Numerical Simulation of the Nonlinear Flow Properties in Self-Affine Aperture-Based Fractures

Abstract

In order to study the effect of fracture geometry on the nonlinear flow properties in aperture-based fractures, a fractal model based on the self-affinity is proposed to characterize the three-dimensional geometry of rough-walled fractures. By solving the N–S (Navier–Stokes) equation directly, the relationships between the Forchheimer-flow characteristics, fractal dimension, and standard deviation of the aperture have been obtained. The Forchheimer equation is validated to describe the nonlinear relationship between flow rate and pressure gradient. For lower flow rate, the influence of the fractal dimension almost can be ignored, but the linear coefficient increases and the hydraulic aperture decreases with increasing standard deviation of the aperture, respectively. For larger flow rate, the nonlinear coefficient increases with the growth of the standard deviation of the aperture and fractal dimension. Thus, an empirical relationship between the nonlinear coefficient, fractal dimension, and standard deviation of aperture is proposed. In addition, the critical Reynolds number decreases with the increase of the standard deviation of the aperture and the fractal dimension, and the numerical results are generally consistent with the experimental data.