## 3 Coupled Fluid-Solid Simulation Model
### Simulation Process and Parameters  
In the DEM model, the contact between the soil particles is set as the linear contact bond model. The particle-to-particle bond breaks after reaching the bond strength and degrades into the linear contact model. Parameter calibration was carried out by shear tests, and the detailed parameters of the models corresponding to the three different soil types are shown in Table 2. It should be noted that the shear and tensile strength of the contacts in NSC is zero, so the linear contact bond model degrades to the linear contact model. The linear contact bond model for NSC was chosen but the contact strength is set to zero to make it consistent with the other two types of soil. From the tests, it was observed that the range of soil erosion where tunnel leakage was induced was smaller than the size of the model test box (1000 mm \(\times\) 1000 mm). Therefore, to effectively utilize the computational resources, the size of the calculated soil model was decided based on the region of stratum displacement that occurred in the test. Considering the results of the model tests of the three different strata, the dimensions of the model are 800 mm in width and 660 mm in height. Based on the above considerations, the DEM model and the FDM model were established. The FDM model is shown in Figure 11a. The elements were divided uniformly, the length of each element side was 20 mm, and the surrounding area of the tunnel was adjusted according to its shape. A total of 1320 FDM elements were generated. The water pressure boundary conditions were set according to the model tests. The tunnel opening was set as the outlet boundary. Therefore, the pressure boundary at the opening was set to zero. In addition, the pressure at the surface was set to zero. The other surface (i.e., the sides and bottom) was set as walls, and the pressure was set according to the actual water pressure distribution, as shown in Figure 11b. The water pressure distribution in the numerical model is identical to the conditions of the physical model tests to ensure that the numerical simulation accurately replicates the results of the model tests. In the model tests, as illustrated in Figure 1a depicting the model box design, water supply to the soil originates from water tanks on both sides. Upon filling these tanks, the water pressure on both sides exhibits a gradient distribution under \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Soil Type** & **NSC** & **USC** & **SSC** \\ \hline \multirow{6}{*}{Contacts} & Friction coefficient & 0.9 & 0.8 & 0.8 \\  & Normal stiffness (N/m) & \(5.0\times 10^{7}\) & \(10.0\times 10^{7}\) & \(10.0\times 10^{7}\) \\  & Shear stiffness (N/m) & \(2.5\times 10^{7}\) & \(5.0\times 10^{7}\) & \(5.0\times 10^{7}\) \\  & Tensile strength (force) (N) & 0 & 36 & 40 \\  & Shear strength (force) (N) & 0 & 36 & 40 \\  & Particle density (kg/m\({}^{3}\)) & & 2700 & \\ \multirow{2}{*}{Particles} & Damping & & 0.7 & \\  & Porosity of soils & & 0.4 & \\ \multirow{2}{*}{Fluid} & Density (kg/m\({}^{3}\)) & & 1000 & \\  & Fluid bulk modulus (Pa) & & \(2.0\times 10^{9}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the model. gravity, increasing gradually from the water surface to the bottom. Due to the presence of permeable plates separating the soil from the water tanks, the water pressure at the sides of the soil matches the distribution of water pressure in the tanks, thus demonstrating the gradient change as shown in Figure 11b. If the DEM particles are modeled according to the actual particle size, the number of particles in the model is too large for effective computation. Therefore, it was considered to amplify the particles without affecting the properties of the soil. [PERSON], et al.