## 4 Simulation Results and Analysis  
### Influence of Distance  
Numerical simulation studies of different distances were conducted, as outlined in Table 2. The horizontal and vertical displacements of the surrounding rock at the 20 m cross-section are depicted in Figures 3-5, respectively. These figures show that when the distance is 5 m, 10 m, 15 m, and 20 m, respectively, the horizontal displacements of the left hance of the surrounding rock are 9.80 mm, 9.52 mm, 9.05 mm, and 8.17 mm, while the vertical displacements of the vault are 12.29 mm, 11.96 mm, 11.45 mm, and 10.09 mm. Compared to 5 m, the horizontal displacements of the left hance decreased by 2.8%, 7.6%, and 16.6%, respectively, and the crown displacements decreased by 2.6%, 6.8%, and 18.0%. The above data indicate that as the distance increases, both the horizontal and vertical displacements of the surrounding rock decrease continuously. The deformation of the left hance and the settlement displacement of the crown of the surrounding rock decrease accordingly. At 5 m, the deformation and settlement displacement of the left hance and crown reach their maximum values, indicating the poorest stability of the surrounding rock. From the above-mentioned rock displacement changes, it is evident that the deformation and settlement are most pronounced as the distance increases from 5 m to 10 m. Since this study neglects the impact of loosened rock mass zones, the actual displacements will be slightly larger than those monitored here. However, the overall trend in displacement development will be similar [40].  
To study the stress variation characteristics of the surrounding rock around the tunnel, the stress distribution of surrounding rock at various monitoring points under different distances at the 20 m section is shown in Figure 6. In Figure 6, it is evident that due to the goal cavity being close to the left hance, the left side is more affected by the goal than the right side. The horizontal stress reaches its maximum value at the arch foot, while the vertical stress peaks at the hance. When the distance is 5 m, 10 m, 15 m, and 20 m, respectively, the horizontal stress at the left arch foot of the surrounding rock is 2.31 MPa, 2.25 MPa, 2.11 MPa, and 2.05 MPa, while the vertical stress at the left hance is 1.85 MPa, 1.79 MPa, 1.75 MPa, and 1.66 MPa. Compared to a distance of 5 m, the horizontal stress at the left arch foot decreased by 2.6%, 8.6%, and 11.3%, respectively, and the vertical stress at the crown decreased by 3.2%, 5.4%, and 10.3%. Therefore, it can be concluded that as the distance increases, both the horizontal and vertical stresses in the tunnel decrease, indicating a lighter collapse degree at the tunnel crown and greater stability of the surrounding rock. In this paper, the displacement and stress changes in the tunnel surrounding rock under the influence of a goal at different distances are similar to the results of [PERSON]'s physical model verification [40].  
Figure 7 shows the maximum axial force of the initial support of the tunnel under different distances of the goaf. The initial support axial force is positive under tension and negative under compression. In Figure 7, it is observed that the initial support axial forces are higher at the tunnel arch than at the arch springing and haunch. Specifically, at the tunnel arch, the maximum initial support axial forces are 4556 kN, 4432 kN, 4345 kN, and 4275 kN, respectively. Thus, it can be concluded that the maximum variation in the initial support axial force occurs when the distance ranges from 5 m to 10 m. Specifically, when the distance is 5 m, the initial support axial force reaches its maximum, indicating greater instability of the support structure.  
Figure 4: Vertical displacement of surrounding rock of tunnel under different distances of goals (Units: m).  
Figure 3: Horizontal displacement of surrounding rock of tunnel under different distances of goals (Units: m).