## 5 Engineering Application of the Proposed Method
### Adaptive Analysis of the Proposed Analytical Model in Qingdao Jiaozhou Bay Subsea Tunnel  
In order to further analyze the adaptability of the analytical solutions in the Qingdao Jiaozhou bay subsea tunnel, an additional comparison of the analytical solutions to numerical solutions was conducted. The simulation domain 200 m in width and 120 m in depth was considered here, and the boundary conditions used for the seepage analysis are shown in Figure 12. The surrounding rock parameters of NG and fiducial values of RG in Table 4 were adopted. The GRCs obtained by the numerical model using the reinforced parameters, natural parameters, and reinforcement thickness \(t_{a}\) of 5 m were compared with those by the analytical model, respectively. The results of the three conditions are shown in Figure 13. In general, the GRCs obtained with both methods are in close agreement. While the GRCs obtained by using the natural parameters without reinforcement, are of the maximum relative difference (nearly 9%) of both methods. GRCs obtained by using the reinforced parameters, assuming the entire ground was reinforced, are of the minimum relative difference (nearly 2%). Note that the case of a reinforcement thickness of 5 m corresponding to the practical condition, are of the median relative difference. In addition, according to Reference [17], when the seepage effect for tunnels with weakened rock was taken into account, the predicted GRCs at the tunnel crown obtained from the elasto-plastic analytical solution agreed well with those of the numerical analysis, even when the tunnel depth was twice the tunnel diameter. Hence, the present model can be applied to the Jiaozhou bay tunnel from a practical point of view. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Zone**} & \multirow{2}{*}{\begin{tabular}{c} **Elastic** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Cohesion** \\ **(MPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Internal** \\ **Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Permeability** \\ **Coefficient (cm/s)** \\ \end{tabular} } \\ \hline \multirow{2}{*}{RG} & NG & 1.2 & 0.4 & 0.06 & 25 & \(1.2\times 10^{-4}\) \\ \cline{2-6}  & \
\begin{tabular}{c} Fiducial value \\ Range \\ \end{tabular} & 3 & 0.34 & 0.12 & 31 & \(6\times 10^{-5}\) \\ \cline{2-6}  &
\begin{tabular}{c} Range \\ \end{tabular} & 1.2–4.8 & 0.28–0.4 & 0.06–0.18 & 25–37 & \(3\times 10^{-6}\)\(\sim\)\(1.2\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mechanical parameters of the surrounding rock for the Jiaozhou bay tunnel through f\({}_{3-2}\). Figure 12: Numerical model of the Qingdao Jiaozhou bay subsea tunnel. Since the relationships between a system property \(P\) and the factors are very complex, to simplify the calculations, Equation (13) was modified to:  
\[S_{k}(a_{k})=\left|\frac{1}{m-1}\sum_{i=1}^{m-1}\left(\frac{P_{k}^{i}-P_{k}^{i+1 }}{a_{k}^{i}-a_{k}^{i+1}}\frac{a_{k}^{i+1}}{P_{k}^{i+1}}\right)\right|, \tag{14}\]  
where \(m\) is the number of considered values for each factor \(a_{k}\), \(P_{k}^{i}\) is the system eigenvalue pertinent to \(a_{k}^{i}\). To examine the relationship between the tunnel radial displacement and each reinforced parameter, different variation ranges of each parameter were considered for analysis as displayed in Table 4, in which the reinforced thickness varied from 1 m to 7 m. When studying the influence of a certain parameter on the tunnel radial displacement, the reinforced thickness was set to be 5 m and the permeability coefficient was set to be the maximum value, while the other parameters were set to be the minimum values.