## 4 Model Verification  
As this project is still in the foundation treatment phase, there are no monitoring data available for model verification during tunnel excavation. Therefore, an empirical formula to predict surface settlement was used. [PERSON] [15] proposed that ground settlement follows a normal distribution pattern by summarizing a substantial amount of field-measured settlement data caused by tunnel construction at that time. The surface settlement troughs induced by single-track tunnel excavation are calculated using the [PERSON] empirical formula, which is presented as follows:  
\[s(x)=\frac{AV_{l}}{i\sqrt{2\pi}}exp\biggl{(}-\frac{x^{2}}{2i^{2}}\biggr{)} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Types** & **Conditions** & **Schemes** \\ \hline \multirow{6}{*}{Support pressures \(\mathrm{P_{w}}\) (G-0)} & \multirow{2}{*}{ZH-0} & Equal to 0.9 times the horizontal static water and soil pressure of the tunnel axis \\  & & Equal to 1.0 times the horizontal static water and soil pressure of the tunnel axis \\  & & Equal to 1.1 times the horizontal static water and soil pressure of the tunnel axis \\  & & Equal to 1.2 times the horizontal static water and soil pressure of the tunnel axis \\ \hline \multirow{3}{*}{Grouting pressures \(\mathrm{P_{g}}\) (G-0)} & \multirow{3}{*}{ZJ-0} & Equal to 0.9 times the vertical stress of the vault stratum \\  & & ZJ-1 & Equal to 1.0 times the vertical stress of the vault stratum \\  & & ZJ-2 & Equal to 1.1 times the vertical stress of the vault stratum \\  & & ZJ-3 & Equal to 1.2 times the vertical stress of the vault stratum \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Main parameters of shield. Figure 6: Setting of data recording points. where \(A\) is the excavation area; \(s(x)\) is the surface settlement at a distance x from the tunnel centerline; \(i\) is the distance from the symmetry center of the settlement curve to the inflection point of the curve; and \(V_{I}\) is the stratum loss ratio. Based on the superposition principle, the formula for calculating the surface settlement trough induced by the excavation of a twin-tunnel is obtained as follows:  
\[s(x)=\frac{A_{1}V_{I1}}{i_{1}\sqrt{2\pi}}exp\left[-\frac{(x-D/2)}{2i_{1}^{2}} \right]+\frac{A_{2}V_{I2}}{i_{2}\sqrt{2\pi}}exp\left[-\frac{(x+D/2)}{2i_{2}^{2 }}\right] \tag{2}\]  
where \(A_{1}\) and \(A_{2}\) are the cross-sectional areas of the first and second tunnels, respectively; \(V_{I1}\) and \(V_{I2}\) are the stratum loss ratios caused by the construction of the first and second tunnels, respectively; \(i_{1}\) and \(i_{2}\) are the widths of the settlement troughs induced by the construction of the first and second tunnels, respectively; and \(D\) is the distance between the centers of the two tunnels. A verification was conducted using the non-arch culvert bridge section in the s-s' profile (as shown in Figure 4). The distance from the tunnel axis to the surface at this cross-section is approximately 17.8 m. For the verification of a single tunnel, the natural stratum loss ratio was set at \(V_{I}\) = 0.7%, the coefficient was set at \(k\) = 0.6, and the settlement trough width coefficient was set at \(I\) = \(kZ_{0}\) (\(i\) = 10.68) based on regional engineering experience. Since the twin-line tunnels have the same burial depth and diameter, the parameter values for the two tunnels are the same as those for the single tunnel. The comparison of settlement trough results is shown in Figure 7. It can be observed from Figure 7 that the maximum subsidence of the settlement trough is located above the left-line tunnel during the excavation of a single tunnel. When the right-line tunnel is excavated, the centerline of the settlement trough gradually shifts away from the centers of both twin-line tunnels. Additionally, the empirical formula matches well in terms of magnitude and trend with the results of the numerical simulation, indicating the accuracy and validity of the numerical model.