## 3 Materials and Methods
### Data Analysis  
Coal permeability was estimated utilizing known models that describe the stress-permeability relationship. The [PERSON] et al. (1992) model was designed to analyze laboratory coal permeability, measured under constant confining pressure conditions [33]. The absolute coal permeability under constant confining pressure conditions is determined from the following expression (Equation (2)):  
\[k=k_{0}\cdot e^{-3C_{f}(P_{c0}-P_{0})} \tag{2}\]  
where \(k\) is the coal permeability, mD; \(k_{0}\) is the permeability at the reference confining pressure \(P_{c0}\), mD; \(P_{c}\) is the confining pressure, MPa; and \(P_{c0}\) is the reference confining pressure, MPa. [PERSON] and [PERSON] (2006) developed a model that described the dependence of coal permeability on stress conditions. The model was designed to analyze the laboratory coal permeability obtained under biaxial or hydrostatic confining pressures [34]. Coal compressibility is not constant [27]. For the conditions of a variable compression pressure and constant pore pressure, the coal permeability is defined as (Equation (3))  
\[k=k_{0}\cdot e^{-3C_{0}\cdot\frac{1-e^{-4(P_{c}-P_{0})}}{a}} \tag{3}\]  
A coal seam is typically described by a matchstick geometry model in order to develop theoretical models that address stress--permeability and porosity--permeability relationships. For the isotropic case, the fracture width and coal matrix block size are the same  
Figure 3: A simplified scheme of the studied fractured coal samples, the direction of gas filtration in laboratory tests, and the location of surfaces along which the coal microstructural analysis was performed. **—**—plane perpendicular to the face and butt cleat direction, \(\beta\)—plane of the core end surface. \begin{table}
\begin{tabular}{c c} \hline \hline
**Range of Measurement of Linear Dimensions, \(\mu\)m** & **From 0.5 to 2000** \\ \hline The relative value of the standard deviation of the random component & \\ of the error (considering the contribution of the analyzer software), \%: & \\ -with an increase up to \(\times\)500 inclusive & 0.2 \\ -with an increase over \(\times\)500 & 0.4 \\ Relative error of measurement (considering the contribution of the & \\ analyzer software), \%: & \\ -with an increase up to \(\times\)500 inclusive & \(+/-0.25\) \\ -with an increase over \(\times\)500 & \(+/-0.65\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The main characteristics of the Mineral C7 analyzer [32]. for any coal clear system. For the anisotropic case, these properties are not the same for all coal clear systems. In the anisotropic case, a coal seam is described by an impermeable high (coal matrix) with two regular vertical fracture sets (face and butt cleats). Within a single set, every fracture has the same orientation and width, and there is a constant spacing between fractures. Simultaneously, the fracture width (\(a_{f}\), \(a_{b}\)) and spacing between fractures (\(A_{f}\), \(A_{b}\)) are not consistent among the various cleat systems. The coal permeability was estimated for two sections perpendicular to the face and butt cleat direction: perpendicular to the butt cleat direction (\(k_{f}\)) and perpendicular to the face cleat direction (\(k_{b}\)). Coal cleats are much more permeable than the coal matrix, so the coal matrix permeability is neglected in the calculations [9]. According to earlier works [18; 35], the permeability of a parallel regular fracture system is determined as (Equation (4))  
\[k_{fr}={a_{fr}}^{3}/12A_{fr} \tag{4}\]  
where \(k_{fr}\) is the permeability of the fracture set, D; \(a_{fr}\) is the fracture width, \(\mu\)m; and \(A_{fr}\) is the spacing between adjacent fractures, \(\mu\)m.  
Considering the accepted coal seam model, the permeabilities perpendicular to the butt and face cleat directions are calculated as (Equation (5)) [35]  
\[ \begin{array}{l}k_{f}={a_{f}}^{3}/12A_{f}\\ k_{b}={a_{b}}^{3}/12A_{b}\end{array}