## 4 Closed-Form Solutions and Convergence Deformation of the Weilai Tunnel
### Sensibility Analysis  
The relevant parameters are examined in order to validate the proposed closed-form solutions. Parameters such as \(m_{b},\;s\), and \(k_{\phi}\) are calculated using Equations (44) and (45). Other parameters characterizing the surrounding rocks of the tunnel are shown in Table 8.  
On the basis of the GSI basic chart [50; 51; 52], the GSI values for argillaceous sandstone and sandstone are estimated. The GSI values of sandstone vary from 30 to 60, and those of argillaceous sandstone from 20 to 40. Note that the existing reality is put forward in order to evaluate the rock parameters [53; 54; 55]. In fact, GSI values are low for argillaceous sandstone, while those of pure sandstone are somewhat appreciable. Thereby, different values of GSI are employed for thorough sensibility analysis. Note that the sensibility analysis is carried out mainly considering the creep parameters of the surrounding rocks. The results are presented in Figures 21-24. They mainly describe the time-dependent deformation of the tunnel. Specifically, instant elastic creep deformation, primary creep deformation, and steady creep deformation are displayed in these figures. Initially, owing to the quick effects of rock excavation, the tunnel deformation increases quickly, and this corresponds to the instant elastic deformation. The latter is followed by the primary creep deformation, where there is a decreasing rate in the deformation evolution. Subsequently, the secondary creep deformation happens at a constant rate. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**GSI** & **\(P_{0}\) (MPa)** & **\(\sigma_{c}\) (MPa)** & **\(\sigma_{s}\) (MPa)** & **\(\phi^{(*)}\)** & **S** & **\(m_{b}\)** \\ \hline
20 & 10 & 30 & 12 & 47 & 0 & 0.06 \\
40 & 10 & 35 & 12 & 1.2 & 0.00035 & 0.804 \\
60 & 10 & 35 & 12 & 2.8 & 0.0048 & 2.084 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Parameter characteristics of the surrounding rocks. Figure 21: Time-dependent deformation of the tunnel considering different values of G\({}_{1}\) and GSI. Figure 22: Time-dependent deformation of the tunnel considering different values of G\({}_{2}\) and GSI. Crep parameters have significant influence on the evolution of creep deformation. When these parameters (for instance \(G_{1},\ G_{2},\ G_{p},\ \eta_{3}\)) decrease, there is increased time-dependent deformation of the tunnel. This conforms well to the real situation. Thereby, Figures 21-24 give the long-term trend in the creep deformation of the host rocks. The relation between the tunnel wall deformation and the depth of the plastic zones is also studied. Figures 25-28 illustrate this relation. They reveal that the deformation of the surrounding rock diminishes as the distance from the tunnel wall augments. Initially, the decrease in the deformation of the host rocks is rapid. Afterwards, the deformation keeps decreasing at a constant rate. Consequently, no matter the GSI values, the maximum creep deformation can be found at the tunnel wall, in accordance with the actual situation. It should be noted that the tunnel deformation increases according to the tendencies of the plastic radius. More specifically, the increase of tunnel wall deformation is mainly caused by the increase of the deformation. Figure 23: Time-dependent deformation of the tunnel considering different values of \(G_{p}\) and GSI. Figure 24: Time-dependent deformation of the tunnel considering different values of \(\eta_{3}\) and GSI. The strongest tunnel deformations are associated with the greatest plastic radii, which are mainly generated by the lowest GSI values. Thereby, the large deformations of deep-buried tunnels are inevitable when soft rocks are initially weakened with poor mechanical properties. Mitigating tunnel wall deformations is of primary importance to limit the development of rock creep, and guarantee long-term safety and stability.