Mathematical Model of Carbon Dioxide Injection into a Porous Reservoir Saturated with Methane and Its Gas Hydrate

In this paper, the process of methane replacement in gas hydrate with carbon dioxide during CO2 injection into a porous medium is studied. A model that takes into account both the heat and mass transfer in a porous medium and the diffusion kinetics of the replacement process is constructed. The influences of the diffusion coefficient, the permeability and extent of a reservoir on the time of full gas replacement in the hydrate are analyzed. It was established that at high values of the diffusion coefficient in hydrate, low values of the reservoir permeability, and with the growth of the reservoir length, the process of the CH4-CO2 replacement in CH4 hydrate will take place in the frontal regime and be limited, generally, by the filtration mass transfer. Otherwise, the replacement will limited by the diffusion of gas in the hydrate.


Introduction
Recently, special attention has been paid to the problem of developing natural gas hydrate deposits due to their wide spread in nature and the large reserves of hydrocarbons contained in such deposits. There are several main methods of methane recovery from a methane hydrate deposit: lowering the pressure and increasing the temperature of the deposit to the point of the hydrate decomposition, and injecting organic or brine solutions that promote the gas hydrate dissociation [1][2][3][4]. The mathematical modeling of the development of gas hydrate deposits using these methods is considered in [5][6][7][8][9][10][11]. The reduction of atmospheric emissions of carbon dioxide is very important in reducing the anthropogenic impact on climate. One way to solve this problem is the capture of industrial carbon dioxide gas and its further injection into geological formations for long-term storage. Mathematical models of the processes of heat and mass transfer in extended porous reservoirs upon carbon dioxide injection have been formulated; in particular, in [12][13][14].
One of the important environmental problems that needs to be addressed is the reduction of carbon dioxide emissions into the atmosphere. One of the methods of CO 2 utilization is its underground conservation in the gas hydrate state [15]. The numerical study of the processes of heat and mass transfer, accompanied by hydrate formation, when injecting gas into a porous reservoir, is carried out, for example, in [16][17][18][19]. the kinetic parameter, is independent of time. Therefore, the model of Kim-Bishna does not take into account the deceleration of the driving force of the substitution process with the increasing thickness of the carbon dioxide layer over time.
In this paper, we present a mathematical model for the exchange of CH 4 with CO 2 in gas hydrate, taking into account both the kinetics and heat and mass transfer in a natural reservoir. At the same time, when describing the kinetics of the replacement process, the dependence of the rate of diffusion mass transfer of carbon dioxide in the gas hydrate on the growth of the thickness of the carbon dioxide gas monohydrate layer is taken into account. Mainly, attention is paid to the study of the degree of influence of the replacement kinetics and heat and mass transfer in the reservoir on the intensity of the methane recovery process from the gas hydrate at different parameters of the reservoir.

Problem Statement
The continuous process of CO 2 injection into a porous medium is studied in a one-dimensional linear approximation [37]. The problem is considered in a one-dimensional approximation; i.e., the presented mathematical model has a limited area of applicability. It is valid only for reservoirs with a sufficiently large thickness and width. In addition, the presented model corresponds to the simplest case of carbon dioxide injection, when injection occurs continuously for a long time and with a constant pressure. This choice is due to the fact that the present work aims to identify only the most important, general and universal laws of the process of replacement of methane in CH 4 hydrate with carbon dioxide, which are independent of the specific method of the carbon dioxide injection into a gas hydrate deposit. Figure 1 shows the temperature and pressure conditions for the existence of hydrates CH 4 and CO 2 . The ranges of pressures and temperatures corresponding to the stable existence of CO 2 and CH 4 hydrates are below curves 1 and 2, respectively. In this paper, it is supposed that the substitution process will be realized under thermobaric conditions corresponding to the existence of CH 4 hydrate and CO 2 hydrate, as well as the existence of carbon dioxide in the gaseous state (darkened area in the phase diagram). does not take into account the deceleration of the driving force of the substitution process with the increasing thickness of the carbon dioxide layer over time.
In this paper, we present a mathematical model for the exchange of CH4 with CO2 in gas hydrate, taking into account both the kinetics and heat and mass transfer in a natural reservoir. At the same time, when describing the kinetics of the replacement process, the dependence of the rate of diffusion mass transfer of carbon dioxide in the gas hydrate on the growth of the thickness of the carbon dioxide gas monohydrate layer is taken into account. Mainly, attention is paid to the study of the degree of influence of the replacement kinetics and heat and mass transfer in the reservoir on the intensity of the methane recovery process from the gas hydrate at different parameters of the reservoir.

Problem Statement
The continuous process of CO2 injection into a porous medium is studied in a one-dimensional linear approximation [37]. The problem is considered in a one-dimensional approximation; i.e., the presented mathematical model has a limited area of applicability. It is valid only for reservoirs with a sufficiently large thickness and width. In addition, the presented model corresponds to the simplest case of carbon dioxide injection, when injection occurs continuously for a long time and with a constant pressure. This choice is due to the fact that the present work aims to identify only the most important, general and universal laws of the process of replacement of methane in CH4 hydrate with carbon dioxide, which are independent of the specific method of the carbon dioxide injection into a gas hydrate deposit. Figure 1 shows the temperature and pressure conditions for the existence of hydrates CH4 and CO2. The ranges of pressures and temperatures corresponding to the stable existence of CO2 and CH4 hydrates are below curves 1 and 2, respectively. In this paper, it is supposed that the substitution process will be realized under thermobaric conditions corresponding to the existence of CH4 hydrate and CO2 hydrate, as well as the existence of carbon dioxide in the gaseous state (darkened area in the phase diagram). As methane hydrate is less stable than carbon dioxide hydrate, under certain conditions, carbon dioxide molecules will replace methane molecules in gas hydrate without releasing free water [14,15].
Let the hydrate deposit at the initial moment of time consist of three components, namely the skeleton of a porous medium, methane and its hydrate.
As methane hydrate is less stable than carbon dioxide hydrate, under certain conditions, carbon dioxide molecules will replace methane molecules in gas hydrate without releasing free water [14,15].
Let the hydrate deposit at the initial moment of time consist of three components, namely the skeleton of a porous medium, methane and its hydrate. Let us write the mass conservation equations for CO 2 and CH 4 [38][39][40][41][42]: For the skeleton, hydrate and gas, the subscripts sk, h and g are used, respectively; the subscripts i = dc, mt refer to the parameters of CO 2 and CH 4 ; S g is the gas saturation; m is the reservoir porosity; υ g(i) and ρ 0 g(i) (i = dc, mt) are the partial velocities and densities of the components of mixture carbon dioxide and methane; and J g(mt) and J g(dc) are the intensities of the CH 4 release from hydrate and the CO 2 transition to hydrate.
Depending on various factors, such as the composition and state of the gas, the pressure and temperature of the medium, the gas hydrate can have a diverse structure: from solid (ice-like) to porous (snow-like). In the case of the hydrate porous structure, during the replacement process, a mixed gas hydrate can be formed with a continuous distribution of the concentrations of CH 4 and CO 2 in the volume of the hydrate. However, in our work, we will consider the case when the hydrate has a continuous (ice-like) structure, so there is a clear boundary between CO 2 and CH 4 hydrates. Then, the value of volume hydrate saturation will be equal to where S h(i) is used for the saturation of pores with methane hydrate (i = mt) and the carbon dioxide hydrate (i = dc), where S g + S h = 1.
As the gas hydrate is immovable, we obtain where G (i) and ρ 0 h(i) (i = dc, mt) are the relative mass of gas in the hydrate and the hydrate density, respectively.
We write the equation of H 2 O mass conservation in the hydrate in the following form: Considering the initial state (S hd = 0, S hm = S hm0 ) in integrating this equation, we find the dependence of the phase saturation of CH 4 hydrate and CO 2 hydrate: Note that the density of carbon dioxide hydrate is greater than the density of methane hydrate: ρ 0 h(dc) = 1115 kg/m 3 , ρ 0 h(mt) = 910 kg/m 3 [2]. In the hydrates of CO 2 and CH 4 in the equilibrium state, there are about six molecules of H 2 O per gas molecule. It is considered that the average mass fraction of CO 2 and CH 4 in the hydrate are equal to G (dc) = 0.28 and G (mt) = 0.12, respectively. Therefore, the following relation is performed with good accuracy: Energies 2020, 13, 440

of 17
There is no release of H 2 O from the gas hydrate composition in the process of the CH 4 -CO 2 exchange [20,21]. Hence, taking into account Equation (3), the following relation is valid for the intensities J g(dc) and J g(mt) : where M (dc) and M (mt) are the molar mass of carbon dioxide and methane. For the gas mixture as a whole, we introduce the mass-average velocity: Adding up the Equation (1), and then taking into account Equations (4) and (5), we obtain To describe the mixture filtration and the diffusion mixing of its components, we will use the Darcy law and Fick's law, respectively: where D g is the diffusion mixing coefficient of the components of gas mixture; and w g(i) (i = dc, mt) is the diffusion rate of the components of the methane and carbon dioxide mixture. We assume that the mixture of methane and carbon dioxide is a calorically perfect gas and obeys the Dalton law: where R (dc) and R (mt) are the specific gas constant of carbon dioxide and methane. The equation for the internal energy of the system has the following form: Here, the designation of the specific thermal conductivity and heat of the j-th phase are λ j and c j (j = g, h) respectively; the designation of the specific heat of CH 4 hydrate decomposition and CO 2 hydrate formation are l h(mt) and l h(dc) ; the designations λ and ρc are used for the thermal conductivity and specific volume heat capacity.
The system of relations is supplemented by the following conditions: Energies 2020, 13, 440 6 of 17 where L is the linear reservoir size; T 0 and p 0 are the initial reservoir temperature and pressure; and T e and p e are the temperature and pressure of the CO 2 injected into the layer.

Description of the CH 4 -CO 2 Replacement Kinetics in the Hydrate
We accept that the rate of substitution of methane in the composition of gas hydrate with CO 2 is limited by the diffusion of gas through the layer of CO 2 hydrate formed between the gas phase and CH 4 hydrate. The diffusing carbon dioxide with an average density of ρ g(dc) in the hydrate will be called mobile, and the gas in the hydrate with a mass concentration G i is immobile. Note that in connection with the adopted Equation (4), it becomes unnecessary to solve the inverse diffusion problem for the methane transfer through the hydrate layer of carbon dioxide.
We consider the following limiting scheme. We will assume that the skeleton of a porous medium is a set of channels with a cylindrical shape of radius a. In the presented model, it is assumed that CH 4 hydrate completely covers the channel walls. For this case, CH 4 hydrate will be located in the ring-shaped layer between r = a and r = a (dc) , and CO 2 hydrate in the layer between r = a (dc) and r = a g . Then, we obtain For the process of the transfer of a diffusing gas through a layer of CO 2 hydrate, we will use the diffusion equation [43,44]: where D h(dc) is the diffusion coefficient of CO 2 in its hydrate; ρ g(dc) is the average carbon dioxide density, which is dissolved in its hydrate. Under the boundary conditions, the quasi-stationary solution of Equation (13) is as follows: where ρ g(dc)s is the mobile carbon dioxide density in the CO 2 hydrate for a saturated state. The value of the mass of mobile carbon dioxide passing per unit of time through a unit contact surface area between CO 2 and CH 4 hydrates is equal to Substituting Equation (15) into Equation (16), we obtain The CO 2 consumption intensity J g(dc) is associated to the flow j g(dc) : J g(dc) = A h j g(dc) , where A h is the total specific surface area of contact between CO 2 and CH 4 hydrates (i.e., the surface area of contact Energies 2020, 13, 440 7 of 17 between the hydrates per unit volume of a reservoir). For this value in the framework of the adopted scheme, we obtain where n (dc) is the number of channels with a diameter of 2a (dc) in the volume unit. Using the analogy with Henry's law, we will assume that the saturated concentration of free carbon dioxide in the composition of carbon dioxide hydrate is proportional to the partial pressure of carbon dioxide in the gas phase; i.e., where Γ (dc) is Henry's parameter. Then, for the intensity J g(dc) of the carbon dioxide consumption for the formation of its hydrate per volume unit of porous medium, using Equations (17) and (18), we can write We introduce a new empirical parameter, which will be called the reduced diffusion coefficient for carbon dioxide in a gas hydrate: Substituting Equation (19) and the first equation from Equation (9) into the last formula, we obtain In [45], based on the experimental data processing, the values of the methane diffusion coefficient through the gas hydrate layer were found to be of the order of 10 −13 m 2 /s at the temperature of 274 K. By analogy with liquids, the diffusion coefficient D h(dc) and Henry's parameter Γ (dc) can be regarded as sufficiently conservative values (for example, weakly dependent on the value p g(dc) ). Since the relative change in temperature for the processes under consideration is small, the reduced diffusion coefficient D will be considered constant. It should be mentioned that many factors can influence the intensity of the hydrate decomposition and formation, such as the introduction of various surfactants, the exposure of the shock and electromagnetic waves [46][47][48][49][50][51][52][53], etc. The empirical parameter D depends both on the hydrate structure and on the porous medium skeleton features.
We transform the above equations, using the independent variables ρ 0 g(dc) , ρ 0 g , T and S h(dc) . The first equation from Equation (1), taking into account Equations (7) and (8), can be written as As the second equation, we use Equation (6), which, taking into account Equation (7), will take the form The heat influx in Equation (10), taking into account Equations (4) and (7), can be written in the form Energies 2020, 13, 440 8 of 17 To close this system, it is enough to add one more differential equation following from the first equation in Equation (2): For other variable parameters included in Equations (23)-(26), taking into account Equations (12), (20) and (22), we obtain Based on Equation (9) for pressure, we can get In accordance with the above assumptions, the permeability of the gas mixture and the radius of porous tubules will be specified as where k is the absolute permeability coefficient of the porous skeleton. We supplement the system of Equations (23)-(29) with the boundary and initial conditions: The system of Equations (23)- (29) with the boundary and initial conditions in Equation (30) was solved by the finite difference method.  Table 1 [11,16,19,39,54]. Note that the temperature fields in a reservoir will largely depend on the porous reservoir material, since it affects the values of thermal conductivity and specific volumetric heat capacity. The averaged values of the coefficients of thermal conductivity and specific volumetric heat capacity, characteristic of most reservoirs, are used in the work. The process will approach the frontal mode of hydrate substitution in the reservoir with a decrease in permeability. When implementing this mode, the reservoir is conventionally divided into two zones: the near one, containing carbon dioxide hydrate and the gas mixture CO 2 -CH 4 , and the far one, containing methane and its hydrate. In Figure 2, we can observe that the frontal mode is limited primarily by filtration mass transfer, and the role of diffusive mass transfer weakens.

Calculation Results
We used the approximate analytical solutions presented in [31] to test the mathematical model. In [31], approximate analytical solutions were obtained for the problem of decomposition of methane gas hydrate during the injection of gaseous CO 2 into a gas-hydrated formation based on the method of converting to a self-similar variable. replacement with carbon dioxide in the hydrate saturated sand sample was studied. The initial methane hydrate saturation was 10%, the sample porosity was 34%, and the initial temperature in the reactor was 275 K.      Figure 4 shows a comparison of the results of numerical study and experimental data, given in [55], for changes in temperature over time. As can be seen from Figure 4, energy is released due to the replacement; after the completion of the CH4-CO2 replacement process, the sample temperature returns to the initial one. It is worth noting the satisfactory agreement between the experimental data and the calculation results using the proposed model.  is the equilibrium temperature of phase transitions at the current pressure in the reservoir. It is shown that the injection of the CO2 into the reservoir leads to the increase of carbon dioxide partial pressure in the porous medium and to the initiation of the CH4 hydrate to CO2 hydrate transition process. The surface of the methane displacement by CO2 reaches the right border of porous medium in a time of about 1 min. Subsequently, methane is displaced, which was recovered from the gas hydrate as a result of the replacement process. The process of CH4-CO2 exchange in the methane hydrate occurs with the reservoir heating. We also performed a comparison with the experimental data presented in [55]. In this work, the process of the replacement of methane in CH 4 hydrate with CO 2 in a dispersed medium is considered, and the dynamics of temperature changes for this process is presented. The experiment was carried out in two stages. The first stage was a test study of the methane hydrate formation in a quartz sand sample partially saturated with water. At the second stage of the experiment, the process of methane replacement with carbon dioxide in the hydrate saturated sand sample was studied. The initial methane hydrate saturation was 10%, the sample porosity was 34%, and the initial temperature in the reactor was 275 K. Figure 4 shows a comparison of the results of numerical study and experimental data, given in [55], for changes in temperature over time. As can be seen from Figure 4, energy is released due to the replacement; after the completion of the CH 4 -CO 2 replacement process, the sample temperature returns to the initial one. It is worth noting the satisfactory agreement between the experimental data and the calculation results using the proposed model.  Figure 4 shows a comparison of the results of numerical study and experimental data, given in [55], for changes in temperature over time. As can be seen from Figure 4, energy is released due to the replacement; after the completion of the CH4-CO2 replacement process, the sample temperature returns to the initial one. It is worth noting the satisfactory agreement between the experimental data and the calculation results using the proposed model.  [55]. The solid line is the numerical study results according to the proposed model, while the points are the experimental data. Figure 5 represents the parameter distributions along the length of the reservoir at different of time instants at k0 = 10 −16 m 2 , D = 10 −16 m 2 /s, Т0 = 274 K, p0 = 3 MPa; Ts(p) is the equilibrium temperature of phase transitions at the current pressure in the reservoir. It is shown that the injection of the CO2 into the reservoir leads to the increase of carbon dioxide partial pressure in the porous medium and to the initiation of the CH4 hydrate to CO2 hydrate transition process. The surface of the methane displacement by CO2 reaches the right border of porous medium in a time of about 1 min. Subsequently, methane is displaced, which was recovered from the gas hydrate as a result of the replacement process. The process of CH4-CO2 exchange in the methane hydrate occurs with the reservoir heating.  It is shown that the injection of the CO 2 into the reservoir leads to the increase of carbon dioxide partial pressure in the porous medium and to the initiation of the CH 4 hydrate to CO 2 hydrate transition process. The surface of the methane displacement by CO 2 reaches the right border of porous medium in a time of about 1 min. Subsequently, methane is displaced, which was recovered from the gas hydrate as a result of the replacement process. The process of CH 4 -CO 2 exchange in the methane hydrate occurs with the reservoir heating. The specific (expressed per unit of the reservoir cross-sectional area) mass of the conserved CO2 Qg(dc) and the specific mass of the CH4 extracted from the reservoir Qg(mt) over a period of time T can be determined as follows: Figure 6 shows the time dependences of the specific mass flow rates of the injected and extracted carbon dioxide, the mass flow rate of the extracted methane, the specific mass of carbon dioxide buried in the reservoir in the gas hydrate state, and the specific mass of the produced methane. It is evident that, at the initial stage, free methane is displaced from the reservoir. In the second stage, CH4 in the hydrate is replaced by CO2. As a result of the process of CO2 injection into the reservoir, about 35 kg/m 2 of methane is extracted and about 115 kg/m 2 of CO2 is stored. The mass flow rate of the gas passing through a unit cross-sectional area of the reservoir is equal to q g(dc)e = ρ 0 g(dc)e mS g υ ge , q g(dc) = ρ 0 g(dc) mS g υ g , q g(mt) = ρ 0 g(mt) mS g υ g where q g(dc)e is the specific mass flow rate of the injected CO 2 ; and q g(dc) and q g(mt) are the specific mass flow rates of CO 2 and CH 4 at the reservoir outlet. The specific (expressed per unit of the reservoir cross-sectional area) mass of the conserved CO 2 Q g(dc) and the specific mass of the CH 4 extracted from the reservoir Q g(mt) over a period of time T can be determined as follows: Figure 6 shows the time dependences of the specific mass flow rates of the injected and extracted carbon dioxide, the mass flow rate of the extracted methane, the specific mass of carbon dioxide buried in the reservoir in the gas hydrate state, and the specific mass of the produced methane. It is evident that, at the initial stage, free methane is displaced from the reservoir. In the second stage, CH 4 in the hydrate is replaced by CO 2 . As a result of the process of CO 2 injection into the reservoir, about 35 kg/m 2 of methane is extracted and about 115 kg/m 2 of CO 2 is stored. Energies 2020, 13, x FOR PEER REVIEW 12 of 17 Figure 6. Evolution with time of the specific mass flow rates of the CO2 injected into the layer qg(dc)e and produced from the reservoir qg(dc), the produced methane qg(mt), the specific mass of the conserved CO2 Qg(dc) and the specific mass of the produced CH4 Qg(mt).
The influence of various parameters on the time trep of the complete replacement of CH4 in hydrate with CO2 was studied. Figure 7 shows the dependence of the time trep of complete replacement on the absolute reservoir permeability k; the dependence is constructed for two values of the reduced diffusion coefficient D. It can be seen that, at low values of the reservoir permeability, the reduced diffusion coefficient has little effect on the time of transition of CH4 hydrate in CO2 hydrate in the entire reservoir. This is due to the fact that, in the case with low values of the porous medium permeability k, the CH4-CO2 replacement process is limited by the mass transfer due to filtration. At high values of the reservoir permeability, the rate of the methane replacement in the CH4 hydrate with carbon dioxide significantly depends on the reduced diffusion coefficient magnitude D (Figure 7). We note that the increase in the replacement time, in the case of the growth in the porous media permeability k at the high values of the reservoir permeability, is due to the decrease in the specific contact surface between the CO2 hydrate and CH4 hydrate with an increased value of k. Figure 6. Evolution with time of the specific mass flow rates of the CO 2 injected into the layer q g(dc)e and produced from the reservoir q g(dc) , the produced methane q g(mt) , the specific mass of the conserved CO 2 Q g(dc) and the specific mass of the produced CH 4 Q g(mt) .
The influence of various parameters on the time t rep of the complete replacement of CH 4 in hydrate with CO 2 was studied. Figure 7 shows the dependence of the time t rep of complete replacement on the absolute reservoir permeability k; the dependence is constructed for two values of the reduced diffusion coefficient D. It can be seen that, at low values of the reservoir permeability, the reduced diffusion coefficient has little effect on the time of transition of CH 4 hydrate in CO 2 hydrate in the entire reservoir. This is due to the fact that, in the case with low values of the porous medium permeability k, the CH 4 -CO 2 replacement process is limited by the mass transfer due to filtration. At high values of the reservoir permeability, the rate of the methane replacement in the CH 4 hydrate with carbon dioxide significantly depends on the reduced diffusion coefficient magnitude D (Figure 7). We note that the increase in the replacement time, in the case of the growth in the porous media permeability k at the high values of the reservoir permeability, is due to the decrease in the specific contact surface between the CO 2 hydrate and CH 4 hydrate with an increased value of k. The dependence of the time trep of the total substitution of CH4 by CO2 in the CH4 hydrate on the hydrate saturated reservoir length is shown in Figure 8. It can be seen that, for short-length reservoirs with a decrease in permeability, the rate of hydrate of CH4 methane to the hydrate of CO2 transfer increases. This is due to the fact that, in this case, the process of the CH4-CO2 replacement is limited by diffusion, and a decrease in permeability corresponds to an increase in the specific contact surface between the hydrate and the gas. It is also worth noting that, in the case of a short-length reservoir, the complete CH4-CO2 replacement occurs at relatively short time intervals ( Figure 8). As the length of the reservoir increases, the process of transition of CH4 hydrate to CO2 hydrate in the reservoir becomes increasingly dependent on the filtration mass transfer, which, in turn, depends significantly on the porous medium permeability. Therefore, in the case of long reservoirs, the time for the complete replacement of methane in the CH4 hydrate with carbon dioxide decreases with increasing the reservoir permeability.   The dependence of the time t rep of the total substitution of CH 4 by CO 2 in the CH 4 hydrate on the hydrate saturated reservoir length is shown in Figure 8. It can be seen that, for short-length reservoirs with a decrease in permeability, the rate of hydrate of CH4 methane to the hydrate of CO 2 transfer increases. This is due to the fact that, in this case, the process of the CH 4 -CO 2 replacement is limited by diffusion, and a decrease in permeability corresponds to an increase in the specific contact surface between the hydrate and the gas. It is also worth noting that, in the case of a short-length reservoir, the complete CH 4 -CO 2 replacement occurs at relatively short time intervals (Figure 8). As the length of the reservoir increases, the process of transition of CH 4 hydrate to CO 2 hydrate in the reservoir becomes increasingly dependent on the filtration mass transfer, which, in turn, depends significantly on the porous medium permeability. Therefore, in the case of long reservoirs, the time for the complete replacement of methane in the CH 4 hydrate with carbon dioxide decreases with increasing the reservoir permeability. The dependence of the time trep of the total substitution of CH4 by CO2 in the CH4 hydrate on the hydrate saturated reservoir length is shown in Figure 8. It can be seen that, for short-length reservoirs with a decrease in permeability, the rate of hydrate of CH4 methane to the hydrate of CO2 transfer increases. This is due to the fact that, in this case, the process of the CH4-CO2 replacement is limited by diffusion, and a decrease in permeability corresponds to an increase in the specific contact surface between the hydrate and the gas. It is also worth noting that, in the case of a short-length reservoir, the complete CH4-CO2 replacement occurs at relatively short time intervals (Figure 8). As the length of the reservoir increases, the process of transition of CH4 hydrate to CO2 hydrate in the reservoir becomes increasingly dependent on the filtration mass transfer, which, in turn, depends significantly on the porous medium permeability. Therefore, in the case of long reservoirs, the time for the complete replacement of methane in the CH4 hydrate with carbon dioxide decreases with increasing the reservoir permeability.  Figure 9 shows the dependence of the time of full replacement trep on the reservoir length; the dependence is constructed for two values of the reduced diffusion coefficient D. It is evident that, as the reservoir length increases, the influence of the kinetics on the time of full transition of CH4 hydrate to CO2 hydrate decreases, and the role of the filtration mass transfer increases.  Figure 9 shows the dependence of the time of full replacement t rep on the reservoir length; the dependence is constructed for two values of the reduced diffusion coefficient D. It is evident that, as the reservoir length increases, the influence of the kinetics on the time of full transition of CH 4 hydrate to CO 2 hydrate decreases, and the role of the filtration mass transfer increases. Energies 2020, 13, x FOR PEER REVIEW 14 of 17

Conclusions
The process of methane replacement in gas hydrate with carbon dioxide during CO2 injection into a porous medium was studied. The calculations show that, for the process of CO2 injection into the gas hydrate reservoir, two stages can be distinguished. The first stage is specified by the beginning of the CH4-CO2 replacement process; in the second stage, methane, formed as a result of replacement, is extracted from the hydrate saturated reservoir.
It was established that at high values of the reduced diffusion coefficient of carbon dioxide in its hydrate, low values of the reservoir permeability, and with the growth of the reservoir length, the process of the CH4-CO2 replacement in CH4 hydrate will take place in the frontal regime. In this regime, the replacement process is determined primarily by the intensity of filtration mass transfer in the reservoir. Otherwise, the process of the CH4-CO2 replacement will occur in the volume area of the reservoir. In this case, the process is determined by the intensity of gas diffusion in hydrate. For the case of highly permeable porous media (the hydrate formation process is limited by the diffusion of gas in the hydrate), an increase in the methane replacement time in the CH4 hydrate with carbon dioxide is shown, which is caused by a decrease in the specific surface of contact between CH4 hydrate and CO2 hydrate with increasing reservoir permeability. The presented mathematical model is more universal and allows us to consider the simultaneous influence of two factors (diffusion mechanism, filtration mass transfer) on the process under study.

Conclusions
The process of methane replacement in gas hydrate with carbon dioxide during CO 2 injection into a porous medium was studied. The calculations show that, for the process of CO 2 injection into the gas hydrate reservoir, two stages can be distinguished. The first stage is specified by the beginning of the CH 4 -CO 2 replacement process; in the second stage, methane, formed as a result of replacement, is extracted from the hydrate saturated reservoir.
It was established that at high values of the reduced diffusion coefficient of carbon dioxide in its hydrate, low values of the reservoir permeability, and with the growth of the reservoir length, the process of the CH 4 -CO 2 replacement in CH 4 hydrate will take place in the frontal regime. In this regime, the replacement process is determined primarily by the intensity of filtration mass transfer in the reservoir. Otherwise, the process of the CH 4 -CO 2 replacement will occur in the volume area of the reservoir. In this case, the process is determined by the intensity of gas diffusion in hydrate. For the case of highly permeable porous media (the hydrate formation process is limited by the diffusion of gas in the hydrate), an increase in the methane replacement time in the CH 4 hydrate with carbon dioxide is shown, which is caused by a decrease in the specific surface of contact between CH 4 hydrate and CO 2 hydrate with increasing reservoir permeability. The presented mathematical model is more universal and allows us to consider the simultaneous influence of two factors (diffusion mechanism, filtration mass transfer) on the process under study.