CC BY
6
MSC 93X30
DOI: 10.14529/ mmp230207
MODELLING THE PROCESS OF NON-EQUILIBRIUM HYDRATE FORMATION IN A POROUS RESERVOIR
M.K. Khasanov1, S.R. Kildibaeva1, M.V. Stolpovskiy2
1Ufa University of Science and Technology, Sterlitamak, Russian Federation 2Ufa State Petroleum Technological University, Ufa, Russian Federation E-mail: hasanovmk@mail.ru, freya.13@mail.ru, s-maxim.pmm@mail.ru
This paper presents a solution to the flat-dimensional problem of gas hydrate formation in a porous medium. Highly permeable reservoirs are considered, as a result of which it is assumed that the process accompanied by the transition of gas into the hydrate composition is nonequilibrium. Based on the numerical solution, the influence of injection pressure and formation permeability on the peculiarities of phase transitions process has been studied. It is shown that with an increase in the injection pressure, both the maximum possible temperature of the system and the length of the hydrate-containing region increase. It has been found that the maximum temperature realized in the system, depending on the permeability of the reservoir, has a non-monotonic character. The influence of the initial temperature of the porous reservoir on the dynamics of phase transitions has been studied.
Keywords: gas hydrate; phase transitions; nonequilibrium process; porous medium; kinetics.
Introduction
Natural deposits of hydrates are found at the bottom of seas and oceans, as well as in permafrost. The main aspect of the study of hydrates is the fact that the gas reserves contained in them are large enough and can provide the need for it for a very long period of time. According to estimates, the content of natural gas in gas hydrates is estimated at about 1015 m3 [1,2]. Another feature of the study is the fact that gas hydrates can accumulate a sufficiently large amount of industrial gases at sufficiently low economic costs [3, 4].
Experimental studies in the field of hydrate formation are presented, in particular, in [5,6]. In these papers studies of gas hydrate formation were carried out in samples of small size and usually under thermal and barostatic conditions. Therefore, in these studies, due to the small size of the samples, the process of gas hydrate formation is limited only by the kinetics of the process. The works [7-9] consider mathematical models of the formation and decomposition of methane gas hydrates in porous reservoirs under thermal and depression effects. Studies on the replacement of methane in the hydrate composition of carbon dioxide are presented in [10,11].
This paper presents a mathematical model of gas injection into a solid water-saturated porous medium, accompanied by a process of phase transitions in a nonequilibrium regime. This is based on the fact that the gas-solid gas hydrate transition process is considered for highly permeable formations, where in addition to heat and mass transfer, the kinetics of the process must be taken into account. Note that in contrast to [7-9], in the present work, the mathematical model takes into account the real properties of gas (to describe the dependence of the methane supercompressibility factor on pressure and temperature, the Latonov-Gurevich equation is used). It is shown that gas injection in the reservoir can
be divided into three characteristic regions. In addition, the effect of injection pressure and formation permeability on the maximum temperature realized in the system and the extent of the area saturated with gas hydrate has been studied. It is also shown that as the initial formation temperature increases, the extent of the region containing gas hydrate decreases.
1. Problem Statement
For small drops in pressure and temperature, values of the equilibrium pressure ps as a function of the temperature T are approximated by [12]
Ps = Pso • exp((T - Tso)/T), (1)
where T* is the empirical parameter. The system of basic equations for the real gas is the following [13]:
d(p„S„) д . .
m-—— + — (mSgVgPg) = ~Jg,
dSi T dSh
дТ n (дТ др\ д f дТ\ n dp T r (2)
PC— + PgCglllSgVg — + Eg = A'T" + РдСдГПЗдЩ-^ + JhL h ,
дЬ g \ дх g дх) дх\дх) дЬ
kg др ^g дх'
mSgVg = - — 7Г-, P = ZgPgRgT.
Here t is the time; m is the porosity reservoir; Si, pi, Ji (i = l, h,g) are the pore volume saturation, density and intensity of the i-th phase consumption, respectively; vg, , are the speed, permeability and dynamic viscosity of methane, respectively; Lh is the specific heat of hydrate formation; zg is the methane supercompressibility coefficient; Rg is the reduced methane constant; pc and A are the specific volume heat capacity and thermal conductivity of the reservoir. Since the main contribution to the values of pc and A is made by the corresponding parameters of the porous medium skeleton, we consider them as constant values. Hereinafter, the lower indices i = l,h,g refer to water, hydrate, and methane, respectively.
Let us represent the dependence of the gas supercompressibility coefficient zg on pressure and temperature as follows [14]:
/ t \ \ p/pc
0,4-lgi-J +0,73J + 0, lp/pc, (3)
where pc and Tc are the critical pressure and temperature of CH4, respectively. Thus, for the coefficients of differential throttling and adiabatic cooling of gas, we can write:
1 TfdzA 1
e9 =---,Vg =--
g pg Cg zA dT J p pg Cg g
dZg
where the derivative ( ^^ ) is taken at constant pressure
Taking into account that a solid hydrate is a two-component system with a mass concentration of gas G, in order to relate the values of the rates of water flow Ji and gas Jg for hydrate formation with the value of the intensity Jh , we can write the following relations:
Ji = (1 - G) Jh, Jg = GJh. (4)
From the second and third equations of system (2), taking into account (4), we can obtain:
mpi (Si - Sio) = -mph (1 - G) Sh. (5)
In deriving (5), it is assumed that in the initial state, the formation is saturated with CH4 and H2O, the volume fraction of which is equal to Sl0, and there is no hydrate. The volume fraction of gas hydrate in this case is equal to zero. From expression (5), we can get a limit on the maximum possible value of hydrate saturation Sh max realized at full transition of water (Sl = 0) in the hydrate state:
Sh max pi Sio/ (ph (1 - G)).
Dependence of the phase permeability coefficient on the current gas saturation Sg and the absolute permeability k0 of the reservoir is as follows [15]:
kg = k0Sg .
Intensity of gas consumption per unit volume for hydrate formation, according to [16], is as follows:
Jg = Mg KAs (f - feq ) . (6)
In (6), Mg is the mass of 1 mol of hydrate-forming gas CH4; K is the reaction parameter of decomposition or hydrate formation; As is the surface area of the hydrate formation; feq and f is the CH4 fugacity, determined for both equilibrium and for current system pressure. In the case of hydrate formation, according to [16], we assume that K = 5, 875 ■
10~12mol/(m2 ■ Pa ■ s), As = \jm ^^^-(SgSiSh)2^3. In the case of nonequilibrium decomposition of a solid hydrate K = K0exp (—^ff), where K0 = 3,6 • 104mol/(m2 ■
Pa ■ s), AE/R = 9752,73K-As = \]mHS£SlfSh.
We find the fugacity f of the real gas, corresponding to the pressure p, on the basis of the expression
p
ln/ = lnpst+ / ^dp, (7)
p
Pst
where pst = 105 Pa.
Initial value conditions and conditions on the left boundary of the reservoir are represented in the form
t = 0 : p = po, T = To, Sh = 0, Si = Sio;
(8)
x = 0 : p = pw, T = Tw. As conditions on the right boundary of the reservoir, we use conditions of the form
dT
x = L : p = po, —— = 0. (9)
dx
When solving the resulting system of equations, the following parameters are given: the skeleton porosity m, the water densities pl and the gas hydrate ph, the absolute reservoir permeability ko, the dynamic gas viscosity p,g, the specific heat of hydrate formation Lh, the reduced gas constant Rg, the gas heat capacity cg, the relative mass content gas in gas hydrate G, the specific volumetric heat capacity pc and the thermal conductivity coefficient A of the system. In this case, the following unknown functions are found: the pressure p, the temperature T, the hydrate saturation Sh and the water saturation Sl. These unknowns are functions of the coordinate x and the time t. An implicit finite-difference method was used to solve the system of differential equations [17]. For this, a uniform space-time grid was introduced.
2. Analysis of Results
Fig. 1 shows the distribution of the main parameters saturation of the porous medium for the case of gas injection under pressure pw = 8 MPa and the temperature Tw = 278 K in a porous formation with the absolute permeability k0 = 10-13m2 for the time moment t = 0, 38 h. For the other parameters characterizing the system, the following values were taken: po = 5 MPa, To = 280 K, Sio = 0,15 m = 0, 2, A = 2 W/(m • K), pc = 2, 5 • 106 J/(K • m3), pi = 1000 kg/m3, ph = 900 kg/m3, G = 0,13, Lh = 5 • 105 J/kg, Tso = 280 K, T = 10 K, pso = 5, 5 MPa, ^ = 10-5 Pa • s, M = 16 • 10-3 kg/mol, Rg = 520
J/(kg • K ), cg = 1560 J/(kg • K ),
l = 50 m. From Fig. 1 shows that the hydrate saturation of near area takes the maximum possible value corresponding to the full transition of water to the hydrate state. The pores of the far region contain gas and water. The intermediate region separating the above-mentioned areas is saturated with gas and water. In this area, hydrate saturation varies from maximum to zero. In the far region, only gas and water are present in the pores.
Fig. 2 shows the dependence of the maximum temperature realized in the porous medium (fragment a) and the coordinate of the area occupied by gas hydrate (fragment b) on the injection pressure pw for the time moment t =10 min. The initial reservoir pressure was assumed to be p0 = 4 MPa. As follows from the fragment a, the maximum temperature of the porous medium increases with increasing injection pressure. This is explained due to increasing pressure of injection and, consequently, an increase in release of latent heat of phase transition.
Fig. 1. Distribution of pressure, temperature and hydrate saturation of the porous medium
Fragment b of Fig. 2 also shows that with an increase in the gas pressure, the extent of the region, the extent of the area containing gas hydrate increases. This can be explained by the fact that with increasing values of pw, the conditions corresponding to hydrate formation are satisfied in most of the reservoir.
5 6 7 8 9^'MPa 5 6 7 8 9 Pw, MPa
Fig. 2. Dependence of the maximum temperature (a) and the coordinate of the area occupied by the hydrate (b) on the injection pressure
Fig. 3 shows the dependence of the maximum possible temperature realized in the porous medium(fragment a) and the coordinate of the area occupied by hydrate (fragment b) on the formation permeability coefficient for the time moment t =10 min. The initial pressure and injection pressure were assumed to be p0 = 4 MPa and pw = 6 MPa, respectively. As follows from fragment a, the maximum possible temperature of a porous medium, depending on of the system depending on formation permeability has a nonmonotonic character. At the same time, in the region of low k0, with an increase in permeability, the maximum possible temperature of the porous medium increases. This is explained by the fact that with increasing permeability increases the intensity of hydrate formation and, consequently, the growth of system temperature. In this case, the adiabatic effect, associated with the real gas properties, appear rather weak. In the region of high values of permeability with growth of k0 value the maximum temperature of the system is observed to decrease. In this case, the diabetic effect plays a more significant role, even in the case of an increase in the intensity of phase transitions. This leads to a decrease in the maximum possible system temperature with an increase in reservoir permeability.
From fragment b of Fig. 3. it follows that an increase in the permeability coefficient of a solid formation leads to a corresponding increase in the extent of the area occupied by the gas hydrate. This can be explained by the fact that with an increase in the value of ko, the area of propagation of the pressure disturbance also increases, which leads to a significant increase in the area occupied by the gas hydrate.
Fig. 4 shows the time dependence of the coordinate of the area occupied by gas hydrate. Lines 1 and 2 correspond to the initial temperature T0 = 280 K and T0 = 280, 5 K. The other parameters of the porous medium correspond to those in Fig. 1. From Fig. 4 it follows that in the initial moments of time after gas injection the length of the area occupied by gas hydrate increases to its maximum value, and then it practically does not change. With
increasing initial temperature of the system, the maximum extent of the area occupied by gas hydrate decreases. This is due to the fact that the conditions of gas hydrate existence are realized in smaller areas from the left boundary of the porous medium.
Fig. 3. Dependence of the maximum temperature (a) and the coordinate of the area occupied by hydrate (b) on formation permeability
Fig. 4. Time dependence of the coordinate of the conditional boundary between the far field and intermediate regions
Conclusion
The paper presents a mathematical model of gas injection into a porous medium, accompanied by the process of phase transitions in a nonequilibrium regime. The distributions of the main parameters of the porous medium are constructed. It is shown that as a result of gas injection in the reservoir, three characteristic regions in which gas, hydrate and water are in different states can be distinguished. In addition, the effect of injection pressure and formation permeability on the system temperature and the intensity of phase transitions has been studied. It was shown that as the initial formation temperature increases, the length of the area saturated with gas hydrate decreases.
Acknowledgments. The study was carried out at the expense of the grant of the Russian
Science Foundation no. 21-79-10227, https://rscf.ru/project/21-79-10227/.
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Received February 2, 2023
УДК 532.546:536.421 БЭТ: 10.14529/mmp230207
МОДЕЛИРОВАНИЕ ПРОЦЕССА НЕРАВНОВЕСНОГО ГИДРАТООБРАЗОВАНИЯ В ПОРИСТОМ ПЛАСТЕ
М.К. Хасанов1, С.Р. Кильдибаева1, М.В. Столповский2
1 Стерлитамакский филиал Уфимского университета науки и технологий, г. Стерлитамак, Российская Федерация
2Уфимский государственный нефтяной технический университет, г. Уфа, Российская Федерация
В работе представлено решение плоскоодномерной задачи об образовании газогидрата в пористом пласте, изначально насыщенном метаном и водой. Рассматриваются высокопроницаемые пласты, вследствие чего полагается, что процесс гидратообразова-ния является неравновесным. Система основных уравнений представляет собой законы сохранения масс, тепла и Дарси, которая дополняется заданием кинетики гидрато-образования, а также начально-граничными условиями. Кроме того, представленная математическая модель учитывает параметры, которые приближены к реальным характеристикам системы: реальные свойства газа, эффекты Джоуля-Томсона и адиабатического сжатия и т.д. На основе численного решения исследовано влияние давления инжекции и проницаемости пласта на особенности протекания процесса гидратообра-зования. Показано, что с ростом давления инжекции возрастает как максимальная температура системы, так и протяженность области, насыщенная газогидратом. Установлено, что максимальная температура, реализуемая в системе, в зависимости от проницаемости пласта имеет немонотонный характер. Исследовано влияние исходной температуры пористого пласта на динамику образования газогидрата. Показано, что с увеличением исходной температуры пласта происходит уменьшение протяженности области, содержащей в своем составе газогидрат, уменьшается.
Ключевые слова: газовый гидрат; неравновесный процесс; кинетика.
Исследование выполнено за счет гранта Российского научного фонда № 21-79-10227, https://rscf.ru/project/21-79-10227/.
Литература
1. Boswell, R. Current Perspectives on Gas Hydrate Resources / R. Boswell, T.S. Collett // Energy and Environmental Science. - 2011. - V. 4. - P. 1206-1215.
2. Zhao Jiafei. Analysing the Process of Gas Production for Natural Gas Hydrate Using Depressurization / Zhao Jiafei, Zhu Zihao, Song Yongchen, Liu Weiguo, Zhang Yi, Wang Dayong // Applied Energy. - 2015. - V. 142. - P. 125-134.
3. Anshits, A. Possibilities of SO2 Storage in Geological Strata of Permafrost Terrain / A. Anshits, N. Kirik, B. Shibistov // Advances in the Geological Storage of Carbon Dioxide. - 2006. - V. 65. - P. 93-102.
4. Lee Yohan. CH4 Recovery and CO2 Sequestration Using Flue Gas in Natural Gas Hydrates as Revealed by a Micro-Differential Scanning Calorimeter / Lee Yohan, Kim Yunju, Lee Jaehyoung, Lee Huen, Seo Yongwon // Applied Energy. - 2015. - V. 150. - P. 120-127.
5. Chuvilin, E.M. Formation and Accumulation of Pore Methane Hydrates in Permafrost: Experimental Modelling / E.M. Chuvilin, D.A. Davletshina // Geosciences. - 2018. - V. 8. -Article ID: 467. - 15 p.
6. Bello-Palacios, A. Experimental and Numerical Analysis of the Effects of Clay Content on CH4 Hydrate Formation in Sand / А. Bello-Palacios, S. Almenningen, P. Fotland, G. Ersland // Energy and Fuels. - 2021. - V. 35, № 12. - P. 9836-9846.
7. Хасанов, M.K. Исследование режимов образования газогидратов в пористой среде, частично насыщенной льдом / М.К. Хасанов // Теплофизика и аэромеханика. - 2015. -Т. 22, № 2. - С. 255-266.
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9. Musakaev, N.G. The Mathematical Model of the Gas Hydrate Deposit Development in Permafrost / N.G. Musakaev, M.K. Khasanov, S.L. Borodin // International Journal of Heat and Mass Transfer. - 2018. - V. 118. - P. 455-461.
10. Khasanov, M.K. Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and its Hydrate / M.K. Khasanov, N.G. Musakaev, M.V. Stolpovsky, S.R. Kildibaeva // Mathematics. -2020. - V. 8, № 9. - Article ID: 1482. - 15 p.
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14. Шагапов, В.Ш. Математическая модель течения природного газа в трубопроводах с учетом диссоциации газогидратов / В.Ш. Шагапов, Н.Г. Мусакаев, Р.Р. Уразов // Инженерно-физический журнал. - 2008. - T. 81, № 2. - С. 271-279.
15. Баренблатт, Г.И. Движение жидкостей и газов в природных пластах / Г.И. Баренблатт, В.М. Ентов, В.М. Рыжик. - М.: Недра, 1982.
16. Sun Xuefei. Kinetic Simulation of Methane Hydrate Formation and Dissociation in Porous Media / Sun Xuefei, K.M. Kishore // Chemical Engineering Science. - 2006. - V. 61, № 11. -P. 3476-3495.
17. Khasanov, M.K. Mathematical Model for Carbon Dioxide Injection into Porous Medium Saturated with Methane and Gas Hydrate / M.K. Khasanov, M.V. Stolpovsky, I.K. Gimaltdinov // International Journal of Heat and Mass Transfer. - 2018. - V. 127. -P. 21-28.
Марат Камилович Хасанов, кандидат физико-математических наук, доцент, кафедра прикладной информатики и программирования, Стерлитамакский филиал Уфимского университета науки и технологий (г. Стерлитамак, Российская Федерация), hasanovmk@mail.ru.
Светлана Рустамовна Кильдибаева, кандидат физико-математических наук, доцент, научный сотрудник, Стерлитамакский филиал Уфимского университета науки и технологий (г. Стерлитамак, Российская Федерация), freya.13@mail.ru.
Максим Владимирович Столповский, кандидат физико-математических наук, доцент, кафедра физики, Уфимский государственный нефтяной технический университет (г. Уфа, Российская Федерация), s-maxim.pmm@mail.ru.
Поступила в редакцию 2 февраля 2023 г.
