## 3 Analysis of Results
### Pressure Boundary Conditions  
Figure 9 shows the variation in the grout diffusion distance over time. The horizontal axis represents the azimuth angle corresponding to the measurement points, while the vertical axis represents the volume fraction of grout. At \(t=0\) s, grouting begins, and at this time, except for the grouting port, all measurement points have a \(\phi\) value of 0, indicating that the pores are completely filled with air. At \(t=0.6\) s, the \(\phi\) value near the grouting port approaches 1, indicating that the voids near the grouting port have been completely filled with grout. Between \(t=1.2\) s and \(t=2.4\) s, the curve as a whole moves toward the straight line \(\phi=1\), meaning that the grout gradually fills the gaps. At \(t=3\) s, the \(\phi\) value at most locations reaches 1. Only at the starting and ending positions does \(\phi\) remain between 0 and 1, indicating that the grout has diffused throughout the entire backfilling layer. However, small amounts of air still remain at the top and bottom of the tunnel. This behavior suggests that while the grout has filled the majority of the backfilling layer, some air pockets remain in the less accessible regions. Figure 8: Local analysis of pressure and velocity field of grout in the backfilling layer: (**a**) pressure contour; (**b**) velocity contour; (**c**) streamline plot. Figure 9: Grout filling condition in the backfilling layer under pressure boundary conditions. (**a**) \(\alpha=30^{\circ}\); (**b**) \(\alpha=90^{\circ}\); (**c**) \(\alpha=150^{\circ}\). Figure 10 illustrates the dissipation of grouting pressure within the gravel backfilling layer. The phase interface is defined at \(\phi\) = 0.5, and its position is indicated by a dashed line in Figure 10. The highest pressure occurs at the grouting port, and the pressure gradually decreases from the grout inlet to the grout outlet, eventually approaching 0 kPa, in line with the expected boundary condition. As shown in Figure 10a, when the grout injection port is located at the 30\({}^{\circ}\) position, the dissipation process of the injection pressure can be divided into two parts--from 0\({}^{\circ}\) to 30\({}^{\circ}\) and from 30\({}^{\circ}\) to 180\({}^{\circ}\)--with the injection port serving as the boundary. In the 0\({}^{\circ}\) to 30\({}^{\circ}\) range, the grout diffuses toward the upper outlet. Due to its proximity to the fluid outlet and the short distance traveled, the pressure dissipates quickly, with the pressure change following a nearly linear trend. In the 30\({}^{\circ}\) to 180\({}^{\circ}\) range, the grout diffuses toward the lower outlet. Since this part of the grout is farther from the fluid outlet and must travel a greater distance through a large volume of the porous medium, the dissipation of fluid pressure is slower, resulting in a more gradual pressure change. The distribution of fluid pressure corresponds to the distance over which the grout diffuses, with the fluid pressure typically decreasing to zero at the interface between the grout and air. Figure 10a illustrates the diffusion of grout when the injection port is at the 30\({}^{\circ}\) position. At this point, the grout primarily diffuses downward, and the diffusion speed remains relatively steady. Every 0.6 s, the interface between the grout and air moves downward by approximately 20\({}^{\circ}\). For instance, at 0.6 s, the fluid pressure drops to zero at the 90\({}^{\circ}\) position, and the grout has diffused to the 100\({}^{\circ}\) position; at 1.2 s, the pressure drops to zero at the 120\({}^{\circ}\) position, and the grout has diffused to the 130\({}^{\circ}\) position. Notably, when the injection port is positioned at 150\({}^{\circ}\) (Figure 9c), the grout primarily diffuses upward. In this case, the grout initially maintains the same diffusion speed, but the rate of diffusion slows down in the later stages. Figure 11 shows the variation in the grout flow rate under pressure boundary conditions.