# Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and Its Hydrate

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}and hydrate, the second region is saturated with methane and water, the third contains methane and hydrate. The main features of mathematical models that provide a consistent description of the considered processes are investigated. It was found that at sufficiently high injection pressures and low pressures at the right reservoir boundary, the boundary of carbon dioxide hydrate formation can come up with the boundary of methane gas hydrate decomposition. It is also shown that at sufficiently low values of pressure of injection of carbon dioxide and pressure at the right boundary of the reservoir, the pressure at the boundary of hydrate formation of carbon dioxide drops below the boiling pressure of carbon dioxide. In this case, for a consistent description of the considered processes, it is necessary to correct the mathematical model in order to take into account the boiling of carbon dioxide. Maps of possible solutions have been built, which show in what ranges of parameters one or another mathematical model is consistent.

## 1. Introduction

_{2}utilization can be solved by using this method.

_{4}hydrate to CO

_{2}hydrate during carbon dioxide injection can occur in two different modes [14,15,16,17]. A characteristic feature of the first mode is the absence of free water release. In the second mode, there is an intermediate zone of methane hydrate dissociation to water and gas, and the subsequent formation of CO

_{2}hydrate from water and carbon dioxide. In this case, the released heat can accelerate the decomposition of methane gas hydrate. Experiments were carried out to study the formation of CO

_{2}hydrates in [18,19,20,21,22]. It should be noted that experimental works, which study these processes, were carried out in examples of small size. Due to the small size of the samples, the processes of formation and decomposition of gas hydrate are limited by kinetic mechanisms. The possibility of replacement of CH

_{4}by CO

_{2}in gas hydrate upon the injection of carbon dioxide is due to the negative value of the Gibbs free energy of this reaction (i.e., the negative value of the change in the Gibbs free energy) [23]. The thermodynamic conditions of the CH

_{4}–CO

_{2}replacement reaction are maintained even without the additional influence of external factors. For cases when natural reservoirs of real length are considered, the limitation of formation of gas hydrate CO

_{2}and decomposition of CH

_{4}gas hydrate are determined not by the kinetics of the phase transition processes, but by heat and mass transfer in the reservoir, as well as the release and absorption of heat during phase transitions. These do not allow comparing the results of mathematical modeling of the formation and decomposition of gas hydrates in extended natural reservoirs with the experimental data available to date. Therefore, in our opinion, it is necessary to examine mathematical models describing the studied processes.

_{2}. In [24,25,26,27], approximate analytical solutions are given for replacing CH

_{4}hydrate with CO

_{2}when CO

_{2}is injected into a reservoir. In these works, self-similar solutions for a semi-infinite reservoir are constructed. These self-similar solutions can adequately describe only the initial process. The solution is consistent only until the pressure perturbation reaches the reservoir right boundary. In the future, over time, the effect on the course of the process will completely depend on the conditions on the right boundary. In [28], the case of mathematical modeling of CO

_{2}injection in a gaseous state into a natural reservoir containing methane and its gas hydrate was considered. A similar problem was considered in [29], but instead of CO

_{2}in the gaseous state, carbon dioxide in the liquid state was supplied. However, in [29], only the case when the recovery of methane from the hydrate occurs in the replacement mode and is not accompanied by dissociation of methane gas hydrate is studied in detail. In the present work, unlike in [29], the injection mode of liquid CO

_{2}reservoir into a gas hydrate of finite length, accompanied by dissociation of CH

_{4}gas hydrate and the subsequent formation of CO

_{2}hydrate, is studied in detail.

## 2. Mathematical Model

_{2}through the left border (x = 0) of the reservoir is considered. The temperature T

_{e}and injection pressure p

_{e}are selected in the phase diagram Figure 1 in such a way that they correspond to the conditions for the existence of CO

_{2}in the liquid state and the conditions for the existence of CO

_{2}hydrate [30].

_{e}> p

_{0}(for t ≥ 0).

_{2}starts the decomposition of methane hydrate. The next stage is associated with the formation of CO

_{2}hydrate, and its formation requires released water and carbon dioxide. In [31], it was shown that in natural porous layers, filtration (convective) transfer significantly dominates over diffusion transfer. In addition, currents in natural reservoirs are laminar, and the viscosity of carbon dioxide exceeds the viscosity of methane. In this regard, in this paper we will assume the stability of the front of displacement of methane by carbon dioxide.

^{−13}m

^{2}). In this case the process of filling the reservoir with injected carbon dioxide will proceed rather slowly, and since there was no carbon dioxide in the initial state, the process of conversion of CH

_{4}hydrate to CO

_{2}hydrate will also occur very slowly (i.e., limited by the rate of filtration). Therefore, for the considered process (when the reservoir length L > 100 m), times of the order of several tens of days are of practical interest. At such long times, which significantly exceed the characteristic time of the process kinetics (which, according to experimental data, is on the order of several hours), the kinetic mechanisms no longer limit the process of the transition of CH

_{4}hydrate to CO

_{2}hydrate. Therefore, considering sufficiently large time values (on the order of several days or ten days), the kinetics of the processes of formation and decomposition of gas hydrates can be neglected. In this regard, the constructed model is valid only for large time scales, relatively long reservoir length (L ≥ 100 m) and low permeability (k < 10

^{−13}m

^{2}).

_{2}and its hydrate. In the intermediate (second) region, only methane and water are present in the reservoir. In the far (third) region, in addition to methane, its hydrate is present in the pores Figure 2. Methane hydrate decomposition is present only between the 2 and 3 boundaries, respectively, on the moving frontal surface. In this case, the formation of CO

_{2}hydrate occurs only at the border between 1 and 2 regions.

_{j}, S

_{j}, υ

_{j}, k

_{j}, C

_{j}and μ

_{j}are density, saturation, speed, phase permeability, specific heat capacity and dynamic viscosity of carbon dioxide (j = c) and methane (j = m) respectively; R

_{m}is the gas constant of methane; β is the compressibility factor of CO

_{2}; k

_{0}is the absolute permeability; ρC and λ are the specific volumetric heat capacity and system thermal conductivity coefficient.

_{c}on its saturation S

_{c}, the following relation is used [25,37]:

_{c}is the carbon dioxide saturation. The saturation of carbon dioxide in the first region is equal to:

_{hc}is the saturation for carbon dioxide gas hydrate.

_{c}on the hydrate saturation of S

_{hc}, the following relation is used:

_{m}on its saturation S

_{m}, the following relation is used:

_{m}is the methane saturation.

_{m}

_{(3)}in the third region is equal to:

_{m}in the third region on the saturation of methane gas hydrate ν, the following relation is used:

_{2}and CH

_{4}on the border between the first and second areas are:

_{c}, ρ

_{hc}, S

_{hc}are the mass concentrations in hydrate of carbon dioxide, density and saturation of carbon dioxide gas hydrate; ${\dot{x}}_{(n)}$ is the speed of movement of the near boundary of phase transitions. Hereinafter, to describe the thermophysical characteristics at the boundary between 1 and 2 regions, the index n is used.

_{hc}is the heat of the carbon dioxide gas hydrate formation, T

_{(i)}and p

_{(i)}are the temperature and the pressure; index i = 1 and i = 2 belong to the first and second area; S

_{l}and ρ

_{l}are the pore saturation and density of water. The pressure and temperature at the boundary between 1 and 2 areas are assumed to be continuous.

_{m(i)}, T

_{(i)}and p

_{(i)}are pore saturation, temperature and pressure of methane, index i = 3 belong to the third area; G

_{m}, ${\rho}_{hm}$, ν, L

_{hm}are the mass concentrations in hydrate of methane, density, saturation and heat of the methane gas hydrate formation; ${\dot{x}}_{(d)}$ is the speed of movement of the far border of phase transitions. The pressure and temperature on this surface are related by the phase equilibrium condition for methane and its hydrate [30]:

_{2}hydrate formation moves along the coordinate grid by exactly one step in one step by time. This time step is calculated by iterations. The distributions of pressure and temperature in the far and intermediate regions, as well as the position of the phase transition boundary x = x

_{(d)}, are determined based on the end-to-end counting method.

_{e}= 282 K, T

_{0}= 274 K, p

_{0}= 3.5 MPa, p

_{e}= 4 MPa, k

_{0}= 10

^{−16}m

^{2}, ρC = 2 × 10

^{6}J/(K∙m

^{3}), λ = 2 W/(m∙K). For these parameters, the following values of the self-similar coordinates of the phase transition boundaries were obtained in the work of [24] (p. 744, Figure 5):

## 3. Calculations, Results and Discussion

_{2}. For the value of the initial pressure of the system, the values p

_{0}= 3.1 MPa (a) and 2.9 MPa (b) are assumed.

_{2}hydrate formation with increasing injection pressure. It also follows from Figure 4 that with an increase in the injection pressure, the coordinate of the hydrate reservoir boundary also increases, and the coordinate of the dissociation boundary of methane hydrate increases slightly. This is due to the increase in the filtration rate of carbon dioxide with increasing injection pressure. The rate of displacement of methane by carbon dioxide has a major effect on the rate of hydrate formation. An increase in the velocity of this boundary leads to a decrease in the influence on it of a hotter left boundary and, accordingly, to a decrease in its temperature. According to Figure 4, at high pressures, the near boundary of phase transitions can catch up with the far boundary. In this case, methane in the gas hydrate is replaced by carbon dioxide at the only phase transition boundary. In addition, at sufficiently low injection pressures, the pressure at the near boundary may fall below the equilibrium boiling pressure. Therefore, in this case, it is necessary, in addition to the area containing liquid CO

_{2}and its gas hydrate, to introduce into consideration a region saturated with gaseous CO

_{2}and its gas hydrate. Thus, at low values of the injection pressure of p

_{e}in the reservoir, an additional region saturated with gaseous carbon dioxide and its hydrate will additionally appear. At high injection pressures, a CO

_{2}gas hydrate formation will occur in the replacement mode in the absence of a region containing methane and water. For average injection pressures, two different cases are possible, depending on the pressure at the right boundary of the reservoir, p

_{0}. At relatively high values of p

_{0}, the regime is implemented according to the scheme shown in Figure 2. At relatively low values of p

_{0}, the formation of CO

_{2}hydrate will occur in the replacement mode, but is accompanied by boiling of liquid carbon dioxide.

_{e}= 3.7 MPa for case (a) and 3.64 MPa for case (b). Figure 5 shows the increase in temperature at the boundary of the formation of CO

_{2}gas hydrate with increasing pressure at the right boundary p

_{0}. Also, according to Figure 5, as the pressure on the right border p

_{0}decreases, the coordinates of both the near and far boundaries of the phase transition increase. This is due to the fact that the rate of CO

_{2}hydrate formation boundary is determined by the rate of methane displacement by carbon dioxide. Thus, the filtration rate of carbon dioxide increases with increasing pressure drop in the reservoir. An increase in the velocity of this boundary leads to a decrease in the influence on it of a hotter left boundary and, respectively, to a decrease in its temperature. As a consequence, according to Figure 5, at low values of p

_{0}, it is possible to merge the boundaries of phase transitions and therefore it is also possible to merge the formation of CO

_{2}hydrate in the replacement mode. Also, in the case of low values of p

_{0}, the pressure at the near phase transition boundary can fall below the equilibrium boiling pressure. Thus, as follows from Figure 5, at low pressures at the right boundary of the reservoir, the formation of CO

_{2}gas hydrate occurs in the replacement mode with the formation of a region saturated with gaseous CO

_{2}and its gas hydrate. At high values of p

_{0}, the formation of CO

_{2}gas hydrate occurs according to the scheme shown in Figure 2.

_{2}. For the pressure of injection, the values are p

_{e}= 3.75 MPa (a) and 3.65 MPa (b). From here and on, the pressure at the right boundary of the reservoir was assumed to be p

_{0}= 3 MPa. As follows from Figure 6, with an increase in the permeability of the reservoir, the phase transitions may merge and produce CO

_{2}gas hydrate in the replacement mode. Also, according to Figure 6, at relatively high injection pressures, the process of forming CO

_{2}gas hydrate will occur without forming a region saturated with gaseous CO

_{2}and its gas hydrate. At low injection pressures, the process will be accompanied by boiling of liquid carbon dioxide.

_{*}, separating modes with boiling and without boiling of carbon dioxide (dashed line) and modes with and without dissociation of methane gas hydrate (solid line) on the permeability of the porous medium (a), the pressure value at the right boundary of the reservoir (b), from the initial temperature of the reservoir (c) and from the injection temperature (d). The dashed line corresponds to the equilibrium boiling pressure of liquid CO

_{2}. Figure 7 shows four regions. In region I, there is no dissociation of CH

_{4}hydrate and the CO

_{2}boiling mode is present. This mode is implemented at high values of injection pressure. In region II, solutions are located without dissociation of CH

_{4}hydrate and with CO

_{2}boiling. This mode is typical for highly permeable reservoirs, for reservoirs with low pressure at the right boundary, also when there is a low temperature of injected liquid and low initial reservoir temperature. Region III corresponds to the mode of the process with the dissociation of CH

_{4}hydrate and without boiling of CO

_{2}. This mode is typical for low-permeable reservoirs and high values of pressure at the right reservoir boundary. This mode is also implemented at high values of the injection temperature and of the initial temperature of the reservoir. In region IV, solutions are located with the dissociation of CH

_{4}hydrate and with boiling of CO

_{2}. This mode is implemented at low values of injection pressure.

## 4. Conclusions

_{2}into a reservoir of finite length, accompanied by the decomposition of CH

_{4}gas hydrate and the formation of CO

_{2}gas hydrate. The dependences of the phase transition boundaries coordinates that determine the rate of dissociation of CH

_{4}gas hydrate and the formation of CO

_{2}gas hydrate, as well as the pressure and temperature values at these boundaries on the pressure of carbon dioxide injected, as well as the permeability and initial pressure of the reservoir, were obtained. It was established that with increasing injection pressure and decreasing pressure at the right boundary of the reservoir, the temperature decreases at the boundary of carbon dioxide gas hydrate formation and the coordinate of the carbon dioxide gas hydrate formation increases. It is shown that at sufficiently low pressures at the right reservoir boundary and high injection pressures, the formation boundary of carbon dioxide gas hydrate can catch up with the decomposition boundary of methane gas hydrate. This corresponds to the mode of the process with the replacement of CH

_{4}with CO

_{2}in hydrate without releasing free water. It was also found that at sufficiently low values of injection pressure and pressure at the right reservoir boundary, the pressure at the boundary of carbon dioxide gas hydrate formation drops below the boiling pressure of carbon dioxide. These values of pressure at the right boundary and injection pressure can correspond both to the existence of a region saturated with the products of decomposition of methane gas hydrate, and its absence. Hydrate decomposition can be accompanied by boiling and occur without it. It has been established that the mode with the dissociation of CH

_{4}hydrate to water and gas is realized in low-permeable reservoirs, as well as at relatively high values of the initial reservoir temperature and injection temperature. It was established that the mode with boiling of liquid carbon dioxide is characteristic for relatively low values of pressure at the right boundary of the reservoir and CO

_{2}injection pressure.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Phase diagrams of “CO

_{2}-H

_{2}O” (

**a**) and “CH

_{4}-H

_{2}O” (

**b**) systems [29]. The blue curve is a two-phase liquid-vapor equilibrium line for CO

_{2}. The black and red curves are the lines of three-phase equilibrium “CH

_{4}-H

_{2}0-hydrate CH

_{4}” and “CO

_{2}-H

_{2}O-hydrate CO

_{2}”, respectively.

**Figure 3.**Comparison of numerical solutions (solid curve) and approximate analytical solutions (dashed curve). (

**a**)—the coordinate of the near boundary of phase transitions; (

**b**)—the coordinate of the far boundary of phase transitions.

**Figure 4.**Dependencies of pressures, temperatures and coordinates of phase transitions on carbon dioxide injection pressure. p

_{0}= 3.1 MPa (

**a**) and 2.9 MPa (

**b**).

**Figure 5.**Dependencies of pressure values, temperatures and coordinates of phase transition boundaries on pressure at the right reservoir boundary. p

_{e}= 3.7 MPa (

**a**) and 3.64 MPa (

**b**).

**Figure 6.**Dependencies of pressure values, temperatures and coordinates of phase transition boundaries from reservoir permeability. p

_{e}= 3.75 MPa (

**a**) and 3.65 MPa (

**b**).

**Figure 7.**Dependences of the values p

_{*}on the porous medium permeability k

_{0}(

**a**), on the pressure value at the right reservoir boundary p

_{0}(

**b**), on the initial reservoir temperature T

_{0}(

**c**) and on the injection temperature T

_{e}(

**d**).

Variables | Symbol | Value | Unit |
---|---|---|---|

Porosity | φ | 0.2 | - |

Initial hydrate saturation | ν | 0.2 | - |

Permeability | k_{0} | 2 × 10^{−16} | m^{2} |

Reservoir length | L | 100 | m |

Initial temperature | T_{0} | 274 | K |

Injection temperature | T_{e} | 280 | K |

Specific volumetric heat capacity | ρC | 2.0 × 10^{6} | J/(K∙m^{3}) |

System thermal conductivity coefficient | λ | 2 | W/(m∙K) |

Mass concentrations of carbon dioxide in hydrate | G_{c} | 0.28 | - |

Mass concentrations of methane in hydrate | G_{m} | 0.13 | - |

Gas constant of methane | R_{m} | 520 | J/(kg∙K) |

Density of carbon dioxide hydrate | ρ_{hc} | 1100 | kg/m^{3} |

Density of methane hydrate | ρ_{hm} | 900 | kg/m^{3} |

Density of water | ρ_{l} | 1000 | kg/m^{3} |

Density of carbon dioxide | ρ_{c}_{0} | 890 | kg/m^{3} |

Compressibility factor of carbon dioxide | β | 10^{−8} | Pa^{−1} |

Dynamic viscosity of carbon dioxide | µ_{c} | 10^{−4} | Pa∙s |

Dynamic viscosity of methane | µ_{m} | 10^{−5} | Pa∙s |

Heat of the formation of carbon dioxide hydrate | L_{hc} | 3.51 × 10^{5} | J/kg |

Heat of the formation of methane hydrate | L_{hm} | 4.48 × 10^{5} | J/kg |

Specific heat capacity of carbon dioxide | C_{c} | 2600 | J/(kg∙K) |

Specific heat capacity of methane | C_{m} | 1560 | J/(kg∙K) |

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**MDPI and ACS Style**

Khasanov, M.K.; Musakaev, N.G.; Stolpovsky, M.V.; Kildibaeva, S.R. Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and Its Hydrate. *Mathematics* **2020**, *8*, 1482.
https://doi.org/10.3390/math8091482

**AMA Style**

Khasanov MK, Musakaev NG, Stolpovsky MV, Kildibaeva SR. Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and Its Hydrate. *Mathematics*. 2020; 8(9):1482.
https://doi.org/10.3390/math8091482

**Chicago/Turabian Style**

Khasanov, Marat K., Nail G. Musakaev, Maxim V. Stolpovsky, and Svetlana R. Kildibaeva. 2020. "Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and Its Hydrate" *Mathematics* 8, no. 9: 1482.
https://doi.org/10.3390/math8091482