)
(A) C + (306, 500)
For decane,
and K C +
m + 0.3796 ) 1.485w * 0.1644w 2 ) 0.01667w 3 .
i
i
i+1
ȍx B .
N
and B L +
i
i
i+1
15
TABLE B-30—FUGACITY CALCULATION RESULTS (PROBLEM 18)
Component
i
yi
fvi
(psia)
Ki +yi /xi
xi
fLi
(psia)
fLi /fvi
85.3847
0.28650
Iteration 1 (Wilson K–Value Estimate)
C1
0.58262
0.02370
C4
0.41186
0.46695
24.5823
0.882021
298.023
C10
0.00553
0.50935
0.010854
C1
0.57165
0.08117
7.04293
294.517
279.596
0.94934
C4
0.41277
0.46224
0.892986
148.515
148.117
0.99732
C10
0.01557
0.45659
0.034107
C1
0.57115
0.08542
6.6861
294.392
292.992
0.99524
C4
0.41258
0.46326
0.89059
148.363
148.288
0.99950
C10
0.01628
0.45132
0.03607
C1
0.57114
0.08583
6.6543
294.394
294.253
0.99952
C4
0.41254
0.46345
0.890144
148.344
148.332
0.99992
C10
0.01633
0.45072
0.036227
C1
0.57114
0.08587
6.6511
294.396
294.381
0.99995
C4
0.41253
0.46348
0.890073
148.342
148.34
0.99999
C10
0.01633
0.45065
149.526
151.385
1.06778
1.01243
3.35554
3.14253
Iteration 2 (Successive Substitution)
2.89097
3.05736
1.05755
Iteration 3 (Successive Substitution)
3.01459
3.02765
1.00433
Iteration 4 (Successive Substitution)
3.02324
3.02426
1.00034
Iteration 5 (Successive Substitution)
0.036239
3.02377
3.02385
1.00003
Iteration 6 (GDEM Promotion)
C1
0.57114
0.08588
6.65071
294.397
294.397
1.00000
C4
0.41253
0.46349
0.890061
148.342
148.342
1.00000
C10
0.01633
0.45064
0.03624
Recall that the compositions y i and x i result from the phase-split
calculation based on feed composition z i and the current K-value estimates. For the initial K-value estimates and resulting compositions
from the phase-split calculation, EOS Constants A and B are
0.5
A L + (0.02370)(0.02370)[(0.04906)(0.04906)] (1 * 0)
0.5
) (0.02370)(0.46695)[(0.04906)(0.4544)] (1 * 0)
0.5
) (0.02370)(0.50935)[(0.04906)(2.435)] (1 * 0)
0.5
) (0.46695)(0.02370)[(0.4554)(0.04906)] (1 * 0)
0.5
) (0.46695)(0.46695)[(0.4554)(0.4554)] (1 * 0)
0.5
) (0.46695)(0.50935)[(0.4554)(2.435)] (1 * 0)
0.5
) (0.50935)(0.02370)[(2.435)(0.04906)] (1 * 0)
0.5
) (0.50935)(0.46695)[(2.435)(0.4544)] (1 * 0)
0.5
) (0.50935)(0.50935)[(2.435)(2.435)] (1 * 0)
+ 1.252,
B L + (0.02370)(0.02701) ) (0.46695)(0.07308)
) (0.50935)(0.1922) + 0.1327,
A v + 0.1725,
3.02379
ƪ(0.1812) 3 * (1 * 0.1327)(0.1812) 2ƫ
) ƪ1.252 * 3(0.1327) * 2(0.1327)ƫ(0.1812)
2
* ƪ1.252(0.1327) * (0.1327) * (0.1327)
2
and Z 3v * (1 * B v) Z 2v ) ǒA v * 3 B 2v * 2 B vǓ Z v
* ǒA v B v * B 2v * B 3v Ǔ + 0.
16
3
ƫ
+ 0.0005 [ 0.
Fugacity Calculations. Fugacity values of each component for
each phase are calculated with Eq. 4.23,
f
ln p +ln f + Z * 1 * ln(Z * B) *
ln
Z-Factor Calculation. With the EOS constants for each phase,
the Z factor (i.e., volume solution to the cubic EOS) can be solved.
Eq. 4.20 is used for each phase separately.
* ǒA L B L * B 2L * B 3LǓ + 0
1.00000
The solutions to these two equations with Constant A and B values
calculated in the previous section are Z L + 0.1812 and
Z v + 0.8785.
We can check, for example, the liquid solution by substituting
Z L + 0.1812 into Eq. 4.20 together with A L + 1.252 and
B L + 0.1327.
and B v + 0.04690.
Z 3L * ǒ1 * B L Ǔ Z 2L ) ǒA L * 3 B 2L * 2 B LǓ Z L
3.02379
and ln
ƪ
Z) ǒ1 ) Ǹ2Ǔ B
Z) ǒ1 ) Ǹ2Ǔ B
ƫ
A
2 Ǹ2 B
fi
B
+ ln f i + i (Z * 1) * ln(Z * B)
B
yi p
)
A
2 Ǹ2 B
ǒ
Bi 2
*
B
A
ȍyA
N
j
j+1
Ǔƪ
ij
ln
Z) ǒ1 ) Ǹ2Ǔ B
Z) ǒ1 ) Ǹ2Ǔ B
ƫ
,
. . . . . . . . . . . . . . . . . . . . (4.23)
PHASE BEHAVIOR
TABLE B-31—PHASE STABILITY TEST RESULTS (PROBLEM 18)
Component
i
yi
zi
fyi
(psia)
Ki
fzi
(psia)
S+fzi /fyi
Vapor–Like Stability Test: Ki +yi /zi *
C1
0.66910
0.50
1.3540
1,053
1,066
1.0118
C4
0.30930
0.42
0.7450
194.7
197.0
1.0118
C10
0.02166
0.08
0.2740
2.712
2.744
1.0118
C1
0.31870
0.50
1.5430
1,048
1,066
1.0168
C4
0.47670
0.42
0.8664
193.7
197.0
1.0168
C10
0.20460
0.08
0.3846
2.699
2.744
1.0168
Liquid–Like Stability Test: Ki +zi /yi **
*Unstable; converged solution, SV =1.0118, 12 iterations.
**Unstable; converged solution, SL =1.0168, 6 iterations.
TABLE B-32—CONVERGED FLASH SOLUTION (PROBLEM 18)
Component
i
Initial K Values From Stability Test
Ki +(yi )v /(yi )L
yi
xi
Ki =yi /xi
fvi
(psia)
fLi
(psia)
C1
2.08907
0.629843
0.330082
1.90814
1,019.52
1,019.52
C4
0.645515
0.348699
0.513307
0.67932
210.076
210.076
C10
0.10537
0.021457
0.156611
0.13701
2.26859
2.26859
giving the results in Table B-30. Component fugacities are clearly
not equal within an acceptable tolerance; e.g., ( f v) C 10 + 1.068 psia
and ( f L) C 1 + 3.355 psia. K values are then updated with the fugacity ratio, f Lńf v, as a correction term.
K i(n)1) + K i(n)
f Li(n)
.
f vi(n)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.48)
This type of simple K-value update is called successive substitution,
and for decane the second K-value estimate is given by
+ K (1)
K (2)
C
C
10
10
f (1)
L ,C
f (1)
v ,C
10
10
+ (0.01085) 3.355 + (0.01085)(3.142) + 0.0341.
1.068
After the first GDEM promotion, convergence was achieved, resulting in vapor-phase mole fraction of F v + 0.853401. K values were
K C1 + 6.65071, K C4 + 0.890061, and K C10 + 0.03624. Table
B-30 gives the phase compositions.
b. Table B-31 gives the phase-stability test results at 1,500 psia and
280°F. Results from the converged solutions of the vapor- and liquidlike tests are shown. Both stability tests indicated that the feed composition was unstable and would therefore split into two (or more)
phases. The vapor-like test required 12 iterations to converge, including two GDEM promotions. The liquid-like stability test required six
iterations to converge, including one GDEM promotion.
Because two unstable solutions were found, the two-phase flash
calculation was initialized with K values based on the two incipientphase compositions found in the stability tests; i.e., K i + (y i) vń(y i) L .
With these initial estimates, the two-phase flash calculation converged in eight iterations, including one GDEM promotion. The final
vapor-phase mole fraction was F v + 0.566844. Note how close the
final converged K values are to the initial estimates from the stability
test. Table B-32 gives the results.
Problem 19
Problem. The following are calculated phase properties from the
flash calculation at 500 psia and 280°F in Problem 18.
M L + 111.7 lbmńlbm mol,
M v + 35.46 lbmńlbm mol,
EXAMPLE PROBLEMS
v L + 2.721 ft 3ńlbm mol,
and v v + 13.837 ft 3ńlbm mol.
These molar volumes include the effect of a slight shift in volume
by use of volume translation. What is the phase molar volume and
liquid density without volume translation?
Solution. The volume shift, c i, for each component is calculated from
EOS constants b i and the volume translation ratios, s i, given in Problem 18. Eq. 4.21 gives the b i values for the PR EOS.16
R 2T 2
a + W oa p c a , where W oa + 0.45724;
c
RT
b + W ob p c , where W ob + 0.07780;
c
a + ƪ1 ) mǒ1 * ǸT rǓƫ ;
2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . . (4.21)
This gives
b C + 0. 07780(10.7315)(343.0)ń(667.8)
1
+ 0.4288 ft 3ńlbm mol,
b C + 1.160 ft 3ńlbm mol,
4
b C + 3.053 ft 3ńlbm mol,
10
c C + (* 0.1595)(0.4288) + * 0.06840 ft 3ńlbm mol
1
c C + (* 0.0675)(1.1603) + * 0.07832 ft 3ńlbm mol,
4
and c C + (0.0655)(3.053) + 0.2000 ft 3ńlbm mol.
10
From Eq. 4.25,
ȍx c
N
v L + v LEOS *
i i
i+1
ȍy c .
N
and v v + v vEOS *
i i
. . . . . . . . . . . . . . . . . . . . . . . (4.25)
i+1
17
TABLE B-33—RECOMBINED SEPARATOR WELLSTREAM MOLAR COMPOSITION AND CONSISTENCY CHECK OF
SEPARATOR K VALUES WITH THE STANDING18 LOW-PRESSURE K-VALUE CORRELATION (PROBLEM 20)
zi
xi
Reported
yi
CO2
4.01
1.12
3.84
3.84
2.087
N2
0.85
0.03
0.80
0.80
3.394
C1
89.83
10.68
85.12
85.16
2.606
8.42
8.41
C2
2.88
2.56
2.86
2.86
1.543
1.14
1.13
C3
1.30
3.86
1.45
1.45
0.811
0.289
0.337
i-C4
0.32
2.60
0.46
0.45
0.346
0.121
0.123
n-C4
0.43
5.31
0.72
0.72
0.180
0.0884
0.0810
i-C5
0.15
3.88
0.37
0.37
*0.256
0.0390
0.0387
C5
0.11
4.16
0.35
0.35
*0.391
0.0303
0.0264
Calculated
Fi
Standing
Reported
3.18
3.58
37.0
28.3
C6
0.07
7.58
0.52
0.51
*0.859
0.0126
0.0092
C7+
0.05
58.22
3.51
3.48
*2.010
0.00145
0.00086
Total
gC
100.00
100.00
100.00
100.00
0.778
0.778
7)
MC
7)
M
0.7783
135
98
135
135
100.2
18.6
23.4
23.4
With liquid compositions calculated in Problem 18 at 500 psia and
280°F, the molar volume without volume translation, v LEOS , is given
by
v LEOS + 2.721 ) [(0.08588)(* 0.0684) ) (0.46349)
(* 0.07832) ) (0.45064)(0.2000)]
+ 2.721 ) (* 0.006 * 0.0363 ) 0.0901),
+ 2.769 ft 3ńlbm mol.
The molecular weight of liquid is needed to convert from molar
volume to density.
M L + (0.08588)(16.04) ) (0.46349)(58.12)
) (0.45064)(142.29) + 92.44 lbmńlbm mol,
ò L + 94.44ń2.769 + 33.38 lbmńft .
3
Problem 20
Problem. Separator samples were collected during a production test
from the discovery well of a gas-condensate reservoir. Use the Hoffmann-Crump-Hocott 17 (HCH) K-value plot (Eqs. 3.155 and 3.156)
to check the consistency of measured separator compositions. Plot
the data together with the low-pressure Standing18 K-value correlation line given by Eq. 3.161. Also recombine the separator samples
to check the reported wellstream composition (laboratory recombined values can be in error). Finally, calculate the Watson characterization factor of the C 7) component.
Solution. Table B-33 gives reported separator compositions; calculated K values from the ratio of separator-gas to separator-oil molar
compositions, K i + y ińx i; and finally, the recombined wellstream
composition, z i.
Separator conditions are 390 psig and 52°F. The HCH variable F i
is given by Eq. 3.156, with b i and T bi values given in Table 3.3.
Methane, for example, has an F i value given by
b i + 300 cycleń°R,
T bi + 94°R,
and F i + 300 ƪ1ń94 * 1(52 ) 460)ƫ + 2.606,
where modified values of b i and T bi are given by Standing (instead
of values given by Eq. 3.156).
The K-value pressure product for methane is given by
K i p sp + ǒ89.83ń10.68Ǔ(390 ) 14.7) + (8.411)(404.7),
+ 3, 404 psia.
which is plotted vs. F i + 2.606 on semilog paper (Fig. B-1).
The Standing low-pressure K-value correlation is plotted together
with the measured K-values on Fig. B-1. From Standing’s18 correlation, Slope A 0 and Intercept A 1 are
K i + p1 10 ǒ A0 ) A1 Fi Ǔ ,
. . . . . . . . . . . . . . . . . . . . . . . (3.161a)
sp
F i + b iǒ1ńT bi * 1ńTǓ, . . . . . . . . . . . . . . . . . . . . . . . (3.161b)
b i + logǒ p cińp scǓńǒ1ńT bi * 1ńT ciǓ ,
which gives
18
Ki
Component
A 0( p) + 1.2 ) ǒ4.5
. . . . . . . . . . . . . . (3.161c)
10 *4Ǔ p ) ǒ15
10 *8Ǔ p 2 ,
. . . . . . . . . . . . . . . . . . (3.161d)
A 1ǒ pǓ + 0.890 * ǒ1.7
10
*4
Ǔ p * ǒ3.5
10 *8Ǔ p 2,
. . . . . . . . . . . . . . . . . . . (3.161e)
nC
7)
+ 7.3 ) 0.0075T ) 0.0016p,
bC
7)
+ 1, 013 ) 324n C
and T bC
7)
7)
+ 301 ) 59.85n C
. . . . . . . . . . . . . (3.161f)
* 4.256n 2C
7)
7)
,
* 0.971n 2C
. . . . . . . (3.161g)
,
7)
. . . . . (3.161h)
giving
A 0 + 1.2 ) ǒ4.5
10 *4Ǔ(404.7) ) ǒ15
10 *8Ǔ(404.7)
2
+ 1.407
and A 1 + 0.890 * ǒ1.7
* ǒ3.5
10 *4Ǔ(404.7)
10 *8Ǔ(404.7) + 0.8155.
2
The methane K value from the Standing correlation is, for example,
K C + ǒ1ń404.7Ǔ10 [1.407)(0.8155)(2.606)] + 8.415,
1
which can be compared with the measured value of 8.411.
PHASE BEHAVIOR
The Standing C 7) K value requires calculating b C and T bC
7)
7)
from separator conditions.
nC
+ 7.3 ) 0.0075(52) ) 0.0016(404.7)
7)
y w + y ow A g A s ,
+ 8.34 (approximate carbon number),
bC
7)
+ 1, 013 ) (324)8.34 * (4.256)(8.34)
Solution. From Fig. 9.29, the temperature for hydrate formation of
a 0.7-gravity gas at 1,000 psia is about 69°F. From Eq. 9.23, the water content in the gas at 1,000 psia and 69°F is
* 1.117 ln p ) 16.44 ,
+ 3, 419 cycle-°R,
T bC
2
7)
and F C
+ 301 ) (59.85)(8.34) * (0.971)(8.34) + 732.6°R,
7)
7)
. . . . . . . . . . . . . . . . . (9.23b)
g g * 0.55
Ag + 1 )
ǒ1.55
10 4 Ǔg g T *1.446 * ǒ1.83
+ 3, 419ƪ(1)ń(732.6) * (1)ń(52 ) 460)ƫ + * 2.01.
+ ǒ1ń404.7Ǔ10
[1.407)(0.8155)(*2.01)]
+ 0.00145,
which can be compared with the measured value of K C + 0.050
7)
B 58.22 + 0.00086.
From Fig. B-1 the measured K-value data plot as a straight line
almost coincident with the Standing correlation. This indicates that
the measured compositions are probably consistent.
Recombination is made on the basis of separator gas/oil ratio,
R sp with Eq. 6.8. Separator-oil density and molecular weight are
both required for the recombination calculation, and most laboratories use the Standing-Katz8 density correlation to estimate ò osp on
the basis of separator-oil composition (oil molecular weight is calculated from Eq. 6.9). For this sample, the separator properties and
gas mole fraction, F gsp, are given by M osp + 100.2 lbm/lbm mol
and ò osp + 45.28 lbm/ft3.
ǒ
F gsp + 1 )
2, 130ò osp
M osp R sp
Ǔ
*1
,
. . . . . . . . . . . . . . . . . . . . (6.8)
ƪ
ƫ
+ 0.94102 .
* ǒ1.83
ln y ow +
ƫNj + 0.9972,
and y w + (0.9972)(0.000546) + 0.000544.
From Eq. 9.24, the solution water/gas ratio is given by
yw
[ 135y w ,
1 * yw
. . . . . . . . . . . . . . . . . . . (9.24)
which gives r sw + (47, 300)(0.000544)ń(1 * 0.000544) + 25.7
lbm/MMscf.
At 15°F and 1,000 psia, the water content is
10 *5 ,
EXAMPLE PROBLEMS
*1.446
+ 0.000546,
y w + 8.61
Problem 21
Problem. A refrigeration/expansion process is used to reduce water
and condensate content of a gas stream. The well effluent arrives at
the separator at 2,000 psia and 155°F. It is cooled in the separator
and heat exchanger. Separator pressure is 1,000 psia, separator-gas
gravity is 0.70 (air+1), and separator-gas rate is 65 MMscf/D.
What is the minimum temperature upstream of the choke to prevent hydrate formation in the separator?
What is the water content of the separator gas?
How much water in lbm/D must be removed from the separator
gas if sales specifications call for a maximum dewpoint of )15°F
at 1,000 psia?
*1.288
10 4Ǔ(0.7)(69)
* 1.117 ln(1, 000) ) 16.44,
y ow
10 *5 ,
which is quite close to the reported value of 85.12% (as it should be).
Occasionally, because of entry errors to the recombination computer program or possibly because of inconsistent recombination GOR
used in the laboratory, reported wellstream compositions may not
be the same as those calculated with Eqs. 6.7 through 6.9. In these
situations, contact the laboratory about the inconsistency. It may
even be worthwhile to request a preliminary report of the separator
and recombined compositions before completing pressure/volume/
temperature study.
. . . . . . . . . . . . . . . . (9.23e)
(0.05227)(1, 000) ) 142.3 ln(1, 000) * 9, 625
69 ) 460
y ow + 8.61
1
,
. . . . . . . . . . . . . . . . . . (9.23d)
10 *9ǓC 1.44
.
s
10 4Ǔ(69)
z i + F gsp y i ) ǒ1 * F gspǓ x i .
z C + (0.94102)(89.83) ) (1 * 0.94102)(10.68) + 85.16% ,
ǓC s ,
NJ
A g + 0.9996,
For methane, this is
*6
A g + 1 ) (0.7 * 0.55)ńƪǒ1.55
The wellstream composition is calculated from Eq. 6.7.
. . . . . . . . . . . . . . . . . . . . . (6.7)
*1.288
This gives
which yields
*1
10
and A s + 1 * ǒ3.92
r sw + 135
(2, 130)(45.28)
F gsp + 1 )
(100.2)ń(15, 357)
10 4 T Ǔ
. . . . . . . . . . . . . . . . . . (9.23c)
A s + 1 * ǒ2.22
This yields
KC
0.05227p ) 142.3 ln p * 9, 625
T ) 460
ln y ow +
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.23a)
and r sw + 4.0 lbmńMMscf.
At a separator-gas rate of 65 MMscf/D, the water removal capacity
must then be (25.7*4.0)(65)+1,445 lbm/D.
Problem 22
Problem. Estimate gas solubility for the reservoir brine in Problem
21 at reservoir conditions of 4,050 psia and 255°F. Also estimate
brine density, compressibility, and FVF. The reservoir gas yields 13
STB/MMscf (MMscf of separator gas) of a 69°API stock-tank condensate. Brine salinity is 36,200 ppm total dissolved salts. Assume
separator conditions are 1,000 psia and 80°F.
Solution. The reservoir (wellstream hydrocarbon) specific gravity
is given by
gw +
g g ) 4, 580 r p g o
.
1 ) 133, 000 r p ǒ gńM Ǔ o
. . . . . . . . . . . . . . . . . . (3.55)
However, we need to estimate the amount and specific gravity of
the gas coming from separator condensate at 1,000 psia and 80°F using Eqs. 3.62 through 3.64.
g g) + A 2 ) A 3 R s) ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.62)
19
where A 2 + 0.25 ) 0.2g API and A 3 + * ǒ3.57
R s) +
and g g +
10 *6Ǔg API ;
A1 A2
; . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.63)
ǒ1 * A 1 A 3Ǔ
g g1 R s1 ) g gs1 R s)
.
R s1 ) R s)
. . . . . . . . . . . . . . . . . . (3.64)
Brine density at standard conditions is given by Eq. 9.14, with
T sc + 60°F [289 K].
v woǒ p sc,T Ǔ +
1
+ A 0 ) A 1w s ) A 2w 2s ,
ò woǒ p sc,T Ǔ
where A 0 + 5.916365 * 0.01035794T
) ǒ0.9270048
This gives
A 1 + 1, 152,
10 *5ǓT 2
* 1, 127.522T *1 ) 1, 00674.1T *2 ,
A 2 + 1.63,
10 *4 ,
A 3 + * 2.46
A 1 + * 2.5166 ) 0.0111766T * ǒ0.170552
10 *4ǓT 2 ,
and A 2 + 2.84851 * 0.0154305T ) ǒ0.223982
10 *4ǓT 2 ,
g g) + 0.985 (air + 1),
. . . . . . . . . . . . . . . . . . . . (9.14)
R s) + 2, 620 scfńSTB,
r p + 1ńƪ1ńǒ13
yielding
A 0 + 1.00106,
10 *6Ǔ ) 2, 620ƫ + 12.6 STBńMMscf,
A 1 + * 0.7112,
and g g + 0.711 (air + 1) .
From Fig. 9.2, the gas solubility of a 0.65°API gravity gas in pure
water at 4,000 psia and 250°F is about 19 scf/STB. Pure methane
solubility in pure water at reservoir conditions can be estimated
from Eq. 9.6.
x C + 10 *6
1
ƪȍǒȍ Ǔ ƫ
3
3
i+0
j+0
A i jT j p i , . . . . . . . . . . . . . . . . (9.6)
v w + 0.9756 cm 3ńg,
At reservoir temperature (397 K) and standard pressure, brine
density, ò ow, is also given by Eq. 9.14, which results in
A 0 + 1.0642,
x C + 10 *3ƪ * 0.0256 ) (0.00107)(4, 050) * ǒ9.59
1
(4, 050) ) ǒ3.98
2
or R osw + 7, 370NJǒ2.73
10 *12Ǔ(4, 050) ƫ + 2.73
3
10 *8Ǔ
10 *3
10 *3ǓƫNj
10 *3Ǔńƪ1 * ǒ2.73
+ 0.1813 * ǒ7.692
) ǒ2.6614
10 *6ǓT 2 * ǒ2.612
10 *4Ǔ(255)ǒ2.6614
10 *6Ǔ(255)
2
10 *9Ǔ(255) + 0.115.
3
This can be corrected for specific gravity with Eq. 9.11, but we neglect the correction for simplicity. The resulting gas solubility of the
brine is then
xg
R sw
[ x o + 10 *kscs [ 10 *ǒ17.1
R osw
g
10 *6Ǔ k sC s
c *wǒ p, T Ǔ + ǒ A 0 ) A 1 p Ǔ
10 *6Ǔ (0.115)(36,000)ƫ
.
,
where A + 10 6ƪ0.314 ) 0.58w s ) ǒ1.9
*ǒ1.45
,
. . . . . . . . . (9.9)
A 0 + 0.289
10 *4ǓT
10 *6ǓT 2ƫ
10 6 ,
A 1 + 8.656,
and c *w + 3.09
10 *6 psi *1.
The FVF of brine without dissolved gas at atmospheric pressure
is given by
B ow +
ò wǒ p sc, T scǓ
v oǒ p sc, TǓ
+ w
, . . . . . . . . . . . . . . . . . (9.13)
o
v wǒ p sc, T scǓ
ò wǒ p sc, TǓ
yielding B ow + 1.025ń0.9646 + 1.063 bbl/STB.
From Eq. 9.18, the FVF of brine at reservoir pressure and temperature without dissolved gas is
resulting in
R sw + (17.5)10 ƪ*ǒ17.1
*1
yielding
10 *9ǓT 3 ,
which gives
* ǒ2.612
v ow + 1.0367 cm 3ńg,
and A 1 + 8 ) 50w s * 0.125w sT , . . . . . . . . . . . . . . . . . (9.17)
10 *4ǓT
. . . . . . . . . . . . . . . . . . (9.10)
k s + 0.1813 * ǒ7.692
A 2 + 0.253,
Compressibility of brine without solution gas is given by
This compares with 22 scf/STB from Fig. 9.1. We assume that
R osw + 17.5 for this gas in pure water (from a plot of 19 scf/STB at
g g + 0.65 and 22 scf/STB at g g + 0.55).
Reduction in solubility resulting from salinity can be estimated
from the Setchenow correction (Eqs. 9.9 and 9.10).19 For methane,
the Setchenow constant is
1*NaCl
A 1 + * 0.768,
and ò ow + 0.9646 gńcm 3.
+ 20. 2 scfńSTB.
20
10 *6 + 0.0362,
w s + 36, 200
and ò w + 0.9756 gńcm 3 .
yielding
(k s) C
A 2 + 0.2601,
B *wǒ p,
ǒ
TǓ + B owǒ p sc, TǓ 1 )
A1
p
A0
Ǔ
ǒ1ńA1Ǔ
,
. . . . . . . . . . (9.18)
PHASE BEHAVIOR
giving
B *w + 1.063ƪ1 ) (4, 050)(8.656)ńǒ0.289
10 6Ǔƫ
ǒ*1ń8.656Ǔ
+ 1.049 bblńSTB.
With the Dodson-Standing20 corrections for compressibility and
FVF as a function of gas solubility (Eqs. 9.19 and 9.20, respectively),
the brine volumetric properties including gas solubility effect are
1.5Ǔ
B wǒ p, T, R swǓ + B *wǒ p, TǓǒ1 ) 0.0001 R sw
. . . . . . . . (9.19)
and c wǒ p, T, R swǓ + c *wǒ p, TǓ ǒ1 ) 0.00877 R swǓ , . . . . . . . (9.20)
which give
B w + (1.049)ƪ1 ) (0.0001)ǒ17.5 1.5Ǔƫ + 1.057 bblńSTB
and c w + ǒ3.09
+ 3.55
10 *6Ǔƪ1 ) (0.00877)ǒ17.5 Ǔƫ
10 *6 psi *1.
References
1. Hall, K.R. and Yarborough, L.: “A New EOS for Z-factor Calculations,”
Oil & Gas J. (18 June 1973) 82.
2. Yarborough, L. and Hall, K.R.: “How to Solve EOS for Z–factors,” Oil
& Gas J. (18 February 1974) 86.
3. Sutton, R.P.: “Compressibility Factors for High-Molecular-Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September.
4. Lucas, K.: Chem. Ing. Tech. (1981) 53, 959.
5. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Reservoir Fluids From Their Compositions,” JPT (October 1964) 1171;
Trans., AIME, 231.
6. Wichert, E. and Aziz, K.: “Compressibility Factor of Sour Natural
Gases,” Cdn. J. Chem. Eng. (1971) 49, 267.
7. Wichert, E. and Aziz, K.: “Calculate Z’s for Sour Gases,” Hydro. Proc.
(May 1972) 51, 119.
8. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,
AIME (1942) 146, 140.
9. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,” U.S.
Dept. of Commerce, Washington, DC (1929) 97.
10. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, SPE, Richardson, Texas (1981).
EXAMPLE PROBLEMS
11. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: “The Viscosity of Natural
Gases,” JPT (August 1966) 997; Trans., AIME, 237.
12. Chew, J.N. and Connally, C.A.: “A Viscosity Correlation for Gas-Saturated Crude Oils,” Trans., AIME (1959) 216, 23.
13. Wilson, G.M.: “A Modified Redlich-Kwong EOS, Application to General Physical Data Calculations,” paper 15c presented at the 1969 AIChE
Natl. Meeting, Cleveland, Ohio.
14. Muskat, M. and McDowell, J.M.: “An Electrical Computer for Solving
Phase Equilibrium Problems,” Trans., AIME (1949) 186, 291.
15. Rachford, H.H. and Rice, J.D.: “Procedure for Use of Electrical Digital
Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium,” JPT (October 1952) 19; Trans., AIME, 195.
16. Peng, D.Y. and Robinson, D.B.: “A New-Constant Equation of State,”
Ind. & Eng. Chem. (1976) 15, No. 1, 59.
17. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants
for a Gas-Condensate System,” Trans., AIME (1953) 198, 1.
18. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratios
of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT
(September 1979) 1193.
19. Pawlikowski, E.M. and Prausnitz, J.M.: “Estimation of Setchenow
Constants for Nonpolar Gases in Common Salts at Moderate Temperatures,” Ind. Eng. Chem. Fund. (1983).
20. Dodson, C.R. and Standing, M.B.: “Pressure, Volume, Temperature and
Solubility Relations for Natural Gas-Water Mixtures,” Drill. & Prod.
Prac., API (1944) 173.
SI Metric Conversion Factors
°API 141.5/(131.5)°API)
+g/cm3
atm 1.013 250*
E)05 +Pa
bbl 1.589 873
E*01 +m3
cp 1.0*
E*03 +Pa@s
ft 3.048*
E*01 +m
E*02 +m3
ft3 2.831 685
°F (°F*32)/1.8
+°C
°F (°F)459.67)/1.8
+K
gal 3.785 412
E*03 +m3
lbm 4.535 924
E*01 +kg
lbm mol 4.535 924
E*01 +kmol
psi 6.894 757
E)00 +kPa
E*01 +kPa*1
psi*1 1.450 377
°R 5/9
+K
ton 9.071 847
E*01 +Mg
*Conversion factor is exact.
21
Appendix C
EquationĆofĆState Applications
This appendix presents two examples of fluid characterization with
an equation of state (EOS). The examples treat the gas condensate
and the oil discussed in Chap. 6, Good Oil Co. Wells 7 and 4, respectively. Details of developing a complete fluid characterization are
given for the gas-condensate fluid, including the splitting of C 7)
into five fractions, determining volume-translation coefficients for
the C 7) fractions, and estimating methane through C 7) binary interaction parameters (BIP’s). The resulting characterization is the
starting point for EOS predictions and, particularly, the simulation
of pressure/volume/temperature (PVT) experiments.
GasĆCondensateĆFluid Characterization
The characterization is developed for the Peng-Robinson1 EOS (PR
EOS) on the basis of the C 7) characterization suggested in Chap.
5 with five C 7) fractions. First, predictions are made without modifying the EOS parameters. Then, the measured dewpoint is matched
by modifying the BIP between methane and all C 7) fractions. Finally, constant-volume-depletion (CVD) data are matched by modifying the characterization with three regression parameters.
C7 + Molar Distribution. The first step in the C 7) characterization
is to split the heptanes-plus component into five fractions by use of
the Gaussian quadrature model in Chap. 5. In the absence of experimental true-boiling-point data, the following parameters are assumed: a+1, h+90, and N+5, with M C7)+143 and g C7)+0.795.
The value selected for heaviest fraction molecular weight, M N, is
somewhat higher than the recommended value of M N + 2.5M C7)
+ 2.5(143) + 358. Instead, we use M N + 500, which allows us to
develop a better characterization (particularly the tail-like behavior
of the liquid-dropout curve). The modified b * term is
b *+ ǒ M N * h ǓńX N + (500 * 90)ń(12.6408) + 32.435,
where X 5 is taken from Table 5.6.
The d parameter is calculated from Eq. 5.30.
d + exp
ǒ
a b*
MC
7)
*h
Ǔ
*1 ,
. . . . . . . . . . . . . . . . . . . (5.30)
giving
d + expNJ(1)(32.435)ń[(143 * 90) * 1]Nj + 0.67840.
Table C-1 gives values of f(X) for each fraction, according to
Eq. 5.31, together with calculated mole fractions and molecular
weights based on quadrature points and weighting factors,
X i and W i , respectively.
EQUATION-OF-STATE APPLICATIONS
zi + zC
7)
[W i f( X i)],
Mi + h ) b* Xi ,
and fǒ X Ǔ +
ǒ X Ǔ a*1 ǒ1 ) ln dǓ a
.
G(a)
dX
. . . . . . . . . . . . . . . . . (5.31)
For the first fraction,
X 1 + 0.263560,
W 1 + 0.52175561,
(0.263560)
fǒX 1Ǔ +
G(1)
(1*1)
[1 ) ln(0.67840)]
(0.67840)
1
0.263560
+ 0.677878,
z 1 + 6.85(0.52175561)(0.677878) + 2.4228,
and M 1 + 90 ) 32.435(0.263560) + 98.55.
C7 + Specific Gravities and Boiling Points. Given mole fractions
and molecular weights of the fractions, specific gravities are estimated with the Søreide2 correlation.3
g i + 0.2855 ) C f ǒ M i * 66 Ǔ
0.13
,
. . . . . . . . . . . . . . . (5.44)
where the characterization factor, C f , is modified to ensure that the
calculated C 7) specific gravity equals the measured value of
g C 7)+0.795.
ǒgC7)Ǔ
exp
+
zC
7)
MC
7)
ȍ
N
.
. . . . . . . . . . . . . . . . . . . . . (5.37)
ǒz i M ińg iǓ
i+1
By trial and error, C f + 0.28927 is found to satisfy Eq. 5.37; Table
C-2 gives the results. For the first fraction,
g F + 0.2855 ) 0.28927(98.55 * 66)
1
0.13
+ 0.7404.
Normal boiling points are calculated from the Søreide correlation.
T b + 1, 928.3 * ǒ1.695
exp ƪ * ǒ4.922
) ǒ3.462
10 5Ǔ M *0.03522 g 3.266
10 *3Ǔ M * 4.7685 g
10 *3Ǔ Mgƫ ,
. . . . . . . . . . . . . . . . . . . . (5.45)
1
TABLE C-1—GAUSSIAN QUADRATURE METHOD TO SPLIT C7+
INTO FIVE FRACTIONS FOR RESERVOIR GAS CONDENSATE
C7+
Fraction
i
Quadrature
Point
Xi
Quadrature
Weight
Wi
f(Xi )
Mole
Fraction
zi
Molecular
Weight
Mi
Mass
mi +zi Mi
1
0.263560
0.52175561
0.677878
2
1.413403
0.39866681
1.059051
2.4228
98.55
238.8
2.8921
135.84
3
3.596426
0.07594245
2.470516
392.9
1.2852
206.65
265.6
75.7
4
7.085810
0.00361176
9.567521
0.2367
319.83
5
12.640801
0.00002337
82.58395
0.0132
500.00
6.8500
143.00*
Total
6.6
979.5
*Equals 979.5/6.85.
TABLE C-2—PROPERTIES OF C7+ FRACTIONS FOR RESERVOIR GAS CONDENSATE
C7+
Fraction
i
Molecular
Weight
Mi
Mass
mi +zi Mi
Specific
Gravity
gi *
Ideal
Volume
V+zi Mi /gi
Boiling Point
Tb
°R
1
98.55
238.8
0.7407
322.5
674.1
2
135.84
392.9
0.7879
498.6
793.9
3
206.65
265.6
0.8358
317.8
972.7
4
319.83
75.7
0.8796
86.1
1,175.5
5
500.00
6.6
0.9226
7.2
1,386.3
143.00
979.5
0.7950
1,232.1
*Water+1.
TABLE C-3—TWU4 METHOD FOR CALCULATING CRITICAL PROPERTIES OF C7+ FRACTIONS
FOR RESERVOIR GAS CONDENSATE
Component
i
Tb
(°R)
TcP
a
gP *
g*
DgT
fT
Tc
(°R)
1
674.1
978.7
0.3112
0.6908
0.7404
*0.2195
0.003224
1,004.3
2
793.9
1,102.9
0.2802
0.7304
0.7879
*0.2498
0.003599
1,135.1
3
972.7
1,268.7
0.2333
0.7705
0.8358
*0.2783
0.003965
1,309.6
4
1,175.5
1,434.6
0.1807
0.8005
0.8796
*0.3267
0.004754
1,490.2
5
1,386.3
1,589.5
0.1278
0.8201
0.9226
*0.4008
0.006209
1,670.5
vcP
(ft3/lbm mol)
Dgv
fv
vc
(ft3/lbm mol)
pcP
(psia)
Dgp
fp
pc
(psia)
1
6.90
*0.2471
*0.0085
6.4475
393.8
*0.0245
0.00256
441.4
2
9.33
*0.2947
*0.0114
8.5142
314.2
*0.0283
0.00294
362.7
3
14.15
*0.3424
*0.0152
12.5336
220.0
*0.0321
0.00504
266.9
4
21.69
*0.4124
*0.0217
18.2317
142.5
*0.0388
0.01028
191.2
5
32.14
*0.5103
*0.0328
24.7141
87.2
*0.0499
0.02037
140.4
*Water+1.
which, for the first fraction, gives
T b + 1, 928.3 * ǒ1.695
exp ƪ * ǒ4.922
) ǒ3.462
10 5Ǔ(98.55)
*0.03522
(0.7404)
3.266
10 *3Ǔ(98.55) * 4.7685 (0.7404)
10 *3Ǔ(98.55)(0.7404)ƫ + 674.1°R .
C7 + Critical Properties. Critical properties T c and p c are calculated from the Twu4 correlations (Eqs. 5.68 through 5.78). Table
C-3 shows the calculations from left to right, in the order required
to solve the rather tedious Twu correlations.
Acentric factor is calculated from the Lee-Kesler5 correlation.
2
w+
) A 3 ln T br ) A 4 T br6
* lnǒ p cń14.7Ǔ ) A 1 ) A 2 T *1
br
A 5 ) A 6 T *1
) A 7 ln T br ) A 8 T br6
br
. . . . . . . . . . . . . . . . . . . . (5.60)
for reduced normal boiling points T br + T bńT c t 0.8. The KeslerLee6 correlation,
w + * 7.904 ) 0.1352K w * 0.007465K 2w
) 8.359T br ) ǒ1.408 * 0.01063 K wǓT *1
br ,
. . . . . . . (5.61)
is used to calculate higher reduced boiling points [making use of the
/g)].
Watson characterization factor defined by Eq. 5.34, ( K w + T 1ń3
b
Table C-4 shows the results.
PHASE BEHAVIOR
m + 0.7941,
TABLE C-4—CALCULATION OF ACENTRIC FACTOR FOR
C7+ FRACTIONS OF RESERVOIR GAS CONDENSATE
Component
i
a + 1.4955,
Tb /Tc
Kw
[(°R)1/3]
w
1
0.671
11.842
0.2864
and b + 1.8997 ft 3ńlbm mol.
2
0.699
11.752
0.3881
3
0.743
11.855
0.5754
By trial and error, the value of c that gives p+14.7 psia from Eq.
4.19 is
4
0.789
11.998
0.8313
5
0.830
12.086
1.1185
a + 1.7995
c + 0.06151 ft 3ńlbm mol
or s + cńb + (0.06151)ń(1.8997) + 0.0324 .
Volume-Translation Parameters. Volume-translation parameters,
s i, for pure components through C 6 are taken from Table 4.3. Values
of s i for the C 7) fractions are determined to ensure that the EOS
characterization for each separate C 7) fraction correctly calculates
a density at standard conditions that is consistent with the specific
gravity of that fraction. The actual molar volume at standard conditions, v + Mń(62.37g) in ft3/lbm mol, is equal to the EOS-calculated molar volume, v EOS (without volume translation), less the volume-translation parameter, c,
Table C-5 gives results for the other fractions.
BIP’s. The BIP’s between nonhydrocarbons and hydrocarbons are
taken from Table 4.1. The modified Chueh-Prausnitz7 equation,
k ij
+ 2.1340 ft 3ńlbm mol.
The correct c is determined when v EOS and EOS Constants a and b
in the PR EOS,
v 1ń3
ci
)
v 1ń3
cj
Ǔ ȳȴȧ
,
. . . . . . . . . . . . . . . . (5.79)
. . . . . . . . . . . . (4.19)
with v c in ft3/lbm mol. For the first fraction, v cF1 + 6.508 ft3/lbm mol.
By use of the same approximate relation for methane, v c + 1.447
ft3/lbm mol and k ij for this pair is
kC
Z * (1 * B)Z ) ǒ A * 3B * 2B Ǔ
2
ǒ
B
2v 1ń6
v 1ń6
ci
cj
v ci [ 0.4804 ) 0.06011M i ) 0.00001076M 2i ,
calculate a pressure of 14.7 psia at T + T sc . The EOS constants are
calculated from Eqs. 4.20 through 4.22.
3
ȱ
+ Aȧ1 *
Ȳ
is used for methane/ C 7) pairs with A+0.18 and B+6. For use with
this correlation, hydrocarbon critical volumes should be estimated
with the following approximate correlation.
v + v EOS * c + (98.55)ń[(62.37)(0.7404)]
a
,
p + RT *
v * b v(v ) b) ) b(v * b)
10 5 psia-ft 3ńlbm mol
1ńF 1
+ 0.18 * 0.18
2
ƪ
2(1.447)
(1.447)
1ń6
1ń3
(6.508)
1ń6
) (6.508)
1ń3
ƫ
6
+ 0.0301.
Table C-6 gives other methane/ C 7) BIP’s, and Tables C-7 and C-8
summarize the PR EOS fluid characterization.
Z * ǒAB * B 2 * B 3Ǔ + 0
and Z c + 0.3074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20)
R 2T 2
a + W oa p c a , where W oa + 0.45724;
c
RT
b + W ob p c , where W ob + 0.07780;
c
a + ƪ1 ) mǒ1 * ǸT rǓƫ ;
2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . . (4.21)
m + 0.3796 ) 1.485w * 0.1644w ) 0.01667w .
2
3
. . . . . . . . . . . . . . . . . . . . (4.22)
for wu0.49. This results in
T r + TńT c + (60 ) 460)ń(1, 004.3) + 0.5174,
EOS Predictions
With the PR EOS characterization given in Tables C-7 and C-8, a
dewpoint pressure of 3,535 psia is predicted at reservoir temperature of 186°F; this is approximately 500 psi lower than the measured
value. Figs. C-1 through C-4 show calculated EOS results. The liquid-dropout data are seriously overpredicted at pressures from
2,500 to 3,500 psia. Otherwise, the predictions are quite reasonable.
Wellstream compositions are acceptable, being somewhat too rich
at 3,500 psia and somewhat too lean at 2,900 psia.
Matched Dewpoint Pressure
Multiplying the BIP’s between methane and all C 7) fractions by a
factor of 2.09 matches the measured dewpoint pressure of 4,015 psia.
Figs. C-1 through C-4 present calculated CVD results. The predicted
PVT data are not very good; in particular, the liquid-dropout curve at
3,515 psia is overpredicted (21.2% vs. the measured value of 3.3%)
and equilibrium-gas C 7) compositions are severely underpredicted.
TABLE C-5—CALCULATION (CHECK) OF VOLUME-TRANSLATION PARAMETERS, s, FOR C7+
FRACTIONS IN RESERVOIR GAS CONDENSATE
i
v + Mńò *
Tr+Tsc /Tc
m
a
b
Guess c
vEOS+v + c
pcalc
(psia)
s+c/b
105
a
1
2.1340
0.5174
0.7941
1.4955
1.7995
1.8997
0.06151
2.1955
14.7
0.0324
2
2.7644
0.4578
0.9325
1.6941
3.1686 105
2.6127
0.14435
2.9087
14.4
0.0552
3
3.9644
0.3968
1.1829
2.0671
6.9940
105
4.0964
0.44038
4.4048
14.5
0.1075
4
5.8295
0.3487
1.5100
2.6189
1.6022
106
6.5089
1.00387
6.8334
14.8
0.1542
3.4744
106
9.9361
1.58485
10.2745
14.5
0.1595
5
8.6897
0.3111
1.8582
3.3189
* ò + 62.37g.
EQUATION-OF-STATE APPLICATIONS
3
parameters were chosen, and a sum-of-squares (SSQ) function was
minimized with a nonlinear regression algorithm. The SSQ function
is defined as
TABLE C-6—CALCULATION OF METHANE/C7+ BIP’s FOR
RESERVOIR GAS CONDENSATE
Approximate vc
(ft3/lbm mol)
Component
Methane
1.447
—
Fraction 1
6.509
0.0301
Fraction 2
8.844
0.0416
Fraction 3
13.362
0.0582
Fraction 4
20.806
0.0763
Fraction 5
33.225
0.0945
ȍr ,
M
Methane kij
F SSQ +
2
i
i
where M+total number of measured data included in the regression. The residuals, r i, are defined in terms of experimental data,
d xi ; calculated data, d ci ; and weight factors, w i . For dewpoint pressure and Z factors,
ǒd
ri +
Regression of CVD Data
The measured CVD data, including dewpoint pressure, were then
matched by modifying parameters in the original EOS (the characterization with predicted dewpoint of 3,535 psia). Three regression
xi
Ǔ
* d ci
wi .
d xi
For relative oil volumes, V ro , and cumulative gas produced, n pńn,
r i + (d xi * d ci)w i .
All weight factors, w i , are set to unity.
TABLE C-7—FINAL PR EOS CHARACTERIZATION FOR RESERVOIR GAS CONDENSATE
Component
z
Tc
(°R)
M
pc
(psia)
w
vc
(ft3/lbm mol)
Zc
g*
Tb
(°R)
s+c/b
N2
0.0018
44.01
547.6
1,070.6
0.2310
1.505
0.2742
0.5072
350.4
*0.0577
CO2
0.0013
28.01
227.3
493.0
0.0450
1.443
0.2916
0.4700
139.3
*0.1752
C1
0.6192
16.04
343.0
667.8
0.0115
1.590
0.2884
0.3300
201.0
*0.1651
C2
0.1408
30.07
549.8
707.8
0.0908
2.370
0.2843
0.4500
332.2
*0.1070
C3
0.0835
44.10
665.7
616.3
0.1454
3.250
0.2804
0.5077
416.0
*0.0848
i-C4
0.0097
58.12
734.7
529.1
0.1756
4.208
0.2824
0.5631
470.6
*0.0686
C4
0.0341
58.12
765.3
550.7
0.1928
4.080
0.2736
0.5844
490.8
*0.0686
i-C5
0.0084
72.15
828.8
490.4
0.2273
4.899
0.2701
0.6247
541.8
*0.0410
C5
0.0148
72.15
845.4
488.6
0.2510
4.870
0.2623
0.6310
556.6
*0.0410
C6
0.0179
86.18
913.4
436.9
0.2957
5.929
0.2643
0.6640
615.4
*0.0154
F1
0.024227
98.55
1,004.4
441.5
0.2864
6.447
0.2640
0.7405
674.1
0.0322
F2
0.028921
135.84
1,135.1
362.7
0.3882
8.514
0.2535
0.7879
793.9
0.0552
F3
0.012852
206.65
1,309.6
266.9
0.5756
12.535
0.2380
0.8357
972.7
0.1075
F4
0.002367
319.83
1,490.2
191.1
0.8316
18.236
0.2179
0.8796
1,175.5
0.1544
F5
0.000132
500.00
1,670.4
140.3
1.1188
24.725
0.1935
0.9224
1,386.4
0.1599
*Water+1.
TABLE C-8—BIP’s FOR FINAL PR EOS CHARACTERIZATION OF
RESERVOIR GAS CONDENSATE
N2
4
N2
0
CO2
CO2
C1
C2
C3
i-C4
0
0
C1
0.025
0.105
0
C2
0.010
0.130
0
0
C3
0.090
0.125
0
0
0
i-C4
0.095
0.120
0
0
0
0
C4
i-C5
C5
C6
F1
F2
F3
F4
C4
0.095
0.115
0
0
0
0
i-C5
0.100
0.115
0
0
0
0
0
0
C5
0.110
0.115
0
0
0
0
0
0
0
C6
0.110
0.115
0
0
0
0
0
0
0
0
F1
0.110
0.115
0.030
0
0
0
0
0
0
0
0
F2
0.110
0.115
0.042
0
0
0
0
0
0
0
0
0
F3
0.110
0.115
0.058
0
0
0
0
0
0
0
0
0
F4
0.110
0.115
0.076
0
0
0
0
0
0
0
0
0
0
0
F5
0.110
0.115
0.095
0
0
0
0
0
0
0
0
0
0
0
F5
0
0
0
PHASE BEHAVIOR
Fig. C-1—CVD liquid-dropout behavior for gas-condensate example comparing measured, EOS-predicted, and dewpointmatched calculations.
Fig. C-2—CVD Z-factor behavior for gas-condensate example
comparing measured, EOS-predicted, and dewpoint-matched
calculations.
Fig. C-3—CVD equilibrium-gas C7+ behavior for gas-condensate
example comparing measured, EOS-predicted, and dewpointmatched calculations.
Fig. C-4—CVD equilibrium-gas C7+ molecular-weight behavior
for gas-condensate example comparing measured, EOS-predicted, and dewpoint-matched calculations.
The total number of data is 17 and includes one saturation pressure, six Z factors, five relative oil volumes, and five cumulative gas
productions. Because the number of data is somewhat limited, only
three regression parameters are used. Initially, before parameters
have been changed, three data contribute most to the SSQ function:
p d , Z d , and V ro at 2,915 psia. Each is approximately 25% of the
total SSQ. The initial SSQ is approximately (F SSQ) i + 0.05.
Regression I. The first regression uses the following three regression parameters: P 1, the multiplier to BIP’s between methane and
all C 7) fractions; P 2, the multiplier to T c for all C 7) fractions; and
P 3, the multiplier to p c for all C 7) fractions. Fig. C-5 shows the reduction in the SSQ function at each iteration. The final SSQ value
is approximately 4% of the initial value (0.002/0.05). Six iterations
were required to find a minimum. Practically, however, the minimum was located after four iterations, with only small parameter adjustments made during the last two iterations. The final parameters
are P 1 + 4.34, P 2 + 0.910, and P 3 + 0.849.
Figs. C-6 and C-7 show the change in the multipliers at each iteration. The BIP multipliers increase monotonically to a value of
approximately 4.3, resulting in C 1 through C 7) BIP’s ranging from
0.13 to 0.40. The large BIP values are outside the range of what is
probably acceptable because they generally should not exceed
approximately 0.3 for the PR EOS.
C 7) critical temperatures decreased almost monotonically to
approximately 10% less than the initial values. C 7) critical pressures increased during the first iterations, then finally decreased to
EQUATION-OF-STATE APPLICATIONS
Fig. C-5—Reduction in SSQ function for regression cases with
three different sets of parameters to match measured gas-condensate PVT data.
approximately 15% below the starting values. At Iteration 3, the
minimum SSQ was almost reached, but the multiplier to critical
pressures was approximately 1.0. During the last three iterations,
the multiplier was reduced to 0.85 without any significant reduction
5
pc
Tc
Fig. C-6—Variation in C1 through C7+ BIP multipliers used in Regressions I and II to match measured gas-condensate PVT data.
Fig. C-7—Variation in C7+ critical-property multipliers used in
Regression I to match measured gas-condensate PVT data.
Fig. C-8—CVD liquid-dropout behavior for gas-condensate example comparing measured and EOS Regression I calculations.
Fig. C-9—CVD Z-factor behavior for gas-condensate example
comparing measured and EOS Regression I calculations.
Fig. C-10—CVD equilibrium-gas C7+ behavior for gas-condensate example comparing measured and EOS Regression I calculations.
in the SSQ function. This indicates that C 7) critical pressures are
probably not very important when matching PVT data and that
another parameter could be chosen instead.
Figs. C-8 through C-11 show calculated results for the CVD experiment. Dewpoint pressure was overpredicted by only 8 psi
6
Fig. C-11—CVD equilibrium-gas C7+ molecular-weight behavior
for gas-condensate example comparing measured and EOS Regression I calculations.
(0.2%) despite the relative low weight factor used (a factor of 10 or
more is commonly used). Also, the experimental liquid dropout of
3.3% at 3,515 psia was 4.9% with the modified characterization, a
very good match.
PHASE BEHAVIOR
Fig. C-12—Variation in the two C7+ critical-temperature multipliers used in Regression II to match measured gas-condensate
PVT data.
Fig. C-13—CVD liquid-dropout behavior for gas-condensate example comparing measured and EOS Regression II calculations.
Fig. C-14—CVD Z-factor behavior for gas-condensate example
comparing measured and EOS Regression II calculations.
Fig. C-15—CVD equilibrium-gas C7+ behavior for gas-condensate example comparing measured and EOS Regression II calculations.
Regression II. The second regression uses the following three
regression parameters: P 1, the multiplier to BIP’s between methane and all C 7) fractions; P 2, the multiplier to T c for C 7) fractions F 1 through F 3 ; and P 3, the multiplier to T c for C 7) fractions F 4 and F 5. Fig. C-5 shows the reduction in the SSQ
function at each iteration. The final SSQ function value is
approximately 3% of the initial value (0.017/0.05). Four iterations were required to find a minimum. The final parameters are
P 1 + 2.29, P 2 + 0.932, and P 3 + 1.047.
Figs. C-6 and C-12 show the change in the multipliers at each iteration. The BIP multipliers converged to a value of approximately
2.3, resulting in C 1 through C 7) BIP’s ranging from 0.07 to 0.22.
These BIP values are reasonable for the PR EOS.
C 7) critical temperatures for fractions F 1 through F 3 decreased to approximately 7% less than the initial values. C 7) critical temperatures for fractions F 4 and F 5 increased, fluctuating
from 2 to 10% above the initial values, finally converging to a 5%
increase. Figs. C-13 through C-16 show calculated results for the
CVD experiment.
This regression gives an excellent match of almost all measured
PVT data, including the data used in the regression and equilibriumgas compositions and properties that were not included in the regression. Dewpoint pressure was overpredicted by 8 psi (0.2%),
which is sufficiently close, although a larger weight factor (e.g., 10)
would force the calculated dewpoint to match the measured value
almost exactly. On the other hand, the experimental accuracy of
dewpoint pressure is less than 0.2% and further refinement with a
EQUATION-OF-STATE APPLICATIONS
Fig. C-16—CVD equilibrium-gas C7+ molecular-weight behavior
for gas-condensate example comparing measured and EOS Regression II calculations.
larger weight factor is probably not justified. Finally, the measured
liquid dropout of 3.3% at 3,515 psia was calculated to be 4.9%, also
a very good match.
7
Fig. C-17—Variation in the three C7+ critical-temperature multipliers used in Regression III to match measured gas-condensate
PVT data.
Fig. C-18—CVD liquid-dropout behavior for gas-condensate
example comparing measured and EOS Regression III calculations.
Measured
Calculated
Fig. C-19—CVD Z-factor behavior for gas-condensate example
comparing measured and EOS Regression III calculations.
Fig. C-20—CVD equilibrium-gas C7+ behavior for gas-condensate example comparing measured and EOS Regression III calculations.
Regression III. Results almost as good as those for Regression II are
achieved by fitting only critical temperatures of the C7+ fractions,
namely multipliers to T c(F 1, F 2), T c(F 3, F 4), and T c(F 5), with
the final parameters being P 1 + 0.915, P 2 + 1.023, and
P 3 + 1.239 (Fig. C-17). The converged F SSQ + 0.0026 is 5% of the
initial value. The C 1 through C 7) BIP’s are the same as those used
in the prediction, ranging from 0.03 to 0.095. Calculated dewpoint is
4,044 psia (0.7%), and liquid dropout at 3,515 psia is 4.9% compared
with the measured value of 3.3%. Figs. C-18 through C-21 compare
calculated and measured results for the CVD experiment.
Comparing Different Fluid Characterizations. More analysis is
needed to determine whether any real difference exists between the
fluid characterizations determined in Regressions II and III. For
depletion calculations, the results are almost identical. For gas cycling, however, they may provide quite different results.
When limited PVT data are available to tune an EOS (as in this
example), it usually is good practice to evaluate two or three “equally good” characterizations. As in our example, different modifying
parameters might be used. Alternative EOS’s can also be tried [e.g.,
the Soave-Redlich-Kwong EOS8 (SRK EOS) with the Pedersen et
al.9 fluid characterization as a starting point]. Each fluid characterization can then be evaluated with the results from compositional
simulation of the reservoir process being studied.
8
Fig. C-21—CVD equilibrium-gas C7+ molecular-weight behavior
for gas-condensate example comparing measured and EOS Regression III calculations.
Generating Modified Black-Oil PVT Data. Figs. C-22 through
C-24 present modified black-oil PVT properties calculated with the
various characterizations discussed earlier. Figs. C-22 and C-23
give oil properties R s (solution gas/oil ratio) and B o [oil formation
PHASE BEHAVIOR
Fig. C-22—Modified black-oil PVT property solution gas/oil ratio,
Rs , vs. pressure for gas-condensate example for three EOS
models: dewpoint-match only and Regressions I and II.
Fig. C-23—Modified black-oil PVT property saturated-oil FVF,
Bo , vs. pressure for gas-condensate example for three EOS
models: dewpoint-match only and Regressions I and II.
volume factor (FVF)]. Note that these oil properties do not increase
monotonically, as is usually exhibited by reservoir oils. The reason
is that the first condensate that drops out just below the dewpoint is
relatively “heavy” compared with the condensate that drops out at
lower pressures.
For example, the fluid characterization from Regression II yields
a stock-tank-oil (STO) gravity of 45°API produced from the reservoir condensate at the dewpoint, a 50°API STO produced from the
reservoir condensate at 3,515 psia, and a 53°API STO produced
from the reservoir condensate at 3,000 psia. The corresponding
solution gas/oil ratios at dewpoint, 3,515, and 3,000 psia are 1,500,
1,880, and 2,100 scf/STB, respectively, and oil FVF’s are 1.835,
2.109, and 2.319 bbl/STB, respectively.
This behavior is typical for gas condensates with a “tail” on the
liquid-dropout curve; i.e., the retrograde condensation is small in a
pressure interval just below the dewpoint (approximately 500 psi in
this example), with the start of a more rapid increase in retrograde
condensation occurring at some lower pressure (at approximately
3,500 psia in this example). In the region of the tail retrograde behavior, produced reservoir gas has only slight changes in composition during depletion because only small amounts of the heaviest
components are being lost from the original reservoir gas. This
should be reflected in the EOS characterization by only slight decrease in C 7) composition. The behavior should also be reflected
by modified black-oil PVT property r s (solution oil/gas ratio) of the
reservoir gas. Solution oil/gas ratio should decrease only slightly in
the region of the tail-like retrograde condensation.
ReservoirĆOil Fluid Characterization
The second example treats the oil in Chap. 6, Good Oil Co. Well 4.
This is a slightly volatile oil with a bubblepoint of 2,600 psi at 220°F,
an initial solution gas/oil ratio of 750 scf/STB, and a bubblepoint oil
FVF of 1.45 RB/STB. In this example, we look at two EOS characterizations. The first characterization uses the PR EOS with the
Søreide2 and Whitson10 methods for developing three C 7) fractions. This approach is basically the same as that used for the gas
condensate presented earlier. The second characterization uses the
SRK EOS with the Pedersen et al.9 method for characterizing the
Fig. C-24—Modified black-oil PVT property solution oil/gas ratio, rs , vs. pressure for gas-condensate example for three EOS
models: dewpoint-match only and Regressions I and II.
Fig. C-25—CVD-based cumulative condensate recovery vs.
pressure for gas-condensate example for three EOS models:
dewpoint-match only and Regressions I and II.
EQUATION-OF-STATE APPLICATIONS
Referring again to the fluid characterization from Regression II,
calculated r s decreases only slightly from 136 STB/MMscf at the
dewpoint to 122 STB/MMscf at 3,515 psia. Compared with larger
decreases in r s at lower pressures (e.g., to 88 STB/MMscf at 3,015
psia), the slight decrease in r s predicted from dewpoint to 3,515 psia
appears very reasonable. Calculations from Regressions I and III
also show similar r s behavior. Fig. C-25 summarizes the effect of
treating the tail-like retrograde behavior properly with an EOS fluid
characterization. The figure plots cumulative stock-tank condensate
produced during depletion on the basis of modified black-oil PVT
data ( r s) generated with the fluid characterizations discussed previously. In particular, the characterization based only on fitting the
dewpoint pressure is compared with the the fluid characterizations
determined in Regressions I and II. The effect on condensate recovery is clear from the comparison.
9
TABLE C-9—COMPOSITIONS OF RESERVOIR OIL AND EQUILIBRIUM GAS AND K VALUES
AT 2,600-psia BUBBLEPOINT PRESSURE AND 220°F
Bubblepoint-Oil Composition
Component
PR EOS
SRK EOS
Equilibrium-Gas Composition
PR EOS
SRK EOS
K Values at Bubblepoint
PR EOS
SRK EOS
N2
0.16
0.16
0.52
0.59
3.28
3.66
CO2
0.91
0.91
1.31
1.43
1.44
1.57
C1
36.47
36.47
77.13
76.97
2.11
2.11
C2
9.67
9.67
10.16
10.57
1.05
1.09
C3
6.95
6.95
4.87
4.95
0.70
0.71
i-C4
1.44
1.44
0.77
0.78
0.54
0.54
C4
3.93
3.93
1.85
1.82
0.47
0.46
i-C5
1.44
1.44
0.51
0.50
0.36
0.35
C5
1.41
1.41
0.46
0.44
0.33
0.31
C6
4.33
4.33
1.00
0.94
0.23
0.22
F1
15.91
19.07
1.35
0.97
0.085
0.051
F2
14.28
9.31
0.0623
0.0358
0.0044
0.0038
F3
3.11
4.91
0.000050
0.000110
0.000016
0.000022
TABLE C-10—PR EOS CHARACTERIZATION OF RESERVOIR OIL WITH SØREIDE-WHITSON C7+ METHOD2,3,10
Component
M
Tc
(°R)
pc
(psia)
w
s+c/b
ăąg*ąă
Tb
(°R)
vc
(ft3/lbm mol)
Zc
N2
28.01
227.3
493.0
0.0450
*0.1930
0.4700
139.3
1.443
0.2916
CO2
44.01
547.6
1,070.6
0.2310
*0.0820
0.5072
350.4
1.505
0.2742
C1
16.04
343.0
667.8
0.0115
*0.1590
0.3300
201.0
1.590
0.2884
C2
30.07
549.8
707.8
0.0908
*0.1130
0.4500
332.2
2.370
0.2843
C3
44.10
665.7
616.3
0.1454
*0.0860
0.5077
416.0
3.250
0.2804
i-C4
58.12
734.7
529.1
0.1756
*0.0840
0.5631
470.6
4.208
0.2824
C4
58.12
765.3
550.7
0.1928
*0.0670
0.5844
490.8
4.080
0.2736
i-C5
72.15
828.8
490.4
0.2273
*0.0610
0.6247
541.8
4.899
0.2701
C5
72.15
845.4
488.6
0.2510
*0.0390
0.6310
556.6
4.870
0.2623
C6
86.18
913.4
436.9
0.2957
*0.0080
0.6640
615.4
5.929
0.2643
F1
120.08
1,086.6
397.1
0.3419
0.0403
0.7750
746.2
8.333
0.2838
F2
255.96
1,401.5
230.0
0.6866
0.1255
0.8618
1,070.9
17.562
0.2685
F3
545.00
1,707.3
137.0
1.2213
0.1326
0.9354
1,424.3
28.250
0.2113
*Water+1.
three C 7) fractions. Both EOS characterizations predict the measured PVT data reported in Chap. 6 (Tables 6.2 through 6.7) reasonably well. The characterizations are not modified by regression in
this example (possibly an interesting exercise for the reader). The
two characterizations are presented first. Calculated results are then
compared with measured data reported in Chap. 6. Finally, a study
of modified black-oil PVT properties is given.
Peng-Robinson1 Characterization.The methods presented for the
gas condensate in the Gas-Condensate-Fluid Characterization section (see also Sec. 5.6) were used to develop a fluid characterization
for this reservoir oil. Three C 7) fractions, determined with the
Gaussian quadrature approach, were used. Table C-9 gives mole
fractions of the reservoir oil, Table C-10 gives component properties, and Table C-11 provides BIP’s. Volume translation was used
to ensure accurate volumetric predictions.
Soave-Redlich-Kwong Characterization.8 For comparison, the
Pedersen et al.9 characterization procedure (Sec. 5.6) was used to
develop an EOS description of the same reservoir oil. The split of
the C 7) fraction is made by use of an exponential distribution to
C 80, then regrouping in subfractions with approximately equal mass
10
fractions. Tables C-9, C-12, and C-13 give the resulting composition and properties.
TABLE C-11—BIP’s FOR PR EOS CHARACTERIZATION OF
RESERVOIR OIL
Component
N2
N2
0.000
CO2
C1
CO2
0.000
0.000
C1
0.025
0.105
0.000
C2
0.010
0.130
0.000
C3
0.090
0.125
0.000
i-C4
0.095
0.120
0.000
C4
0.095
0.115
0.000
i-C5
0.100
0.115
0.000
C5
0.110
0.115
0.000
C6
0.110
0.115
0.000
F1
0.110
0.115
0.035
F2
0.110
0.115
0.063
F3
0.110
0.115
0.092
PHASE BEHAVIOR
TABLE C-12—SRK EOS CHARACTERIZATION WITH PEDERSEN et al.9 C7+ METHOD FOR RESERVOIR OIL
Tc
(°R)
pc
(psia)
vc
(ft3/lbm mol)
Zc
1.443
0.2916
s+c/b
493.0
0.0450
*0.0080
547.6
1,070.6
0.2310
0.0830
0.5072
350.4
1.505
0.2742
343.0
667.8
0.0115
0.0230
0.3300
201.0
1.590
0.2884
30.07
549.8
707.8
0.0908
0.0600
0.4500
332.2
2.370
0.2843
44.10
665.7
616.3
0.1454
0.0820
0.5077
416.0
3.250
0.2804
i-C4
58.12
734.7
529.1
0.1756
0.0830
0.5631
470.6
4.208
0.2824
C4
58.12
765.3
550.7
0.1928
0.0970
0.5844
490.8
4.080
0.2736
i-C5
72.15
828.8
490.4
0.2273
0.1020
0.6247
541.8
4.899
0.2701
C5
72.15
845.4
488.6
0.2510
0.1210
0.6310
556.6
4.870
0.2623
0.2643
M
N2
28.01
227.3
CO2
44.01
C1
16.04
C2
C3
ăąg*
Tc
(°R)
w
Component
0.4700
139.3
C6
86.18
913.4
436.9
0.2957
0.1470
0.6640
615.4
5.929
F1
133.98
1,079.5
354.8
0.5935
0.1535
0.7899
802.3
9.561
0.2928
F2
258.05
1,307.1
232.5
0.9030
0.1422
0.8577
1,075.7
16.734
0.2774
F3
468.63
1,615.2
199.4
1.2322
*0.0422
0.9247
1,369.0
26.756
0.3078
TABLE C-13—BIP’s FOR SRK EOS CHARACTERIZATION
OF RESERVOIR OIL
Component
N2
N2
0.000
CO2
C1
CO2
C1
0.000
0.000
0.000
0.020
0.150
0.000
C2
0.060
0.150
0.000
C3
0.080
0.150
0.000
i-C4
0.080
0.150
0.000
C4
0.080
0.150
0.000
i-C5
0.080
0.150
0.000
C5
0.080
0.150
0.000
C6
0.080
0.150
0.000
F1
0.080
0.150
0.000
F2
0.080
0.150
0.000
F3
0.080
0.150
0.000
Analyzing EOS Results. Fig. C-26 plots oil density vs. pressure.
EOS predictions are accurate at the bubblepoint, somewhat too high
at undersaturated conditions, and significantly overpredicted at low
pressures. Overall, the predictions are quite good, particularly in the
important pressure regions. Fig. C-27 shows the differential oil volume factor, B od. The EOS predictions are similar, slightly overpredicting the undersaturated oil compressibility and overpredicting
the shrinkage of oil at lower pressures.
A useful graphical analysis for undersaturated oil behavior is a
log-log plot of oil relative volume, B odńB od, b, vs. the pressure ratio
pńp b (Fig. C-28). The slope of this plot yields Constant A
( A+*slope), where instantaneous oil compressibility is given by
c o + Ańp ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)
and cumulative oil compressibility, c o (used in material-balance
equations), is given by
A lnǒ p ńp Ǔ .
co + p *
i
p
i
. . . . . . . . . . . . . . . . . . . . . . . . . . (C-1)
The constant A is a characteristic value for an oil reservoir with
constant bubblepoint pressure. With Eqs. 3.107 and C-1, the
constant allows easy calculation of undersaturated oil compressibility at any reservoir pressure. For reservoirs with bubblepoint variation, A correlates well with bubblepoint pressure (approximately
linear and increasing with bubblepoint).
For this example, the plot in Fig. C-28 indicates that measured
data intercept the relative oil volume ratio, V ro + B odńB od,b, at a
EQUATION-OF-STATE APPLICATIONS
Fig. C-26—DLE oil-density behavior for the reservoir-oil example comparing measured and EOS predictive models based on
the PR and SRK EOS’s that use the Søreide-Whitson method2,3,10 and the Pedersen et al.9 method, respectively, for characterizing C7+ fractions.
value of logǒ pńp bǓ + 0.015, corresponding to pńp b + 1.03. Such
an intercept indicates that the reported bubblepoint pressure is
approximately 3% too low. Forcing measured data through
log(V ro) + 0 should be done only if it results in a linear trend
through all the data. For this example, it is very difficult to honor the
reported bubblepoint pressure and still have a linear plot that passes
through most of the reported oil relative volume data.
EOS results confirm that a linear trend with a zero intercept
should be expected. The PR EOS has a slope of 0.070, slightly less
than the SRK EOS slope of 0.074. The measured data have a slope
(with nonzero intercept) of 0.059. Then, resulting oil compressibilities at initial pressure of 5,000 psia are
(c o) meas + 0.059ń5, 000 + 11.8
10 *6 psi *1,
(c o) PREOS + 0.070ń5, 000 + 14.0
and (c o) SRKEOS + 0.074ń5, 000 + 14.8
10 *6 psi *1,
10 *6 psi *1.
Cumulative oil compressibilities are given by
(c o) meas + ƪ0.059ń(5, 000 * 2, 650)ƫ ln(5, 000 * 2, 650)
11
Measured intercept implies reported pb too low.
Fig. C-27—DLE differential-oil FVF behavior for the reservoir-oil
example comparing measured and EOS predictive models
based on the PR/Søreide-Whitson method2,3,10 and the
SRK/Pedersen et al.9 method.
+ 15.9
10 *6 psi *1,
(c o) PREOS + 18.9
and (c o) SRKEOS + 19.9
10 *6 psi *1,
10 *6 psi *1.
Returning to Fig. C-27, the SRK EOS seems to underpredict differential oil volume factors more than the PR EOS. Practically, however, the two characterizations predict nearly identical oil shrinkage. This is seen in Fig. C-29, which shows the oil volume ratio,
V ro + B odńB od, b, vs. pressure. This ratio gives a true measure of the
reservoir-oil shrinkage during depletion, whereas the ratio B od is
misleadingly related to the “meaningless” residual oil volume. We
highly recommend that the ratio B odńB od, b be used as “data” in regression (instead of B od directly) to ensure accurate oil shrinkage
from the EOS without also having to fit the residual oil volume at
standard conditions. [The residual oil is of no practical interest because it will never be produced to the surface and probably never ex-
Fig. C-29—DLE oil-shrinkage behavior for reservoir-oil example comparing measured and EOS predictive models based on
the PR/Søreide2 method and the SRK/Pedersen et al.9 method.
12
Fig. C-28—DLE undersaturated-oil-volume (compressibility) behavior for reservoir-oil example comparing measured and EOS
predictive models based on the PR/Søreide-Whitson method2,3,10 method and the SRK/Pedersen et al.9 method.
isted in the reservoir. Furthermore, the experimental procedure used
in reducing the pressure from the last stage of depletion (approximately 150 psi) to standard pressure and reservoir temperature involves bleeding the system down slowly. This bleeding is a nonequilibrium process that cannot really be simulated with a PVT
package (it can be estimated by a series of 5 to 10 additional depletion stages, starting at the lowest reported depletion stage)].
Fig. C-30 shows the differential solution gas/oil ratio vs. pressure and indicates that both EOS characterizations overpredict the
measured data by 5 to 10%. Correcting this deviation from measured data may lead to unnecessary and severe changes in the EOS
characterization. What is really important to predict are (1) the
separator flash gas/oil ratio (GOR) and (2) the cumulative gas
coming out of solution during depletion.
Table C-14 shows the calculated and measured separator data.
Interestingly, calculated separator gas/oil ratios are 1 to 2% lower
than measured data. That is, the differential GOR’s are consider-
Fig. C-30—DLE solution gas/oil ratio behavior for reservoir-oil
example comparing measured and EOS predictive models
based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
PHASE BEHAVIOR
TABLE C-14—MEASURED AND CALCULATED TWO-STAGE SEPARATOR TEST RESULTS
FOR RESERVOIR OIL
GOR
(scf/STB)
Measured
Stage 1 (315 psia and 75°F)
Stage 2 (14.7 psia and 60°F)
Total or at bubblepoint
ąăgg *
Bo
(bbl/STB)
gAPI
(°API)
549
246
795
0.704
1.286
0.884
1.148
1.007
1.495
40.1
PR EOS Characterization
Stage 1
Stage 2
Total or at bubblepoint
Percent deviation
559
219
778
*2.1
0.707
1.272
0.866
*2.0
1.129
1.006
1.483
*0.8
40.1
0.0
SRK EOS Characterization
Stage 1
Stage 2
Total or at bubblepoint
Percent deviation
569
216
785
*1.2
0.712
1.270
0.865
*2.1
1.124
1.006
1.494
*0.1
39.0
*2.6
*Air+1.
ably overpredicted, while the separator gas/oil ratios are only
slightly underpredicted. Clearly, the separator gas/oil ratio predictions are accurate enough, satisfying the first requirement given
previously. However, the question is how well the EOS characterizations estimate cumulative gas coming out of solution. Fig.
C-31, which plots ǒR sd,b * R sdǓńB od, b vs. pressure, shows this. The
figure indicates that the measured data are somewhat overpredicted by both EOS’s (the two characterizations give very similar
results). Although the overprediction is not excessive, these data
[ ǒR sd,b * R sdǓńB od, b] could be used in regression (together with oil
shrinkage data B odńB od,b) to improve the EOS characterization.
Fig. C-32 shows the gas specific gravity of equilibrium gas released during the differential-liberation experiment (DLE). The EOS
characterizations predict the measured data accurately, with slight underestimation at the two highest pressures. Laboratory gas specific
gravities may be difficult to measure accurately because of sampling
procedures that can result in loss of liquids during transfer from the
PVT cell to the sampler. Such errors would tend to result in specific
gravities that are too low, the opposite of what Fig. C-32 shows.
A problem that may arise in fitting reservoir gas specific gravities
with an EOS is the choice and number of components used to describe the C 7) fraction. Often the lightest EOS C 7) fraction consti-
Fig. C-31—DLE cumulative-released-gas behavior for reservoiroil example comparing measured and EOS predictive models
based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
EQUATION-OF-STATE APPLICATIONS
tutes most of the total C 7) material in the calculated reservoir gas
phase (in certain pressure regions). If this component is too heavy
or too light compared with the actual C 7) material of the reservoir
gas, it will cause the EOS-calculated gas specific gravity to be too
heavy or too light. For this example, the Soave-Redlich-Kwong
characterization with 12 C 7) fractions gave basically the same gas
specific gravities (within 1%) for all pressures down to 200 psia.
Gas specific gravity usually is not important in reservoir engineering calculations of oil reservoirs, particularly if gas Z factors are
predicted accurately. However, because the equilibrium-gas specific gravity indirectly reflects the gas composition (and thus the liquid
yield from the reservoir gas), it may be important for gas-condensate and volatile-oil reservoirs where a significant amount of stocktank-liquid production comes from the reservoir gas phase.
Fig. C-33 shows the equilibrium-gas-phase Z factor. At pressures
just below the bubblepoint the PR EOS predicts the measured data accurately, while the SRK EOS predicts the data somewhat better at intermediate and lower pressures. Neither EOS predicts the general
shape of the measured Z-factor curve. As an independent check of the
EOS Z factors, the Standing-Katz11 correlation (Eq. 3.42) was used
with specific gravities from the PR EOS results and with the Sutton12
Fig. C-32—DLE released (equilibrium) -gas specific-gravity behavior for reservoir-oil example comparing measured and EOS
predictive models based on the PR/Søreide2 and SRK/Pedersen
et al.9 methods.
13
Fig. C-33—DLE released (equilibrium) -gas Z-factor behavior for
reservoir-oil example comparing measured and EOS predictive
models based on the PR/Søreide2 and SRK/Pedersen et al.9
methods; Standing-Katz11 Z factors calculated on the basis of
PR gas compositions also shown.
Fig. C-34—DLE oil-viscosity behavior for reservoir-oil example
comparing measured and Lohrenz-Bray-Clark13 (LBC) viscosity
model (regressed Vc )/predictive PR EOS/Søreide2 and SRK
EOS/Pedersen et al.9 methods.
pseudocritical properties (Eq. 3.47). Fig. C-33 presents the StandingKatz Z factors as open circles. The results are closest to the SRK EOS
Z factors, which is not surprising. The SRK EOS usually gives better
gas volumetric properties than the PR EOS for methane-rich systems.
Fig. C-34 presents the oil viscosities. Measured values are
compared with calculated values by use of the Lohrenz-BrayClark13 correlation, with compositions and densities from EOS results. Experimental oil viscosities are difficult to obtain with an accuracy of more than approximately 5 to 10%, so the results
presented here are acceptable. To obtain these calculated results,
the critical volumes of C 7) fractions were modified by regressing
on measured oil viscosities and reported (calculated) gas viscosities. The default critical volumes were increased 10 to 20% to obtain the match. The modifications to C 7) critical volumes differ
for every reservoir system, mainly because the Lohrenz-BrayClark correlation is strongly dependent on both critical volumes
and oil densities. The modifications are usually less when oil densities are accurately predicted by the EOS.
A useful plot for correlating oil viscosities measured at different
laboratories is oil viscosity vs. density (Fig. C-35). Reservoir oils
from the same reservoir should have a unique viscosity/density relationship. (One exception would be if a reservoir exhibited compositional gradients characterized by variation in relative oil paraffinicity/aromaticity.) Because most laboratories measure oil density
accurately (i.e., consistently from one laboratory to another), erroneous viscosity data from a laboratory will plot parallel to the reservoir’s correct viscosity/density relation, shifted by a more or less
constant amount.
Reported gas viscosities, even though they are calculated with a
correlation (on the basis of measured specific gravities), should be
accurate within 5% or less. Therefore, including gas viscosities in
the viscosity regression ensures that critical volumes of the C 7)
Oil Density, lbm/ft3
Fig. C-35—DLE oil-viscosity vs. -density behavior for reservoir-oil
example comparing measured and EOS predictive models based
on PR/Søreide2 and SRK/Pedersen et al.9 methods.
14
Fig. C-36—DLE oil- and gas-viscosity behavior for reservoir-oil
example comparing measured and LBC viscosity model13 (regressed Vc )/predictive PR EOS/Søreide2 and SRK EOS/Pedersen et al.9 methods.
PHASE BEHAVIOR
Fig. C-37—Modified black-oil PVT property solution gas/oil ratio,
Rs , for reservoir-oil example comparing measured/converted
and EOS predictive models based on the PR/Søreide2 and SRK/
Pedersen et al.9 methods.
Fig. C-38—Modified black-oil PVT property saturated-oil FVF,
Bo , for reservoir-oil example comparing measured/converted
and EOS predictive models based on the PR/Søreide2 and SRK/
Pedersen et al.9 methods.
fractions are not modified unrealistically (i.e., to the point where gas
viscosities are no longer predicted accurately). Fig. C-36 shows gas
and oil viscosities together for this reservoir system.
Generating Black-Oil PVT Data. In this section, we consider calculation of modified black-oil PVT properties (Chap. 7). We look
at the problems involved in generating consistent black-oil PVT
properties for a reservoir with a gas cap in equilibrium with an underlying reservoir oil and try to determine whether black-oil PVT
properties are the same for the gas cap and reservoir oil.
Several other questions are also raised. How accurate are reservoir phase densities calculated from black-oil PVT data? What surface gravities should be chosen? How do differential data corrected
with separator flash data (Eqs. 6.32 and 6.33) compare with results
from the Whitson-Torp14 method? And finally, how should modified black-oil PVT data be extrapolated for saturation conditions
above the original saturation condition?
Gas Cap and Reservoir-Oil PVT. The Whitson-Torp method was
used to develop modified black-oil PVT for the reservoir oil with the
two EOS characterizations presented previously. A DLE was simulated where the equilibrium oil and equilibrium gas from each stage
of depletion was passed separately through a two-stage separator
(300 psia at 75°F and 14.7 psia at 60°F). Figs. C-37 through C-40
present the results for the reservoir oil for the two characterizations
as solid lines.
The reservoir was then considered to have a gas cap. The gas-cap
composition was taken from the bubblepoint calculation (Table C-9).
This gas was depleted by a CVD experiment, where the equilibrium
gas and equilibrium oil from each stage of depletion was passed separately through a two-stage separator under the same conditions as in
the previous paragraph. Figs. C-37 through C-40 present the results
for the two characterizations for the reservoir gas as dashed lines.
As Figs. C-37 through C-39 show, significant differences in modified black-oil PVT data exist for the two characterizations. Signif-
Fig. C-39—Modified black-oil PVT property solution oil/gas ratio,
rs , for reservoir-oil example comparing EOS predictive models
based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
Fig. C-40—Modified black-oil PVT property dry gas FVF, Bgd ,
for reservoir-oil example comparing EOS predictive models
based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
EQUATION-OF-STATE APPLICATIONS
15
icant differences are also seen between the PVT properties generated from the reservoir oil and the reservoir gas. The difference in
PVT properties calculated for the two EOS characterizations seems
considerably larger than the differences in predictions of measured
PVT data. This is because the differences in black-oil PVT data lie
mainly in the gas-phase properties, which are not well-defined by
the experimental PVT data. Comparison of equilibrium-gas compositions (Table C-9) supports the differences seen in Figs. C-37
through C-39. Table C-9 shows that more C 7) material is predicted
by the PR EOS for the bubblepoint equilibrium gas.
The significant differences in black-oil PVT properties calculated
from the reservoir oil and gas cause a real dilemma. First, most reservoir simulators require that saturated R s and B o data increase
monotonically with pressure. From Figs. C-37 and C-38, we see that
only the reservoir-oil PVT data satisfy this requirement. This leads
to the question of how use of the reservoir-oil PVT properties in the
gas cap would affect reservoir performance? The answer can only
be found by comparing black-oil with compositional simulations.
Another concern is choosing the surface oil and gas gravities.
These gravities are used together with the pressure-dependent blackoil properties to calculate reservoir phase densities (Eq. 7.6). Typically only one oil gravity and one gas gravity can be provided to a reservoir simulator. If phase densities are important, then care must be
taken to choose the surface gravities that give the best reservoir phase
densities, particularly in the range of pressures most important to the
reservoir recovery mechanisms. For this example, the oil specific gravities range from 0.715 (from the reservoir gas) to 0.825 (from the
reservoir oil) and the gas specific gravities range from 0.88 to 0.91.
Figs. C-37 and C-38 show the black-oil properties B o and R s calculated with Eqs. 6.32 and 6.33 on the basis of conversion of differential-liberation data by use of separator test results. For this particular oil, the traditional conversions are not bad, somewhat
overpredicting B o and R s. For more volatile oils, the difference can
be much more significant.
For a reservoir system that is initially undersaturated, the fluid can
become saturated at a pressure greater than the initial saturation
condition. For example, the reservoir oil in this example might produce at a low flowing bottomhole pressure that results in a high gas
saturation near the wellbore. During a shut-in period, the pressure
increase near the wellbore will saturate the free gas developed during production. If the R s vs. pressure curve increases only to the initial bubblepoint and remains constant at higher pressures, the gas
would stop dissolving in the oil at the initial bubblepoint. To ensure
that free gas continues to dissolve into the oil at higher pressures, the
R s curve must be extrapolated to higher pressures.
One approach to developing an extension to the R s curve is to
add a small amount of equilibrium gas (evolved at the original bubblepoint pressure) to the original oil. A new bubblepoint is determined for the new mixture. The separator gas/oil ratio is also determined, thereby providing a new point on the R s vs. (bubblepoint)
pressure curve. This procedure can be continued at increasing bubblepoint pressures until the initial reservoir pressure is reached.
Alternatively, equilibrium gas from each new bubblepoint can be
used to generate the next mixture. This approach is often used to
estimate the PVT properties for a reservoir that exhibits bubblepoint variation with depth.
Two problems may arise when generating an extension of the R s
curve. Either a maximum in bubblepoint pressure may be reached
that is less than the initial reservoir pressure or a dewpoint instead
of a bubblepoint may be calculated, indicating that the procedure
has passed through a critical condition. In either of these situations, completing the extension of the R s to the initial pressure is
not possible. If successful, this method generates an extension to
the original R s curve that may become flat or even exhibit a decreasing slope at higher pressures.
Immiscible gas injection into an undersaturated oil reservoir defines a second situation that requires an extrapolated R s curve. The
development of the extrapolated R s for this situation is somewhat
different. Here, the injection gas should be used to determine mixtures with increasing bubblepoints and GOR’s. A swelling test with
the injection gas can be simulated to obtain the necessary mixtures
for extending the R s curve.
16
The same two problems that can occur with the equilibrium-gas
procedure also can occur with this method. Namely, that a maximum can be reached below the initial pressure and that transition
through a critical mixture can result in a dewpoint condition. A richer injection gas tends to cause both problems, whereas leaner injection gas may avoid the problems (depending somewhat on the degree of undersaturation). Extension of the R s curve with this method
usually results in a relatively steep increase in R s at increasing pressures. Leaner injection gas results in steeper curves.
Caution should be used in modeling a gas-injection process with
modified black-oil PVT properties, particularly when significant
phase behavior effects are expected (e.g., vaporization and swelling).
The Cook et al.15 method (Chap. 7) for modifying black-oil PVT
properties for vaporizing immiscible gas injection processes and
compositional simulation are alternatives that can be considered.
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11. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,
AIME (1942) 146, 140.
12. Sutton, R.P.: “Compressibility Factors for High-Molecular Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September.
13. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of
Reservoir Fluids From Their Compositions,” JPT (October 1964)
1171; Trans., AIME, 231.
14. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion
Data,” JPT (March 1983) 610; Trans., AIME, 275.
15. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A Beta-Type Reservoir
Simulator for Approximating Compositional Effects During Gas Injection,” SPEJ (October 1974) 471.
SI Metric Conversion Factors
°API
bbl
ft3
°F
lbm mol
psi
psi*1
°R
141.5/(131.5)°API)
+g/cm3
1.589 873
E*01 +m3
2.831 685
E*02 +m3
(°F*32)/1.8
+°C
4.535 924
E*01 +kmol
6.894 757
E)00 +kPa
1.450 377
E*01 +kPa–1
5/9
+K
PHASE BEHAVIOR
Appendix D
Understanding Laboratory
Oil PVT Reports
M.B. Standing
Introduction
The subject of how to read and make proper use of information contained in laboratory pressure/volume/temperature (PVT) reports
has not been treated adequately in course texts. This is borne out by
comments of students in a basic petroleum engineering course, who
find the subject one of the most difficult to understand in the whole
course. I hope the following discussion of the why and wherefore
of a typical PVT report will be helpful. The discussion pertains to
Report RFL 10641 on the Raleigh field contained in this section.
Sample pages of this report are given at the end of this appendix.
Purpose of the Report
First, the form of data presentation in the report developed because
of its use in material-balance calculations. Some of the tabular information is set up to satisfy that need. Second, the report should
cover all past, present, and future situations that might require calculations. To do this with a minimum of tables and curves, the data
are normalized to a reference state and only data for the reference
state are given. The petroleum engineer must then “work back”
from the reference state to a particular situation.
Third, the laboratory tests are carried out on the basis of two different thermodynamic processes being under way at the same time.
These are (1) flash equilibrium separation of gas and oil in the surface
traps during production and (2) differential equilibrium separation of
gas and oil in the reservoir during pressure decline. As a consequence,
the report gives both flash and differential data and it becomes necessary to be able to shift between the two sets of data. Finally, the report
gives data on the particular sample obtained. This may not be the
proper “average” of all the fluid in the reservoir, and slight adjustment
of the data may be necessary at a later time. Therefore, some detail
is given to the manner of obtaining the sample and the conditions that
exist at the sampling time. Also, the compositional analysis of the
sample is given so that equilibrium calculations can be made for
conditions other than those studied in the laboratory.
With these generalities in mind, we now consider specific data
presented in the Raleigh report. The surface flash separation data are
considered first, followed by the reservoir differential data. We then
consider how to convert certain differential data to equivalent flash
data. Page numbers refer to the pages of report.
UNDERSTANDING LABORATORY OIL PVT REPORTS
Separator Tests of Reservoir Fluid Sample
(Report Page 5)
As we show later, one form of the material-balance equation is an
equality between the expansion of the original reservoir oil (between the initial pressure and any subsequent pressure) and the volume voidage that has occurred down to the subsequent pressure.
The separator test data on Page 5 of the report, which shows the
quantity of surface gases and stock-tank oil (STO) that results when
1 bbl of bubblepoint oil is flashed through certain surface trap sequence, allows computation of voidages. The tabulation also gives
the oil gravity (°API) of the STO and, in some instances, the gravity
of gas coming from the primary trap.
Cols. 1 and 2 give the pressure/temperature condition of the surface trap tests that were investigated. These should be specified by
the reservoir engineer at the time the test is planned so that they will
apply to future field operations. Referring to the bottom line of data,
the surface situation modeled here is a two-stage separation [i.e., a
primary trap operating at 200 psig and 73°F, followed by a stock
tank operating at 14.7 psia (0 psig) and 73°F].
When 1 bbl of bubblepoint oil (defined in Footnote 2 as oil saturated at 3,236 psig and 258°F) is flashed (processed) through this
trap arrangement, the STO amounts to 0.5974 bbl and has a quality
of 48.5°API (Cols. 6 and 5). The formation volume factor (FVF)
of the bubblepoint oil, B ob, is therefore 1/0.5974+1.674 bbl/bbl
STO (Col. 7).
Cols. 3 and 4 show the surface gas/oil ratio from the trap and tank.
The primary trap ratio is 875 ft3/bbl STO, and the tank vapors
amount to 134 ft3/bbl STO. The solution gas/oil ratio at bubblepoint
conditions (3,236 psig and 258°F), R sb , is 875)134+1,009 ft3/bbl
STO when flashed through this surface trap arrangement. As this
table shows, R sb, B ob, and oil gravity all vary with the trap pressure/
temperature situation. Surface-gas gravity does also, but usually is
reported only for the single-stage atmosphere flash. To calculate
reservoir voidage properly, the measured STO and the produced gas
have to be handled according to the information in this table. However, note that these data always refer only to the bubblepoint oil as
the reference fluid. Determination of FVF’s for other reservoir
fluids requires additional information.
1
FVF, volume/residual volume
Fig. D-1—FVF for oil and gas and for total (oil plus gas) system as a function of pressure above
and below bubblepoint.
Fluid Properties at Pressures
Lower Than Bubblepoint Pressure
We now consider situations at reservoir pressures greater than the
bubblepoint pressure. We first look at the FVF, then at the fluid density, and then the compressibility of the fluid. Cols. 1 and 2 of the
Reservoir Fluid Sample Tabular Data on Page 3 give the pressure/volume relations of the original fluid at 258°F. Note that the data are presented in terms of a unit volume at the bubblepoint condition. Col. 2
gives the volume of the system at pressure p per unit system volume
at 3,236 psig and 258°F. These are listed as relative volumes (i.e., relative to the bubblepoint).
Consider the FVF of the original oil in the reservoir. On Page 1,
we see that the original pressure (listed as last reservoir pressure under well characteristics) was 5,783 psig at *12,650 ft. Thus, if we
want the oil FVF at 5,783 psig, we obtain it by multiplying the FVF
at the bubblepoint by the relative volume (to the bubblepoint),
V rel + V RńV ob . We multiply because
Bo +
V V
VR
+ ob R
Vo
V o V ob
. . . . . . . . . . . . . . . . . . . . . . . . . . (D-1)
and the reference bubblepoint oil volume cancel out. Therefore, B oi,
the initial FVF, is 1.674 0.9424+1.577 when the 200-psig primary trap is involved. It will be different if another trap pressure is used.
Reservoir oil density at pressures greater than 3,236 psig also
make use of the relative-volume data of Col. 2, Page 3. The added
information is the density of the bubblepoint oil. This is always given in the summary data on Page 2 of the report. We see here that the
specific volume at the bubblepoint v^ ob + 0.02772 ft3/lbm. This
comes from direct weight/volume measurements on the sample in
the PVT cell. If now we wish the density, ò oi, of the initial reservoir
oil, we have
ò o + ^1 + ^1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-2)
v oi
v ob V rel
and ò oi +
1
0.02772
0.9424
+ 38.3 lbmńft 3 .
. . . . . . . (D-3)
Compressibility of reservoir oil at pressures higher than the bubblepoint is also obtainable from the relative-volume data. Recall
that the definition of compressibility is
ǒ Ǔ.
c o + 1 dV
V dp
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-4)
T
It makes no difference whether the volume units in the equation are
relative volumes to the bubblepoint, to FVF’s, or to specific volume
values. To evaluate c o at pressure p, it is only necessary to differentiate the p * V rel data in Cols. 1 and 2 graphically to get dV/d p at the
2
pressure and divide by V rel . A less accurate value can be obtained
by the assumption
ǒ Ǔ
DV rel
co + 1
V rel Dp
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-5)
T
For example, to get c o at 4,500 psig by use of relative-volume values of 500 psi on each side of 4,500 psig
co +
+
(0.9562 * 0.9781)
1
(0.9562 ) 0.9781)ń2 (5, 000 * 4, 000)
0.0219 + 22.7
1
0.9671 1, 000
10 *6 volńvol-psi.
Note that Page 2 of the report lists some compressibility numbers.
These are not the same as those indicated earlier because they are
changes in volume (in the pressure interval indicated) per unit volume at the lower pressure. For example, the value of 22.33 10*6
for the 5,000- to 4,000-psi interval is obtained as
(0.9562 * 0.9781)
1
+ 22.39
0.9781 (5, 000 * 4, 000)
10 *6.
The compressibility data on Page 2 are set up in this manner because
of the way they are used in one form of material balance.
Total FVF of Original Oil
at Less Than Bubblepoint Pressure
We have seen that, to calculate the FVF of the oil at pressures higher
than bubblepoint, we multiply the bubblepoint FVF times the relative volume given in Col. 2, Page 3. Obviously, if we multiply
B ob by V rel at pressures less than p b, we also get an FVF. In fact, we
get the total FVF, B t, of the original system. That is, at p t p b , we
will have two phases and B t is the volume relation of both gas and
liquid phases in equilibrium at pressure p (Fig. D-1).
We mentioned earlier that one form of the material balance makes
use of the expansion of the original oil between the initial system
pressure and any subsequent pressure. This expansion is given by
E o + N(B t * B oi) ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-6)
where N+initial stock-tank barrels in the reservoir and
(B t * B oi) +the expansion per unit STO; therefore, E o +expansion (in barrels) of the original oil system. Sometimes the expansion
equation is written
E o + N(B t * B ti). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-7)
At p u p b , whether the FVF is considered to be total FVF or an oil
FVF makes no difference, it is the same thing. For example, see Eq.
4.4 of Ref. 2 or Eq. 8.17 of Ref. 3.
PHASE BEHAVIOR
Initial
bubblepoint
oil
Fig. D-2—PVT cell volume vs. pressure during differential vaporization test (showing oil shrinkage) and incremental liberated gas volumes (bȀ–b and cȀ–c) at pressures below bubblepoint.
DifferentialĆLiberation Tests
Up to this point, we have considered what happens when reservoir
fluid comes to the surface and is separated into surface gas and oil
products. We modeled flash equilibrium conditions because we believe that the action going on in the trap is essentially one where the
whole system entering the trap immediately separates into two components, trap gas and trap liquid. This constitutes the elements of a
flash separation.
The standard PVT report includes data referred to as “differential
data.” These are gas-solubility and phase-volume data taken in a
manner to model what some people believe happens to the oil phase
in the reservoir during pressure decline. The argument that differential-liberation tests model the subsurface behavior comes primarily
from two things.
1. Reservoir pressure changes are not as violent or as large as the
pressure changes that occur when entering surface traps. The subsurface changes are more gradual and might be considered to be a series
of infinitesimal changes.
2. Because of the relative permeability characteristics of reservoir
rock/fluid systems, the gas phase moves toward the well at a faster
rate than the liquid phase. As a result, the overall composition of the
entire reservoir system is changing.
These two ideas promote the idea that a test procedure modeled
on a differential process should be used to study subsurface behavior. Because of experimental limitations and time/cost considerations, a laboratory cannot perform a true differential procedure.
Instead, it performs a series of stepwise flashes at the reservoir temperature (usually about 10) beginning at the bubblepoint. Of course,
the greater the number of steps, the more closely the true differential
process is modeled.
The differential data are reported in the last three columns of Page
3. Supplementary differential-release data are given on Page 4. Note
that the three columns are headed “Differential Liberation at
258°F.” The best way to understand these data is to explain how the
values are obtained. The laboratory starts with a known volume of
the original system in the PVT cell, which may be of the order of 100
to 200 cm3. The volume at the bubblepoint pressure (3,236 psig in
this instance) is determined accurately because it is a reference for
all subsequent measurements.
Referring to Page 3, we see that the first pressure step was to 2,938
psig. At this pressure, the original system will be in two phases. Its
volume would be at bȀ on Fig. D-2. The first step in altering the overall system composition is made at 2,938 psig by removing the gas
phase from the PVT cell while maintaining constant pressure. The
quantity of gas removed is determined by collecting it in a calibrated
container. The volume that the gas phase occupied in the cell is deUNDERSTANDING LABORATORY OIL PVT REPORTS
termined by the amount of mercury injected during the removal process. Also, the gas gravity is measured on the sample bleedoff. The
volume of liquid remaining in the cell is at Point b on Fig. D-2.
This procedure is repeated by taking the 2,938-psig saturated
liquid to 2,607 psig (Point cȀ) and removing a second batch of gas
at that pressure. Again the volume of the displaced gas in the cell
at 2,607 psig is determined along with the gravity of the removal
gas. The volume of liquid phase remaining after the second gas-removal step is illustrated by Point c in Fig. D-2. This process of removing batches of equilibrium gas continues until the cell pressure
at the last displacement is 0 psig. The differential data on Page 3
show 11 equilibrium removals, all at 258°F. The final volume of
liquid phase remaining in the cell at 0 psig and 258°F is corrected
by thermal-expansion tables (or by cooling the cell) to 0 psig and
60°F. This 0-psig/60°F liquid is called residual oil. Note that residual oil and STO are not the same thing. They are both products of
the original oil in the system but are developed by different pressure/temperature routes.
Once residual oil has been reached, the data obtained are recalculated and presented on the basis of a unit barrel of residual oil. The
cumulative amount of gas removed from the cell (liberated from
solution) at each pressure step is given as a gas/oil ratio(GOR). Col.
4 shows that 183 ft3/bbl residual oil was liberated between 3,236 and
2,938 psig and 362 ft3/bbl residual oil was liberated between 2,938
and 2,607 psig. By the time 0 psig and 258°F had been reached, the
original system had liberated 1,518 ft3/bbl residual oil. Col. 5 shows
the amount of gas in solution at the various pressures. This is the difference of the 1,518 ft3 total liberated and the amount liberated between the original bubblepoint pressure and that pressure. For example, the solution gas/oil ratio at 2,938 psig is 1,518*183+1,335
ft3/bbl residual oil.
At this point, be sure that you understand why the solution gas/oil
ratio determined from surface flash and from differential removal
will be different. It is because the processes for obtaining residual
oil and STO from bubblepoint oil are different. The first is a multiple
series of flashes at the elevated reservoir temperature, and the second is generally a one– or two-stage flash at low pressures and low
temperature. The quantity of gas released will be different, and the
quantity of final liquid will be different. Also, the quality (gravity)
of the products will be different (compare °API of residual oil with
°API of STO). The only thing that will be the same for the two processes is the total weight of the end products.
Col. 6 gives the relative volumes of the liquid phase measured during the differential liberation of gas. Note that these are volumes at
pressure p per unit volume of residual oil. Again, these relative volumes must not be confused with FVF’s because FVF’s are specified
per barrel of STO. Note on Page 3 that relative volumes start at 1.000
3
Fig. D-3—Differential vaporization and flash-corrected solution gas/oil ratio vs. pressure above
and below bubblepoint pressure.
at 0 psig/60°F and that the value of 1.109 at 0 psig/258°F is the thermal expansion of 42.2°API residual oil from 60 to 258°F. At pressures higher than 3,236 psig (the original bubblepoint), the system composition remained constant. Therefore, the relation of the relative oil
volume at pupb to the bubblepoint value, 2.075, must be the same
as the relative-volume numbers in Col. 2 (e.g., 1.948/2.075+0.9387
at 6,000 psig).
The data on Page 4 are differential liberation data that refer to the
oil and gas phases in the reservoir at 258°F. Col. 2 shows that the
gravity of the 183 ft3/bbl residual oil liberated between 3,236 and
2,938 psig was 0.870. The next batch between 2,938 and 2,607 psig
(362*183+179 ft3/bbl residual oil) was 0.846. The gas deviation
(compressibility) factor of the first liberated gas was 0.886 at 2,938
psig. The oil density at 2,938 psig/258°F was 0.5905 g/cm3.
Once you understand the basic difference between flash and differential data as given in the standard PVT report, proceed to calculation of flash solubilities and oil FVF’s at less than the bubblepoint from the differential data.
Calculation of Flash Solubility
From Differential Solubilities
The laboratory report requires calculation of flash solubility data
rather than providing it because the laboratory does not know what
trap pressures will be used in the field during its producing life.
Instead, the laboratory concentrates on providing sufficient data to
handle any normal situation by simple data conversions.
First, consider the solubility data we have.
1. Differential solubility data at the bubblepoint state (3,236
psig/258°F) and at 11 pressures less than the bubblepoint pressure.
The bubblepoint value is 1,518 ft3/bbl residual oil. All fluids at pressures greater than pb have this amount of gas.
2. Flash solubility of the bubblepoint oil for four different surface
trap situations. These vary from 1,206 ft3/bbl STO for a single flash
to atmospheric pressure to 1,009 ft3/bbl STO for a 200-psig primary-trap-tank situation. Fig. D-3 shows these.
We now wish to determine the “flash-converted” values (i.e., the
amount of gas obtained at the surface when a unit of saturated reservoir oil at less than 3,236 psig is flashed through a surface trap setup). To illustrate, we use the 200-psig-primary/0-psig-tank situation
at the reservoir pressure of 2,301 psig. Looking at the differentialliberation data in Col. 4, Page 3, we see that 506 ft3 of gas has come
out of solution per barrel of residual oil when the pressure declined
from 3,236 to 2,301 psig. In other words, we can say that the
2,301-psig saturated oil contains less gas by this amount. If this liquid were taken to the surface and processed through the traps, it
would also show somewhat less gas solubility than the 1,009 ft3/bbl
STO that the bubblepoint oil shows; however, it would not be 506
ft3 less because we are on a different oil base.
4
If we let (DR s) diff be the liberated gas/oil ratio by differential vaporization, (DR s) diff + (R sb) diff * ( R s) diff , we can convert this to a
(DR s) flash as follows.
Vg
V g V or
+
,
V or V ob
V ob
V g V ob
Vg
+
,
V ob V o
Vo
Vg
+ (DR s) flash ,
Vo
V or
+ 1 ,
V ob
2.075
and
V ob
+ 1.674 RBńSTB ,
Vo
. . . . . . . . . . . . . . . . . . . . . . (D-8)
where V g is in cubic feet and V o , V ob, and V or are in barrels.
Therefore,
(DR s) flash + (DR s) diff 1.674
2.075
2301
and (R s) flash + 1, 009 * (DR s) flash + 1009 * 506 1.674
2.075
+ 1, 009 * 408 + 601 scfńSTB .
. . . . . . . . (D-9)
This can be generalized as
(R s) flash + ǒR sbǓ flash * (DR s) diff
B ob
.
ǒV obńV or Ǔ
. . . . . . . . . (D-10)
At times, this relationship will yield negative values of (DR s) flash at
low pressures. This is not inconsistent with the physics of the situation because a saturated oil at a high reservoir temperature but a low
pressure may give off no gas when processed through the cooler surface traps. We would expect then to get a (DR s) flash of zero at some
finite value of p (Fig. D-3).
Calculation of FVF's
From Relative Volumes
We now consider FVF’s at pressures lower than bubblepoint. We have
the full relative-volume curve of the saturated-liquid phase in terms
of residual oil. Fig. D-4 shows this. The bubblepoint state has a relative oil volume of 2.075 bbl/bbl residual oil. We also have the FVF
at the bubblepoint state, B ob, with a value of 1.674 bbl/bbl STO. We
can see that the relative oil volume and the FVF at pressure p can be
related by transferring to the common point, the bubblepoint.
Let V ońV or +relative volume of saturated oil at pressure p in bbl/
bbl residual oil. Then,
PHASE BEHAVIOR
Nonphysical:
neglect
Fig. D-4—Differential vaporization and flash-corrected oil FVF vs. pressure above and below
bubblepoint pressure.
vs. pressure can be used to the point where R s + 0, with R s + 0 at
lower pressures. The corresponding B o at R s + 0 is taken from the
linear trend of flash-corrected B o vs. flash-corrected R s (Fig. D-5).
V o V or
V
+ o,
V or V ob
V ob
V o V ob
V
+ o,
V ob V o
Vo
References
V
and o + B o ,
Vo
with all volumes in barrels. Therefore,
B o + V ońV or
B ob
.
ǒV ońV orǓ
b
1. “Raleigh Field PVT Report,” Report RFL 1064, Core Laboratories Inc.,
Houston (1958).
2. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroluem Reservoir Engineering: Physical Properties, McGraw-Hill Book Co. Inc. New York
City (1960).
3. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,
Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).
SI Metric Conversion Factors
At 2,301 psig we would have
B o + 1.787 1.674 + 1.442 RBńSTB.
2.075
As in the previous instance, we must give special consideration
to the low range of pressure. If we apply the above equation to the
0 psig/258°F point, we get B o + 1.109(1.674ń2.75) + 0.895. This
is an absurdity because any oil at 258°F must have a B o of close to
1.1 owing to thermal expansion. Therefore, as Fig. D-4 shows, we
draw the oil FVF curve into the ordinate at the value of thermal expansion shown by the differential curve. Alternatively, the R s curve
°API
bbl
cp
ft
ft3
°F
lbm
psi
141.5/(131.5)°API)
+g/cm3
1.589 873
E*01 +m3
1.0*
E*03 +Pa@s
3.048*
E*01 +m
2.831 685
E*02 +m3
(°F*32)/1.8
+°C
4.535 924
E*01 +kg
6.894 757
E)00 +kPa
*Conversion factor is exact.
Nonphysical
extrapolation
Fig. D-5—Flash-corrected oil FVF vs. flash-corrected solution gas/oil ratio showing “normal”
linear behavior (in particular, the nonphysical behavior at low pressures because of the approximate nature of the traditional differential-to-flash correction.
UNDERSTANDING LABORATORY OIL PVT REPORTS
5
The California Co.
Box 713
Brookhaven, Mississippi
Attention: Mr. O.H. Fennell
Subject: Reservoir Fluid Study
Central Oil Co. No. 5-2 Well
Raleigh Field
Smith County, Mississippi
Our File Number: RFL 1064
Gentlemen:
Subsurface fluid samples were collected from the Central Oil Co. No. 5-2 well on March 14, 1958, by a representative of Core Laboratories Inc. The results of fluid studies performed with these samples are transmitted to you in the following report.
The saturation pressure of the fluid was determined to be 3,236 psig at the reservoir temperature of 258°F. This value is considerably
less than the static reservoir pressure measured immediately before sampling and indicates that the reservoir exists in a highly undersaturated condition. The presence of a column of water in the bottom of the tubing necessitated obtaining the samples approximately
900 ft above the producing formation. Because the reservoir was highly undersaturated, the pressure in the tubing at the point of sampling was still well above the measured saturation pressure. The data presented in this report are felt to be representative of the reservoir fluid and may be applied to calculations without adjustment.
Differential pressure depletion of the fluid at the reservoir temperature of 258°F evolved 1,518 scf gas/bbl residual oil with an accompanying FVF of 2.075 bbl saturated fluid/bbl residual oil. Under similar depletion conditions, the viscosity of the fluid was measured
from pressures exceeding reservoir pressure to atmospheric pressure. The viscosity of the fluid decreased to 0.093 cp at saturation
pressure, then increased to a maximum of 0.700 cp at atmospheric pressure.
To determine the effects of changes in surface separation pressure on the produced fluid, flash vaporization tests were performed at
four operating pressures and atmospheric temperature. The tests indicate the optimum separator pressure to be approximately 150 psig
with near optimum conditions as low as 100 psig.
Again it was a pleasure to cooperate with you by performing this study. Should any questions arise or if we may assist you further,
please do not hesitate to call.
Very truly yours,
P.L. Moses,
Operations Supervisor
Core Laboratories Inc.
Reservoir Fluid Div.
PLM:ds
3 cc—Addressee
3 cc—Mr. C. L. Pickett
The California Co.
Natchez, Mississippi
3 cc—Mr. E. J. Deu Pree
The California Co.
New Orleans, Louisiana
6
PHASE BEHAVIOR
Page
File
Company The California Co.
Data Sampled March 14, 1958
Well
County
Smith
State
Mississippi
Central Oil Co. No. 5-2
Field State
Raleigh
1
of
11
RFL 1064
FORMATION CHARACTERISTICS
Formation Name
Date First Well Completed
Original Reservoir Pressure
Original Produced Gas/Oil Ratio
Production Rate
Separator Pressure and Temperature
Oil Gravity at 60°F
Datum
Original Gas Cap
Hosston
,
, 19
psi at
1,100
psi,
49
ft
ft3/bbl
B/D
°F
°API
ft subsea
None
WELL CHARACTERISTICS
Elevation
Total Depth
Completion Date
Tubing Size and Depth
Productivity Index
Last Reservoir Pressure
Date
Reservoir Temperature
Status of Well
Pressure Gauge
Normal Production Rate
Gas/Oil Ratio
Separator Pressure and Temperature
Base Pressure
Well Making Water
438 ft DF
12,770 PBD
ft
12,732–12, 752,12, 758–12,765 ft
2
in. to
12,704 ft
B/D-psi at
B/D
5,783
psi at
12,650 ft
March 14
,1958
256*
°F at
2,650 ft
Shut-In 27 Hours
Amerada (DO)
B/D
1,100
ft3/bbl
psi,
°F
psi Abs.
% Cut
SAMPLING CONDITIONS
Sampled at
Status of Well
Gas/Oil Ratio
Separator Pressure and Temperature
Tubing Pressure
Casing Pressure
Core Laboratories Engineer
Type Sampler
11,800 ft
Shut-In 27 Hours**
psi,
2,128
ft3/bbl
°F
psi
psi
LBB
Wofford
REMARKS:
*Temperature extrapolated to midpoint of perforations, 258°F.
**Before sampling, well was flowed at successive rates of 127, 115, and 103 B/D. The well was then shut in
for 24 hours.
UNDERSTANDING LABORATORY OIL PVT REPORTS
7
of
11
Page
2
File
RFL 1064
Well Central Oil Co. No. 5-2
VOLUMETRIC DATA OF Reservoir Fluid SAMPLE
1. Saturation pressure (bubblepoint pressure)
3,236
psi at 258°F.
2. Thermal expansion of saturated oil at 6,000 psi+ V at 258°F + 1.13094 .
V at 73°F
3. Compressibility of saturated oil at reservoir temperature: vol/vol-psi.
From 6,000 psi to 5,000 psi+18.32 10*6
From 5,000 psi to 4,000 psi+22.33 10*6
From 4,000 psi to 3,236 psi+28.64 10*6
4. Specific volume at saturation pressure: ft3/lbm 0.02772 at 258°F.
8
PHASE BEHAVIOR
Page
3
of
11
File
RFL 1064
Well Central Oil Co. No. 5-2
Reservoir Fluid SAMPLE TABULAR DATA
Differential Liberation at 258°F
Gas/Oil Ratio
Gauge
Pressure
(psi)
6,000
5,500
5,300
5,000
4,590
4,500
4,100
4,000
3,800
3,720
3,600
3,500
3,400
3,390
3,300
3,236
3,200
3,141
3,110
3,094
3,039
2,969
2,938
2,882
2,800
2,792
2,640
2,607
2,448
2,301
2,300
2,237
2,024
1,903
1,825
1,800
1,665
1,505
1,501
1,300
1,261
1,092
1,078
900
800
761
686
656
518
346
310
200
97
0
Pressure/Volume
Relation at 258 °F
Relative Volume of Oil
and Gas
V/Vob
Viscosity*
of Oil
at 258 °F
(cp)
0.9387
0.9471
Liberated/bbl
Residual Oil
In Solution/bbl
Residual Oil
0.119
Relative Oil
Volume
Vo /Vor
1.948
1.965
0.113
0.9562
1.984
0.107
0.9666
2.006
0.102
0.9781
0.9833
2.030
2.040
0.099
0.9888
0.9918
0.9948
2.052
2.058
2.064
0.096
0.9979
1.0000
1.0047
1.0128
0.093
0
1,518
2.071
2.075
183
1,335
1.970
362
1,156
1 .867
506
1,012
1.787
670
848
1.698
815
703
1.624
957
561
1.544
1,089
429
1.472
1,209
309
1.399
0.095
1.0192
1.0273
1.0387
1.0534
0.104
1.0697
1.1025
1.1517
0.118
1.2177
1.3003
1.3997
0.134
1.4994
1.6244
0.155
1.8717
2.1540
2.5475
0.179
2.9926
3.4741
4.3966
0.220
0.700
1,262
1,328
1,518
256
190
0
at 60°F =
1.367
1.311
1.109
1.000
*Viscosity measurement made with differential-liberation procedure that is a separate experiment from the differential-liberation test used to measure PVT data.
V+volume at given pressure; Vob +volume at bubblepoint pressure at the specified temperature, and Vor +residual oil volume at 14.7 psi absolute pressure and 60°F.
Gravity of residual oil+42.2°API at 60°F.
UNDERSTANDING LABORATORY OIL PVT REPORTS
9
Page
4
File
RFL 1064
of
11
Well Central Oil Co. No. 5-2
Supplementary Differential-Liberation Data
Pressure
(psig)
3,236
2,938
2,607
2,301
1,903
1,505
0
Oil Density
(g/cm3)
0.5773
0.5905
0.6055
0.6179
0.6326
0.6455
0.7340
Gas Gravity
0.870
0.846
0.833
0.830
0.835
1.532
Deviation Factor
Z
0.886
0.879
0.878
0.884
0.897
Page
File
5
of
11
RFL 1064
Well Central Oil Co. No. 5-2
SEPARATOR TESTS OF Reservoir Fluid SAMPLE
GOR1
Separator
Pressure
(psi gauge)
Temperature
(°F)
Separator
0
50
100
200
75
74
75
73
1,206
1,011
950
875
Stock Tank
Stock-Tank
Gravity
(°API at 60°F)
Shrinkage
Factor
Vor /Vob 2
Formation
Volume Factor
Vob /Vor 3
Flashed Gas
Specific
Gravity
0
35
68
134
45.6
48.1
48.5
48.5
0.5456
0.5872
0.5949
0.5974
1.833
1.703
1.681
1.674
0.942
1Separator
and stock-tank gas/oil ratio in cubic feet of gas at 60°F and 14.7 psi absolute per barrel of STO at 60°F.
factor, Vor /Vob , is barrels of STO at 60°F per barrel of saturated oil at 3,236 psi gauge and 258 °F.
3FVF, V /V , is barrels of saturated oil at 3,236 psi gauge and 258°F per barrel of STO at 60°F.
ob or
2Shrinkage
This table provides results of four separate two-stage separator tests. The first two columns of data give the primary-separator conditions. In all tests, the second
(final) separator is at standard (stock-tank) conditions. For example, conditions for the first two-stage separator test are (1) psp 1+0 psig and Tsp 1+75°F and (2)
psp 2+0 psig and Tsp2 +60°F, with total Rsb +1,206)35+1,241 scf/STB, Bob +1.833, gAPI+45.6°API, and gg +0.942.
10
PHASE BEHAVIOR
Page
6
of
File
RFL 1064
Company The California Co.
Data Sampled March 14, 1958
Well
County
Smith
State
Mississippi
Central Oil Co. No. 5-2
Field State
Raleigh
11*
HYDROCARBON ANALYSIS OF RESERVOIR FLUID SAMPLE
Component
wt%
mol%
Nitrogen
0.18
0.51
Methane
9.54
45.21
Ethane
2.80
7.09
Propane
2.67
4.61
iso–butane
1.29
1.69
n–butane
2.15
2.81
iso–pentane
1.47
1.55
n–pentane
1.91
2.01
Hexanes
5.01
4.42
Heavier
72.30
28.91
0.68
1.19
100.00
100.00
Carbon Dioxide
Density at 60°F
(g/cm3)
°API
at 60°F
Molecular
Weight
0.8142
42.1
190
Core Laboratories Inc.
Reservoir Fluid Div.
P. L. Moses,
Operations Supervisor
*Pages 7 through 11 of the original report are graphical representations of the tabular data in Pages 3 and 4.
UNDERSTANDING LABORATORY OIL PVT REPORTS
11
Nomenclature
a+ numerical constant(s) used in equations;
dimensional equation-of-state (EOS) constant
describing molecular attractive forces,
psia/(ft3-lbm mol)2
a i+ EOS constant of Component i
A+ numerical constant(s) used in equations;
dimensionless EOS constant describing molecular
attractive forces
A aq+ dimensionless EOS constant for aqueous phase in
hydrocarbon/water system
A H + intermediate variable used for selecting
pseudocomponents defined by the logarithm of C 7
K value
A HC+ EOS dimensionless constant for hydrocarbon phase
in hydrocarbon/water system
A ij + intermediate terms in Newton-Raphson solution of
the Michelsen two-phase isothermal flash (Eq. 4.58)
A L + intermediate variable used for selecting
pseudocomponents defined by the logarithm of
maximum K value in a mixture
b+ inverse FVF, b=1/B, L3/L3, scf/ft3 or STB/RB;
dimensional EOS constant describing molecular
repulsive forces, L3/n, ft3/lbm mol
b g+ inverse gas FVF, L3/L3, scf/ft3
b gd+ inverse dry-gas FVF, L3/L3, scf/ft3
b gw+ inverse wet-gas FVF, L3/L3, scf/ft3
b i+ Hoffmann et al. K-value correlation parameter (Eq.
3.156) for Component i; EOS constant of
Component i
b o+ inverse oil FVF, L3/L3, STB/RB
B+ FVF, L3/L3, RB/STB or ft3/scf; dimensionless EOS
constant describing molecular repulsive forces
+ wet-gas FVF gas, L3/L3, ft3/scf
B gd+ dry-gas FVF, L3/L3, ft3/scf
B *gd+ modified dry-gas FVF, L3/L3, ft3/scf
B gw+ B g+wet-gas FVF, L3/L3, ft3/scf
B ij + intermediate terms in Newton-Raphson solution of
the Michelsen two-phase isothermal flash (Eq. 4.57)
B o+ oil FVF, L3/L3, RB/STB
B *o+ modified oil FVF, L3/L3, RB/STB
B ob+ oil FVF at bubblepoint (saturated) conditions,
L3/L3, RB/STB
B od+ differential oil FVF, L3/L3, RB/residual bbl
B osp+ separator-oil FVF, L3/L3, RB/STB bbl
NOMENCLATURE
B t+ total (gas plus oil) FVF of gas/oil system, L3/L3,
RB/STB
B ti + B t at initial reservoir pressure, L3/L3, RB/STB
Btw + total (gas plus water) FVF of gas/water system,
L3/L3, RB/STB
B w+ gas-saturated brine FVF, L3/L3, RB/STB
B ow+ brine FVF at atmospheric pressure and reservoir
temperature without solution gas, L3/L3, RB/STB
B *w+ brine FVF at reservoir pressure and temperature
without solution gas, L3/L3, RB/STB
c+ isothermal compressibility, Lt2/m, psi*1;
dimensionless EOS volume-translation constant
(volume shift), L3/n, ft3/lbm mol
c+ cumulative (average) compressibility, Lt2/m, psi*1
c g + gas isothermal compressibility, Lt2/m, psi*1
c gw+ total (gas plus water) isothermal compressibility of
gas/water system, Lt2/m, psi*1
c i+ EOS volume-translation (“shift”) constant, ft3/lbm
mol; c i+1/( K i * 1) in Muskat-McDowell
phase-split algorithm (Eq. 4.39)
c o + oil isothermal compressibility, Lt2/m, psi*1
c sw+ salt concentration, molality
c w+ saturated-brine isothermal compressibility,
Lt2/m, psi*1
c *w+ brine isothermal compressibility without solution
gas, Lt2/m, psi*1
c wv + salt concentration, molarity
C f+ Søreide specific gravity correlation characterization
factor (Eq. 5.44)
C i+ molar concentration, n/L3, lbm mol/ft3;
hydrate-former constant
C og+ conversion from stock-tank condensate (condensed
from a reservoir gas) to equivalent surface gas,
L3/L3, scf/STB
C oo+ conversion from stock-tank oil (produced from
a reservoir oil) to equivalent surface gas,
L3/L3, scf/STB
C sv+ salt concentration in water, ppm by volume
C sw+ salt concentration in water, ppm by weight
d ci + calculated Data i used in least-squares regression
d TP+ tangent-plane distance, L
d xi + experimental Data i used in least-squares
regression
D CO w+ CO2/water binary-diffusion coefficient, L2/t, ft2/sec
2
1
D ij + binary-diffusion coefficient, L2/t, ft2/sec
D oij + low-pressure binary-diffusion coefficient,
L2/t, ft2/sec
D im+ effective diffusion coefficient of Component i in a
mixture, L2/t, ft2/sec
e i+ intermediate terms in Newton-Raphson solution of
the Michelsen two-phase isothermal flash (Eq. 4.52)
E g+ gas expansion term used in generalized gas/oil
material balance, L3/L3, scf/STB
E g+ average expansion term used in generalized gas/oil
material balance, L3/L3, scf/STB
E o+ oil expansion term used in generalized gas/oil
material balance, L3/L3, STB/STB
E o+ average oil expansion term used in generalized
gas/oil material balance, L3/L3, STB/STB
f+ generic function; pure-component fugacity,
m/Lt2, psia
f+ pure-component fugacity, m/Lt2, psia
f eqi + final converged-solution equilibrium fugacities in a
two-phase flash, m/Lt2, psia
f i + fugacity of Component i in a mixture, m/Lt2, psia
f i + fugacity of Component i in a mixture, including
gravity potential, L/mt2, psia
f Li+ fugacity of Component i in the liquid phase,
m/Lt2, psia
f M+ parameter in Twu correlation for molecular weight
f pc+ Twu correlation parameter for critical pressure
f Tc+ Twu correlation parameter for critical temperature
f vc + Twu correlation parameter for critical volume
f vi+ Component i fugacity in the vapor phase,
m/Lt2, psia
f yi+ Component i fugacity in an incipient
(saturation-pressure calculation) or (phase-stability
test) trial phase, m/Lt2, psia
f zi+ Component i fugacity in the overall (feed) mixture,
m/Lt2, psia
F+ sum-of-squares function
F+ proportioning factor
F EOS+ generic representation of an EOS function
F i+ characterization factor in Hoffman et al. K-value
correlation
F g+ f v +mole fraction of wellstream or overall mixture
in the gas phase
F gg+ mole fraction of reservoir gas that remains gas at
surface conditions
F gsp+ mole fraction of wellstream that is gas in the
primary separator
F oo+ volume fraction of total stock-tank oil that comes
from the reservoir oil
Fosp + mole fraction of wellstream that is oil in the
primary separator
g *+ normalized Gibbs energy
g c+ mass-to-force conversion factor
g *mix+ overall-mixture normalized Gibbs energy
g *x+ liquid-phase normalized Gibbs energy
g *y+ vapor- or incipient-phase normalized Gibbs energy
g *z + feed-composition normalized Gibbs energy
(considered as a single phase)
G+ original gas in place, L3, scf
G d+ original dry gas in place, L3, scf
G mix+ mixture Gibbs energy
G p+ cumulative gas produced, L3, scf
G pd+ cumulative dry gas produced, L3, scf
G pw+ cumulative wet gas produced, L3, scf
G w+ original wet gas in place, L3, scf
G z + overall-composition Gibbs energy (considered as a
single phase)
2
h+ depth, L, ft; Rachford-Rice function in phase-split
calculation
h ref+ reservoir reference depth, L, ft
H g+ surface-gas gross-heating value, Btu/scf
H i+ component gross-heating value, Btu/scf;
Henry’s constant
i+ carbon number
I+ constant in Eq. 4.64
J a + Jacoby aromaticity factor, fraction
J ij+ Jacobian terms in Newton-Raphson solution of
Michelsen two-phase isothermal flash (Eq. 4.55)
k+ permeability, L2, md
k ij+ EOS binary-interaction parameter between
Component Pair i-j
k ijaq+ binary-interaction parameter for Component Pair i-j
in aqueous phase in a water/hydrocarbon system
k ijHC + binary-interaction parameter for Component Pair i-j
in nonaqeous phase in a water/hydrocarbon system
k rg+ gas relative permeability
k ro+ oil relative permeability
k s+ Setchenow constant, molarity (mol/kg)
K i+ y ińx i+equilibrium ratio (K value) of Component i
K i(vs)+ equilibrium ratio of Component i in a vapor/solid
system
K w+ Watson characterization factor, T1/3, °R1/3
L+ total liquid yield, L3/L3, gal/Mscf
L i+ liquid yield of Component i, L3/L3, gal/Mscf
m+ mass, m, lbm or g; correlating function in
correction term a for EOS Constant A
m g+ gas mass, m, lbm
m o+ oil mass, m, lbm
m SRK+ function in correction term a for Constant A in the
Soave-Redlich-Kwong EOS (Pedersen et al.
charaterization procedure (Eq. 5.80)
m s + salt mass, m, g
m t+ total-system mass, m, lbm
m ow + pure-water mass, m, g
M+ molecular weight, m/n, lbm/lbm mol
M air+ air molecular weight, m/n, lbm/lbm mol
M b+ boundary molecular weight in gamma distribution
model, m/n, lbm/lbm mol
M Cn)+ C n) molecular weight, m/n, lbm/lbm mol
M C7 + molecular weight of C 7, m/n, lbm/lbm mol
M C7)+ C 7) molecular weight, m/n, lbm/lbm mol
M g+ gas molecular weight, m/n, lbm/lbm mol
M g + surface-gas molecular weight, m/n, lbm/lbm mol
M N+ heaviest C 7) fraction molecular weight,
m/n, lbm/lbm mol
M o+ oil molecular weight, m/n, lbm/lbm mol
M o+ stock-tank oil molecular weight, m/n, lbm/lbm mol
M osp+ molecular weight of separator oil, m/n, lbm/lbm
mol
M P+ molecular weight of paraffin hydrocarbons, m/n,
lbm/lbm mol
n+ moles, n, lbm mol
n c+ number of types of cavities per water molecule in
hydrate crystal lattice, n
n g+ moles of gas, n, lbm mol
n g+ moles of surface gas, n, lbm mol
n L + moles of liquid phase, n, lbm mol
n o+ moles of oil, n, lbm mol
n o+ moles of stock-tank oil, n, lbm mol
n ow + moles of pure water, n, mole
n v+ moles of vapor phase, n, lbm mol
N+ original oil in place, L3, STB; total number of
components, n; last component in a mixture
N C + C 7) approximate carbon number in Standing’s
7)
low-pressure K-value correlation, n
PHASE BEHAVIOR
N H+
N L+
N p+
N sp+
p+
p b+
p cP +
p c+
+
7)
p d+
p i+
p K+
p pc+
p pc+
pc C
p pcHC +
p pr +
pr+
p ref+
p R+
ps+
p sc+
p sp +
p sp1, p sp2+
p st+
p v+
p vw+
p vpw+
p wf+
p(M)=
P+
P c+
Pg +
Po +
P 0+
P 1+
q g, q g +
q gg+
q go+
q o, q o +
q og+
q oo+
Q+
Q cum+
Q d+
Q Mi+
*
+
QMi
Q Wi+
Q zi+
r+
r+
number of heavy ( C 7)) pseudocomponents, n
number of light pseudocomponents, n
cumulative oil produced, L3, STB
number of separator stages, n
pressure, m/Lt2, psia
bubblepoint pressure, m/Lt2, psia
critical pressure of paraffin hydrocarbons,
m/Lt2, psia
critical pressure, m/Lt2, psia
critical pressure of C 7), m/Lt2, psia
dewpoint pressure, m/Lt2, psia
initial pressure, m/Lt2, psia
convergence pressure, m/Lt2, psia
pseudocritical pressure, m/Lt2, psia
pseudocritical pressure adjusted for
nonhydrocarbon content, m/Lt2, psia
pseudocritical pressure of hydrocarbon components
only in a gas, m/Lt2, psia
pseudoreduced pressure, dimensionless
reduced pressure, dimensionless
reference pressure, m/Lt2, psia
average reservoir pressure, m/Lt2, psia
saturation pressure, m/Lt2, psia
pressure at standard conditions, m/Lt2, psia
separator pressure, m/Lt2, psia
primary- and secondary-separator pressure,
m/Lt2, psia
stock-tank pressure, m/Lt2, psia
vapor pressure, m/Lt2, psia
water/brine vapor pressure, m/Lt2, psia
pure-water vapor pressure, m/Lt2, psia
wellbore flowing pressure, m/Lt2, psia
density function of the gamma probability molar
distribution
parachor
capillary pressure, m/Lt2, psi
surface-gas-“component” parachor
stock-tank-oil-“component” parachor
integral (area) of p(M) from h to the
molecular-weight boundary M b
integral (area) of Mp(M) from h to the
molecular-weight boundary M b
total surface-gas production rate, L3/t, scf/D
production rate of surface gas from reservoir gas,
L3/t, scf/D
production rate of surface gas from reservoir oil,
L3/t, scf/D
total stock-tank-oil production rate, L3/t, STB/D
production rate of stock-tank condensate from
reservoir gas, L3/t, STB/D
production rate of stock-tank oil from reservoir oil,
L3/t, STB/D
generic for cumulative production in the
constant-volume-depletion experiment; variable in
saturation pressure algorithm; parameter in gamma
distribution model
cumulative production quantity from
constant-volume-depletion table (produced from
dewpoint pressure)
cumulative production quantity from initial to
dewpoint pressure
cumulative molecular weight, m/n, lbm/lbm mol
normalized cumulative molecular weight variable,
m/n, lbm/lbm mol
cumulative weight fraction
cumulative mole fraction
radius, L
Residual i used in least-squares regression
NOMENCLATURE
r+
r e+
r og+
r p+
r s+
r *s +
r sd +
r w+
R+
R go+
R i+
R p+
Rs +
R *s +
R sd +
R sdb+
R sp +
R spw+
R sp1+
R sw +
R swg+
R s1 +
R s)+
R gg+
R go+
s+
s i+
S+
S g+
SL+
S o+
S v+
S w+
S 0+
T+
DT+
Tb+
T bF +
T br+
T c+
T cP+
Tc C +
7)
T ij+
T pc+
T *pc+
T pcHC+
T pr+
T r+
T rpw+
T sc+
T sp+
average pore radius, L
well external drainage radius, L, ft
oil/gas ratio, L3/L3, STB/scf or STB/MMscf
total producing oil/gas ratio, L3/L3, STB/scf or
STB/MMscf
solution oil/gas ratio, STB/scf or STB/MMscf
modified solution oil/gas ratio, L3/L3, STB/scf or
STB/MMscf
solution oil/gas ratio at dewpoint pressure, STB/scf
wellbore radius of a well, L, ft
universal gas constant+10.73146 psia-ft3/
°R-lbm mol
GOR, L3/L3, scf/STB
fugacity ratio variable for Component i
total producing GOR, L3/L3, scf/STB
solution gas/oil ratio, L3/L3, scf/STB
modified solution gas/oil ratio, L3/L3, scf/STB
differential solution gas/oil ratio, L3/L3,
scf/residual bbl
differential solution gas/oil ratio at bubblepoint,
L3/L3, scf/residual bbl
separator-gas/oil ratio, L3/L3, scf/separator bbl
solution gas/water (pure) ratio, L3/L3, scf/STB
GOR of first-stage separator, L3/L3,
scf/separator bbl
solution gas/water (brine) ratio, L3/L3, scf/STB
solution water/gas ratio, L3/L3, STB/scf or
STB/MMscf
GOR from first-stage separator, L3/L3, scf/STB
solution gas/oil ratio of first-stage separator oil,
L3/L3, scf/STB
surface-gas specific-gravity ratio
stock-tank-oil specific-gravity ratio
skin factor, dimensionless
c ińb i+dimensionless volume-translation (“shift”)
variable used in EOS
sum of mole numbers (fugacity ratio) in
phase-stability test; gamma distribution
model variable
gas saturation, fraction
sum of mole numbers in liquid phase
(phase-stability test)
oil saturation, fraction
sum of mole numbers in vapor phase
(phase-stability test)
water saturation, fraction
specific-gravity correlation variable
temperature, T, °F or °R
hydrate-forming point, T, °F
normal boiling point at 1 atm, T, °R
normal boiling point at 1 atm, T, °F
reduced normal boiling point
critical temperature, T, °R
critical temperature of paraffin hydrocarbons, T, °R
C7+ critical temperature, T, °R
low-pressure diffusion-coefficient-equation
parameter between Component Pair i-j
pseudocritical temperature, T, °R
pseudocritical temperature adjusted for
nonhydrocarbon content, T, °R
hydrocarbon-component pseudocritical temperature
in a gas, T, °R
pseudoreduced temperature
reduced temperature
reduced temperature of pure water
standard condition temperature, T, °F or °R
separator temperature, T, °F
3
T sp1, T sp2+ primary- and secondary-separator
temperature,T, °F
T st+ stock-tank temperature, T, °F
u i+ component molar velocity, n/t, lbm mol/sec
Du i+ logarithm of fugacity ratios used in GDEM
promotion algorithm
v+ molar volume, L3/n, ft3/lbm mol
v c+ critical molar volume, L3/n, ft3/lbm mol
v cP+ critical molar volume of paraffin hydrocarbons,
L3/n, ft3/lbm mol
v g + gas molar volume, L3/n, ft3/lbm mol
v g + gas molar volume at standard conditions, L3/n,
v g^ +379 scf/lbm mol
v *Mi+ modified molar volume, L3/n, ft3/lbm mol
v pc+ pseudocritical molar volume, L3/n, ft3/lbm mol
v pr+ pseudoreduced molar volume
v~r+ reduced molar volume+Vr
v^ + specific volume, L3/m, ft3/lbm
~
v^ w+ specific volume of brine, L3/m, cm3/g
v^ *w+ brine specific volume at reservoir pressure and
temperature without solution gas, L3/m, cm3/g
~
v^ wsc+ brine specific volume at standard pressure and
reservoir temperature without solution gas,
L3/m, cm3/g
V+ volume, L3, ft3 or bbl
V+ average volume, L3, ft3 or bbl
V c+ critical volume, C 2) L3, ft3 or bbl
V C + ideal-solution liquid volume of C 2) components
2)
V C + ideal-solution liquid volume of C 3) components
3)
V cell+ original cell volume at saturation pressure in a PVT
experiment, L3, ft3
V g+ gas volume, L3, ft3 or bbl
V g+ surface-gas volume, L3, scf
V o+ oil volume, L3, ft3 or bbl
V o+ stock-tank oil volume, L3, STB
V ob+ bubblepoint oil volume, L3, ft3 or bbl
V oi+ initial oil volume, L3, ft3 or bbl
V or + residual oil volume, L3, ft3 or bbl
V or + residual oil volume at reservoir temperature from
differential-liberation experiment, L3, residual bbl
V osp+ separator-oil volume, L3, bbl
V pHC+ hydrocarbon pore volume (HCPV), L3,
ft3 or res bbl
V r + reduced volume, L3, ft3 or bbl
V R+ reservoir oil volume, L3, ft3 or bbl
V ro + oil volume/oil volume at saturation pressure
V rt+ total (gas)oil) volume relative to
saturation volume
V s + reservoir oil volume at saturation pressure,
L3, ft3 or bbl
V t+ total (gas)oil) volume, ft3 or bbl
V w+ water volume, L3, ft3 or bbl
w i+ weight fraction
w g+ surface-gas weight fraction
w o+ stock-tank-oil weight fraction
W i+ Gaussian quadrature weight factor
x+ coordinate direction
x g, x g+ surface-gas-“component” mole fraction in
reservoir oil
x i+ Component i mole fraction in oil phase
x MEOH+ methanol inhibitor mole fraction
x o, x o+ stock-tank-oil-“component” mole fraction in
reservoir oil
x vi+ Component i volume fraction
X i+ Gaussian quadrature point
y+ Hall-Yarborough Z-factor correlation
reduced-density parameter
4
y g, y g+ surface-gas-“component” mole fraction in
reservoir gas
y i+ Component i mole fraction in gas phase or
incipient phase
y j i+ fraction (probability) of Type j molecule occupying
Type i cavity
y o, y o+ stock-tank-condensate-“component” mole fraction
in reservoir gas
y w+ water mole fraction in reservoir gas
y pw+ pure-water mole fraction in gas phase
Y+ function for smoothing two-phase (gas/oil)
volumetric data below bubblepoint during
constant-composition-expansion experiment
Y a + Yarborough aromaticity factor, fraction
Y i+ Component i mole number
z Cn + mole fraction of first carbon number component in
a C n) fraction
z C + C 6 mole fraction in overall mixture
6
z C + C 7 mole fraction in overall mixture
7
z C + C 7) mole fraction in overall mixture
7)
z i+ Component i mole fraction in overall mixture
z ref+ reservoir mole fraction at reference depth
Z+ compressibility, or “deviation,” factor
Z c+ critical Z factor
Z d + dewpoint pressure Z factor
Z L+ liquid-phase Z factor
Z R+ Rackett Z factor for calculating saturated
liquid densities
Z v+ vapor phase Z factor
Z 2 + two-phase Z factor
a+ correction term to Constant A in EOS’s;gamma
distribution model parameter; Hall-Yarborough
equation parameter for the Standing-Katz Z-factor
chart; Twu property correlation parameter
a w+ Constant A correction term in Peng-Robinson EOS
for water/brine
b+ Constant B correction term in the
Zudkevitch-Joffe-Redlich-Kwong EOS; parameter
in the gamma distribution model; solution vector in
Newton-Raphson solution of the Michelsen
two-phase isothermal flash (Eq. 4.54)
b *+ parameter in the modified gamma distribution
model used with Gaussian quadrature
g+ specific gravity, air+1 or water+1
g API+ (141.5/go )*131.5, oil gravity, °API
g C + C 7) specific gravity, water+1
7)
g g , g g + gas specific gravity, air+1
g , g g + total average gas specific gravity, air+1
g
g gc + corrected separator gas specific gravity for Vazquez
correlations, air+1
g gg+ specific gravity of surface gas from reservoir
gas, air+1
ggHC+ gas specific gravity of hydrocarbon components in
a gas mixture, air+1
g go+ specific gravity of surface gas from reservoir
oil, air+1
g g1+ first-stage separator-gas specific gravity, air+1
g g)+ specific gravity of gas released from first-stage
separator oil, air+1
g o , g o + stock-tank oil specific gravity, water+1
g og+ specific gravity of stock-tank condensate from
reservoir gas, water+1
g oo+ specific gravity of stock-tank oil from reservoir
oil, water+1
g P + specific gravity of paraffin hydrocarbons, water+1
g w+ wellstream (reservoir gas) specific gravity, air+1;
brine specific gravity, water+1
Dg M+ parameter in the Twu molecular-weight correlation
PHASE BEHAVIOR
Dg P+ parameter in the Twu critical-pressure correlation
Dg T + parameter in the Twu critical-temperature
correlation
Dg v+ parameter in the Twu critical-volume correlation
G+ gamma function
d+ parameter in the modified gamma distribution
model used with Gaussian quadrature
D+ deviation
e+ parameter used in the Wichert-Aziz
nonhydrocarbon correction method for
pseudocritical properties
e/k+ Leonard-Jones 12-6 potential parameter, K
h+ gamma distribution model parameter (minimum
molecular weight), m/n, lbm/lbm mol
q+ generic symbol for any component property; Twu
property correlation parameter
Q+ generic property of “grouped” Pseudocomponent I,
where I contains “original” Components i (iŮI);
e.g., molecular weight MI (Eqs. 5.82 through 5.94)
l 1, l 2 + eigenvalues
m+ dynamic viscosity, m/Lt2, cp
m g+ gas viscosity, m/Lt2, cp
m gsc+ low-pressure gas viscosity at specified temperature,
m/Lt2, cp
m i+ low-pressure gas viscosity of Component i at
specified temperature, m/Lt2, cp
m o+ oil viscosity, m/Lt2, cp
m ob+ bubblepoint (saturated) oil viscosity, m/Lt2, cp
m oD+ dead (degassed) oil viscosity at standard pressure
and specified temperature, m/Lt2, cp
m w+ water viscosity, m/Lt2, cp
m pw+ pure-water viscosity at standard pressure and
specified temperature, m/Lt2, cp
(m pw) 20°C+ pure-water viscosity at standard pressure and 20°C,
m/Lt2, cp
m *w+ water/brine viscosity at standard pressure and
specified temperature, m/Lt2, cp
m cwH+ water chemical potential of water in filled hydrate,
m/Lt2, psia
m cwMT+water chemical potential in empty hydrate,
m/Lt2, psia
m 1, m 2 + GDEM-promotion eigenvalue parameters
D m CO + low-pressure gas-viscosity correction for CO2
2
D m H S+ low-pressure gas-viscosity correction for H2S
2
D m N + low-pressure gas-viscosity correction for N2
2
n+ kinematic viscosity, L2/t, cSt
c+ Lucas gas-viscosity correlation parameter, cp*1
c T+ Thodos (Lohrenz-Bray-Clark) gas viscosity
correlation parameter, cp*1
ò+ mass density, m/L3, lbm/ft3 or g/cm3
ò air+ air density, m/L3, lbm/ft3
ò C + C 1 apparent pseudoliquid density at standard
1
conditions, m/L3, lbm/ft3
ò C + C 2 apparent pseudoliquid density at standard
2
conditions, m/L3, lbm/ft3
NOMENCLATURE
òC
+ C 2) pseudoliquid density at standard conditions,
2)
m/L3, lbm/ft3
ò g+ gas density, m/L3, lbm/ft3
ò g+ surface-gas density, m/L3, lbm/ft3
ò ga+ separator-gas apparent pseudoliquid density,
m/L3, lbm/ft3
ò i + liquid density of Component i at standard
conditions, m/L3, lbm/ft3
ò o+ oil density, m/L3, lbm/ft3
ò o+ stock-tank oil density, m/L3, lbm/ft3
ò ob+ bubblepoint oil density, m/L3, lbm/ft3
ò osp + separator-oil density, m/L3, lbm/ft3
ò M+ molar density, n/L3, lbm mol/ft3
ò Mc+ critical molar density, n/L3, lbm mol/ft3
ò Msc+ low-pressure molar density, n/L3, lbm mol/ft3
ò pij+ partial density of surface Phase i produced from
reservoir Phase j, m/L3, lbm/ft3
ò po+ pseudoliquid density, m/L3, lbm/ft3
ò pr+ pseudoreduced density
ò r+ reduced density
ò ref+ reference density (air or water), m/L3, lbm/ft3
ò sL+ saturated-liquid density, m/L3, lbm/ft3
ò w+ saturated-brine density, m/L3, g/cm3
ò *w+ water/brine density at reservoir pressure and
temperature without solution gas, m/L3, g/cm3
ò wsc+ brine density at standard pressure and reservoir
temperature without solution gas, m/L3, g/cm3
Dò p + density/pressure correction for Standing-Katz oil
density correlation, m/L3, lbm/ft3
Dò T+ density/temperature correction for Standing-Katz
oil density correlation, m/L3, lbm/ft3
Dò wH+ density difference between water/brine and the
hydrocarbon phase, m/L3, g/cm3
s+ interfacial tension (IFT), m/t2, dynes/cm
s lim + limiting hydrocarbon/water IFT at Dò wH+0,
m/t2, dynes/cm
s go + gas/oil IFT, m/t2, dynes/cm
s i j+ Leonard-Jones 12-6 potential parameter, Å
s wH+ water/hydrocarbon IFT, m/t2, dynes/cm
t+ sheer stress, m/Lt2, psi
f+ porosity
f i+ fugacity coefficient for Component i; generalized
weighting factor for mixing rule
(f i) w + fugacity coefficient for Component i in brine
(f i) pw + fugacity coefficient for Component i in pure water
w+ acentric factor
W a, W b+ constants in cubic EOS’s
W oa, W ob+ numerical constants in cubic EOS’s
W i j+ low-pressure diffusion-coefficient-correlation
parameter
Superscripts
o+ low pressure
5
Author Index
A
Abbott, M.M., 48, 49, 66
Abdul-Majeed, G.H., 38, 45
Abou-Kassem, J.H., 24, 44
Abramowitz, M., 86
Abu-Khamsin, S.A., 38, 45
Agarwal, R., 67
Ahmed, T., 45
Al-Khafaji, A.H., 36, 38, 45
Al-Marhoun, M.A., 38, 45
Alani, G.H., 3, 33–35, 45
Amirijafari, B., 144, 160
Amyx, J.W., 44, 108
Andersen, T.F., 86, 208
Austad, T., 69, 70, 72, 73, 78, 86
Auxiette, G., 140
Aziz, K., 24, 25, 37, 44, 45, 66, 177, 192, 224
B
Baker, L.E., 56–59, 66, 141
Bardon, M.F., 79, 86
Bass, D.M. Jr., 44, 108
Bath, P.G.H., 67
Batycky, J.P., 84, 87
Beal, C., 36–38, 45
Bedrikovetsky, P.G., 64, 67
Beggs, H.D., 24, 30, 35–38, 44, 45
Behrens, R.A., 84, 87
Belery, P., 40, 44, 64, 65, 67
Benedict, M., 4, 80, 86
Benham, A.L., 126, 131, 132, 141
Bergman, D.F., 36, 37
Beu, K.L., 3
Bhagia, N.S., 3
Bicher, L.B. Jr., 1, 3
Boe, A., 120
Borthne, G., 117, 120
Bray, B.G., 45, 72, 86, 175–77, 192, 206, 208, 224
Brill, J.P., 24, 44
Brinkman, F.H., 42, 46
Brown, G.G., 3, 13, 17, 108
AUTHOR INDEX
Brulé, M.R., 73, 83, 86
Buchanan, R.D. Jr., 141
Bucklin, R.W., 157, 161
Burrows, D.B., 45
C
Campbell, J.M., 43, 44, 46, 144, 152, 153, 155, 157, 160, 161
Canfield, F.B., 43, 46
Carlson, H.A., 3
Carr, N.L., 26, 27, 45
Carroll, J.J., 158
Carson, D.B., 2, 154, 155, 161
Cavett, R.H., 80, 81, 86
Chaback, J.J., 64, 67
Chaperon, I., 140
Chen, C.-J., 161
Chew, J.N., 37, 38, 45, 183, 192
Chien, M.C.H., 45
Chierici, G.L., 146, 160
Cho, S.J., 104, 108
Chorn, L.G., 86
Chou, J.C.S., 143, 146, 159
Christman, P.G., 140
Christoffersen, K., 44, 45
Chueh, P.L., 83, 84, 87, 195, 208
Civan, F., 108
Clark, C.R., 45, 72, 86, 175–77, 192, 206, 208, 224
Clark, G.C., 17, 46
Clark, N.J., 94, 108
Clever, H.L., 145, 160
Coats, K.H., 4, 44, 65, 66, 67, 84, 85, 86, 87, 113, 119
Collins, A.G., 147, 160
Connally, C.A., 37, 38, 45, 183, 192
Cook, A.B., 127, 128, 141
Cook, R.E., 119, 208
Correia, R.J., 160
Costain, T.G., 133, 140, 141
Craft, B.C., 35, 45, 86, 108, 158, 213
Cragoe, C.S., 26, 29, 44, 113, 120, 178, 192
Creek, J.L., 64, 67
Cronquist, C., 16, 17, 119
Crowe, A.M., 66
1
Crump, J.S., 45, 86, 108, 189, 192
Culberson, O.L., 143, 159, 160
Cullick, A.S., 34, 45
D
da Silva, F.V., 40, 44, 64, 67, 120
Dake, L.P., 108, 119
Dalen, V., 66
Daubert, T.E., 77–82, 86
David, R.A., 2, 17
de Jong, L.N.J., 67
Deaton, W.M., 154, 161
Delclaud, J., 44
Dempsey, J.R., 26, 45
DeRuiter, R.A., 72, 86
Dindoruk, B., 141
Dixon, T.N., 119
Dodson, C.R., 97, 108, 143, 145, 147–49, 151, 159, 192
Donnelly, H.C., 3
Donohoe, C.W., 141
Dougherty, E.L. Jr., 64, 67
Dowden, W.E., 141
Dranchuk, P.M., 24, 44
Drickamer, H.G., 64, 67
Drohm, J.K., 108, 119, 120
E
Eakin, B.E., 45, 192
Earlougher, R.C. Jr., 171
Edmister, W.C., 12, 17, 42, 46, 66, 81, 86
Eilerts, C.K., 1, 3, 11, 14, 17, 26, 28, 44, 45
Ely, J.F., 44
Enick, R.M., 151, 158
Erbar, J.H., 70, 86
Ericksen, 161
F
Faissat, B., 64, 67
Farshad, F.F., 30, 36, 38, 45
Fayers, J.F., 141
Fetkovich, M.D., 120
Fetkovich, M.J., 44, 158
Fevang, Ø., 17
Fick, A., 21, 44
Firoozabadi, A., 39, 45, 68–72, 77, 86, 149, 158, 160
Fiskin, J.M., 3
Flaitz, J.M., 108
Forgarasi, M., 45
Fowler, W.N., 67
Francis, R.J., 17, 46
Fredenslund, A., 66, 67, 86, 87, 208
Freze, R., 66, 87
Frost, E.M., 154, 161
Fuller, G.G., 66
Fussell, D.D., 141
G
Gaddy, V.L., 159
Galimberti, M., 43, 44, 46
Gardner, J.W., 141
2
Gibbs, J.W., 1, 2, 49, 52–64, 67
Glasø, O., 29, 30, 36, 37, 43, 45, 46
Glass, E.D., 46
Golan, M., 120
Gold, D.K., 44, 45
Golding, B.H., 28, 45, 86
Goldthorpe, W.H., 108, 119, 120
Gonzalez, M.H., 26, 45, 182, 192
Goodrich, J.H., 140
Goodwill, D., 108
Gorell, S.B., 140
Gouel, P.L., 64, 67
Govier, G.W., 45
Graue, D.J., 136, 141
Griewank, A.K., 45
H
Haaland, S., 86
Haas, J.L. Jr., 150, 158
Hachmuth, K.K., 2
Hadden, S.T., 45
Hall, K.R., 23, 24, 44, 82, 86, 175, 177, 192, 223
Haman, S.E.M., 50, 66
Hammerschmidt, E.G., 151, 157, 161
Hankinson, R.W., 34, 45
Hanley, H.J.M., 44
Harvey, A.H., 161
Harvey, M.T., 140
Hassoon, S.F., 45
Hawkins, M., 35, 45, 86, 108, 158, 213
Heidemann, R.A., 66
Hicks, B.L., 3
Hildebrand, M.A., 161
Hinds, R.F., 17, 105, 108
Hirschberg, A., 63, 67
Hocott, C.R., 45, 86, 108, 149, 160, 189, 192
Hoffmann, A.E., 45, 86, 108, 189, 192, 220
Holder, G.D., 158, 161
Holland, C.J., 145, 160
Holm, L.W., 125, 137, 138, 140, 141
Holt, T., 63, 67
Hooper, H.H., 158
Hou, Y.C., 48, 66
Hutchinson, C.A. Jr., 124, 140
J
Jacoby, R.H., 78, 79, 86, 119, 208, 221
Jennings, J.W., 44, 45
Jensen, F., 141
Jensen, J.I., 66
Jhaveri, B.S., 51, 52, 66, 83, 87
Joffe, J., 47, 50, 66, 83, 87, 223
John, V.T., 156, 161
Johns, R.T., 128, 141
Josendal, V.A., 125, 137, 138, 140
Jossi, J.A., 38, 45
K
Kattan, R.R., 45
Katz, D.L., 1–3, 5, 9–11, 13, 14, 16, 17, 23–26, 28, 30–34, 38, 39, 44, 45,
68–72, 77, 86, 90, 108, 140, 143, 148, 149, 151, 152, 154, 155,
159–61, 177, 179, 181, 183, 190, 192, 205, 206, 208, 223, 224
PHASE BEHAVIOR
Kawanaka, S., 140, 141
Kay, W.B., 11, 13, 17, 19, 24, 25, 38, 40, 44, 85, 87, 93, 108, 140
Kelm, C.H., 140
Kennedy, G.C., 3, 141
Kennedy, H.T., 1, 3, 28, 33–35, 45, 108
Kennedy, J.T., 141
Kesler, M.G., 24, 44, 71, 79, 80–84, 86, 194, 208
Kestin, J., 147, 160
Khalifa, H.E., 160
Khan, S.A., 38, 45
Kistenmacher, H., 151, 161
Klins, M.A., 140
Kniazeff, V.J., 118, 119
Kobayashi, R., 3, 45, 143, 156, 159, 161
Kobe, K.A., 160
Koch, H.A. Jr., 124, 140
Kuenen, J.P., 11, 17
Kumar, K.H., 47, 66
Kunzman, W.J., 141
Kuo, S.S., 132, 140
Kurata, F., 2, 10, 11, 17, 28, 45
Kutasov, I.M., 146, 160
Kwong, J.N.S., 1, 4, 47, 48–51, 63, 64, 66, 221, 223
McLeod, H.D. Jr., 149, 153, 155, 161
McRee, B.C., 141
Mehra, R.K., 66
Mehta, B.R., 156, 161
Merrill, R.C. Jr., 87
Metcalfe, R.S., 64, 67, 125, 127, 128, 136, 138, 140, 141
Michel, S., 158
Michelsen, M.L., 44, 46, 47, 54, 55, 57–67, 141, 151, 156, 161,
220, 221, 223
Mohamed, R., 158
Monger, T.G., 140, 141
Monroe, R.R., 3, 45
Monroy, M.R., 45
Montel, F., 64, 67
Morris, R.W., 67
Moses, P.L., 17, 108, 141
Muckleroy, J.A., 1, 2
Mullen, N.B., 3
Muller, H.G., 152, 161
Munck, J., 156, 161
Murphy, G.B., 86
Muskat, M., 1, 4, 53, 63, 64, 66, 67, 183, 185, 186, 192, 220
N
L
Lacey, W.N., 3, 63, 67, 140
Lasater, J.A., 29, 30, 45
Lawsa, W.F., 161
Lee, A.L., 26, 45, 182, 192
Lee, B.I., 12, 17, 24, 44, 66, 71, 79–81, 83, 84, 86, 194, 208
Lee, R.L., 161
Lee, S.T., 84, 85, 87
Lein, C.L., 141
Leshikar, A.G., 45
Li, Y.-K., 59, 61, 66, 67, 84, 86, 87, 158
Lindeberg, E., 67
Little, J.E., 3
Lo, T.S., 119, 120
Lohrenz, J., 17, 33, 38, 43, 45, 46, 72, 86, 175–77, 192, 206, 208, 224
Long, G., 146, 160
Lucas, K., 27, 28, 38, 45, 175, 176, 182, 192, 224
Ludecke, D., 161
Luks, K.D., 66, 141
M
MacAllister, D.J., 72, 86
Macleod, D.B., 38, 39, 45
Maddox, R.N., 70, 86
Madrazo, A., 31, 45
Makogon, Y.F., 161
Malone, R.D., 161
Mannan, M., 44
Mansoori, G.A., 141
Markham, A.E., 160
Martin, J.J., 47, 48, 51, 66
Mather, A.E., 158
Matthews, T.A., 2, 25, 44
Mayer, E.H., 108
McAuliffe, J.C., 66
McCain, W.D. Jr., 26, 44, 45, 86
McDowell, J.M., 1, 4, 53, 66, 183, 185, 186, 192, 220
McKetta, J.J. Jr., 143, 147, 148, 159, 160
AUTHOR INDEX
Nagy, Z., 49, 66, 171
Naville, S.A., 118, 119
Nectoux, A., 44
Nelson, E.F., 86
Nemeth, L.K., 3, 28, 45
Newley, T.M.J., 87
Ng. H.-J., 156, 161
Ng, H.-Y., 66
Nghiem, L.X., 61, 62, 66, 67, 87, 158
Nielsen, R.B., 157, 161
Nishio, M., 66
Nokay, R., 39, 45, 80, 86
Novosad, Z., 133, 140, 141
Nuttaki, R., 158
O
O’Brian, L.J., 137, 141
O’Leary, 141
Olds, R.H., 3, 28, 45
Olson, C.R., 108
Opfell, J.B., 46
Organick, E.I., 28, 45, 86
Orr, F.M. Jr., 125, 126, 128, 140, 141
P
Panagiotopoulos, A.Z., 151, 161
Papadopoulos, K.D., 161
Park, S.J., 141
Parks, A.S., 3, 108
Parrish, W.R., 156, 161
Patel, P.D., 141
Patel, V.C., 66
Patton, C.C., 158
Pawlikowski, E.M., 145, 160, 192
Pebdani, F.N., 45
Pedersen, K.S., 49, 66, 67, 83, 84, 86, 87, 200–08, 221
Peneloux, A., 48, 51, 64, 66, 83, 87
3
Peng, D.Y., 1, 4, 47, 50, 51, 63, 64, 66, 83, 86, 124, 140, 150, 160, 185,
192, 193, 202, 208, 223
Perschke, D.R., 141
Peterson, A.V., 141
Pierce, A.C., 66
Pitzer, K.S., 81, 82, 86, 147, 160
Poettman, F.H., 3
Polling, B.E., 44, 66, 171
Pope, G.A., 141
Powers, J.E., 161
Prausnitz, J.M., 44, 66, 83, 84, 86, 87, 145, 156, 158, 160, 161, 171, 192,
195, 208
R
Rachford, H.H., 52–55, 66, 183, 184, 186, 192, 221
Rackett, H.G., 34, 39, 45, 223
Ramesh, A.B., 119, 208
Ramey, H.J. Jr., 39, 40, 44, 45, 149, 160
Rao, V.K., 79, 86
Rasmussen, P., 161
Ratkje, S.K., 67
Rauzy, E., 66, 87
Raynal, M., 108
Razsa, M.J., 86
Reamer, H.H., 3, 108
Redlich, O., 1, 4, 47, 48, 49, 50, 51, 63, 64, 66, 221, 223
Reese, D.E., 44, 158
Reid, R.C., 28, 44, 49, 66, 82, 86, 151, 161, 171
Renner, T.A., 40, 45, 141
Reudelhuber, F.O., 17, 105, 108
Riazi, M.R., 71, 77, 78, 79, 80, 81, 82, 86
Rice, J.D., 52, 53, 54, 55, 66, 183, 184, 186, 192, 221
Riemens, W.G., 63, 67
Risnes, R., 66
Robinson, D.B., 1, 4, 47, 50, 51, 63, 64, 66, 83, 86, 124, 140, 150, 156,
160, 161, 185, 192, 193, 202, 208, 223
Robinson, J.R., 36, 37, 45
Rochon, J., 44
Roess, L.C., 80, 86
Rogers, P.S.Z., 147, 160
Roland, C.H., 2, 43, 44, 46
Rosman, A., 45
Rowe, A.M. Jr., 44, 45, 46, 143, 146, 159
Rubin, L.C., 4, 80, 86
Russell, M.P.M., 67
Rzasa, M.J., 1–3, 44, 46
Shirkovskiy, A.I., 49, 66, 171
Sibbald, L.R., 140, 141
Sicking, J.N., 42, 46
Sigmund, P.M., 40, 45
Silvey, F.C., 108
Simon, R., 38, 45, 136, 141
Singleterry, C.C., 2, 17
Siu, A., 87
Skjaeveland, S., 120
Skjold-Jorgensen, S., 161
Sloan, E.D., 155, 156, 161
Smart, G.T., 65, 66, 67
Soave, G., 1, 4, 47, 49, 50, 51, 63, 64, 66, 200, 202, 205, 208, 221
Song, K.Y., 156, 161
Søreide, I., 67, 79, 83, 86, 120, 144, 145, 150, 158, 193, 201–08
Spencer, G.C., 141
Spivak, A., 119
Stalkup, F.I. Jr., 130, 138, 140
Standing, M.B., 1–3, 23–27, 29–39, 42–46, 90, 91, 94, 95, 108, 143, 145,
147–49, 151, 159, 160, 172, 177, 179–83, 189, 190, 192, 205, 206,
208, 221, 223, 224
Starling, K.E., 1, 4, 24, 44, 47, 66, 83, 86, 108, 161
Stegun, I.A., 86
Stephenson, R.E., 4
Stiel, L.I., 38, 45
Sutton, R.P., 24, 25, 30, 36, 38, 44, 45, 160, 175, 178, 192, 205, 208
T
Takacs, G., 24, 44
Tang, D.E., 119, 120
Teja, A.S., 66
Terry, R.E., 86
Thodos, G., 38, 45, 141, 224
Thomassen, P., 66, 67, 86, 87, 208
Thomson, G.H., 34, 45
Tindy, R., 108
Torp, S.B., 42–44, 46, 105, 108, 111, 112, 119, 207, 208
Trainer, R.P., 3, 45
Trangenstein, J.A., 66
Trekell, R.E., 155, 161
Trengove, R., 108, 120
Trube, A.S., 35, 45
Trujillo, D.E., 140, 141
Turek, E.A., 67, 141
Twu, C.H., 82, 83, 86, 194, 208, 221, 223, 224
U
S
Saeterstad, T., 161
Sage, B.H., 1, 3, 28, 45, 63, 67, 108, 140
Salman, N.H., 45
Saltman, W., 2, 39, 45
Sandler, S.I., 84, 86, 87
Savidge, J.L., 44
Schaafsma, J.G., 3, 140
Schlijper, A.G., 84, 87
Schmidt, G., 52, 66
Schrader, 64, 67
Schroeder, G.M., 66
Schroeter, J.P., 156, 161
Schulte, A.M., 4, 63, 67
Sepehrnoori, K., 141
Shelton, J.L., 134, 135, 140, 141
4
Unruh, C.H., 3, 161
Usdin, E., 66
V
van der Burgh, J., 67
van der Waals, J.D., 1, 2, 24, 33, 44, 47, 48, 51, 66
Vazquez, M., 30, 35, 36, 38, 45
Vink, D.J., 2, 17
Vogel, J.L., 32, 45, 67, 126, 141
von Stackelberg, M., 152, 161
W
Walter, C.J., 141
PHASE BEHAVIOR
Wang, Y., 128, 141
Watson, K.M., 77, 78, 81, 86
Webb, G.B., 4, 80, 86
Webster, D.C., 3
Wehe, A.H., 147, 148, 160
Weinaug, C.F., 3, 38, 39, 45
Wenzel, H., 52, 66
Wheaton, R.J., 64, 67
Whiting, R.L., 44, 108
Whitson, C.H., 4, 17, 26, 30, 42–46, 54, 65–67, 73, 74, 76, 78, 79, 83–86,
105, 108, 111, 112, 119, 120, 144, 145, 150, 158, 201–04, 207, 208
Wichert, E., 24, 25, 44, 177, 192, 224
Wiebe, R., 159
Wilcox, W.I., 2, 154, 161
Wilke, C.R., 40, 45
Williams, B., 3
Wilson, G.M., 42, 46, 53, 54, 58, 66, 183–87, 192
Wilson, K., 141
Woods, R.W., 119
Wu, R.S., 84, 87
AUTHOR INDEX
Y
Yale, W.D., 3
Yarborough, L., 4, 23, 24, 32, 44, 45, 47, 50, 66, 79, 82, 86, 126, 134, 135,
141, 175, 177, 192, 223
Yellig, W.F., 125, 136, 138, 140
Young, L., 66
Young, L.C., 4
Youngren, G.K., 51, 52, 66, 83, 87, 119, 120
Ypma, J.G.J., 67
Yu, A.D., 141
Z
Zana, E., 45
Zhou, D., 126, 141
Zick, A.A., 4, 53, 55, 63, 67, 119–21, 124, 126, 128, 132, 133, 140
Zudkevitch, D., 47, 50, 66, 83, 87, 223
5
Subject Index
A
Absolute zero, 167
Acentric factor, 81, 162, 163, 194
Air density, 167
Alani-Kennedy method, 33
Algorithms, 53
Flash calculations, 53
Gravity/chemical equilibrium, 64
Michelsen stability test, 57
Minimum miscibility pressure, 127
Newton-Raphson, 53, 72
Vapor/liquid equilibrium (VLE), 47, 139
Alkanes, 7, 38, 82
American Soc. for Testing Materials
(ASTM), 69
Antifreezes, 157
API Research Projects, 6
Aromaticity factor, 78, 79
Asphaltene
Chemical structure, 9
Precipitation, 134, 139
Atomic mass, 18
Aziz correlation, 37
B
Beggs-Robinson correlation, 37
Bergman correlation, 37
Binary-interaction parameters (BIP’s), 49, 150,
164, 193, 195, 202
Binary mixtures
Critical locus, 122
Gibbs energy surfaces, 56–61
Phase equilibria, 56
p-T phase envelope, 54
Black oil, 13, 15
Composition, 6
PVT formulations
Modified, 110, 116, 200, 207
Traditional, 109, 116
PVT properties, 20, 109
Boiling points, 162, 163
Correlation, 81
Heptanes-plus fractions, 79, 193
n-alkanes, 7, 8
North Sea condensate, 69
Bottomhole oil, 88
Brines
Composition, 143
Gas/oil ratio, 21
Properties (Problem 22), 190
SUBJECT INDEX
Bubblepoint curves, 11, 13, 15
Bubblepoint oil
FVF, 35
Viscosity, 37
Bubblepoint pressure, 29, 210
Calculation (Problems 12 and 16), 180, 183
C
Campbell’s calculation methods, 155
Carbon dioxide, 25
Diffusivity, 40
Flooding, 121
Hydrate-formation conditions, 161
Injection, 135
MMP correlation, 138
Physical properties, 134, 135
Slim-tube displacement, 125
Carbon-12 standard, 18
Carr correlation, 27
Chemical compounds, 18
Chemical potential, 47
Clathrates, 151, 156
Color change, 125
Component fractions, 19
Compositional correlation, 38
Compositional gradients, 63
Compositional relations, 114
Compressibility
Brine, 145, 190
Gas, 23
Gas (Problem 6), 174
Isothermal, 20, 210
Saturated oil, 36, 94
Undersaturated oil, 35
Computer programs
CSMHYD, 156
GAMSPL, 74, 75, 165
UNIQUAC, 156
Constant composition expansion (CCE), 88
Gas condensate, 94
Oil, 93
Constant volume depletion (CVD), 97
Consistency check, 105
Gas condensate, 102–07, 195–200
Convergence pressure, 44, 184
Calculation (Problem 17), 184
Correlations, 18
Corresponding states theory, 18
Cricondenbar, 11
Cricondentherm, 11
Critical constants, 162, 163
Critical pressure, 80, 82
Critical properties, 18
Rules for calculation, 84
Critical temperature, 80, 82
Critical volume, 82
Crude oil
California, 29, 35
Composition and properties, 6
h series, 9
Hydrocarbon classes, 8
Simulated distillation, 72
Crystallography, of hydrates, 152
D
Dead stock-tank oil
Saturation with CO2, 136
Viscosity, 36
Density, 19
Air, 167
Brines, 190
Carbon dioxide, 134, 135, 138
Conversion factors, 168
Gas, 22
Liquid, 162
Oil, 30, 34, 113, 130, 179
Partial, 118
Water, 167
Depletion reservoirs, 12, 15, 97, 110
Dewpoint curves, 11, 13, 15
Dewpoint pressure, 28, 102, 131, 195
Diatomic compounds, 18
Differential liberation expansion (DLE), 88
Laboratory procedure, 95
Oil sample, 98–101
Oil volumetric properties, 126
Raleigh report, 211
Reservoir oil, 203–07
Diffusion coefficients, 21, 38, 40
Conversion factors, 171
Distillation, 68
Dry gas, 13
Composition, 6
FVF, 111, 113
E
Earth’s gravitational acceleration, 167
Elements, chemical, 18
Equations of state
Applications, 193
Composition calculations, 115
1
Critical-properties estimation, 82
Cubic, 47, 151
Matching to measured data, 65
Multiple-contact PVT experiments, 126
Oil-gravity calculations, 112
Peng-Robinson, 50, 150, 156, 164, 185, 193,
196, 202
Predictions, 195
Redlich-Kwong, 48
Slim-tube profiles, 123, 132
Soave-Redlich-Kwong, 49, 83, 164, 200, 202
Solubility predictions, 150–54
Ternary system (problem 18), 185
Two-phase flash algorithm, 53
van der Waals, 48
Water/hydrocarbon systems, 146
Zudkevitch-Joffe-Redlich-Kwong, 50
Ethane
Density, 30
p-V diagram, 11
Ethane/n-heptane system, 11, 13, 122
F
Fick’s law, 21
Field shrinkage factor, 88
Flash calculations, 52, 53, 89, 184, 212
Problem 17, 184
Formation volume factor (FVF)
At less than bubblepoint pressure, 210, 212
Brine, 146, 191
Bubblepoint oil, 35, 91
Carbon dioxide, 135
Dry gas, 111, 113
Gas, 20, 109
Nitrogen/oil mixture, 129
Oil, 20, 99, 109, 111
Separator oil, 90
Total, 28
Water, 20
Water/brine, 145
Wet gas, 113
Formation-water properties, 142
Fugacity, 49
Calculation (Problem 18), 187
G
Galimberti-Campbell method, 43
Gamma-distribution model, 73
Gas
Composition, 114
Density, 22, 113
FVF, 20, 109, 113
Gravity, 22, 111
High-sulfur-content (Problem 2), 172
Phase behavior, 5
Properties, pseudocritical, 24
Problem 8, 177
Properties and correlations, 18
Properties (Problem 1), 172
Volumetric properties, 5, 22
Gas cap, 207
Gas chromatography, 68, 70, 89, 130
Gas-condensate
Boiling points, 69
Composition, 6
Constant composition expansion, 94, 97
Constant volume depletion, 102–07
Effect of nitrogen, 129
Fluid characterization, 193
Isotherms, 14
Material-balance calculations, 92
MBO properties, 118
p-T diagrams, 13
2
PVT analysis, 88
Retrograde region, 14
Stepwise-regression procedure, 85
Gas constant, 167
Gas cycling, 130
Gas injection
Methods, 121
Modifications, 119
Gas mixtures, 22
Gas/oil ratio (GOR), 13, 21, 109
CO2/oil system, 136
Separator test, 88, 91, 100
Gasoline properties (Problem 3), 173
Gas solubility, 143
Calculation (Problem 22), 190
Gas viscosity, 26
Calculation
Problem 7, 175
Problem 14, 182
Gaussian quadrature functions, 77, 193
General dominant eigenvalue method
(GDEM), 54
Gibbs energy surfaces, 56–61
Gibbs free energy, 52
Gibbs phase rule, 8
Gravity/chemical equilibrium (GCE), 63
Greek alphabet, 166
Gross heating value, 162, 163
H
Hammerschmidt’s equation, 157
Henry’s law, 143
Heptanes-plus (C7+) fractions
Acentric factor, 195
Boiling points, 193
Characterization, 68
Critical properties, 194
Gas cycling, 130
Liquid-dropout curve, 108
Molar distribution, 193
Pseudocritical properties, 25
Single carbon number, 71
Specific gravity, 193
Hoffman method, 41
Hydrate formation
Calculation methods, 154–56, 190
Calculation (Problem 21), 190
Hydrates, 151
Crystallography, 152
Inhibition, 157
Phase diagrams, 153
Hydrocarbons
Component properties, 162–63
Crude oil, 8
Heavy, 121, 138
Hydrate-former constants, 155, 160
Intermediate, 121, 129, 132
Light, 121, 138, 145, 172
/nonhydrocarbon component pairs, 164
Parachors, 39
p-T diagrams, 9, 13
Ternary system, 122–24
/water systems, 142
Hydrogen sulfide, 25, 172
I
Ideal gas law, 22
Ideal liquid yield, 162, 163
Immiscible CO2/oil behavior, 136
Inflow-performance relations (IPR’s), 116, 117
Inspection-properties estimation, 77
Interfacial tension (IFT), 21, 38
Methane/water system, 149
Water/brine/hydrocarbon systems, 149
Isothermal gravity/chemical equilibrium, 64
J
Jacobian matrix, 55
Jacoby aromaticity factor, 78
Joule-Thompson expansion, 154, 158, 159
K
Katz-Carson charts, 154, 155
Kay’s mixing rule, 19, 24, 25, 93
K values
Black oil, 114
Calculation (Problem 15), 183
Correlations, 40
Hydrate formation, 154
MMP calculations, 127
Nonhydrocarbon, 43
Reservoir oil/gas, 116, 202
Standing low-pressure, 43
L
Laboratory experiments
Differential liberation expansion, 95
Slim-tube displacements, 124
True-boiling-point, 68–70, 73
Laboratory reports
General information sheet, 88
Oil PVT, 209
Langmuir adsorption theory, 156
Lasater equation, 29
Lean-gas injection, 128
Lee-Gonzalez correlation, 26
Liquefied petroleum gases (LPG’s), 121, 131
Liquid chromatography, 70
Liquid density, 162, 163
Liquid-dropout curves, 12, 15, 102, 104
Lucas correlation, 28
M
Macleod relation, 39
Mass, 18
Conversion factors, 169
Mass fractions, 19
Mass spectroscopy, 70
Material-balance relations, 53, 92,
108, 117, 209
Methane
/brine system, 149
/butane/decane system, 122
/C7+ BIP’s, 196
Density, 30
/hydrocarbon mixtures, 11
Maximum content determination, 132
/NaCl brine mixtures, 151
/propane/water mixtures, 157
Solubility in water, 143
/water system, 149, 154
Methane-rich injection gases, 128, 129
Methanol, 157, 161
Michelsen stability test, 57, 62, 185
Midvolume-point method, 69
Minimum miscibility pressure (MMP),
122, 125, 127
Miscibility, 122
CO2/oil behavior, 137
Temperature range, 138
Miscible displacement projects
Enriched-gas miscible drive, 131
PHASE BEHAVIOR
Vaporizing-gas miscible drive, 129
Mixing rules, 19, 84
Molality, 143
Molar density, 19
Molar distribution, 70
Exponential distributions, 72
Gamma-distribution model, 73
Molarity, 143
Molar mass, 18
Molar volume, 19, 22, 188
Calculation (Problem 19), 188
Mole, 18
Conversion factors, 171
Molecular mass, 18
Molecular weights, 18, 162, 163
Correlations, 82
Cumulative, 76
Gas-condensate example, 197
Heptanes-plus fractions, 70, 73
Mole fractions, 114, 148
Multicell vaporization model, 127
Multicomponent mixtures
Pseudoternary diagrams, 124
Rachford-Rice function, 53
Multiphase behavior
CO2/oil, 139
Enriched-gas injection, 134
Multistage separation, 91, 111
N
n-alkanes
Boiling point, 7, 8
Parachors, 38
Natural gas, 147
Composition, 6, 155
Correlations for PVT properties, 22
Hydrate-formation conditions, 156
Joule-Thompson expansion, 154, 158, 159
Quadruple points, 157
Natural gas liquids (NGL’s), 21, 130
Newton-Raphson algorithm, 53, 72
Nitrogen
Effect on dewpoint pressure, 131
Injection gas, 124
/NaCl-brine system, 154
/oil mixture, 127, 129, 130
Nitrogen-rich injection gases, 129
Nomenclature, 2, 18, 220
North Sea gas condensate
K-value correlation, 43
Simulated distillation, 72, 73
Specific gravity, 78
TBP distillation, 69, 70
North Sea oils, 29
Gamma density function, 74
O
Oil
Composition, 114
Constant composition expansion, 94
Differential liberation expansion, 95, 98–101
FVF, 111
General information sheet, 89, 90
Gravity, 20
Gross heating value, 94
Near-critical, 6
Phase behavior, 5
Properties and correlations, 18
PVT analysis, 88
Separator test, 93
Volumetric behavior, 5
Oil compressibility
Saturated oil, 36
SUBJECT INDEX
Undersaturated oil, 35, 94
Oil density, 30, 113
Alani-Kennedy method, 34
Differential liberation expansion, 203
Nitrogen/oil mixture, 130
Oil/gas ratio (OGR), 13, 21
Oil/gas/water systems, 145
Oil mixtures, 29
Oil viscosity, 36, 100, 101, 130
Calculation (problem 14), 182
Effect of CO2, 137
Reservoir-oil example, 206
Pressure correction, 32
Separator gas, 34
Temperature correction, 33
Pseudoternary diagrams, 124, 131
Q
Quadruple points, 153, 157
Quaternary diagrams, 124
R
P
Parachors, 38
Paraffinicity, 29
Paraffins, normal, 82
Paraffins/naphthalenes/aromatics (PNA’s), 70
Partial-density formulation, 118
Peng-Robinson equation, 50, 150, 156, 164,
185, 193, 196, 202
Perturbation expansions, 82
Petroleum compounds, 5
Petroleum-refinery products, 6
Petroleum residue, 73
Phase behavior
Conversion factors, 168–71
Gas systems, 5
Historical review, 1
Oil systems, 5
Phase diagrams
Carbon dioxide, 135
Hydrates, 153
Multicomponent systems, 11
Simple systems, 8
Single-component systems, 9
Two-component systems, 10
Phase envelope, 12, 15
Phase equilibria, 47
Binary mixtures, 56
In gravity field, 63
Phase stability, 55
Physical constants, 167
Pressure, conversion factors for, 169
Pressure/temperature diagrams
Depletion experiments, 15
Ethane/n-heptane system, 13
Gas-cap fluid, 16
Gas-condensate system, 13
Hydrocarbon binaries, 13
NaCl brine, 150
Phase envelope, 54
Pure fluids and mixtures, 12
Pure water, 150
Single-component system, 9
Pressure/volume diagrams
Ethane, 11
Pure component, 48
Pure fluids and mixtures, 12
Pressure/volume/temperature (PVT) diagrams
Below bubblepoint, 96
Black oil, 20, 109, 116, 200, 207
Conventional measurements, 88
Gas cap, 207
Laboratory reports, 209
Multicontact experiments, 126
Pure compound, 10
Reservoir oil, 207
Problems, example, 172
Pseudocomponents, 84, 124
Pseudocritical properties, 24, 175, 177
Pseudoization, 85
Pseudoliquid density
Chart for calculating, 31
Oil, 180
Rachford-Rice equation, 52
Radial-flow equation, 116
Raleigh field report, 209–19
Rate equations (IPR’s), 116, 117
Recommendations
Heptane-plus characterization, 83
Laboratory report, 88
Recoveries, 97
Calculated, 98, 107
Corrections, 102
Gas injection, 121
Normal temperature separation, 99
Plant products, 101
Slim-tube, 132
Stock-tank oil, 103
Recovery-pressure curves, 125
Redlich-Kwong equation, 48
Reduced properties, 18
Regression parameters, 196
Reservoir fluids
At less than bubblepoint pressure, 210
Characterization, 201
Classification, 12
Composition, 5
Compressibility, 20
FVF, 20
Grouping and averaging properties, 83
Reservoir gas, 110
Reservoir mixtures, 19
Reservoir oil, 109
Density calculation (Problems 10 and 11),
179, 180
/gas mixtures, 127
Slim-tube displacement, 125
Reservoir voidage (Problem 9), 178
Reservoir water, 142
Residual oil saturation (ROS), 121, 133
Retrograde condensation, 11, 108, 126, 129
S
Salinity, 142
Correction, 144
Salts
Concentrations, 143
Gas solubility, 145
Sample analysis
Bottomhole oil, 88
Gas-condensate, 92, 102–07, 195
Oil, 93
Subsurface fluid, 215
Saturated oil, 16
Compressibility, 94
Rate equation, 117
Saturation-pressure calculation, 62
Separator gas
Composition, 43, 89
Pseudoliquid density, 34
Water content, 190
Separator-oil composition, 89, 174
Separator test, 91, 189, 205
Raleigh report, 209
3
Well-effluent composition (Problem 4), 174
Setchenow relation, 144
Simulated distillation, 70, 72
Single carbon number (SCN), 69–72, 77
SI standards, 18
SI system units, 163, 166
Slim-tube displacements, 122, 124, 138
Soave-Redlich-Kwong equation, 49, 83
Sodium chloride brine, 142
Solubility
Carbon dioxide, 136, 140
Differential, 212
Gas in water/brine system, 143, 153
Methane in water, 154
Natural gas in brines, 153
Salinity correction, 144
Water in natural gas, 147
Solution gas, 142
Solution gas/oil ratio, 21, 111, 143
Solution oil/gas ratio, 112, 113
Søreide correlations, 79, 193, 202
Specific gravity, 19
Components, 162, 163
Gas, 22, 111
Heptanes-plus fractions, 70, 79, 193
Oil, 20, 78
Reservoir-oil example, 205
Stock-tank oil, 112
Wellstream, 25
Specific volume, 19
Standard atmosphere, 167
Standing-Katz method, 30, 179
Standing-Katz Z-factor chart, 23, 177
Standing’s correlations, 24, 29, 37, 43, 90, 180
Stepwise regression, 85
Stock-tank oil, 109
Cabin Creek, 137
Gravity, 26
PVT data, 96
Recovery, 103
Slim-tube displacements, 137–39
True-boiling-point distillation, 69
Viscosity, 36
/wellstream ratio, 26
Subsea oil and gas (Problem 13), 181
Subsurface sampling, 214
Sulfur-rich gas (Problem 2), 172
Sum-of-squares (SSQ) function, 196
Surface gravity, 26, 111
Surface-separator calculations, 40, 43
Surface-separator gas, 109, 110
Surface tension, 171
Sutton correlations, 24, 175
Swelling
Dead stock-tank oil, 136
Reservoir oil by CO2, 137
4
Conversion factors, 170
Test, 126
T
Temperature correction, 33
Temperature scale conversions, 167
Ternary systems, 122, 183, 185
Thermodynamic properties, 47
Toluene, 83
Total dissolved solids (TDS), 142
True-boiling-point (TBP) analysis, 68–70, 73
Two-phase flash calculation, 52
U
Undersaturated oil
Compressibility, 35, 94, 204
Radial-flow equation, 116
Viscosity, 38
Units, 2, 18, 162
Universal gas constant, 22, 164
Universal oil products (UOP) factor, 77
V
van der Waals EOS, 48
van der Waals-Platteeuw model, 156
Vapor/liquid equilibrium (VLE) algorithms,
47, 139
Vapor/liquid/liquid (VLL) behavior, 121, 139
Vazguez-Beggs correlations, 30, 35
Viscosity, 21
Bubblepoint oil, 37
Carbon dioxide, 134
Conversion factors, 170
Correlation, 72
Gas, 26, 175, 182
Gas-free oil, 37
Oil, 36, 100, 101, 182
Oil/nitrogen mixture, 130
Undersaturated oil, 38
Water/brine, 147
Volatile oil, 13, 15, 109–11
Composition, 6
Gamma density function, 74
Volume fractions, 19
Volume-translation parameters, 51, 195
Volumetric behavior
Calculation, 47
CO2-rich stock-tank oil, 137
Gas systems, 5
Oil systems, 5
Two-phase systems, 93
Volumetric properties, 19, 22, 216
W
Water
Content of separator gas (Problem 21), 190
Density, 167
FVF, 146
/hydrocarbon systems, 142
Reservoir, 142
Solubility in methane/NaCl-brine mixture,
151
Solubility in natural gas, 147
Triple point, 167
Vapor in equilibrium with hydrates, 156
Water-alternating-gas (WAG) ratio, 121, 129
Water/brine
Compressibility, 145
FVF, 145
Viscosity, 147
Water/brine/hydrocarbon systems, 149
Water/ethane system, 157, 161
Water/hydrocarbon systems
EOS predictions, 150
Watson characterization factor, 29, 77, 81, 189
Calculation (Problem 20), 189
Weight factors, 77
Weight fractions, 19, 76, 143
Well production test (Problem 5), 174
Wellstream composition, 88, 91, 116, 174
Wellstream specific gravity, 25
Wet gas, 13, 16
Composition, 6
CVD data, 105
FVF, 113
Whitson-Torp method, 111, 112
Wichert-Aziz correlations, 25
Wilson equation, 42, 53
Y
Yarborough aromaticity factor, 79
Z
Z factor, 22
Calculation (problem 18), 97, 187
Carbon dioxide, 134
Correlations, 23, 177
Gas-condensate example, 197
Reservoir-oil example, 206
van der Waals equation, 48
Zudkevitch-Joffe-Redlich-Kwong equation, 50
PHASE BEHAVIOR