## 4 Discussion
### Prediction and Application of Supporting Time  
An essential application of the convergence-confinement method is to determine the timing of tunnel support. To determine the supporting time of tunnel in Xigeda strata, a combination of fitting formula given above and convergence-confinement method was used. And a convergence-confinement calculation model suitable for Xigeda strata was established. Firstly, based on the theory of virtual support force [37], the longitudinal deformation of Xigeda stratum tunnel under different excavation load release rates is obtained using FLAC3D numerical simulation software. Then, the displacement rate of rock around the tunnel was used as the criterion for judging the timing of support construction. This method holds that, in the initial stage of excavation load release, the deformation of surrounding rock increases with the release of excavation load, and the two are approximately linear. With the rise in excavation release rate, the deformation of surrounding rock will increase sharply at a particular stage, which can be regarded as the best construction time of primary support [29].  
The timing of support construction is calculated with 20% and 25% water content as typical working conditions. Figures 9 and 10 show the longitudinal deformation law of the tunnel under the two working conditions. The figures indicate that under 20% water content, the deformation increment increases when the load releasing rate reaches 60%, and the displacement release rate corresponding to the supporting time is 0.496. Then, \(\omega=20\) and \(u_{r}/u_{\max}=0.496\) are substituted into Formula (9) to obtain \(x/R=0.431\), \(x=2.67\) m, that is, the support should be applied at 2.67 m behind the excavation. Similarly, under 25% water content, the support should be applied at 1.46 m behind the excavation face.  
Figure 10: Displacement increment and displacement release rate of 25% water content. (**a**) Deformation Increment; (**b**) displacement release rate.  
Figure 9: Displacement increment and displacement release rate of 20% water content. (**a**) Deformation increment; (**b**) displacement release rate.  
on the calculation background of the project, taking the distance from the tunnel face as the control index, the supporting time under typical working conditions is given. At a water content of 20%, the support should be applied at 2.67 m behind the excavation face. At a water content of 25%, the support should be applied at 1.46 m behind the excavation face. The construction practice shows that the supporting time proposed in this paper can reduce the stress of the primary support structure to a certain extent, which is conducive to the stability and safety of the structure.  
Conceptualization, Z.Z.; methodology, T.T.; software, T.T.; validation, Y.L.; formal analysis, T.T.; investigation, T.T.; resources, Y.L.; data curation, T.T.; writing--original draft preparation, T.T.; writing--review and editing, Z.Z.; visualization, Y.L.; supervision, Z.Z.; project administration, Z.Z.; funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Science Foundation of China (No. 52074042).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflict of interest.