## 5 Conclusions  
1. Based on the Bingham fluid constitutive equation, this paper establishes the grouting diffusion analytical model in a single rough fracture under constant-pressure control. As the roughness increases, the grouting diffusion velocity and the maximum diffusion distance decreased. The diffusion will tend to stagnate.  
2. By carrying out the grouting diffusion test in single rough fractures, the \"quasi-elliptical\" grouting diffusion mode under the influence of roughness is revealed. Furthermore, the roughness-corrected grouting diffusion analytical formula characterized by the saw-tooth density is established.  
The slurry diffusion rate shows a decreasing trend with the increase in roughness. The equivalent openness for the grouting diffusion process becomes smaller with the increase in the saw tooth density. Under the same conditions of the saw-tooth density, the equivalent openness exhibits a non-linear increase with the increasing angles.  
3. Based on the FEM calculation software COMSOL Multiphysics, the Bingham-Papanastasiou rheological model is introduced to establish an elaborative numerical simulation model for the Bingham fluid diffusion process in a single rough fracture. Through numerical simulation analysis, the temporal and spatial distribution characteristics of the velocity field and pressure field in the grouting diffusion process are obtained. The pressure attenuation in the x-direction is a step-type under different conditions of saw tooth combinations. Additionally, the attenuation gradient increases with the increase in the saw tooth density. The grouting diffusion velocity decreases with the increase in the saw tooth density under different conditions.  
4. Based on the isotropic roughness unit, the roughness correction factor of this analytical model of grouting diffusion is expressed quantitatively with respect to the relative height and relative inclination angle. When the relative height is greater than 0.4 and the relative inclination angle is greater than 40\({}^{\circ}\), the correction factor is lower than 0.5, and the correction effect is more obvious.  
Conceptualization, W.D., C.D., and Q.Z.; methodology, C.D.; software, C.D.; validation, W.D., C.D., and Q.Z.; formal analysis, C.D.; investigation, C.D.; resources, W.D.; writing--original draft preparation, C.D.; writing-review and editing, W.D.; supervision, W.D. and Q.Z. 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 (No. 51578408), the National Natural Science Foundation of China (No. 41977227), and the Key Technology Research of Safety during Highway Tunnel Construction under Complex Condition in Mountain Area in East Yunnan Province. This research has been supported by the National Natural Science Foundation of China (No. 51578408), the National Natural Science Foundation of China (No. 41977227), and the Key Technology Research of Safety during Highway Tunnel Construction under Complex Condition in Mountain Area in East Yunnan Province. The abovementioned support is greatly acknowledged.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Roughness** & **20\({}^{\circ}\)** & **40\({}^{\circ}\)** & **60\({}^{\circ}\)** & **80\({}^{\circ}\)** \\
**Correction Factor (\(\mathbf{\updownarrow}\))** & & & 1 & \\ \hline
0 & & & 1 & \\
0.2 & 0.884 & 0.884 & 0.876 & 0.872 \\
0.4 & 0.862 & 0.514 & 0.506 & 0.242 \\
0.6 & 0.868 & 0.510 & 0.302 & 0.240 \\
0.8 & 0.864 & 0.470 & 0.242 & 0.196 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The roughness correction factor under different roughness elements.