## 3 Numerical Simulation
### Analysis of Results under Different Conditions
#### (3) Relationship between Displacement and Cohesion  
The relationship between displacement and cohesion is illustrated in Figure 6. From the simulation results, it can be observed that cohesion also significantly influences both the X and Z displacements of the surrounding rock of the tunnel. As the cohesion increases, both the X and Z displacements exhibit an approximately linear decrease. When the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Baseline Value** & **Water Level (m)** & **Baseline Value** & **Cohesion (kPa)** & **Baseline Value** & **Friction Angle (\({}^{\circ}\))** \\ \hline \(c=21.0\) kPa & 218.0 & 23.1 & & 22.0 \\ \(c=21.0\) kPa & 217.0 & \(h=217.0\) m & 21.0 & \(h=217.0\) m & 20.0 \\ \(\varphi=20.0^{\circ}\) & 216.0 & \(\varphi=20.0^{\circ}\) & 18.9 & \(c=21.0\) kPa & 18.0 \\  & 215.0 & 16.8 & & 16.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variation of model parameters. Figure 3: Model calculation results. (**a**) Vertical direction (Z direction), (**b**) horizontal direction (X direction). 16 parameters. If each parameter were to be inversely computed one by one, the workload would be immense and the calculation challenging. Therefore, it is necessary to conduct a sensitivity analysis on various factors affecting the surface deformation to identify sensitive factors, thereby determining primary and secondary mechanical parameters. Sensitivity analysis assesses parameters of varying sensitivity to ensure the reliability and accuracy of numerical computations. Sensitivity analysis is a fundamental method within the realm of systems analysis. It assumes that a system consists of system characteristics P and its n factors, represented as \(P=f(x1,\ x2,\ x3,\ \ldots,\ xn)\). Starting with a baseline state \(P^{\star}=f(x1^{\star},\ x2^{\star},\ x3^{\star},\ \ldots,\ xn^{\star})\), each factor is varied within its possible range of values to study and predict the impact of these factor variations on the system characteristic \(P\). If a small change in xi can cause a significant change in \(P\), it indicates that \(P\) is sensitive to xi, and xi is considered a highly sensitive parameter of system characteristic \(P\); otherwise, it is a low sensitivity parameter. To facilitate comparative analysis of sensitivity between factors, it is necessary to normalize the factors in the system. The definition of sensitivity \(S_{k}\) is delineated as follows [18,19]:  
\[S_{k}=\frac{\left|\Delta P\right|}{P}/\frac{\left|\Delta x_{k}\right|}{x_{k}} =\left|\frac{\Delta P}{\Delta x_{k}}\right|\cdot\frac{P}{x_{k}}\qquad\qquad k =1,2,\cdots,n \tag{1}\]  
When the value of \(\frac{\left|\Delta x_{k}\right|}{x_{k}}\) is small, \(S_{k}\) can be approximately represented as the following:  
\[S_{k}=\left|\frac{d\Delta P}{d\Delta x_{k}}\right|\cdot\frac{P}{x_{k}}\qquad \qquad k=1,2,\cdots,n \tag{2}\]  
where \(S_{k}\) denotes the sensitivity of factor \(x_{k}\), where \(k=1,\ 2,\ \ldots,\ n\); \(\left|\frac{\Delta P}{P}\right|\) represents the relative change rate of the system characteristic; \(\left|\frac{\Delta x_{k}}{x_{k}}\right|\) indicates the relative change rate of a certain factor. Only one factor is varied at a time while keeping other factors constant. The baseline values and variation ranges for each factor are provided in Table 3. An analysis is conducted on the influence of water level (\(h\)), cohesion (\(c\)), and friction angle (\(\varphi\)) on surface deformation. The surface deformation values under each set of parameters are obtained. Using the sensitivity defined in Equation (1), the sensitivity of each parameter is calculated, as shown in Table 4.  
From Table 4, it can be observed that among the three parameters in the tunnel model, the sensitivity to water level variation is remarkably high at 3.7899, indicating it has the most significant impact on surface deformation. Following water level, the friction angle \(\varphi\) exhibits considerable sensitivity, while cohesion c is the least sensitive.