On the range of validity of parabolic models for fluid flow through isotropic homogeneous porous media

TL;DR

This work rigorously delineates the range of validity for parabolic, weakly compressible porous-flow models used with Lattice-Boltzmann methods by extending the volume-averaging derivation to compressible 3D flow and by explicitly listing the underlying assumptions. It shows how the commonly neglected hydraulic dispersion term can be reconciled with an effective viscosity correction (a Smagorinsky-like form) and clarifies how REV-based porous-flow models relate to Euler-Euler multiphase formulations under specific conditions. The paper provides clear closures for momentum transfer (Darcy-Forchheimer terms and dispersion) and discusses how REV descriptions map onto multiphase frameworks, thereby strengthening the theoretical foundation for LBM in porous media. Overall, it outlines practical criteria for applying parabolic porous-flow models and highlights when nonlocal and memory effects become significant, impacting the interpretation of simulated viscous behavior and model fidelity in REV-scale simulations.

Abstract

Lattice-Boltzmann methods are established mesoscopic numerical schemes for fluid flow, that recover the evolution of macroscopic quantities (viz., velocity and pressure fields) evolving under macroscopic target equations. The approximated target equations for fluid flows are typically parabolic and include a (weak) compressibility term. A number of Lattice-Boltzmann models targeting, or making use of, flow through porous media in the representative elementary volume, have been successfully developed. However, apart from two exceptions, the target equations are not reported, or the assumptions for and approximations of these equations are not fully clarified. Within this work, the underlying assumption underpinning parabolic equations for porous flow in the representative elementary volume, are discussed, clarified and listed. It is shown that the commonly-adopted assumption of negligible hydraulic dispersion is not justifiable by clear argument - and in fact, that by not adopting it, one can provide a qualitative and quantitative expression for the effective viscosity in the Brinkman correction of Darcy law. Finally, it is shown that, under certain conditions, it is possible to interpret porous models as Euler-Euler multiphase models wherein one phase is the solid matrix.