работа

MSC 93A30
DOI: 10.14529/mmp180206
MATHEMATICAL MODEL OF GAS HYDRATE OF HYDROGEN
SULFIDE FORMATION DURING ITS INJECTION
INTO A NATURAL LAYER
M.Ê. Khasanov1 , G.R. Rakova2
Sterlitamak branch of Bashkir State University, Sterlitamak, Russian Federation,
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russian Federation
E-mail: hasanovmk@mail.ru, rakova_guzal@mail.ru
1
2
The mathematical model of liquid hydrogen sulde injection into the semi-innite
porous layer saturated with the oil and water accompanied by H2 S gas hydrate formation
is presented here. We considered the case when the hydrate formation occurs at the frontal
border and the oil displacement's front by hydrogen sulde is ahead of this boundary.
Solutions for pressure and temperature in every layer's area are built by help of the selfsimilar variable formation method. The values of the parameters of the moving interphase
boundaries are found as the result of the iteration procedure. The coordinate dependence
of phase boundaries on the injection pressure was studied on the basis of the obtained
solutions. We have established that for the existence of solution with two dierent interphase
boundaries, the injection pressure must be above a certain limiting value. The dependence of
the limiting value of pressure on the initial temperature of the layer at dierent temperatures
of the injected hydrogen sulphide is constructed. The results of the calculations showed that
the constructed mathematical model with three areas in the reservoir gives an adequate
description of the process at high injection pressures, the temperature of the injected
hydrogen sulde and the initial temperature of the layer.
Keywords: mathematical model; self-similar variable solution; porous medium;
ltration; gas hydrates; hydrogen sulde.
Introduction
One of the methods for the emission's reducing of hydrogen sulde generated by
industrial facilities into the atmosphere is its underground disposal in the exhausted
hydrocarbon deposits [1, 2]. As there is a risk of the leaking of the recyclable gases in
the form of a uid to the surface at their long-term underground storage we consider the
possibility of their transformation into the gas hydrate state which allows the storing of
the same gas number at much lower pressures compared to its free state [3, 4].
As any technological ideas need to be backed up by the relevant calculations based on
the theoretical models, the construction of the adequate mathematical models of hydrate
formation in the natural layers is an actual task. The mathematical models of hydrate
formation in the porous layers saturated with methane and water are represented in the
works [5-8]. The mathematical model of H2 S hydrate formation in the layers saturated
with oil and water during injection of uid hydrogen sulde are represented in the work
[9]. But in the above mentioned works the task is achieved due to its simple determination
when the gas hydrate formation occurs at the border coinciding with the oil replacement's
border by hydrogen sulde. The mathematical model of H2 S gas hydrate formation is
represented in this work the hydrate formation at the front border that doesn't coincide
with the oil replacement's border by the uid hydrogen sulde.
Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå
è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2018. Ò. 11,  2. Ñ. 7382
73
M.K. Khasanov, G.R. Rakova
1. Problem Statement
The conditions of gas hydrate of hydrogen sulde existence are represented in the
phase diagram (Fig. 1) [6]. The curve line gh determines in this diagram the three-phase
balancing of water, gas hydrogen sulde and its gas hydrate, the curve line lh shows the
balancing of water, uid hydrogen sulde and its gas hydrate, the curve line lg represents
two-phase balancing of uid and gas hydrogen sulde. Suitably the gas hydrate of hydrogen
sulde is to the left of the curve lines gh and lh. All four indicated phases are balanced in
the quadrupole point Q (TQ =302,6 Ê and pQ =2,24 ÌPà).
p, ÌPà
lh
10
5
Q
lg
1
0.5
0.1
273
gh
T, Ê
283
293
303
Fig. 1. Phase diagram of H2 S-H2 0 system
Imagine the semi-nite horizontal porous layer (occupied semi-areas x>0) is saturated
with water source saturation Sw0 and oil in the initial moment. We assume that the
initial temperature of the layer T0 is higher than the temperature TQ of the accordance
quadrupole point. Therefore the problem in question the initial position of the layer doesn't
meet the formation conditions of the gas hydrate of hydrogen sulde. Suppose that the
uid hydrogen sulde is pumped through the border (x=0) and the pe pressure as well as
the Te temperature are in accordance with the conditions of the gas hydrate of hydrogen
sulde.
The initial and border conditions are Sw = Sw0 , T = T0 , p = p0 (x ≥ 0) when t = 0,
and the border conditions are T = Te , p = pe (t > 0) when x = 0.
We consider the model with the oil piston displacement by hydrogen sulde as well as
the case when the value of the initial water saturation of the layer isn't higher than 0,2
(i.e. the water is immobile) in the work. Also we consider the time scales that are much
higher than the characteristic time of the process of hydrate formation kinetics. So we can
suppose that hydrate formation occurs at the front border which doesn't coincide with
the oil displacement border by the hydrogen sulde. Therefore three characteristic areas
appear in the layer in accordance with this case. The pores are saturated with hydrogen
sulde and its hydrate in the rst (nearest) area, the water and the hydrogen sulde are
in the second (intermediate) area, the pores saturated with oil and water are in the third
(remote) one. Accordingly, there are two movable interfacial surfaces: between the rst and
the second areas where the water transforms into the gas hydrate state completely (front
hydrate formation) and between the second and the third areas where the oil displacement
by hydrogen sulde occurs (displacement front).
74
Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 2, pp. 7382
ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ
2. Basic Equations
Let's assume the following simplifying assumptions: the porosity of the layer m
is constant, the body of the porous medium, gas hydrate as well as the water are
incompressible and immobile. H2 S gas hydrate is a double-component system with G mass
concentration of the hydrogen sulde. We assume that the uid hydrogen sulde and the
oil are weakly compressible uids. Basic equation system describing in the one-dimensional
case the ltration and heat transfer process in the porous medium represents the laws of
mass energy conservation, Darcy law and equation of state [10]:

∂
∂

(ρi mSi ) + ∂x
(ρi mSi υi ) = 0,
 ∂t


ρc ∂T + ρ c mS υ ∂T = ∂ (λ ∂T ) ,
i i
i i ∂x
∂t
∂x
∂x
(1)
∂p
k
i

mS
υ
=
−
,
i
i

µi ∂x


ρ = ρ (1 + β (p − p )) .
i
0i
i
0
Where t is time; x is coordinate; m is porosity; p is pressure; T -is temperature; low indexes
i=s,l refer respectively to the parameters of hydrogen sulde and oil; ρi is density; ki is
phase permeability; υi is actual average speed; ci is specic heat capacity; µi is dynamic
viscosity; βi is compressibility factor; ρc è λ are eective values of the volumetric heat
capacity and thermal conductivity of the layer saturated. Since the main contribution to
the value ρc and λ includes the corresponding parameters of the rock we assume them as
the constant values.
The phase coecient dependence of the ki permeability on the S(i) saturation and
absolute permeability k0 we dene as follows: ki = k0 Si (i = s, l).
The total transition of the water into the hydrate state takes place on the surface
x = x(n) dividing the rst and the second areas. Therefore, according to the conditions of
the mass heat balance at this boundary we have:
)
(
ks(1) ∂p(1) ks(2) ∂p(2)
ρh G ρh (1 − G)
•
−
(2)
+
= mSh
+
− 1 x (n) ,
µs ∂x
µs ∂x
ρ0s
ρw
•
•
mSh ρh (1 − G)x (n) = mSw0 ρw x (n) ,
(3)
∂T(1)
∂T(2)
•
−λ
= mSh ρh Lh x (n) ,
(4)
∂x
∂x
where the value x(n) is the front motion speed of the gas hydrate formation H2 S, G is
mass concentration of the hydrogen sulde in the hydrate, Lh is specic heat of H2 S gas
hydrate formation, Sw0 is initial water saturation, ρw is water density. Low index n refers
to the parameters at the border, dividing the rst and the second areas, the low indexes 1
and 2 refer to the parameters of the rst and the second areas accordingly. We assume the
temperature at this border as continuous and equal to the temperature of the quadrupole
point TQ .
The oil displacement by the hydrogen sulde takes place on the surface x = x(d)
dividing the second and the third areas. Therefore we have the subject to the conditions
of the oil and the hydrogen sulde mass balance as well as the heat's one at this border:
λ
−
ks(2) ∂p(2)
•
= m (1 − Sw0 ) x (d) ,
µs ∂x
Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå
è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2018. Ò. 11,  2. Ñ. 7382
(5)
75
M.K. Khasanov, G.R. Rakova
−
kl ∂p(3)
•
= m (1 − Sw0 ) x (d) ,
µl ∂x
(6)
∂T(2)
∂T(3)
−λ
= 0,
∂x
∂x
(7)
λ
•
where the value x(d) is the front motion speed of the oil by the hydrogen sulde. Low index
d refers to the parameters at the border, dividing the second and the third areas and the
low index 3 refers to the parameters of the third area.
The pressure and temperature will be considered continuous variables at both borders.
On the basis of the system's equations (1), the equations for the piezoconductivity
and the temperature conductivity will be written in the form:
(
)
∂p(i)
∂p(i)
(p) ∂
,
(8)
= χ(i)
∂t
∂x ∂x
∂T(i)
∂
= χ(T )
∂t
∂x
(p)
where χ(1) =
χ(T ) =
λ
,
ρc
X(1)
(
)
∂T(i)
∂x
∂p(i) ∂T
1
+ χ(T ) X(i)
,
2
∂x ∂x
(9)
ks(1)
ks(2)
kl
(p)
(p)
, χ(2) =
, χ(3) =
,
µs m (1 − Sh ) βs
µs m (1 − Sw0 ) βs
µl m (1 − Sw0 ) βl
ρ0s cs ks(1)
ρ0s cs ks(2)
ρ0l cl kl
=2
, X(2) = 2
, X(3) = 2
.
λµs βs
λµs βs
λµl βl
3. Self-Similar Solution
/√
We introduce self-similar variable: ξ = x
χ(T ) t. For this variable the equations (8),
(9) for the piezoconductivity and temperature conductivity will take the form in every
area:
(
)
dp(i)
d dp(i)
−ξ
= 2η(i)
,
(10)
dξ
dξ
dξ
(
)
dT(i)
dp(i) dT(i)
d dT(i)
−ξ
= X(i)
+2
,
(11)
dξ
dξ dξ
dξ
dξ
(p)
where η(i) = χ(i) /χ(T ) .
After integrating the solutions (10), (11) the solutions for the pressure and temperature
distribution in every area can be written in the form:
(
pe − p(n)
p(1) = p(n) +
(
T(1) = T(n) +
(
∫
exp
0
Te − T(n)
) ξ∫(n)
∫
(
exp
2
− 4ηξ(1)
(
exp
ξ
ξ(n)
(
)
2
exp − 4ηξ(1) dξ
ξ
ξ(n)
0
76
) ξ∫(n)
2
− ξ4
2
− ξ4
,
)
dξ
(12)
)
− X(1) p(1) dξ
)
,
− X(1) p(1) dξ
Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 2, pp. 7382
ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ
(
p(n) − p(d)
p(2) = p(d) +
T(n) − T(d)
ξ
exp − 4η(2) dξ
) ξ∫(d)
∫
(
ξ(d)
exp
ξ(n)
(
p(d) − p0
p(3) = p0 +
(
∫∞
(
exp
T(d) − T0
) ∫∞
ξ
T(3) = T0 +
∫∞
) ∫∞
ξ
exp
(13)
)
− X(2) p(2) dξ
,
− X(2) p(2) dξ
(
exp
(
2
− 4ηξ(3)
2
− 4ηξ(3)
)
dξ
)
,
dξ
( 2
)
exp − ξ4 − X(3) p(3) dξ
(
ξ(d)
2
− ξ4
)
2
− ξ4
exp
ξ(d)
,
)
ξ2
ξ
T(2) = T(d) +
(
)
2
exp − 4ηξ(2) dξ
(
∫
ξ(d)
ξ(n)
(
) ξ∫(d)
2
− ξ4
)
(14)
.
− X(3) p(3) dξ
On the basis of the ratios (2), (4) taking into account the decisions (12), (13) we get the
equations for coordinate's determining for the hydrate formation's front ξ(n) and parameter
values on it:
( 2 )
( 2 )
)
(
(
)
ξ
ξ(n)
p(n) − pe exp − 4η(n)
p(d) − p(n) exp − 4η(2)
(1)
−
k
= KSh ξ(n) ,
(15)
ks(2)
s(1)
ξ(d)
ξ(n)
(
)
(
)
∫
∫
2
2
exp − 4ηξ dξ
exp − 4ηξ dξ
(2)
ξ(n)
(
(T(n) −Te ) exp
(
ξ(n)
∫
0
exp
ξ2
(n)
− 4 −X(1) p(n)
2
− ξ4 −X(1) p(1)
(1)
0
)
dξ
)
(
−
(T(d) −T(n) ) exp
ξ(d)
∫
ξ(n)
ξ2
(n)
− 4 −X(2) p(n)
)
( 2
exp − ξ4 −X(2) p(2) dξ
)
=
(16)
mρh Lh
Sh ξ(n) ,
2ρc
T(n) = TQ ,
(17)
(
)
where K = mµs χ(T ) ρρhs0G + ρh (1−G)
−1 .
ρw
Similarly, on the basis of the ratios (5) to (7) taking into account the decisions (13)
and (14) we get the equations system to determine the oil displacement front's coordinate
by hydrogen sulde ξ(d) and parameter values on it:
( 2 )
)
(
ξ
p(n) − p(d) exp − 4η(d)
(2)
= mµs χ(T ) (1 − Sw0 ) ξ(d) ,
(18)
ks(2)
ξ(d)
(
)
∫
2
exp − 4ηξ(2) dξ
=
ξ(n)
Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå
è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2018. Ò. 11,  2. Ñ. 7382
77
M.K. Khasanov, G.R. Rakova
(
kl
p(d) − p0
∫∞
ξ(d)
)
( 2 )
ξ
exp − 4η(d)
(3)
(
)
2
exp − 4ηξ(3) dξ
( 2
)
(
)
ξ(d)
T(d) − T(n) exp − 4 − X(2) p(d)
∫
ξ(d)
ξ(n)
(
ξ2
)
exp − 4 − X(2) p(2) dξ
(
−
= mµl χ(T ) (1 − Sw0 ) ξ(d) ,
)
(
T0 − T(d) exp −
∫∞
ξ(d)
(
exp
2
− ξ4
2
ξ(d)
4
(19)
)
− X(3) p(d)
)
− X(3) p(3) dξ
= 0.
(20)
The system of boundary equations (15) to (20) is presented in the work as following.
At the beginning the zero approximation of the desired values at the front of hydrate
formation is given. Further, solving the equation (18), we nd the p(d) value (as a function
ξ(d) ), substituting that into equation (19) we get the transcendental equation for nding
ξ(d) . Have solved this equation (by the half division method) we determine the value ξ(d)
(and respectively p(d) ), then from (20) we dene T(d) . Further, substituting (17) into the
equation (16) we get the transcendental equation for nding ξ(n) . Solving this equation
(by the method of half-division), we determine new approximation of value ξ(n) . Then,
solving equation (15), we nd a new approximation of value p(n) . As the result of cyclic
repetition of the described iterative procedure we obtain a sequence of approximate values
which converges to the target values of the boundary parameters. As a result of cyclic
repetition of the described iterative procedure, we obtain a sequence of approximate values
Y(k) = (ξ(n) , ξ(d) , p(n) , p(d) , T(d) ). This sequence converges within the metric:
(
( (k+1)
)
(k+1)
(k)
(k+1)
(k)
ρ Y
, Y (k) = max ξ(n) − ξ(n) , ξ(d) − ξ(d) ,
(k+1)
(k)
(k)
(k) )
(k+1)
(k+1)
p(n)
p(n)
p(d)
T(d)
p(d)
T(d)
−
−
−
,
,
.
p0
p0
p0
p0
T0
T0
To establish the time of the termination of the iterations, the following condition was
used:
(
)
ρ Y (k+1) , Y (k) < 0, 001.
4. Result of Calculation
The Fig. 2 shows the dependence of the coordinate fronts of the hydrate formation H2 S
(curve line1) and the oil displacement by the liquid hydrogen sulde (curve line2) on the
injection pressure. Hereinafter, unless otherwise specied for the parameters characterizing
the system, the following values are adopted: m = 0, 1, Sw0 = 0, 2, p0 = 8 ÌPà,
Te =290 Ê, T0 =305 Ê, k0 = 10−11 m2 , G = 0,24, βs = 3· 10−9 Pà−1 , βl = 1·10−9 Pà−1 ,
λ = 2 w/(ì·Ê), ρc = 2·106 J/(Ê·kg), µs = 2·10−4 Pà·s, µl = 2·10−3 Pà· s, ρh = 1003 kg/m3 ,
ρw = 1000 kg/m3 , ρ0s =890 kg/m3 ,ρ0l =900 kg/m3 , cs = 1800 J/(Ê·kg), cl = 1900 J/(Ê·kg),
Lh = 4,1·105 J/kg.
According to the Fig. 2 the speed of the rst front is almost independent on the
injection pressure and the speed of the second front increases with its growth. This is due
to the fact that the front speed of the oil displacement by hydrogen sulde is limited by
78
Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 2, pp. 7382
ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ
4
x ( n, d )
2
3
2
1
1
pe , MPà
0
8.2
8.4
8.6
8.8
9.0
Fig. 2. Dependence of the border coordinates of hydrate formation H2 S (curve line 1) and
oil displacement by hydrogen sulde (curve line 2) on the injection pressure
8.50
p* , Ì P à
8.45
2
8.40
1
8.35
8.30
303
T0 , Ê
304
305
306
307
Fig. 3. The dependence of the limiting pressure on the initial temperature in the layer at
Te =290 Ê (curve line 1) and 285 Ê (curve line 2)
the intensity of mass transfer in the layer that is proportional to the generated in the layer
pressure drop according to Darcy's law. And the front speed of the hydrate formation is
limited primarily by the dissipation of heat released during the phase transition, i.e. the
intensity of heat transfer in the layer.
It is also seen that for the suciently low values of injection pressure the coordinates of
both fronts are aligned, i.e. the front speed of the oil displacement by the liquid hydrogen
sulde is reduced to the speed of the hydrate formation front. Because the original hydrogen
sulde in the layer was absent, that in this mode, the speed of movement of united fronts
will be limited by the supply of hydrogen sulde, i.e. by the injection pressure.
Thus, there is some value of pressure (called limiting pressure) above which the mode
with two dierent interfacial borders is implemented under consideration in this work.
There are given in the Fig. 3 the dependences of the injection pressure corresponding to
the alignment of the coordinates of both fronts, on the initial layer temperature at dierent
injection temperatures Te =290 Ê (curve line 1) è 285 Ê (curve line 2). According to the
Fig. 3 the given pressure value decreases while increasing the initial layer's temperature
and temperature of hydrogen sulde injected. In other words, the mode with two dierent
interphase bounders is realized at suciently high layer's temperatures and the hydrogen
sulde injected. This is due to the fact that the intensity of heat removal at the border
Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå
è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2018. Ò. 11,  2. Ñ. 7382
79
M.K. Khasanov, G.R. Rakova
of hydrate formation decreases with an increase layer's temperature and the injection
and, accordingly, the rate of hydrate formation front decreases. Thus, the front of oil
displacement by hydrogen sulde outpaces the front of H2 S gas hydrate formation even
at low values of injection pressure at high temperatures of the layer and hydrogen sulde
injected.
Acknowledgements. This work was supported by the Russian Foundation for Basic
Research and the Republic of Bashkortostan (project No. 17-48-020123).
References
1. Machel H.G. Geological and Hydrogeological Evaluation of the Nisku Q-Pool in Alberta,
Canada, for H2 S and/or CO2 Storage. Oil and Gas Science and Technology, 2005, vol. 60,
pp. 5165. DOI: 10.2516/ogst:2005005
2. Xu T., Apps J.A., Pruess K., Yamamoto H. Numerical Modeling of Injection and Mineral
Trapping of CO2 with H2 S and SO2 in a Sandstone Formation. Chemical Geology, 2007, vol.
24, no. 34, pp. 319346. DOI: 10.1016/j.chemgeo.2007.03.022
3. Byk S.Sh., Makogon Yu.F., Fomina V.I.
1980.
Gazovye gidraty
[Gas Hydrates]. Moscow, Khimiya,
4. Duchkov A.D., Sokolova L.S., Ayunov D.E., Permyakov M.E. Assesment of Potential of West
Siberian Permafrost for the Carbon Dioxide Storage. Earth's Cryosphere, 2009, vol. 13, no 4,
pp. 6268.
5. Gimaltdinov I.K., Khasanov M.K. Mathematical Model of the Formation of a Gas
Hydrate on the Injection of Gas into a Stratum Partially Saturated with Ice.
Journal of Applied Mathematics and Mechanics, 2016, vol. 80, no. 1, pp. 5764.
DOI: 10.1016/j.jappmathmech.2016.05.009
6. Shagapov V.Sh., Rakova G.R., Khasanov M.K. On the Theory of Formation of Gas Hydrate
in Partially Water-Saturated Porous Medium when Injecting Methane. High Temperature,
2016, vol. 54, no. 6, pp. 858866. DOI: 10.1134/S0018151X16060171
7. Shagapov V.Sh., Chiglintseva A.S., Belova S.V. On the Theory of Formation of a Gas Hydrate
in a Heat-Insulated Space Compacted with Methane. Journal of Engineering Physics and
Thermophysics, 2017, vol. 90, no. 5, pp. 11471161. DOI: 10.1007/s10891-017-1669-8
8. Shagapov V.Sh., Chiglintceva A.S., Belova S.V. Mathematical Modelling of Injection
Gas Hydrate Formation into the Massif of Snow Saturated the Same Gas. Proceedings
of
the
Mavlyutov
Institute
of
Mechanics,
2016. vol. 11, no. 2, p. 233239.
DOI: 10.21662/uim2016.2.034
9. Khasanov M.K. Injection of Liquid Hydrogen Sulde in a Layer Saturated with Oil and Water.
Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy,
2017, vol. 3, no. 2, pp. 7284.
10. Shagapov V.Sh., Khasanov M.K., Musakaev N.G., Ngoc Hai Duong. Theoretical
Research of the Gas Hydrate Deposits Development Using the Injection of Carbon
Dioxide. International Journal of Heat and Mass Transfer, 2017, vol. 107, pp. 347357.
DOI: 10.1016/j.ijheatmasstransfer.2016.11.034
Received February 20, 2018
80
Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 2, pp. 7382
ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ
ÓÄÊ 519.6+532.546
DOI: 10.14529/mmp180206
ÌÀÒÅÌÀÒÈ×ÅÑÊÀß ÌÎÄÅËÜ ÎÁÐÀÇÎÂÀÍÈß ÃÀÇÎÃÈÄÐÀÒÀ
ÑÅÐÎÂÎÄÎÐÎÄÀ ÏÐÈ ÅÃÎ ÈÍÆÅÊÖÈÈ Â ÏÐÈÐÎÄÍÛÉ ÏËÀÑÒ
Ì.Ê. Õàñàíîâ1 , Ã.Ð. Ðàôèêîâà2
Ñòåðëèòàìàêñêèé ôèëèàë Áàøêèðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà,
ã. Ñòåðëèòàìàê, Ðîññèéñêàÿ Ôåäåðàöèÿ
2
Èíñòèòóò ìåõàíèêè èì. Ð.Ð. Ìàâëþòîâà îáîñîáëåííîå ñòðóêòóðíîå
ïîäðàçäåëåíèå Óôèìñêîãî ôåäåðàëüíîãî èññëåäîâàòåëüñêîãî öåíòðà
Ðîññèéñêîé àêàäåìèè íàóê, ã. Óôà, Ðîññèéñêàÿ Ôåäåðàöèÿ
1
Ïðåäñòàâëåíà ìàòåìàòè÷åñêàÿ ìîäåëü èíæåêöèè æèäêîãî ñåðîâîäîðîäà â ïîëóáåñêîíå÷íûé ïîðèñòûé ïëàñò, íàñûùåííûé íåôòüþ è âîäîé, ñîïðîâîæäàþùåéñÿ îáðàçîâàíèåì ãàçîãèäðàòà H2 S. Ðàññìîòðåí ñëó÷àé, êîãäà ãèäðàòîîáðàçîâàíèå ïðîèñõîäèò íà
ôðîíòàëüíîé ãðàíèöå, à ôðîíò âûòåñíåíèÿ íåôòè ñåðîâîäîðîäîì îïåðåæàåò äàííóþ
ãðàíèöó. Ìåòîäîì ñâåäåíèÿ ê àâòîìîäåëüíîé ïåðåìåííîé ïîñòðîåíû ðåøåíèÿ äëÿ äàâëåíèÿ è òåìïåðàòóðû â êàæäîé èç îáëàñòåé ïëàñòà. Çíà÷åíèÿ ïàðàìåòðîâ ïîäâèæíûõ
ìåæôàçíûõ ãðàíèö íàéäåíû êàê ðåçóëüòàò èòåðàöèîííîé ïðîöåäóðû. Íà îñíîâå ïîëó÷åííûõ ðåøåíèé èññëåäîâàíà çàâèñèìîñòü êîîðäèíàò ìåæôàçíûõ ãðàíèö îò äàâëåíèÿ
èíæåêöèè. Ïîêàçàíî, ÷òî äëÿ ñóùåñòâîâàíèÿ ðåøåíèé ñ äâóìÿ ðàçëè÷íûìè ìåæôàçíûìè ãðàíèöàìè âåëè÷èíà äàâëåíèÿ èíæåêöèè äîëæíà áûòü âûøå íåêîòîðîãî ïðåäåëüíîãî çíà÷åíèÿ. Ïîñòðîåíà çàâèñèìîñòü ïðåäåëüíîãî çíà÷åíèÿ äàâëåíèÿ èíæåêöèè
îò íà÷àëüíîé òåìïåðàòóðû ïëàñòà ïðè ðàçíûõ çíà÷åíèÿõ òåìïåðàòóðû èíæåêòèðóåìîãî ñåðîâîäîðîäà. Ðåçóëüòàòû ðàñ÷åòîâ ïîêàçàëè, ÷òî ïîñòðîåííàÿ ìàòåìàòè÷åñêàÿ
ìîäåëü ñ òðåìÿ îáëàñòÿìè â ïëàñòå äàåò àäåêâàòíîå îïèñàíèå ïðîöåññà ïðè âûñîêèõ
çíà÷åíèÿõ äàâëåíèÿ èíæåêöèè, òåìïåðàòóðû çàêà÷èâàåìîãî ñåðîâîäîðîäà è íà÷àëüíîé
òåìïåðàòóðû ïëàñòà.
Êëþ÷åâûå ñëîâà: ìàòåìàòè÷åñêàÿ ìîäåëü; àâòîìîäåëüíîå ðåøåíèå; ïîðèñòàÿ
ñðåäà; ôèëüòðàöèÿ; ãàçîãèäðàòû; ñåðîâîäîðîä.
Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå ÐÔÔÈ è Ðåñïóáëèêè Áàøêîðòîñòàí (ïðîåêò  17-48-020123 ð− à).
Ëèòåðàòóðà
1. Machel, H.G. Geological and Hydrogeological Evaluation of the Nisku Q-Pool in Alberta,
Canada, for H2 S and/or CO2 Storage / H.G. Machel // Oil and Gas Science and Technology. 2005. V. 60. P. 5165.
2. Xu, T. Numerical Modeling of Injection and Mineral Trapping of CO2 with H2 S and SO2 in a
Sandstone Formation / T. Xu, J.A. Apps, K. Pruess, H. Yamamoto // Chemical Geology. 2007. V. 24,  34. P. 319346.
3. Áûê, Ñ.Ø. Ãàçîâûå ãèäðàòû / Ñ.Ø. Áûê, Þ.Ô. Ìàêîãîí, Â.È. Ôîìèíà. Ì.: Õèìèÿ,
1980.
4. Äó÷êîâ, À.Ä. Îöåíêà âîçìîæíîñòè çàõîðîíåíèÿ óãëåêèñëîãî ãàçà â êðèîëèòîçîíå Çàïàäíîé Ñèáèðè / À.Ä. Äó÷êîâ, Ë.Ñ. Ñîêîëîâà, Ä.Å. Àþíîâ, Ì.Å. Ïåðìÿêîâ // Êðèîñôåðà
Çåìëè. 2009. Ò. 13,  4. Ñ. 6268.
5. Ãèìàëòäèíîâ, È.Ê. Ìàòåìàòè÷åñêàÿ ìîäåëü îáðàçîâàíèÿ ãàçîãèäðàòà ïðè èíæåêöèè ãàçà
â ïëàñò, ÷àñòè÷íî íàñûùåííûé ëüäîì / È.Ê. Ãèìàëòäèíîâ, Ì.Ê. Õàñàíîâ // Ïðèêëàäíàÿ
ìàòåìàòèêà è ìåõàíèêà. 2016. Ò. 80,  1. Ñ. 8090.
Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå
è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2018. Ò. 11,  2. Ñ. 7382
81
M.K. Khasanov, G.R. Rakova
6. Øàãàïîâ, Â.Ø. Ê òåîðèè îáðàçîâàíèÿ ãàçîãèäðàòà â ÷àñòè÷íî âîäîíàñûùåííîé ïîðèñòîé ñðåäå ïðè íàãíåòàíèè ìåòàíà / Â.Ø. Øàãàïîâ, Ã.Ð. Ðàôèêîâà, Ì.Ê. Õàñàíîâ //
Òåïëîôèçèêà âûñîêèõ òåìïåðàòóð. 2016. Ò. 54,  6. Ñ. 911920.
7. Øàãàïîâ, Â.Ø. Ê òåîðèè ïðîöåññà îáðàçîâàíèÿ ãàçîãèäðàòà â çàìêíóòîì òåïëîèçîëèðîâàííîì îáúåìå, îïðåññîâàííîì ìåòàíîì / Â.Ø. Øàãàïîâ, À.Ñ. ×èãëèíöåâà, Ñ.Â. Áåëîâà
// Èíæåíåðíî-ôèçè÷åñêèé æóðíàë. 2017. Ò. 90,  5. Ñ. 12081222.
8. Øàãàïîâ, Â.Ø. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå íàãíåòàíèÿ ãèäðàòîîáðàçóþùåãî ãàçà â
ñíåæíûé ìàññèâ, íàñûùåííûé òåì æå ãàçîì / Â.Ø. Øàãàïîâ, À.Ñ. ×èãëèíöåâà, Ñ.Â. Áåëîâà // Òðóäû Èíñòèòóòà ìåõàíèêè èì. Ð.Ð. Ìàâëþòîâà Óôèìñêîãî íàó÷íîãî öåíòðà
ÐÀÍ. 2016. Ò. 11,  2. Ñ. 233239.
9. Õàñàíîâ, Ì.Ê. Èíæåêöèÿ æèäêîãî ñåðîâîäîðîäà â ïëàñò, íàñûùåííûé íåôòüþ è âîäîé / Ì.Ê. Õàñàíîâ // Âåñòíèê Òþìåíñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà. Ôèçèêîìàòåìàòè÷åñêîå ìîäåëèðîâàíèå. Íåôòü, ãàç, ýíåðãåòèêà. 2017. Ò. 3,  2. Ñ. 7284.
10. Shagapov, V.Sh. Theoretical Research of the Gas Hydrate Deposits Development Using the
Injection of Carbon Dioxide / V.Sh. Shagapov, M.K. Khasanov, N.G. Musakaev, Ngoc Hai
Duong // International Journal of Heat and Mass Transfer. 2017. V. 107. P. 347357.
Ìàðàò Êàìèëîâè÷ Õàñàíîâ, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, äîöåíò, êàôåäðà ≪Ïðèêëàäíàÿ èíôîðìàòèêà è ïðîãðàììèðîâàíèå≫, Ñòåðëèòàìàêñêèé ôèëèàë
Áàøêèðñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà (ã. Ñòåðëèòàìàê, Ðîññèéñêàÿ Ôåäåðàöèÿ), hasanovmk@mail.ru.
Ãóçàëü Ðèíàòîâíà Ðàôèêîâà, êàíäèäàò ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ó÷åíûé ñåêðåòàðü, Èíñòèòóò ìåõàíèêè èì. Ð.Ð. Ìàâëþòîâà îáîñîáëåííîå ñòðóêòóðíîå ïîäðàçäåëåíèå Óôèìñêîãî ôåäåðàëüíîãî èññëåäîâàòåëüñêîãî öåíòðà Ðîññèéñêîé àêàäåìèè
íàóê (ã. Óôà, Ðîññèéñêàÿ Ôåäåðàöèÿ), rakova− guzal@mail.ru.
Ïîñòóïèëà â ðåäàêöèþ 20 ôåâðàëÿ 2018 ã.
82
Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 2, pp. 7382