## 5 The Initial Water Inrush Coefficient of Inter-Layer Rock Is: K=\(K_{0}\)
### Solution of Fluid-Solid Coupling Model of Inter-Layer Rock
#### 5.1.1 Solution of Flat Plate Flows Water Inrush Model  
The unidirectional flow of incompressible fluid water in the boundary waters inrush fracture can be expressed by mass conservation and the N-S equation. The N-S equation includes fluid pressure, viscous force, body force, unsteady inertia force, and convective inertia force. It is generally believed that the flow of common fluid in the fracture is approximately steady motion. The body force is equivalent to the pressure on calculation, and the unsteady inertia force is neglected. The equation is simplified as:  
\[\frac{\partial u}{\partial x}+\frac{\partial y}{\partial v}=0, \tag{23}\]  
\[\rho\bigg{(}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \bigg{)}=-\frac{\partial p}{\partial x}+\mu\bigg{(}\frac{\partial^{2}u}{ \partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\bigg{)}, \tag{24}\]  
Figure 16: The flow chart of solving the water inrush model of inter-layer rock. \[\rho\bigg{(}u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}\bigg{)}=- \frac{\partial p}{\partial y}+\mu\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}+ \frac{\partial^{2}v}{\partial y^{2}}\bigg{)}. \tag{25}\]  
In the formula, \(u,v\) is the flow velocity of water in the \(x,y\) direction of the fracture; \(\mu\) is the viscosity of fluid water; \(\rho\) is the density of water; \(p\) is pressure. Taking the water-inrush fracture boundary in Figure 17, the fracture surface of the water-inrush fracture is a rough surface, so it is approximately regarded as symmetrical distribution; only half of the water-inrush fracture surface is discussed. In Figure 17, \(b_{i}\) is the distance from any point on the boundary water inrush fracture surface to the middle line of the fracture, \(i=1,2,3,\ldots\ldots,n\). If the water inrush fracture width \(e\) can be regarded as the mean of all fracture widths, the roughness fluctuation of the water inrush fracture surface \(b_{i}\) is much smaller than the width of the water inrush fracture; that is,  
The flow velocity \(v\) of water in the \(y\) direction of the boundary water irnush fracture is much smaller than the flow velocity \(u\) in the \(x\) direction; that is, \(v\ll u\), so the flow velocity \(v\) in the \(y\) direction can be ignored. Equation (25) can be transformed into  
\[\frac{\partial p}{\partial y}=0,\text{ as}p=p(x). \tag{30}\]  
Based on the ratio of inertial force and viscous force in the analysis Formula (25) of the N-S mass conservation equation, we have  
\[\frac{\left|\text{inertia force}\right|}{\left|\text{cohesion}\right|}=\frac{ \rho\left|u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right| }{\mu\left|\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y ^{2}}\right|}=\frac{\rho\left|u\frac{\partial u}{\partial x}\right|}{\mu\left| \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}} \right|}\sim\frac{\rho}{\mu}\frac{U\frac{U}{U}}{\frac{U}{U^{2}}+\frac{U}{c^{2 }}}=\frac{\rho\text{I}e}{\mu}\frac{\delta}{\delta^{2}+1}=\text{Re}\delta \tag{31}\]  
where \(\text{Re}\) is the Reynolds coefficient. When water flows in the boundary water irnush fracture, \(\text{Re}=\rho\text{I}e/\mu\). If Equation (30) meets the condition \(\text{Re}\delta\ll 1\), the inertia force of water flow can be ignored, and Equation (25) can be transformed into  
\[-\frac{\partial p}{\partial x}+\mu\frac{\partial^{2}u}{\partial y^{2}}=0. \tag{32}\]  
The cubic law of water flow in the boundary water-irnush fracture can be obtained by integrating the cross-sectional area along the \(y\) direction of the boundary water-irnush fracture and combining the Formula (30):  
\[q=\int\limits_{e}udy=-\ \frac{e^{3}}{12\mu}\frac{\partial p}{\partial x}=-\ \frac{e^{3}}{12\mu}\frac{\Delta p}{L}. \tag{33}\]  
In the formula, \(\Delta p\) is the pressure difference at both ends of the boundary water irnush fracture. \(e\) is the distance from the flat approximate plate surface to the fracture center line.