## 3 Validation of the Analytical Solutions  
### Comparison to Numerical Simulation Results  
To verify the correctness of the proposed analytical solutions, the predicted results by the numerical simulations conducted by FLAC3D are first compared with those obtained by the analytical approach. The fluid-solid coupling model was used here [23], as both the NG and RG are ideal elastic-plasticity bodies that obey the Mohr-Coulomb criterion. Since the analytical model and the applied loads are both centrosymmetric, 1/4 of the spatial domain was adopted to speed up the calculations, and the boundary conditions for this situation are displayed in Figure 5. To eliminate the influence of the boundary effect, the tunnel radius \(r_{0}\) was taken as 1 m, and the boundary range radius \(R_{0}\) was selected as \(20r_{0}\). The in situ stress of \(p_{0}=3.5\) MPa and the seawater pressure of \(p_{a}=p_{e}=1.5\) MPa were used in the suggested models. Since paths 4, 6, and 9 contain all the configurations and transition modes, the selection of these paths has enough pertinence and representativeness. In the numerical calculations, the stress control method was exploited such that the acting internal stress on the tunnel was reduced in 40 steps by 2.5%. The seepage field of the model was calculated first, then the radial displacement and the plastic radius were calculated at each step. The parameters selected in the calculations are given in Table 2.  
Figure 5: The developed numerical model introduced into FLAC3D: (**a**) details of the used mesh in the numerical method, (**b**) boundary conditions. Based on Table 1 and Figure 3 in combination with Reference [20], the stepwise determinations of the consecutively encountered configurations are provided in Table 3. Figure 6 shows the ground reaction curves (GRCs) for the abovementioned three paths obtained by the analytical solutions and the numerical analysis. The plotted results revealed that there was a reasonably good agreement between the analytical results and those predicted by the numerical approach, which implies that the hypotheses and derivations are reliable from a practical point of view under the explained conditions. Since the plastic zones generated in numerical models are affected by meshing factors, plastic radii cannot be accurately obtained. While the development of plastic zones is consistent with the analytical model. Furthermore, the tunnel displacement and plastic radii increased significantly with the decrease in \(\lambda\), indicating that support should be applied in time to control the development of displacement and plastic zones. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**Case I (\(r_{a}\) = 1.5 m)**} & \multicolumn{2}{c}{**Case II (\(r_{a}\) = 1.5 m)**} & \multicolumn{2}{c}{**Case III (\(r_{a}\) = 1.2 m)**} \\  & \multicolumn{2}{c}{**Processing Path 4**} & \multicolumn{2}{c}{**Processing Path 6**} & \multicolumn{2}{c}{**Processing Path 9**} \\ \cline{2-7}  & **NG** & **RG** & **NG** & **RG** & **NG** & **RG** \\ \hline Elastic modulus (GPa) & 0.8 & 2.4 & 0.8 & 2.4 & 0.8 & 2.4 \\ Poisson’s ratio & 0.4 & 0.3 & 0.4 & 0.3 & 0.4 & 0.3 \\ Cohesion (MPa) & 0.1 & 1.0 & 0.3 & 0.8 & 0.2 & 2.2 \\ Internal friction angle (\({}^{\circ}\)) & 10 & 20 & 10 & 20 & 10 & 25 \\ Dilation angle (\({}^{\circ}\)) & 20 & 20 & 25 & 10 & 20 & 10 \\ Permeability coefficient & \(6\times 10^{-4}\) & \(3\times 10^{-4}\) & \(5\times 10^{-4}\) & \(2\times 10^{-4}\) & \(5\times 10^{-4}\) & \(2\times 10^{-4}\) \\ (cm/s) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used for the sake of validation.