## 2 Theoretical Analysis
### Analytical Model for Key Strata (KS)  
The occurrence of floor water inrush after mining excavations mainly depends on the capacity of the undamaged impermeable layer to resist pressure, particularly the key strata. Although groundwater outburst accidents in some coalmines frequently occur at the stope floor close to the coal wall of the working face and that close to the rear goaf [16; 18], the underlying mechanism remains unclear. Furthermore, the correlation between vertical failure characteristics and the stability of the seam floor should be elucidated. In order to further determine the inrush locations and associated risk of the post-mining floor, the remaining waterproof layer beneath the goaf was taken out from Figure 6a. For comparison, the analytical models of both the horizontal and inclined key strata were established simultaneously, as shown in Figure 6b, where \(S\) and \(L\) are the strike and inclined lengths of the model, corresponding to the width of the coal wall and the advancing distance of reaching a critical water in-rush coefficient, respectively, and \(h_{\text{k}}\) is the normal thickness of the model. The models shown in Figure 6c,d mainly consider the influence of roof caving and groundwater on the key strata, and their mechanical loads are expressed as follows:  
\[ \begin{array}{rcl}Q(x,y)&=&\gamma(H+h_{a}+h_{\text{b}}-h_{\text{k}})-\frac{ \gamma Hx}{5}\\ P(x,y)&=&P_{0}+\rho gy\sin\alpha\end{array}  \tag{8}\]  
where \(Q(x,y)\) denotes the equivalent load associated with the roof collapse and floor failure, \(P(x,y)\) indicates the hydraulic pressure in the aquifer, \(h_{\text{a}}\) represents the mining failure depth of the floor, \(h_{\text{b}}\) refers to the thickness of the waterproof layer, and the other parameters are the same as those defined above. According to the distributed characteristics of boundary composite loads, Equation (9) can be used to represent the deflection of the horizontal and inclined key strata. \[ \begin{array}{rcl}w(x,y)&=&\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{ \infty}a_{mn}x\sin^{2}\left(\frac{m\pi x}{5}\right)\sin^{2}\left(\frac{n\pi y} {L}\right)&\text{(horizontal strata)}\\ w(x,y)&=&\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}a_{mn}xy\sin^{2} \left(\frac{m\pi x}{5}\right)\sin^{2}\left(\frac{n\pi y}{L}\right)&\text{( inclined strata)}\end{array}  \tag{9}\]  
Figure 6: Analytical models for waterproof key strata. (**a**) Position of the waterproof layer; (**b**) horizontal or inclined key strata models; (**c**) profile A-A along the horizontal model; (**d**) profile A-A along the inclined model. In Equation (9), \(a_{nm}\) denotes the deflection coefficient, which can be solved by using the variational method [43], and m and n are integers and typically have a positive value. By substituting Equation (9) into Equation (10), we can obtain the main stress components \(\sigma_{x}\), \(\sigma_{y}\) and \(\tau_{xy}\). \[\sigma_{x} = -\frac{Ez}{1-\mu^{2}}(\frac{\partial^{2}w}{\partial x^{2}}+\mu \frac{\partial^{2}w}{\partial y^{2}})\] \[\sigma_{y} = -\frac{Ez}{1-\mu^{2}}(\frac{\partial^{2}w}{\partial y^{2}}+\mu \frac{\partial^{2}w}{\partial x^{2}}) \tag{10}\] \[\tau_{xy} = -\frac{Ez}{1+\mu}(\frac{\partial^{2}w}{\partial x\partial y})\]  
where \(E\) and \(\mu\) denote the elastic modulus and Poisson's ratio of the waterproof key strata, respectively. Based on the theorem of rock mechanics, the failure criterion for water inrush in the horizontal or inclined key layers can then be determined, as evident in Equation (11).