On the Inertia Effects on the Darcy Law: Numerical Implementation and Confrontation of Micromechanics-Based Approaches

Abstract

In this paper, we investigate the nonlinear deviation of the Darcy law in the domain of high pressure gradient. Classically, the (linear) Darcy law can be deduced from asymptotic homogenization approaches and the numerical resolution of the Stokes flow problem on the unit cell of the porous medium. At high-speed steady flow of a fluid, nonlinear effects on the macroscopic filtration law arise and are accounted by considering the convection term in the Navier–Stokes equation. These nonlinear effects has been often studied in asymptotic homogenization framework by expanding the solution in power series at low Reynolds number. This has two advantages: (i) The Navier–Stokes problems are replaced by a chain of linear problems with source terms which depend on the solution at lower order, and (ii) the macroscopic nonlinear filtration law is derived in the form of a polynom. We develop a Fast Fourier Transform (FFT)-based numerical algorithm to compute the solution of this elementary problems and to compute the higher-order permeability tensors in connection with the morphology of the porous medium. The results are then compared to the solution of the full Navier–Stokes problem by means of finite element method (FEM) which allows evaluating the capacity of the expansion method to account for the nonlinear effects. We determine the convergence radius of the polynomial series, and we give the limit of the series expansion method in terms of the Reynolds number.

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