## 2 Numerical Model for Earth Pressure Balance (EPB) Shield Tunnelling Simulation
### Ground Formation with Coupled Discrete Element Method-Finite Difference Method (DEM-FDM)  
The DEM-FDM coupling program can generate a ground formation for tunnelling using the following codes for PFC3D and FLAC3D: (a) one-dimensional structural element coupling, (b) wall-zone coupling, and (c) ball-zone coupling. This study mainly adopts the wall-zone coupling scheme. This method allocates the wall elements of PFC3D along the zone faces of FLAC3D. The created walls allow the contact force and moment applied by discrete elements on the walls to be transmitted to gridpoints of the zones. Conversely, information about the zone gridpoints (e.g., force, displacement, and velocity) can be delivered not only to the wall elements but also to ball elements. The program guide gives further details of the coupling methods [38,39]. To reproduce the initial stress condition in the ground model based on wall-zone coupling, the model has to be gravitationally stabilized before installation of the EPB model. The ground is formulated and stabilized as follows. 1. Wall compaction. Within the specified DEM domain, after the ball elements are generated using the conditions determined by preliminary calibration (Table 5), 6 walls (12 facets) are installed around the ball assembly. Excluding the side where tunnelling \begin{table}
\begin{tabular}{c c c} \hline \hline
**Calibrated Contact Parameters of Adhesive Rolling Resistance Linear Contact Model** & **Value** \\ \hline Normal stiffness, \(k_{\text{in}}\) [MN/m] & 4.5 \\ Shear stiffness, \(k_{\text{s}}\) [MN/m] & 3 \\ Friction coefficient, \(\mu\) [\(\cdot\)] & 0.07 \\ Reference gap, \(\beta_{\text{r}}\) [mm] & 0 \\ Effective modulus, \(E^{\ast}\) [MN/m\({}^{2}\)] & 10 \\ Rolling friction coefficient, \(\mu_{\text{r}}\) [-] & 0.03 \\ Normal critical damping ratio, \(\beta_{\text{r}}\) [-] & 0.05 \\ Shear critical damping ratio, \(\beta_{\text{s}}\) [-] & 0.05 \\ Maximum attractive force, \(F_{0}\) [N] & 0.01 \\ Attraction range, \(D_{0}\) [mm] & 2.5 \\ \hline \hline \multicolumn{3}{c}{**Calibrated Mohr–Coulomb Failure Criterion Parameters**} \\ \hline
**Parameter** & **Calibrated Value** & **Target Value** & **Error [\%]** \\ \hline Internal friction angle, \(\varphi\) [\({}^{\circ}\)] & 27.22 & 27.3 & 0.29 \\ Cohesion, \(c\) [kPa] & 26.17 & 26.3 & 0.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calibrated parameters after large-scale triaxial tests. Figure 5: Results of large-scale triaxial tests for calibration of contact parameters: (**a**) stress–strain curve; (**b**) Mohr–Coulomb failure envelope. is initiated and the bottom wall, the designated loads are applied through five walls (one top and four side walls) with a servo-mechanism. On the top wall, vertical earth pressure is loaded considering the cover depth. On the four side walls, the average lateral earth pressure is applied considering the lateral earth pressure coefficient at rest (\(K_{0}\) = 0.5; Figure 6a,b). 2. Wall-zone compaction. Zone elements are generated around wall-compacted balls to simulate the rest domain of the ground formation. Along zone faces of the FDM and DEM domain boundaries (i.e., wall-zone boundaries), walls are created by wrapping zone faces. After that, input parameters (Table 5) are applied to zones and time-stepping is conducted with gravitational acceleration until the convergence criterion is satisfied. During wall-zone compaction, a quarter of the target elastic modulus (\(E\) = 6 MPa) is applied as zones should be sufficiently deformed to guarantee efficient transmission of stress into the DEM domain. Upon completion of ground stabilization, the elastic modulus is modified to the designated value (\(E\) = 24 MPa; Figure 6c,d). Figure 6: Ground formation modeling procedure: (**a**) wall compaction (front view); (**b**) wall compaction (side view); (**c**) wall–zone compaction (front view); (**d**) wall–zone compaction (sectional view).