# Seepage Characteristics Study of Single Rough Fracture Based on Numerical Simulation
## 3 Results and Discussion
### Effect of Rough Element Shape and Density on Permeability Coefficient  
In the study of groundwater dynamics, especially in the solution of flow equation, it is generally considered that the permeability coefficient \(K\) is a constant. However, when evaluating the scale effect of dispersion, it was found that the permeability coefficient \(K\) varied with the scale of interest, i.e., \(K\) affected by scale effect. The ideal smooth parallel plates fracture can be considered as uniform medium, and the fracture permeability coefficient has no scale effect. However, the surface of the actual fracture is rough and uneven. Due to the existence of wall roughness elements, the roughness and bending degree of the fracture surface are difficult to predict. The flow movement in the real single fracture has strong heterogeneity, so the scale effect of permeability coefficient \(K\) is also reasonable. According to the theory of hydrodynamics, in the fully developed turbulent flow, the relation between the average velocity and the hydraulic gradient is as follows:  
\[V^{2}=KJ \tag{11}\]  
where \(V\) is average velocity, \(K\) is permeability coefficient of fracture media, and \(J\) is hydraulic gradient. Hydraulic gradient \(J\) can be expressed as:  
\[J=\frac{H_{2}-H_{1}}{L} \tag{12}\]  
where \(H_{1}\) and \(H_{2}\) are the corresponding piezometric head values at points 1 and 2, respectively, and \(L\) is the horizontal distance between points 1 and 2. Then, the expression of permeability coefficient \(K\) can be obtained from Equations (11) and (12):  
\[K=\frac{V^{2}}{H_{1}-H_{2}}L \tag{13}\]  
Assuming that the seepage flow is one-dimensional, Equation (13) can be used to obtain the permeability coefficient \(K\). This assumption is reasonable in the ideal smooth parallel plates fracture, but considering the influence of the roughness of the fracture wall on the flow, many small vortices will occur in the flow after bypassing the roughness elements, and they will hinder the seepage movement. Therefore, the two-dimensional method should be used to study the seepage movement in rough fractures. Although Equation (13) may no longer be suitable for studying the scale effect of permeability coefficient \(K\) in rough fractures, it can still be used to study the correlation between permeability coefficient \(K\) and the horizontal distance \(L\) from its corresponding point to the inlet, so as to indirectly study the scale effect of permeability coefficient \(K\). In this paper, the simulation was carried out under the condition that the inlet velocity was set to 0.1 m/s in the fractures with triangular, rectangular, and sinusoidal roughness element shapes and density of 4, 5, and 6, respectively, and the observation points on the symmetrical center line were selected. Under the conditions of different roughness element densities, the relation in each observation point calculated by Equation (13) between \(K\) and \(L\) was determined. The results are shown in Table 4.  
As shown in Table 4, the relationship between the permeability coefficient \(K\) and the corresponding horizontal distance \(L\) can be expressed as a linear function of \(K=\alpha+\beta L\). Here, \(\alpha\) represents the intercept value, and its variation range is \(-0.521\)--\(-0.126\); \(\beta\) represents the slope of \(L\), which varies from 5.99-20.05. Parameters \(\alpha\) and \(\beta\) are related to fracture aperture, fracture roughness, and hydraulic gradient. In order to illustrate the influence of roughness element density on the scale effect of permeability coefficient \(K\), the simulation results are presented in Figure 11.  
As shown in Figure 11, the \(K\)-\(L\) relation diagrams under the conditions of different densities and shapes of roughness elements are drawn respectively. It can be seen that the farther away from the inlet, the greater the permeability coefficient, which indirectly proves that the permeability coefficient \(K\) has a scale effect. The closer to the outlet, the faster the \(K\) value increases.