## 2 Numerical Computation Considering Fluid-Solid Coupling
### Discrete Element Computational Model and Mesh Division  
As shown in Figure 1, a fault-tunnel-surrounding rock computational model is established ahead of the tunnel face. Selecting the center of the tunnel as the coordinate origin, the cross-section of tunnel is configured as a straight wall semi-circular arch with dimensions mirroring those of the Xianglushan Tunnel in central Yunnan, standing at 6 m in height and spanning 6.5 m. The model edge length method is 60 m, 40 m, and 40 m, respectively. The bottom interface of the model is set as a fixed interface, while the other interfaces are set as free interfaces. The model initializes with the overburden rock mass weight corresponding to a burial depth of 500 m, serving as the initial geostress on the upper boundary. The water pressure increased from 1 MPa to 2.5 MPa, and a simulation experiment was conducted every 0.5 MPa for tunnel excavation under five different working conditions. Figure 1 shows the tunnel excavation approaching the rich-water fault model. The mechanical properties of the of surrounding rock of tunnel and joints are detailed in Tables 1 and 2, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Volumetric** & **Bulk Modulus** & **Shear Modulus** & **Poisson’s** & **Cohesion \(c\)** & **Internal Friction** \\  & **Weight \(\gamma\) (kN/m\({}^{3}\))** & **K (GPa)** & **_G (GPa)_ & **Ratio \(\
u\)** & **(MPa)** & **Angle \(\varphi\) (\({}^{\circ}\))** \\ \hline Rock mass & 2000 & 1.44 & 19.4 & 0.25 & 0.35 & 28.6 \\ Fault fracture zone & 1800 & 0.48 & 6.4 & 0.25 & 0.12 & 28.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical parameters of the rock masses (modified from [PERSON] et al. [27]).  
The 3 DEC 5.20 software implements a range of elastoplastic constitutive relationships. For the block constitutive model, the Mohr-Coulomb failure criterion is employed, effectively capturing both elastic and plastic deformations within the rock mass. This model integrates fluid-structure interaction through a coupled approach, wherein the fluid pressure acting on the fault is dynamically updated as the excavation progresses. The interaction between the fluid and the solid matrix is governed by the effective stress principle, ensuring that changes in water pressure directly influence the stress state within the surrounding rock and faults. The joint constitutive model uses contact elasticity and plasticity under Coulomb slip failure, with shear and tensile responses determined by cohesion, tensile strength, and residual friction values.  
Additionally, 3 DEC features a robust Discrete Fracture Network (DFN) generation capability. The DFN module treats embedded fracture groups in the rock mass as discrete, finite-planar fractures, with each fracture approximated as circular. The geometry of these fractures in the DFN model is influenced by independent statistical distributions of various parameters, including fracture size (radius), orientation, and location. The principles of model mesh discretization are illustrated in Figure 2. This detailed methodological approach enhances the model's accuracy in simulating the complexities of fluid-structure interactions, particularly in the context of varying water pressures in fault zones.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Normal Stiffness** & **Shear Stiffness \(k\)s** & **Internal Friction** & **Cohesive \(c\) (MPa)** & **Tensile Strength** \\  & \(k_{n}\) (GPa) & (GPa) & **Angle \(\varphi\) (\({}^{\circ}\))** & & \(\sigma_{\mathbf{b}}\) (MPa) \\ \hline Joint & 18.6 & 6.2 & 30 & 0.5 & 0.45 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical parameters of the joint (modified from [PERSON] et al. [27]).  
Figure 1: Three-dimensional discrete element numerical computation model: (**a**) fault-tunnel-surrounding rock system; (**b**) excavation model of tunnel approaching the water-rich fault.  
Figure 2: Three-dimensional discrete element numerical model meshing.