## 3 Simulation Results and Analysis  
### Discussion: Evolution Characteristics of Displacement Field under Different Water Pressures  
The present study analyzes the minimum safe thickness under four different water pressure scenarios within the fault rupture zone, assuming constant parameters such as a burial depth of 500 m, a fault dip angle of 70\({}^{\circ}\), and a fault rupture zone width of 30 m. The water pressures considered are increasing from 1 MPa to 2.5 MPa. The failure condition is defined as a sudden increase in the maximum unbalanced stress, leading to uncontrolled and continuously increasing displacements at the tunnel face. This ultimately results in the ejection of rock blocks and the occurrence of a tunnel inrush. The evolution of the displacement field of the anti-inrush body at the tunnel working face during the excavation process is shown in Figure 3. The numbers in different colors in the figure represent the magnitude of the displacements. The displacement contour plots shown in Figure 3 represent the displacement states of the model at the point of equilibrium following each excavation stage. From the observations in Figure 3, it is evident that under the same fault water pressure conditions, the magnitude of displacements within the anti-inrush body increased significantly after the fourth excavation stage. This indicates that as the tunnel progresses towards the water-bearing fault, the displacements of the anti-inrush body at the tunnel working face, which were initially induced solely by the unloading due to excavation, gradually transition to being influenced by both the unloading and the water pressure from the upstream water-bearing fault. Consequently, the impact of the upstream water-bearing fault on the stability of the anti-inrush body becomes increasingly pronounced. Under the water pressures of 1 MPa and 1.5 MPa, the displacement contour plots after the sixth excavation stage depict the displacement patterns at the point of evident instability and failure of the tunnel working face. However, for the 2 MPa and 2.5 MPa water pressure, the instability and failure of the tunnel working face was observed after the fifth excavation stage. The minimum required thicknesses of the anti-inrush body at the tunnel working face were determined to be 3-4 m for the water pressures of 1 MPa and 1.5 MPa, and 4-5 m for the 2 MPa and 2.5 MPa water pressure. Furthermore, it was observed that under the same excavationstage, the maximum lateral displacement of the anti-nirush body at the tunnel working face gradually increased with the increase in fault water pressure. These findings suggest that as the fault water pressure increases, the minimum safe thickness of the anti-nirush body required for the tunnel to traverse the water-bearing fault increases. Figure 4 presents the displacement variation curves at the center of the tunnel working face. As shown in Figure 4, the trends of displacement changes at the tunnel working face center are generally consistent across different water pressures as the excavation progresses. As the tunnel advances towards the water-bearing fault, the lateral displacement response at the tunnel working face becomes increasingly pronounced, and the growth rate remains relatively consistent. From Figure 4, it can be observed that, at the same coordinate location, the displacement response continuously increases with the rise in water pressure. Moreover, this change becomes more evident as the tunnel working face approaches the fracture zone of the fault. As the water pressure increases, the displacement is more susceptible to sudden changes when the thickness between the tunnel working face and the fault remains the same, indicating a higher risk of sudden water intrusion.