## 2 Analytical Model  
The characteristics of the fracture distribution in rock mass directly affect the flow process of grouting diffusion and the reinforcement effect. The actual joints in rock mass are generally rough fractures rather than smooth plate fractures, as shown in Figure 1. In order to obtain a more effective formula to reflect the actual diffusion effect, it is necessary to correct the fracture openness by more precise methods. At present, there are many ways based on different theoretical formulas to get the value of equivalent fracture openness. There is variability in the fracture openness at each location. The original average openness \(b_{O}\) can be obtained by mathematical statistics. \[b_{O}=\ \frac{1}{N}\sum_{i=1}^{N}b_{Oi} \tag{1}\]  
In Equation (1), \(b_{Oi}\) is the fracture openness of the \(i\)th measurement point, and \(N\) is the number of measurements. The grouting diffusion process in the rough fracture is difficult to be solved directly based on the structural characteristics of the irregular fracture surface. Therefore, by introducing roughness correction coefficients \(\xi\), the rough fracture original average openness is corrected to the equivalent fracture openness \(b\), as shown in Figure 1. Set \(b=\xi b_{O}\). Then, based on the established plate fracture model characterized by the equivalent fracture openness \(b\), the grouting diffusion model can be deduced. In the field of grouting engineering during tunnel construction, the cementitious slurry, one of the most widely used slurry material, is often judged to be a Bingham fluid. The diffusion diagram of a Bingham's fluid during laminar movement in parallel fractures is shown in Figure 2. Assume that the roughness-corrected equivalent fracture openness is \(b\). There is a flow core zone and initial shear stress in the movement process of the Bingham flow type slurry. The flow core's height is \(h_{0}\), the initial shear stress is \(\tau_{0}\), the shear stress at the junction position between the slurry and above and below the fracture surface is \(\tau_{w}\), the constant grouting pressure is \(P_{in}\), and the diffusion radius is \(r\). According to the Bingham flow behavior, the peak surface of grouting diffusion is a quasi-trapezoidal shape. The diffusion velocity of the flow core zone is large, and \(\tau_{w}\geq\tau_{0}\). \[\tau_{w}=\frac{dp}{dr}\frac{b}{2},\ \ \ \tau_{0}=\frac{dp}{dr}h_{0} \tag{2}\]  
\[u=-\frac{dp}{dr}\times\frac{b^{2}}{12\mu}\left[1-3\frac{h_{0}}{b}+4\left( \frac{h_{0}}{b}\right)^{3}\right] \tag{3}\]  
Equation (3) determines the average grouting diffusion velocity \(u\), where \(\mu\) is the viscosity. Figure 1: Schematic diagram of rough fracture and equivalent plate fracture. Figure 2: Calculation diagram of the Bingham fluid diffusion in a two-dimensional parallel fracture. When grouting during practical tunnel construction, the slurry spreads and fills around and through the grouting anchor, in which the grouting hole acts as the center. The diffusion diagram is shown in Figure 3. The grouting flow \(q\) in the fracture at any moment can be expressed as Equation (4). \[q=2\pi rub \tag{4}\]  
\[b_{O}=\ \frac{1}{N}\sum_{i=1}^{N}b_{Oi} \tag{1}\]0  
Equation (5) shows the relationship between the grouting flow, diffusion radius, and pressure at any time under constant-pressure-controlled conditions, which can be obtained from Equations (2), (3) and (4), respectively. These three properties are all variables. To solve the abovementioned problem, the grouting pressure gradient among the three variables needs to be eliminated by introducing a boundary condition. For the above problem, [PERSON] obtained the solution by mathematical methods [21]. In this paper, Equation (6) is obtained based on the Gustafson Model [22] for the grouting diffusion distance and time in a rough fracture under controlled constant pressure by introducing the roughness correction factor.