## 5 Conclusions  
In this study, CT scans were employed to investigate the void distribution of actual fractures under different fracture dislocation distance states. Spatial variations of void during the fracture dislocation distance process were analyzed based on parameters such as average mechanical aperture, relative spatial fractal dimension, and fracture surface roughness angle distribution. Subsequently, numerical fluid flow simulations were conducted based on the fracture void models obtained from CT experiments at different fracture dislocation distances. The impact of fracture dislocation on fracture permeability and the nonlinear behavior of fluid flow was analyzed. The main conclusions are as follows:  
1. Fracture dislocation significantly influences aperture distribution. The aperture frequency distribution obtained from CT experiments exhibits a normal distribution. As the fracture dislocation distance increases, the numerical range of the local aperture gradually transitions from relatively narrow to broader, and the maximum local aperture of fractures increases. The average mechanical aperture (\(e_{\text{m}}\)) growth follows a logarithmic function and is closely related to the roughness angle of the fracture surface. 2. The average mechanical aperture and the distribution of local apertures during fracture dislocation affect the complexity of void connectivity. With increasing fracture dislocation distance, the relative spatial fractal dimension (\(D_{\Delta}\)) of fractures shows a logarithmic relationship, indicating a decrease in void connectivity and a reduction in the complexity of the fluid flow pathways. 3. The fracture permeability increases nonlinearly with the increase in dislocation distance. Under the same injection flow rate, the seepage flow in the fracture changes from linear flow to nonlinear flow, and the dominant flow linear flow in dislocation fracture is very obvious. When the dislocation distance reaches 5 mm, nonlinear behaviors such as eddy currents occur, and the influence range of eddy currents gradually expands with the increase in dislocation distance. The fracture surface morphology structure as the boundary of seepage has a significant influence on the seepage behavior. It is also found that the change in dislocation fracture permeability with dislocation distance is closely related to the change in spatial fractal dimension. 4. With the increase in dislocation distance, the critical hydraulic gradient \(J_{\text{c}}\) and the inertia coefficient B, which distinguish linear and nonlinear flows, show a power function decrease and gradually tend to a stable value, indicating that the fluid in the fracture is more likely to produce nonlinear flow with the increase in dislocation distance. The degree of distribution of the cavity is gradually reduced, and the fluid presents a dominant flow, which enhances the nonlinear ability of fluid in the fracture. The inertia coefficient B and the critical hydraulic gradient \(J_{\text{c}}\) in the Forchheimer equation can quantitatively describe the nonlinear seepage characteristics of the fracture. In summary, this study reveals the influence of fracture dislocation on fluid flow characteristics through a combination of CT experiments and numerical simulations. This research holds crucial significance for understanding the nonlinear behavior of fracture flow and the impact of fracture dislocation on permeability, providing a more profound perspective for fracture flow studies. Further studies are still needed on the effects of different fracture morphologies and the seepage-flow coupling of fracture shear processes. Conceptualization, J.Z. ; methodology, J.Z. and T.D. ; software, T.D. and Y.W. ; validation, T.D., T.M. and Y.W. ; formal analysis, T.D. ; writing--original draft preparation, J.Z., T.D. and Z.Y. ; writing--review and editing, J.Z. All authors have read and agreed to the published version of the manuscript. This research is supported by funding from the National Natural Science Foundation of China (Grant Nos.