Reservoir Permeability Evolution during the Process of CO₂-Enhanced Coalbed Methane Recovery

Abstract

In this study, we have built a dual porosity/permeability model through accurately expressing the volumetric strain of matrix and fracture from a three-dimensional method which aims to reveal the reservoir permeability evolution during the process of CO₂-enhanced coalbed methane (CO₂-ECBM) recovery. This model has accommodated the key competing processes of mechanical deformation and adsorption/desorption induced swelling/shrinkage, and it also considered the effect of fracture aperture and effective stress difference between each medium (fracture and matrix). We then numerically solve the permeability model using a group of multi-field coupling equations with the finite element method (FEM) to understand how permeability evolves temporally and spatially. We further conduct multifaceted analyses to reveal that permeability evolution near the wells is the most dramatic. This study shows that the farther away from the well, the gentler the evolution of permeability. The evolution of reservoir permeability near the injection well (IW) and the production well (PW) are very different, due to the combined effects of effective stress changes and gas adsorption and desorption. Furthermore, adsorption is the main controlling factor for the change of permeability for regions near the IW, while the change in effective stress is the main cause for the change in permeability near the PW. Increasing the injection pressure of CO₂ will cause the reservoir permeability to evolve more quickly and dynamically.

1. Introduction

Coalbed methane (CBM) recovery from coal seams, as an unconventional resource, is a growing contributor to the energy supply for the energy-hungry world [1,2,3]. Carbon dioxide (CO₂) geological storage through reducing the risk of CO₂ migration to the surface is becoming an important method for effective control of greenhouse gas [4,5,6,7]. CO₂-enhanced coalbed methane (CO₂-ECBM) recovery is a technology that increases CBM production while controlling greenhouse gas by injecting CO₂ into CBM reservoirs, which has already been applied in several field projects [8,9,10]. At the end of 20th century, a company named Amoco first carried out pilot field of ECBM recovery in the San Juan basin of the United States [11]. The results showed that methane (CH₄) production is increased by five times as compared to conventional production methods [9,11]. Then, a series of field ECBM recovery pilot projects have been executed in North America, Poland, China, and Japan [10,11,12]. The implementation of these projects has proved the feasibility of the CO₂-ECBM recovery technology.

The CBM reservoir can be considered as a typical fractured sorbing media where the majority of gas is stored in the matrix in the adsorbed state and only free gas exists in fractures [13,14,15]. Hence, during the process of CO₂-ECBM recovery, mass transfer between the coal matrix and fractures occurs in the reservoir. Early laboratory studies have demonstrated that that coal can adsorb approximately twice as much CO₂ (in mole) by volume as CH₄ [12]. As a result, the competitive adsorption capacity will help CO₂ replace CH₄ adsorbed in the coal matrix during the process of CO₂-ECBM. A multi-scale schematic to characterize the CO₂-ECBM recovery process is shown in Figure 1. However, one of the technical challenge is that the process of adsorption/desorption of the binary gas will result in the swelling/shrinkage of the coal matrix [16,17,18,19]. Another unavoidable fact is that the pore pressure of the coal seam will change during the process of CO₂-ECBM recovery which can directly affect the effective stress [20,21,22]. These two factors will cause the pore volume and fracture aperture to change dynamically, and ultimately result in the dynamic evolution of reservoir porosity and permeability, which are the key parameters for predicting CBM production and CO₂ storage.

Some previous studies have been carried out mainly focusing on the dynamic porosity/permeability of fractured sorbing media like the CBM reservoir to determine the complex and dynamic of evolution characteristics of reservoir porosity/permeability. The following references are representative studies conducted in this context. Seidle et al. [23] explored the relationship between permeability and horizontal stress under uniaxial conditions using the matchstick geometry model. Seidle and Huitt [24] established a permeability model based on experimental measurements but did not consider the mechanical deformation process. Palmer and Mansoori [25] developed a widely used model incorporating both mechanical and adsorption/desorption processing effects based on uniaxial deformation condition. Cui and Bustin [17] considered the effect of normal stress and derived a stress-dependent permeability model. Robertson and Christiansen [26] proposed a permeability model based on the assumption that the axial stress is equal to the radial stress. Connell et al. [27] analyzed the permeability evolution of coal under tri-axial strain and stress conditions by introducing a matrix deformation correction factor. Zhang et al. [28] established a general porosity/permeability model based on the poroelasticity theory. In their model, the pore volume change induced by the change in the effective stress and gas adsorption were both considered. Liu and Rutqvist [29] considered the matrix-fracture interactions in coal and derived a permeability model for coal. In their model, the matrix was connected by a “matrix bridge”. Furthermore, the “internal swelling coefficient” concept was introduced to quantify matrix-fracture interactions. Wu et al. [30] assumed that the strain induced by mechanics is negligible with respect to the strain induced by adsorption/desorption of the coal matrix and built a dual-porosity/permeability model of the CBM reservoir, in that the pore pressure in the matrix and that in the fracture system are assumed equal.

There is no doubt that these models have significantly improved our understanding of the change of porosity and permeability for multi-porosity media like CBM reservoirs. However, several issues still remain unaddressed, including the fact that most of the previous studies do not accurately consider all the physical parameters involved in the evolution of porosity/permeability, such as the accurate independent expression of effective stress of the matrix and fracture, fracture aperture, and actual volumetric strain of matrix and fracture. In most previous models, the matrix and fracture effective stress were not separately accommodated. However, Robertson [31] noted that the permeability of fractures was approximately eight orders of magnitude higher than that of the matrix. This permeability difference results in a fracture gas pressure that is generally different from the matrix gas pressure. To accurately characterize the evolution of permeability, the difference in gas pressure in matrix and in fracture-induced effective stress difference in these two systems should be considered. In many previous studies, the researchers established the permeability model by using the method of representative element volume (REV) [1,7,21]. But due to the fact that the volume of the REV is very small, they often replace the matrix volume and fracture volume with the length of themselves when the volumetric strain of the REV is expressed. Our previous research, which considered the truly three-dimensional REV volume, has shown that this simplification will certainly bring certain errors, especially for reservoir rock with large compressibility [32]. Furthermore, most previous studies have not considered the effect of the fracture aperture when calculating the volumetric strain of multi-porosity media since the fracture aperture is very small. Moreover, previous studies have shown that the matrix plays a central role in gas storage, whereas the fracture system is the primary gas flow migration channel [21,33]. Therefore, in most previous permeability models, fracture permeability behavior was the primary focus. However, the porosity/permeability of the matrix has a critical impact on the long-term production of gas. Thus, it is important that the permeability evolution model includes both fractures and matrix. Furthermore, for some studies aimed at reservoir response characteristics in the process of CO₂-ECBM recovery, the focus is often on the permeability evolution at a certain location of the reservoir or the average permeability of the reservoir. As we know, the adsorption and desorption behaviors and pore pressure evolution behaviors in the production well (PW) and injection well (IW) are not the same. Hence, using one point or an average of reservoir permeability evolution to investigate the evolution behavior of an entire reservoir during the process of CO₂-ECBM recovery is not profound enough. It is the purpose of this research, to consider volumetric strain induced by the mechanical process and adsorption/desorption process accurately and to establish a porosity/permeability model for the CBM reservoir to reveal the reservoir permeability evolution during the process of CO₂-ECBM recovery.

In this paper, we consider the CBM reservoir as dual porosity media that consists of matrix and fracture as shown in Figure 2, which is modified from a previous study [34]. Based on the combination of mechanical deformation and adsorption/desorption-induced swelling/shrinkage, and considering the fracture aperture and effective stress difference of matrix and fracture, a dual porosity/permeability model is established through accurately expressing deformation of each medium from a three-dimensional method to reveal the reservoir permeability evolution during the process of CO₂-ECBM recovery. Then, by substituting the porosity permeability evolution model into the fluid transport equation and combining the reservoir deformation equations, a set of coupled partial differential equations (PDE) is finally formed and solved by the finite element method (FEM). Finally, we have carried out some analyses on the evolution of reservoir permeability during the process of CO₂-ECBM.

2. Modeling

This section will focus on modeling the dynamic porosity and permeability of the CBM reservoir. The models will be used in the governing equations which include the reservoir deformation equation and binary gas transport equations. Besides the above two steps, an FEM model will also be built. They are all aimed to accurately describe the reservoir permeability evolution in the process of CO₂-ECBM recovery. These derivations are based on the following assumptions:

(1)

The CBM reservoir exhibits isothermal behavior, the process of gas adsorption/desorption only occurs in the matrix and obeys the role of Langmuir isothermal behavior [35].

(2)

The CBM reservoir is saturated with mixture gas that contains only CH₄ and CO₂ (a water phase is not included in the model).

(3)

The CBM reservoir is considered as a dual-porosity media consisting of matrix and fractures. Each medium is homogeneous and isotropic.

(4)

The deformation of the CBM reservoir is infinitesimal.

(5)

The gas flow satisfies Darcy’s law in the matrix and the fracture system.

2.1. Dynamic Porosity and Permeability

In this section, we focus on modeling the porosity and permeability of the CBM reservoir by accurately considering the combined effect of the mechanical and adsorption/desorption during the process of CO₂-ECBM recovery.

2.1.1. Dynamic Porosity and Permeability of Fractures

According to the effective stress principle for multi-porous media [36,37,38], the effective stress of the matrix and the fracture system of the CBM reservoir can be written as:

$$\{\begin{array}{l}{\sigma }_{em}=\sigma -(\alpha p_m+\beta p_f)\\ {\sigma }_{ef}=\sigma -\beta p_f\end{array}$$

(1)

where subscripts m and f represent the matrix and the fracture system, respectively; p is the gas pressure; σ is the average principal stress and can be expressed as Equation (2) [39]; α is the effective stress coefficient of the matrix; and β is the effective stress coefficient of the fracture system [40], and can be expressed as Equation (3):

$$\sigma ={\sigma }_{kk}/3=({\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33})/3$$

(2)

$$\{\begin{array}{l}\alpha =1-\frac{K}{K_m}\\ \beta =1-\frac{K}{K_f}\end{array}$$

(3)

In the above equation, K_m and K_f are the bulk modulus of the coal matrix and the fracture system [41], K is the bulk modulus of the dual porosity media. The expression for K is defined as:

$$K=\frac{E}{3(1-2v)}$$

(4)

In that, v is the Poisson ratio of the dual porosity media, E is the elastic modulus of the dual porosity media.

Based on the Dalton’s law, the gas pressure (Pa) of a binary mixture of nonreactive gases in the coal matrix and the fracture system can be defined as [42,43]:

$$\{\begin{array}{l}{p}_m=p_{m1}+p_{m2}\\ p_f=p_{f1}+p_{f2}\end{array}$$

(5)

where subscript 1 represents CH₄, and subscript 2 represents CO₂.

The desorption induced shrinkage strain during gas production can be expressed as:

$$\Delta {\epsilon }_s={\epsilon }_s-{\epsilon }_{s0}$$

(6)

where [7,21,22,44]:

$$\{\begin{array}{l}{\epsilon }_s=\frac{{\displaystyle \sum {\epsilon }_{Li}{b}_i{p}_{mi}}}{1+{\displaystyle \sum b_i{p}_{mi}}}\\ {\epsilon }_{s0}=\frac{{\displaystyle \sum {\epsilon }_{Li}{b}_i{p}_{mi0}}}{1+{\displaystyle \sum b_i{p}_{mi0}}}\end{array}$$

(7)

and ε_s is the strain induced by desorption with subscripted 0 representing the initial state; ε_L is the Langmuir-type strain coefficient, b_i = 1/P_Li and given by the extended Langmuir model [43].

The length of the REV is assumed as s, then,

$$s=a+b$$

(8)

After the initial equilibrium state, and based on the accurate consideration of the effective stress in the coal matrix and the fracture system, the volumetric strain of the REV can be expressed as:

$$\Delta {\epsilon }_v=-\frac{a^3}{s^3{K}_1}\Delta {\sigma }_{em}-\frac{s^3-a^3}{s^3{K}_f}\Delta {\sigma }_{ef}+\frac{a^3}{s^3}\Delta {\epsilon }_s$$

(9)

The first item on the right of the equal sign of the above equation means the degree of contribution of the coal matrix volumetric strain induced by the change in effective stress of the coal matrix to the volumetric strain of the REV. The second item means the degree of contribution of the fracture system volumetric induced by the change in effective stress in the fracture to the volumetric strain of the REV. The third item represents the degree of contribution of matrix shrinkage caused by gas desorption.

The following equation can be written based on Equations (1) and (9):

$$\Delta {\epsilon }_v=-\frac{a^3}{s^3{K}_m}[\Delta \sigma -(\alpha \Delta p_m+\beta \Delta p_f)]-\frac{s^3-a^3}{s^3{K}_f}(\Delta \sigma -\beta \Delta p_f)+\frac{a^3}{s^3}\Delta {\epsilon }_s$$

(10)

Therefore, we can rewrite the change of effective stress in the fracture by:

$$\Delta \sigma -\beta \Delta p_f=\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)$$

(11)

The fracture deformation can be obtained from the following equation as:

$$\Delta b=-\frac{b_0}{3K_f}(\Delta \sigma -\beta \Delta p_f)$$

(12)

Then the change of fracture porosity can be got base on the definition of fracture porosity as:

$$\frac{{\varphi }_f}{{\varphi }_{f0}}=1+\frac{\Delta b}{b_0}=1-\frac{1}{3K_f}\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)$$

(13)

Based on the Cubic Law [45], we can finally write the permeability ratio for fracture system of dual porosity media like coal as:

$$\frac{k_f}{k_{f0}}={(\frac{{\varphi }_f}{{\varphi }_{f0}})}^3=[1-\frac{1}{3K_f}\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m){]}^3$$

(14)

2.1.2. Dynamic Porosity and Permeability of Matrix

Similar to Equation (11), the change of effective stress in the matrix can be written as:

$$\Delta \sigma -(\alpha \Delta p_m+\beta \Delta p_f)=\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)-\alpha \Delta p_m$$

(15)

After the initial equilibrium state, and considering the shrinkage induced by gas desorption, the volumetric strain of the matrix can be expressed as:

$$\begin{array}{l}{\epsilon }_{mv}=-\frac{\Delta \sigma -(\alpha \Delta p_m+\beta \Delta p_f)}{K_m}+\Delta {\epsilon }_s=\\ -\frac{1}{K_m}[\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)-\alpha \Delta p_m]+\Delta {\epsilon }_s\end{array}$$

(16)

When the volume of the matrix changes, the volume of pore in the matrix will change accordingly. We assume that pores in the matrix have the same strain as the matrix due to gas desorption [46,47]. Based on this assumption the volumetric strain of the pores in the matrix can be derived from:

$$\begin{array}{l}{\epsilon }_{mpv}=-\frac{\Delta \sigma -(\alpha \Delta p_m+\beta \Delta p_f)}{K_{mp}}+\Delta {\epsilon }_s=\\ -\frac{1}{K_{mp}}[\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)-\alpha \Delta p_m]+\Delta {\epsilon }_s\end{array}$$

(17)

where K_mp is the bulk modulus of the pores in the matrix and can be derived as [38]:

$$K_{mp}=\frac{K_m{\varphi }_m}{\alpha }$$

(18)

As the porosity of matrix can be written as:

$${\varphi }_m=\frac{V_{mp}}{V_m}$$

(19)

$$d{\varphi }_m=d\left(\frac{V_{mp}}{V_m}\right)=\frac{V_{mp}}{V_m}\left(\frac{d{V}_{mp}}{V_{mp}}-\frac{d{V}_m}{V_m}\right)$$

(20)

Therefore,

$$d{\varphi }_m={\varphi }_m(d{\epsilon }_{mpv}-d{\epsilon }_{mv})$$

(21)

Combining Equations (16)–(21), the change of matrix porosity can be derived as:

$$\frac{{\varphi }_m}{{\varphi }_{m0}}=1+\frac{{\varphi }_{m0}-\alpha }{K_m{\varphi }_{m0}}[\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)-\alpha \Delta p_m]$$

(22)

Therefore, the permeability ratio for the matrix of dual porosity media like coal as:

$$\frac{k_m}{k_{m0}}={(\frac{{\varphi }_m}{{\varphi }_{m0}})}^3=\{1+\frac{{\varphi }_{m0}-\alpha }{K_m{\varphi }_{m0}}[\frac{1}{\frac{a^3}{s^3{K}_m}+\frac{s^3-a^3}{s^3{K}_f}}(\frac{a^3}{s^3}\Delta {\epsilon }_s-\Delta {\epsilon }_v+\frac{1}{s^3}\frac{a^3}{K_m}\alpha \Delta p_m)-\alpha \Delta p_m]{\}}^3$$

(23)

After establishing the dynamic porosity and permeability model, the field equations should be illustrated, which may include the deformation characteristics of the CBM reservoir and the government equations of binary gas transport. In the following, we will demonstrate those necessary field equations during the process of CO₂-ECBM recovery.

2.2. Governing Equations

A group of field equations for coal deformation and gas transport are defined in this section. And these field equations are coupled through the porosity/permeability model established in the above section for the dual porosity media like the CBM reservoir.

2.2.1. Deformation Equation

The deformation of dual porosity media has been widely studied, and the Navier-type equation for the dual-porosity model can be expressed as [48,49,50]:

$$G{u}_{i,jj}+\frac{G}{1-2v}{u}_{k,kj}+f_i=\alpha p_{m,i}+\beta p_{f,i}+K{\epsilon }_{s,i}$$

(24)

where G is the shear stiffness.

2.2.2. Binary Gas Transport

The mass balance equation of the each component of gas (CH₄/CO₂) can be expressed for a static medium, that incorporates the convective and dispersion modes of transport but involves the interchange between adsorbed gas and free gas as [10,21,42,43,51,52]:

$$\frac{\partial m_i}{\partial t}+\nabla ·(v_g·{\rho }_i)+\nabla (-D_i\nabla m_{iF})=Q_{si}$$

(25)

where the gas content of a component gas i is m_i which includes both the free-phase and adsorbed gas; v_g is the vector of convective velocity; ρ_i is the gas density in the matrix or the fracture; D_i is the diffusion coefficient; m_iF represents the gas content of the free state in the matrix or the fracture; Q_si is the gas source or sink.

The mass of each component of gas (CH₄/CO₂) present in a unit of the matrix and fracture system can be expressed as follows [43,52].

For the matrix,

$$m_{mi}=\underset{free-gas}{\underbrace{{\varphi }_m{\rho }_{mi}}}+\underset{adsorbed-gas}{\underbrace{{\rho }_c{\rho }_{ai}\frac{V_{Li}{b}_{imi}}{1+{\displaystyle \sum b_i{p}_{mi}}}}}={\varphi }_m{p}_{mi}\frac{M_i}{RT}+{\rho }_c{p}_{ai}\frac{M_i}{RT}\frac{V_{Li}{b}_{imi}}{1+{\displaystyle \sum b_i{p}_{mi}}}$$

(26)

where ρ_ai is the density of gas at standard conditions, and ρ_c is the coal density.

For the fracture,

$$m_{fi}={\varphi }_f{\rho }_{fi}=\frac{{\varphi }_f{p}_{fi}{M}_i}{RT}$$

(27)

The vector of convective velocity v_g can be expressed as follows (the effect of gas gravity is neglected) [9,42,43].

For the matrix,

$$v_{gm}=-\frac{k_m}{{\mu }_m}\nabla p_m=-\frac{k_m}{{\mu }_m}\nabla (p_{m1}+p_{m2})$$

(28)

For the fracture,

$$v_{gf}=-\frac{k_f}{{\mu }_f}\nabla p_f=-\frac{k_f}{{\mu }_f}\nabla (p_{f1}+p_{f2})$$

(29)

In the above two equations, μ_m and μ_f are the gas viscosity of a nonpolar binary gas mixture in the matrix and fracture, respectively. They can be written as [43]:

$${\mu }_m=\frac{{\mu }_1}{1+\frac{p_{m2}}{p_{m1}}\sqrt{\frac{M_2}{M_1}}}+\frac{{\mu }_2}{1+\frac{p_{m1}}{p_{m2}}\sqrt{\frac{M_1}{M_2}}}$$

(30)

$${\mu }_f=\frac{{\mu }_1}{1+\frac{p_{f2}}{p_{f1}}\sqrt{\frac{M_2}{M_1}}}+\frac{{\mu }_2}{1+\frac{p_{f1}}{p_{f2}}\sqrt{\frac{M_1}{M_2}}}$$

(31)

Combining Equations (26)–(29), binary gas transport in the matrix can be expressed as Equations (32) and (33).

For CH₄,

$$\begin{array}{l}[{\varphi }_m+{\rho }_c{p}_{a1}\frac{V_{L1}{b}_1(1+b_2{p}_{m2})}{(1+{\displaystyle \sum b_i{p}_{mi}{)}^2}}]\frac{\partial p_{m1}}{\partial t}-{\rho }_c{p}_{a1}\frac{V_{L1}{b}_1{b}_2{p}_{m1}}{(1+{\displaystyle \sum b_i{p}_{mi}{)}^2}}\frac{\partial p_{m2}}{\partial t}+p_{m1}\frac{\partial {\varphi }_m}{\partial t}+\\ \nabla (-\frac{k_m}{{\mu }_m}{p}_{m1}\nabla p_m)+\nabla (-D_1{\varphi }_m\nabla p_{m1})=-w(p_{m1}-p_{f1})\end{array}$$

(32)

where $w=8[1+(2/a^2)]k_m/{\mu }_m$ is the transfer coefficient between the matrix and the fracture.

And for CO₂,

$$\begin{array}{l}[{\varphi }_m+{\rho }_c{p}_{a2}\frac{V_{L2}{b}_2(1+b_1{p}_{m1})}{(1+{\displaystyle \sum b_i{p}_{mi}{)}^2}}]\frac{\partial p_{m2}}{\partial t}-{\rho }_c{p}_{a2}\frac{V_{L2}{b}_1{b}_2{p}_{m2}}{(1+{\displaystyle \sum b_i{p}_{mi}{)}^2}}\frac{\partial p_{m1}}{\partial t}+p_{m2}\frac{\partial {\varphi }_m}{\partial t}+\\ \nabla (-\frac{k_m}{{\mu }_m}{p}_{m2}\nabla p_m)+\nabla (-D_2{\varphi }_m\nabla p_{m2})=-w(p_{m2}-p_{f2})\end{array}$$

(33)

Binary gas transport in fracture system can be expressed as Equations (34) and (35).

For CH₄,

$$p_{f1}\frac{\partial {\varphi }_f}{\partial t}+{\varphi }_f\frac{\partial p_{f1}}{\partial t}+\nabla ·(-\frac{k_f}{{\mu }_f}{p}_{f1}\nabla p_f)+\nabla (-D_1{\varphi }_f\nabla p_{f1})=w(p_{m1}-p_{f1})$$

(34)

and for CO₂,

$$p_{f2}\frac{\partial {\varphi }_f}{\partial t}+{\varphi }_f\frac{\partial p_{f2}}{\partial t}+\nabla ·(-\frac{k_f}{{\mu }_f}{p}_{f2}\nabla p_f)+\nabla (-D_2{\varphi }_f\nabla p_{f2})=w(p_{m2}-p_{f2})$$

(35)

After establishing the constitutive relationships for the binary gas transport and the deformation of the CBM reservoir, we will solve these field equations including the dynamic porosity and permeability using the FEM.

2.3. Model Implementation

The above mathematical model will be solved using the PDE module of COMSOL Multiphysics (a commercial FEM software). Next, we will describe the FEM model, which includes the solution domain, initial conditions, boundary conditions, and the main parameters in this simulation.

A production area of 400 m × 400 m defined to represent the CBM reservoir is shown in the left of Figure 3. The wells have a diameter of 0.1 m. It is a regular five-spot pattern with the IW in the center and the PW in the four corners of the square reservoir. Due to the symmetry of the configuration, only one-quarter (top right corner) of the production area is simulated as shown in the right of Figure 3. For the geomechanical boundary conditions of the simulation area, all four sides are confined in the normal direction (in COMSOL Multiphysics, the symmetric boundary condition is equivalent to a roller condition, and they both mean that the displacement is zero in the direction perpendicular (normal) to the boundary, but the boundary is free to move in the tangential direction) while the PW and IW are unconfined. For the binary gas flow, no flow conditions are applied to the boundaries except for the IW and PW. The initial gas pressure in the reservoir is 2.5 MPa. The initial pressures of CH₄ and CO₂ are set according to their composition. CH₄ has an initial pressure of 2.3 MPa, and based on partial pressures, the initial pressure of CO₂ is 0.2 MPa. A constant pressure of 0.1 MPa is applied to the PW and a constant pressure of 4MPa is applied to the IW. The fracture spacing and aperture are assumed as a = 0.01 m and b = 0.0002 m, respectively. The simulation time is 10,000 days. Meshing is as shown on the right in Figure 3. The remaining main parameters are listed in

Table 1. Main input parameters for the simulation model.

Symbol	Value	Mean	Unit
εL1	0.0128	Langmuir volume strain of CH4	-
εL2	0.0237	Langmuir volume strain of CO2	-
VL1	0.0256	Langmuir volume of CH4	m3/kg
VL2	0.0477	Langmuir volume of CO2	m3/kg
PL1	2.07 × 106	Langmuir pressure of CH4	Pa
PL2	1.38 × 106	Langmuir pressure of CO2	Pa
μ1	1.15 × 10−5	Gas viscosity of CH4	Pa·s
μ2	1.60 × 10−5	Gas viscosity of CO2	Pa·s
D1	3.6 × 10−12	Diffusion coefficient of CH4	m2/s
D2	5.8 × 10−12	Diffusion coefficient of CO2	m2/s
φm0	0.04	Intrinsic porosity of matrix	-
km0	1.0 × 10−17	Intrinsic permeability of matrix	m2
φf0	0.003	Intrinsic porosity of fracture	-
kf0	1.0 × 10−15	Intrinsic permeability of fracture	m2
E	4 × 109	Young’s modulus of coal	Pa
Km	12 × 109	Bulk modulus of matrix	Pa
Kf	1.5 × 108	Bulk modulus of fracture	Pa
v	0.32	Poisson’s ratio	-

, which are mainly extracted from the literatures [51,53,54].

3. Results and Analysis

In this section, we will analyze the simulation results to reveal the evolution of reservoir permeability from different angles. We first analyze the permeability evolution of simulation area at different times. Then as shown in Figure 3, we select one group of locations that include two points on the diagonal line named as Point A (50, 50) and Point B (150, 150) to analyze how permeability evolves at these two points during the whole simulation period. After these analyses, another group of locations are selected closer to the wellheads and denoted as Point A′ (20, 20) and Point B′ (180, 180), respectively. Next we made a comparative analysis between those two groups. Finally, we adjust the IW pressure and analyze the influence of IW pressure on the evolution of reservoir permeability.

3.1. Permeability Evolution in the Whole Simulation Area

Here we analyze the matrix and fracture permeability separately from the reservoir scale. Matrix permeability and fracture permeability evolutions at the different times of 10th, 100th, 1000th, 2000th, 5000th, and 10,000th day are shown in Figure 4 and Figure 5, respectively. It can be seen that permeability evolution during the process of CO₂-ECBM recovery is rather complicated.

3.1.1. Matrix Permeability Evolution

On the 10th day, the decrease in effective stress due to the increase in pore pressure caused by the injected gas will expand the pores and cause the increase of permeability near the IW. Conversely, the increase in effective stress due to the decrease in pore pressure caused by the produced gas will compress the pores and cause the decreased permeability near the PW.

On the 100th day, compared with the 10th day, the permeability in the vicinity of the IW decreased significantly, while the permeability in the vicinity of the PW has not significant changed. The reason for this phenomenon is because of gas adsorption/desorption in the matrix. To be specific, the coal matrix near the IW has begun to adsorb the injected CO₂, which directly results in the expansion of the matrix and the decrease in pore volume and fracture aperture, eventually resulting in a decrease in permeability. Conversely, the coal matrix near the gas PW has begun to desorb CH₄, which directly results in the shrinkage of the matrix and the increase of the pore volume and fracture aperture, eventually resulting in an increase in permeability. And, the increase in permeability induced by desorption offsets the decrease in permeability caused by the increase in effective stress, and ultimately results in no significant change in permeability near the gas PW.

When the time passes to the 1000th day, the range of decrease in permeability due to adsorption has greatly expanded to occupy the lower left corner of the entire model. While, for the upper right part of the simulation area, the increase in permeability caused by desorption is barely maintained and the permeability does not drop rapidly in this area. On the 2000th day, the permeability of the matrix in the upper right part of the simulation area is further reduced.

Approaching the 5000th day, it can be seen that the permeability in the upper right part of the simulation area is already lower than the lower left corner. This phenomenon will become more and more obvious with the increase of time.

3.1.2. Fracture Permeability Evolution

The fracture permeability evolution is similar to that of the matrix. But the biggest difference between them is that the evolution of fracture permeability is more sensitive to the mechanical properties of the fracture.

On the 10th day, the decrease in effective stress due to the increase of gas pressure caused by the injected gas will expand the fracture aperture and eventually cause an increase in permeability near the IW. Conversely, the increase of effective stress due to the decrease of gas pressure caused by the produced gas will compress the fracture aperture and eventually cause a decrease in permeability near the PW.

As we have discussed before, as time goes by, the role of gas adsorption and desorption gradually becomes prominent and exceeds the effect of effective stress changes. Specifically, for gas adsorption near the IW, the decrease in fracture aperture and permeability exceed the increase in permeability caused by the reduction of effective stress; the final permeability shows a decreasing trend. However, the increase in fracture aperture and permeability induced by gas desorption near the gas PW cannot make up for the reduction in permeability caused by the increase of effective stress, and the final permeability also shows a trend of decreasing. This phenomenon becomes more and more profound on the 100th, 1000th, and 2000th days as shown in Figure 5. On the 5000th day, it can be seen that the permeability in the upper right part of the simulation area is already lower than that in the lower left corner.

3.2. Permeability Evolutions at Different Locations

In this section, we focus on the evolutions of matrix permeability and fracture permeability at the two representative points named Point A and Point B. First of all, we understand the key parameters that affect the porosity/permeability evolutions during the process of CO₂-ECBM recovery. As described in the previous section, the key factors that control the evolutions of porosity and permeability can be illustrated by Figure 6. Because the volumetric strain induced by the change of effective stress is difficult to show for coupled processes, next we will mainly analyze the evolution of pore pressure in the matrix, the pore pressure in fracture, the shrinkage/swelling induced by desorption/adsorption of gas in matrix, and the gas partial pressure in matrix.

The matrix permeability and fracture permeability evolutions as a function of time at the two different points in the simulation area are shown in Figure 7 and Figure 8, respectively. It can be seen intuitively that the trends of permeability changes in the matrix system are similar to that in the fracture system but their magnitudes are significantly different. The overall trend is a decline in permeability over time. Three stages (I, II, and III) can be divided based on the numerical values between Point A and Point B for both the matrix and the fracture system.

3.2.1. The First Stage (I)

At the first stage (I), the permeability at point A is larger than that of Point B: Point A is near the IW, while Point B is near the PW. As shown in Figure 9 and Figure 10, pore pressure will initially increase at Point A and decrease at Point B, which directly causes a decrease in effective stress at Point A and an increase in effective stress at Point B and in turn increases the pore volume and fracture aperture at Point A, while both are decreased at Point B. Therefore, in the first stage, the changes of pore pressure boost the permeability at point A and inhibit permeability at Point B.

However, this trend does not remain for a long time. For point A, the increased pore pressure breaks the previous equilibrium state of adsorption/desorption and the coal matrix begins to adsorb gas, as shown in Figure 11, which causes the swelling of the solid component of matrix. This process will decrease the pore size of the matrix and reduce the fracture aperture, which then can reduce the permeability. Please note at the same time, the increase in pore pressure can also increase permeability based on the effective stress principle. Therefore, a competitive relationship occurs at point A in the first stage. When the effect of the adsorption process exceeds that of the change in effective stress, the permeability of Point A decreases instead of increasing.

For Point B, the permeability behavior is opposite to that of Point A, specifically, the decreased pore pressure in the matrix promotes gas desorption. This process causes matrix shrinkage and increases the pore volumes and fracture aperture, which directly leads to permeability enhancement. However, the effect of the desorption process does not exceed the effect of the change in effective stress which can be seen from Figure 7 and Figure 8; there is no obvious rising stage of permeability at Point B, which can be explained from Figure 11 and Figure 12. In the first stage (I), the desorbed of gas at Point B does not cause a large shrinkage of the matrix because the main desorbed gas is CH₄ which has a small Langmuir volume strain constant compared with CO₂. These processes eventually result in a gradual crossover of the permeability ratio at point A and the permeability ratio at Point B, and then start the second stage (II).

3.2.2. The Second Stage (II)

At the second stage (II), the permeability at Point A is lower than permeability at Point B for both the matrix and fracture. But as shown from Figure 7 and Figure 8, the permeability difference between Point A and Point B is firstly increased and then decreased in the matrix and fracture in this stage. For the early period of this stage, the permeability evolution at Point A and Point B are the continuations of the end of the first stage. As Figure 12 shows, with the time goes, the injected gas transports to Point B, the partial pressure of CO₂ increases dramatically while the partial pressure of CH₄ decreases gently, which results in that the gas pressure tends to increase at Point B, as shown in Figure 9 and Figure 10. If the process of gas adsorption/desorption is not included in the analysis, the increased pore pressure would enhance permeability at Point B. However, as we can see from Figure 11, the swelling induced by adsorption at Point B gradually catches up with that of Point A. Furthermore, the total gas pressure at Point B is smaller than that at Point A. These two processes lead to a higher decline rate in permeability at Point B than that at Point A, which finally results in a gradual crossover of the permeability ratio line at Point A and the permeability ratio line at Point B, and then starts the third stage (III).

3.2.3. The Third Stage (III)

At the third stage (III), the general trend is that the permeability at Point A is slightly higher than that at Point B. This situation can be seen as a continuation of the second stage (II). To analyze its causes, we still need to start from the perspective of effective stress and adsorption/desorption of gas. On the one hand, the swelling induced by gas adsorption for Point A and Point B at this stage is very similar, as shown in Figure 11. One the other hand, the pore pressure in fracture and matrix at this stage at these two points always has a differential as shown in Figure 9 and Figure 10. This differential makes sure that the effective stress at Point A is lower than that of Point B. Hence, if we only consider the permeability change induced by the changes in effective stress, the permeability at Point A should be larger than that at Point B. If we combine the effects of effective stress and the effects of gas adsorption/desorption, the final result will not change significantly since the effects of gas adsorption/desorption at these two points are very similar. Therefore, the finally result is that the permeability at Point A is slightly higher than that at Point B.

3.3. Some Other Influencing Factors

3.3.1. Permeability Evolution Rules at Other Locations

Through comparing permeability evolution at Point A (50, 50) and Point B (150, 150), we find that the evolution of permeability in different locations in a reservoir is not always the same and is a function of the distance from the wells. In this section, a group of locations are selected closer to the wellheads and denoted as Point A′ (20, 20) and Point B′ (180, 180), respectively. We then made a comparative analysis between those four locations. The permeability evolutions at those points of the matrix and fracture are shown in Figure 13 and Figure 14, respectively. It is obvious that the matrix and fracture permeability evolution are similar but with different magnitudes.

Numerical results show that the farther the distance between two points in a group, the greater the gap in their permeabilities. This is because the farther the two points are located, the closer they are to the wells and the more dramatic the evolution of permeability. For the group closer to the wells, the permeability evolution starts and completes the stages of the entire process earlier.

3.3.2. Permeability Evolution under Different Injection Pressures

As discussed before, the key factors that affect the permeability evolution are the changes in effective stress and the strain induced by gas adsorption or desorption. To change a factor that can influence not only the effective stress but also the gas adsorption/desorption, the most suitable parameter is pore pressure in the reservoir. As the flowing bottom hole pressure of the PW is already very small, we adjusted the injection pressure of CO₂ to 6 MPa for the IW and compared it with the situations of 4 MPa injection pressure. The matrix permeability and fracture permeability evolutions at Point A and Point B under those two injection pressures are shown in Figure 15 and Figure 16, respectively.

Permeability evolution of matrix and fracture are also similar under the higher injection pressure condition. Higher injection pressure causes the permeability to evolve more dynamically. If we divide the whole process into three stages as before, we find that the group with higher injection pressure starts and completes those stages of the entire process earlier than the group with lower injection pressure.

3.4. Schematic to Explain the Mechanisms

A schematic to summarize our previous analyses for the complex evolution of permeability during the process of CO₂-ECBM recovery at different distances from the wells is shown in Figure 17. In the schematic, the innermost circle represents the matrix pore. The size of it represents the porosity/permeability of the matrix as the relationship between porosity and permeability described by the cubic law in this study. The thickness of the ring between the outermost circle and the second circle represents the fracture aperture and permeability. The size of the outermost circle is roughly the result of the combination of the strain caused by the change in the effective stress and the swelling/shrinkage caused by the adsorption/desorption. Finally, in the four locations shown in the figure, the left two points only represent their relative distance from the IW, and the right two points represent only their relative distance from the PW.

4. Conclusions

This study has built a dual porosity/permeability model through accurately expressing the volumetric strain of the matrix and fracture from a three-dimensional method which aims to reveal the reservoir permeability evolution during the process of CO₂-ECBM recovery. This model has accommodated the key competing processes of mechanical deformation and adsorption/desorption-induced swelling/shrinkage of the CBM reservoir and it also considered the effect of fracture aperture and effective stress difference between fracture and matrix. We then numerically solved the permeability model using a group of multi-field coupling equations with FEM to understand how permeability evolves temporally and spatially in a CBM reservoir. From our multifaceted analyses of the results, the following conclusions can be drawn.

(1)

The evolution of reservoir permeability near the IW and the PW are very different. The reason for this is because the combined effect of effective stress changes and gas adsorption and desorption. Therefore, when analyzing the evolution of reservoir permeability during the process of CO₂-ECBM recovery, it is not enough to only consider the evolution of the reservoir average permeability or the evolution of the permeability at a certain location in the reservoir.

(2)

Since the Langmuir volumetric strain constant of CO₂ is greater than that of CH₄, the swelling due to the adsorption of CO₂ is greater than the shrinkage due to desorption of CH₄. As a result, adsorption is the main factor for the change of permeability for regions near the IW, while the change in effective stress is the main cause for the change in permeability for regions the PW. Therefore, the overall trend of the evolution of permeability of the entire reservoir shows a downward trend.

(3)

Permeability evolution near the wells is the most dramatic. The farther away from the well, the gentler the evolution of permeability during the process of CO₂-ECBM recovery. When we increase the injection pressure of CO₂, the reservoir permeability evolution becomes quicker and more dynamic.