1. Introduction

2. Materials and Methods

Table 1. Industrial analysis and R_o max of the coal samples

Figure 1. Overall layout of true triaxial permeability tester.

Figure 1. Overall layout of true triaxial permeability tester.

Figure 2. Sample holder of true triaxial permeability tester.

Figure 2. Sample holder of true triaxial permeability tester.

Figure 3. HM bedding plane is parallel to gas inlet direction.

Figure 3. HM bedding plane is parallel to gas inlet direction.

Figure 4. WY¹ bedding plane is parallel to gas inlet direction.

Figure 4. WY¹ bedding plane is parallel to gas inlet direction.

Figure 5. WY² bedding plane is perpendicular to gas inlet direction.

Figure 5. WY² bedding plane is perpendicular to gas inlet direction.

3. Results

Figure 6. Change law of permeability with gas pressure.

Figure 6. Change law of permeability with gas pressure.

4. Discussion

4.1. Gas pressure sensitivity analysis

In the quantitative exploration of the response of permeability to gas pressure, the relative change in coal permeability caused by each reduction in unit gas pressure under constant stress was defined as the sensitivity coefficient of permeability to gas pressure C_p:

$$C_p=-\frac{1}{k_0}\frac{\partial k}{\partial p},$$

(1)

where C_p is the gas pressure sensitivity coefficient, mPa^-1; $\partial k$ is the permeability change in coal, μm²；$\partial p$ is the change in gas pressure, mPa; and k₀ is the initial permeability value, indicating coal permeability when the gas pressure is 0.5 mPa.

Figure 7 shows the gas pressure sensitivity coefficient of coal sample permeability under the experimental conditions of this study. The sensitivity coefficient and gas pressure data were fitted in accordance with a power function relationship, and the degree of fit was above 90% (Table 2). The sensitivity coefficient C_p decreased with an increase in gas pressure, decreased faster before 0.8 mPa, and tended to be stable after 0.8 mPa. According to the relationship between desorption amount and gas pressure, when gas pressure is low, the increment of CH₄ adsorption by coal particles is large [16,17,18,19]. In addition, the slippage effect is more pronounced in the low-pressure region. Therefore, the sensitivity coefficient C_p of the permeability to the gas pressure in the low-pressure zone was relatively large. When the gas pressure was high, the increase in CH₄ adsorption by the coal particles tended to be stable, and the effect of matrix expansion on permeability was small. Therefore, C_p is small in the high-pressure area. Also shown in Figure 7 is that the sensitivity coefficient of the bedding plane perpendicular to the seepage direction was less than that of the bedding plane parallel to the seepage direction.

Figure 7. Relation curve between sensitivity coefficient and gas pressure.

Figure 7. Relation curve between sensitivity coefficient and gas pressure.

Table 2. Fitting of sensitivity coefficient of coal permeability

Table 2. Fitting of sensitivity coefficient of coal permeability

Coal sample	Fitting equation	Fit R2
HM	Cp = 0.48095p-2.87622	0.91913
WY1	Cp = 0.60954p-2.47003	0.92759
WY2	Cp = 0.22574p-3.66777	0.93473

4.2. Mathematical model of coal permeability

4.2.1. Mathematical model

According to the literature, the volume deformation of coal owing to desorption or adsorption has a linear relationship with the amount of gas adsorption [20,21,22]:

$${\epsilon }_V=\epsilon \times V,$$

(2)

where ε_V is the deformation of gas adsorption or desorption; ε is the volumetric strain coefficient, taking the value of 7.4×10^-4 g/cm³; and V is the adsorption amount of gas. The gas adsorption capacity V can be obtained from the Langmuir adsorption curve [23,24,25]:

$$V=\frac{V_{L}p}{p+p_L},$$

(3)

where V and V_L denote the gas adsorption capacities at pressures p and p_L, respectively.

Chikatamarl‘s gas adsorption test shows that the volume strain of coal caused by gas adsorption is proportional to the amount of gas adsorbed. According to rock mechanics, the stress and strain of the deformation of the coal body absorbing gas can be expressed as [26,27,28,29]

$${\sigma }_{i,j}=\frac{E}{1+v}\left({\epsilon }_{i,j}+\frac{v}{1-2v}{\epsilon }_b{\delta }_{i,j}\right)+\alpha p{\delta }_{i,j}+K{\epsilon }_V{\delta }_{i,j},$$

(4)

where εV is the deformation of gas adsorption or desorption, which can be obtained from formula (2); E is Young's modulus; υ is Poisson's ratio; K is the bulk modulus of elasticity, with α=1-K/K_s, K_s being the modulus of the solid matrix. The bulk modulus K is often several orders of magnitude larger than the pore bulk modulus K_p of coal, where ${\delta }_{i,j}$, the Kronecker delta, is zero when i≠j and one when i=j, and the Einstein summation convention is followed. The available permeability is [30,31,32]

$$k=k_0\exp \left\{-\frac{3}{K_p}\left[\left(\sigma -{\sigma }_0\right)-\left(p-p_0\right)\right]\right\},$$

(5)

where K_p is the pore bulk modulus, K_p = Kφ, and p₀ is the initial gas pressure.

Assumptions are that if ε_xx =ε_yy = 0 and α = 1, σ_xx and σ_yy can be expressed as

$${\sigma }_{xx}={\sigma }_{yy}=\frac{v}{1-v}{\sigma }_{zz}+\frac{1-2v}{1-v}p+\frac{1-2v}{1-v}K{\epsilon }_V.$$

(6)

Change in stress (σ− σ₀) can be expressed as

$$\sigma -{\sigma }_0=\frac{2\left(1-2v\right)}{3\left(1-v\right)}\left[\left(p-p_0\right)+K\left({\epsilon }_V-{\epsilon }_{V0}\right)\right].$$

(7)

Substituting formula (7) into formula (5) and combining formulas (2) and (3), we obtained the mathematical model of the permeability change with gas pressure:

$$\begin{array}{cc}k=& k_0\mathrm{e}\mathrm{x}\mathrm{p}\{\frac{3\Delta p}{\frac{E}{\left(1-2v\right)}{\varphi }_0}·[\frac{1+v}{1-v}-\frac{2E\epsilon V_L{p}_L}{3\left(1-v\right)\left(p_L+p_0\right)\left(p+p_L\right)}\left]\right\}\end{array}.$$

(8)

4.2.2. Experimental verification

By using the basic parameters of the experimental coal samples (Table 3), the theoretical model curves of the permeability of the three coal samples were calculated using formula (8) and compared with the experimental permeability data (Figure 8, Figure 9 and Figure 10). The relationship between the permeability value of coal and gas pressure conformed to the power function, and the fitting degree R² exceeded 99% (Table 4). The theoretical model and test fitting curves exhibited a high degree of agreement. Coal permeability samples decreased with the increase in gas pressure, and the rate of decrease was fast in the low-pressure section of 0.8 MPa, and slow in the high-pressure section of 0.8~1.5 MPa.

Table 3. Basic parameters of experimental coal samples.

Table 3. Basic parameters of experimental coal samples.

Coal sample	Elastic modulus/GPa	Poisson's ratio	Initial permeability/mD	Initial porosity	Adsorption gas constant/m·t-1	Adsorption gas constant/MPa
HM	1.5	0.35	0.04272	0.09	15.25	4.35
WY1	1.7	0.34	0.03834	0.06	24.31	3.89
WY2	1.6	0.35	0.03049	0.04	30.58	4.41

Figure 8. Comparison of theoretical model and experimental data of HM.

Figure 8. Comparison of theoretical model and experimental data of HM.

Figure 9. Comparison of theoretical model and experimental data of WY^1.

Figure 9. Comparison of theoretical model and experimental data of WY^1.

Figure 10. Comparison of theoretical model and experimental data of WY^2.

Figure 10. Comparison of theoretical model and experimental data of WY^2.

Table 4. Fitting of coal sample permeability and gas pressure.

Table 4. Fitting of coal sample permeability and gas pressure.

Coal sample	Fitting equation	Fit R2
HM	k = 0.01012p-1.83137	0.99402
WY1	k = 0.00931p-1.93214	0.99274
WY2	k = 0.00753p-1.97026	0.99962

As shown in Figure 8 and Figure 10 and Table 5, the theoretical values and experimental data present the same change trend; the difference between the two was mainly concentrated in the low-pressure area below 0.8 MPa, where the theoretical value was lower than the experimental data. The reason for this may be that the theoretical model does not consider the slippage effect, which is more evident in the low-pressure region [33,34,35,36,37]. The maximum absolute error between the theoretical value and experimental data reached 0.00448 mD, and the maximum relative error reached 13.8%. However, overall, the data demonstrate that the basic change trends of the theoretical values and experimental data are consistent. Therefore, according to the basic parameters of the coal samples and the theoretical permeability model, the corresponding permeability under a certain gas pressure can be calculated. The permeability data obtained directly from the experiment truly reflected the change in permeability with gas pressure. However, these experiments only explain the permeability change rule of the experimental coal samples and not coal samples from other places in the coal seam, which is not conducive to engineering applications. Moreover, experimental coal samples are always limited, and testing each location in the coal seam is impossible. The mathematical model of permeability based on desorption-adsorption is critical to understanding the influence mechanism of gas adsorption and gas pressure change on permeability from a microscopic level. This specific expression is obtained using a direct empirical formula and mechanical derivation and has a certain universality, but the key parameters in the mathematical model remain unclear. Therefore, the permeability at different locations in the coal seam can be calculated using this mathematical model.

Table 5. Error analysis between experimental data and theoretical value of permeability.

Table 5. Error analysis between experimental data and theoretical value of permeability.

Coal sample	Pressure/MPa	Permeability/mD		Absolute error/mD	Relative error/%
		experimental data	theoretical value
HM	0.50	0.03570	0.03248	0.00322	9.0
	1.05	0.00944	0.00982	0.00038	4.0
	1.55	0.00617	0.00532	0.00085	13.8
WY1	0.50	0.03582	0.03134	0.00448	12.5
	1.01	0.00875	0.00961	0.00086	9.8
	1.51	0.00648	0.00577	0.00071	10.9
WY2	0.50	0.03028	0.02830	0.00198	6.5
	1.00	0.00737	0.00687	0.00050	6.7
	1.50	0.00329	0.00299	0.00030	9.1

4.3. Numerical simulation of gas extraction

COMSOL is a finite element analysis software that considers the coupling of multiple physical fields and has achieved good application results in many fields. This study used COMSOL to simulate the coupling of the mechanical field of coal reservoirs and the seepage field of coalbed methane, considering the adsorption/desorption of coalbed methane on the surface of coal seams, diffusion in pores, and the seepage process in fractures. The simulated geometric modeling is a square area, with the length and width set to 40m, the height set according to the thickness of the coal seam at 6 m, and the drilling radius set to 0.1 m with reference to the actual parameters. The established geometric model is shown in Figure 11. The various parameters for the simulation process refer to the onsite measured values and results in the literature [38,39,40,41]. The parameter settings for each module are listed in Table 6.

Table 6. Simulation parameters of gas extraction.

Table 6. Simulation parameters of gas extraction.

Parameter name	Value [unit]	Describe
E	2650 [MPa]	Elastic modulus of coal
ν	0.35	Poisson's ratio of coal
ρ	1280 [kg/m³]	Coal seam density
k0	0.035 [mD]	Initial permeability
φ0	0.06	Initial porosity
μ	1.00E-5 [Pa·s]	Methane dynamic viscosity
VL	0.025 [m³/kg]	Langmuir constant
PL	4 [MPa]	Langmuir pressure
VM	22.4 [L/mol]	Molar volume of methane under standard conditions
R	8.413510 [J/mol/K]	Gas state constant
T	293 [K]	Sample temperature
M	16 [g/mol]	Gas molecular mass of methane
Fx	15 [MPa]	Confining pressure in X direction
Fy	15 [MPa]	Confining pressure in Y direction
Fz	15 [MPa]	Axial pressure in Z direction
r	0.1 [m]	Borehole radius
p0	0.5, 0.7, 0.9, 1.1, 1.3, 1.5 [MPa]	Initial pressure
pb	0.08 [MPa]	Suction negative pressure
t	10 [d]	Adsorption time

Figure 11. Geometric model for numerical simulation of gas extraction.

Figure 11. Geometric model for numerical simulation of gas extraction.

A linear elastic model was adopted for the solid mechanics module of the simulation. The left, front, and lower interfaces of the model were set as sliding boundaries; thus, the normal upward displacement of the interface was zero. The right, rear, and upper interfaces of the model were set as pressure boundaries, with a pressure of 15 MPa. They simulated the X- and Y-axis stresses of the confining pressure in physical experiments and the Z-axis stress of the axial pressure.

The gas diffusion process in the coal seam pores was simulated using general partial differential equations. The diffusion source term f is represented by the following formula:

$$f=-\frac{V_M(u-p){(u+P_L)}^2}{t{V}_{L}RT{P}_L\mathsf{\rho }+\mathrm{t}{\phi }_0{V}_M{(u+P_L)}^2}$$

(9)

where u represents the gas pressure in the pore, and p represents the gas pressure in the fracture.

The seepage of coalbed methane in fractures was simulated using Darcy's law. The initial air pressures of the model were 0.5, 0.7, 0.9, 1.1, 1.3, and 1.5 MPa. The negative pressure of the gas extraction hole was set to 0.08 MPa. Permeability k is a function of the gas pressure, and its functional relationship was obtained by fitting the experimental data, set according to the fitting function in Table 4. The expression is as follows:

k = 0.00931p^-1.93214

(10)

The numerical simulation results showed (Figure 12) that with an increase in gas pressure, the efficiency of coal seam gas extraction gradually decreased. When the gas pressure was 0.5 MPa, the daily gas production was approximately 6000 m³. When the gas pressure increased to 1.5 MPa, the daily gas production decreased to 3000 m³. In addition, in the low-pressure zone below 0.9 MPa, gas production changed slightly with each increment of 0.2 MPa of gas pressure. However, in the high-pressure zone above 0.9 MPa, gas production changed significantly with each 0.2 MPa of gas pressure. This is also consistent with the law that the permeability of the coal sample changes with the gas pressure obtained from the experiment; that is, the sensitivity coefficient to gas pressure is high in the low-pressure area and low in the high-pressure area.

Figure 12. Comparison of coalbed gas extraction efficiency under different gas pressure.

Figure 12. Comparison of coalbed gas extraction efficiency under different gas pressure.

5. Conclusions

References

1. Li, N.; Fang, L.; Huang, B.; Chen, P.; Cai, C.; Zhang, Y.; Liu, X.; Li, Z.; Wen, Y.; Qin, Y. Characteristics of Acoustic Emission Waveforms Induced by Hydraulic Fracturing of Coal under True Triaxial Stress in a Laboratory-Scale Experiment. Minerals 2022, 12, 104. [Google Scholar] [CrossRef]
2. Song, W.; Yao, J.; Li, Y.; Sun, H.; Yang, Y. Fractal models for gas slippage factor in porous media considering second-order slip and surface adsorption. Int. J. Heat. Mass. Tran. 2018, 118, 948–960. [Google Scholar] [CrossRef]
3. Liu, S.; Tang, S.; &amp, *!!! REPLACE !!!*; Yin, S.; Yin, S. Coalbed methane recovery from multilateral horizontal wells in southern qinshui basin. Advances in Geo-Energy Research 2018, 2(1), 34–42. [Google Scholar] [CrossRef]
4. Ye, D.; Liu, G.; Gao, F.; Xu, R.; Yue, F. A multi-field coupling model of gas flow in fractured coal seam. Advances in Geo-Energy Research 2021, 5(1), 104–118. [Google Scholar] [CrossRef]
5. Tian, Y.; Yu, X.; Li, J.; Yin, X.; Neeves, K.B.; Wu, Y.S. Scaling law for slip flow of gases in nanoporous media from nanofluidics, rocks, and pore-scale simulations. Fuel 2018, 1, 1–23. [Google Scholar] [CrossRef]
6. Li, W.; Jiang, B.; Zhu, Y. Impact of tectonic deformation on coal methane adsorption capacity. Adsorpt. Sci. & Techno. 2019, 37, 698–708. [Google Scholar] [CrossRef]
7. Cai, J.; Wei, W.; E. I.; Hu, X.; Liu, R.; Wang, J. Fractal characterization of dynamic fracture network extension in porous media. Fractals 2017, 25, 1750023. [CrossRef]
8. Chen, Z.; Li, G.; Wang, Y.; Li, Z.; Chi, M.; Zhang, H.; Tian, Q.; Wang, J. An Experimental Investigation of the Gas Permeability of Tectonic Coal Mineral under Triaxial Loading Conditions. Minerals 2022, 12, 70. [Google Scholar] [CrossRef]
9. Liu, C.; Yin, G.; Li, M.; Shang, D.; Deng, B.; Song, Z. Deformation and permeability evolution of coals considering the effect of beddings. Int. J. Rock. Mech. Min. 2019, 117, 49–62. [Google Scholar] [CrossRef]
10. Liu, J.; Song, Z.; Yang, C.; Li, B.; Ren, J.; Xiao, M. True triaxial experimental study on the influence of axial pressure on coal permeability. Front. Earth. Sc-switz. 2022, 10, 1024483. [Google Scholar] [CrossRef]
11. Xiong, J.; Liu, X.; Liang, L. Application of adsorption potential theory to methane adsorption on organic-rich shales at above critical temperature. Environ. Earth. Sci. 2018, 77, 1–18. [Google Scholar] [CrossRef]
12. Liu, J.; Gao, J.; Zhang, X.; Jia, G.; Wang, D. Experimental study of the seepage characteristics of loaded coal under true traxial conditions. Rock. Mech. Rock. Eng. 2019, 52, 2815–2833. [Google Scholar] [CrossRef]
13. Liu, J.; Song, Z.; Li, B.; Ren, J.; Chen, F.; Xiao, M. Experimental Study on the Microstructure of Coal with Different Particle Sizes. Energies 2022, 15, 4043. [Google Scholar] [CrossRef]
14. Liu, Y.; Yin, G.; Li, M.; Zhang, D.; Deng, B.; Liu, C.; Lu, J. Anisotropic mechanical properties and the permeability evolution of cubic coal under true triaxial stress paths. Rock. Mech. Rock. Eng. 2019, 52, 2505–2521. [Google Scholar] [CrossRef]
15. Xie, J.; Gao, M.; Yu, B.; Zhang, R.; Jin, W. Coal permeability model on the effect of gas extraction within effective influence zone. Geomech. Geophys. Geo. 2015, 1, 15–27. [Google Scholar] [CrossRef]
16. Yu, B.; Liu, C.; Zhang, D.; Zhao, H.; Li, M.; Liu, Y.; Li, H. Experimental study on the anisotropy of the effective stress coefficient of sandstone under true triaxial stress. J. Nat. Gas. Sci. Eng. 2020, 84, 103651. [Google Scholar] [CrossRef]
17. Liu, Y.; Li, M.; Yin, G.; Zhang, D.; Deng, B. Permeability evolution of anthracite coal considering true triaxial stress conditions and structural anisotropy. J. Nat. Gas. Sci. Eng. 2018, 52, 492–506. [Google Scholar] [CrossRef]
18. Letham, E.A.; Bustin, R.M. The impact of gas slippage on permeability effective stress laws: implications for predicting permeability of fine-grained lithologies. Int. J. Coal. Geol. 2016, 167, 93–102. [Google Scholar] [CrossRef]
19. Lu, J.; Yin, G.; Deng, B.; Zhang, W.; Li, M.; Chai, X. . & Liu, Y. Permeability characteristics of layered composite coal-rock under true triaxial stress conditions. J. Nat. Gas. Sci. Eng. 2019, 66, 60–76. [Google Scholar] [CrossRef]
20. Zimmerman, R.W. Coupling in poroelasticity and thermoelasticity. Int. J. Rock. Mech. Min. 2000, 37, 79–87. [Google Scholar] [CrossRef]
21. Neuzil, C.E. Hydromechanical coupling in geologic processes. Hydrogeol. J. 2003, 11, 41–83. [Google Scholar] [CrossRef]
22. Palciauskas, V.V.; Domenico, P.A. Characterization of drained and undrained response of thermally loaded repository rocks. Water. Resour. Res. 1982, 18, 281–290. [Google Scholar] [CrossRef]
23. Cao, P.; Liu, J.; Leong, Y.K. General gas permeability model for porous media: bridging the gaps between conventional and unconventional natural gas reservoirs. Energ. fuel. 2016, 30, 5492–5505. [Google Scholar] [CrossRef]
24. Si, L.; Li, Z.; Yang, Y. Coal permeability evolution with the interaction between nanopore and fracture: Its application in coal mine gas drainage for Qingdong coal mine in Huaibei coalfield, China. J. Nat. Gas. Sci. Eng. 2018, 56, 523–535. [Google Scholar] [CrossRef]
25. Yu, P.; Liu, Y.; Wang, J.; Kong, C.; Gu, W.; Xue, L. . & Jiang, L. A new fracture permeability model: Influence of surrounding rocks and matrix pressure. J. Pet. Sci. Eng. 2020, 193, 107320. [Google Scholar] [CrossRef]
26. Duan, M.; Jiang, C.; Gan, Q.; Li, M.; Peng, K.; Zhang, W. Experimental investigation on the permeability, acoustic emission and energy dissipation of coal under tiered cyclic unloading. J. Nat. Gas. Sci. Eng. 2020, 73, 103054. [Google Scholar] [CrossRef]
27. Duan, M.; Jiang, C.; Gan, Q.; Zhao, H.; Yang, Y.; Li, Z. Study on permeability anisotropy of bedded coal under true triaxial stress and its application. Transport. Porous. Med. 2020, 131, 1007–1035. [Google Scholar] [CrossRef]
28. Duan, M.; Jiang, C.; Xing, H.; Zhang, D.; Peng, K.; Zhang, W. Study on damage of coal based on permeability and load-unload response ratio under tiered cyclic loading. Arab. J. Geosci. 2020, 13, 1–11. [Google Scholar] [CrossRef]
29. Zhu, W.C.; Wei, C.H.; Liu, J.; Qu, H.Y.; Elsworth, D. A model of coal–gas interaction under variable temperatures. Int. J. Coal. Geol. 2011, 86, 213–221. [Google Scholar] [CrossRef]
30. Liu, Q.; Li, Z.; Wang, E.; Niu, Y.; Kong, X. A dual-permeability model for coal under triaxial boundary conditions. J. Nat. Gas. Sci. Eng. 2020, 82, 103524. [Google Scholar] [CrossRef]
31. Gao, Q.; Liu, J.; Huang, Y.; Li, W.; Shi, R.; Leong, Y.K.; Elsworth, D. A critical review of coal permeability models. Fuel 2022, 326, 125124. [Google Scholar] [CrossRef]
32. Liu, T.; Lin, B.; Yang, W. Impact of matrix–fracture interactions on coal permeability: model development and analysis. Fuel 2017, 207, 522–532. [Google Scholar] [CrossRef]
33. Li, B.; Ren, J.; Liu, J.; Liu, G.; Lv, R.; Song, Z. Diffusion and Migration Law of Gaseous Methane in Coals of Different Metamorphic Degrees. Int. J. Heat. Technol. 2019, 37, 1019–1030. [Google Scholar] [CrossRef]
34. Liu, J.; Song, Z.; Li, B.; Ren, J.; Chen, F.; Xiao, M.; Yang, C. Study on the Destructive Effect of Small Faults on Coal Pore Structure. Geofluids 2022, 2022. [Google Scholar] [CrossRef]
35. Ren, J.; Song, Z.; Li, B.; Liu, J.; Lv, R.; Liu, G. Structure feature and evolution mechanism of pores in different metamorphism and deformation coals. Fuel 2021, 283, 119292. [Google Scholar] [CrossRef]
36. Zhou, Y.; Li, Z.; Yang, Y.; Zhang, L.; Si, L.; Kong, B.; Li, J. Evolution of coal permeability with cleat deformation and variable Klinkenberg effect. Transport. Porous. Med. 2016, 115, 153–167. [Google Scholar] [CrossRef]
37. Zhu, W.; Liu, L.; Liu, J.; Wei, C.; Peng, Y. Impact of gas adsorption-induced coal damage on the evolution of coal permeability. Int. J. Rock. Mech. Min. 2018, 101, 89–97. [Google Scholar] [CrossRef]
38. Wang, C.; Zhang, J.; Zang, Y.; Zhong, R.; Wang, J.; Wu, Y. . & Chen, Z. Time-dependent coal permeability: Impact of gas transport from coal cleats to matrices. J. Nat. Gas. Sci. Eng. 2021, 88, 103806. [Google Scholar] [CrossRef]
39. Cai, J.; Wood, D.A.; Hajibeygi, H.; Iglauer, S. Multiscale and multiphysics influences on fluids in unconventional reservoirs: Modeling and simulation. Ager 2022, 6, 91–94. [Google Scholar] [CrossRef]
40. Zimmerman, R.W.; Somerton, W.H.; King, M.S. Compressibility of porous rocks. J. Geophys. Res-sol. Ea. 1986, 91, 12765–12777. [Google Scholar] [CrossRef]
41. Liu, G.; Liu, J.; Liu, L.; Ye, D.; Gao, F. A fractal approach to fully-couple coal deformation and gas flow. Fuel 2019, 240, 219–236. [Google Scholar] [CrossRef]