## 3 Numerical Simulations
### Numerical Modeling Procedure  
In this paper, the modeling process consists of two stages: continuous excavation of the tunnel through the river and coupled solution of the groundwater seepage and stress fields. In COMSOL Multiphysics, excavation could be implemented using the \"activation\" function, as shown in Figure 4. The principle was to set an activation multiplier for the part of the model that needs to be excavated. When the excavation operation was required, the elasticity matrix of the computational node was multiplied by the activation multiplier and the density was multiplied by the square of the activation multiplier, thus causing the loss of the bearing capacity at that location and achieving the excavation effect. In order to control the progress of the tunnel excavation, this can be achieved by controlling the length of the tunnel excavation (defined as depth_1). Then, depending on the length at which the node was calculated with respect to the size of the parameter, it was decided whether the node was activated or not, and the excavation progress can be controlled simply by changing the value of depth_1. In order to achieve continuous excavation, the result of the previous excavation step was assigned as the initial condition for the next excavation step.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**The Strata** & **Filling Density \(\rho\)** & **Bulk Modulus K (MPa)** & **Shear Modulus K (MPa)** & **Porosity** & **Permeability** & **c (kPa)** & **\(\varphi\) (\(\diamond\))** \\ \hline Miscellaneous fill & 2.01 & 6.72 & 4.23 & 0.09 & \(7.09\times 10^{-14}\) & 24 & 16 \\ Silt clay and silt & 2.05 & 6.38 & 3.83 & 0.1 & \(9.45\times 10^{-12}\) & 18 & 23 \\ Round gravel & 2.2 & 33.33 & 25 & 0.3 & \(1.12\times 10^{-11}\) & 2 & 35 \\ Moderately weathered & 2.45 & 393.80 & 415 & 0.225 & \(1.77\times 10^{-13}\) & 1.2 & 38 \\ argillaceous siltstone & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The main parameters of the strata.  
The excavation procedure was as follows:  
1. In the first step, the pressure expressions on the grid points at different elevations were obtained based on the water level of 29 m on the bank side and the riverbed side. The water level pressure added to the numerical model takes into account the effect of water level changes in the Liuyang River. The gravity of the water body in the river was applied to the surface of the riverbed as an applied uniform load.
2. In the second step, the ground stress equilibrium was performed for the whole model. For the finite element model involving the joint action of groundwater and geotechnical body, the initial ground stress field was obtained by coupling the gravity field with the seepage field for equilibrium. At this point, the initial deformation was zero because the deformation has been completed before the construction of the new tunnel.
3. In the third step, the \"activation\" function was used to simulate tunnel excavation. The numerical simulation took into account four factors: the tunnel excavation surface spacing, the tunnel depth, the tunnel spacing and the water level on the riverbed side. The tunnel boring process is in the x-direction of the model (see Figure 3a) and is simulated using a stepwise excavation method with a step length of 10 m. The newly excavated part of prestressing is zero, and the remaining part of prestressing is the stress field after excavation stability in the previous step.