## 6 Conclusions  
The determination of reasonable ground reinforcement has a significant impact on the surrounding rock stability of subsea tunnels. In the present study, a mechanical model for the surrounding rock accounting for ground reinforcement and seepage force was proposed. Based on multi-objective design theory, an optimization method of ground reinforcement parameters was developed. The method was then applied to the Jiaozhou bay subsea tunnel project in Qingdao. The main results are summarized as follows:  
1. By taking into account the ground reinforcement and seepage effect, the analytical solutions to six configurations with different distributions of plastic zone(s) were provided, which were then validated by numerical simulations as well as existing methods. It was shown that seepage effect has significant influence on tunnel radial displacement, which indicates the benefit of the proposed model in predicting mechanical behavior of subsea tunnels.
2. Under the given conditions, the main factors affecting the stability of the surrounding rock were identified through sensitivity analysis of the ground reinforcement parameters, which are the internal friction angle followed by the cohesive force and the reinforcement thickness according to the sensitiveness. Nevertheless, the significant effect of each parameter on the displacement is within a certain range, as the parameter increases to a certain extent, its influence would be weak enough to be ignored rationally. Therefore, there exists an economical and reasonable value for each reinforcement parameter.
3. For reasonable design of reinforcement parameters in subsea tunnels, the tunnel radial displacement and the seepage quantity should be taken into account. Aiming at this multi-objective programming problem, the calculations were performed based on the stratified sequencing method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Thickness (m)** & **Elastic modulus (GPa)** & **Internal Friction Angle (\({}^{\circ}\))** & **Cohesion (MPa)** \\ \hline Sensitivity & 1.61 & 0.14 & 4.61 & 2.72 \\ Optimal value & 31 & 1.8 & 5 & 0.12 \\ Surplus value & 1 & 0.4 & 0.3 & 0.032 \\ Actual value & 30 & 3 & 5 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Sensitivities of different reinforcement parameters.  
By application of this approach to the Jiaozhou bay tunnel, it was revealed that the predicted factors were consistent with the actual values, indicating the reliability of the proposed method.  
Conceptualization, Z.S. and D.Z.; methodology, Z.S. and Q.F.; validation, Z.S., D.Z. and Q.F.; formal analysis, Z.S.; investigation, Q.F.; writing--original draft preparation, Z.S.; writing--review and editing, Z.S. and Q.F.; project administration, D.Z.; funding acquisition, D.Z.  
This research was funded by the National Natural Science of China, grant number 51738002 and National key R&D Program of China, grant number 2017 YFC0805401.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.