## 3 Case Analysis and Model Verification
### Model Verification
In contrast, the error and root mean square error obtained in this study by considering the temporal variation in grout viscosity were smaller (5.29% and 9.83 kPa). Therefore, considering the temporal variation in grout viscosity can significantly improve the accuracy of the model. The variation of the grout pressure mainly depends on the radial dissipation of the grout pressure. With the increasing of grout diffusion distance, more dissipation of the grout pressure occurs [33]. The grout diffusion distance \(D\) after grout pressure dissipation for 1 h is shown in Figure 8.  
The result considering the temporal variation in grout viscosity shows a significant decrease in diffusion distance. The results of the grout diffusion distance \(D\), diffusion radius \(R_{s}\), and grout dissipation pressure after grout pressure dissipation for 1 h are listed in Table 4.  
Figure 7: Theoretical values and field measurements of the grout pressure during longitudinal and radial diffusion phase (1 h). When the temporal variation in grout viscosity is considered during the calculation process, the \(R_{s}\) at the top (\(\theta=0^{\circ}\)), the sides (\(\theta=90^{\circ}\)), and the bottom (\(\theta=180^{\circ}\)) of the tunnel are 5.00 m, 5.03 m, and 5.04 m, respectively. The radial dissipation value of the grout pressure is 27.2 kPa, 41.2 kPa, and 44.3 kPa, respectively. Compared with the results considering the temporal variation in grout viscosity, the results show a larger diffusion range and greater dissipation pressure without considering the temporal variation in grout viscosity. With the increase in the grout viscosity, the grout diffusion distance and dissipation pressure are reduced [42]. Therefore, the temporal variation in grout viscosity has an important influence on the radial dissipation process of the grout pressure. The variation of the diffusion radius \(R_{s}\) with the diffusion time is shown in Figure 9a. The variation of the radial dissipation value of the grout pressure \(\Delta P\) with the diffusion radius \(R_{s}\) is shown in Figure 9b. With the increase in diffusion time, the grout diffusion radius shows a non-linear increase, and the increasing rate gradually slows down. In addition, as the diffusion time  
Figure 8: The grout diffusion distance \(D\) after grout pressure dissipation for 1 h.  
Figure 9: The relationship between the diffusion radius \(R_{s}\) and diffusion time (**a**), and the radial dissipation value of the grout pressure \(\Delta P\) (**b**). \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\theta\)/\({}^{\circ}\) & \(D\)/cm & \(R_{s}\)/m & \(\Delta P\)/kPa \\ \hline Considering temporal variation in grout viscosity & 0 & 27.7 & 5.00 & 27.2 \\ \hline Ignoring temporal variation in grout viscosity & 90 & 30.4 & 5.03 & 41.2 \\ \hline Ignoring temporal variation & 180 & 31.0 & 5.04 & 44.3 \\ \hline Ignoring temporal variation in grout viscosity & 0 & 33.2 & 5.06 & 46.8 \\ \hline \end{tabular}
\end{table} 
Table 4: The grout diffusion distance \(D\), diffusion radius \(R_{s}\), and grout dissipation pressure after 1 h.  
increases, the influence of the temporal variation in grout viscosity on the grout diffusion radius is more significant. An increase in the diffusion radius leads to an increase in the radial dissipation value of the grout pressure. When considering the temporal variation in grout viscosity, the pressure dissipation is faster owing to the increasing viscosity. Compared with the grout pressure variation during the circumferential distribution and radial dissipation process of the grout pressure, the grout pressure dissipation values during the longitudinal filling process are not related to the cross-sectional location of the grout. The dissipation value of the grout pressure is about 4.9 kPa within 1 h. Therefore, the grout viscosity variation has little effect on the longitudinal grout pressure.