## 2 Materials and Methods
### Experimental Principle and Scheme  
Suppose there is a rock with a cross-sectional area A and a length L, which is clamped in the core holder, as shown in Figure 3. The flow rate Q is measured by passing a fluid having a viscosity of \(\mu\) through the center of the rock at a pressure difference \(\Delta\)P. The experiment proves that the volume flow rate Q through the core per unit time is proportional to the pressure difference \(\Delta\)P core cross-sectional area A and inversely proportional to the core length L and the viscosity of the fluid \(\mu\):  
\[\text{Q}\propto\frac{\text{Q}}{\mu}\frac{\Delta\text{P}}{\text{L}}\text{ or Q}=\text{K}\frac{\text{Q}}{\mu}\frac{\Delta\text{P}}{\text{L}}. \tag{1}\]  
Equation (1) is the so-called \"Darcy equation.\" It can be seen from the Equation (1) that A and L are the geometrical dimensions of the rock and \(\Delta\)P is the external condition. When the external conditions, square sizes, and fluid properties are constant, the fluid throughput Q depends on the proportionality constant K. We refer to K as the permeability of the rock, then Equation (1) can be rewritten as:  
\[\text{K}=\frac{\text{Q}\mu\text{L}}{\Delta\text{AP}} \tag{2}\]  
The previous discussion was based on the incompressible fluid (liquid). The gas permeability we designed was based on the compressed fluid (liquid), so the \"Darcy equation\" needs to be modified to apply it. Figure 3: Permeability calculation model. Figure 2: Three-axis creep-seepage test system for coal and rock. (**a**) Image of the triaxial cell; (**b**) schematic diagram of the three-axis creep-seepage test system. It is well known that the characteristics of compressible gas are that when the pressure increases, the fluid can be compressed; when the pressure is lowered, the fluid expands; when the temperature is constant, the expansion of the fluid obeys [PERSON]'s law. If considering the simplest flat line seepage, the inlet pressure is set as P\({}_{1}\), and the outlet pressure is set as P\({}_{2}\). When the pressure changes from P\({}_{1}\) to P\({}_{2}\), the volume of the gas necessarily changes, so the flow rate also varies. Therefore, it is necessary to consider substituting the average volumetric flow rate into the Darcy equation. If the gas expansion is regarded as an isothermal process, according to [PERSON]'s law:  
\[\mathrm{Q_{1}P_{1}=Q_{2}P_{2}\cdots=P_{0}Q_{0}=\overline{PQ}} \tag{3}\]  
\[\overline{Q}= \frac{\mathrm{P_{0}Q_{0}}}{\overline{P}}\ \ \overline{P}=\frac{ \mathrm{P_{1}+P_{2}}}{2} \tag{4}\]  
\[\overline{Q}= \frac{2\mathrm{P_{0}Q_{0}}}{\mathrm{P_{1}+P_{2}}} \tag{5}\]  
where: \(\overline{P}\) is the mean pressure, MPa; \(\overline{Q}\) is the average volume flow under average pressure, mL/s; P\({}_{0}\) is the atmospheric pressure, MPa; Q\({}_{0}\) is the gas flow at atmospheric pressure, mL/s. From the above analysis, the correction of the Darcy formula of the compressible fluid is:  
\[\mathrm{K=}\frac{2\mathrm{P_{0}Q_{0}u_{g}L}}{\mathrm{A}\left(\mathrm{P_{1}^{2 }-P_{2}^{2}}\right)} \tag{6}\]  
where \(\mathrm{\mu_{g}}\) is the viscosity of the gas, Pa s.  
To simulate the permeability evolution with repeated increase and decrease of the stress on the post-strength coal seam in the process of coal mining, the test scheme adopted in this test was as follows: When the gas pressure was 0.7 MPa (this value was based on the geological conditions of the Da-ping Coal Mine), the confining pressure was 2 MPa, 3 MPa, and 4 MPa, and the axial stress was loaded until the coal samples were damaged. Then, the permeability evolution in the process of unloading axial stress and confining pressure and the permeability evolution in the process of loading axial stress to the secondary damage were measured. To make the experiment method clearer, we list the experimental scheme in Table 1. It should be noted that to determine the permeability of coal samples, we had to measure the gas flow rate.