## 3 Three-Dimensional Finite Element Analysis Model  
### Numerical Model  
The size of the model is 120 m in length, 60 m in width, and 40 m in height. From top to bottom, the soil consists of silty clay 21, mucky-silty clay 23, and silty clay 51. The thickness of silty clay 21 is 4 m, and the thickness of mucky-silty clay 23 and silty clay 51 can vary. The soil element adopted the three-dimensional eight-node linear element C3D8P with pore pressure, and the number of meshes is 119,200, as shown in Figure 2.  
Figure 1: Geological profile. Figure 2: Schematic diagram of stratum grid division. This study used the homogeneous circle method to simulate the tunnel, which indicated a homogeneous cylinder was used to simulate the tunnel, and the elastic modulus of concrete was reduced to equate the weakening effect of the joints and bolts on the stiffness. The length of the tunnel is 120 m, with a total of 100 rings. The buried depth of the tunnel is 14.7 m, the outer and inner diameters of the tunnel are 6.6 m and 5.9 m, respectively. Moreover, the excavation gap is 0.13 m, and the influence of grouting material was simulated by a grouting equivalent layer with a thickness of 0.13 m. Moreover, both the tunnel lining and the grouting equivalent layer adopted the three-dimensional eight-node non-coordinating linear element C3D8I, and the number of grids is 14,800, as shown in Figure 3.  
Both the tunnel excavation process and the long-term consolidation process adopted fluid-solid coupling analysis steps, and used the \"model change\" function to realize soil excavating and lining construction. The disturbance of shield excavation can be simulated by the stress releasing method and the convergent displacement method. In this paper, the convergent displacement method was used to simulate the shield excavation. The convergent displacement method was to preset the convergent displacement during excavation, which can highlight the main contradiction and reflect the disturbance of shield construction. Due to the existence of soil loss volume, the soil movement mode at the excavation profile can be simplified as the soil at the excavation profile is converging to the periphery of the shield. As shown in Figure 4, by assuming that the convergent displacement of the top of the excavation profile is \(\delta_{\text{max}}\), the convergent displacement of the bottom is zero, and the center of the shield periphery is the convergent point of the displacement, the soil displacement converging mode was obtained. In this case, the excavation diameter of the shield machine used is 6.86 m, and the diameter of the shield periphery is 6.81 m, which adopted the diameter of the shield tail. Consequently, the convergence displacement at any angle can be calculated by Equation (1) proposed by [PERSON] (2016). \[\delta=\frac{\delta_{\text{max}}}{2(R-R_{0})}\bigg{(}\sqrt{R^{2}-2R(R-R_{0}) \cos\Bigl{(}\theta+\frac{\pi}{2}\Bigr{)}+\left(R-R_{0}\right)^{2}}-R_{0}\bigg{)} \tag{1}\]  
where \(\delta_{\text{max}}\) is the convergent displacement of the vault (m); \(\delta\) is the convergent displacement at any polar angle (m); \(R\) is the excavation radius of the shield machine (m); \(R_{0}\) is the outer radius of the shield tail(m); \(\theta\) is the polar angle. Figure 3: Schematic diagram of grid division of tunnel and grouting layer. The following methods were used to simulate the shield construction process. First, the support pressure of the tunnel face in the front was added. Then, the \"model change\" function, which can kill the soil corresponding to a ring segment, was used to simulate the excavation in the front. At the same time, the lining and grouting layer at the rear was activated, and the grouting pressure to simulate the effect of simultaneous grouting was applied.