## 1 Introduction
### Determination of Numerical Model and Simulation Parameters for Coal Samples Containing Gas  
(1) Mathematical model of failure, deformation, and the failure of gas-containing coal  
In the RFPA\({}^{\text{2D}}\) gas plate software, assuming that the failure and deformation failure of gas-containing coal is the test of monomer coal, the permeability-damage coupling equation [39] can be adopted, and the elastic modulus of the damage element can be espressed as follows  
\[E=(1-D)E_{0} \tag{7}\]  
Figure 4: Fitting curves of the mechanical properties of coal samples under different gas pressures. (**a**) Compressive strength and gas pressure; (**b**) elastic modulus and gas pressure; (**c**) Poisson’s ratio and gas pressure. where \(E\) is the elastic modulus of the damaged element; \(E_{0}\) is the elastic modulus of the element without damage; \(D\) is the damage variable. In addition, these parameters are assumed scalars. The Moore-Coulomb criterion is adopted for the coal failure criterion, i.e.,  
\[\sigma_{1}-\sigma_{3}\frac{1+\sin\phi}{1-\sin\phi}\geq f_{c} \tag{8}\]  
where \(\sigma_{1}\) is the maximum principal stress; \(\sigma_{3}\) is the minimum principal stress; \(\varphi\) is the Angle of internal friction; \(f_{c}\) is uniaxial compressive strength. When the shear stress reaches the more-Coulomb loss threshold, the damage variable \(D\) is expressed as follows  
\[D= \begin{cases}\quad 0,\;\varepsilon<\varepsilon_{c0}\\ 1-\frac{f_{c}}{\varepsilon_{0}\varepsilon},\;\varepsilon_{c0}\leq\varepsilon \end{cases}  \tag{9}\]  
where \(f_{c}\) is the residual compressive strength; \(\varepsilon\) is the dependent variable; \(\varepsilon_{c0}\) is the maximum compressive strain. Damage will cause the gas permeability coefficient of coal material to increase sharply, that is, the change of the gas permeability coefficient can be described as follows  
\[\lambda= \begin{cases}\lambda_{0}\text{e}^{-\beta(\sigma_{1}-ap)},&D=0\\ \xi\lambda_{0}\text{e}^{-\beta(\sigma_{1}-ap)},&D>0\end{cases}  \tag{10}\]  
where \(\lambda\) is the air permeability coefficient; \(\lambda_{0}\) is the initial air permeability coefficient; \(\xi\) is the increasing rate of permeability coefficient; \(\alpha\) is gas pressure coefficient; \(\beta\) is permeability coupling coefficient. (2) Establishment of numerical model for deformation and failure of gas-containing coal  
The stress-strain characteristics and mechanical parameters of coal samples under different gas pressures in the process of loading failure were studied by using the RFPA2D numerical simulation software. The main variable controlled by numerical simulation was gas pressure, and the model of coal samples containing gas was established, as shown in Figure 5.  
(3) The numerical simulation process, initial and boundary conditions of deformation, and failure of gas-containing coal  
Figure 5: Initial model of coal sample containing gas. It can be seen from Table 3 that with the increase in gas pressure, the loading force (compressive strength) of the coal sample decreases from 792.1 N to 565.5 N, and its compressive strength decreases by 28.61%. The loading force (compressive strength) of the coal sample is negatively correlated with gas pressure, and the greater the gas pressure, the smaller the loading force required for the coal sample to reach peak failure. With the increase in gas pressure, the elastic modulus of the coal sample decreases from 10.963 MPa to 8.865 MPa, and its elastic modulus decreases by 19.14%. There is a negative correlation between the elastic modulus of coal sample and gas pressure, and the greater the gas pressure, the smaller the elastic modulus of coal sample. This is consistent with the test's results.