## 5 The Initial Water Inrush Coefficient of Inter-Layer Rock Is: K=\(K_{0}\)
### Solution of Fluid-Solid Coupling Model of Inter-Layer Rock
#### 5.1.2 Solution of Fracture Flow Water Inrush Model
The continuity equation of network waters inrush fracture is  
\[\frac{\partial(\rho\phi_{k})}{\partial t}+\frac{\partial}{ \partial x}\bigg{(}k_{f}\frac{\partial p}{\partial x}\bigg{)}+W=0, \tag{71}\] \[\frac{\partial(\rho\phi_{k})}{\partial t}+\frac{\partial}{ \partial y}\bigg{(}k_{f}\frac{\partial p}{\partial y}\bigg{)}+W=0, \tag{72}\]  
where \(\rho\) is the density of water, kg/m\({}^{3}\); \(\phi_{k}\) is the porosity of inter-layer rock, %; \(k_{f}\) is the water inrush rate of fracture, mD; \(W\) is the sink-source term. \[\frac{\partial(\rho\phi_{k})}{\partial t}=\rho\frac{\partial\phi_{k}}{ \partial t}+\phi_{k}\frac{\partial\rho}{\partial t}. \tag{73}\]  
Taking pressure as a variable of density \(\rho\) and porosity \(\phi_{k}\) of inter-layer rock, then Equation (73) can be transformed into  
\[\rho\frac{\partial\phi_{k}}{\partial t}+\phi_{k}\frac{\partial\rho}{\partial t }=\rho\frac{\partial\phi_{k}}{\partial p}\frac{\partial p}{\partial t}+\phi_{ k}\frac{\partial\rho}{\partial p}\frac{\partial p}{\partial t}. \tag{74}\]  
Assuming the compressibility factor \(\chi_{f}=(1/\rho)(\partial\rho/\partial p)\) of the fluid, the above equation can be obtained. \[\frac{\partial(\rho\phi_{k})}{\partial t}=\rho\bigg{(}\phi_{k}\chi_{f}+\frac{ \partial\phi_{k}}{\partial p}\bigg{)}\frac{\partial p}{\partial t}=\rho S \frac{\partial p}{\partial t}. \tag{75}\]  
In formula (75), \(S\) is the water storage coefficient, which is generally considered to be caused by the change in inter-layer rock strata. Substituting Equation (75) into Equations (71) and (72), there is  
\[\rho S\frac{\partial p}{\partial t}+\frac{\partial}{\partial x}\bigg{(}k_{f} \frac{\partial p}{\partial x}\bigg{)}+W=0 \tag{76}\]  
Equation (76) is the fluid-solid coupling water inrush model of water inrush fracture and water formed by inter-layer rock fracture. Based on the above coupling model of fracture deformation and stress of inter-layer rock, the fracture state model of water accumulation in the upper part of inter-layer rock in network water inrush fractures is obtained, and the equation of fracture flow water inrush model is solved. \[\text{water inrush model}:\rho S\frac{\partial p}{\partial t}+\frac{ \partial}{\partial x}\bigg{(}k_{f}\frac{\partial p}{\partial x}\bigg{)}+W=0 \tag{77}\] \[\rho S\frac{\partial p}{\partial t}+\frac{\partial}{\partial x} \bigg{(}k_{f}\frac{\partial p}{\partial x}\bigg{)}+W=0\] \[\text{Interlayer rock deformation model}:(\lambda+G)u_{ij}+Gu_{ij}+f_{i}=0(i,j= 1,2,3)\] \[\text{Water inrush fracture deformation model}:\left\{ \begin{array}{l} \sigma_{n}{}^{\prime}=k_{n}\varepsilon_{n}b,\varepsilon_{n}=\delta_{n}/b\\ \delta_{s}{}^{\prime}=k_{s}\varepsilon_{s}b,\varepsilon_{s}=\delta_{s}/b\\ \text{Effective stress model}:\left\{\begin{array}{l}\sigma_{n}{}^{\prime}= \sigma_{n}-\beta p_{f}\\ \sigma_{s}{}^{\prime}=\sigma_{s}\\ \end{array}\right.\\ \text{Water inrush rate equation}:k_{f}=\frac{\beta^{2}}{12/RC^{5}}\end{array} \right\}.\]  
Initial conditions: assuming that the initial pressure and effective pressure are \(p_{0}\) and \(\sigma_{0}\) respectively, the gob water and inter-layer rock density is \(\rho_{1}\) and \(\rho_{s}\), respectively. Boundary conditions: constant pressure or constant flow on the interface \(S\) of the inter-layer rock mass. \[p\big{|}_{s}=p_{0}(x,t),\text{or }Q=Q_{0}(x,t). \tag{78}\]The boundary displacement is quantified as:  
\[\overrightarrow{u}\Big{|}_{s}=\overrightarrow{u_{0}}(x,t). \tag{79}\]  
According to the above solution conditions, the variables \(u\) and \(p\) of the above fluid-solid coupling model are solved. The key parameters such as water inrush velocity, flow rate, strain, and stress are further solved to solve the fracture flow water inrush model.