## 2 Numerical Model Establishment and Validation
### Numerical Model Setup  
FLAC3D 5.0, a finite difference analysis software, is frequently adopted to simulate the ground response caused by pipe-jacking construction [27]. In numerical simulations of the pipe-jacking process, two primary control methodologies are commonly used: the stress control method [28] and the displacement control method [29]. For the stress control method, pipe-soil friction and jacking force must be assigned at the outset of the simulation, typically derived from various empirical or theoretical equations. Conversely, since the jacking direction and displacement are well-defined in the design of pipe jacking, it is sufficient to input only the actual jacking speed to ensure the accuracy of the simulation results in the displacement control method. Therefore, this study selects the displacement control method for simulation analysis. A FLAC 3D model was built based on the case analysis, as depicted in Figure 3. Except for the steel pipeline, which was modeled by shell elements, all other components were represented using solid elements. It should be noted that the tail void is assumed to be uniformly filled with lubricant, which is modeled as a solid element using the \"equivalent layer\" method [24; 25; 26]. Consequently, direct contact between the pipe and soil was precluded. To realistically simulate the pipe-lubricant-soil interaction during the jacking process, specific soil-pipe and pipe-lubricant contact conditions were defined. The model's cross-sectional dimensions were defined as 80 m in width and 40 m in height, with a jacking length of 60 m. To avoid boundary effects, the left and right boundaries were set at a distance equal to about 5 times the box culvert width from the tunnel boundary [29]. Meanwhile, the model was constrained to the corresponding horizontal displacement on all sides. The top surface was simulated as a free surface, whereas the bottom was fixed using rollers. Moreover, the soil was assumed to be homogeneous and modeled by the Mohr-Coulomb failure criterion, while other components, including \"the equivalent layer\", were treated as isotropic elastic bodies. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Soil** & \
\begin{tabular}{c} **Young’s** \\ **Modulus (MPa)** \\ \end{tabular} & **Poisson’s Ratio** &
\begin{tabular}{c} **Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} & **Cohesion (MPa)** & **Density (kg/m\({}^{3}\))** \\ \hline Gravelly clay & 19.4 & 0.4 & 20.7 & 0.058 & 2000 \\ Clay & 16.9 & 0.35 & 11.7 & 0.042 & 1830 \\ \hline \end{tabular}
\end{table} 
Table 1: The soil mechanical properties in the study case. Figure 2: The detailed parameters of the validated case. It is assumed that the simulation process mirrors the actual construction process, with the monitoring scheme designed based on the real project conditions (Figure 1), as delineated by the white lines (Line 1 and Line 2) in Figure 3. Additionally, special attention is given to Point O, which denotes the intersection of the monitoring lines. Firstly, the existing structure was included in the initial equilibrium step, and the displacements from this step were reset to zero to eliminate the interference of soil consolidation and the existing structure on the results. Following this, 19 steel pipelines were sequentially jacked in an L-shaped arrangement to form the new pipe roof (Figure 3, from 1 to 9). Finally, the new tunnel was completed. Throughout the jacking process, the consolidation of the lubricant injected into the tail void was considered, with its mechanical properties evolving over time, as detailed in Table 2. Additionally, a control group was established to investigate the influence of lubricant stiffness on surface settlement, featuring a lubricant with a Young's modulus of 0.23 MPa, a Poisson's ratio of 0.4, and a density of 2100 kg/m\({}^{3}\). The properties of the contact were based on [PERSON]'s direct-shear experiments, with cohesion and friction angles of 2.66 kPa and 9.89\({}^{\circ}\), respectively [30].