## 7 Conclusions  
Based on the theoretical analysis, experimental comparison and other methods, the displacement effect of grout on water in grouting is analyzed in this paper. The column penetration and diffusion mechanism, as well as the pressure attenuation calculation formula of the Bingham fluid considering the displacement effect, are obtained and discussed. The main conclusions are summarized as follows:  
1. Based on the rheological equation and the steady-state column penetration continuity equation of Bingham fluid, the column penetration diffusion mechanism of Bingham fluid considering the displacement effect is proposed. Compared with the existing penetration grouting experiments, the grout diffusion radius obtained by the penetration grouting mechanism of Bingham fluid considering the displacement is closer to the experimental value than that of Bingham fluid without considering the displacement effect.
2. The influence of the groundwater pressure on the grout diffusion radius considering the displacement effect in the grout diffusion process is more obvious than that without considering the influence of the displacement effect. The larger the grouting pressure, penetration coefficient of porous media and water-cement ratio of grout, the larger the diffusion radius of grout.
3. Using computer programming technology and the Comsol Multi-physics platform, a three-dimensional numerical simulation program for the penetration grouting mechanism of Bingham fluid, considering and without considering the displacement, is obtained. The rationality of the simulation program is verified with model experiments, which can provide support for grouting construction design.  
Figure 11: Comparison between the simulation results of the two grouting theoretical models; the blue and yellow colors, respectively, represent the simulation results of the grouting theoretical model without considering the displacement effect and while considering the displacement effect.  
**Author Contributions:** Conceptualization, Z.Y.; methodology, Z.Y., D.Z. and C.L.; formal analysis, Z.Y., D.Z. and Z.Z.; writing--original draft, D.Z. and C.L.; writing--review and editing, D.Z., C.L. and Z.Z.; software, C.L.; funding acquisition, Z.Y. and Y.Y.; supervision, Z.Y., Y.Z. and Y.Y.; validation, Z.Y., Y.Z. and Y.Y.; visualization, D.Z., Z.Z., Y.D. and M.Z.; resources W.X., N.H., X.B. and D.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 41861134008), the [PERSON] academician workstation of Yunnan Province (Grant No. 202105 AF150076), the Key R&D Program of Yunnan Province (Grant No.202003 AC100002) and the General Program of basic research plan of Yunnan Province (Grant No.202001 AT070043).  
**Institutional Review Board Statement:** Not applicable.  
**Infrared Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Acknowledgments:** The authors acknowledge the financial supports of the National Natural Science Foundation of China (Grant No. 41861134008), the [PERSON] academician workstation of Yunnan Province (Grant No. 202105 AF150076), the Key R&D Program of Yunnan Province (Grant No.202003 AC100002) and the General Program of basic research plan of Yunnan Province (Grant No.202001 AT070043), and we acknowledge the technical support of [PERSON], [PERSON] and X.L.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.