## 2 Methodology
### Discrete Fracture Network Model  
Based on peridynamics, the fracture is abstracted as a discrete network with a certain length, tendency, dip angle, and width, as shown in Figure 3.  
The single micro-fracture of coal-rock mass is simplified into a two-parallel plate structure, and the fluid flow in the fracture follows the Navier--Stokes (N--S) equation [14; 15; 16]:  
\[\frac{\partial\mathrm{u_{i}}}{\partial\mathrm{t}}=F_{\mathrm{i}}-\frac{1}{ \rho}P_{\mathrm{i}}+\
u u_{i,j} \tag{24}\]  
Figure 1: Schematic of calculation model of pore microunit in isotropic homogeneous medium. Figure 3: Schematic of plane fissure water flow. Figure 2: Verification results of numerical simulation of pore microunit deformation at different synchronous displacements. In the formula, \(u_{i}\) represents the component of fluid velocity, \(F_{i}\) represents the external force, \(\rho\) represents the density of fluid, \(P_{i}\) represents the component of fluid pressure, and \(\
u\) represents the kinematic viscosity coefficient of fluid. When the fluid velocity in the fracture is small, the fluid motion state can be regarded as laminar flow. Since the width of the fracture is constant, the velocity \(u_{x}\) at each point in the \(x\) direction is constant. At the same time, the velocity is low, and the velocity head can be ignored. The total head is the sum of the pressure head and the position head, so the above formula is  
\[\frac{\mathrm{d}^{2}u_{x}}{\mathrm{d}z^{2}}=\frac{1}{\rho v}\frac{\mathrm{d}p }{\mathrm{d}x}=\frac{g}{v}\frac{\mathrm{d}p}{\mathrm{d}x} \tag{25}\]  
In the water flow of parallel plate fissure, when \(z=\pm d/2\) and \(u_{x}=0\), the velocity distribution can be obtained in the following form:  
\[u_{x}=\frac{g(d^{2}-4z^{2})}{8v}J \tag{26}\]  
In the formula,  
\(J=-\frac{\mathrm{d}h}{\mathrm{d}x}\) represents the hydraulic gradient. The flow through the fracture can be integrated to obtain  
\[q=\int\limits_{-\dfrac{d}{2}}^{\dfrac{d}{2}}u_{x}\mathrm{d}z=\frac{gd^{3}J}{1 2v} \tag{27}\]  
Rewrite the above equation in the form of [PERSON]'s law:  
\[q=k_{f}\frac{gdJ}{v} \tag{28}\]  
In the formula,  
\(k_{f}\) is the permeability of the fracture, which is proportional to the width of the fracture. \[k_{f}=\frac{d^{2}}{12} \tag{29}\]  
Based on the law of conservation of mass, the basic equations of near-field dynamics of seepage in discrete fracture networks can be derived:  
(1) Uniform type (\(\omega=1\))  
\[\frac{\partial\mathrm{u_{i}}}{\partial\mathrm{t}}=F_{\mathrm{i}}-\frac{1}{ \rho}P_{\mathrm{i}}+\
u u_{i,j} \tag{24}\]0  
(2) Linear model (\(\omega=1-\left\|\xi_{\mathrm{x}_{i}\mathrm{x}}\right\|/\delta\))  
\[\frac{\partial\mathrm{u_{i}}}{\partial\mathrm{t}}=F_{\mathrm{i}}-\frac{1}{ \rho}P_{\mathrm{i}}+\
u u_{i,j} \tag{24}\]1  
(3) Countdown type (\(\omega=1/\left\|\xi_{\mathrm{x}_{i}\mathrm{x}_{i}}\right\|\))\[\frac{\partial\mathrm{u_{i}}}{\partial\mathrm{t}}=F_{\mathrm{i}}-\frac{1}{ \rho}P_{\mathrm{i}}+\
u u_{i,j} \tag{24}\]2  
In the formula,  
\(P_{f}\) represents the fluid density function, \(\varphi_{f}\) represents the ratio of fracture space volume to particle volume, \(\Phi_{f}\) represents the water head of particles on the fracture surface, \(\mu_{f}\) represents the fluid viscosity, \(k_{f}\) represents the fracture permeability, and \(R_{f}\) represents the source and sink terms of the fracture. The uniform network discrete fracture seepage model was selected as the object, and the numerical analysis of the unit with a single fracture in the interior was verified by experimental verification under the action of a rapid stretching load. As shown in Figure 4, a micro-element containing a single fracture is selected from the microscopic local area of a two-dimensional network's discrete fracture seepage model. The origin of the \(xOy\) coordinate system for this micro-element is the midpoint of the contained fracture. The \(x\) direction is horizontally 50 mm long, and the \(y\) direction is vertically 50 mm long.