## 3 Mathematical Equations of the Conceptual Model
### Governing Equations of Gas Flow in the Coal Matrix and Cleats  
The gas flow in coal matrix and cleat system can be expressed with mass balance equations as:  
\[\frac{\partial m}{\partial t}+\
abla\cdot(\rho_{g}\mathbf{q_{g}})=Q_{S} \tag{7}\]  
where \(\rho_{g}\) is the gas density, \(\mathbf{q_{g}}\) the [PERSON]'s velocity vector, \(Q_{S}\) is the gas source, t is the time, and \(m\) is the gas content, including free-phase gas and adsorbed gas. Based on the assumption that gas sorption takes places only in the matrix system, the gas content in the matrix and cleats is defined as:  
\[m_{m}=\phi_{m}\rho_{gm}+\rho_{n}\rho_{c}\frac{V_{L}p_{m}}{p_{m}+P_{L}} \tag{8}\]  
\[m_{f}=\phi_{f}\rho_{gf} \tag{9}\]  
where \(\rho_{gm}\) and \(\rho_{gf}\) are the gas density in the matrix and cleats, respectively, \(\phi_{m}\) is the porosity of the matrix system, \(\phi_{f}\) is the porosity of the cleat system, \(\rho_{n}\) is the gas density under standard conditions; and \(\rho_{c}\) is the coal density. The density of an ideal gas is calculated by:  
\[\rho_{g}=\frac{M_{g}}{RT}p \tag{10}\]  
where \(M_{g}\) is the molecular weight of the gas and \(R\) is the universal gas constant.  
The mass between the matrix and cleat system is conditioned by a smooth pressure gradient that can be written as [35]:  
\[Q_{S}=\frac{\rho_{gm}k_{m}\psi}{\mu}\Big{(}p_{m}-p_{f}\Big{)} \tag{11}\]  
where \(\mu\) is dynamic viscosity of CH\({}_{4}\), \(k_{m}\) is the matrix permeability, \(\psi=4(1/a_{x}^{2}+1/a_{y}^{2})\) is the shape factor, and \(a_{x}\) and \(a_{y}\) are the matrix spacing.  
Darcy's law is used to describe gas flow in coal seams. The Darcy velocity is given by:  
\[\mathbf{q_{g}}=-\frac{k}{\mu}\
abla p \tag{12}\]  
where \(k\) is the permeability of the coal and gravity is ignored. Substituting Equations (8)-(12) into Equation (7) produces the final governing equations of gas flow in the matrix and cleat system:  
\[\frac{\partial m_{m}}{\partial t}+\
abla\!\left(-\frac{k_{m}}{\mu}\rho_{gm} \
abla p_{m}\right)=-\frac{\rho_{m}k_{m}\psi}{\mu}\Big{(}p_{m}-p_{f}\Big{)} \tag{13}\]\[\phi_{f}\frac{\partial\rho_{fg}}{\partial t}+\rho_{gf}\frac{\partial\phi_{f}}{ \partial t}+\
abla\!\left(-\frac{k_{f}}{\mu}\rho_{gf}\
abla p_{f}\right)=\frac{ \rho_{m}k_{m}\psi}{\mu}\!\left(p_{m}-p_{f}\right) \tag{14}\]  
The dynamic porosity for the matrix can be expressed as [22]:  
\[\phi_{m}=\frac{1}{1+S}[(1+S_{0})\phi_{m0}+\alpha(S-S_{0})] \tag{15}\]  
where \(S=\varepsilon_{v}+\frac{p_{m}}{K_{s}}-\varepsilon_{s}-\alpha_{T}T\), \(S_{0}=\varepsilon_{v0}-\frac{p_{m0}}{K_{s}}-\varepsilon_{L}\frac{p_{m0}}{p_{ m0}+P_{L}}-\alpha_{T}T_{0}\).  
The cubic relationship of permeability and porosity is widely used for the matrix [20].  
\[\frac{k}{k_{0}}=\left(\frac{\phi}{\phi_{0}}\right)^{3} \tag{16}\]  
where \(\phi_{0}\) and \(k_{0}\) are the initial porosity and initial permeability of the cleat system, respectively.  
We substitute Equation (15) into Equation (16) to obtain:  
\[k_{m}=k_{m0}\!\left(\frac{1}{1+S}\!\left[(1+S_{0})+\frac{\alpha}{\phi_{m0}}( S-S_{0})\right]\right)^{3} \tag{17}\]  
where \(k_{m0}\) is the initial matrix permeability at the initial pressure \(p_{m0}\) and matrix porosity \(\phi_{m0}\).  
We consider the permeability anisotropy in terms of swelling/shrinkage, thermal expansion, and coal deformation for a two-dimension case with two orthogonal sets of fractures. Thus, the directional permeability \(k_{fx}\) and \(k_{yy}\) of the cleat system becomes [24]:  
\[\frac{k_{fi}}{k_{f0}}=\left[1+\frac{2(1-R_{m})}{\phi_{f0}}(\Delta\varepsilon_{ j}-\frac{1}{3}\alpha_{T}\Delta T-\frac{1}{3}\Delta\varepsilon_{s})\right]^{3},i\
eq j \tag{18}\]  
where \(i,j=x\), \(y\) for two-dimensional case, \(R_{m}=E/E_{S}\) is elastic modulus reduction ratio and \(\phi_{f0}\) and \(k_{f0}\) are the initial porosity and initial permeability of the cleat system, respectively.