Constraints of Pore-Bulk Strain Ratio and Interference Time on the Evolution of Coal Permeability during CO2 Injection

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China Laboratory of Mine Cooling and Coal-heat Integrated Exploitation, China University of Mining and Technology, Xuzhou 221116, China Beijing Key Laboratory for Precise Mining of Intergrown Energy and Resources, China University of Mining and Technology, Beijing 100083, China School of Engineering, The University of Western Australia, WA 6009, Australia


Introduction
CO 2 is the main contributor to the greenhouse effect. CO 2 sequestration in deep nonexploitable coal seams can simultaneously reduce CO 2 emissions and enhance coalbed methane recovery (CO 2 -ECBM), which has great potential for development [1][2][3][4]. The study of coal permeability evolution is the main problem to enhance gas recovery of reservoir [5][6][7]. When the gas pressure drops to the desorption point, the adsorbed gas enters into pores, the matrix shrinks, and the pores' volume increases [8][9][10]. In the process of gas injection, solid-gas coupling leads to significant differences in permeability, and its evolution process is highly complex.
In the past few decades, several coupling models have been proposed to interpret the permeability change during CO 2 injection. Based on the Langmuir model, a finite element model was proposed to investigate the coupling of sorption-induced deformation and coal seam gas flow process [11]. A multiphase flow model for the gas injection process considering the coupling of matrix deformation and heat conduction process was proposed [12]. Then, the single-porosity elastic model to study the evolution process of porosity and permeability of CO 2 gas injection under in situ stress, taking into account the deformation and flow process of micropore and fracture, was revised [13,14]. A dual porosity model considering the statistical properties of geological parameters, which can introduce relatively large errors, was proposed [15]. The effects of directional compaction, single fracture, and anisotropic expansion on coalbed methane reservoirs through a dual porosity model were discussed [16]. Based on the multiple porosity model, the relationship between matrix strain and macro permeability of coal by introducing the strain splitting function was studied [17]. Two nonequilibrium models with two different nonlinear adsorption isotherms, i.e. nonlinear Freundlich and Langmuir adsorption isotherms, were developed [18]. The change of fracture porosity and permeability caused by coal matrix shrinkage into the dual pore model to study the quasisteady flow in unconventional reservoirs was introduced [19,20]. Although these models have been used to interpret and match some experimental data, the interaction between coal matrix and fracture induced by the adsorption-induced strain has not been understood well.
Previous model conclusions [15,21] are inconsistent with experimental data of huge changes in coal permeability caused by coal swelling. The experimental data are sensitive to the change of stress and pore pressure, especially for low permeability coal seams. Using transient pulse attenuation technology, it is found that the compressibility of face cleats is considerably sensitive to the sorption-induced swelling/shrinkage and offers significant effects on the coal permeability [22]. The adsorption and desorption energy of CO2 gas significantly affects the volume change of microscopic and macroscopic pores in coal, resulting in the expansion/contraction and permeability fluctuation of coal [23].Previous models usually only studied the ultimate equilibrium permeability, but neglected its evolution process. Studies [14,22,23] show that for low permeability coal seams, it often takes a long time for permeability to evolve to the final equilibrium state during gas injection. To solve this problem, a series of research was conducted [24][25][26] to study the change process of the intermediate nonequilibrium state in the process of coalbed methane exploitation. In recent years, a new functional relationship between coal permeability and fracture aperture was defined [27], and the evolution process of coal permeability from the initial equilibrium state to final equilibrium state was analyzed. However, the study on the division of fracture and matrix-induced strain by adsorption usually adopts the method of single or discrete fracture explicitly embedded into the models.
In this study, the explicit approach as adopted in previous studies is transformed into a continuous approach. The explicit method uses geometric methods to characterize the fracture pore structure, such as digital core method and fracture discrete network method. However, due to its high computational requirements, the analysis of large-scale multifield coupling problems is still limited [5,6]. In the process of gas adsorption, coal will undergo uneven deformation, which will lead to the change of throat and channel length in the internal pores; so, the pore-bulk strain ratio is introduced to characterize this change process. Under the coupling effect of stress, the change of pore pressure in coal will lead to the change of permeability. The pore volume strain ratio and interference time proposed in this work can quantitatively characterize the gas adsorption process and the accompanying nonuniform evolution process from the initial state to the final state. Thus, a new multifield coupling model of nonequilibrium permeability including fracture pore dual scale is established. Compared with the previous models, this new coupling model captures the influence of local transient behavior of coal on gas flow from the perspective of interference time. The results of this study are reported in the following sections.

Governing Equations
Coal seam is a typical dual porosity medium under the action of in situ stress [28][29][30][31]. The deformation caused by the adsorption process is one of the main inducements of coal seam permeability. Firstly, considering the pore adsorption strain ratio, a dual porosity model is established and coupled to the solid deformation equation under the influence of effective stress. The following assumptions are made for coal deformation and gas seepage process: (1) The deformation process of coal meets linear elasticity (2) The effect of anisotropy is not considered (3) Gas viscosity coefficient is a constant (4) Gas in the pores is saturated 2.1. Permeability Model. The ratio of pore strain and bulk strain induced by adsorption can be expressed as [32]: where ε sp is the pore strain induced by sorption, ε sb is the bulk strain of coal induced by sorption, and ψ is the function of ε sb . According to Taylor expansion, Eq.(1) can be expressed as where γ = ∂ψ/∂ε sb j ε sb =0 . Experimental results [15] show that the sorption-induced strain of pore is larger than that of bulk, and the two strains can be regarded as equal only when they are in the equilibrium state. At the beginning of gas injection, the pressure in fracture is greater than that in matrix. So, gas permeates into the matrix, and the amount of adsorbed gas in the matrix increases. Then, the volume expands, and the fracture volume decreases as shown in Figure 1. We can draw the conclusion that the sign of the fracture strain is opposite to that of the matrix strain. In addition, coal porosity is usually very small (0:001 < ϕ < 0:03); so, the sorption-induced strain of matrix is approximately equal to that of bulk. Then, the sigh of the sorption-induced fracture strain is opposite to that of bulk, i.e., at the initial state γ 0 < 0. The initial value of γ can be expressed as [33]: where C f is the coal compressibility, E is the Young's modulus of coal, and ν is the Poisson ratio of coal. The characteristic interference time τ 1 is determined by the onset of pressure interference between two hydro fractures spaced 2d (as shown in Figure 2) apart and defined by [34]: λ i is the hydraulic diffusivity of gas under the initial reservoir pressure and temperature (p i , T), which can be measured by experiment. Pore structure, temperature, pressure, and gas composition have an important influence on the hydraulic diffusion coefficient, but here we ignore the change of temperature and pressure, assuming that the gas is an ideal gas, and considering the homogeneous reservoir; so, the hydraulic diffusion coefficient λ i is considered to be a constant.
When coal reaches the final equilibrium state, the strain ratio γ = 1, which means pore strain induced by sorption, is equal to the bulk strain. In combination with equation (3), we assume that the adsorption strain ratio changes linearly from the initial state to the final equilibrium state. As shown in Figure 3, γ can be expressed as The change rate of pore volume can be expressed as [11,35]: where V = V p + V s , V is total volume of porous coal, V p is the pore volume of porous coal, and V s is solid volume of porous coal. σ is the mean compressive stress, α is the Biot coefficient, p is the gas pressure. ε s is the gas sorption-induced strain, K is the bulk modulus of coal, and K p is the bulk modulus of pores.

Geofluids
where ϕ is the porosity of coal. Combined Eq.(6)-(10), we obtain Substitute where K s is the bulk modulus of coal grains. Then, is the initial gas pressure, ε s0 is initial gas sorptioninduced strain, p L is the Langmuir pressure coefficient, and ε L is the Langmuir volumetric strain constant; so, we obtain Find the partial derivatives of the equations on both sides of Eq. (11): The governing equations of gas flow can be expressed as where V L is the Langmuir volume coefficient, k is the permeability, μ is the gas viscosity, and Q s is source term of gas flow, Q s = ωðp f − p m Þ. Experiments [11,12] show that the relationship between porosity and permeability of coal can be expressed as Combined Eq. (16) with Eq. (17), we can obtain the permeability of the pore system and fracture system: where subscript m denotes matrix and f denotes fractures.

Governing Equation of
Coal Deformation. The structure of the dual porosity model is shown in Figure 4. The equilibrium equation of matrix and fracture can be expressed as where σ ij is the component of stress tensor, and f i is the component of body force. The relationship between coal deformation and stress induced by adsorption can be expressed as [36] where G is the elastic shear modulus, σ kk = σ 11 + σ 22 + σ 33 , and δ ij is the Kronecker delta. According to the Langmuir equation, the sorption-induced strain can be expressed as Combined Eq. (19) with Eq. (20), the governing equation of coal deformation can be expressed as [37,38] 2.3. Governing Equation of Gas Flow. According to the state of occurrence, gas in the coal reservoir can be divided into two parts: the adsorption phase and the free phase. The total mass can be described by the following equations: 4 Geofluids where m m and m f , respectively, denote the mass of gas in matrix and fracture. ρ gm and ρ gf , respectively, denote the gas density in matrix and fracture. ρ ga denotes the gas density in standard condition. ρ c denotes coal density. Based on the conservation law of mass, the governing equation of gas mass can be expressed as where t is the time. ρ g is the gas density, and q g is Darcy's flow velocity. Neglecting the effect of gravity, Darcy's flow can be expressed as where _ q g represents the derivation of q g . Substituting Eqs. (23), (24), and (26) into Eq.(25), the mass conservation equation of the gas can be expressed as [11,39,40] where p a is the atmospheric pressure, p a = 101:325kPa.

Coupled Governing Equation.
The dual porosity model proposed in this paper considers the pore strain induced by adsorption as a linear relationship with the bulk strain. The coal deformation governing equation is rewritten as There is no deformation in the coal at initial pressure, and the porosity evolution equation in the gas injection process is where ε v = ε 11 + ε 22 + ε 33 is the volume strain of coal. Substituting Eq. (8) for Eq. (5), we obtain the coal bed gas flow equation of the pore system and fracture system: Distance between two hydro fractures spaced, 2d m ð Þ 8 Cleat compressibility, C f MPa −1 À Á 3:2 × 10 −7 P outlet = p a (∂Ω 1 )

Model Verifications against Experimental Data
The process of gas seepage in dual porosity media is highly nonlinear both in time and space. The governing equations of gas seepage in dual porosity media are partial differential equations, which are difficult to obtain analytical solutions. In this paper, the multiphysical coupling software COM-SOL is used for numerical solution. In order to verify the correctness of the model, we compare the simulation results with the experimental data [41].The material property parameters used in the calculation model are shown in Table 1.
The simulation model is a square domain with the size of 0:076 m × 0:038 m. CO 2 is injected at the bottom of the model, and the outlet is the upper boundary. The gas injection pressure varies from 6 MPa to 9 MPa. The bottom boundary is hinge constraint, and the upper, left, and right boundaries are uniformly distributed, as shown in Figure 5. The numerical results are compared with two sets of laboratory data (confining pressure 24 MPa and 20 MPa, respectively), as shown in Figures 6 and 7. The two experimental cases are simulated as follows: the permeability evolution process of each experiment from the initial equilibrium state to the final equilibrium state is calculated (Figures 8 and 9). The red lines in Figures 6 and   1E Figure 10: Permeability evolution process of fracture system of model I.  Figure 11: Permeability evolution process of pore system of model I.
8 Geofluids 8 represent the upper limit of permeability simulation results under different injection pressures, and the green lines represent the lower limit of permeability simulation results. It is found that the experimental results may not reach the equilibrium state of permeability evolution. According to Figures 6 and 7, the coupling model proposed in this paper is in good agreement with the experimental data. According to Figures 8 and 9, the evolution of permeability in CO 2 gas injection can be divided into four stages: (1) when CO 2 is injected into coal, the pore space becomes larger, and the permeability increases with the increase of fracture pressure. (2) The gas pressure in fracture is larger than that in matrix. The gas begins to diffuse from fracture to matrix. The volume of the matrix expands, the volume of the fracture shrinks, and the permeability of coal decreases. (3) With the gradual diffusion of gas into the matrix, only local expansion deformation has an impact on the permeability of coal, and the permeability increases gradually, and (4) the pressure of the matrix and pore is in equilibrium with each other, the coal mass is uniformly expanded, and the coupling process is stable.
From Figures 6 and 7, it can be seen that coal permeability is affected by the injection pressure. Figures 8 and  9 correspond to the evolution of permeability with time under different injection pressures of Figures 6 and 7, respectively. Each point of the measured data in Figures 6  and 7 corresponds to the specific location of the corresponding curve in Figures 8 and 9. Because the curve is not monotonous, the measured data mostly correspond to the two points of the curve, as shown in Figure 8. When the gas injection reaches the final stable state, the permeability value is the upper limit of the permeability curve. Figures 6 and 7 show that the measured permeability values lie between the upper and lower limits, indicating that the measured permeability values may not reach the final equilibrium state, which may due to the low porosity of coal samples. The permeability evolution process of the fracture and pore system of model I is not a monotone line, which can be seen in Figures 10 and 11.

Permeability Tests under the Condition of Constant
Confining Stress. The boundary conditions of the numerical model are shown in Figure 5. The gas parameters of dual porosity coal are shown in the following table, which comes from previous experiments (Robertson E.P. and Christiansen R.L., 2005). The size of the calculation model is 0:1 m × 0:05 m. The lower boundary is fixed hinge constraint, and the upper, left, and right boundaries are subjected to uniform pressure 7 MPa. There is no mass transfer on the left boundary and the right boundary. CO 2 is injected from the bottom of the model and outflows from the upper boundary. The inlet pressure is 4.8 MPa, and outlet pressure is atmospheric pressure. The parameters of coal and gas for simulation are shown in Table 2.
The permeability evolution process of the coal fracture system and pore system is shown in Figures 5 and 6. Accord-ing to Figures 5 and 6, during the process of gas injection, the permeability change rule of the pore system and fracture system is similar, which includes four stages: increasing, decreasing, then gradually restoring, and finally balancing. The permeability of the pore system is changing more obviously. The reason is that the porosity of pore system is small, and the small change of pore volume will have a more obvious impact on its permeability.
In this paper, a dual porosity model considering the adsorption strain ratio is proposed. The effects of three main factors on permeability evolution during gas injection are analyzed, including (1) Adsorption strain ratio (2) Interference time

(3) Gas injection pressure
The simulation parameters of different cases are shown in Table 3. The evolution of permeability with different adsorptive strain ratios during gas injection is shown in Figures 12-14. According to Figures 12-14 below, although the adsorption strain ratio does not affect the permeability after the final equilibrium state of gas injection, the lower limit of permeability decreases obviously when the absolute value of the adsorption strain ratio increases.
The comparison of the evolution process of permeability with the interference time is shown in Figure 13. According to Figure 13, we obtain that the difference of permeability is mainly the lower limit of the permeability evolution process Distance between two hydro fractures spaced, 2d m ð Þ 7 Cleat compressibility, C f MPa −1 À Á 1:04 × 10 -7  When the confining pressure is maintained at 7 MPa, the evolution of coal permeability with gas injection pressure is shown in Figure 14. It can be seen from Figure 14  that with the increase of the injection pressure, the lower limit of permeability value gradually decreases, and the upper limit gradually increases in the process of coal permeability evolution. The upper limit is at the final equilibrium state of the curve.

Permeability Tests under the Condition of Three Roller
Constrains. The boundary condition of model II is shown in Figure 15. The property parameters are shown in Table 4. Evolution of the coal fracture and pore permeability under the condition of three roller constrains is shown in Figures 16 and 17. According to Figures 16 and 17, it can be seen that the fracture system and pore system of coal of model II in the process of gas injection also include four stages: first rising, decreasing, restoring, and finally balancing. The change of permeability of pore system is more obvious.   Cleat compressibility, C f MPa −1 À Á 7:5 × 10 -8

Geofluids
The simulation parameters of different cases are shown in Table 5.
The influence of coal adsorption strain ratio γ on the permeability evolution process in the CO 2 injection process of model II is shown in Figure 18. According to Figure 18, the permeability after final equilibrium is the same under different adsorption strain ratios. The lower limit of permeability is proportional to γ.
The evolution process of the permeability of model II varies with the interference time is shown in Figure 19. It can be seen from Figure 19 that with the increase of coal  Figure 16: Permeability evolution process of the fracture system of model II.  Figure 17: Permeability evolution process of the pore system of model II.  Geofluids interference time, the lower limit of the permeability curve gradually decreases; that is, the change of coal porosity is more obvious. However, due to the constant gas injection pressure, when the coupling process of coal deformation and gas permeability is eventually balanced, the permeability of coal with different interference times is unchanged. The evolution process of permeability under different gas injection pressures is shown in Figure 20. It can be seen from 13 Geofluids Figure 20 that the upper limit of permeability is at the final equilibrium state, which is proportional to the injection pressure, while the lower limit of permeability value is inversely proportional to the injection pressure. The permeability under different injection pressures after equilibrium has little difference. The main reason is that the free expansion deformation is limited by the roller constraints; so, the model cannot expand freely.

Conclusions
A nonequilibrium coal permeability model is developed to quantify the impact of local transient behaviors on gas flow triggered by gas injection. Based on the results of this study, the following conclusions can be drawn: (1) The permeability of coal is restricted by the initial and ultimate pore-body strain ratio and the disturbance time. The equilibrium permeability is determined by the initial and the limit pore volume strain ratio, and the evolution time from the initial to the limit equilibrium permeability is determined by the interference time (2) It is assumed in the literature that the current permeability data are measured under equilibrium conditions, but are distributed within the permeability range described by the equilibrium value. This suggests that the equilibrium hypothesis of those experiments may not be valid (3) The coal permeability from the initial value to the final value all experienced a similar evolution stage. The injection pressure leads to an initial increase, and the fracture diffuses to the periphery, leading to a gradual decrease. The diffusion region expands from the fracture to the whole matrix region, leading to recovery, and finally reaches the equilibrium permeability

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The author(s) declare(s) that they have no conflicts of interest. 14 Geofluids