## 4 Comparison with Numerical Results  
The Jia Jiaioshan Tunnel (Chongqing, China) proposed for the China Chengdawan high-speed railway passes through the deep slow flow zone of soluble rock (Figure 6). The permeability coefficient of the surrounding rock in the water-rich high-pressure section is 1.0 \(\times\) 10\({}^{-6}\)-9.0 \(\times\) 10\({}^{-6}\) m/s. The predicted maximum water pressure is 3 MPa and the maximum water inflow is 32.5 \(\times\) 10\({}^{4}\) m\({}^{3}\)/d. A parallel adit is proposed to assist the construction of the tunnel and drain during operation to reduce the water pressure on the tunnel lining. The cross-sectional dimensions of the tunnel and parallel adit are shown in Figure 7. According to the equal permeable area principle (equal circumference of the outer profile of the section), the Jia Jiaioshan Tunnel and the proposed parallel adit are equivalent to two circular sections with radii of 6.9 m and 3.9 m, respectively. The crown position of the tunnel remains unchanged in the equivalent process. Buried depth of tunnel and water table and position parameters of parallel adit relative to the tunnel are shown in Table 1.  
Figure 6: Hydrogeological profile of the Jia Jiaioshan Tunnel.  
The following calculations are based on the parameters of the Jia Jiaioshan Tunnel (Table 1). In order to verify the accuracy of the semi-analytical results, a finite element model is established using ABAQUS (2020) software. The basic assumptions and calculation parameters of the numerical model are consistent with the analytical model. The boundary conditions of the numerical model are set as follows: The total head at the bottom and both sides of the model is 0. The upper part of the model boundary is the water table, and the presse head is 0. The drainage volume is set at the boundary both the tunnel and parallel adit. The numerical model is 9600 m wide and 2000 m high to simulate a semi-infinite domain with a deeply buried tunnel and a parallel adit (Figure 8).  
Figure 9 shows the variation of the maximum additional hydraulic gradient \(\mid\!\!\!\!\mid\mathrm{i}_{jk}\!\mid\!\!\!\!\mid\mathrm{max}\) at the parallel adit and tunnel boundary during the iteration. Horizontal coordinates represent the iterative steps, i.e., 1-1 represents the first step of the first iteration, 1-2 represents the second step of the first iteration. The additional radial hydraulic gradient decreases quickly, indicating that the analytical algorithm has a fast convergence speed. After two iterations, \(\mid\!\!\!\!\mid\mathrm{i}_{p,1}\!\mid\!\!\!\!\mid\mathrm{max}/\mathrm{i}_{p}=0.00796\%\). Termination condition (11) is met, and the iteration  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(K\)/(m\(\cdot\)s\({}^{-1}\)) & \(d_{w}\)/m & \(d_{l}\)/m & \(r_{l}\)/m & \(r_{p}\)/m & \(S_{h}\)/m & \(S_{v}\)/m \\ \hline \(5\times 10^{-6}\) & 80 & 360 & 6.9 & 3.9 & 14 & 1.40 \\ \hline \end{tabular}
\end{table}
Table 1: Calculation parameters.  
Figure 8: Numerical model of the drainage schemes for the tunnel and parallel adit.  
Figure 7: Cross-section of the proposed Jia Jiaioshan Tunnel and parallel adit.  
ends. Theoretical calculations show that the convergence rate is related to the relative positions of the parallel adit and tunnel, burial depth, and drainage volume.  
Figure 10 shows the pressure head distribution around the tunnel. The left half of the figure is the numerical result, and the right half is the semi-analytical result. The semi-analytical solution is consistent with the numerical solution, which verifies the semi-analytical solution. The numerical results are slightly larger than the semi-analytical results because the numerical model is not a true semi-infinite domain. The larger the range of the model, the closer the semi-analytical solution is to the numerical solution.