## 2 Materials and Methods
### Experimental Procedure
#### 2.2.3 Gas Seepage Tests of Stressed Coal Samples
Therefore  
\[2p\frac{\partial p}{\partial x}=\frac{\partial p^{2}}{\partial x}=const.= \frac{p_{1}^{2}-p_{2}^{2}}{L}=2\overline{p}\frac{p_{1}-p_{2}}{L} \tag{7}\]  
where \(\overline{p}\) represents the algebraic mean value of gas pressure, i.e., \(\overline{p}=\frac{1}{2}(p_{1}+p_{2})\). Then, substitute Equation (7) into Equation (5), and we obtain  
\[k=\frac{2\mu QpL}{A(p_{1}^{2}-p_{2}^{2})} \tag{8}\]  
According to the gas state equation, it is easy to obtain  
\[k=\frac{2\mu Q_{2}p_{2}L}{A(p_{1}^{2}-p_{2}^{2})} \tag{9}\]  
Because of \(pQ=p_{1}Q_{1}=p_{2}Q_{2}=\overline{p}\overline{Q}\). The above Equation (9) suggests the commonly used method for computing the permeability in steady-state tests of coal containing gas. Byintroducing the algebraic mean gas pressure, the calculated value becomes more stable and tends to approach a constant under different experimental conditions (when the structural characteristics of the coal remain unchanged). Therefore, the calculation of permeability in this paper is all conducted using this equation. Based on this, gas seepage tests of stressed coal samples were conducted via the self-designed Thermo-Hydro-Mechanical coupled triaxial experimental system, housed within the State key laboratory of coal mine disaster dynamics and control of Chongqing University. The experimental system is schematically shown in Figure 1, which comprises a power pump unit, compressed gas cylinder, test chamber, control unit, and flow measurement unit. Additionally, a measuring cylinder was used to confirm gas emission by observing bubbles. This experimental system accommodates a maximum axial load of 1000 kN, confining a pressure of 60 MPa and a gas pressure of 6 MPa, with a measurement accuracy of \(\pm 1\%\) for all three parameters. Axial loading employs displacement closed-loop control. The apparatus enables comprehensive evaluation of seepage in porous media under static/dynamic loading-unloading cycles, replicating complex geo-environmental conditions including variable in situ stresses, gas pressure, and temperatures. Both prepared raw coal samples and briquette coal samples shaped under different forming pressures were utilized for gas seepage tests, with various test conditions including axial stress, confining pressure, gas pressure, and gas type being set. The detailed test protocol can be found in Table 2, and the specific test steps are outlined as follows: \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Coal Sample**} & **Axial Stress** & **Confining Pressure** & **Gas Pressure** & **Gas Type** & **Sample Forming** \\  & \(\sigma_{1}\) (MPa) & \(\sigma_{2}\)=\(\sigma_{3}\) (MPa) & \(p\) (MPa) & **Gas Type** & **Pressure** \\  & & & & & \(\sigma_{m}\) (MPa) \\ \hline Raw coal samples & & 2, 4, 6 & & CO\({}_{2}\) & – \\  & & 4, 6, 8, 10 & 2, 4 & & N\({}_{2}\) & CH\({}_{4}\) & 50, 70, 100, 130 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Protocol of the seepage tests of gassy coal samples. Figure 1: Thermo–Hydro–Mechanical coupled tri-axial experimental system. 1 Position a coal sample in the test chamber, ensuring its end surfaces are in smooth pre-contact with the axial indenters. To facilitate gas flow exclusively along the axial direction within the coal sample, it should be encased cylindrically with heat shrink tubing and securely fastened with steel hoops. 2 Mount the extensometer to measure the coal sample's circumferential deformation and then seal the test chamber. 3 Initiate operation of both the power pump unit and the control unit, filling the test chamber with hydraulic oil to provide confining pressure. 4 Input target axial stress and confining pressure into the program control system based on test conditions, then load to reach these specified conditions. 5 With the loading state unchanged and the outlet sealed, inject test gas into the chamber for the specimens to adsorb until equilibrium is reached.