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
where
kg 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
The average pore pressure p is determined by the inlet and outlet gas pressures, i.e., , where pin and pout 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 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
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.
Combining Equations (3) and (4) results in
It could also be written in the form of the fracture width change and matrix dimension change
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
where
E is the elastic modulus;
v is Poisson’s ratio; and
σx,
σy, and
σz 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
In recognizing the strains of a matrix block as the change in dimension divided by the original dimension, strains can also be expressed as
Substituting Equation (9) into Equation (8) produces
The increase in the fracture opening due to a single effect of matrix shrinkage can be expressed as [
19]
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]
where
Smax and
pL are sorption constants.
Smax is the Langmuir strain, which represents the matrix strain at infinite adsorption pressure.
pL is the Langmuir pressure, which represents the pressure when the adsorption strain is
Smax/2.
Similar to Equation (9), the swelling strain induced by gas adsorption can be written as
Substituting Equation (13) into Equation (12) produces
The increase in the fracture opening due to a single effect of matrix swelling resulting from gas adsorption can be expressed as [
19]
The effective stress of a porous medium is expressed as
where
σ is the effective stress;
σt 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]
where
Kv is the bulk modulus and
Ks 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,
The cleat compressibility
Cf was defined by Amyx et al. (1960) [
41] as Equation (19)
Combining Equations (4), (18) and (19) produces
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
Similarly, the actual deformation of a fracture opening can be obtained by combining Equations (11), (15) and (20).
Substituting Equations (21) and (22) into Equation (6) yields an equation for permeability changes with respect to other parameters:
where
Combining Equations (23) and (4) produces
McKee et al. (1988) [
18] noted that the cleat compressibility
Cf is a function of the effective stress. According to Mckee and coworkers,
Cf can be expressed as
where
C0 is the cleat compressibility, and α is the declining rate of cleat compressibility.
The cleat compressibility
Cf expressed as Equation (26) is modified as Equation (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
The subscript “0” represents the initial state of each parameter.
Combining Equations (25) and (28) produces
Equation (29) is the expression of permeability model that considers the gas slippage effect.
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.