A Novel Permeability Model of Coal Considering Gas Slippage and Gas Sorption-Induced Strain

Abstract

As an unconventional natural gas, coalbed methane (CBM) has been recognized as a significant fuel and chemical feedstock that should be recovered. Permeability is a key factor that controls CBM transport in coal. The slippage effect is an influential phenomenon that occurs during gas penetration processes, especially in low-permeable media. Apparent permeability may differ greatly from intrinsic permeability due to gas slippage. However, the gas slippage effect has not been considered in most analytical permeability models. Based on the cubic law, a new analytical model suited for the permeability analysis of coal under different stress conditions is derived, taking into consideration gas slippage and matrix shrinkage/swelling due to gas desorption/adsorption. To enhance its application, the model is derived under constant hydrostatic stress and pore pressure. The new analytical model is then compared with the existing models, and its reliability is verified by the comparison between the analytical prediction and the experimental permeability data under different stress conditions.

1. Introduction

As a porous adsorption medium, coal is both the source rock and reservoir rock for coalbed methane (CBM). To exploit CBM, water in a reservoir is drained first to reduce the reservoir pressure. After the fluid pressure decreases below the critical desorption pressure level, CBM begins to desorb and diffuse into cleats or fractures and then penetrates into the shaft. Permeability is a key controlling factor of gas transport in coal and gas production through which CBM recovery rates and economic benefits are influenced.

To investigate the permeability characteristics of coal, many permeability models have been derived by numerous researchers since Gray (1987) [1,2,3,4,5,6,7,8,9,10,11,12] proposed the first one. Seidle and Huitt (1995) [13] derived a permeability model for coal that only considers the effects of matrix expansion and shrinkage. Palmer and Mansoori (1996 [14], 1998 [4]) considered the comprehensive effects of pore pressure and matrix expansion and shrinkage on changes in porosity. Based on the linear elasticity mechanics of an isotropic porous medium, a commonly applied permeability model was derived by Shi and Durucan (2004 [5], 2005 [6]). Liu and Rutqvist (2010) [15] accommodated the role of swelling strains, not only over contact bridges, but also across non-contact areas between these bridges. Liu et al. (2011) [16] improved their former permeability model by applying a “free expansion plus push back” approach, whereby coal was allowed to expand freely due to gas sorption and was then pushed back by the applied effective stress to the original constrained conditions. In addition, anisotropic characteristics have been considered in some permeability models [17]. However, these models are not suitable for the investigation of permeability evolution under laboratory conditions. Permeability experiments often cause variable changes in confining, axial, and air pressures. In addition, gas adsorption/desorption could cause the sample to expand/shrink [8]. Therefore, some permeability models that are better suited for laboratory permeability analysis are proposed [18,19,20]. The McKee model and the Robertson model were modified in Zou et al. (2016) [8].

Klinkenberg (1941) [21] proposed the gas slippage effect, which describes the phenomenon that when gas is flowing through porous media, the migration velocity of gas molecules at the wall hole is not zero. The slippage effect can be described as

$$k_g=k\left(1+\frac{4c\lambda }{\gamma }\right)$$

(1)

where k_g is the apparent permeability under the pore pressure p; k is the Klinkenberg permeability or the absolute permeability; c is a scale factor; λ is the mean free path of gas molecules; and γ is the average pore radius.

Experiments have shown that the average pore radius is inversely proportional to the average pore pressure p [8,21,22,23,24,25]; therefore, Equation (1) can also be expressed as

$$k_g=k\left(1+\frac{b}{p}\right), b=\frac{4c}{\gamma }\lambda p$$

(2)

The average pore pressure p is determined by the inlet and outlet gas pressures, i.e., $p=\left(p_{in}+p_{out}\right)/2$, where p_in and p_out represent the inlet and outlet pressures, respectively; and b is the slippage factor.

The gas slippage effect has a strong impact on penetration, especially in less permeable media. The impact of the slippage effect on the permeability could be interpreted by the Knudsen number Kn. The Knudsen number is commonly used to classify flow regimes in small pores [24,25]. This number is defined as the ratio of the molecular mean free path to a characteristic length, such as pore size.

Table 1. Knudsen number and flow regime classifications for porous media [26].

Flow Regime	Knudsen Number	Model to Be Applied
Continuum flow	Kn < 0.01	Darcy’s equation for laminar flow and Forchheimer’s equation for turbulent flow
Slip flow	0.01 < Kn < 0.1	Darcy’s equation with Klinkenberg or Knudsen’s correction
Transition flow	0.1 < Kn < 10	Darcy’s law with Knudsen’s correction or Burnett’s equation with slip boundary conditions
Free molecular flow	Kn > 10	Knudsen’s diffusion equation alternative methods are DSMC and lattice Boltzmann

shows the gas flow regimes with different Knudsen number ranges [26]. Within the transition flow regime, the slip flow and the diffused flow co-exist. For low-permeability media (within the slip flow regime and transition flow regime), permeability models that neglect the gas slippage effect are not sufficiently accurate.

Based on the cubic law, which has been widely applied to describe permeability changes with respect to porosity changes [4,14,27], a new analytical model suited for permeability analysis of coal under different stress conditions is derived considering the effects of matrix shrinkage/swelling due to gas desorption/adsorption and gas slippage.

2. Derivation of the Permeability Model

Coal cleats take two forms: face cleats and butt cleats. These cleats are often normal to the bedding and are perpendicular to one another [28,29]. A conceptual schematic of a coal fracture or cleat system is shown in Figure 1, and the matrix blocks are often surrounded by fractures or cleats. Coalbed methane molecules can be adsorbed within the pores and onto the surface of a matrix block or can exist in a fracture or cleat system in a free state.

The cubic law proposed by Reiss (1980) [30], which has been widely applied to describe permeability changes with respect to porosity changes [4,9,10,27,31], is used for the derivation of the permeability model in the present study. It can be expressed as

$$\frac{k}{k_0}={\left(\frac{\varphi }{{\varphi }_0}\right)}^3$$

(3)

where ϕ₀ is the original porosity, and k₀ is the original permeability.

An equation that describes coal porosity with respect to the matrix dimension and fracture width was proposed by Robertson and Christiansen (2006) [19] and is expressed as Equation (4). This equation assumes that the fracture width η is far smaller than the matrix block dimension δ.

$$\varphi =\frac{3\eta }{\delta }$$

(4)

Combining Equations (3) and (4) results in

$$\frac{k}{k_0}={\left(\frac{3\eta }{\delta }/\frac{3{\eta }_0}{{\delta }_0}\right)}^3$$

(5)

It could also be written in the form of the fracture width change and matrix dimension change

$$\frac{k}{k_0}={\left[\left(\frac{\Delta \eta }{{\eta }_0}+1\right)/\left(\frac{\Delta \delta }{{\delta }_0}+1\right)\right]}^3$$

(6)

On a large scale, coal is anisotropic due to its layered characteristics. While in a matrix block, however, coal behaves in a more isotropic manner, i.e., the linear strain in all directions is equivalent to one-third of the volumetric strain [3]. It is assumed that elastic deformation occurs in a matrix block during the gas penetration process [5,6,15,27,31]. Based on Hooke’s law, strains occurring in different directions of a matrix can be expressed as

$$\{\begin{array}{c}\Delta {\epsilon }_x=\frac{1}{E}\left[\Delta {\sigma }_x-\nu \left(\Delta {\sigma }_y+\Delta {\sigma }_z\right)\right]\\ \Delta {\epsilon }_y=\frac{1}{E}\left[\Delta {\sigma }_y-\nu \left(\Delta {\sigma }_x+\Delta {\sigma }_z\right)\right]\\ \Delta {\epsilon }_z=\frac{1}{E}\left[\Delta {\sigma }_z-\nu \left(\Delta {\sigma }_x+\Delta {\sigma }_y\right)\right]\end{array}$$

(7)

where E is the elastic modulus; v is Poisson’s ratio; and σ_x, σ_y, and σ_z are effective stresses in different directions.

The stresses of different principal directions for coal are equivalent when coal is subjected to hydrostatic stress. Elastic strains can be written as

$$\Delta {\epsilon }_x=\Delta {\epsilon }_y=\Delta {\epsilon }_z=\frac{1}{E}\Delta \sigma \left(1-2\nu \right)$$

(8)

In recognizing the strains of a matrix block as the change in dimension divided by the original dimension, strains can also be expressed as

$$\Delta {\epsilon }_x=\Delta {\epsilon }_y=\Delta {\epsilon }_z=-\frac{\Delta {\delta }_1}{{\delta }_0}$$

(9)

Substituting Equation (9) into Equation (8) produces

$$\Delta {\delta }_1=-\frac{{\delta }_0}{E}\Delta \sigma \left(1-2\nu \right)$$

(10)

The increase in the fracture opening due to a single effect of matrix shrinkage can be expressed as [19]

$$\Delta {\eta }_1=-\Delta {\delta }_1=\frac{{\delta }_0}{E}\Delta \sigma \left(1-2\nu \right)$$

(11)

Several studies have demonstrated that coal matrix deformation is significantly influenced by gas adsorption or desorption [3,31,32,33,34]. Shrinkage/swelling strain due to gas desorption/adsorption can be expressed as [2,13,35]

$$\Delta {\epsilon }_s=S_{\max }\left(\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}\right)$$

(12)

where S_max and p_L are sorption constants. S_max is the Langmuir strain, which represents the matrix strain at infinite adsorption pressure. p_L is the Langmuir pressure, which represents the pressure when the adsorption strain is S_max/2.

Similar to Equation (9), the swelling strain induced by gas adsorption can be written as

$$\Delta {\epsilon }_s=\frac{\Delta {\delta }_2}{{\delta }_0}$$

(13)

Substituting Equation (13) into Equation (12) produces

$$\Delta {\delta }_2={\delta }_0{S}_{\max }\left(\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}\right)$$

(14)

The increase in the fracture opening due to a single effect of matrix swelling resulting from gas adsorption can be expressed as [19]

$$\Delta {\eta }_2=-\Delta {\delta }_2=-{\delta }_0{S}_{\max }\left(\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}\right)$$

(15)

The effective stress of a porous medium is expressed as

$$\sigma ={\sigma }_t-\beta p$$

(16)

where σ is the effective stress; σ_t is the total stress; p is the average pore pressure; and β is the effective stress coefficient, where 0 ≤ β ≤ 1. The effective stress coefficient could be determined by the following equation [36,37,38]

$$\beta =1-K_v/K_s$$

(17)

where K_v is the bulk modulus and K_s represents the bulk modulus of solid grain material.

We all know that the effective stress coefficient is less than 1 [8,39,40]. However, most current permeability models assume an effective stress factor of 1, which leads to an overestimation of gas permeability for the same stress change [28].

It is assumed that the effective stress coefficient of coal does not change with pore pressure and confining stress; therefore,

$$\Delta \sigma =\Delta {\sigma }_t-\beta \Delta p$$

(18)

The cleat compressibility C_f was defined by Amyx et al. (1960) [41] as Equation (19)

$$C_f=-\frac{1}{{\varphi }_0}\cdot \frac{\Delta \varphi }{\Delta \sigma }$$

(19)

Combining Equations (4), (18) and (19) produces

$$\Delta {\eta }_3=-{\eta }_0{C}_f\left(\Delta {\sigma }_t-\beta \Delta p\right)$$

(20)

The actual deformation of the coal matrix during gas permeation is the sum of the deformations caused by external stress and gas desorption/adsorption. Combining Equation (10) with (14) produces the total deformation of the coal matrix

$$\begin{gathered} \Delta {\delta }_t=\Delta {\delta }_1+\Delta {\delta }_2 \\ =-\frac{{\delta }_0}{E}\left(\Delta {\sigma }_t-\beta \Delta p\right)\left(1-2\nu \right)+{\delta }_0{S}_{\max }\left(\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}\right) \end{gathered}$$

(21)

Similarly, the actual deformation of a fracture opening can be obtained by combining Equations (11), (15) and (20).

$$\begin{gathered} \Delta {\eta }_t=\Delta {\eta }_1+\Delta {\eta }_2+\Delta {\eta }_3 \\ =\frac{{\delta }_0}{E}\left(\Delta {\sigma }_t-\beta \Delta p\right)\left(1-2\nu \right)-{\eta }_0{C}_f\left(\Delta {\sigma }_t-\beta \Delta p\right)-{\delta }_0{S}_{\max }\left(\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}\right) \end{gathered}$$

(22)

Substituting Equations (21) and (22) into Equation (6) yields an equation for permeability changes with respect to other parameters:

$$\frac{k}{k_0}={\left[\frac{\left(\Delta {\sigma }_t-\beta \Delta p\right)\left[{\delta }_0\left(1-2v\right)-E{\eta }_0{C}_f\right]-E{\delta }_0{S}_{\max }\Delta p^{\prime }+E{\eta }_0}{-{\eta }_0\left(\Delta {\sigma }_t-\beta \Delta p\right)\left(1-2v\right)+E{\eta }_0{S}_{\max }\Delta p^{\prime }+E{\eta }_0}\right]}^3$$

(23)

where

$$\Delta p^{\prime }=\frac{p}{p+p_L}-\frac{p_0}{p_0+p_L}$$

(24)

Combining Equations (23) and (4) produces

$$\frac{k}{k_0}={\left[\frac{\left(\Delta {\sigma }_t-\beta \Delta p\right)\left[3-6v-C_f\right]-3E{S}_{\max }\Delta p^{\prime }+E{\varphi }_0}{-{\varphi }_0\left(\Delta {\sigma }_t-\beta \Delta p\right)\left(1-2v\right)+E{\varphi }_0{S}_{\max }\Delta p^{\prime }+E{\varphi }_0}\right]}^3$$

(25)

McKee et al. (1988) [18] noted that the cleat compressibility C_f is a function of the effective stress. According to Mckee and coworkers, C_f can be expressed as

$$C_f=\frac{C_0}{\alpha \left(\Delta {\sigma }_t-\Delta p\right)}\left\{1-\exp \left[-\alpha \left(\Delta {\sigma }_t-\Delta p\right)\right]\right\}$$

(26)

where C₀ is the cleat compressibility, and α is the declining rate of cleat compressibility.

The cleat compressibility C_f expressed as Equation (26) is modified as Equation (27).

$$C_f=\frac{C_0}{\alpha \left(\Delta {\sigma }_t-\beta \Delta p\right)}\left\{1-\exp \left[-\alpha \left(\Delta {\sigma }_t-\beta \Delta p\right)\right]\right\}$$

(27)

The gas slippage effect is considered in our model. Rearranging Equation (2) produces the relationship between the apparent permeability ratio and the absolute permeability ratio, which is expressed as

$$\frac{k_g}{k_{g0}}=\frac{k}{k_0}·\frac{p+b}{p_0+b_0}·\frac{p_0}{p}$$

(28)

The subscript “₀” represents the initial state of each parameter.

Combining Equations (25) and (28) produces

$$\frac{k_g}{k_{g0}}={\left[\frac{\left(\Delta {\sigma }_t-\beta \Delta p\right)\left[3-6v-C_f\right]-3E{S}_{\max }\Delta p^{\prime }+E{\varphi }_0}{-{\varphi }_0\left(\Delta {\sigma }_t-\beta \Delta p\right)\left(1-2v\right)+E{\varphi }_0{S}_{\max }\Delta p^{\prime }+E{\varphi }_0}\right]}^3·\frac{p+b}{p_0+b_0}·\frac{p_0}{p}$$

(29)

Equation (29) is the expression of permeability model that considers the gas slippage effect.

3. Results and Discussion

Long flame coal from the Hunchun coalfield in the Jilin province of China was used. All samples had dimensions of Φ 50 × 100 mm. The properties of studied coal have been introduced in detail in our previous research [8]. Figure 2 shows the low-permeability rock test system. The testing method is a steady-state method. The high-pressure nitrogen cylinder provides a stable gas pressure on the upper surface of the coal sample. The loading error of the rock test system is less than 0.5%. The gas penetration rate is measured at the outlet end after forming a steady-state flow under the differential gas pressure.

3.1. Model Parameters

The mechanical properties of coal have been introduced in our previous research [8]. The elastic modulus of coal samples is approximately 2.0 GPa, and Poisson’s ratio is 0.31.

Two types of coal porosities were considered. The coal fracture porosity was obtained by a density testing method, which is expressed as

$$\varphi =\left[\left({\rho }_t-{\rho }_a\right)/{\rho }_t\right]\times 100\%$$

(30)

where ρ_t is the density of the coal matrix, and ρ_a is the apparent density of the coal sample. The fracture porosities of coal samples that were drilled at different depths in the Hunchun coalfield are listed in

Table 2. Fracture porosities of coal samples drilled at different depths in the Hunchun coalfield.

Drill Hole No.	Coal Seam No.	Sample Depth/m	Porosity/%
2711	19#	248.68–248.90	11.49
2711	19a#	254.56–255.30	10
2711	20#	287.00	10.56
2313	23#	117.90–119.35	14.63
704	23#	666.90	12.67
704	21#	615.40–616.90	8.39
704	23a#	657.70–658.70	5.67
K-4	20#	361.00–363.20	6.38
K-4	23#	422.46–425.06	10.39
1508	20#	447.95–448.65	11.81
Average	-	-	10.20

.

The mercury injection test is carried out to determine the pore size distribution characteristics of coal matrix and coal matrix porosity, and the results are shown in Figure 3. The average porosity of the coal matrix determined by the mercury injection test is 0.11.

A Langmuir stress of P_L = 2.86 MPa was obtained via sorption tests. Robertson et al. (2005 [35], 2006 [19]) obtained an extremely small Langmuir strain of Anderson and Gilson coals with an average value of 0.0025 via N₂ adsorption [8]. An average compressibility change rate of α = 0.44 MPa⁻¹ was also obtained by combining the results of McKee et al. (1988) [18].

Several investigations show that the effective stress coefficient is less than 1 [19,20,31,39,40,41,42]. Walsh (1981) [43] found that β = 0.9 for a rock mass containing a polished joint, and Kranzz (1979) [44] found that β = 0.56. Through our previous work, an effective stress coefficient of our coal is 0.53 [8].

3.2. Hydrostatic Stress Remains Constant

The permeability of the coal samples subjected to different gas pressure levels was tested via N₂ under different hydrostatic stresses. The confining stress remains constant, and only the pore pressure changes. In this case, we assume that the gas slippage factor remains constant b = b₀. If the gas slippage factor varies with the pore pressure, the formula will become extremely complicated and impractical. The change in σ_t is zero when the hydrostatic stress applied to a coal sample remains constant, i.e.,

$$\Delta {\sigma }_t=0$$

(31)

By substituting Equation (31) into Equation (29) and by adding Equation (27), we arrive at a permeability model that considers the gas slippage effect when hydrostatic stress remains constant during experiments.

$$\frac{k_g}{k_{g0}}={\left[\frac{-3\alpha \beta \Delta p\left(1-2v\right)-E{\varphi }_0{C}_0\left[1-\exp \left(\alpha \beta \Delta p\right)\right]-3E\alpha S_{\max }\Delta p^{\prime }+E\alpha {\varphi }_0}{\alpha {\varphi }_0\left[\beta \Delta p\left(1-2v\right)+E{S}_{\max }\Delta p^{\prime }+E\right]}\right]}^3·\frac{p+b}{p_0+b_0}·\frac{p_0}{p}$$

(32)

The permeability model that neglects the gas slippage effect can be expressed as

$$\frac{k}{k_0}={\left[\frac{-3\alpha \beta \Delta p\left(1-2v\right)-E{\varphi }_0{C}_0\left[1-\exp \left(\alpha \beta \Delta p\right)\right]-3E\alpha S_{\max }\Delta p^{\prime }+E\alpha {\varphi }_0}{\alpha {\varphi }_0\left[\beta \Delta p\left(1-2v\right)+E{S}_{\max }\Delta p^{\prime }+E\right]}\right]}^3$$

(33)

3.2.1. Results Obtained When Considering the Gas Slippage Effect and Discussion

The permeability model (Equation (32)) is fitted with experimental results under the hydrostatic stress of 8 MPa. Figure 4 shows the fitting result, and the parameters obtained from the fitting are listed in

Table 3. Parameters obtained from the fitting.

Parameters	Sample M1	Sample M2	Sample M3	Average
Cleat compressibility C0 (MPa−1)	0.14	0.124	0.154	0.14
Slippage factor b (MPa)	0.289	0.182	0.21	0.23

.

To evaluate the validity of the model parameters, the slippage factors are also obtained by fitting the slippage effect using Equation (2) using the same experimental results. The process for fitting the slippage effect is shown in Figure 5. The permeability increases linearly with the reciprocal of the pore pressure. The slippage factors obtained from the experimental permeability data are within a range of 0.21~0.32 MPa, with an average value of 0.26 MPa. This value approaches the average value of the slippage factors that were obtained by the model fitting.

The Knudsen number is defined as the ratio of a molecular mean free path to a characteristic length, such as pore size. According to the classification of flow regimes [24,25], a flow is considered a continuum for K_n < 0.001, and the system is considered a free molecular flow for K_n > 10. The intermediate values of 0.001 < K_n < 0.1 are representative of a slip flow regime, and those within the range of 0.1 < K_n < 10 are associated with a transition flow regime. Within the transition flow regime, the slip flow and the diffused flow co-exist. The molecular mean free path or N₂ in this paper is assumed at the constant of 3.8 × 10⁻⁸ m, and the pore size distribution under different pressures is determined by the mercury injection test (see Figure 3). The Knudsen numbers at different average pore pressures are calculated, and the results are presented in Figure 6. The Knudsen numbers are within the range of 0.04~0.29, belonging to the slip flow regime and the transition flow regime.

To further verify the conclusions, the model parameters shown in Table 3 are added to Equation (32), and the experimental results when the hydrostatic stresses are 10 MPa, 12 MPa, and 14 MPa are predicted. A comparison of the calculated results and the experimental ones is shown in Figure 7. Well-matching results are found, which indicates that coal permeability could be predicted reliably by our model.

Figure 8 shows the model calculation results under a hydrostatic stress of 8 MPa and a broader range of pore pressure levels, and Figure 9 shows the results of sample M1 under different hydraulic stresses. The permeability follows a hook-shaped path with increasing average pore pressure. Considering the gas slippage effect, the gas permeability will decrease when the pore pressure increases with a lower pore pressure range. However, the effects of gas slippage decline with increasing gas pressure [23] and approximately diminish when the gas pressure increases beyond 2.0 MPa. At this point, the mean free path of gas molecules (diameter of approximately 0.98 Å) is far lower than the aperture of coal cleats (3–40 um) [29]. Within the higher pore pressure range, the coal permeability is mainly influenced by the effective stress [8]. As shown in Figure 8 and Figure 9, the permeability increases with increasing pore pressure when the pore pressure changes within a higher scale.

3.2.2. Results Obtained When Neglecting the Gas Slippage Effect and Discussion

The permeability model that neglects the gas slippage effect (Equation (33)) is fitted with experimental results under the hydrostatic stress 8 MPa. The fitting process is shown in Figure 10. The permeability increases exponentially with an increase in pore pressure. Without considering the slippage effect, the permeability model cannot achieve better matching results from experimental data; therefore, the permeability model that does take into consideration the gas slippage effect is more accurate for coal with low permeability.

3.2.3. Comparison with Other Permeability Models

In our previous study [8], the McKee (1988) [18] and the Robertson (2006) [19] models were modified as follows: ① the effective stress coefficient in the effective stress item of the model equation is changed from 1 to β (0 < β < 1); and ② the gas slippage effect is considered.

The modified McKee permeability model is expressed as:

$$\frac{k_g}{k_{g0}}=\frac{p+b}{p_0+b_0}·\frac{p_0}{p}\exp \left\{-\frac{3C_{f0}}{\alpha }\left[1-e^{-\alpha \left(\Delta {\sigma }_t-\beta \Delta p\right)}\right]\right\}$$

(34)

and the modified Robertson and Christiansen permeability model:

$$\frac{k_g}{k_{g0}}=\frac{p+b}{p_0+b_0}·\frac{p_0}{p}\exp \left\{3C_{f0}\frac{1-e^{\alpha \beta \Delta p}}{-\alpha }+\frac{9}{{\varphi }_0}\left[\frac{1-2\nu }{E}\Delta p-\frac{S_{\max }{p}_L}{\left(p_0+p_L\right)}\ln \left(\frac{p+p_L}{p_0+p_L}\right)\right]\right\}$$

(35)

Figure 11, Figure 12 and Figure 13 compare the calculated results of the three permeability models under a hydrostatic stress of 12 MPa and the corresponding experimental data. All three permeability models achieve good matching with the experimental permeability data. For a wider pore pressure range, the permeability calculated with three models all follow a hook-shaped path with increasing pore pressure.

3.2.4. Model Verification with CO₂ Permeability Experimental Data

Coal differs from other porous media [45,46], and the gas permeability varies with the gases tested. The different adsorbability of N₂, CO_2, and CH₄ to the coal matrix results in a significant difference of gas permeability under the same pressure [40,47]. Pini et al. (2009) [48] observed that the swelling of coal due to CO₂ was larger than that due to N₂. Pan and Connell (2011) [17] measured bituminous coal swelling strains caused by CH₄, N₂, and CO₂ perpendicular and parallel to the bedding direction. The Gilson coal (bituminous coal) permeability was measured in the laboratory for pure CO₂ gas by Robertson (2005) [35], and their results are used to verify our permeability model in this paper. The mechanical properties and swelling parameters are directly obtained from Robertson (2005) [35], as shown in

Table 4. Mechanical properties and swelling parameters obtained from Robertson (2005) [35].

Parameters	Poisson’s Ratio	Elastic Modulus (GPa)	Langmuir Strain	Langmuir Stress (MPa)	Cleat Compressibility (MPa−1)
Values	0.35	1.38	0.015	3.83	0.041

. The modified McKee and Robertson models are also calculated for comparison. The model calculation results and laboratory experimental data are shown in Figure 14. Because different models require different parameters, the calculation results for the three permeability models vary greatly. However, a relatively good match is obtained between our model results and the experimental results.

3.3. Pore Pressure Remains Constant

The change in p is zero when gas pressure is unchanged, i.e.,

$$\Delta p=0$$

(36)

By substituting Equation (36) into Equation (25) and by adding Equation (27), we arrive at a permeability model with a pore pressure that remains constant during the experiments expressed as

$$\frac{k}{k_0}={\left[\frac{-\alpha {\varphi }_0\Delta {\sigma }_t\left(1-2v\right)-E{C}_0\left[1-\exp \left(-\alpha \Delta {\sigma }_t\right)\right]+E\alpha }{\alpha \Delta {\sigma }_t\left(1-2v\right)+E\alpha }\right]}^3$$

(37)

The permeability k in Equation (37) is the intrinsic permeability. Coal permeability under different hydrostatic and gas pressure levels is tested via N₂. Experimental results are published in our former research paper [8]. The intrinsic permeability is obtained by fitting the slippage effect (Equation (2)) with the laboratory permeability data, which are listed in

Table 5. Coal intrinsic permeability under different hydrostatic stresses (×10⁻² mD).

Samples	Hydrostatic Stress (MPa):
	8	10	12	14
M1	2.09	0.84	0.57	0.21
M2	3.06	1.26	0.67	0.26
M3	1.10	0.59	0.32	—

.

A comparison between the experimental data and the model calculation results is shown in Figure 15, which shows a good matching result. This validates the rationality of our model.

4. Conclusions

Based on the cubic law, which has been widely applied to describe permeability evolution with respect to porosity changes and considering gas slippage and matrix shrinkage/swelling due to gas desorption/adsorption, a new analytical model suited for permeability analysis of coal under different stress conditions is derived. To enhance its application, the permeability model is derived under constant hydrostatic stress and pore pressure, respectively. Comparisons between the calculated results and N₂ and CO₂ experimental permeability data show good matching results and thus indicate that coal permeability can be reliably predicted from our permeability model.

When the slippage effect is considered, the permeability follows a hook-shaped path with increasing average pore pressure. The gas slippage effect leads to a decline in permeability when the pore pressure increases within a lower pore pressure range. However, the effects of gas slippage decline with increasing gas pressure. Compared with an analytical permeability model that neglects the gas slippage effect, the model that does consider the gas slippage effect achieves better matching results from the experimental data. Therefore, the latter is more accurate for coal permeability analysis.