## 3 Coupled Fluid-Solid Simulation Model  
### Description of the Coupled Fluid-Solid Model  
When simulating seepage erosion, consideration was given to the characteristics of the mixed soil composed of sand and clay. For soil predominantly consisting of sand, the erosion process involves the detachment, transportation, and clogging of particles [19; 20; 21; 22; 23], driven by hydraulic forces. These phenomena were simulated with the coupled fluid-solid model, using FDM to calculate fluid dynamics and DEM to simulate soil particles. However, with the addition of clay, they adhere to the surface of sand particles, affecting the contact behavior between sand particles. These clay particles are lost during seepage erosion. Degradation caused by seepage-induced erosion of clay is considered and addressed in Section 3.2. Since only the damage phenomena can be obtained from the model test and the differences in the mechanism of erosion cannot be analyzed in depth, the seepage erosion in different strata is further analyzed by numerical simulation. From the test results, it can be seen that the seepage erosion caused by tunnel leakage is mainly concentrated in the vertical plane perpendicular to the tunnel axis where the seepage point is located, so a two-dimensional coupled flow-solid model is established for simulation. FDM and DEM are coupled in the numerical model, in which the soil is simulated by DEM and fluid is simulated by FDM. The forces applied to the soil particles in the flow field meet Equation (1):  
\[\frac{\partial\overset{\rightarrow}{u}}{\partial t}=\frac{\overset{ \rightarrow}{f}_{mech}+\overset{\rightarrow}{f}_{fluid}}{m}+\overset{ \rightarrow}{g} \tag{1}\]  
Figure 8: Phenomena of soil C: (**a**) leakage at the tunnel top; (**b**) soil remains stable after 120 min. where \(\overrightarrow{u}\) is the particle velocity, \(m\) is the particle mass, \(\overrightarrow{f}_{fluid}\) is the hydraulic force applied on the particle, \(\overrightarrow{f}_{mech}\) is the sum of the external forces acting on the particle (externally exerted force and contact force between the particles), and \(\overrightarrow{\text{g}}\) is the acceleration of gravity. The force \(\overrightarrow{f}_{fluid}\) acting on the particles is mainly composed of the drag force generated by the fluid [24], which can be calculated according to the following equation:  
\[f_{drag}=V_{i}\gamma_{w}i \tag{2}\]  
where \(V_{i}\) is the particle volume, \(\gamma_{w}\) is the unit weight of water, and \(i\) is the hydraulic gradient. To realize the bidirectional coupling between fluid and soil, the influence of the soil on the flow field is reflected through the change of permeability coefficient. For granular materials, the permeability coefficient can be obtained by converting the porosity, which is based on the Kozeny-Carman equation [25]:  
\[k=\frac{1}{C_{s}S_{s}^{2}T^{2}}\frac{\gamma_{w}}{\mu}\frac{e^{3}}{1+e} \tag{3}\]  
Here, \(C_{s}\) is the shape factor of the granular material, \(S_{s}\) is the surface area per unit volume of the soil, T is the tortuosity factor, \(\mu\) is the dynamic viscosity, and \(e\) is the void ratio. To realize the bidirectional coupling calculation, the DEM and FDM joint calculation method is adopted in the model, as shown in Figure 9. The DEM analysis was performed with the commercial DEM software PFC 5.0 [26]. The FDM calculation was performed with the commercial FDM software FLAC 5.01 [27]. The influence of the flow field on particle behavior is simulated by calculating seepage forces in the FDM model. These seepage forces are then applied to the particles in the DEM model to simulate the effect of the flow field on particles. Conversely, the impact of particles on the flow field is achieved by altering the permeability coefficient in the FDM model. Soil particles undergo movement, resulting in changes in their porosity, which is then converted into a permeability coefficient.