## 4 Results
### Relation between Friction Resistance Coefficient and Reynolds Number  
Figure 7 shows the relationship between the friction resistance coefficient and the Reynolds number. It can be seen from the scattered points in the figure that the friction resistance coefficient decreased with the increase in the Reynolds number under the normal pressure of 0.1 to 0.4 MPa. In this study, according to the frictional resistance coefficient model shown in Table 1 and Equation (5), the frictional resistance coefficient was inversely proportional to the Reynolds number. The scattered data were fitted with the inverse proportional function. It can be seen from the fitted curve in Figure 7 that the Reynolds number and frictional coefficient of the fluid were different under conditions with the same hydraulic pressure and normal pressure. The fracture roughness affected the relationship between the frictional resistance coefficient and the Reynolds number. When the JRC of the fracture was 11.69, the roughness of the fracture was relatively tiny. The Reynolds number changed in a \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Hydraulic Pressure**} & \multicolumn{6}{c}{**Normal Pressure (MPa)**} \\ \cline{2-7}  & **0.10** & **0.15** & **0.20** & **0.25** & **0.30** & **0.40** \\ \hline
0.20 & 0.88 & 0.89 & 0.90 & 0.91 & 0.94 & 0.95 \\
0.30 & 0.78 & 0.81 & 0.86 & 0.90 & 0.93 & 0.95 \\
0.40 & 0.76 & 0.80 & 0.84 & 0.90 & 0.93 & 0.94 \\
0.50 & 0.76 & 0.77 & 0.82 & 0.88 & 0.93 & 0.93 \\
0.60 & 0.73 & 0.77 & 0.81 & 0.85 & 0.91 & 0.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rate of decline in seepage flow under different water pressure and normal pressure conditions. Figure 6: Relationship between joint roughness coefficient and seepage flow under different normal pressure and water pressure values. (**a**) Normal stress = 0.10 MPa, (**b**) Normal stress = 0.15 MPa, (**c**) Normal stress = 0.20 MPa, (**d**) Normal stress = 0.25 MPa, (**e**) Normal stress = 0.30 MPa, (**f**) Normal stress = 0.40 MPa. small range, between 10 and 300, and the frictional resistance coefficient of the fluid was also slight, between 0 and 10. When the JRC was 19.64, the roughness of the fracture was relatively large. The Reynolds number varied widely between 200 and 1000, and the frictional resistance coefficient of the fluid also varied significantly between 10 and 100. To quantitatively analyze the influence of fracture roughness on the relationship between the Reynolds number and the friction resistance coefficient, the relationship curve between the friction resistance coefficient, the Reynolds number, and roughness was established. Figure 8 shows the fitting curve between the JRC and \(\mathrm{Re}\cdot f/96\) under the normal pressure of 0.1 to 0.4 MPa. The relationship between \(\mathrm{Re}\cdot f/96\) and JRC is demonstrated in Attachment 1. \(\mathrm{Re}\cdot f/96\) increased non-linearly with the increase in JRC. When the JRC was less than 14.86, the range of \(\mathrm{Re}\cdot f/96\) values varied slightly between 0 and 30. The fitted curve was parallel to the abscissa and formed an approximately straight  
Figure 7: Relationship between the Reynolds number and the frictional resistance coefficient under different normal stresses and different JRC values. (**a**) Normal stress = 0.10 MPa, (**b**) Normal stress = 0.15 MPa, (**c**) Normal stress = 0.20 MPa, (**d**) Normal stress = 0.25 MPa, (**e**) Normal stress = 0.30 MPa, (**f**) Normal stress = 0.40 MPa. line. When the JRC was greater than 14.86, \(\text{Re}\cdot f/96\) was affected by normal pressure, and the fitting curve deviated from the straight line with the increase in JRC. The smaller the normal pressure, the greater the degree of deviation of the fitting curve. According to the fitting curve in Figure 8, the frictional resistance coefficient model of the fluid considering the fracture roughness is obtained as follows:  
\[f=\frac{96}{\text{Re}}\Big{[}1+a\big{(}\text{JRC}\big{)}^{b}\Big{]} \tag{12}\]  
where \(f\) is the friction resistance coefficient.