## 3 Theoretical Study
### Model Verification  
In order to verify the rationality of the permeability model, it was used to fit the experimental data. The rationality of the permeability model was ascertained through an analysis of the results. The basic parameters are listed in Table 3.  
The permeability data fitting included two parts. In the first part, the permeability model was used to fit the permeability data of four stress paths under a hydrostatic stress state. This included the fitting of permeability data where the gas pressures were the same and the average stress was increasing, and the fitting of permeability data where the average stresses were the same and the gas pressure was increasing. There were four fitting groups in each situation. In the second part, an initial condition was selected, and the permeability changes under the four stress paths were fitted. Here, the initial conditions chosen were an initial radial stress and a gas pressure of 4 MPa and 2 MPa, respectively. According to the previous analysis, the slow increase and sharp increase in the permeability were mainly due to the plastic damage and destructive cracks, respectively. Therefore, the fitting interval was from the initial hydrostatic permeability (permeability in the hydrostatic stress state) to the minimum permeability. The fitting results for the first part are shown in Figures 13 and 14.  
The fitting results showed that under the conditions where the gas pressures were the same and the average stress was increasing, the fitness values for the experimental data were 0.9725, 0.9612, 0.9545, and 0.9578. Under the conditions where the average stresses were the same and the gas pressure was increasing, the fitness values reached 0.9998, 0.9887, 0.9189, and 0.9963.  
In the second part, the variation in coal permeability under the four stress paths were fitted under the conditions that the radial stress was 4 MPa and the gas pressure was 2 MPa. The fitting results were shown in Figure 15.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Elastic modulus (MPa) & \(E\) & 2960 \\ Poisson’s ratio & \(v\) & 0.4 \\ Bulk modulus (MPa) & \(K\) & 4933.33 \\ Bulk modulus of coal skeleton (MPa) & \(K_{s}\) & 14,800 \\ Initial porosity & \(\phi_{0}\) & 0.04 \\ Initial permeability (mD) & \(k_{0}\) & 0.2 \\ Langmuir volumetric strain & \(\varepsilon_{L}\) & 0.004 \\ Langmuir pressure (MPa) & \(P_{L}\) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical parameters of coal samples.  
Figure 14: Fitting results of permeability data under hydrostatic stress state: (**a**) CTC; (**b**) ATC; (**c**) UCP; (**d**) AUCP. (\(\sigma_{3}\) = 4 MPa).  
Figure 13: Fitting results of permeability data under hydrostatic stress state: (**a**) CTC; (**b**) ATC; (**c**) UCP; (**d**) AUCP. (P = 2 MPa).  
From the fitting results in Figure 15, under the four stress paths, the fitness values for the experimental data reached 0.9873, 0.9695, 0.9395, and 0.9459.  
The fitting results for the two parts showed that the theoretical calculations were in good agreement with the experimental data, and the overall fitting result was relatively good, which proved that the permeability model was reasonable.