# Numerical Simulation on the Leakage-Induced Collapse of Segmental Tunnels
## 3 Numerical Model
### Soil Model  
#### 3.1.1 Description of the Coupled Fluid-Solid Model  
In this numerical model, the finite difference method (FDM) and the discrete element method (DEM) were integrated, where the DEM simulates the behavior of the soil particles while FDM simulates the fluid behavior. The soil is represented by rigid particles in DEM simulation, with their movements being independent of each other. The transmission of force between particles occurs through contacts. [PERSON]'s second law governs the relationship between force (or moment) and particle motion, where motion results from contact forces and other physical forces acting on the particles. Contact forces between particles adhere to the force-displacement law, which updates contact forces based on relative motion at contact points. Specifically, the interparticle forces can be determined according to the distance between particles and the contact law. Next, the resultant force on each DEM particle will be determined. Then, the movement of each particle will be calculated using [PERSON]'s second law, and the position of the particle will be updated after each calculation step. This approach ensures the accurate simulation of mechanical behaviors among particles, supported by previous studies demonstrating the effectiveness of DEM models in capturing soil deformation behavior [26; 27]  
The forces exerted on the soil particles within the flow field follow Equation (1):  
\[\frac{\partial\overset{\rightarrow}{u}}{\partial t}=\frac{\overset{ \rightarrow}{f}_{mech}+\overset{\rightarrow}{f}_{fluid}}{m}+\overset{ \rightarrow}{g} \tag{1}\]  
where \(\overset{\rightarrow}{u}\) is the particle velocity, \(m\) is the particle mass, \(\overset{\rightarrow}{f}_{fluid}\) is the hydraulic force applied on the particle, \(\overset{\rightarrow}{f}_{mech}\) is the sum of the external forces acting on the particle (externally exerted force and contact force between the particles), and \(\overset{\rightarrow}{g}\) is the acceleration of gravity. The force \(\overset{\rightarrow}{f}_{fluid}\) applied on the particles is mainly composed of the drag force generated by the fluid [28], which can be calculated according to the following equation:  
\[\overset{\rightarrow}{f}_{drag}=V_{i}\gamma_{w}i \tag{2}\]  
where \(V_{i}\) is the particle volume, \(\gamma_{w}\) is the unit weight of water, and \(i\) is the hydraulic gradient. To achieve bidirectional coupling between the fluid and soil, the soil's influence on the flow field was accounted for by adjusting the permeability coefficient. In granular materials, the permeability coefficient can be determined by converting the porosity. The Kozeny-Carman equation was adopted in the model [29]:  
\[k=\frac{1}{C_{s}S_{s}^{2}T^{2}}\frac{\gamma_{w}}{\mu}\frac{e^{3}}{1+e} \tag{3}\]  
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. The DEM and FDM joint calculation process can be described as follows. First, the strata model and fluid model are established, respectively, in DEM and FDM. Then, the fluid mesh and node information are transferred to the DEM program. Subsequently, the FDM software computes the flow field, transmitting the resulting pressure distribution and velocity to the DEM software. Next, the DEM software reads the fluid field, applieshydraulic forces to particles, and conducts DEM calculations. After completing the calculations, the DEM software reads the porosity of each fluid unit and converts it into an updated permeability coefficient. This coefficient is then returned to the FDM software to update the flow field parameters. In this coupling process, the coupling time was set to 0.01 s, and it was assumed that the fluid field remained unchanged during this brief period [24]. The condition of termination in this model is the number of calculation steps [24,30]. These steps are iterated until the calculation steps reach the required number.