# Seepage Characteristics Study of Single Rough Fracture Based on Numerical Simulation
## 2 Materials and Methods  
### Theoretical Background  
The current research on fracture seepage is mainly based on the LCL:  
\[q=\frac{g\overline{e}^{3}}{12\mu}J \tag{1}\]  
where: \(q\) is discharge per unit width, mm\({}^{2}\)/s; \(g\) is gravitational acceleration, m/s\({}^{2}\); \(e\) is fracture aperture, mm; \(\mu\) is the kinematic viscosity coefficient of water, mm\({}^{2}\)/s; \(J\) is the hydraulic gradient, dimensionless. Hydraulic gradient refers to the ratio of head loss along the seepage path to the length of the seepage path; it can be understood as the mechanical energy lost by the water flow to overcome the friction resistance through the infiltration path of unit length; or the driving force that makes water flow at a certain velocity to overcome friction; or is the head drop per unit distance along the flow direction in the aquifer (the ratio of the water level difference at any two points to the distance between the two points).  
For the establishment, development, and improvement of LCL, many researchers had introduced absolute roughness \(\Delta\) and relative roughness \(\delta\), and established the relations among discharge per unit width \(q\), fracture aperture \(e\) and fracture roughness, so that LCL could be applied to study the seepage in rough fractures.  
Lomize [19]:  
\[\text{Flow of laminar}:\ q=\frac{g(\overline{e})^{3}}{12\mu}J\frac{1}{1+6(\frac{ \Delta}{\overline{e}})^{1.5}} \tag{2}\]  
\[\text{Flow of turbulent}:\ q=\overline{e}\sqrt{gJ\overline{e}}[2.6+5.1\log_{10} \left(\frac{2\Delta}{\overline{e}}\right)^{-1}] \tag{3}\]  
Louis [20]:  
\[\text{Flow of laminar}:\ q=\frac{g(\overline{e})^{3}}{12\mu}J\frac{1}{1+8.8( \frac{\Delta}{2\overline{e}})^{1.5}} \tag{4}\]  
\[\text{Flow of turbulent}:\ q=4\overline{e}\sqrt{gJ\overline{e}}\log_{10}[1.9( \frac{\Delta}{2\overline{e}})^{-1}] \tag{5}\]  
[PERSON] and [PERSON] [21]:  
\[q=\frac{g(\overline{e})^{3}}{12\mu}J\frac{1}{1+0.6(\frac{\sigma_{e}}{\overline {e}})^{1.2}} \tag{6}\]  
[PERSON] et al. [22]:  
\[q=\frac{g(\overline{e})^{3}}{12\mu}J^{m}\frac{1}{1+1.2(\frac{\Delta}{\overline {e}})^{-0.75}} \tag{7}\]  
The above formulas can be generalized as [11; 23]:  
\[q=C\frac{g}{12\mu}(\widetilde{e})^{n}J^{m}\frac{1}{1+\epsilon\delta^{\eta}} \tag{8}\]  
In the above equations: \(q\) is discharge per unit width, mm\({}^{2}\)/s; \(\overline{e}\) is average fracture aperture, mm; \(\widetilde{e}\) is the representative fracture aperture, mm; \(g\) is gravitational acceleration, m/s\({}^{2}\); \(\mu\) is kinematic viscosity coefficient of water, mm\({}^{2}\)/s; \(J\) is hydraulic gradient, dimensionless; \(C\) is the unit conversion parameter to ensure that the dimensions on both sides of the equation are consistent; \(\sigma_{e}\) is mean square deviation of fracture aperture, mm\({}^{2}\); \(\Delta\) is absolute roughness of fracture, i.e., the height of roughness element \(a\) in Figure 1, mm; \(\delta\) is relative roughness, \(\delta=\Delta/e_{m}\), i.e., ratio of roughness element height \(\Delta\) to fracture aperture \(e_{m}\), dimensionless; \(n\), \(m\), \(\eta\) are indices of fracture aperture \(e\), hydraulic gradientand relative roughness \(\Delta\), respectively, and their values obtained by different researchers are quite different; \(\varepsilon\) is a coefficient of \(\delta\).