Mathematical Modeling of a Non-Isothermal Flow in a Porous Medium Considering Gas Hydrate Decomposition: A Review

Abstract

Deposits of natural gas hydrates are some of the most promising sources of hydrocarbons. According to studies, at the current level of natural gas consumption, the traditional reserves will last for about 50 years, and the gas hydrate deposits will last for at least 250 years. Therefore, interest in the study of gas hydrates is associated first of all with gas production from gas hydrate deposits. Additionally, gas hydrates are widely studied for solving practical problems, such as transportation and storage of natural gas, utilization of industrial gases and environmental and technological disasters associated with gas hydrates. When solving practical problems related to gas hydrates, in addition to laboratory and field studies, mathematical modeling is also widely used. This article presents the mathematical models of non-isothermal flow in a porous medium considering the decomposition of gas hydrate. The general forms of the mass conservation equations, Darcy’s law and the energy conservation equation are given. The article also presents derivations of the equations for taking into account the latent heat of phase transitions and non-isothermal filtration parameters for the energy conservation equation. This may be useful for researchers to better understand the construction of the model. For the parameters included in the basic equations, various dependencies are used in different works. In all the articles found, most often there was an emphasis on one or two of the parameters. The main feature of this article is summarizing various dependencies for a large number of parameters. Additionally, graphs of these dependencies are presented so that the reader can independently evaluate the differences between them. The most preferred dependencies for calculations are noted and explained.

1. Introduction

Gas hydrates are crystalline, solid substances consisting of water molecules linked by hydrogen bonds, thereby forming cavities, and molecules of other substances contained in these cavities [1,2]. In nature, such substances are most often methane, ethane, propane and carbon dioxide. The formation of mixed hydrates containing various substances in the cavities is also possible. The crystalline structure and filling of the gas hydrates’ cavities depend on the guest molecule and the conditions of the hydrate formation. The composition of gas hydrates is usually determined by the expression M·nH₂O, where n is the number of water molecules per one guest molecule. One cubic meter of methane hydrate contains about 160 m³ of gas reduced to standard conditions (273.15 K and 101,325 Pa) and about 0.8 m³ of water. Figure 1 shows a cavity containing a methane molecule and the physical appearance of gas hydrates [3].

Gas hydrates form at particular pressures and temperatures. Figure 2 shows phase equilibrium curves for the “H₂O–CH₄” system [4].

In some cases, the effect of self-preservation of gas hydrates is possible [1,5]: if at T < 0 °C a gas hydrate begins to decompose, then an ice film forms on its surface. After this, the film reaches a certain critical thickness, and further decomposition almost completely stops.

The existing deposits of gas hydrates are one of the most promising sources of hydrocarbons [6,7,8,9]. According to BP’s Statistical Review of World Energy 2020, at the end of 2019, the amount of the world’s conventionally recoverable natural gas reserves is 198.8 trillion m³ [10]. According to “Resources to Reserves 2013”, the amount of gas in gas hydrates is estimated to be from 1000 to 20,000 trillion m³ [11]. Consequently, at the current level of natural gas consumption of 4.04 trillion m³ [12], conventional reserves will last for about 50 years, and gas from hydrates will last for at least 250 years. According to studies, about 20% of the land in the permafrost zone and up to 90% of the bottom of the seas and oceans are in the gas hydrate stability zone (this is the zone in which the thermodynamic conditions are above the h-i-g_CH4 and h-w-g_CH4 curves, Figure 2) [1,13,14,15,16]. So far, deposits of gas hydrates have been discovered along the coasts of Eurasia, North and South America, Australia and Japan; and at the depths of the Black, Caspian and Mediterranean seas, Lake Baikal, etc. [17,18]. The first industrial production of gas from gas hydrates was carried out in 1969 in Russia at the Messoyakha gas field, where the extraction of free natural gas led to the release of gas from gas hydrates [19]. In 2002, during the JAPEX/GCS/JNOC Mallik 5L-38 research program in Canada, a small-scale field test was carried out to extract gas from hydrates with a pressure reduction method [20,21,22,23]. After that, in 2007–2008, a full-scale production test was carried out using the Aurora/JOGMEC/NRCan Mallik 2L-38 program [24,25,26,27]. At the beginning of 2013, gas was produced from a hydrate formation at a depth of 300 m below the seabed near the island of Honshu in Japan, where about 119 thousand m³ of methane was produced in 6 days; production was consequently halted due to equipment becoming clogged with sand [28]. To develop existing gas hydrate deposits, a number of researchers proposed to use the following methods: depression (pressure reduction); the thermal method (temperature increase); the introduction of inhibitors; and the replacement of methane in the composition of the hydrate with carbon dioxide [29]. The most promising method of gas production from gas hydrate deposits is considered to be lowering the bottomhole pressure below the equilibrium pressure of hydrate formation for a given reservoir temperature (i.e., below the h-i-g_CH4 and h-w-g_CH4 curves, Figure 2) to initiate the process of gas hydrate decomposition [30]. After that, the released free gas can be recovered by conventional gas-production methods. Additionally, replacement of CH₄-CO₂ in the composition of gas hydrate has been considered a promising method for extracting methane from gas hydrate deposits in the last two decades [31,32,33,34,35,36,37,38,39,40]. This is possible due to the fact that carbon dioxide hydrate is thermodynamically more stable with the same parameters (pressure and temperature) than methane hydrate. Carbon dioxide replaces the methane in the hydrate, after which the released methane is extracted. This method allows one, in addition to extracting energy carriers, to solve the problem of carbon dioxide utilization. In addition to the main methods mentioned above, the decomposition of gas hydrate can occur with exposure to high-frequency electromagnetic radiation [41].

The study of gas hydrates is also necessary to prevent possible environmental and technological disasters. For example, in the development of fields in the northern regions, sudden gas outbursts can occur due to the decomposition of gas hydrates during drilling and well operations [1,42,43]. According to some researchers, climate warming in the Arctic latitudes can lead to a rapid release of gas from the decomposition of natural gas hydrates, which might explain the presence of funnel craters found in the Yamal and Krasnoyarsk regions of Russia [16]. On the shelf of the Arctic seas, the presence of so-called gas flares (fountains of gas bubbles) was observed, which reached a diameter of 1 km; and the probable cause of this phenomenon is the decomposition of underwater gas hydrates [44]. When methane is released from gas hydrates into the atmosphere, an increase in the greenhouse effect is possible, which can lead to an acceleration of the warming process and the decomposition of natural gas hydrate deposits.

It is possible to use gas hydrates in the transportation and storage of various gases, their purification and separation. At the end of the last century, it was discovered that even at atmospheric pressure, hydrates can be relatively stable due to the self-preservation effect described above. This is a huge advantage to storing and transporting gases in the form of gas hydrates [45]. In Japan, in 2003, Mitsui Engineering and Shipbuilding (MES) for the first time developed a project for the transportation and storage of natural gas in the form of hydrate granules, and in 2009, along with Chugoku Electric Power, built the first plant for the production of natural gas hydrate with a capacity of 5 tons/day [46]. A concept for the production of gas hydrates in the northern regions of Russia using natural conditions has been developed. Experimental results have been obtained about the creation of methane and ethane hydrates under free-convection conditions in closed-type reactor chambers [47]. One of the urgent environmental problems is considered in [48,49], where the utilization of sulfur dioxide produced by metallurgical companies is studied. The storage of sulfur dioxide in underground facilities is associated with the risks of its possible leakage, and the injection of this gas into natural reservoirs would allow its storage in a stable gas hydrate form. This circumstance provides reliable storage of sulfur dioxide for a relatively low price [50].

In the gas transportation industry, gas hydrates are considered to have a negative effect, since the formation of a solid phase in wells and pipelines leads to deterioration in their hydraulic capacity and even complete blockage of the pipes [2,51,52,53]. Therefore, it is necessary to prevent the formation of gas hydrates in equipment used in the oil and gas industry. The following methods may be used: gas purification from water vapor, the use of inhibitors, and gas heating. The most common of these methods is the injection of organic inhibitors (methanol, ethanol, etc.). Their presence reduces the activity of water and shifts the conditions of gas hydrate phase equilibrium to lower temperatures and high pressures [54].

To solve the problems of natural gas production, and the transportation and storage of gases in the form of gas hydrates, and to study environmental, geological and technological problems associated with gas hydrates, it is necessary to build mathematical models that take into account the main features of the studied processes. Multivariate calculations using such mathematical models make it possible to make fairly accurate forecasts of the process development and reduce the amounts of experimental and field data required. This contributes to the implementation of effective technological and engineering solutions to problems associated with gas hydrates.

This paper presents the general equations of a mathematical model of non-isothermal filtration of gas and water in a porous medium considering the decomposition of gas hydrate. To construct this model, widely known equations of continuum mechanics are used, which the existing well-tested hydrodynamic simulators are based on. The works of various researchers were reviewed to determine the individual parameters included in this model. Methods for calculating these parameters are described, and the most preferred of them are indicated. Such models allow one to study some general fundamental regularities of gas production from gas hydrate deposits.

2. Mathematical Modeling

The basic equations of mathematical models of non-isothermal flow in a porous medium considering the formation or decomposition of gas hydrates are presented in this section. Different sources use different symbols for different parameters. Hence, for the convenience of general perception, unified designations will be used, which may differ from those of the original sources.

The general form of the equations for conservation of masses of gas, water, hydrate and porous skeleton can be written as [55]:

$$\begin{gathered} \frac{\partial }{\partial t}\left(\phi S_j{\rho }_j\right)=-\mathrm{div}\left(\phi S_j{\rho }_j{\overrightarrow{v}}_j\right)+{\displaystyle \sum _k{J}_{kj}}, \frac{\partial }{\partial t}\left(\left(1-\phi \right){\rho }_{sk}\right)=0, \\ {\displaystyle \sum _j{S}_j}=1, J_{kj}=-J_{jk}, {\displaystyle \sum _j{\displaystyle \sum _k{J}_{kj}}}=0 \end{gathered}$$

(1)

Here and below, the subscripts j, k = g, w and h refer to gas, water and hydrate, respectively; the subscript sk refers to the porous-medium skeleton; t is time, s; φ is porosity; S_j is the saturation of pores with the j-th phase; ρ_j is the density of the j-th phase, kg/m³; ${\overrightarrow{v}}_j$ is the speed of the j-th phase, m/s; J_kj is the intensity of mass transition from the k-th phase to the j-th phase in a unit of volume per unit of time, kg/(m³·s). The term on the equation’s left side is the change in the j-th phase mass, the first term on the right side is the mass transfer of the j-th phase, and the second term on the right side is phase transitions. The hydrate is considered immobile (${\overrightarrow{v}}_h=\overrightarrow{0}$), and for it, the first term on the right side is equal to zero. The porous-medium skeleton is assumed to be immobile; its mass is constant.

Darcy’s law is used to describe the motion of phases in a porous medium:

$$\phi S_j{\overrightarrow{v}}_j={\overrightarrow{w}}_j=-\frac{k_{rj}}{{\mu }_j}\mathit{k}\cdot \left(\mathrm{grad}{p}_j-{\rho }_j\overrightarrow{g}\right)$$

(2)

Here, ${\overrightarrow{w}}_j$ is the filtration velocity vector of the j-th phase, m/s; k is the permeability tensor of the porous medium—the use of the tensor allows us to consider the permeability at each point in different directions, m²; k_rj is the relative permeability of the j-th phase; µ_j is the dynamic viscosity of the j-th phase, Pa·s; p_j is the pressure of the j-th phase, Pa; $\overrightarrow{g}=\left(0,0,-g\right)$ is the acceleration of gravity vector considering that vertical axis is directed upwards, m/s².

The first law of thermodynamics is used as the basic equation of the energy conservation [56]:

$$d\left(U+E\right)=\delta A^{\left(e\right)}+\delta Q^{\left(e\right)}$$

(3)

Here, U is the internal energy of some volume of saturated porous medium, J; E is the energy in the field of potential forces (in our case, in the field of gravity), J; δA^(e) is the elementary work of external forces, J; δQ^(e) is the elementary amount of heat received by the considered volume, J.

The left side of Equation (3) can be written as:

$$d\left(U+E\right)=\frac{\partial }{\partial t}\left({\displaystyle \sum _j\phi S_j{\rho }_j{u}_j}+(1-\phi ){\rho }_{sk}{u}_{sk}+{\displaystyle \sum _j\phi S_j{\rho }_{j}gz}+(1-\phi ){\rho }_{sk}gz\right)dxdydzdt$$

(4)

where u is the internal energy of a unit mass, J/kg; g is the value of the acceleration of gravity, m/s²; x, y, and z are Cartesian coordinates, m.

The work of external forces δA^(e) is the sum of the work of pressure forces δA^(e)_pres and the work of gravity δA^(e)_grav:

$$\begin{gathered} \delta A^{\left(e\right)}{}_{pres}=-\mathrm{div}\left({\displaystyle \sum _j{p}_j{\overrightarrow{w}}_j}\right)dxdydzdt, \delta A^{\left(e\right)}{}_{grav}=-\mathrm{div}\left({\displaystyle \sum _j{\rho }_j{\overrightarrow{w}}_{j}gz}\right)dxdydzdt, \\ \delta A^{\left(e\right)}=-\mathrm{div}\left({\displaystyle \sum _j{p}_j{\overrightarrow{w}}_j}+{\displaystyle \sum _j{\rho }_j{\overrightarrow{w}}_{j}gz}\right)dxdydzdt. \end{gathered}$$

(5)

The amount of heat received by the considered volume can be written as:

$$\delta Q^{\left(e\right)}=-\mathrm{div}\left(\overrightarrow{q}\right)dxdydzdt$$

where $\overrightarrow{q}$ is the heat flux vector (J/(m²·s)), which is the sum of the heat transfer with heat conduction ${\overrightarrow{q}}_{cond}$ and convection ${\overrightarrow{q}}_{conv}$:

$${\overrightarrow{q}}_{cond}=-\lambda \mathrm{grad}T, {\overrightarrow{q}}_{conv}={\displaystyle \sum _j{\rho }_j{\overrightarrow{w}}_j{u}_j}, \overrightarrow{q}=-\lambda \mathrm{grad}T+{\displaystyle \sum _j{\rho }_j{\overrightarrow{w}}_j{u}_j}$$

Then, the expression for the amount of heat will be:

$$\delta Q^{\left(e\right)}=-\mathrm{div}\left(-\lambda \mathrm{grad}T+{\displaystyle \sum _j{\rho }_j{\overrightarrow{w}}_j{u}_j}\right)dxdydzdt$$

(6)

By transforming Equations (3)–(6) considering the mass conservation Equation (1), the following general form of the energy conservation equation can be obtained:

$$\begin{gathered} \rho c\frac{\partial T}{\partial t}=\mathrm{div}\left(\lambda \mathrm{grad}T\right)-{\displaystyle \sum _j\left({\rho }_j{\overrightarrow{w}}_j{c}_j\left(\mathrm{grad}T+{\epsilon }_j\mathrm{grad}{p}_j-\frac{\overrightarrow{g}}{c_j}\right)\right)} \\ +{\displaystyle \sum _j\left(\phi S_j{\rho }_j{c}_j{\eta }_j\frac{\partial p_j}{\partial t}+p_j\frac{\partial }{\partial t}\left(\phi S_j\right)\right)}-{\displaystyle \sum _j\left(i_j{\displaystyle \sum _k{J}_{kj}}\right)} \end{gathered}$$

(7)

Here, ρc is the volumetric heat capacity of a saturated porous medium, J/(m³·K); T is temperature, K; λ is the thermal conductivity of the saturated porous medium, W/(m·K); c is the isobaric specific heat capacity, J/(kg·K); ε is the Joule–Thomson coefficient (differential throttling factor), K/Pa; η is the coefficient of adiabatic cooling, K/Pa; i = u + p/ρ is the specific enthalpy, J/kg. Equation (7) accounts for the change in the temperature of the “porous-medium-skeleton–fluids” system being determined by the following factors: thermal conductivity (the first term on the right side of the equation), convective heat transfer (the second term), the action of the Joule–Thomson effect (the third term), the conversion of gravitational potential energy into heat (the fourth term), the effect of adiabatic cooling (the fifth term), the change in energy due to changes in porosity and saturations (the sixth term) and the release of latent heat of the gas hydrate decomposition or formation (the seventh term).

Next, the parameters for Equations (1), (2) and (7) will be determined:

2.1.

Porosity;

2.2.

Densities of gas, water, hydrate and porous-medium skeleton;

2.3.

Permeability;

2.4.

Relative permeabilities;

2.5.

Pressures of phases;

2.6.

Phase transition intensity;

2.7.

Equilibrium pressure and temperature;

2.8.

Latent heat of phase transitions in the energy conservation equation;

2.9.

Non-isothermal filtration parameters (Joule–Thomson effect and adiabatic cooling).

2.1. Porosity

In [57], the following expression to determine porosity is given:

$$\begin{gathered} \frac{\partial \phi }{\partial t}=\frac{1}{E\left(S_h\right)}\frac{\left(1+\nu \right)\left(1-2\nu \right)}{1-\nu }\frac{\partial p_t}{\partial t}, \\ E\left(S_h\right)=\left(0.3+1.35S_h\right)\cdot {10}^9, \\ {p}_t=\left(S_g{p}_g+S_w{p}_w\right)/\left(S_g+S_w\right), \end{gathered}$$

(8)

where E(S_h) is the Young’s modulus of the porous medium considering the presence of hydrate, Pa; ν is Poisson’s ratio of the porous medium (for example, ν = 0.15 [57], ν = 0.35 [58]); p_t is the total pore pressure, Pa. The change in porosity with time depending on the change of the total pore pressure and hydrate saturation are shown in Figure 3. When constructing Figure 3, the value of Poisson’s ratio ν = 0.15 was used.

Figure 3 shows that the change in porosity is mostly affected by the change in the total pore pressure (~10⁻²) and slightly affected by hydrate saturation (~10⁻³). The order of magnitude for porosity is around 10⁻¹. Additionally, in several works [17,41,43,59,60,61,62,63,64,65] the change in porosity with time is neglected and is assumed constant: φ = const.

2.2. Densities of Gas, Water, Hydrate and Porous Medium Skeleton

For a gas, the following equation of state can be used:

$${\rho }_g=\frac{p_g}{z_g{R}_{g}T}$$

(9)

where R_g = R/M_g is the specific gas constant, J/(kg·K); R is the universal gas constant, J/(mol·K); M_g is the molar mass of the gas, kg/mol; z_g is the gas compressibility factor, which can be determined by the following expression [66]:

$$z_g={\left(0.4\cdot \lg \left(\frac{T}{T_c}\right)+0.73\right)}^{\frac{p}{p_c}}+0.1\frac{p}{p_c}$$

(10)

where T_c and p_c are the critical gas temperature (K) and pressure (Pa). This equation is suitable both for a single-component gas and for gas mixtures consisting of hydrocarbon gases, carbon dioxide and nitrogen. In a number of works [57,60,61,63,67,68], the gas is considered ideal, and it is assumed that z_g = 1.

The densities of water, gas hydrate and the porous-medium skeleton are considered constant in several works [17,41,43,57,59,60,61,62,64]:

$${\rho }_w=\mathrm{const}, {\rho }_h=\mathrm{const}, {\rho }_{sk}=\mathrm{const}$$

If the change in porosity is taken into account (8), then, according to mass conservation, Equation (1), the density of the porous-medium skeleton can be calculated as:

$${\rho }_{sk}=\frac{1-{\phi }_0}{1-\phi }{\rho }_{sk0}$$

where φ₀ and ρ_sk0 are known initial values of porosity and density for the porous-medium skeleton.

2.3. Permeability

To calculate the permeability of a porous medium containing a gas hydrate, the following equation can be used [17,41,57,69,70,71]:

$$k=k_0{\left(1-S_h\right)}^N, N\ge 0$$

Here, k₀ is the porous medium’s permeability in the absence of hydrate, m². The exponent N depends on the pore-filling type of gas hydrates (Figure 4).

For pore-filling-type gas hydrates with circular cross-sections, the exponent N can be taken as equal to 5; in simple cubic packing, N can be taken equal to 25 [72]. According to the experimental data from [73,74,75] for the pore-filling type gas hydrate, the upper and lower limits of the exponent N are 15 and 3, respectively. The calculations in [57] were carried out with N equal to 8. In [70,76] N = 4 was used.

To model a specific gas hydrate field, it is necessary to conduct an experimental study to determine the parameter N, which will be the best approximated value of permeability depending on hydrate saturation.

2.4. Relative Permeabilities

To determine the relative permeabilities for gas and water, various equations are used in different works (

Table 1. Calculation of relative permeabilities for gas and water.

Equations	* Parameters Example	Graphic Representation
[57,58,70,77,78,79] $k_{rg}={\left(1-S_e\right)}^{\frac{1}{2}}\cdot {\left[1-S_e^{\frac{1}{m}}\right]}^{2m}$ $k_{rw}=S_e^{\frac{1}{2}}\cdot {\left[1-{\left(1-S_e^{\frac{1}{m}}\right)}^m\right]}^2$ $S_e=\frac{S_w-S_{wr}}{1-S_{gr}-S_{wr}}$, if Se < 0 then Se = 0, if Se > 1 then Se = 1.	m = 0.85 Sgr = 0.05 Swr = 0.2
[28,69,70,78,80,81,82,83,84,85,86] $k_{rg}={\left(1-S_e\right)}^{n_g}$ $k_{rw}={\left(S_e\right)}^{n_w}$ $S_e=\frac{S_w-S_{wr}}{1-S_{gr}-S_{wr}}$, if Se < 0 then Se = 0, if Se > 1 then Se = 1.	ng = 2 nw = 3 Sgr = 0.05 Swr = 0.2
[17,56,64] $k_{rg}=\{\begin{array}{l}0,\hspace{1em}0\le S_g\le S_{gr};\\ {\left(\frac{S_g-S_{gr}}{1-S_{gr}}\right)}^n\left(4-3S_g\right),\hspace{1em}\mathrm{otherwise}.\end{array}$ $k_{rw}=\{\begin{array}{l}0,\hspace{1em}0\le S_w\le S_{wr}.\\ {\left(\frac{S_w-S_{wr}}{1-S_{wr}}\right)}^n,\hspace{1em}\mathrm{otherwise}.\end{array}$	n = 3.5 Sgr = 0.1 Swr = 0.2 Sg = 1 − Sw − Sh Sh = 0

* m, n, n_g, n_w are the empirical parameters; S_gr, S_wr are the residual gas and water saturations.

). The equations in the first two lines are written for two-phase filtration; that is, they do not take into account the presence of the third phase—gas hydrate. Therefore, when using them, a contradiction is possible: at S_w = 0.6, S_h = 0.35 and S_g = S_gr = 0.05, the value k_rg ≈ 0.2—i.e., the gas is considered mobile, although its saturation is equal to the residual, which cannot be. Thus, for the correct calculation of the relative phase permeabilities for gas and water, it is necessary to develop dependencies that take into account the presence of a third gas hydrate phase. The last row of Table 1 shows the equations from [17,56,64], adapted to take into account the presence of gas hydrate in a porous medium. Therefore, when carrying out calculations, these equations are likely to give more accurate results.

To model a specific gas hydrate deposit, it is necessary to conduct experiments and determine specific types of relative permeability curves, as it is when modeling oil and gas fields.

2.5. Pressures of Phases

Capillary pressure p_c is related to gas and water pressures as:

$$p_c=p_g-p_w$$

To determine the capillary pressure, various equations are used in different works (

Table 2. Calculation of capillary pressure.

Equations	* Parameters Example	Graphic Representation
[57,58,65,77,78,79,81,82,83,84,85] $p_c=p_{c0}{\left(S_e^{-\frac{1}{m}}-1\right)}^{1-m}$ $S_e=\frac{S_w-S_{wr}}{1-S_{gr}-S_{wr}}$, if Se < 0 then Se = 0, if Se > 1 then Se = 1.	pc0 = 10 kPa m = 0.85 Sgr = 0.05 Swr = 0.2
[69,78,86,87] $p_c=p_{c0}{S}_e^{-\frac{1}{\lambda }}$ $S_e=\frac{S_w-S_{wr}}{1-S_{gr}-S_{wr}}$, if Se < 0 then Se = 0, if Se > 1 then Se = 1.	pc0 = 5 kPa λ = 4 Sgr = 0.05 Swr = 0.2

* p_c0, m, λ are the empirical parameters; S_gr, S_wr are the residual gas and water saturations.

). The papers [77,86] present graphs of the experimental values of capillary pressure as functions of water saturation. The qualitative forms of these experimental curves are in better agreement with the curve obtained using the dependence from the first line of Table 2. Therefore, their use in calculations is preferable.

The paper [86] presents data of capillary pressure for various samples of a porous medium. From [86], it can be seen that the order of magnitude of capillary pressure is 10³–10⁴ Pa. The order of magnitude of pressure in gas hydrate deposits is 10⁶–10⁷ Pa, which is 2–4 orders of magnitude more than the capillary pressure magnitude. Therefore, in a number of works [17,41,48,49,59,60,62,64,67,71,88], the capillary pressure is neglected and p_c = 0 is assumed, and only one total pressure is used: p = p_g = p_w.

2.6. Phase-Transition Intensity

The following equations can be written for the phase transition intensities:

$$\begin{gathered} J_{hg}=-\frac{\partial }{\partial t}\left(\phi {\rho }_h{S}_{h}G\right), J_{hw}=-\frac{\partial }{\partial t}\left(\phi {\rho }_h{S}_h\left(1-G\right)\right), \\ {J}_h=J_{gh}+J_{wh}=-J_{hg}-J_{hw}=\frac{\partial }{\partial t}\left(\phi {\rho }_h{S}_h\right), \\ G=\frac{M_g}{M_h}=\frac{M_g}{n{M}_w+M_g}, \end{gathered}$$

where G is the mass concentration of gas in the hydrate; M_j (j = g, w, h) is the molar mass of gas, water and hydrate, respectively, kg/mol; n is the hydration number; J_hg, J_hw and J_h are intensities of gas and water released from the hydrate, and the rate of gas hydrate formation, respectively—kg/(m³·s).

A phase transition can be considered as a thermodynamic equilibrium or non-equilibrium (considering the kinetics of phase transition). To consider the kinetics of gas hydrate decomposition, the following equations can be used [89]:

$$\begin{gathered} J_{hg}=M_g{k}_d{A}_s\left(f_{eq}-f_g\right), J_{hw}=n{M}_w{k}_d{A}_s\left(f_{eq}-f_g\right), \\ {J}_h=-M_h{k}_d{A}_s\left(f_{eq}-f_g\right), \\ {k}_d=k_{d0}\exp \left(-\frac{\Delta E}{RT}\right). \end{gathered}$$

Here, k_d is the molar specific rate of gas hydrate decomposition, mol/(m²·Pa·s); k_d0 is the intrinsic rate constant of gas hydrate decomposition, mol/(m²·Pa·s); it does not depend on temperature, pressure or the geometry of the gas hydrate particle; ΔE is the activation energy, J/mol; R is the universal gas constant, J/(mol·K); T is temperature, K; A_s is the total surface area of decomposing hydrate particles per unit volume, m²/m³; f_eq and f_g are gas fugacity at equilibrium pressure at the current temperature and gas fugacity at current pressure and temperature, respectively—Pa. For an ideal gas, the fugacity f is equal to the pressure p, and for a real gas, it can be calculated using the following expression:

$$\ln f=\ln p_{st}+{\displaystyle \underset{p_{st}}{\overset{p}{\int }}\frac{z_g}{p}dp}$$

where p_st is the pressure at which the value of the compressibility factor z_g ≈ 1—the pressure at which gas can be considered almost ideal. It can be assumed that p_st = 10⁵ Pa.

For methane hydrate, the following values are usually used:

Kim et al. (1987) [89]: k_d0 = 1.24·10⁵ mol/(m²·Pa·s); ΔE/R = 9400 ± 545 K.

Clarke and Bishnoi (2001) [90]: k_d0 = 3.6·10⁴ mol/(m²·Pa·s); ΔE/R = 9752.73 ± 29.27 K.

Referencing was performed in [90] to [89], and it was noted that the values from [90] are more accurate, so it is preferable to use them.

Table 3. Equations for calculating A_s.

Equations	Parameters Example	Graphic Representation
[28] $A_s=A_{s0}{\left(\frac{S_h}{S_{h0}}\right)}^{2/3}$	As0 = 105 m2/ms Sh0 = 0.6
[79] $A_s=\phi S_h{A}_{s0}$	As0 = 3·105 m2/m3 φ = 0.3
[70] $A_s=\phi S_h\sqrt{\frac{{\phi }^3{\left(1-S_h\right)}^3}{2k}}$ $k=k_0{\left(1-S_h\right)}^N$	φ = 0.3 k0 = 10−13 m2 N = 4
[76] $A_s={\left(S_g{S}_w{S}_h\right)}^{\frac{2}{3}}\sqrt{\frac{{\left({\phi }_e\right)}^3}{2k}}$ ${\phi }_e=\phi \left(S_g+S_w\right)$ $k=k_0\frac{{\phi }_e}{\phi }{\left(\frac{{\phi }_e\left(1-\phi \right)}{\phi \left(1-{\phi }_e\right)}\right)}^N$	φ = 0.3 k0 = 10−13 m2 N = 4 Sg = 0.1 Sw = 1 − Sg − Sh

presents the equations for calculating A_s from different works. The expressions presented in the last two rows of this table are more preferable, since they do not contain the free experimental parameter A_s0.

2.7. Equilibrium Pressure and Temperature

The following empirical equation can be used for equilibrium pressure and temperature [17,41,60,61,62,64,68,70,88,91]:

$$T=T_{*}\left(A\ln \left(p/p_{*}\right)+B\right)$$

where A and B are empirical coefficients; p_∗ = 1 Pa; T_∗ = 1 K.

The empirical equation for the phase equilibrium of methane hydrate presented in Figure 2 can be represented in the following form [4]:

$$\ln \left(\frac{p}{p_{*}}\right)=A_0+\frac{A_1}{T/T_{*}}+\frac{A_2}{{(T/T_{*})}^2}$$

(11)

Here, p is in MPa, T is in K; p_∗ = 1 MPa; T_∗ = 1 K. The coefficients are given in

Table 4. Coefficients for the phase equilibrium curves of methane hydrate, Equation (11) [4].

Curve	Temperature Interval, K	Coefficients
h-i-gCH4	[178.2, 273]	A0 = ln(2.6) − A1/273 − A2/2732 A1 = −2768 A2 = 69,900
h-w-gCH4	[273, 302]	A0 = ln(2.6) − A1/273 − A2/2732 A1 = −61,987 A2 = 7,526,200
i-w	In [4], it was assumed that for all considered pressure values, the freezing point of water is 273 K.
Q	Quadrupole point where all four phases (methane hydrate, ice, water and methane) coexist. p = 2.6 MPa; T = 273 K.

.

2.8. Latent Heat of Phase Transitions in the Energy Conservation Equation

The last term of Equation (7) can be written as:

$${\displaystyle \sum _j\left(i_j{\displaystyle \sum _k{J}_{kj}}\right)}=i_g\left(J_{gg}+J_{wg}+J_{hg}\right)+i_w\left(J_{gw}+J_{ww}+J_{hw}\right)+i_h\left(J_{gh}+J_{wh}+J_{hh}\right)$$

(12)

The phase-transition intensities of substances into themselves are equal to zero:

$$J_{gg}=J_{ww}=J_{hh}=0$$

If the solubility of gas in water and the evaporation (condensation) of water are not considered, then:

$$J_{gw}=J_{wg}=0$$

so (12) can be rewritten as:

$$\sum _j\left(i_j\sum _k{J}_{kj}\right)=i_g{J}_{hg}+i_w{J}_{hw}+i_h\left(J_{gh}+J_{wh}\right)$$

(13)

It is known that during phase transitions, latent heat is released or absorbed, which can be written as follows:

$$i_g{J}_{gh}+i_w{J}_{wh}=i_h\left(J_{gh}+J_{wh}\right)+\frac{\partial }{\partial t}\left(\varphi S_h{\rho }_h{L}_h\right)$$

or considering that J_kj = −J_jk:

$$i_g{J}_{hg}+i_w{J}_{hw}+i_h\left(J_{gh}+J_{wh}\right)=-\frac{\partial }{\partial t}\left(\phi S_h{\rho }_h{L}_h\right)$$

where L_h is the latent specific heat of hydrate formation or decomposition, J/kg. Then (13) can be rewritten in the following form:

$${\displaystyle \sum _j\left(i_j{\displaystyle \sum _k{J}_{kj}}\right)}=-\frac{\partial }{\partial t}\left(\phi S_h{\rho }_h{L}_h\right)$$

2.9. Non-Isothermal Filtration Parameters (Joule–Thomson Effect and Adiabatic Cooling)

To determine the Joule–Thomson coefficient (differential throttling coefficient), the following relationship can be used [92]:

$$\epsilon =\frac{1}{\rho c}\left(1-\rho T{\frac{\partial \left(1/\rho \right)}{\partial T}|}_p\right)$$

(14)

After the gas density from (9) is substituted into this equation:

$${\epsilon }_g=-\frac{1}{{\rho }_g{c}_g}\frac{T}{z_g}{\frac{\partial z_g}{\partial T}|}_p$$

the partial derivative of gas compressibility factor with respect to temperature at constant pressure can be obtained using the Latonov-Gurevich Equation (10):

$${\frac{\partial z_g}{\partial T}|}_p=\frac{0.4p}{p_{c}T\ln 10}{\left(0.4\cdot \lg \left(\frac{T}{T_c}\right)+0.73\right)}^{\frac{p}{p_c}-1}$$

Then, the Joule–Thomson coefficient for the gas can finally be written as:

$${\epsilon }_g=-\frac{0.4p}{{\rho }_g{c}_g{z}_g{p}_c\ln 10}{\left(0.4\cdot \lg \left(\frac{T}{T_c}\right)+0.73\right)}^{\frac{p}{p_c}-1}$$

If the gas is considered ideal, then:

$${\frac{\partial z_g}{\partial T}|}_p=0$$

and then:

$${\epsilon }_g=0$$

If the density of a substance is assumed to be constant (ρ = const), then it can be substituted into Equation (14), and the Joule–Thomson coefficient for an incompressible fluid can be obtained:

$$\epsilon =\frac{1}{\rho c}\left(1-\rho T{\frac{\partial \left(1/\rho \right)}{\partial T}|}_p\right)=\frac{1}{\rho c}\left(1-0\right)=\frac{1}{\rho c}$$

The adiabatic cooling coefficient is determined from this expression:

$$\eta =\frac{1}{\rho c}-\epsilon$$

Then, for a real and ideal gas, it can be written as:

$${\eta }_g^{real}=\frac{1}{{\rho }_g{c}_g}-{\epsilon }_g \mathrm{and} {\eta }_g^{ideal}=\frac{1}{{\rho }_g{c}_g}.$$

For a substance which density is assumed to be constant (ρ = const), the adiabatic cooling coefficient is equal to zero:

$$\eta =\frac{1}{\rho c}-\epsilon =\frac{1}{\rho c}-\frac{1}{\rho c}=0$$

These parameters affect the temperature, and accordingly, the process of gas hydrate decomposition. Thus, more accurate results are obtained.

3. Conclusions

To solve practical problems related to gas hydrates, it is necessary to build mathematical models which take into account the main features of the studied processes. Such mathematical models are based upon multiphase media mechanics equations, where the system of basic equations includes the equations of continuity and conservation of momentum and energy. The paper presented the equations of conservation of mass, momentum and energy for non-isothermal flow of gas and water in a porous medium considering the formation or decomposition of gas hydrate. A model can be developed by extending these equations to the case of multicomponent filtration. For example, the gas phase can contain various gases as components (methane, carbon dioxide, etc.), whereas the liquid phase may contain dissolved gas and salts; hydrates of various gases may be present.

To carry out calculations for actual gas hydrate fields, it is first and foremost necessary to conduct field studies similar to those carried out for traditional oil and gas fields by determining the reservoir properties (porosity, saturation, permeability, relative permeability, etc.). Second, it is also necessary to carry out studies to determine the dependences of these parameters on hydrate saturation. It is also important to conduct experiments to determine the empirical parameters for the gas hydrate—in particular: composition, kinetics of gas hydrate decomposition, phase equilibrium curve, etc.

This paper presents several equations and empirical ratios used in the construction of mathematical models in different works. These equations can be used to identify the general fundamental laws of processes occurring during non-isothermal filtration of gas and water considering the decomposition of gas hydrate.

Analysis of the works devoted to the gas extraction from gas hydrate deposits showed that it is additionally necessary to solve the following problems:

• determination of relative phase permeabilities for gas and water, taking into account the presence of gas hydrate in the pores;
• gas hydrate decomposition at negative temperatures, when the hydrate decomposes into gas and ice;
• accounting for the self-preservation of gas hydrates in gas hydrate deposits.

One of the promising methods for extracting methane from gas hydrate deposits is the CH₄-CO₂ replacement method. In this regard, it is additionally necessary to solve the following problems:

• filtration of two-component gas (methane and carbon dioxide) and water;
• the solubility of carbon dioxide in water;
• filtration of gas, water and liquid carbon dioxide;
• replacement process: thermodynamic equilibrium or taking into account the kinetics of formation or decomposition of gas hydrates.