## 2 Methodology
### Unified Single-Exponential Discriminant Function (\(T_{js}\))  
Existing scholars have conducted certain research on the prediction of water hazards in roadways, but they have not yet performed dual analysis and prediction from both geometric space and rock mass damage perspectives simultaneously [17; 18; 19; 20; 21; 22; 23; 24]. In the \"geometric damage\" dual-control water inrush prediction model under the PD framework for tunneling, the distance between the working face and the concealed water is denoted as \(D_{JS}\). If a water-probing borehole at the face of the excavation detects the boundary of the concealed water, with the depth of the borehole being \(L_{T}\) and the angle to the horizontal  
Figure 10: Schematic of the relationship between crack propagation and key fracture during the destruction of water-resisting zone.  
plane being \(\beta_{T}\), then the distance between the face of the excavation and the concealed water can be expressed as  
\[\begin{cases}y_{JS}=L_{T}\cos\beta_{T}\\ z_{JS}=L_{T}\sin\beta_{T}\end{cases} \tag{33}\]  
Whether the water diversion channel is completely open can be given by the condition that the water-resisting zone (\(y_{g}\), \(z_{g}\)) is completely destroyed:  
\[\begin{cases}y_{g}<0\\ z_{g}<0\end{cases} \tag{34}\]  
The range of the water-resisting zone can be obtained by subtracting the distance between the head of the excavation face and the implicit waterbody from the water-induced fracture zone and the excavation-induced fracture zone:  
\[\begin{cases}y_{g}=y_{\text{JS}}-y_{s}-y_{j}\\ z_{g}=z_{\text{JS}}-z_{s}-z_{j}\end{cases} \tag{35}\]  
When the water channel of Equation (34) is fully open, it can be expressed as  
\[\begin{cases}y_{\text{JS}}-y_{s}-y_{j}<0\\ z_{\text{JS}}-z_{s}-z_{j}<0\end{cases} \tag{36}\]  
The above formula is the basic discriminant of the complete effectiveness of the water conduction channel. A unified discriminant function for the effectiveness of the water diversion channel on the roof of roadway excavation is constructed:  
\[T_{JS}=\sqrt{\text{e}}^{\frac{y_{g}+y_{j}}{y_{\text{S}}}-1}\cdot\sqrt{\text{ e}}^{\frac{z_{g}+z_{j}}{y_{\text{S}}}-1}+\sqrt{\kappa}^{1-\text{sign}(y_{s}+y_{j}-y_ {\text{S}})\cdot sign(z_{s}+z_{j}-z_{\text{S}})} \tag{37}\]  
In the formula, \(T_{JS}\) is the effective coefficient of the water conduction channel in roadway excavation, where \(T_{JS}>0\), and \(\kappa\) is the critical value of peridynamics local damage related to D\({}_{\text{quiculude}}\), \(0<\kappa<1\), approximated by the porosity of coal-rock mass in the absence of local damage.  
The effectiveness judgment function and principles for the water conduction channel in the roadway roof are detailed in Table 1. Based on these principles, during the roadway excavation process, the effective coefficient \(T_{JS}\) of the water diversion channel can be calculated by selecting appropriate basic parameters based on actual conditions, allowing for the assessment of the water diversion channel's effectiveness.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Concrete Circumstances** & **Corresponding Conditions** & **Form of Effective Judgment Function of Water Diversion Channel** & **Principle of Effective Judgment of Water Conduction Channel** \\ \hline The water-resisting zone was completely undamaged & \(y_{g}>0\) & \(T_{JS}<1\). The closer \(T_{JS}\) is to 0, the safer it is, and the water conduction channel is invalid, so it can be considered that there will be no sudden water inrush. \\ \hline \end{tabular}
\end{table}
Table 1: Effectiveness discrimination function and discrimination principle of water conduction channel in roadway roof.