## 2 Methodology
### Geometric Damage Dual-Control Model  
Establish a \"geometric damage\" dual-control water inrush prediction model under the PD framework of the roadway shown in Figure 7. In front of the roof of roadway \(J\), there is an implicit waterbody \(S\). Assuming the distance between the roadway working face and the implicit waterbody is \(D_{JS}=(y_{JS},z_{JS})\), the formation of an effective water-conducting channel depends on whether the waterbody \(S\)'s water-induced fracture zone \((y_{s},z_{s})\) and the roadway \(J\)'s excavation-induced fracture zone \((y_{j},z_{j})\) can fully communicate, that is, whether the damage to the water-resisting zone \((y_{g},z_{g})\) has reached a certain level. (1) The evolution process of water-induced fracture zone  
The evolution process is illustrated in Figure 8. The process begins with the presence of an implicit waterbody that exerts a certain level of pressure on the roof rock \((s_{1})\). This pressure weakens local solid bonds, leading to their fracture and the activation of fluid bonds, forming a softened fracture zone \((s_{2})\). Subsequently, this damage spreads and connects under the continued pressure of the water, ultimately forming a hydraulically fractured zone \((s_{3})\). Figure 7: “Geometric damage” dual-control water inrush prediction model for roadway. The evolution process of the excavation-induced fracture zone  
The evolution process is illustrated in Figure 9. As the excavation progresses, the surrounding rock on the roof is affected by specific excavation techniques and engineering geological conditions: the integrity of the shallow surrounding rock around the roadway is compromised, leading to block separation (forming the loosened fracture zone \(j_{1}\)); the solid bonds of the middle surrounding rock around the roadway are broken, resulting in non-local stress failure (forming the secondary fracture zone \(j_{2}\)); the deep surrounding rock around the roadway experiences damage, with solid bonds breaking or weakening (forming the crack initiation zone \(j_{3}\)). The destruction of the water-resisting zone and the effectiveness of the conduction  
As shown in Figure 10, the destruction process of the water-resisting zone can be divided into three levels: completely undamaged, partially damaged, and completely destroyed. Figure 8: Schematic of the relationship between crack propagation and key fracture during the evolution of water-induced fracture zone. Figure 9: Schematic of the relationship between crack propagation and key fracture during the evolution of excavation-induced fracture zone. The water-resisting zone is completely undamaged (\(y_{g}>0\) and \(z_{g}>0\)):  
As shown in Figure 10a, all solid bonds in the water-resisting zone are intact \(\rightarrow\) local damage, as determined by Equation (14) (i.e., the ratio of the number of broken bonds to the total number of bonds), \(\mathrm{D_{aquiculude}}\) (\(y\),\(t\)) = 0 and \(\mathrm{D_{aquiculude}}\) (\(z\),\(t\)) = 0 \(\rightarrow\) the water-conducting channel has not yet formed \(\rightarrow\) it can be considered that water will not occur suddenly. Part of the water-resisting zone is damaged (\(y_{g}\cdot z_{g}\leq 0\)):  
As shown in Figure 10b, the partial fracture of the solid bond in the water-resisting zone has caused local damage that has not reached the critical condition. Specifically, if 0 \(<\)\(\mathrm{D_{aquiculude}}\)(\(y\),\(t\)) \(<\)\(\mathrm{D_{critical}}\) and 0 \(<\)\(\mathrm{D_{aquiculude}}\)(\(z\),\(t\)) \(<\)\(\mathrm{D_{critical}}\), the current drainage channel is not fully formed, leading to insufficient effectiveness. Therefore, it can be concluded that a sudden water surge is unlikely to occur at this time, although the possibility of a delayed-type sudden water surge cannot be ruled out.