## 3 Experimental Study
### Grouting Diffusion Pattern  
In this test, the saw teeth are uniformly distributed in the vertical direction. Therefore, there is an anisotropy of the roughness in each direction, with the grouting hole acting as the center. After the slurry spreads out, the density of the saw teeth and the climbing angle encountered in each direction are different. As a result, the slurry anisotropically propagates through the rough fracture, and its diffusion distance is related to the roughness in that direction. When calculating the diffusion distance for this experiment, an anisotropic roughness correction factor \(\xi_{\theta}\) needs to be introduced into the analytical grouting diffusion model. \[\frac{dx}{dt}=\frac{\tau_{0}b}{12\mu}\times\frac{3f\Big{(}\frac{x}{r_{0}},\frac {\xi_{\theta}bP_{in}}{2r_{0}\tau_{0}}\Big{)}}{1+\frac{x}{r_{0}}} \tag{13}\]  
In the formula, \(\xi_{\theta}\) is the correction factor of roughness in different directions. It assumes that the diffusion in each direction is independent of each other during the grouting diffusion process. Taking the saw-tooth density of one per 50 mm as an example, as shown in Figure 14, it can be found that, when the slurry spreads along the \(0^{\circ}\), \(20^{\circ}\), \(70^{\circ}\), and \(90^{\circ}\) directions to a certain distance, the numbers of saw teeth encountered during the diffusion process in each direction are 4, 3, 2, and 0, respectively. Additionally, the climbing angle is different. Figure 15 shows the curve of the grouting diffusion distance in each direction changing with time. It can be seen that the grouting diffusion velocity and distance increment in each direction from \(0^{\circ}\) to \(90^{\circ}\) increase with increasing angles. Figure 13: Grouting diffusion area under different test conditions. (**a**) No saw teeth; (**b**) R1; (**c**) R2; (**d**) R3. As the relationship between the grouting diffusion distance and time is known, the test correction factor of 0.63 can be introduced to find that the goodness of fit between the test data and the theoretical solution is high. Then, the equivalent openness of the fracture is obtained by the theoretical model in reverse. In Figure 16, the equivalent openness for the grouting diffusion process becomes smaller with the increase in the saw tooth density. Taking diffusion along the x+ direction as an example, the equivalent openness under conditions R1, R2, and R3 is 3.72, 3.61, and 3.51, respectively. Figure 14: Schematic diagram of the anisotropic grouting diffusion theory. Figure 15: Diffusion distance in all directions under different test conditions. (**a**) R1; (**b**) R2; (**c**) R3. Under the same condition of the saw-tooth density, the equivalent openness exhibits a non-linear increase with the increasing angles. The roughness correction factor \(\xi_{\theta}\) for different directions can be obtained from the equivalent and actual openness. In this study, it is found that the trend of experimental results and theoretical results are almost the same. The roughness correction factor is divided from 0 to 1 with an accuracy of 0.001. Then, the test data is selected and brought into a mathematical calculation procedure. The theoretical curve approximates the test result when the value of correction factor is chosen, so as to obtain the corresponding roughness correction factor. The saw-tooth density \(\kappa\) is defined as shown in Equation (14). \[\kappa=\frac{cos\theta}{l} \tag{14}\]  
In Equation (14), \(l\) denotes the separation distance of the saw teeth, and \(\theta\) denotes the diffusion angle. Roughness correction factor under different saw tooth combination conditions are shown is Figure 17. It can be assumed that there is a linear relationship between the roughness correction factor \(\xi_{\theta}\) and the saw-tooth density in all directions, which can be expressed in the following equation. \[\xi_{\theta}=a\kappa+1 \tag{15}\]  
In Equation (15), \(a\) is the slope of the fitting curve, which can respond to the sensitivity of a change in direction to a particular roughness.