## 2 Materials and Methods
### Governing Equations  
In the elastic and isotropic model of FLAC, strain increments generate stress increments according to the linear and reversible law of [PERSON]:  
\[\Delta\sigma_{ij}=2G\Delta\varepsilon_{ij}+\alpha_{2}\Delta\varepsilon_{kk} \delta_{ij} \tag{1}\]  
where \(\sigma_{ij}\) is the stress, \(\varepsilon_{ij}\) is the strain, \(\delta_{ij}\) is the Kroenecker delta symbol, and \(\alpha_{2}\) is a material constant related to \(K\) and \(G\), as  
\[\alpha_{2}=K-\frac{2}{3}G \tag{2}\]  
where \(K\) is the bulk modulus, \(K=E/3(1-2v)\), \(G\) is the shear modulus, \(G=E/2(1+v)\), \(E\) is [PERSON]'s modulus, and \(v\) is [PERSON]'s ratio. New stress values are then obtained from the relation:  
\[\sigma_{ij}^{N}=\sigma_{ij}+\Delta\sigma_{ij} \tag{3}\]  
The permeability of the coal-rock mass presents extreme heterogeneity changed by numerous factors, among which the surrounding stress and gas pressure induced by the overlying strata dominate the permeability variation [7; 26].  
In [PERSON]'s theory, effective stress is dominated by the surrounding stress and the gas pressure, which can be expressed as follows:  
\[\sigma^{\prime}=\sigma-p \tag{4}\]  
where \(\sigma^{\prime}\) is the effective stress, \(\sigma\) is the ground stress, and \(p\) is the gas pressure.  
The permeability \(k\) decreases rapidly with increased ground stress under the unchanged gas pressure in a true triaxial permeameter, during which the permeability can be expressed by the exponential equation as follows:  
\[k=k_{0}e^{-\beta\sigma^{\prime}} \tag{5}\]  
where \(k_{0}\) is the permeability of the coal sample under the ideal condition without ground stress, and \(\beta\) is the experimental coefficient. Thus, reducing the surrounding stress on the coal seam is considered a reliable way to enhance the coal seam permeability.  
The permeability of the coal seam presents a difference in the vertical layer orientation and the parallel layer orientation as follows:  
\[\begin{cases}k_{\text{\tiny{p}}}=k_{\text{\tiny{p0}}}e^{-\beta_{\text{\tiny{p }}}\sigma^{\prime}}\\ k_{\text{\tiny{p}}}=k_{\text{\tiny{p0}}}e^{-\beta_{\text{\tiny{p}}}\sigma^{ \prime}}\end{cases} \tag{6}\]  
where \(k_{\text{\tiny{p}}}\) and \(k_{\text{\tiny{p}}}\) are the permeability parallel and vertical to the coal bedding, \(k_{\text{\tiny{p0}}}\) and \(k_{\text{\tiny{p0}}}\) are the permeability under the ideal condition without ground stress, \(\beta_{\text{\tiny{p}}}\) and \(\beta_{\text{\tiny{p}}}\) are the experimental coefficients obtained by mechanical experiments. The coal seam permeability of Equation (5) transforms into:  
\[k=\sqrt{k_{\text{\tiny{p}}}\cdot k_{\text{\tiny{p}}}} \tag{7}\]  
The coal seam permeability, when combining Equations (6) and (7), is expressed as follows:  
\[k=\sqrt{k_{\text{\tiny{p0}}}e^{-\beta_{\text{\tiny{p0}}}\sigma^{\prime}}\cdot k _{\text{\tiny{p0}}}e^{-\beta_{\text{\tiny{p0}}}\sigma^{\prime}}} \tag{8}\]  
We conducted the permeability experiment of the specimen made by the target coal seam in the laboratory (Figure 2) to investigate the permeability in the vertical and parallel directions to the coal layer orientation. We obtained the permeability calculation equation suitable for the target coal seam as follows: