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The mathematical model of the gas hydrate deposit development in permafrost
International Journal of Heat and Mass Transfer 118 (2018) 455–461
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The mathematical model of the gas hydrate deposit development in permafrost N.G. Musakaev a,b,c,⇑, M.K. Khasanov a, S.L. Borodin a,b a
Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen, Russia c Industrial University of Tyumen, Tyumen, Russia b
a r t i c l e
i n f o
Article history: Received 20 July 2017 Received in revised form 23 October 2017 Accepted 31 October 2017
Keywords: Gas hydrates Porous medium Filtration Methane Hydrate dissociation
a b s t r a c t This paper presents the mathematical model of the gas extraction from a reservoir initially saturated with methane and its hydrate, under conditions of negative (below 0 °C) initial temperature of the reservoir. The algorithm is proposed and the numerical scheme is constructed which makes it possible to find the main parameters of the nonisothermal filtration flow in a hydrate-saturated reservoir, taking into account the hydrate decomposition to gas and ice. The analysis of the influences of the mass flow rate of gas extraction, the porous medium permeability, the initial reservoir temperature, and the initial hydrate saturation on the regime and rate of the gas hydrate decomposition was carried out. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Natural gas is one of the main energy carriers, for example, in 2014 it accounts for 21.6% of the world’s electricity production, and its share continues to grow [1]. The main component of natural gas is methane (77–99%). The largest reserves of methane are concentrated in the hydrated state. Their total volume (20000 trillion m3) is two orders of magnitude higher than the volume of traditional methane recoverable reserves (250 trillion m3), and methane hydrate contains more than 50% of carbon from total known world hydrocarbon reserves [2]. Thus, given the increasing demand and the largest amount of fossil fuels in nature, methane from gas hydrates is the most promising source of energy [3,4]. The extraction of methane from gas hydrates causes difficulties due to their solid form. Existing methods rely on dissociation, in which gas hydrates decompose to components [2,4]. The main methods for developing gas hydrate deposits include: pressure reduction, heating and inhibitor injection. The technology of replacement of methane in hydrates by carbon dioxide is attracting attention [5–8]. At the pressure drop method of the gas hydrate deposits development, it is necessary that the pressure in the reservoir becomes lower than the equilibrium hydrate decomposi⇑ Corresponding author at: Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen, Russia. E-mail addresses: [email protected] (N.G. Musakaev), [email protected] (M.K. Khasanov), [email protected] (S.L. Borodin). https://doi.org/10.1016/j.ijheatmasstransfer.2017.10.127 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
tion pressure. In this case the gas hydrate begins to decompose to gas and water (ice) absorbing heat. If the hydrate decomposition occurs to gas and ice then the energy costs of the gas hydrate deposits developing are lower than in the case when the hydrate decomposition occurs to gas and water, since the specific heat of the phase transition hydrate M ice and gas (160 kJ/kg) is much lower than the heat of the phase transition hydrate M water and gas (430 kJ/kg) [9,10]. Of course, this option is only applicable for deposits with the thermodynamic conditions allow the existence of ice. The importance of a theoretical study of the pressure drop method for the gas hydrate deposits development is quite obvious. The results obtained in this study significantly reduce the quantity of necessary experimental and field data, and the clear view of the studied processes makes it possible to effectively control them [7,11]. The mathematical models of the process of the gas hydrate decomposition in a porous medium to gas and water (ice) are presented in the works [10,12–17]. In these works analytical solutions of this problem are constructed and, in particular, showed that two regimes are possible: with the frontal surface or with the extended zone of the gas hydrate dissociation. But in the majority of the works analytical solutions were obtained for the case when the phase transitions occurs at the frontal surface, which gives an adequate mathematical description only for a limited range of parameters characterizing the state of the system and the intensity of the
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impact on the reservoir, namely, at the low gas extraction rates and the low permeability of porous media. In this paper, we propose the mathematical model of the gas hydrate decomposition in the case of a negative (below 0 °C) initial reservoir temperature (before the start of well operation). This model takes into account the main features of the process: nonisothermal gas filtration, phase transitions, real gas properties, adiabatic cooling effect, Joule-Thomson effect. Numerical solutions of the problem of the methane hydrate decomposition in a porous medium with both the frontal surface and extended zone of phase transitions are obtained. 2. Mathematical model Let us consider the problem of the gas extraction from the horizontal reservoir of a limited size rk, the top and bottom of which are impenetrable (Fig. 1). We assume that the reservoir is homogeneous and isotropic, and also neglect the influence of the upper and lower boundaries. Then we can assume that the problem is onedimensional and the parameters of filtration flow depend only on the radial coordinate r and time t. Let the reservoir in the initial state is saturated with gas and its hydrate, the pressure p0 and temperature T0 of which correspond to the thermodynamic conditions of the hydrate stability:
t ¼ 0; r 2 ½r w ; r k :
T ¼ T 0 < 0 C; p ¼ p0 ;
Sh ¼ Sh0 ;
Sg ¼ 1 Sh0 ; where p is the pressure; T is the temperature; rw and rk are the well and reservoir radiuses; Sj (j = i, h, g) is the saturation of pores with jth phase (i – ice, h – hydrate, g – gas); Sh0 is the initial hydrate saturation. The coordinate r starts from the central axis of the well. At the left border of the reservoir (r = rw) through the well, which has opened the reservoir for the entire thickness, the gas extraction with a constant mass flow rate Q (per unit of the well height) begins:
t > 0; r ¼ r w :
r
zg l g Rg T @p2 ; ¼Q @r pkS3g
@T ¼ 0: @r
ð2:1Þ
Here k is the reservoir absolute permeability; lg and zg are the dynamic viscosity and the gas compressibility factor; Rg is the specific gas constant. It is also possible to specify another boundary condition, namely: through the well which has opened the reservoir for the entire thickness, gas is extracted at a constant bottomhole pressure pw:
t > 0; r ¼ r w :
@T ¼ 0: @r
ð2:2Þ
At the gas extraction, the pressure in the reservoir begins to decrease and, correspondingly, when this pressure decreases below the equilibrium pressure of hydrate formation, the gas hydrate decomposes. Fig. 2 shows the phase diagram for the H2O-CH4 system. In this diagram, the g-i-h and g-w-h lines determine the three-phase equilibria of the systems ‘‘methane-ice-hyd rate” and ‘‘methane-water-hydrate”, ‘‘gas-water-hydrate”, and line i-w is two-phase ‘‘water-ice” equilibrium. The lines g-i-h and g-wh correspond to the equilibrium parameters of the gas hydrate decomposition to ice and water, respectively. At the gas extraction, the hydrate decomposition takes place due to pressure drop below equilibrium. And three characteristic zones can appear in the reservoir (Fig. 3): the near (the first zone), in this zone the reservoir pores saturated with gas and ice; the intermediate (the second zone), in this zone gas, ice and hydrate are in equilibrium; and the far (the third zone), in this zone the reservoir pores saturated with methane and its hydrate. Accordingly, the gas hydrate dissociation begins at the surface between the second and third zones (r = r(d)), and ends at the boundary between the first and second zones (r = r(n)). The initial hydrate saturation (Sh0) is equal to the hydrate saturation of the third zone and is constant at all points of this zone. The conditions for temperature and pressure at the right boundary are following:
t > 0; r ¼ r k :
@p ¼ 0; @r
@T ¼ 0: @r
At the modelling, we assume the following assumptions: the hydrate is a two-component system with a constant mass concentration of gas G; the temperatures of the porous medium, gas, hydrate and ice at each point of the reservoir are equal; the reservoir porosity m is constant; the porous medium skeleton, gas hydrate and ice are incompressible and immobile. The system of basic equations describing the processes of filtration and heat transfer in a porous medium, representing the laws of conservation of masses (2.3), (2.4) and energy (2.6), Darcy’s law (2.5) and the equation of state for a gas (2.7). This system in the radial case taking into account accepted assumptions has the form [10,18–20]:
@ 1 @ ðq Sg þ qh Sh GÞ þ ðrqg Sg v g Þ ¼ 0; @t g r @r
ð2:3Þ
@ ðq Si þ ð1 GÞqh Sh Þ ¼ 0; @t i
ð2:4Þ
mSg v g ¼
Fig. 1. Scheme of the modeling area.
p ¼ pw < p0 ;
kg @p
lg @r
;
Fig. 2. Phase diagram of the «H2O-CH4» system.
ð2:5Þ
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Fig. 3. Scheme of the problem of the gas hydrate dissociation in a reservoir with a decrease in pressure.
qc
@T @ @T @ @T @p ¼ k þ ðmqh Lh Sh Þ qg cg mSg v g þe @t @r @r @t @r @r @p þ qg cg mSg g ; @t
p ¼ zg qg Rg T;
e¼
ficiently large time scales, the process is limited not by the process kinetics but by the heat transfer in a porous medium [22–24].
ð2:6Þ ð2:7Þ
1 T @zg ; qg cg zg @T p
g¼ X
qc ¼ ð1 mÞqsk csk þ m k ¼ ð1 mÞksk þ m
X
j¼g;i;h
e;
Sj qj cj ;
j¼g;i;h
Sj k j ;
1
qg c g
X
Sj ¼ 1;
j¼g;i;h
where qj and cj (j = g, i, h) are the density and specific heat of the j-th phase; qsk, ksk and csk are the density, thermal conductivity, and specific heat of the porous medium skeleton; vg and lg are the velocity and dynamic viscosity of the gas phase; e and g are the throttling coefficient (the Joule-Thomson coefficient) and the adiabatic coefficient; Lh is the latent heat of the hydrate decomposition; qc and k are the specific volumetric heat capacity and the thermal conductivity of the system ‘‘porous skeleton – saturants”.For the dependence of the permeability coefficient for gas kg on the gas saturation Sg we use the Kozeny formula: 3
kg ¼ kSg :
ð2:8Þ
The gas compressibility factor (zg) can found using the LatonovGurevich equation:
zg ¼ ½0:17376 lnðT=T c Þ þ 0:73
p=pc
þ 0:1 p=pc ;
ð2:9Þ
where Tc and pc are the empirical critical parameters for a hydrocarbon gas. The values of temperature and pressure in the zone of the hydrate decomposition are related by the condition of phase equilibrium [9,10]:
T ¼ T 0 þ T lnðp=ps0 Þ;
ð2:10Þ
where ps0 is the equilibrium pressure corresponding to the initial reservoir temperature T0; T⁄ is the empirical parameter depending on the type of gas hydrate and phases to which it decomposes. It should be noted that the gas hydrate decomposition is accompanied by nonequilibrium processes [11,21,22]. However, the proposed mathematical model does not take into account the kinetics of the process, since the characteristic relaxation times of the nonequilibrium are, as a rule, very small in comparison with the time intervals that are of interest in the problem under consideration [10,16]. Therefore, the proposed mathematical model with an equilibrium scheme is valid for the limiting case, when, due to suf-
3. Computational scheme for the problem solving Let us obtain differential equations for pressure and temperature (the pressure and thermal diffusivity equations). To this end, we express the gas density from Eq. (2.7) and the gas velocity from the Darcy’s Eq. (2.5). Then we substitute obtained expressions for qg and vg to Eqs. (2.3) and (2.6). Then Eqs. (2.3) and (2.6) can be represented as follows:
@p p @T p @zg p @Sg p qh @Sh zg Rg T 1 ¼ þ G þ @t T @t zg @t Sg @t Sg qg @t mSg r " # qg kg @p @ ; r @r lg @r @T qg cg ¼ @t qc
kg @p @T kg þe lg @r @r lg @ @T mqh Lh rk þ @r @r qc
ð3:1Þ
! 2 @p @p 1 1 þ þ mSg g @r @t qc r @Sh : @t
ð3:2Þ
From Eq. (2.4) we have qi ðSi Si0 Þ þ ð1 GÞqh ðSh Sh0 Þ ¼ 0 (Si0 = 0 is the initial ice saturation). Then, taking into account the relaP tionship j¼g;l;h Sj ¼ 1, we can write:
Si ¼
qh ð1 GÞðSh0 Sh Þ; qi
ð3:3Þ
Sg ¼ 1
qh q ð1 GÞSh0 1 h ð1 GÞ Sh : qi qi
ð3:4Þ
As you can see, in the five basic Eqs. (2.9), (3.1)–(3.4) there are six unknowns (p, zg, T, Sh, Si, Sg), therefore the system is not closed, and its closing can be described by the following algorithm. 1. We discretize the space along the radial coordinate and set the parameters values at the initial time. 2. From Eq. (3.1) by the tridiagonal matrix algorithm using the implicit scheme we calculate the pressure at the all space points at the new time step (pnew ; k = 0, 1, . . . , N; N is the number of k steps by the space variable r). At the pressure calculation we iter iter use the iterative values of other unknowns ðT iter k ; ðzg Þk ; ðSh Þk ; iter ðSi Þiter k ; ðSg Þk Þ. 3. From Eq. (2.9) we calculate the gas compressibility factor
ðzg Þnew , using the values pnew and T iter k . k k 4. From Eq. (3.2) by the tridiagonal matrix algorithm using the implicit scheme we calculate the temperature at the new time iter iter step (T new ) using the values pnew ; ðzg Þnew ; ðSh Þiter k k ; ðSi Þk ; ðSg Þk . k k
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5. Then we calculate the hydrate saturation at the new time step ¼ ðSh Þiter ðDSh Þk . The value of ðDSh Þk is the minimum ðSh Þnew k k from the following three numbers: 5.1. ðSh Þiter k 5.2. The amount of hydrate that can decompose as the pressure increases from pnew to the equilibrium pressure of k hydrate formation ph:
ðDSph Þk ¼
ðph pnew Þ qg ðpnew ; T new Þ ðSg Þiter k k k k ; pnew Gqh k
where ph can be founded from the formula (2.10) at ; the gas density is calculated from the equation T ¼ T new k of state (2.7). 5.3. The amount of hydrate that can decompose as the temperato the equilibrium temperature of ture decreases from T new k hydrate formation Th:
ðDSTh Þk ¼
iter iter ðT new T h Þ qcðpnew ; T new ; ðSh Þiter k k k ; ðSg Þk ; ðSi Þk Þ k ; qh Lh m
where Th can be founded from the formula (2.10) at p ¼ pnew : k 6. From Eq. (3.3) we calculate the ice saturation ðSi Þnew , using the k . value ðSh Þnew k , using the 7. From Eq. (3.4) we calculate the gas saturation ðSg Þnew k value ðSh Þnew . k 8. Next we find the maximum relative error between iterative and new parameters, and if it satisfies the specified accuracy, then we go to the next time step, otherwise we pass to the new iteration (to step 2). This algorithm, due to using the absolutely stable implicit scheme, converges at any steps in time and space. The program testing was carried out by comparing the numerical experiment results with the calculations carried out taking into account self-similar solutions from [10]. This comparison showed a good agreement. 4. Calculations results Consider the features of the gas hydrates decomposition to gas and ice during the gas extraction from a reservoir initially saturated with methane and its hydrate. Calculations, unless otherwise specified, were performed with the following parameters [9,10,25– 27]: p0 = 2.8 MPa; T0 = 270.15 K (3°C); rw = 0.1 m; rk = 500 m; m = 0.1; k = 1014 m2; Sh0 = 0.2; pc = 4.599 MPa; Tc = 190.56 K; T⁄ = 30 K; ps0 = 2.54 MPa; csk = 1000 J/(kgK); ch = 2080 J/(kgK); ci = 2060 J/(kgK); ksk = 1.5 W/(mK); kh = 0.45 W/(mK); ki = 2.2 W/ (mK); Rg = 518.3 J/(kgK); qsk = 2000 kg/m3; qh = 900 kg/m3; qi = 900 kg/m3; G = 0.12; Lh = 166 kJ/kg; the spatial coordinate step Dr = 0.01 m at r 2 [rw, 10], Dr = 0.1 m at r 2 (10, 100] and Dr = 1 m at r 2 (100, rk]; the time step Dt = 600 c. Values cg, kg, lg were determined by interpolating tabular data for methane. Fig. 4 shows the evolution over time of the pressure, temperature and hydrate saturation at constant pressure at the well boundary (boundary condition (2.2)) and at a negative initial reservoir temperature (T0 < 0 °C). It can be seen that over time the length of the second (intermediate) zone increases. This is due to the increase over time of the zone size in which the pressure falls below the equilibrium methane hydrate decomposition pressure. The temperature decreases owing to the gas hydrate decomposition. That is, the temperature in the second zone falls below the initial reservoir temperature (Fig. 4), which is due to absorption of the phase transition latent heat. Consequently, in the case of a negative initial reservoir temperature, the gas hydrate dissociation will always occurs to gas and ice.
Fig. 4. The pressure, temperature and hydrate saturation depending on radial coordinate r at the different time points after the well operation starting t. Lines 1 and 2 correspond to t = 10 and 50 days; pw = 2 MPa; r(n)1 and r(n)2 are the coordinates of the boundary between the first and second zones at t = 10 and 50 days, respectively; r(d)1 and r(d)2 are the coordinates of the boundary between the second and third zones at t = 10 and 50 days, respectively.
Fig. 5 shows the distributions along the reservoir length of the pressure, temperature and hydrate saturation for different values of the mass flow rate of gas Q after the 30 days of the well operation. Here and in the future, the condition (2.1) is adopted at the left reservoir boundary. It is seen that, depending on the value of Q, three cases are possible. In the first case, at relatively low values of the mass flow rate of gas extraction (curve 1), there is no decomposition of the hydrate (Sh is constant value over the entire reservoir length) and only free gas is extracted. In the second case, the hydrate saturation at some distance from the well changes abruptly (curve 2), that is the frontal scheme of the gas hydrate decomposition. At large flow rates (and accordingly the pressure gradient) an extended zone of phase transitions occurs (curves 3 and 4). In this zone gas, ice, and hydrate are in equilibrium. Fig. 5 shows that a decrease in Q leads to a decrease in the length of the intermediate zone and to a decrease in the gas hydrate amount that dissociates in this zone. That is, with a decrease in the mass flow rate of gas extraction, the scheme of the process more and more approximates the scheme with the frontal surface of the hydrate decomposition considered in [13–17]. The temperature in the first region decreases as it approaches the well due to the effects of adiabatic cooling and Joule-Thomson. Calculations show that the Joule-Thomson effect makes a greater contribution to the decrease in temperature in the first region (Fig. 6). This can be explained by the fact that the pressure gradient increases with the approach to the well, and the higher it is the higher the temperature decrease due to the Joule- Thomson effect. In the work a study was made of the effects of the initial reservoir parameters on the gas hydrate decomposition regime, namely, with the frontal boundary or with the extended zone of hydrate decomposition.
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459
Fig. 7. Distributions by the radial coordinate r of the pressure and hydrate saturation at the different values of the reservoir permeability k. Lines 1, 2 and 3 correspond to k = 1013, 1014 b 5 1015 m2; Q = 0.003 kg/(ms).
Fig. 5. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the gas mass flow rate Q. Lines 1, 2, 3 and 4 correspond to Q = 0.001, 0.003, 0.005 and 0.007 kg/(ms).
Fig. 6. Distributions by the radial coordinate r of the temperature with and without allowance of the effects of adiabatic cooling and Joule-Thomson. Line 1 – without adiabatic cooling and Joule-Thomson effects, 2 – without Joule-Thomson effect, 3 – without adiabatic cooling effect, 4 – taking into account both effects; Q = 0.007 kg/ (ms).
Fig. 7 shows the distribution along the reservoir length of the pressure and hydrate saturation for different values of the reservoir permeability after the 30 days of the well operation. It can be seen that the regime with an extended zone of the gas hydrate decomposition is characteristic for reservoirs with low permeability. This is explained by the fact that when the permeability of a porous medium is reduced at the fixed mass flow rate of gas extraction, it is necessary to reduce the pressure at the well and, respectively, in the zone close to the well (Fig. 7). Therefore, as the porous medium permeability decreases, the pressure p(n) decreases at the boundary between the first and second zones. And in accordance with Fig. 5 for the regime with the extended zone of hydrate decomposition corresponds the sufficiently low values of p(n) in comparison with the initial pressure p0. The distributions of parameters along the reservoir length for different values of the initial reservoir temperature T0 and the initial reservoir pressure p0 are shown in Figs. 8 and 9. It can be seen that for the fixed mass flow rate of gas extraction, the regime with
Fig. 8. Distributions by the radial coordinate r of the temperature and hydrate saturation at the different values of the initial reservoir temperature T0. Lines 1 and 2 correspond to T0 = 3 and 5 °C.
the extended zone of the methane hydrate decomposition is characteristic for porous media with a higher reservoir temperature T0 and lower initial reservoir pressure p0. That is, this regime of phase transitions is realized in the reservoirs, the initial state of which is close to the conditions of the gas hydrate decomposition. Fig. 10 shows the distribution along the reservoir length of the pressure, temperature and hydrate saturation at the different values of the initial reservoir hydrate saturation Sh0. Depending on the value of Sh0, three cases are possible. The first case (line 1) is characterized by the absence of hydrate decomposition, in the second case, the dissociation of the gas hydrate completely occurs at the frontal boundary (line 2), in the third case, an extended zone of phase transitions (line 3) occurs. This is explained by the fact that the gas permeability of porous medium decreases with increasing hydrate saturation (formula (2.8)). And this, as noted earlier, leads to a decrease in the pressure p(n) at the boundary between the first and second zones. The evolution in time of the specific (per unit of the well height) volume flow of gas at the well bottom at constant pressure at the
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Fig. 11. Change in time t of the specific volumetric gas flow rate at the well bottom at different values of the initial reservoir hydrate saturation Sh0. Lines 1 and 2 correspond to Sh0 = 0.1 and 0.2.
Fig. 12. Change in time t of the specific cumulative mass of the extracted gas at different values of the initial reservoir hydrate saturation Sh0. The solid line corresponds to Sh0 = 0.2, the dashed line corresponds to Sh0 = 0; rk = 1 km.
Fig. 9. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 2.6 and 2.8 MPa.
kJ/kg; p0 = 2.8 MPa; pw = 2.0 MPa), the dashed lines correspond to the case of the positive initial reservoir temperature T0 = 3 °C (the decomposition of hydrate occurs to gas and water, Lh = 430 kJ/kg; p0 = 3.8 MPa; pw = 3.0 MPa); the graphs are plotted for the same pressure drop Dp = 0.8 MPa. It can be seen that when a hydrate decomposes to gas and ice, then larger amount of gas is produced in comparison the decomposition to gas and water, i.e. this mode of gas production from a reservoir is more profitable. Also, in the case of the negative initial reservoir temperature, the hydrate decomposition occurs at lower pressures, which positively affects the safety of the gas production process. From Fig. 11 it can be seen that the specific volume flow decreases with increasing the initial reservoir hydrate saturation Sh0 which is caused by a decrease in the gas phase permeability kg. Fig. 12 shows comparison of the specific (per unit of the well height) accumulated mass of the extracted gas for two cases: 1) the reservoir in the initial state is saturated with methane and its hydrate (solid line); 2) the reservoir initially filled only with the gas in a free state (dashed line). It can be seen that the second case is characterized by a faster rate of the gas extraction, but the reservoir depletion occurs approximately 30 years after the start of the well operation. And total about 2000 t/m of gas is extracted from the reservoir. The first case is characterized by much longer field operation times (100 years) and a significantly larger specific accumulated mass of the extracted gas (7500 t/m). 5. Conclusion
Fig. 10. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the initial reservoir hydrate saturation Sh0. Lines 1 and 2 correspond to Sh0 = 0.2 and 0.3.
left boundary (pw = const) is shown in Fig. 11. The solid lines correspond to the case of the negative initial reservoir temperature T0 = 3 °C (the decomposition of hydrate occurs to gas and ice, Lh = 166
Mathematical modeling has been carried out and the numerical algorithm has been developed that makes it possible to find the main parameters of the nonisothermal filtration flow in the hydrate-saturated reservoir, taking into account the decomposition of methane hydrate to gas and ice. It is established that for the fixed mass flow rate of gas extraction the regime with the extended zone of the hydrate decomposition is characteristic for reservoirs whose initial state is close to the conditions of the gas hydrate decomposition. Calculations show that the regime with
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the extended zone of the gas hydrate decomposition is realized at a high intensity of the gas extraction and in the reservoirs with low permeability and high initial hydrate saturation. At the same time, a decrease in the gas extraction rate leads to decreasing the length of the gas hydrate decomposition region. It has been established that at a fixed pressure drop in a reservoir, the regime of the gas extraction from the hydrate-saturated reservoir in the case of a negative (T0 < 0 °C) initial temperature and the methane hydrate decomposition to gas and ice is more profit than gas removal from the reservoir with a positive initial temperature and corresponding decomposition of methane hydrate to gas and water. Calculations show that the extraction of gas from the reservoir initially saturated with methane and its hydrate may prove to be more profitable than the operation with the deposit saturated only with gas in a free state, due to the greater amount of accumulated mass of produced gas and longer life of the gas-hydrate field. Acknowledgement This work was carried out at the Sterlitamak Branch of Bashkir State University and financially supported by the Russian Science Foundation (project number 17-79-20001). The computer code is developed using the software of the Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS. We thank Industrial University of Tyumen for granting us the access to libraries.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The mathematical model of the gas hydrate deposit development in permafrost N.G. Musakaev a,b,c,⇑, M.K. Khasanov a, S.L. Borodin a,b a
Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen, Russia c Industrial University of Tyumen, Tyumen, Russia b
a r t i c l e
i n f o
Article history: Received 20 July 2017 Received in revised form 23 October 2017 Accepted 31 October 2017
Keywords: Gas hydrates Porous medium Filtration Methane Hydrate dissociation
a b s t r a c t This paper presents the mathematical model of the gas extraction from a reservoir initially saturated with methane and its hydrate, under conditions of negative (below 0 °C) initial temperature of the reservoir. The algorithm is proposed and the numerical scheme is constructed which makes it possible to find the main parameters of the nonisothermal filtration flow in a hydrate-saturated reservoir, taking into account the hydrate decomposition to gas and ice. The analysis of the influences of the mass flow rate of gas extraction, the porous medium permeability, the initial reservoir temperature, and the initial hydrate saturation on the regime and rate of the gas hydrate decomposition was carried out. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Natural gas is one of the main energy carriers, for example, in 2014 it accounts for 21.6% of the world’s electricity production, and its share continues to grow [1]. The main component of natural gas is methane (77–99%). The largest reserves of methane are concentrated in the hydrated state. Their total volume (20000 trillion m3) is two orders of magnitude higher than the volume of traditional methane recoverable reserves (250 trillion m3), and methane hydrate contains more than 50% of carbon from total known world hydrocarbon reserves [2]. Thus, given the increasing demand and the largest amount of fossil fuels in nature, methane from gas hydrates is the most promising source of energy [3,4]. The extraction of methane from gas hydrates causes difficulties due to their solid form. Existing methods rely on dissociation, in which gas hydrates decompose to components [2,4]. The main methods for developing gas hydrate deposits include: pressure reduction, heating and inhibitor injection. The technology of replacement of methane in hydrates by carbon dioxide is attracting attention [5–8]. At the pressure drop method of the gas hydrate deposits development, it is necessary that the pressure in the reservoir becomes lower than the equilibrium hydrate decomposi⇑ Corresponding author at: Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen, Russia. E-mail addresses: [email protected] (N.G. Musakaev), [email protected] (M.K. Khasanov), [email protected] (S.L. Borodin). https://doi.org/10.1016/j.ijheatmasstransfer.2017.10.127 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
tion pressure. In this case the gas hydrate begins to decompose to gas and water (ice) absorbing heat. If the hydrate decomposition occurs to gas and ice then the energy costs of the gas hydrate deposits developing are lower than in the case when the hydrate decomposition occurs to gas and water, since the specific heat of the phase transition hydrate M ice and gas (160 kJ/kg) is much lower than the heat of the phase transition hydrate M water and gas (430 kJ/kg) [9,10]. Of course, this option is only applicable for deposits with the thermodynamic conditions allow the existence of ice. The importance of a theoretical study of the pressure drop method for the gas hydrate deposits development is quite obvious. The results obtained in this study significantly reduce the quantity of necessary experimental and field data, and the clear view of the studied processes makes it possible to effectively control them [7,11]. The mathematical models of the process of the gas hydrate decomposition in a porous medium to gas and water (ice) are presented in the works [10,12–17]. In these works analytical solutions of this problem are constructed and, in particular, showed that two regimes are possible: with the frontal surface or with the extended zone of the gas hydrate dissociation. But in the majority of the works analytical solutions were obtained for the case when the phase transitions occurs at the frontal surface, which gives an adequate mathematical description only for a limited range of parameters characterizing the state of the system and the intensity of the
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impact on the reservoir, namely, at the low gas extraction rates and the low permeability of porous media. In this paper, we propose the mathematical model of the gas hydrate decomposition in the case of a negative (below 0 °C) initial reservoir temperature (before the start of well operation). This model takes into account the main features of the process: nonisothermal gas filtration, phase transitions, real gas properties, adiabatic cooling effect, Joule-Thomson effect. Numerical solutions of the problem of the methane hydrate decomposition in a porous medium with both the frontal surface and extended zone of phase transitions are obtained. 2. Mathematical model Let us consider the problem of the gas extraction from the horizontal reservoir of a limited size rk, the top and bottom of which are impenetrable (Fig. 1). We assume that the reservoir is homogeneous and isotropic, and also neglect the influence of the upper and lower boundaries. Then we can assume that the problem is onedimensional and the parameters of filtration flow depend only on the radial coordinate r and time t. Let the reservoir in the initial state is saturated with gas and its hydrate, the pressure p0 and temperature T0 of which correspond to the thermodynamic conditions of the hydrate stability:
t ¼ 0; r 2 ½r w ; r k :
T ¼ T 0 < 0 C; p ¼ p0 ;
Sh ¼ Sh0 ;
Sg ¼ 1 Sh0 ; where p is the pressure; T is the temperature; rw and rk are the well and reservoir radiuses; Sj (j = i, h, g) is the saturation of pores with jth phase (i – ice, h – hydrate, g – gas); Sh0 is the initial hydrate saturation. The coordinate r starts from the central axis of the well. At the left border of the reservoir (r = rw) through the well, which has opened the reservoir for the entire thickness, the gas extraction with a constant mass flow rate Q (per unit of the well height) begins:
t > 0; r ¼ r w :
r
zg l g Rg T @p2 ; ¼Q @r pkS3g
@T ¼ 0: @r
ð2:1Þ
Here k is the reservoir absolute permeability; lg and zg are the dynamic viscosity and the gas compressibility factor; Rg is the specific gas constant. It is also possible to specify another boundary condition, namely: through the well which has opened the reservoir for the entire thickness, gas is extracted at a constant bottomhole pressure pw:
t > 0; r ¼ r w :
@T ¼ 0: @r
ð2:2Þ
At the gas extraction, the pressure in the reservoir begins to decrease and, correspondingly, when this pressure decreases below the equilibrium pressure of hydrate formation, the gas hydrate decomposes. Fig. 2 shows the phase diagram for the H2O-CH4 system. In this diagram, the g-i-h and g-w-h lines determine the three-phase equilibria of the systems ‘‘methane-ice-hyd rate” and ‘‘methane-water-hydrate”, ‘‘gas-water-hydrate”, and line i-w is two-phase ‘‘water-ice” equilibrium. The lines g-i-h and g-wh correspond to the equilibrium parameters of the gas hydrate decomposition to ice and water, respectively. At the gas extraction, the hydrate decomposition takes place due to pressure drop below equilibrium. And three characteristic zones can appear in the reservoir (Fig. 3): the near (the first zone), in this zone the reservoir pores saturated with gas and ice; the intermediate (the second zone), in this zone gas, ice and hydrate are in equilibrium; and the far (the third zone), in this zone the reservoir pores saturated with methane and its hydrate. Accordingly, the gas hydrate dissociation begins at the surface between the second and third zones (r = r(d)), and ends at the boundary between the first and second zones (r = r(n)). The initial hydrate saturation (Sh0) is equal to the hydrate saturation of the third zone and is constant at all points of this zone. The conditions for temperature and pressure at the right boundary are following:
t > 0; r ¼ r k :
@p ¼ 0; @r
@T ¼ 0: @r
At the modelling, we assume the following assumptions: the hydrate is a two-component system with a constant mass concentration of gas G; the temperatures of the porous medium, gas, hydrate and ice at each point of the reservoir are equal; the reservoir porosity m is constant; the porous medium skeleton, gas hydrate and ice are incompressible and immobile. The system of basic equations describing the processes of filtration and heat transfer in a porous medium, representing the laws of conservation of masses (2.3), (2.4) and energy (2.6), Darcy’s law (2.5) and the equation of state for a gas (2.7). This system in the radial case taking into account accepted assumptions has the form [10,18–20]:
@ 1 @ ðq Sg þ qh Sh GÞ þ ðrqg Sg v g Þ ¼ 0; @t g r @r
ð2:3Þ
@ ðq Si þ ð1 GÞqh Sh Þ ¼ 0; @t i
ð2:4Þ
mSg v g ¼
Fig. 1. Scheme of the modeling area.
p ¼ pw < p0 ;
kg @p
lg @r
;
Fig. 2. Phase diagram of the «H2O-CH4» system.
ð2:5Þ
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Fig. 3. Scheme of the problem of the gas hydrate dissociation in a reservoir with a decrease in pressure.
qc
@T @ @T @ @T @p ¼ k þ ðmqh Lh Sh Þ qg cg mSg v g þe @t @r @r @t @r @r @p þ qg cg mSg g ; @t
p ¼ zg qg Rg T;
e¼
ficiently large time scales, the process is limited not by the process kinetics but by the heat transfer in a porous medium [22–24].
ð2:6Þ ð2:7Þ
1 T @zg ; qg cg zg @T p
g¼ X
qc ¼ ð1 mÞqsk csk þ m k ¼ ð1 mÞksk þ m
X
j¼g;i;h
e;
Sj qj cj ;
j¼g;i;h
Sj k j ;
1
qg c g
X
Sj ¼ 1;
j¼g;i;h
where qj and cj (j = g, i, h) are the density and specific heat of the j-th phase; qsk, ksk and csk are the density, thermal conductivity, and specific heat of the porous medium skeleton; vg and lg are the velocity and dynamic viscosity of the gas phase; e and g are the throttling coefficient (the Joule-Thomson coefficient) and the adiabatic coefficient; Lh is the latent heat of the hydrate decomposition; qc and k are the specific volumetric heat capacity and the thermal conductivity of the system ‘‘porous skeleton – saturants”.For the dependence of the permeability coefficient for gas kg on the gas saturation Sg we use the Kozeny formula: 3
kg ¼ kSg :
ð2:8Þ
The gas compressibility factor (zg) can found using the LatonovGurevich equation:
zg ¼ ½0:17376 lnðT=T c Þ þ 0:73
p=pc
þ 0:1 p=pc ;
ð2:9Þ
where Tc and pc are the empirical critical parameters for a hydrocarbon gas. The values of temperature and pressure in the zone of the hydrate decomposition are related by the condition of phase equilibrium [9,10]:
T ¼ T 0 þ T lnðp=ps0 Þ;
ð2:10Þ
where ps0 is the equilibrium pressure corresponding to the initial reservoir temperature T0; T⁄ is the empirical parameter depending on the type of gas hydrate and phases to which it decomposes. It should be noted that the gas hydrate decomposition is accompanied by nonequilibrium processes [11,21,22]. However, the proposed mathematical model does not take into account the kinetics of the process, since the characteristic relaxation times of the nonequilibrium are, as a rule, very small in comparison with the time intervals that are of interest in the problem under consideration [10,16]. Therefore, the proposed mathematical model with an equilibrium scheme is valid for the limiting case, when, due to suf-
3. Computational scheme for the problem solving Let us obtain differential equations for pressure and temperature (the pressure and thermal diffusivity equations). To this end, we express the gas density from Eq. (2.7) and the gas velocity from the Darcy’s Eq. (2.5). Then we substitute obtained expressions for qg and vg to Eqs. (2.3) and (2.6). Then Eqs. (2.3) and (2.6) can be represented as follows:
@p p @T p @zg p @Sg p qh @Sh zg Rg T 1 ¼ þ G þ @t T @t zg @t Sg @t Sg qg @t mSg r " # qg kg @p @ ; r @r lg @r @T qg cg ¼ @t qc
kg @p @T kg þe lg @r @r lg @ @T mqh Lh rk þ @r @r qc
ð3:1Þ
! 2 @p @p 1 1 þ þ mSg g @r @t qc r @Sh : @t
ð3:2Þ
From Eq. (2.4) we have qi ðSi Si0 Þ þ ð1 GÞqh ðSh Sh0 Þ ¼ 0 (Si0 = 0 is the initial ice saturation). Then, taking into account the relaP tionship j¼g;l;h Sj ¼ 1, we can write:
Si ¼
qh ð1 GÞðSh0 Sh Þ; qi
ð3:3Þ
Sg ¼ 1
qh q ð1 GÞSh0 1 h ð1 GÞ Sh : qi qi
ð3:4Þ
As you can see, in the five basic Eqs. (2.9), (3.1)–(3.4) there are six unknowns (p, zg, T, Sh, Si, Sg), therefore the system is not closed, and its closing can be described by the following algorithm. 1. We discretize the space along the radial coordinate and set the parameters values at the initial time. 2. From Eq. (3.1) by the tridiagonal matrix algorithm using the implicit scheme we calculate the pressure at the all space points at the new time step (pnew ; k = 0, 1, . . . , N; N is the number of k steps by the space variable r). At the pressure calculation we iter iter use the iterative values of other unknowns ðT iter k ; ðzg Þk ; ðSh Þk ; iter ðSi Þiter k ; ðSg Þk Þ. 3. From Eq. (2.9) we calculate the gas compressibility factor
ðzg Þnew , using the values pnew and T iter k . k k 4. From Eq. (3.2) by the tridiagonal matrix algorithm using the implicit scheme we calculate the temperature at the new time iter iter step (T new ) using the values pnew ; ðzg Þnew ; ðSh Þiter k k ; ðSi Þk ; ðSg Þk . k k
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5. Then we calculate the hydrate saturation at the new time step ¼ ðSh Þiter ðDSh Þk . The value of ðDSh Þk is the minimum ðSh Þnew k k from the following three numbers: 5.1. ðSh Þiter k 5.2. The amount of hydrate that can decompose as the pressure increases from pnew to the equilibrium pressure of k hydrate formation ph:
ðDSph Þk ¼
ðph pnew Þ qg ðpnew ; T new Þ ðSg Þiter k k k k ; pnew Gqh k
where ph can be founded from the formula (2.10) at ; the gas density is calculated from the equation T ¼ T new k of state (2.7). 5.3. The amount of hydrate that can decompose as the temperato the equilibrium temperature of ture decreases from T new k hydrate formation Th:
ðDSTh Þk ¼
iter iter ðT new T h Þ qcðpnew ; T new ; ðSh Þiter k k k ; ðSg Þk ; ðSi Þk Þ k ; qh Lh m
where Th can be founded from the formula (2.10) at p ¼ pnew : k 6. From Eq. (3.3) we calculate the ice saturation ðSi Þnew , using the k . value ðSh Þnew k , using the 7. From Eq. (3.4) we calculate the gas saturation ðSg Þnew k value ðSh Þnew . k 8. Next we find the maximum relative error between iterative and new parameters, and if it satisfies the specified accuracy, then we go to the next time step, otherwise we pass to the new iteration (to step 2). This algorithm, due to using the absolutely stable implicit scheme, converges at any steps in time and space. The program testing was carried out by comparing the numerical experiment results with the calculations carried out taking into account self-similar solutions from [10]. This comparison showed a good agreement. 4. Calculations results Consider the features of the gas hydrates decomposition to gas and ice during the gas extraction from a reservoir initially saturated with methane and its hydrate. Calculations, unless otherwise specified, were performed with the following parameters [9,10,25– 27]: p0 = 2.8 MPa; T0 = 270.15 K (3°C); rw = 0.1 m; rk = 500 m; m = 0.1; k = 1014 m2; Sh0 = 0.2; pc = 4.599 MPa; Tc = 190.56 K; T⁄ = 30 K; ps0 = 2.54 MPa; csk = 1000 J/(kgK); ch = 2080 J/(kgK); ci = 2060 J/(kgK); ksk = 1.5 W/(mK); kh = 0.45 W/(mK); ki = 2.2 W/ (mK); Rg = 518.3 J/(kgK); qsk = 2000 kg/m3; qh = 900 kg/m3; qi = 900 kg/m3; G = 0.12; Lh = 166 kJ/kg; the spatial coordinate step Dr = 0.01 m at r 2 [rw, 10], Dr = 0.1 m at r 2 (10, 100] and Dr = 1 m at r 2 (100, rk]; the time step Dt = 600 c. Values cg, kg, lg were determined by interpolating tabular data for methane. Fig. 4 shows the evolution over time of the pressure, temperature and hydrate saturation at constant pressure at the well boundary (boundary condition (2.2)) and at a negative initial reservoir temperature (T0 < 0 °C). It can be seen that over time the length of the second (intermediate) zone increases. This is due to the increase over time of the zone size in which the pressure falls below the equilibrium methane hydrate decomposition pressure. The temperature decreases owing to the gas hydrate decomposition. That is, the temperature in the second zone falls below the initial reservoir temperature (Fig. 4), which is due to absorption of the phase transition latent heat. Consequently, in the case of a negative initial reservoir temperature, the gas hydrate dissociation will always occurs to gas and ice.
Fig. 4. The pressure, temperature and hydrate saturation depending on radial coordinate r at the different time points after the well operation starting t. Lines 1 and 2 correspond to t = 10 and 50 days; pw = 2 MPa; r(n)1 and r(n)2 are the coordinates of the boundary between the first and second zones at t = 10 and 50 days, respectively; r(d)1 and r(d)2 are the coordinates of the boundary between the second and third zones at t = 10 and 50 days, respectively.
Fig. 5 shows the distributions along the reservoir length of the pressure, temperature and hydrate saturation for different values of the mass flow rate of gas Q after the 30 days of the well operation. Here and in the future, the condition (2.1) is adopted at the left reservoir boundary. It is seen that, depending on the value of Q, three cases are possible. In the first case, at relatively low values of the mass flow rate of gas extraction (curve 1), there is no decomposition of the hydrate (Sh is constant value over the entire reservoir length) and only free gas is extracted. In the second case, the hydrate saturation at some distance from the well changes abruptly (curve 2), that is the frontal scheme of the gas hydrate decomposition. At large flow rates (and accordingly the pressure gradient) an extended zone of phase transitions occurs (curves 3 and 4). In this zone gas, ice, and hydrate are in equilibrium. Fig. 5 shows that a decrease in Q leads to a decrease in the length of the intermediate zone and to a decrease in the gas hydrate amount that dissociates in this zone. That is, with a decrease in the mass flow rate of gas extraction, the scheme of the process more and more approximates the scheme with the frontal surface of the hydrate decomposition considered in [13–17]. The temperature in the first region decreases as it approaches the well due to the effects of adiabatic cooling and Joule-Thomson. Calculations show that the Joule-Thomson effect makes a greater contribution to the decrease in temperature in the first region (Fig. 6). This can be explained by the fact that the pressure gradient increases with the approach to the well, and the higher it is the higher the temperature decrease due to the Joule- Thomson effect. In the work a study was made of the effects of the initial reservoir parameters on the gas hydrate decomposition regime, namely, with the frontal boundary or with the extended zone of hydrate decomposition.
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459
Fig. 7. Distributions by the radial coordinate r of the pressure and hydrate saturation at the different values of the reservoir permeability k. Lines 1, 2 and 3 correspond to k = 1013, 1014 b 5 1015 m2; Q = 0.003 kg/(ms).
Fig. 5. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the gas mass flow rate Q. Lines 1, 2, 3 and 4 correspond to Q = 0.001, 0.003, 0.005 and 0.007 kg/(ms).
Fig. 6. Distributions by the radial coordinate r of the temperature with and without allowance of the effects of adiabatic cooling and Joule-Thomson. Line 1 – without adiabatic cooling and Joule-Thomson effects, 2 – without Joule-Thomson effect, 3 – without adiabatic cooling effect, 4 – taking into account both effects; Q = 0.007 kg/ (ms).
Fig. 7 shows the distribution along the reservoir length of the pressure and hydrate saturation for different values of the reservoir permeability after the 30 days of the well operation. It can be seen that the regime with an extended zone of the gas hydrate decomposition is characteristic for reservoirs with low permeability. This is explained by the fact that when the permeability of a porous medium is reduced at the fixed mass flow rate of gas extraction, it is necessary to reduce the pressure at the well and, respectively, in the zone close to the well (Fig. 7). Therefore, as the porous medium permeability decreases, the pressure p(n) decreases at the boundary between the first and second zones. And in accordance with Fig. 5 for the regime with the extended zone of hydrate decomposition corresponds the sufficiently low values of p(n) in comparison with the initial pressure p0. The distributions of parameters along the reservoir length for different values of the initial reservoir temperature T0 and the initial reservoir pressure p0 are shown in Figs. 8 and 9. It can be seen that for the fixed mass flow rate of gas extraction, the regime with
Fig. 8. Distributions by the radial coordinate r of the temperature and hydrate saturation at the different values of the initial reservoir temperature T0. Lines 1 and 2 correspond to T0 = 3 and 5 °C.
the extended zone of the methane hydrate decomposition is characteristic for porous media with a higher reservoir temperature T0 and lower initial reservoir pressure p0. That is, this regime of phase transitions is realized in the reservoirs, the initial state of which is close to the conditions of the gas hydrate decomposition. Fig. 10 shows the distribution along the reservoir length of the pressure, temperature and hydrate saturation at the different values of the initial reservoir hydrate saturation Sh0. Depending on the value of Sh0, three cases are possible. The first case (line 1) is characterized by the absence of hydrate decomposition, in the second case, the dissociation of the gas hydrate completely occurs at the frontal boundary (line 2), in the third case, an extended zone of phase transitions (line 3) occurs. This is explained by the fact that the gas permeability of porous medium decreases with increasing hydrate saturation (formula (2.8)). And this, as noted earlier, leads to a decrease in the pressure p(n) at the boundary between the first and second zones. The evolution in time of the specific (per unit of the well height) volume flow of gas at the well bottom at constant pressure at the
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Fig. 11. Change in time t of the specific volumetric gas flow rate at the well bottom at different values of the initial reservoir hydrate saturation Sh0. Lines 1 and 2 correspond to Sh0 = 0.1 and 0.2.
Fig. 12. Change in time t of the specific cumulative mass of the extracted gas at different values of the initial reservoir hydrate saturation Sh0. The solid line corresponds to Sh0 = 0.2, the dashed line corresponds to Sh0 = 0; rk = 1 km.
Fig. 9. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 2.6 and 2.8 MPa.
kJ/kg; p0 = 2.8 MPa; pw = 2.0 MPa), the dashed lines correspond to the case of the positive initial reservoir temperature T0 = 3 °C (the decomposition of hydrate occurs to gas and water, Lh = 430 kJ/kg; p0 = 3.8 MPa; pw = 3.0 MPa); the graphs are plotted for the same pressure drop Dp = 0.8 MPa. It can be seen that when a hydrate decomposes to gas and ice, then larger amount of gas is produced in comparison the decomposition to gas and water, i.e. this mode of gas production from a reservoir is more profitable. Also, in the case of the negative initial reservoir temperature, the hydrate decomposition occurs at lower pressures, which positively affects the safety of the gas production process. From Fig. 11 it can be seen that the specific volume flow decreases with increasing the initial reservoir hydrate saturation Sh0 which is caused by a decrease in the gas phase permeability kg. Fig. 12 shows comparison of the specific (per unit of the well height) accumulated mass of the extracted gas for two cases: 1) the reservoir in the initial state is saturated with methane and its hydrate (solid line); 2) the reservoir initially filled only with the gas in a free state (dashed line). It can be seen that the second case is characterized by a faster rate of the gas extraction, but the reservoir depletion occurs approximately 30 years after the start of the well operation. And total about 2000 t/m of gas is extracted from the reservoir. The first case is characterized by much longer field operation times (100 years) and a significantly larger specific accumulated mass of the extracted gas (7500 t/m). 5. Conclusion
Fig. 10. Distributions by the radial coordinate r of the pressure, temperature and hydrate saturation at the different values of the initial reservoir hydrate saturation Sh0. Lines 1 and 2 correspond to Sh0 = 0.2 and 0.3.
left boundary (pw = const) is shown in Fig. 11. The solid lines correspond to the case of the negative initial reservoir temperature T0 = 3 °C (the decomposition of hydrate occurs to gas and ice, Lh = 166
Mathematical modeling has been carried out and the numerical algorithm has been developed that makes it possible to find the main parameters of the nonisothermal filtration flow in the hydrate-saturated reservoir, taking into account the decomposition of methane hydrate to gas and ice. It is established that for the fixed mass flow rate of gas extraction the regime with the extended zone of the hydrate decomposition is characteristic for reservoirs whose initial state is close to the conditions of the gas hydrate decomposition. Calculations show that the regime with
N.G. Musakaev et al. / International Journal of Heat and Mass Transfer 118 (2018) 455–461
the extended zone of the gas hydrate decomposition is realized at a high intensity of the gas extraction and in the reservoirs with low permeability and high initial hydrate saturation. At the same time, a decrease in the gas extraction rate leads to decreasing the length of the gas hydrate decomposition region. It has been established that at a fixed pressure drop in a reservoir, the regime of the gas extraction from the hydrate-saturated reservoir in the case of a negative (T0 < 0 °C) initial temperature and the methane hydrate decomposition to gas and ice is more profit than gas removal from the reservoir with a positive initial temperature and corresponding decomposition of methane hydrate to gas and water. Calculations show that the extraction of gas from the reservoir initially saturated with methane and its hydrate may prove to be more profitable than the operation with the deposit saturated only with gas in a free state, due to the greater amount of accumulated mass of produced gas and longer life of the gas-hydrate field. Acknowledgement This work was carried out at the Sterlitamak Branch of Bashkir State University and financially supported by the Russian Science Foundation (project number 17-79-20001). The computer code is developed using the software of the Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS. We thank Industrial University of Tyumen for granting us the access to libraries.
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