An analytical model for saturated nonuniform laminar flow through rough fractures

Abstract

Estimating the flow rate accurately through a single rough fracture poses a fundamental challenge. This study presents an analytical model for single-phase saturated nonuniform laminar flow through rough fractures. The flow analytical model of the fractures is constructed using homotopy thinning methods, average inertia approaches, and shortest path techniques, thereby simplifying the nonuniform flow problem to solving the integral of the incircle aperture function, from which the modified average Darcy velocity considering inertial effects is expressed. The flow equations in the global coordinates are then derived. The validity of the proposed model is verified by comparing it with flow simulations with the Navier–Stokes equations, perturbation solutions, previously corrected Reynolds equations, and experimental flow tests. The results obtained from the proposed model agree very well with those from simulations and experiments. The effective errors Di range within ±4.0% of the simulation results with an arithmetic mean of |Di| equal to 1.03%. As surface roughness increases, the proposed model can effectively capture the inertial behaviors arising from the nonuniform flow field. Compared to the previous corrected Reynolds equations and perturbation solutions, the proposed model demonstrates enhanced accuracy and applicability, introducing a new approach to address nonuniform flow problems in rough fractures with more complex geometries.