## 2 Numerical Simulation Modeling and Boundary Conditions
### Physical Model  
Geological data from a mining face in Shanxi were collected and analyzed. Using the Fluent 17.0 software, a three-dimensional seepage model of a water-injected coal seam as a porous medium was established. We selected a rectangular parallelepiped with dimensions of 100 m \(\times\) 150 m \(\times\) 3 m as the calculation area. A hole with a length of 85 m, a diameter of 75 mm, and a sealed length of 10 m was modeled with the starting face of the injection hole centered 1.5 m above the bottom of the coal seam (i.e., the center of the borehole is located at the center of the side of the coal seam). The mesh software was used to divide the calculation area into an unstructured tetrahedral mesh. According to calculations, the total number of nodes in the unstructured tetrahedral mesh is 49,312 and the total number of elements is 255,360. To simplify the simulation process, the mesh was divided from the radial plane of the borehole as shown in Figure 1:  
[PERSON]'s law: [PERSON]'s law applies to low-speed fluid flow in porous media. The expression for [PERSON]'s law in one-dimensional form is  
\[q=vS=-\frac{\pi r^{2}K}{\mu}\frac{\partial p}{\partial\chi}=\frac{\pi r^{2}K(p _{0}-p_{1})}{\mu L} \tag{1}\]  
In the equation, v represents the seepage velocity, which is the volume of fluid passing through a unit cross-sectional area per unit time, expressed in m\({}^{3}\)/(s m\({}^{2}\)); q is the water injection rate, in m\({}^{3}\)/s; S is the cross-sectional area, in m\({}^{2}\); r is the radial distance from the center of the borehole, in cm; p = p(r,t) is the fluid pressure at a distance r (m) from the borehole and at time t (s), in atm; p\({}_{1}\) is the initial fluid pressure in the original formation, in atm; K is the permeability of the formation, in Darcy; L is the length, in cm (\(\mu\), pa*s).  
Equation of state: The equation of state describes the relationship between the density \(\rho\) of a substance and its pressure p and temperature T. Fluid seepage in the seam can be regarded as taking place under isothermal conditions, so the equation of state only needs to state the relationship between fluid and pressure. The equation of state for a fluid is  
\[\rho=\rho_{0}e^{C(\rho-\rho_{0})} \tag{2}\]  
where C is the compression coefficient, in atm\({}^{-1}\); \(\rho\) is the density of the fluid, in g/cm\({}^{3}\), \(\rho_{0}\) is the value of \(\rho_{0}\) at P = P\({}_{0}\).  
Continuity equation: the continuity equation of radial flow is  
\[\frac{1}{r}\frac{\partial(rv_{t}\rho)}{\partial r}=\frac{\partial(\varphi \rho)}{\partial t} \tag{3}\]  
where the porosity of the formation is written as a %; \(\rho\) is the density of the fluid, in g/cm; \(v_{t}\) is the radial percolation volume velocity of the inflow unit, in cm\({}^{3}\)/(s cm\({}^{2}\)); t is the time, in second; and r is the distance from the borehole axis along the radial direction to a point, in cm.  
Figure 1: Model and mesh division.  
Differential equations: Based on [PERSON]'s law, the equation of state, and the continuity equation, the basic differential equations of coal-rock seepage are deduced.  
\[\frac{\partial^{2}p}{\partial r^{2}}+\frac{1}{r}\frac{\partial p}{\partial r}= \frac{\phi\mu C_{t}}{K}\frac{\partial p}{\partial t} \tag{4}\]  
Boundary conditions: Assume that there is a borehole in an infinite homogeneous medium, and that the borehole is filled with water at the moment t = 0 (s), at a constant injection pressure Pw (atm); there is no liquid pressure (zero) in the whole medium before the injection. These conditions can be represented by the following initial and boundary conditions:  
\[p(t=0,r)=0p(t=0,r=\infty)=0p(t=0,r=r_{w})=p_{w} \tag{5}\]  
Theoretical solutions of the basic differential equations were obtained by the Polubarinova-Kochina method according to the initial and boundary conditions:  
\[p=\frac{p_{w}}{\text{Ei}\left(-\frac{\phi\mu C_{r}r_{w}^{2}}{4 Kt}\right)}\text {Ei}\left(-\frac{\phi\mu C_{r}r_{w}^{2}}{4 Kt}\right) \tag{6}\]