## 2 Theoretical Analysis  
### Hydraulic-Mechanical (HM) Model for Seam Floors  
The extraction of the coal seam above a confined aquifer inevitably result in the floor strata between the correlated aquifer and coal seam being affected by the redistribution of both hydraulic and excavation-induced pressures. The stressed floor may experience damage across a certain area when these pressures reach or exceed the strength threshold of the floor rocks. This can directly reduce the effective thickness of the aquiclude and enhance the risk of groundwater outburst, thus posing a potential safety threat and hindering the efficiency of underground coal production. Consequently, determining the range of floor failure after mining is of great significance. Figure 2 clearly shows the spatial redistribution of mechanical loads on the seam floor above an aquifer. The hydraulic-mechanical models for inclined and horizontal seam floors were constructed based on the A-A section (Figure 2). As evident in Figure 3, \(x\) represents the inclined direction along the coal wall, z indicates the downward direction perpendicular to the seam floor and \(G(x,z)\) refers to any point in the floor strata. According to [PERSON] et al. (2019), A\({}_{1}\) and A\({}_{3}\) denote the concentration areas of stress in the coal body, which can be approximately divided into two parts by different peaks. A\({}_{2}\) belongs to the stress-relaxed area, wherein the abutment pressure in this area is gradually restored owing to the compaction of collapsed roof rocks, and has the value of \(\gamma H_{\text{m}}\). The water pressure in the underlying aquifer serves as an additional mechanical source at the bottom of the seam floor, resulting in the floor rock mass being much more appropriate for expansion into the excavation spaces during the mining process. Consequently, the risk of water inrush increases with increasing groundwater pressure. To simplify the calculation, the models adopted linearly distributed equivalent loads, and the influences of tectonic stress and primary rock stress were not considered in the current study. The function of each load in Figure 3b can be expressed as:  
Figure 2: Spatial redistribution of mechanical loads on the seam floor above an aquifer. A-A represents the section along the inclination. \[F_{1}(x) = k_{1}\gamma H-\frac{(k_{1}-1)\gamma H}{l_{1}}\Big{(}x-\frac{l}{2}-l_{ 2}\Big{)}\qquad\Big{(}l_{2}+\frac{l}{2}\leq x<l_{1}+l_{2}+\frac{l}{2}\Big{)}\] \[F_{2}(x) = \frac{k_{1}\gamma H}{l_{2}}\Big{(}x-\frac{l}{2}\Big{)}\qquad\qquad \qquad\qquad\qquad\Big{(}\frac{l}{2}\leq x<l_{2}+\frac{l}{2}\Big{)}\] \[F_{3}(x) = \gamma H_{\rm m}\qquad\qquad\qquad\qquad\qquad\Big{(}-\frac{l}{2} \leq x<\frac{l}{2}\Big{)}\] \[F_{4}(x) = k_{2}\gamma H-\frac{k_{2}\gamma H}{l_{3}}\Big{(}l_{3}+\frac{l}{2 }+x\Big{)}\qquad\qquad\Big{(}-\frac{l}{2}-l_{3}\leq x<-\frac{l}{2}\Big{)}\] \[F_{5}(x) = \gamma H+\frac{(k_{2}-1)\gamma H}{l_{4}}\Big{(}l_{4}+l_{3}+\frac {l}{2}+x\Big{)}\quad\Big{(}-\frac{l}{2}-l_{3}-l_{4}\leq x<-\frac{l}{2}-l_{3} \Big{)}\] \[F_{6}(x) = P_{0}+\rho gx\sin\alpha\qquad\qquad\Big{(}-\frac{l}{2}-l_{3}-l_{ 4}\leq x\leq l_{1}+l_{2}+\frac{l}{2}\Big{)}\]  
where \(k_{1}\) and \(k_{2}\) are the concentrated coefficients of abutment pressure on both sides of the coal wall, respectively, \(L\) is the width of the working face, \(\gamma\) stands for the bulk density of the floor rock mass, \(H\) is the average mining depth, \(H_{\rm m}\) denotes the caving height of the roof strata due to coal extraction, \(P_{0}\) is the hydraulic pressure in the confined aquifer, \(\rho\) represents the density of groundwater, \(g\) refers to the gravitational acceleration, \(\alpha\) denotes the dip angle of the coal seam, and \(l_{1}\)-\(l_{4}\) correspond to the lengths of mechanical loads, respectively. Figure 3: Hydraulic-mechanical (HM) models for seam floors. (**a**) A case with horizontal strata; (**b**) A case with inclined strata. Figure 4 shows the decomposition of the vertical load acting on the upper boundary in Figure 3b.