## 1 Introduction  
Numerous fractures of different scales exist in subsurface rock mass, which controls engineering rock mass' physical, hydrological, and mechanical properties, thus affecting the deformation of surrounding rock, the internal stress distribution, and the seepage characteristics of the surrounding rock mass. Stress balance changes during rock engineering often result in relative sliding along fracture surfaces, altering the fluid permeation behavior within the fractures. For instance, in deep rock engineering applications such as enhanced geothermal system development [1], unconventional oil and gas reservoir exploitation [2],and carbon dioxide sequestration [3], the injection of high-pressure fluids into deep rock mass is employed to expand interconnected fracture pathways, which ultimately constructs a fracture network to enhance energy recovery or increase the storage capacity of deep rock strata. The injection of high-pressure fluids induces changes in the stress field in the rock mass, leading to relative sliding along the fracture surfaces and consequently impacting the flow characteristics. The complexity of fluid flow within the fractures arises from alterations in the stress environment caused by high-pressure fluid injection. As a result, fluid flow within the fractures transitions from linear to nonlinear states, affecting the solute and heat transfer properties within the fractures [4]. Therefore, investigating the changes in permeability characteristics of fluid flow in rock fractures resulting from slip displacement is important. Studying the deformation and permeability characteristics of a single fracture is a prerequisite for understanding the permeability properties of fractured rock masses [5; 6]. Early research on rock fractures assumed that the upper and lower surfaces of fractures were two smooth parallel plates. A cubic law was derived by simplifying the solution to the Navier-Stokes equation, indicating that the fracture's volume flow rate is proportional to the cube of the pressure gradient and the fracture aperture [7]. However, natural fractures are not simple parallel plate structures; they possess a complex rough structure, heterogeneous void distribution, and tortuous geometric shapes. These factors lead to deviations from the cubic law in permeability [8; 9], especially when fractures undergo deformation and contact influenced by various factors. This prompts changes in fluid flow velocity or direction within the channels, causing nonlinear fluid flow behaviors such as local eddy currents and turbulence [10]. In contrast to the cubic law, the Forchheimer law describes the nonlinear relationship between pressure gradient and volume flow rate and has been verified in various contexts [11; 12; 13]. [PERSON] et al. [14] carried out non-matching flow tests on fractures and found that the Forchheimer equation is highly sensitive to changes in fracture aperture. [PERSON] et al. [15] proposed a quantitative analysis of the global inertial permeability for the nonlinear term in the Forchheimer equation, characterizing non-Darcy flow in porous media. Therefore, the Forchheimer equation provides an accurate model for describing the nonlinear seepage of mismatched fractures and better understanding the seepage characteristics of fractures under complex conditions. For a single fracture, its void distribution exhibits complex characteristics closely related to the roughness structure of the fracture surface [16; 17; 18]. Scholars usually parameterize the roughness structure of fracture surfaces and analyze its influence on the permeability characteristics of fractures. Parameters include joint roughness coefficients and plane fractal dimensions. By wavelet analysis, [PERSON] et al. [19] divided the fracture surface roughness into primary and secondary roughness. They found that the secondary roughness significantly influences the nonlinear behavior of seepage. [PERSON] et al.