Non-Darcian Flow Characterization in Three-Dimensional Rough-Walled Fractures Using Forchheimer and Izbash Equations

This study numerically investigated the applicability of the Forchheimer and Izbash equations for describing non-Darcian flow in three-dimensional (3D) rough-walled rock fractures. High-precision flow simulations were conducted on 52 synthetic 3D rough-walled fractures with varied apertures and surface roughness, generated from 56 combinations after excluding four cases with surface contact, under different hydraulic gradients. The simulation results captured transverse flow, back flow, and non-uniform streamlines on horizontal planes, which cannot be observed in conventional two-dimensional (2D) fracture models. The total eddy volume ratio negatively correlated with the aperture and positively correlated with roughness, and the 3D fractures exhibited a much smaller eddy volume ratio than the 2D fractures. Both equations provided excellent fits to the simulated data, with coefficients of determination R2 > 0.996. Notably, the Forchheimer coefficients showed strong and monotonic correlations with the aperture and roughness and are therefore predictable and characterizable, whereas the Izbash coefficients showed weak and non-monotonic correlations. Since non-negligible prediction errors occurred at low Reynolds numbers when the equations were fitted over the entire flow range, a piecewise fitting strategy was proposed, which reduced the prediction errors of both equations to within 5% across the full range and quantitatively divided the flow into the Darcy, weak inertial, and strong inertial regimes. Double-parameter equations relating the critical Reynolds numbers to the aperture and roughness were then established, allowing the flow regime to be predicted directly from the geometric parameters without additional simulation. These findings facilitate reasonable flow regime division and accurate full-range flow characterization in rock fractures.