## 1 Introduction
[PERSON] quantitatively studied the influence of tortuosity on fluid seepage behavior according to the size of the fracture contact area. The test results showed that the larger the contact area between the fluid and the fracture, the more significant the effect of the fracture tortuosity on the fluid seepage flow [17]. The tortuosity of rough fractures complicates the fluid flow path and the contact area between the fluid and the fracture surface, resulting in additional energy losses and seepage flow losses [18; 19; 20]. The tortuosity of rough fractures is a vital factor affecting fluid seepage behavior. The tortuosity of rough fractures should be considered when studying the seepage characteristics of the fluid in rough fractures. Understanding the influence of the rough fracture tortuosity on fluid flow behavior requires further analysis and research. It can be seen from the above analysis that the roughness and tortuosity of rough fractures lead to a decrease in fluid seepage flow. Researchers have shown that the actual reason for decreasing seepage flow is the friction resistance between the fluid and the rough fracture surface [21]. The friction resistance coefficient is often used to represent the size of the friction resistance [22; 23]. [PERSON] et al. studied the relationship between the frictional resistance coefficient and groundwater seepage flow in fracture channels through laboratory experiments. The results showed that the seepage flow of the fluid decreased non-linearly with the increase in the frictional resistance coefficient [24]. [PERSON] et al. deduced the relationship between the frictional resistance coefficient and seepage flow using numerical simulation experiments. They analyzed the reasons for decreasing seepage flow in rough fractures [25]. [PERSON] et al. established a model of the relationship between seepage flow and the frictional resistance coefficient and studied the influence of frictional resistance on the fluid seepage state and seepage behavior [26]. The frictional resistance changes the seepage flow of the fluid, which significantly affects the seepage behavior of the fluid. However, due to the rough surface and complex geometry of natural rock fractures, the friction resistance cannot be deduced strictly theoretically. Manyscholars have put forward calculation models for the frictional resistance coefficient of fluids, developed through many laboratory tests and numerical simulation experiments [27]. These friction resistance coefficient models are shown in Table 1. In the models, \(f\) is the friction resistance coefficient. \(Re\) is the Reynolds number. \(e\) is the rough fracture aperture. \(\xi\) is the height of the unevenness of the rough fracture (Appendix A). It can be observed in Table 1 that it is not possible to comprehensively analyze the seepage behavior of a fluid using these friction coefficient models, and the following problems require further discussion. Firstly, the friction resistance coefficient model in Table 1 is established considering the Reynolds number, the peak asperity height, and the fracture aperture, but the specific causes of fluid flow reduction in rough fractures cannot be clearly described. Secondly, many studies show that the tortuosity of the fracture affects the flow behavior of the fluid, but the friction resistance coefficient models presented in Table 1 do not take the tortuosity of fractures into account. In addition, these friction resistance coefficient models do not consider the effects of water pressure and normal pressure on fractures in the derivation process. The studies mentioned above show that the roughness and tortuosity of the fracture affect the seepage behavior of the fluid in rough fractures. The frictional resistance between the fluid and the rough fracture surface causes a decrease in the seepage flow.