## 2 Methodology
### Equivalent Continuous Porous Medium Model
In the case of steady-state seepage, the relationship between the near-field dynamic micropermeability and macroscopic permeability of coal-rock mass under different influence functions is derived:  
\[k_{\text{b}}\big{[}x_{\text{i}},x_{\text{j}}\big{]}= \begin{cases}\frac{4k}{ \pi\rho_{\text{i}}^{2}}&(\omega=1)\\ \frac{12k}{\pi\rho_{\text{i}}^{2}}(1-\frac{\left\|\xi_{x_{\text{i}}\mathbf{x}_{ \text{j}}}\right\|}{\delta})&(\omega=1-\frac{\left\|\xi_{x_{\text{i}}\mathbf{x} _{\text{j}}}\right\|}{\delta})\\ \frac{2k}{\pi\rho_{\text{i}}^{2}}\frac{1}{\left\|x_{\text{i}\mathbf{x}_{\text{ j}}}\right\|}&(\omega=\frac{1}{\left\|\xi_{x_{\text{i}}\mathbf{x}_{ \text{j}}}\right\|})\end{cases}  \tag{20}\]On the basis of the above analysis, the basic near-field dynamic equations of the unidirectional flow process in isotropic porous uniform medium can be obtained:  
1. Uniform type (\(\omega\) = 1) \[\frac{\partial}{\partial\mathrm{t}}(\rho[\mathrm{x_{i}}]\varphi[ \mathrm{x_{i}}])=\int_{\mathrm{H_{x}}}\frac{\rho\left[\mathrm{x_{i}},\mathrm{x_{ j}}\right]}{\mu}\frac{4k}{\pi\delta^{2}}\frac{\Phi\left[\mathrm{x_{i}} \right]-\Phi[\mathrm{x_{i}}]}{\left\|\xi_{\mathrm{x_{i}}\mathrm{x_{j}}}\right\| ^{2}}dA_{\mathrm{x_{j}}}+R[\mathrm{x_{i}}]\] (21) 2. Linear model (\(\omega\) = 1 - \left\|\xi_{\mathrm{x_{i}}\mathrm{x_{j}}}\right\|/\delta\) \[\frac{\partial}{\partial\mathrm{t}}(\rho[\mathrm{x_{i}}]\varphi[ \mathrm{x_{i}}])=\int_{\mathrm{H_{x}}}\frac{\rho\left[\mathrm{x_{i}},\mathrm{x _{j}}\right]}{\mu}\frac{12k}{\pi\delta^{2}}(1-\frac{\left\|\xi_{\mathrm{x_{i} }\mathrm{x_{j}}}\right\|}{\delta})\frac{\[PERSON]\left[\mathrm{x_{j}}\right]- \[PERSON][\mathrm{x_{i}}]}{\left\|\xi_{\mathrm{x_{i}}\mathrm{x_{j}}}\right\|^{2}}dA _{\mathrm{x_{j}}}+R[\mathrm{x_{i}}]\] (22) 3. Countdown type (\(\omega\) = 1/\(\left\|\xi_{\mathrm{x_{i}}\mathrm{x_{j}}}\right\|\)) \[\frac{\partial}{\partial\mathrm{t}}(\rho[\mathrm{x_{i}}]\varphi[ \mathrm{x_{i}}])=\int_{\mathrm{H_{x}}}\frac{\rho\left[\mathrm{x_{i}},\mathrm{x _{j}}\right]}{\mu}\frac{2k}{\pi\delta}\frac{1}{\left\|\xi_{\mathrm{x_{i}} \mathrm{x_{j}}}\right\|}\frac{\Phi\left[\mathrm{x_{j}}\right]-\Phi[\mathrm{x_{ i}}]}{\left\|\xi_{\mathrm{x_{i}}\mathrm{x_{j}}}\right\|^{2}}dA_{\mathrm{x_{j}}}+R[ \mathrm{x_{i}}]\] (23)  
Through a Matlab program, a uniform isotropic porous homogeneous medium was selected as the object, and a numerical analysis of slow tensile load on microunits containing a single pore was carried out to verify the experimental results. As shown in Figure 1, a single circular pore micro-element is selected from the microscopic local area of an isotropic homogeneous medium. The origin of the \(xOy\) coordinate system for this micro-element is the center of the circular pore, with horizontal length \(L\) and vertical length \(W\). The vertical boundary is fixed, while the horizontal boundary is subjected to compression or tension loads, with no other fractures inside. The horizontal length \(L\) and vertical length \(W\) of the micro-element are both 50 mm, and the pore diameter \(D\) is 10 mm. Based on the properties of common coal seam surrounding rocks, the elastic modulus is 30 GPa, Poisson's ratio is 0.33, and the density is 2400 kg/m\({}^{3}\). For the boundary conditions, the vertical boundary is fixed, and a tensile load of 200 MPa is applied to the horizontal boundary. In terms of near-field dynamic discrete parameters, there are 100 material points in the \(x\) direction, 106 material points in the y direction (with 3 fixed material points at the top and bottom of the vertical boundary), and 1 material point in the z direction. The spacing \(\Delta\) between material points is 5 \(\times\) 10\({}^{-4}\) m, the volume \(\Delta V\) of a single material point is 1.25 \(\times\) 10\({}^{-10}\) m\({}^{3}\), and the virtual boundary layer volume \(\Delta V\delta\), which is the volume of the upper and lower fixed boundaries, is 3.75 \(\times\) 10\({}^{-8}\) m\({}^{3}\). The neighborhood range \(\delta\) should be greater than 3\(\Delta\), and can be set to 3.014\(\Delta\).