## 2 Numerical Model  
### Preliminary Work for Numerical Model  
The numerical model was based on a previous metro tunnel project in Seoul, completed in 2014. The shield tunnel's total length is 1742 m, and its diameter is \(\sim\)8.0 m. The tunnel was constructed using a 7.73 m EPB shield TBM manufactured by a Japanese company. Table 1 lists the detailed project information. Unlike the TBM machine data, ground information for the previous project was not fully collected. Therefore, the tunneling site chosen for ground modeling comprised weathered soil with a cohesion of 26.30 kPa and an internal friction angle of 27.30\({}^{\circ}\)[24]. Based on soil shear-strength design parameters, three-dimensional direct shear tests were simulated to determine the DEM parameters for each particle and its interactions, to verify its geotechnical design parameters. Table 2 lists the DEM parameters of each particle and its interaction parameters. Given this study's simulation time and purpose, 100-mm single spheres were used. Contact forces were calculated using the Hertz-Mindlin contact model developed by various studies [25; 26; 27; 28; 29; 30] cited elsewhere [31]. Simulated, 3D direct-shear testing estimated the virtual ground model's shear strength. The tests applied each particle's DEM parameters and interactions using two large, frictionless box geometries of size 4 m (width) \(\times\) 4 m (length) \(\times\) 0.75 m (height). Figure 1a shows 26,317 randomly generated particles stacked by gravity. Tests were conducted until the shear strain reached 15% under five different normal load conditions (98, 147, 196, 245, and 294 kPa), with a shear displacement rate of 0.01 m/s (Figure 1b). The servo control system controlled the normal load for coupling work with the DEM simulation. During simulation, the reaction force estimated from the lower box in the shear displacement direction was used to calculate the shear stress, considering the changing contact surface area between two boxes. \begin{table}
\begin{tabular}{c c c} \hline
**Information Type** & **Property** & **Description** \\ \hline \multirow{6}{*}{Project outline} & Location & Seoul \\  & Construction year & 2014 \\  & Tunnel application & Metro \\  & Shield tunnel length & 1274 m \\  & Geology & Mixed ground \\  & & (Alluvial deposits–soft rock) \\ \hline \multirow{6}{*}{TBM machine data} & Shield diameter & 7.73 m \\  & Maximum cutterhead torque & 9500 kN\(\cdot\)m \\  & Cutterhead power & 1440 kW \\  & Maximum thrust & 56,000 kN \\  & Maximum screw conveyor torque & 110 kN\(\cdot\)m \\  & Screw conveyor power & 75 kW \\ \hline \end{tabular}
\end{table} 
Table 1: Project and EPB shield TBM information. \begin{table}
\begin{tabular}{c c c} \hline
**Subject** & **Property** & **Description** \\ \hline \multirow{6}{*}{Particle} & Poisson’s ratio & 0.25 \\  & Particle density & 3462 kg/m\({}^{3}\) \\  & Particle shape & Single sphere \\  & Particle diameter & 100 mm \\  & Shear modulus & 100 MPa \\ \hline \multirow{6}{*}{Interaction between particles} & Coefficient of static friction & 1.0 \\  & Coefficient of rolling friction & 0.2 \\ \cline{1-1}  & Coefficient of restitution & 0.2 \\ \cline{1-1}  & Contact model & Hertz–Mindlin \\ \hline \end{tabular}
\end{table} 
Table 2: DEM particle properties used in this numerical study. The geotechnical design parameters were compared with the numerical simulation results, in Figure 2, based on the Mohr-Coulomb failure criteria and linear regression. The cohesion and internal friction angle estimated from the numerical simulation were about 25.42 kPa and 26.72\({}^{\circ}\), respectively. Overall, the numerically estimated cohesion and internal friction angle were underestimated by about 3.3% and 2.1%, respectively, relative to those based on geotechnical design parameters. For the calibration work, this study set the shear speed to 0.01 m/s, which was much faster than the standard shear displacement rate.