## 6 Conclusions  
In this study, a series of indoor model tests were conducted to analyze the characteristics of the grouting cement returned along the pile body, and the grouting mode with different conditions was revealed. These findings benefit the understanding of the process of grouting diffusion along the pile body for the pile tip post-grouting under different load conditions and grouting conditions and could be useful for potential engineering applications, such as post-grouting piles, tunnels, and the metro train construction process.  
1. The returned height increases non-linearly with increasing grouting pressure when the grout volume is low. With the increase of the grout volume, the returned height with increasing grouting pressure is not monotonic, and the threshold value is approximately 600 kPa. The radial expansion distance with increasing grouting pressure is also not monotonic, and the threshold value is also approximately 600 kPa.
2. The returned height increases with the grout volume increased, but the radial expansion distance is maintained within the range of 5-8 cm for the grouting pressure at 500 kPa and 600 kPa. When the grouting pressure is 700 kPa, the relationship between the returned height and the grout volume is affected by the degree of unloading, but the radial expansion distance is affected by the grout volume.
3. The grout diffusion range in the loading soil is higher than that in the unloading soil and the returned height is greater than the radial expansion distance under the same grouting pressure, grout volume, and the applied normal stress of the soil.
4. Regardless of loading and unloading conditions in the soil stress history, the characteristic of the grouting cement returned along the pile body during the process of the pile tip post-grouting is mainly affected by the grouting pressure and the grout volume, whereas the transition of grouting mode is mainly affected by the grouting pressure.
5. A simplified model to describe the grouting cement diffusion pattern and the corresponding grouting mode of grouting cement returned along the pile body during the process of pile tip post-grouting was proposed. This model can comprehensively reflect the grouting cement diffusion characteristics along the pile body, with different grouting and load conditions taken into account.  
Conceptualization, Y.W. and Y.D.; methodology, Y.W.; software, Y.D.; validation, Y.W., Y.D. and X.Z.; formal analysis, Y.W.; investigation, L.L.; resources, C.Z.; data curation, J.Z.; writing--original draft preparation, Y.W.; writing--review and editing, Y.W.; visualization, Y.D.; supervision, X.Z.; project administration, Y.W.; funding acquisition, Y.W., X.Z. and C.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Natural Science Foundation of Chongqing (CSTB2023 NSCQ-BHX0149), the China Postdoctoral Science Foundation (2023 MD734112), the Special Funding of Chongqing Postdoctoral Research Project (2022 CQBSHTB3051), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202100705), the Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education Tongji University (KLE-TJGE-G2201), the National Natural Science Foundation of China (42377151 and 41672265), and the Cooperation Projects between Undergraduate University in Chongqing and Affiliated Institutes of the Chinese Academy of Sciences (HZ2021009).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data of this study are available from the corresponding author upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.