## 4 Results and Discussion
### Seepage Velocity Field Analysis of Dislocation Fracture
The occurrence of nonlinear phenomena is closely related to the contact state and shape of the fracture surfaces. For example, when the fracture displacement is 3 mm, the contact area of the fracture surface is more significant than that at a 2 mm fracture displacement and exhibits an irregular distribution. The increase in the relative spatial fractal dimension \(D_{\Delta}\) leads to fluid accumulation and pressure at smaller apertures, forming vortices after contact points. This is consistent with the understanding of nonlinear behavior in flow presented by [PERSON] et al. [10]. The right-upward causes the vortex at 40 mm along the Y-axis inclined geometric structure with a \(10^{\circ}\) inclination, and it is influenced by the boundary shape, resulting in a flow direction change and the formation of a vortex. Regarding the deviation of the permeability \(K\) from the growth trend in Figure 11 when the fracture displacement \(d\) is 7 mm, in addition to the increased complexity indicated by the relative spatial fractal dimension \(D_{\Delta}\), the development area of the vortex is much larger than that in the flow velocity field at other fracture displacements. This is because at this fracture displacement, the larger aperture at 40 mm along the Y-axis occupies a larger area. Considering the variation in the fracture flow velocity field with fracture displacement \(d\) in Figure 13, it can be seen that the influence range of the vortex gradually expands with increasing fracture displacement, and it is affected by the boundary shape and fracture surface structure, impacting the development of permeability \(K\). Figure 13: Distribution of flow velocity field under different dislocation distances of fractures. Although the Reynolds number \(Re\) is lower than 300 in Table 1, the fluid velocity fields in Figure 13 show a century non-linear behavior. To better understand the nonlinear changes in the process of fracture seepage, the nonlinear factor E is introduced as in [49]:  
\[E=\frac{\text{BQ}^{2}}{\text{AQ}+\text{BQ}^{2}} \tag{11}\]  
This parameter quantifies the proportion of nonlinear behavior in the overall seepage process, providing a comprehensive perspective for understanding the complexity of fracture seepage. When the nonlinear factor \(E\) is set to 0.1, the fluid transitions from linear to nonlinear states [50], and the corresponding critical hydraulic gradient is defined as the critical hydraulic gradient \(J_{\text{c}}\)[51]. The Forchheimer equation is employed in describing the nonlinear flow characteristics of fracture seepage. The parameters of this equation have distinct meanings, where A represents the intrinsic permeability of the fracture, and B is considered a crucial parameter for describing nonlinear flow [15; 18; 45]. Therefore, an analysis of the variations in the critical hydraulic gradient \(J_{\text{c}}\) and the inertial coefficient B with the fracture displacement is presented in Figure 14.  
Figure 14: The relationship between critical hydraulic gradient and inertia coefficient with fracture dislocation distance: (**a**) variation with the critical hydraulic gradient of dislocation distance; (**b**) variation with the inertial coefficient B of dislocation distance [47]. Figure 14 shows that with the increase in fracture displacement \(d\), the critical hydraulic gradient \(J_{\text{c}}\) and the inertial coefficient B exhibit a gradually decreasing power-law relationship. The rate of decrease gradually slows down, and the nonlinear factor \(E\) under different fracture displacements is consistently greater than 0.1. This indicates that fracture seepage predominantly manifests as nonlinear seepage in this study. Figure 14b illustrates the variation in the inertial parameter B of the Forchheimer equation with fracture displacement \(d\). The curve's form aligns with the results of other researchers and is smaller than the seepage under shear conditions when the fracture experiences stress.