# Back Analysis of Geotechnical Engineering Based on Data-Driven Model and Grey Wolf Optimization
## 5 Numerical Example and Application
### Application: Goupitan Experimental Tunnel  
Goupitan Hydropower Station is a landmark west-to-east transmission project located on the Wujiang River in Guizhou Province, China [27]. According to the preliminary design of the underground powerhouse, the right bank tailwater tunnel and the construction diversion tunnel pass through the soft clay rock mass. An experiment tunnel with a buried depth of 70 m was excavated to understand the rheological characteristics of clay rock. The tunnel size was 2 m in width and 2 m in height, respectively. Some monitoring points are set up to obtain the deformation of surrounding rock during tunnel excavation. Rock stratum \(S_{2h}^{1-2}\) and \(S_{2h}^{1-1}\) are located approximately 3 m below and 30 m above the tunnel, respectively (Figure 11). The 4# and 6# borehole of 7 m depth is set at the position of 11.6 m in the tunnel, where the 4# borehole is horizontal, the 5# borehole is 45\({}^{\circ}\) inclined, and the 6# borehole is vertical. Five monitoring points numbered 1-5 were arranged at a depth of 0, 1, 2, 4, and 6 m (Figure 11). Table 2 lists the deformation of the 3-day, 5-day, and 11-day at each monitoring point. The developed back analysis framework was utilized to identify the rheological properties of the rock mass based on the 3 rd, 5 th, and 11 th day monitored displacements  
Figure 11: Goupitan experimental tunnel and monitored points. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Time (Day)**} & \multicolumn{2}{c}{**Displacement (mm)**} \\ \cline{2-3}  & **4\#** & **6\#** \\ \hline
3 & 2.558 & 1.778 \\
5 & 3.789 & 2.377 \\
11 & 4.531 & 2.685 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The monitoring displacement in the monitored borehole. Figure 10: The convergence process of GWO. at 4# and 6# boreholes. For the developed back analysis, a physical model is critical to identify the rheological mechanical properties of the rock mass. In this study, the rheological properties of the rock mass include the shear modulus \(G_{1}\) and viscosity \(\eta_{1}\) for the Kelvin model, the shear modulus \(G_{2}\) and viscosity \(\eta_{2}\) for the Maxwell model. In geotechnical engineering, it is not easy to determine an analytical solution for use. Although the numerical method is commonly used to understand the rheological mechanism and deformations behavior of the rock mass, it is time-consuming in practical and large-scale geotechnical engineering. In this study, a ROM surrogate model, which replaced the numerical method, was used to improve the efficiency of back analysis. A uniform design was utilized to construct a group of 42 samples, and Fast Lagrangian Analysis of Continua (FLAC) software was utilized to solve the displacement of the tunnel wall in the rock mass. Rock masses \(S_{2h}^{1-1}\) and \(S_{2h}^{1-2}\) were regarded as [PERSON]'s material [28; 29], and their rheological mechanical properties were identified by the developed method. Table 3 lists the range of unknown rheological mechanical parameters based on laboratory tests and a field survey. Figure 12 indicates the predicted displacement comparison by the ROM and the numerical model along 4# and 6# boreholes, respectively. The calculated deformations by the ROM were found very close to the deformation computed by the numerical solution. It proves that the ROM surrogate model characterized well the nonlinear mapping relationship between unknown rheological properties and deformation of the tunnel wall in the rock mass. Hence, it could replace the numerical method in the back analysis. According to the back analysis theory and the proposed method, the rheological parameters of \(S_{2h}^{1-1}\) and \(S_{2h}^{1-2}\) are determined based on the displacement of the surrounding rock mass on the 3 rd, 5 th and 11 th days of the 4# and 6# boreholes (Table 4). The developed method identified the rheological mechanical properties in a reasonable way. Compared  
Figure 12: Displacement comparison of monitoring borehole.