A Corrected Cubic Law for Single-phase Laminar Flow through Rough-walled Fractures

Hydraulic properties of natural fractures are essential parameters for the modeling of fluid flow and transport in subsurface fractured porous media. The cubic law, based on the parallel-plate concept, has been traditionally used to estimate the hydraulic properties of individual fractures. This upscaling approach, however, is known to overestimate the fractures hydraulic properties. Dozens of methods have been proposed in the literature to improve the accuracy of the cubic law. The relative performance of these various methods is not well understood. In this work, a comprehensive review and benchmark of almost all commonly used cubic law-based approaches in the literature, covering 43 methods is provided. We propose a new corrected cubic law for incompressible, single-phase laminar flow through rough-walled fractures. The proposed model incorporates corrections to the hydraulic fracture aperture based on the flow tortuosity and local roughness of the fracture walls. We identify geometric rules relative to the local characteristic of the fracture and apply an efficient algorithm to subdivide the fracture into segments, accordingly. High-resolution simulations for Navier-Stokes equations, computed in parallel, for synthetic fractures with various ranges of surface roughness and apertures are then performed. The numerical solutions are used to assess the accuracy of the proposed model and compare it with the other 43 approaches, where we demonstrate its superior accuracy. The proposed model retains the simplicity and efficiency of the cubic law but with pronounced improvement to its accuracy. The data set used in the benchmark, including more than 7500 fractures, is provided in open-access.