## 5 Conclusions  
In this study, two boundary conditions were defined to simulate the grouting process under different injection positions, thereby revealing the flow behavior of grout within the backfilling layer. The simulation results provide a scientific basis for optimizing the layout of grouting points and for the rational control of injection pressure and flow rate. This contributes to improving the uniformity and quality of grouting, reducing construction risks, and offering valuable references for subsequent numerical simulations and on-site monitoring. The main conclusions are as follows:  
1. When grout diffuses downward, the diffusion speed is relatively fast, and the transition zone between the grout and air is longer. In contrast, when grout diffuses upward, the diffusion speed is slower, and the transition zone is shorter. Under the same time conditions, the grouting effect is better when the grouting port is at 90\({}^{\circ}\) compared to the cases at 30\({}^{\circ}\) and 150\({}^{\circ}\).
2. Under the pressure boundar y condition, the grout flow rate increases rapidly during the initial stage and reaches its peak at approximately 0.2 s. The grouting position has a noticeable effect on the peak flow rate: when grouting is applied at the upper part of the segment, the peak flow rate is higher, reaching approximately 0.25 m\({}^{2}\)/s, whereas grouting at the middle and lower parts results in a lower peak, around 0.18 m\({}^{2}\)/s. Subsequently, the flow rate gradually decreases and stabilizes, entering a steady-state phase.
3. Under velocity boundary conditions, the grouting pressure remains approximately stable during the initial stage of the process. As grouting progresses, the pressure gradually increases. The position of the grouting port has a significant influence on the pressure variation: the lower the port is positioned, the faster the pressure increases. At 2 s into the grouting process, the grouting pressures at the upper, middle, and lower ports are 16.92 kPa, 25.40 kPa, and 31.54 kPa, respectively. The abnormally high grouting pressure may be attributed to locally dense regions, which increase flow resistance and consequently result in elevated grouting pressure.
4. The analytical formula developed in this study can predict the variation in grouting pressure to a certain extent. Compared to the predicted values, the simulated pressures are lower in the early stage of grouting and higher in the later stage. Porosity, grout viscosity, and injection velocity all have significant effects on grouting pressure. As porosity decreases or injection velocity increases, the discrepancy between the numerical simulation results and the theoretical predictions becomes more pronounced,  
Figure 21: Laboratory grouting pressure measurements with fixed injection rates.  
indicating that the existing formula provides greater accuracy and applicability under low injection velocity conditions.  
Conceptualization, Y.L.; Methodology, Y.L.; Software, X.L.; Formal analysis, X.L. and Y.Z.; Resources, D.C.; Data curation, Y.Z. and D.C.; Writing--original draft, D.C.; Writing--review & editing, L.C.; Visualization, Y.Z. and L.C.; Project administration, Y.L.; Funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
Authors [PERSON], [PERSON] and [PERSON] were employed by the company China Railway 15 th Bureau Group Underground Engineering Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The sponsors had no role in the design, execution, interpretation, or writing of the study.