## 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 attach water inrush fracture divides the inter-layer rock layer into \(r\) blocks; the attach water inrush fracture of each inter-layer rock layer can be regarded as a closed polygonal loop, and the algebraic sum of the pressure difference in the two closed loops is zero. \[\overrightarrow{u}\Big{|}_{s}=\overrightarrow{u_{0}}(x,t). \tag{79}\]9  
In the formula, \(L_{i}\) is the length of the first \(i\) water inrush fracture loop; \(I_{i}\) is the pressure gradient of the \(i\) water inrush fracture circuit; \(K_{j}\) is the total number of basic units surrounding the \(j\) polygon. The longitudinal joint waters inrush fractures divide the inter-layer rock strata into irregular basic units, and only one fracture in each unit forms a curved channel. \[Q_{pf}=\frac{\rho CK_{f}}{\mu}(p_{1}-p_{2}).\]0  
In the formula, \(\Delta H\) is the total pressure difference from the inflow boundary of the outflow boundary, \(S_{j}\) is the total number of the basic units of the \(j\) polygon, and \(i\) is the code of the unit path. Equations (96)-(98) are available. \[\left\{ \begin{array}{l}\sum\limits_{N_{j}}Q_{i}=0\\ \sum\limits_{K_{j}}Q_{i}=0\\ \sum\limits_{S_{j}}L_{i}I_{i}=0\end{array} .\right. \tag{99}\]There are \(n+m+r\) equations in the formula, which constitute the control equation of this model. In the calculation process, the loop pressures gradient and node flow are used as the basic variables, which increases the number of unknowns to be solved. If the node pressure \(p_{i}\) is used as the basic variable, the fracture unit diversion coefficient and the pressure gradient are used to represent the node flow, and any joint water irnush fracture unit \(k\) is studied. The node number of the two endpoints is \(i,j\). The width of the fracture is \(b_{k}\), the length is \(L_{k}\), the porosity is \(\Phi_{k}\), and \(\rho\) is the water density. The flow rate of the inter-layer rock stratum unit \(k\) is  
\[q_{k}=-\frac{b_{k}^{2}}{12u}\left(p_{i}-p_{j}\right)/L_{k}. \tag{100}\]  
The node with \(m\) fracture unit convergence is \(i\), it is connected with another \(m\) node \(k_{i}\) through \(m\) units. The flow balance of node \(i\) is studied. Because the node itself does not have the ability to store water, according to the mass conservation equation, the algebraic sum of the total flow at the node is zero; that is,  
\[\sum_{k=1}^{m}q_{k}=0, \tag{101}\]  
\[-\frac{\rho}{12u}\sum_{k=1}^{m}\frac{b_{k}^{2}(p_{i}-p_{ki})}{L_{k}}=0. \tag{102}\]  
In the formula, \(p_{i}\) denotes the water pressure of node \(i\); \(p_{ki}\) denotes the water pressure of another node in the \(k\) unit of the contact node \(i\). If considering the performance of the stored water of each joint water irnush fracture unit, the node has a sink or source, expressed by \(W_{i}\); the above equation can be written as  
\[\sum_{k=1}^{m}\left(\frac{\partial(\rho\Phi_{k})}{\partial t}-\frac{\rho}{12u }b_{k}^{2}\frac{p_{i}-p_{ki}}{L_{k}}\right)+W_{i}=0. \tag{103}\]  
All the nodes studied constitute \(n\) equations for solving the water head of \(n\) nodes; not only the storage capacity of fractures is considered, but also the flow control equation of the fracture network is directly expressed by node pressure. It is convenient to solve by modern calculation methods such as finite element method. In summary, by analyzing the characteristics of the water-surge channels in the inter-layer rock, the boundary water-inrush fractures, network water-inrush fractures, and attach water-inrush fractures can be transformed into the optimized flat plate flow model, fracture flow model, and pore-fracture flow model for the solution, and the analysis of water-inrush mechanism in the inter-layer rock can be carried out through the analysis of the water movement in the water conduction channel in different flow modes, and the constructed coupled flow-consolidation model can realistically respond to the mechanism of the disaster caused by the water surge in the interlayer rock.