## 5 Discussion  
### Frictional Resistance Coefficient Model of Validation  
In the process of a fluid flowing along a rough fracture, on the one hand, the roughness of the fracture affects the relationship between the frictional resistance coefficient and the Reynolds number. On the other hand, the tortuosity of the fracture affects the relationship between the tortuous resistance coefficient and the Reynolds number. When the fluid flows through the rough fracture, the frictional resistance between the fluid and the rough fracture wall prevent the flow of the fluid, which reduce the seepage flow and increase the head loss [52]. Therefore, to further explore the specific reasons for the decrease in fluid flow and the increase in head loss, the relationship among the frictional resistance coefficient, the Reynolds number, and fracture roughness and tortuosity was established in relation to a fluid flowing through rough fractures.  
To verify the rationality of the revised frictional resistance coefficient model, the revised model was compared with the different frictional resistance coefficient models presented in Table 1. The frictional resistance coefficient and the Reynolds number were calculated using experimental data. The modified frictional resistance coefficient model and the frictional resistance coefficient models in Table 1 were compared and analyzed. The results are shown in Figure 10.  
Figure 10: Comparison of the parallel-plate model, Lomize model, Nazridoust model, Zhang model, and the proposed model under different JRC values. (**a**) Normal pressure = 0.10 MPa, (**b**) Normal pressure = 0.40 MPa.  
Figure 10 indicates that the modified frictional resistance coefficient model showed the same variation trend as the frictional resistance coefficient model in Table 1. The calculated modified friction resistance coefficient was consistent with the size of the friction resistance coefficient and matched well with the friction resistance coefficient model in Table 1. However, for different normal pressures, there were differences between the modified frictional resistance coefficient model and the model in Table 1. When the normal stress was 0.10 MPa, the modified frictional resistance coefficient model matched the model proposed by [PERSON]. When the normal pressure was 0.40 MPa, the modified frictional resistance coefficient model corresponded to the model proposed by [PERSON] The modified frictional resistance coefficient model was affected by normal stress. Normal stress affected the change in the aperture of the roughness fracture, and mainly affected the change of parameters \(a\), \(b\), which corresponds to the modified friction coefficient model. The curves of the parallel-plate model and the Lomize model were flat with little variation in their trends and were not affected by the normal stress. With the increase in the Reynolds number, the frictional resistance coefficient hardly changed, and the frictional resistance coefficient was also small, close to 0.  
Compared with the frictional resistance coefficient models presented in Table 1, the modified frictional resistance coefficient was clearly able to express the influence of the roughness and tortuosity of the fracture on the fluid seepage resistance. The modified model can accurately show the effect of fracture roughness and tortuosity on fluid flow behavior, and reasonably reveal the reasons for a decrease in seepage flow. The seepage behavior was comprehensively analyzed by thoroughly considering on-way resistance and local resistance. The modified model was obtained under the action of water pressure and normal pressure, and can thus be applied to analyze the seepage behavior of fractured rock under normal pressure. The modified model was closer to practical applications and more in line with the external environment of the fracture.