## 4 Results
### Relation between Tortuous Resistance Coefficient and Reynolds Number  
Figure 9 shows the relationship between the tortuous resistance coefficient and the Reynolds number with different tortuosity. The tortuous resistance coefficient decreases with the increase in the Reynolds number under normal pressures of 0.10 MPa to 0.40 MPa. The relationship between the Reynolds number and the tortuous resistance coefficient varied with the change in the tortuosity. The tortuosity affected the relationship between the tortuous resistance coefficient and the Reynolds number. When the tortuosity of the  
Figure 8: Relationship between JRC and \(\text{Re}\cdot f/96\) under different normal stresses. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Normal Stress (MPa)** & **Parameter \(a\) (\(\times 10^{-13}\))** & **Parameter \(b\)** & **Fitting Coefficient \(\text{R}^{2}\)** \\ \hline
0.10 & 450,000 & 7.49 & 0.96 \\
0.15 & 16,900 & 8.83 & 0.96 \\
0.20 & 3390 & 9.03 & 0.96 \\
0.25 & 548 & 10.36 & 0.96 \\
0.30 & 14 & 10.77 & 0.97 \\
0.40 & 1.43 & 11.41 & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fitting parameters \(a\), \(b\) and fitting coefficient \(\text{R}^{2}\) of the JRC and \(\text{Re}\cdot f/96\) curve. rough fracture was less than 1.109, the Reynolds number varied from 10 to 700. In the range of 10-700, the tortuous resistance coefficient decreased first and became stable with the increase in the Reynolds number. The tortuous resistance coefficient fell on a straight line parallel to the abscissa. When the tortuosity of the rough fracture was greater than 1.109, the Reynolds number changed within the range of 100-1000. In this range, the tortuous resistance coefficient decreased with the increase in the Reynolds number. The tortuous resistance coefficient of the fluid decreased with the increase of in Reynolds number and was generally reduced significantly. The relationship between the tortuous resistance coefficient and the Reynolds number indicated that the influence of the tortuosity of seepage path on the fluid seepage process cannot be ignored. The tortuosity was introduced into Equation (8) to show the effect of the  
Figure 9: Relation between Reynolds number and tortuous resistance coefficient at different tortuosity under different normal pressures. (**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. tortuosity of the rough fracture on fluid seepage behavior. The expression was obtained as follows:  
\[\lambda=\frac{\tau}{\tau-1}\cdot f \tag{13}\]  
where \(\tau\) is the tortuosity of the rough fracture and \(f\) is the friction resistance coefficient. Equation (13) establishes the relationship among the tortuosity, the frictional resistance coefficient, and the tortuous resistance coefficient. The tortuous resistance coefficient can be calculated based on the frictional resistance coefficient and the tortuosity. The tortuous resistance coefficient model of the fluid considering fracture tortuosity was also obtained:  
\[\lambda=\frac{\tau}{\tau-1}\frac{96}{\text{Re}}\Big{[}1+a(\text{JRC})^{b} \Big{]} \tag{14}\]  
For the rough fractured channel, when the fluid encounters protrusions and obstacles on its seepage path, or when the contact area of the seepage path changes sharply, a vortex form and tortuous resistance are generated. According to the Bernoulli equation [49; 50; 51], fluid in the vortex region causes a decrease in seepage flow and an increase in head loss. In the above analysis, the variation curve of the tortuous resistance coefficient and the Reynolds number reveals the influence of tortuosity on the fluid seepage. The tortuous resistance coefficient model provides a reliable experimental basis for analyzing the loss of fluid seepage flow in the local scope of the seepage path. The tortuous resistance coefficient model can reflect the influence of the tortuosity of a rough fracture on the seepage flow and the local head loss.