## 7 Conclusions  
This study, through theoretical analysis, laboratory experiments, and numerical simulations, has established the permeation grouting mechanism of Bingham fluid in porous media with different particle size distributions. This mechanism effectively explains the  
Figure 16: A comparison of experimental, theoretical, and numerical simulation results.  
\begin{table}
\begin{tabular}{c} **Values Obtained** \\ **from Equation (16)** \\ \end{tabular} & & **Results/cm** \\ \hline
1 & 12.95 & 18.71 & 13.21 & 19.21 & 8.65 \\
2 & 15.12 & 27.39 & 15.56 & 28.95 & 10.24 \\
3 & 13.52 & 24.85 & 13.80 & 25.74 & 9.56 \\
4 & 15.57 & 29.25 & 16.59 & 30.21 & 11.32 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of theoretical analysis, numerical simulation, and experimental results.  
impact of particle size distribution variations in porous media on the diffusion of Bingham fluid. The main conclusions of this research are as follows:  
1. Based on the theory of porous media fractal geometry and the rheological equation of Bingham fluid, a theoretical model for the infiltration grouting of Bingham fluid in porous media with different particle size distributions was established. This model reveals the infiltration grouting mechanism of Bingham fluid in porous media composed of different particle sizes.
2. The diffusion form of [PERSON] fluid in porous media conforms to the theoretical model of spherical seepage grouting. In the process of grouting test, the parallel electrical instrument can be used to control the diffusion form of slurry macroscopically. In the experiment, the diffusion radius of [PERSON] fluid in sand and gravel is 31.8% smaller than the theoretical diffusion radius considering the particle size distribution of sand and gravel. At the same time, the diffusion radius of [PERSON] fluid considering particle size distribution in sand and gravel is smaller than that without considering particle size distribution.
3. The theoretical model and numerical simulation model obtained by considering the different particle size distribution can effectively describe the infiltration and diffusion mechanism of Bingham fluid in sand and gravel with different particle size distribution. The seepage grouting model deduced by considering the particle size distribution and the established numerical simulation model are more consistent with the experimental measurements.  
In conclusion, the presented spherical permeation grouting mechanism for Bingham fluids, considering varying particle size distributions in gravelly soil, provides a more accurate depiction of the permeation and diffusion patterns of Bingham fluids in gravelly soil compared to the mechanism that neglects particle size distribution. The investigation into the influence of different particle size distributions in porous media on permeation grouting, as explored in this study, holds significance for both practical grouting engineering applications and theoretical research.  
The source of the subject, the acquisition of funds, the experimental design, and the first draft were carried out by B.X.; the experimental construction, data analysis and chart drawing were carried out by [PERSON], [PERSON], [PERSON] and [PERSON]. All authors contributed to the study. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the National Natural Science Foundation of China (52204081, 52174105).  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflict of interest.