## 2 Materials and Methods
### Numerical Calculation Model and Method  
A numerical model of the shield tunnelling empty-pushing through the mining method tunnel was established using the MIDAS-GTS finite element calculation software, considering the self-weight stress field for the ground stress field. During the modelling process, certain conditions were optimized and assumed, considering the following basic assumptions: According to the ISRM and other relevant Chinese standards, and concerning the advanced rock mechanics experiment method [26, 27], the mechanical parameters of the soil layer at the construction site are measured to form a geological survey report. In the report, the corresponding physical and mechanical properties of representative soil layers are selected as the simulation parameters in the model, the detailed parameter settings are shown in Table 1. The entire construction process was simulated using the construction step sequence, considering the spatial displacement variation during construction, without considering the time effect. Figure 2: Cross-sectional view of the mining method tunnel. To minimize the effect of boundary constraints on the computational results, the boundary for each direction was established based on similar numerical simulations and experimental experiences for tunnels [28; 29; 30; 31]. According to the principles of the numerical simulation, the boundary should be placed three to five times the diameter of the tunnel section outside the retaining structure. In this model, a distance four times the diameter of the tunnel section was adopted. The overall range of the numerical model was 70 m \(\times\) 50 m \(\times\) 40 m. The main stressed soil layers were simulated, from top to bottom, and their corresponding thicknesses were as follows: compacted fill (8.2 m), residual silty clay (5.2 m), strongly weathered silty mudstone (2.4 m), and moderately weathered silty mudstone (24.3 m extending from the bottom boundary of the model). The net distance between the left and right tunnel lines was 8.4 m. Each tunnel line had a horizontal width of 6.6 m and a vertical width of 6.75 m. For displacement boundary conditions, a free surface was used at the top of the model, and normal fixed constraints were employed for the other five faces in the vertical direction. Specifically, x and y normal fixed constraints were utilized for the horizontal constraints, and a z fixed constraint was applied to the bottom, as shown in Figure 3a. The modified Mohr-Coulomb criterion derived from [PERSON] et al. 's [32] and [PERSON] et al. 's [33] closed solutions for the plastic zone and stress distribution around circular tunnels in the elastic-plastic semi-infinite plane based on the Mohr-Coulomb yield criterion and [PERSON]'s [34] analysis of the Yumo railway project in China was adopted as the failure criterion for the grid elements using practical engineering experience as a basis. Solid elements were employed to simulate each soil layer. The initial support and shield segments were modelled using plate elements, whereas the gravel backfill layer and guide platform were simulated using solid elements. The numerical model corporates 8800 hexahedral mesh elements and 7601 nodes. Mesh size is controlled using a linear gradient, with a mesh size of 0.5 m for initial supports, segments, pea gravel, and soil inside the tunnel. Outside the tunnel, the mesh size for the soil gradually increases linearly with the distance from the tunnel centreline, reaching a maximum mesh size of 5 m.  
To simulate the initial state of natural soil layers during the finite element modelling of foundation pit excavation, it is crucial to balance the initial stress. This implies that the soil model should only contain the initial stress field without any initial displacements. Initially, the soil displacement and stress fields under their own weights were initially calculated. Subsequently, to construct the initial stress field, the displacement zeroing function of MIDAS-GTS was utilized to remove the completed settlement displacement.