## 2 Theoretical Background  
The Navier-Stokes (NS) equation [28; 29; 30] is commonly used to describe fluid flow in rock fractures. In the case of actual applications, however, the NS equation is generally homogenized to [PERSON]'s well-known law by assuming that the rock fracture is a smooth parallel-plate model. Based on [PERSON]'s law, the head loss (\(\Delta P\)) can be calculated as follows [31; 32]:  
\[-\Delta P=\frac{12\mu QL}{we^{3}} \tag{1}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Number** & **Model** & **Expression** & **Seepage State** \\ \hline
1 & Parallel-plate [18] & \(f=\frac{5\mu}{26}\) & Laminar flow \\
2 & [PERSON] and [PERSON] [6] & \(f=\frac{5\mu}{26}\left[1+9.57115\times 10^{-4}(\xi/e)^{1.1172}\right]\) & \(Re<10\) \\
3 & [PERSON], [PERSON], & \(f=\frac{12\mu}{26}(1+0.12 Re^{0.66\mu})\) & \(Re\leq 10\) \\  & [PERSON] [21] & \(f=\frac{5\mu}{26}\left[1+6(\xi/e)^{1.5}\right]\) & Laminar flow \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different frictional resistance coefficient models. where \(L[L]\) is the fracture length; \(e[L]\) is the fracture aperture; \(w[L]\) is the fracture width; \(\mu\big{[}ML^{-1}T^{-2}\big{]}\) is the viscosity of the fluid; \(Q\big{[}L^{3}T^{-1}\big{]}\) is the flow rate of the fluid; and \(\Delta P\big{[}ML^{-1}T^{-2}\big{]}\) is the pressure difference between the inlet and outlet of the fracture, also known as the head loss of the fluid. Due to the complexity of the surface geometry of natural rock fractures, the roughness and tortuosity of the fractures cause the relationship between seepage flow and head loss to deviate from [PERSON]'s law [33; 34]. When the fluid flows through the rock fracture, there are many reasons for the increase in head loss and the decrease in seepage flow. On the one hand, the friction resistance prevents the fluid flow in the rock fracture, causing the fluid seepage to produce a frictional head loss. The magnitude of the frictional head loss can be expressed by the well-known Darcy-Weisbach equation [35]:  
\[\Delta P_{f}=f\frac{L}{D_{h}}\frac{\rho v^{2}}{2} \tag{2}\]  
where \(D_{h}[M]\) is the hydraulic diameter, equal to two times that of the fracture aperture \(e\); \(\rho\big{[}ML^{-3}\big{]}\) is the fluid density; \(v\big{[}LT^{-1}\big{]}\) is the average velocity of the fluid; \(f\) is the frictional resistance coefficient; and \(\Delta P_{f}\big{[}ML^{-1}T^{-2}\big{]}\) is the frictional head loss. On the other hand, when fluid flows through obstacles or bending positions in rock fractures, it easily forms a vortex flow, resulting in additional resistance and local head loss. In a rough rock fracture, the reasons for the local head loss are different. In engineering practice, the regional head loss is usually converted into the frictional head loss, equivalent to the fracture length of \(L_{0}\). The regional head loss is written as follows:  
\[\Delta P_{j}=f\frac{L_{0}}{D_{h}}\frac{\rho v^{2}}{2} \tag{3}\]  
where \(\Delta P_{j}\big{[}ML^{-1}T^{-2}\big{]}\) is regional head loss and \(L_{0}[L]\) is equivalent length. When the fluid flows in a rough fracture channel, the fluid will produce frictional head loss and local head loss. The total head loss is the sum of the frictional head loss and regional head loss, that is:  
\[\Delta P=\Delta P_{f}+\Delta P_{j} \tag{4}\]  
The friction coefficient is a crucial parameter in describing the head loss in the process of fluid seepage, which has been widely used in geological engineering activities such as groundwater and petroleum engineering. As we all know, the frictional resistance coefficient is usually not constant. Its magnitude is mainly related to two key factors; one is the Reynolds number, which represents the seepage state of the fluid, and the other is the fracture roughness, which indicates the complexity of the geometric shape of the fractured channel [36; 37]. The frictional resistance coefficient can be determined by evaluating the Reynolds number of the fluid and the roughness of the fracture.