## 2 Theoretical Analysis of Temporal and Spatial Patterns of Rough Fracture Slurry Pressure
Solving for their integrals, ignoring the higher-order minors, there is an average value of the flow rate of the slurry in the fracture as follows:  
\[\overline{u}=\frac{1}{\mu\Big{(}\frac{\pi l{r_{ll}}^{2}}{u_{ll}}\Big{)}}\bigg{[} \frac{\tau_{0}l}{4}-\frac{l^{2}}{12}\frac{dp}{dr}\bigg{]} \tag{5}\]  
According to the law of mass conservation, the flow rate of slurry in the rock fractures can be expressed as:  
\[q=2\pi l{r}\overline{u} \tag{6}\]  
Combining Equation (4) with Equation (6), the relationship between slurry pressure and diffusion distance is given by:  
\[\frac{dp}{dr}=\frac{12q}{2\pi l^{3}r}\mu\bigg{(}\frac{\pi l{r}^{2}}{u_{ll}} \bigg{)}-\frac{3\tau_{0}}{l} \tag{7}\]  
Integrating Equation (7) and considering the boundary conditions, i.e., \(r=r_{ll}\) and \(p=p_{0}\), with \(p_{0}\) being the external standard atmospheric pressure:  
\[p(r_{ll},t)=\frac{q}{2\pi l}\int_{\tau_{0}}^{\sqrt{\frac{\pi}{\pi l}}}\frac{1 }{r}\mu\bigg{(}\frac{\pi l{r_{ll}}^{2}}{u_{ll}}\bigg{)}dr\frac{3\tau_{0}}{l} \bigg{(}\sqrt{\frac{qt}{\pi l}}-r\bigg{)}+p_{0} \tag{8}\]  
Combining Equation (1) and Equation (8) and integrating yields:  
\[p(r_{t},t)=\frac{qA}{4B\pi l}\bigg{(}\frac{\pi l}{q}\bigg{)}^{B}\Big{[}{r_{ll }}^{2B}-{r_{0}}^{2B}\Big{]}+\ln\frac{r_{0}}{r_{ll}}-\frac{3\tau_{0}}{l}(r_{ll }-r_{0})+p_{0} \tag{9}\]  
The flow rate \(q\) in the equation needs to be modified when it is used in rough cracks; that is, the slurry flow rate \(q^{\prime}\) in rough cracks and the slurry flow rate \(q\) in smooth cracks meet:  
\[q^{\prime}=\lambda q \tag{10}\]  
where is the \(\lambda\) flow correction coefficient. Through numerical analysis and joint roughness coefficient (\(JRC\)) index, He [36] obtained the correction coefficient expression as follows:  
\[z\frac{dp}{dr}=\tau_{0}+\mu\frac{du}{dz} \tag{4}\]0  
\[z\frac{dp}{dr}=\tau_{0}+\mu\frac{du}{dz} \tag{4}\]1 Slurry flow rate distribution. where \(JRC\) is the roughness coefficient of the crack. The grout flow rate \(q\) can be calculated according to its velocity, which is solved by [PERSON]'s equation:  
\[z\frac{dp}{dr}=\tau_{0}+\mu\frac{du}{dz} \tag{4}\]2  
\[z\frac{dp}{dr}=\tau_{0}+\mu\frac{du}{dz} \tag{4}\]3  
where \(p_{1}\) is the air pressure in the grouting pump, MPa; \(p_{2}\) is the pressure value at the outlet, which is 0.1 MPa; the flow velocity of \(v_{1}\) and \(v_{2}\) represent slurry in the grouting pump and grouting hole, m/s, respectively; \(h_{1}\) and \(h_{2}\) are, respectively, the height of the liquid level of the slurry in the grouting pump and the height of the grouting hole, m; \(\rho_{i}\) is the slurry density, kg/m\({}^{3}\); \(g\) is the acceleration of gravity, m/s\({}^{2}\). After the combined calculation, the following can be obtained:  
\[z\frac{dp}{dr}=\tau_{0}+\mu\frac{du}{dz} \tag{4}\]4  
Combined with all the above formulas, the spatial and temporal distribution equation of slurry pressure when slurry flows in rough cracks is as follows:  
\[ \begin{array}{ll}p(r_{t},t)&=\frac{qA\cos^{4}(0.802 IRC+3.7412)}{4B\pi l}\Big{(} \frac{\pi l}{l}\Big{)}^{B}\\ &\times\left[{r_{tl}}^{2B}-{r_{0}}^{2B}\right]+\ln\frac{r_{0}}{r_{tl}}-\frac{3 r_{0}}{l}(r_{tl}-r_{0})+p_{0}\end{array}  \tag{15}\]