A Lubrication-Based Solver for Shear-Thinning Flow in Rough Fractures

Abstract

The depth-averaged (2D) lubrication theory is often adopted to simulate Newtonian flow in rough fractures. This approach, which is computationally much less expensive than using 3D CFD solvers, allows addressing large ensembles of stochastic fracture realizations. For creeping flow, the degree of approximation introduced is limited as long as the apertures vary relatively smoothly. We propose the first generalization of this approach addressing the flow of fluids whose rheology, described by the Ellis model, is shear-thinning (ST) above a crossover shear stress and Newtonian (of viscosity μ0) below. The resulting nonlinear Reynolds equation for pressures is solved for a vast range of realistic rheological parameter values using a novel and specifically designed finite volume-based numerical model. The spatial discretization takes inspiration from the graph p-Laplacian to yield a symmetric Newton Jacobian, allowing for a highly efficient inexact implementation of the preconditioned conjugate gradient-based Newton-Krylov method. This is combined with a parameter continuation strategy to increase code robustness and ensure global convergence for flow indices as low as 0.1 with an excellent efficiency. This original solver is used to investigate realistic synthetic rough fracture geometries, which exhibits both self-affinity and a correlation length. The results show that the ST rheology mitigates the effects of aperture heterogeneities, increasing fracture transmissivity by several orders of magnitudes as compared to the Newtonian flow of viscosity μ0 if the imposed macroscopic gradient is sufficiently large, and even rendering the rough fracture up to 10 times more permeable than a smooth fracture of identical mean aperture.