# Seepage Characteristics Study of Single Rough Fracture Based on Numerical Simulation
## 3 Results and Discussion
### Super-Cubic Phenomenon of Seepage in Rough Fractures  
According to Equation (8), there is a power function relation between the ratio of discharge per unit width to hydraulic gradient \(q/\gamma^{m}\) and the average fracture aperture \(\overline{e}\), which can be expressed as:  
\[\frac{q}{J^{m}}=K_{e}(\overline{e})^{n} \tag{10}\]  
where \(K_{e}\) is the parameter; \(n\) is fracture aperture index.  
The relations between \(q/\gamma^{m}\) and average fracture aperture \(\overline{e}\) were drawn according to the simulation results, as shown in Figure 10.  
The fitting equations between \(q/\gamma^{m}\) and the average aperture \(\overline{e}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(A\)\({}^{*}\)} & \multicolumn{2}{c}{**Triangular**} & \multicolumn{2}{c}{**Rectangular**} & \multicolumn{2}{c}{**Sinusoidal**} \\ \cline{2-7}  & **Fitting Equation** & \(\mathbf{R^{2}}\)** ** & **Fitting Equation** & \(\mathbf{R^{2}}\)** ** ** & **Fitting Equation** & \(\mathbf{R^{2}}\)** ** ** ** ** \\ \hline
6 & \(q/J^{m}=0.060(\overline{e})^{4.764}\) & 0.950 & \(q/J^{m}=0.868(\overline{e})^{4.090}\) & 0.919 & \(q/J^{m}=1.342(\overline{e})^{3.409}\) & 0.919 \\
5 & \(q/J^{m}=0.061(\overline{e})^{4.827}\) & 0.963 & \(q/J^{m}=1.218(\overline{e})^{4.177}\) & 0.936 & \(q/J^{m}=1.972(\overline{e})^{3.299}\) & 0.843 \\
4 & \(q/J^{m}=0.124(\overline{e})^{4.554}\) & 0.954 & \(q/J^{m}=2.552(\overline{e})^{4.214}\) & 0.931 & \(q/J^{m}=1.138(\overline{e})^{3.608}\) & 0.940 \\ \hline \end{tabular} \({}^{*}\)\(A\) is ratio of roughness elements spacing to height; ** R\({}^{2}\) is coefficient of determination in fitting.  
\end{table}
Table 3: The fitting equations between \(q/\gamma^{m}\) and the average aperture \(\overline{e}\).  
In the three types of rough fractures, the variation range of coefficient \(K_{e}\) is within the same order of magnitude, and its maximum value is within three times of the minimum value. The curves of different \(A\) values (roughness element density) in Figure 10 are very close to one another or even coincide. Therefore, only the fracture aperture index \(n\) is discussed here. The ratio of discharge per unit width to hydraulic gradient \(q/\overline{m}\) and average fracture aperture \(\overline{e}\) obtained based on the simulation results were fitted through regression analysis. Coefficient of determination and fracture aperture index \(n\) corresponding to different shapes and densities of roughness elements are shown in Table 3.  
In the fracture of triangular roughness elements, the fracture aperture index \(n\) values are 4.554-4.764; in the fracture of rectangular roughness elements, they are 4.090-4.214, and, in the fracture of sinusoidal roughness elements, they are 3.299-3.608. All are greater than 3, i.e., there is a super-cubic relation in symmetrical-uncoupled fractures between the discharge per unit width \(q\) and average aperture \(\overline{e}\). The simulation results also verify [PERSON]'s research conclusion [11].  
When studying the fracture aperture index \(n\), the change of average fracture aperture \(\overline{e}\) affects the value of relative roughness (negative correlation, the increase/decrease of the former will cause the decrease/increase of the latter), and the change of roughness density \(A\) essentially causes the change of relative roughness (positive correlation, the former and the latter increase and decrease at the same time), so it is impossible to qualitatively describe the effect of the change of roughness density \(A\) on the fracture aperture index \(n\) by controlling variables. This is also verified by the relation between the ratio \(q/\overline{m}\) of discharge per unit width to hydraulic gradient and average fracture aperture \(\overline{e}\) in Figure 10: although they increase together on the whole, the increasing trend is difficult to quantify.