## 2 Numerical Model for Earth Pressure Balance (EPB) Shield Tunnelling Simulation  
The numerical modeling of EPB shield tunnelling first needs a virtual ground to be excavated. This is 23 m-deep, homogeneous, weathered granite soil formed above bedrock. The center of the TBM is driven around 18.9 m below the ground level. Table 1 summarizes the ground conditions for the simulation, including the soil type and properties, ground dimensions, and diameter of the excavation area. Figure 1 depicts the model set-up. To apply the DEM-FDM coupling scheme, the ground formation is divided into two domains: one for each method. The DEM domain is modeled with ball and wall elements using PFC3D. Through this domain, the EPB shield model advances with an excavation area 6.2 m in diameter; tunnelling can be conducted by the EPB shield model with a diameter of 6.14 m. Meanwhile, the FDM domain is composed of zone elements (grids) using FLAC3D to surround the DEM domain. This domain reduces the number of ball elements (particles) by substituting them with zone elements. This reduction in the number of ball elements can significantly decrease computational cost. Furthermore, it is possible to apply horizontal in situ stress effectively through a wall-zone coupling method. As mentioned, DEM modeling requires the contact model governing translational and rotational motion of particles to be determined for the input macroscopic strength parameters. In this study, the governing contact model has to contain an adhesion component as the soil type in the excavation area (weathered granite) has weak cohesion. Among the contact models built into the PFC3D program, the bonding-based contact model (linear contact and parallel bond model) is the most widely applied, especially for simulating the attractive force between particles for rock mechanics [40; 41; 42; 43]. However, it cannot reproduce the cohesion of soil, because once adhesion is destroyed by the separation of particles, the attractive interparticle force is not regenerated. Therefore, this study simulates the behavior of cohesive soil using the adhesive rolling resistance linear contact model, which provides the linear elastic, frictional (including rolling), and adhesive interparticle behaviors [44; 45]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline Type of soil & Weathered granite soil \\ Dry unit weight [kN/m\({}^{3}\)] & 19.5 \\ Internal friction angle, \(\varphi\) [\({}^{\circ}\)] & 27.3 \\ Cohesion, \(c\) [kPa] & 26.3 \\ Lateral earth pressure coefficient at rest, \(K_{0}\) [-] & 0.5 \\ Thickness of soil layer [m] & 23 \\ Cover depth [m] & 15.8 \\ Diameter of excavation area [m] & 6.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ground conditions of numerical model for earth pressure balance (EPB) shield tunnelling simulation. Figure 2 outlines this contact model. Figure 2a shows activation and deactivation of contact and adhesion depending on the surface gap (\(g_{s}\)) and specified attraction range (\(D_{0}\)). Figure 2b depicts the components of the contact model, specifically the contact force (\(F_{c}\)) and contact moment (\(M_{c}\)) between two ball elements. The contact force is the summation of linear force (\(F^{l}\)), dashpot force (\(F^{d}\)), and attractive force (\(F^{a}\)) expressed with normal and shear components (labeled with subscripts \(n\) and \(s\), respectively): (\(F_{c}=F^{l}+F^{d}+F^{a}\)); the contact moment is equal to the rolling resistance moment (\(M_{r}\)): (\(M_{c}=M^{r}\)). The contact force and moment depend on the following governing contact parameters: normal stiffness (\(k_{n}\)), shear stiffness (\(k_{s}\)), friction coefficient (\(\mu\)), reference gap (\(g_{r}\)), effective modulus (\(E^{*}\)), rolling friction coefficient (\(\mu_{r}\)), normal critical damping ratio (\(\beta_{n}\)), shear critical damping ratio (\(\beta_{s}\)), maximum attraction force (\(F_{0}\)), and attraction range (\(D_{0}\)). These parameters have to be calibrated before application in the model.