## 2 PMGIM Induction Mechanism  
### Mechanical Analysis  
The primary objective of grouting treatment in water-inrush-prone working faces is to utilize high-pressure slurry to displace water, thereby converting fractured aquifers into relatively impermeable barriers. The grouting treatment process can be categorized into three distinct stages: filling grouting, pressure-boost penetration grouting, and high-pressure expansion. During pressurized seepage grouting and high-pressure expansion grouting, the slurry initially permeates into fractures through seepage, gradually filling the fracture spaces [22]. As the amount of grouting increases and due to the incompressibility of the grout, internal pressure is exerted on the fracture walls, generating tangential tensile stress. When this tangential tensile stress surpasses the combined bonding force of secondary weak surfaces and the original tangential stress, the grout causes these weak surfaces to crack and further extends the fractures. The fractures continue to expand and extend until the grouting pressure can no longer induce further extension. Varying degrees of microseismic events are triggered by fracture initiation, propagation, and extension. In this study, the rock mass is modeled as an elastic semi-infinite body. Assuming the rock mass to be a homogeneous and isotropic elastic medium, the stress state around the grouting hole can be simplified as a plane strain problem. Typically, grouting holes in these projects are circular; therefore, the established model of a flat circular hole is shown in Figure 1. The rock mass surrounding the grouting hole experiences two principal stresses that are unequal and perpendicular to each other in the horizontal direction. Here, \(\delta_{1}\) represents the maximum principal stress, and \(\delta_{3}\) denotes the minimum principal stress. The rock mass around the circular hole is subjected to radial stress (\(\delta_{r}\)), circumferential stress (\(\delta_{\theta}\)), and shear stress (\(\delta_{r\theta}\)) at a distance (\(r\)) from the hole wall to the center of the grouting hole and at an angle (\(\theta\)) with the X-axis. According to elasticity theory, the stress state around an unlined hole in polar coordinates is given by the following:  
\[\left\{ \begin{matrix}\sigma_{r}=\left(1-\frac{r^{2}}{\delta^{2}}\right)\left( \frac{\sigma_{1}+\sigma_{3}}{2}\right)+\left(\frac{\sigma_{1}-\sigma_{2}}{2} \right)\left(1-\frac{4\delta^{2}}{\delta^{2}}+\frac{3r^{4}}{\delta^{4}}\right) \cos 2\theta+\frac{r^{2}}{\delta^{2}}\left(P_{0}+P_{h}\right)\\ \sigma_{\theta}=\left(1+\frac{r^{2}}{\delta^{2}}\right)\left(\frac{\sigma_{1} +\sigma_{3}}{2}\right)-\left(\frac{\sigma_{1}-\sigma_{2}}{2}\right)\left(1+ \frac{3r^{4}}{\delta^{4}}\right)\cos 2\theta-\frac{r^{2}}{\delta^{2}}\left(P_{0}+P_{h}\right)\\ \tau_{r\theta}=-\left(\frac{\sigma_{1}-\sigma_{2}}{2}\right)\left(1+\frac{2r^ {2}}{\delta^{2}}-\frac{3r^{4}}{\delta^{4}}\right)\sin 2\theta\end{matrix} \right. \tag{1}\]  
where \(r\) is the radius of the drilling hole, \(\delta\) is the radial distance, \(P_{0}\) is the grout weight pressure, and \(P_{d}\) is the grouting pump pressure, which is zero during the filling and grouting stage. When \(r=\delta\), the stress state around the hole is as follows:  
\[\left\{ \begin{matrix}\sigma_{r}=P_{0}+P_{h}\\ \sigma_{\theta}=\left(\sigma_{1}-\sigma_{3}\right)-2\left(\sigma_{1}-\sigma_{ 3}\right)\cos 2\theta-P_{0}+P_{h}\\ \tau_{r\theta}=0\end{matrix} \right. \tag{2}\]  
According to the maximum tensile stress failure criterion, the crack propagates along the path of least resistance. Specifically, when \(\theta=0^{\circ}\) or \(\theta=180^{\circ}\), \(\delta_{\theta}=\delta_{\theta min}\), and the crack extends in the direction perpendicular to the minimum principal stress. The mechanical criterion for the grouting pump pressure required for crack initiation is as follows:  
\[P_{h}=3\sigma_{3}-\sigma_{1}+\sigma_{T}-P_{0} \tag{3}\]  
Figure 1: Stress state of the rock mass around the grouting borehole.