Simplified Equivalent Fracture Models Capturing Roughness Heterogeneity Effects on Hydraulic Behavior and Cubic Law Deviation of Rough-Walled Fractures

Fracture roughness is critical to fluid flow behavior in fractured rock masses. However, the mechanism by which such roughness heterogeneity influences fluid flow and amplifies cubic law deviation remains incompletely understood. Current theoretical analyses are focused on uniform roughness or smooth parallel plates, neglecting the roughness heterogeneity in natural fractures. The equivalent fracture geometry models with heterogeneous roughness are established based on the fracture walls of the smooth parallel plates and the sinusoidal profiles in this study. Based on the geometry models and derived from the Navier–Stokes equations, two simplified fracture models are proposed: the equivalent plate–sinusoidal walled and the sinusoidal–sinusoidal walled fracture model, validated via COMSOL Multiphysics. A roughness heterogeneity index Zr is defined to quantify the roughness heterogeneity. The influence of roughness heterogeneity on hydraulic behavior (e.g., fluid flow rate, equivalent hydraulic aperture) is analyzed and compared with those of uniform roughness and smooth parallel plates. Additionally, the influence of roughness heterogeneity on the power–law exponent relationship between fracture mechanical aperture and flow rate is examined. The results indicate that the flow rate and hydraulic aperture decrease with increasing roughness heterogeneity, while the deviation of fluid flow from the cubic law increases. The power–law exponent can be as high as 15.5. This study provides theoretical models for understanding the effects of roughness heterogeneity and a reference for extending flow models to complex fracture morphologies.