## 4 Reliability Analysis of High-Pressure Tunnel
### Sensitivity Analysis  
#### 4.3.1 Sensitivity Analysis of Parameters Under Single Failure Mode  
An in-depth exploration of the influence of random variables on the reliability index under each failure mode will help to assess the risk level of high-pressure tunnel engineering more accurately. The sensitivity comparison analysis curves under each failure mode are shown in Figures 9-11. The results indicate that with the increase in the COV of random variables, the reliability indexes of the three failure modes showed a gradually decreasing trend, which also means that the corresponding failure probability will gradually increase. According to the analysis of standards and relevant studies in the literature [6; 35], compared to the mechanical parameters of surrounding rock, the COV of lining mechanical parameters generally varied less and was generally less than 0.1. In addition, considering that the surrounding rock of the high-pressure tunnel belongs to hard gneiss, the COV of its surrounding rock's mechanical parameters had a small floating space within the set numerical range. From this, it can be seen that within a reasonable range of fluctuation in the COV, the reliability index for each failure mode was between 3.7 and 4.2, which met the reliability standards for hydraulic structures. Figure 10: Sensitivity analysis curve of failure mode 2. Figure 9: Sensitivity analysis curve of failure mode 1. In failure mode 1--surrounding rock convergence mode--the final radial displacement is affected by factors such as surrounding rock characteristics, lining properties, and excavation construction technology. As shown in Table 7 and Figure 9, with the increase in the COV of random variables, the reliability index showed a downward trend to varying degrees. \(E_{2}\) and \(E_{1}\) have the most significant sensitivity, while \(c\) and \(\varphi\) have lower sensitivity. When the COV changed by 0.01, the average fluctuation ranges of the reliability index of the \(E_{2}\) and \(E_{1}\) parameters were about 3% and 2.7%, respectively. The change in \(E_{2}\) will affect the constraint effect of the support structure on the surrounding rock and then indirectly affect the deformation behavior of the surrounding rock. The change in \(E_{1}\) directly affects the stiffness and deformation capacity of the surrounding rock. These two parameter variables play a crucial role in the surrounding rock convergence mode. Failure mode 2--anchor length failure mode--is mainly related to the plastic zone of the tunnel's surrounding rock, and the sensitivity variation law of the reliability index is shown in Table 8 and Figure 10. The calculation results show that the sensitivity of \(E_{1}\) was the strongest, the sensitivity of \(\varphi\) and \(E_{2}\) was the second, and the sensitivity of \(c\) was the smallest. The \(E_{1}\) parameter was critical to ensure the effective fixation of the anchor. In the design and evaluation of the anchor system, the influence of \(E_{1}\) should be given priority, and the role of the two parameters \(\varphi\) and \(E_{2}\) should also be paid attention to. These parameters play a key role in limiting the expansion of the plastic zone, ensuring the stability of the surrounding rock, and improving the safety and economy of the project. \begin{table}
\begin{tabular}{c c c c c} \hline
**COV** & \(E_{1}\) **(GPa)** & **C (MPa)** & \(\varphi\) (\({}^{\circ}\)) & \(E_{2}\) **(GPa)** \\ \hline
0.04 & Inf & 4.4706 & 4.4260 & 5.1993 \\
0.06 & 5.0690 & 4.4652 & 4.3740 & 4.8916 \\
0.08 & 4.9354 & 4.4497 & 4.2959 & 4.5127 \\
0.10 & 4.7341 & 4.4352 & 4.2811 & 4.0176 \\
0.12 & 4.4046 & 4.4129 & 4.2740 & 3.6002 \\
0.14 & 4.0393 & 4.4076 & 4.2694 & 3.2613 \\
0.16 & 3.7004 & 4.4005 & 4.2671 & 2.9915 \\
0.18 & 3.3938 & 4.3669 & 4.2640 & 2.7750 \\
0.20 & 3.1340 & 4.2694 & 4.2611 & 2.6008 \\ \hline \end{tabular}
\end{table} 
Table 7: Changes in COV and reliability indexes for failure mode 1. Figure 11: Sensitivity analysis curve of failure mode 3.