Evolution of Seepage Flow Rate in a Single Rough Fracture at Different Fractal Dimensions

The hydraulic characteristics of a single rough fracture are critical in rock seepage studies, with roughness being the dominant factor influencing fracture hydraulic behavior. In this study, four sets of single rough fractures with fractal dimensions ranging from 1.2 to 1.8 were generated using the Weierstrass–Mandelbrot function, and the evolution of the correction coefficient for rough fracture laminar flow rate based on the classical cubic law with fractal dimension was numerically investigated. The results demonstrate that fractal dimension significantly influences hydraulic aperture and hydraulic gradient along tortuous seepage paths, thereby altering the overall flow rate. For rough fractures with apertures below 2 mm under laminar flow, the maximum variations in the correction coefficient caused by different apertures and pressure differences at identical fractal dimensions were merely 1.07% and 2.09%, respectively. While the correction coefficients of seepage flow rate show minimal variation, maintaining values between 0.67 and 0.72 across the fractal dimension range of 1.2–1.4, it exhibits a rapid decline as the fractal dimension increases from 1.4 to 1.8, with values at 1.8 decreasing by 47.8% and 33.7% compared to those at 1.4 and 1.6, respectively. Finally, a fitting function was developed to characterize the relationship between the correction coefficient and the fractal dimension for single rough fractures. The research findings can serve as a reference for understanding and quantitatively characterizing the seepage properties of rough single fractures.