## 2 Numerical Model for Earth Pressure Balance (EPB) Shield Tunnelling Simulation
### Numerical Model and Analysis  
The EPB shield tunnelling model is completed by installing the EPB shield model within the stabilized ground formation. Before importing the machine model, the initiation of the tunnelling side wall is replaced with a wall with a hole for tunnelling. After that, the shield model is inserted up to the depth of the bulkhead wall, because unless the chamber is filled with prestressed balls, abrupt settlement at the DEM-FDM boundary will occur, and the coupling will be broken. In addition, when importing the EPB shield model, balls overlapping the TBM are eliminated to reduce numerical errors caused by the overlapping. Figure 8 presents the final configuration of the EPB shield tunnelling model with coupled DEM-FDM. The model can simulate EPB shield tunnelling under the designated TBM operational conditions. The penetration rate and rotational speed of the cutter head and screw conveyor can be manually assigned. Moreover, TBM machine data (torque, thrust force, chamber pressure, soil discharge, and ground settlement) can be gathered as the TBM advances. The torque is calculated as the summation of the vector product of the contact force vector and the position vector from the rotation axis of all cutter head elements. The sum of the contact forces in the x-direction of all EPB shield components is regarded as the thrust force. The chamber pressure is estimated as the average stress in the x-direction of the bulkhead wall, as the face pressure is generally estimated with several pressure meters installed on the bulkhead wall. Soil discharge is deduced from the number of ball elements in the model domain by deleting balls discharged outside of the model domain. Lastly, ground settlement is measured by reading the displacement of zone gridpoints. This study considers several analysis cases to examine the applicability of the simulation model for gathering the appropriate machine data. In addition to tunnelling under constant operational conditions, this study focuses on simulating tunnelling while controlling the TBM operational conditions. Before a simulation can control the operational conditions, their appropriate operational ranges have to be defined. Accordingly, those for torque, thrust force, and chamber pressure are determined using the following empirical and theoretical relationships. Equation (1) gives an empirical formula widely adopted by tunnel engineers for estimating the required cutter head torque [48; 49]. \[T=\alpha D^{3}, \tag{1}\]  
where \(T\) is the required torque (kNm), \(\alpha\) is the unitless coefficient of proportionality for the torque, and \(D\) is the diameter of the TBM (m). Figure 8: EPB shield tunnelling model using coupled DEM–FDM: (**a**) front view; (**b**) sectional view. The proportional relationship between required thrust force and TBM area is well established, because the thrust force is generally designed based on the area of the TBM [50]. Therefore, the relationship is as in Equation (2). \[F_{T}=\beta\frac{\pi D^{2}}{4}, \tag{2}\]  
where \(F_{T}\) is the required thrust force (kN) and \(\beta\) is the unitless coefficient of proportionality for the thrust force. When tunnelling with EPB shields, the face pressure is achieved by pressurizing the much within the chamber (i.e., face pressure equals the chamber pressure). Much research has sought to determine the adequate face pressure [51]; conventionally, the chamber pressure should be between Rankine's active lateral earth pressure and the lateral earth pressure at rest for preventing surface heaving or settlement, as shown in Equations (3) and (4). \[P_{c,min}=K_{a}\sigma_{v}{{}^{\prime}}, \tag{3}\]  
\[P_{c,max}=K_{0}\sigma_{v}{{}^{\prime}}, \tag{4}\]  
where \(P_{c,min}\) is the minimum chamber pressure, \(P_{c,max}\) is the maximum chamber pressure, \(K_{a}\) is the Rankine's active lateral earth pressure coefficient, \(K_{0}\) is the lateral earth pressure coefficient at rest, and \(\sigma_{v}{{}^{\prime}}\) is the vertical effective stress.