# Numerical Simulation on the Leakage-Induced Collapse of Segmental Tunnels
## 3 Numerical Model
### Soil Model
#### 3.1.2 Numerical Model Based on the Tests  
Observations from the tests revealed that the area of soil erosion resulting from tunnel leakage was smaller than the dimensions of the model test box (1000 mm \(\times\) 1000 mm). Consequently, to optimize computational resources, the size of the simulated soil model was determined according to the region of strata displacement observed in the tests. The dimensions of the model were set at 800 mm in width and 660 mm in height. With these considerations, both the DEM model and the FDM model were established.  
The FDM model elements were uniformly divided, with each element side measuring 20 mm. Additionally, adjustments were made to the surrounding area of the tunnel based on its shape. In total, 1320 FDM elements were generated. The water pressure boundary conditions were established in accordance with the model tests: the pressure at the strata top and at the tunnel opening were set to 0, while the pressure at the sides and bottom was determined based on the actual water pressure distribution. The pressure distribution is shown in Figure 5a.  
In the DEM soil model, the particle size was enlarged by a factor of 2.5 compared to actual soil particles to enhance computational efficiency. [PERSON] et al. [33] proposed the following equation to describe the particle size effect:  
\[RES=(\frac{L}{R_{min}})(\frac{1}{1+\frac{R_{max}}{R_{min}}}) \tag{4}\]  
where \(L\) is the length of the shortest side of the model, and \(R_{max}\) and \(R_{min}\) are the maximum and minimum particle diameters, respectively. When RES \(\geq\) 10, the size and number of particles in the model have only a small influence on the mechanical properties of the soil. Considering the model size and computational ability, the soil particles in the model were enlarged by a factor of 2.5. The shortest side of the model was taken as the diameter of the shield tunnel \(R\) = 124 mm, the maximum particle diameter of the enlarged model particles was 2.5 mm, and the minimum particle diameter was 0.25 mm. Then, the RES in this model  
Figure 5: FDM and DEM model: (**a**) Water pressure distribution based on FDM model (**a**); (**b**) DEM model.  
is 45, which meets the above requirements and the enlarged particles should not influence the mechanical behavior of the soil. A rigid circular wall was employed to simulate the tunnel. Consistent with the model tests, water and sand inrush were triggered by removing a section of the wall at the corresponding position. Due to the enlarged particle size, the tunnel opening was also enlarged. While the tunnel opening size in the model tests was 2 cm, it was enlarged to 5 cm in the numerical simulation. The strata model is shown in Figure 5b.  
Within the DEM model, the contact among soil particles was defined by the linear contact bond model. Once the particle-to-particle bond reaches its strength limit, it transitions to the linear contact model. Calibration of parameters was conducted through shear tests, and the specific parameters are outlined in Table 2.  
To reflect the response of the tunnel when subjected to the load redistribution induced by the external strata change, the loads at critical moments calculated by the strata model were selected and applied to the tunnel. The contact force of the soil particles to the tunnel was monitored during the strata calculations. Considering the tunnel size and external particle size, the monitoring range of each monitoring point was taken as the average contact force on the tunnel wall over circle segments with a 30\({}^{\circ}\) center angle. In this case, the wall size of the monitoring point range (31.4 mm) was 12.6 times the maximum particle size (2.5 mm), which could reduce the error caused by the particle size effect. Based on this idea, a total of 12 measurement points were arranged outside the tunnel in the strata model, as shown in Figure 6.