## 4 Results and Analyses
### Convergence of the Preceding Tunnel  
The convergence deformation of the lining in the transverse monitoring section of the preceding tunnel after the completion of the following tunnel is depicted in Figure 8. The black line in the figure shows the shape of the lining before excavation of the following tunnel and the lines in other colors show the deformed shape after excavation. In addition, the convergence deformation of lining is defined as the variation of tunnel diameter. The horizontal convergence is the variation of tunnel diameter in the horizontal direction, and the vertical convergence is the variation of tunnel diameter in the vertical direction. A positive value of convergence deformation indicates an increase in the diameter when quantifying the deformation. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Reinforcement Strategy** & **tf/D = 0.00** & **tf/D = 0.05** & **tf/D = 0.10** & **tf/D = 0.15** & **tf/D = 0.20** & **tf/D = 0.30** & **Decrease Rate** \\  & & & & & & & (\%)\({}^{\Omega}\) \\ \hline RR & 27.25/17.96 [(1)] & 21.09/13.21 & 17.11/10.02 & 15.25/8.87 & 14.62/8.06 & 13.99/7.68 & 48.7/57.2 \\ LR & 27.25/17.96 & 18.75/12.91 & 13.22/9.22 & 10.15/7.76 & 8.26/7.21 & 7.32/6.65 & 73.1/63.0 \\ TR & 27.25/17.96 & 17.56/12.56 & 12.87/9.01 & 10.21/7.51 & 8.14/7.14 & 7.21/6.51 & 73.5/63.8 \\ CR & 27.25/17.96 & 20.06/15.63 & 16.25/13.53 & 13.12/12.38 & 11.09/11.67 & 8.02/11.23 & 70.6/37.5 \\ \hline \end{tabular} Note: [(1)] The maximum horizontal displacement/the maximum vertical displacement; [(2)] decrease rate = (Value \(t\)=0.000 \(-\) Value \(t\)=0.300)/Value \(t\)=0.00D.  
\end{table} 
Table 2: The maximum value of horizontal and vertical displacements (in mm). Figure 7: The maximum displacement of the preceding tunnel: (**a**) horizontal displacement; (**b**) vertical displacement. Figure 8a shows the convergence deformation of lining for the cases of RR. With the increase in \(t\), the deformation tends to decrease. The vertical convergence is 22.4, 18.6, 15.0, 12.6, 11.2, and 10.3 mm, and the horizontal convergence values are \(-\)12.6, \(-\)8.8, \(-\)6.2, \(-\)4.1, \(-\)2.5, and \(-\)1.7 mm for \(t\) = 0.0 \(D\), 0.05 \(D\), 0.10 \(D\), 0.15 \(D\), 0.20 \(D\), and 0.30 \(D\), respectively. In the case of \(t\) = 0.0 \(D\) (i.e., the soil is not improved), the tunnel is elongated in the vertical direction while compressed in the horizontal direction and the shape of the lining resembles a standing \"duck egg\" [42,43]. As the value of \(t\) increases, the \"duck egg\" gradually becomes oblique. Figure 8b shows the convergence deformation of the linings for the LR. It can be seen that the variation of the shape with the change of \(t\) is similar to that of the RR. The vertical convergence of the linings are 22.4, 15.6, 12.7, 11.3, 10.5, and 9.7 mm and the horizontal convergence values are \(-\)12.6, \(-\)6.8, \(-\)3.2, \(-\)1.8, \(-\)1.0, and \(-\)0.2 mm for \(t\) = 0.0 \(D\), 0.05 \(D\), 0.10 \(D\), 0.15 \(D\), 0.20 \(D\), and 0.30 \(D\), respectively. As the value of \(t\) increases, the convergence decreases faster in the LR than in the RR. It indicates that the LR has a better restraint effect on convergence deformation of the preceding tunnel lining than the RR. Figure 8c shows the convergence deformation of the lining for the TR. With the increase in \(t\), the convergence deformation of the lining gradually decreases. The tendency is very close to the cases of LR. The vertical convergence values of the linings are 22.4, 12.4, 10.6, 10.1, 9.6, and 9.2 mm and the horizontal convergence values are \(-\)12.6, \(-\)4.6, \(-\)2.2, \(-\)0.6, \(-\)0.3, and \(-\)0.1 mm for \(t\) = 0.0 \(D\), 0.05 \(D\), 0.10 \(D\), 0.15 \(D\), 0.20 \(D\), and 0.30 \(D\), respectively.