# Seepage Characteristics Study of Single Rough Fracture Based on Numerical Simulation
## 3 Results and Discussion  
### Fracture Roughness, Discharge per Unit Width, and Hydraulic Gradient  
In this paper, 2D model was used to simulate the seepage in rough single fracture. The simulation roughness elements' shapes analyzed were triangular, rectangular and sinusoidal. The height of roughness elements, i.e., absolute roughness, was 1 mm, the density of roughness elements \(A\) (\(A=b/a\), where \(a=1\) mm, \(b=4\), 5, 6 mm) was 4, 5, and 6 respectively, and the fracture aperture was 3, 4, 5, 6, and  
Figure 8: _Cont._  
Figure 7: The relations between discharge per unit width \(q\) and hydraulic gradient \(J\) in fractures with rectangular roughness elements of different densities (**a**) \(A=6\), (**b**) \(A=5\), and (**c**) \(A=4\). Equation (8) shows that there is a power exponential function relation between discharge per unit width \(q\) and hydraulic gradient \(J\) as follows:  
\[q=K_{c}J^{m} \tag{9}\]  
where \(K_{c}\) is coefficient of hydraulic gradient; \(m\) is termed hydraulic gradient index (Table S1). According to the simulation results under the above different conditions, the fitting parameters \(K_{c}\) and \(m\) were obtained, as shown in Table 2.  
It can be seen from the simulation results (Figures 6-8) that the discharge per unit width \(q\) is mainly affected by the fracture aperture: a larger fracture aperture results in larger seepage flow. At the same time, the flow is also affected by the roughness element density and relative roughness. Table 2 shows that the variation range of \(m\) value is 0.586-0.934, all less than 1. When the shape and density of the roughness elements remain unchanged, a decrease in fracture aperture will cause an increase of \(m\) value, and it means that, in narrower fractures, the influence of hydraulic gradient is more obvious. When the \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{A}\) *} & \multicolumn{3}{c}{**Triangular**} & \multicolumn{3}{c}{**Rectangular**} & \multicolumn{3}{c}{**Sinusoidal**} \\ \cline{2-10}  & \(\overline{\boldsymbol{z}}\) (mm) ** & \(K_{c}\) & \(m\) & \(\overline{\boldsymbol{z}}\) (mm) ** & \(K_{c}\) & \(m\) & \(\overline{\boldsymbol{z}}\) (mm) ** & \(K_{c}\) & \(m\) \\ \hline \multirow{4}{*}{6} & 2.33 & 4.64 & 0.714 & 1.73 & 6.15 & 0.892 & 1.82 & 5.66 & 0.787 \\  & 3.21 & 8.68 & 0.652 & 2.48 & 152.76 & 0.779 & 2.68 & 100.35 & 0.691 \\  & 4.09 & 139.63 & 0.627 & 3.22 & 214.68 & 0.690 & 3.53 & 165.89 & 0.657 \\  & 4.96 & 185.24 & 0.612 & 3.97 & 255.05 & 0.626 & 4.39 & 206.83 & 0.624 \\  & 5.84 & 239.21 & 0.598 & 4.72 & 305.13 & 0.586 & 5.24 & 270.23 & 0.617 \\ \hline \multirow{4}{*}{5} & 2.24 & 4.70 & 0.726 & 1.57 & 6.20 & 0.910 & 1.65 & 5.48 & 0.795 \\  & 3.10 & 8.88 & 0.661 & 2.29 & 160.04 & 0.799 & 2.48 & 107.87 & 0.709 \\  & 3.97 & 138.72 & 0.633 & 3.00 & 240.27 & 0.723 & 3.32 & 163.41 & 0.660 \\  & 4.83 & 182.05 & 0.607 & 3.71 & 285.63 & 0.657 & 4.15 & 211.22 & 0.634 \\  & 5.69 & 234.72 & 0.593 & 4.43 & 402.73 & 0.641 & 4.98 & 277.11 & 0.616 \\ \hline \multirow{4}{*}{4} & 2.06 & 4.51 & 0.739 & 1.33 & 6.40 & 0.934 & 1.38 & 5.22 & 0.815 \\  & 2.89 & 8.86 & 0.674 & 2.00 & 219.19 & 0.848 & 2.18 & 9.60 & 0.708 \\ \cline{1-1}  & 3.72 & 133.68 & 0.637 & 2.67 & 339.72 & 0.759 & 2.98 & 150.47 & 0.670 \\ \cline{1-1}  & 4.55 & 179.24 & 0.613 & 3.33 & 423.07 & 0.691 & 3.78 & 201.23 & 0.639 \\ \cline{1-1}  & 5.37 & 234.95 & 0.600 & 4.00 & 528.74 & 0.658 & 4.58 & 260.81 & 0.623 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The fitting values of parameters \(K_{c}\) and \(m\). Figure 8: The relations between discharge per unit width \(q\) and hydraulic gradient \(J\) in fractures with sinusoidal roughness elements of different densities (**a**) \(A=6\), (**b**) \(A=5\), and (**c**) \(A=4\). shape, height of the roughness elements, and fracture aperture remain unchanged, a higher density of the roughness elements will also cause an increase of \(m\) value, which means that, in rougher fracture, the influence of hydraulic gradient is also more obvious.