Lattice Boltzmann Method for Fluid Flows in Anisotropic Porous Media with Brinkman Equation

Abstract

The lattice Boltzmann method (LBM) is applied to simulation of fluid flows in anisotropic porous media with the Brinkman equation. The Brinkman equation is recovered from a kinetic equation for the distribution function that has a forcing term to introduce anisotropy of the permeability of the porous media. Since the forcing term contains the drag force proportional to the fluid velocity, the LBM needs matrix calculation to obtain the fluid velocity through the definition of the momentum. The velocity profiles of the LBM show good agreement with analytical solutions for the Poiseuille flow and for the Couette flow filled with anisotropic porous media. The contour lines of the stream functions obtained by the LBM show good agreement with those of the finite difference method (FDM) in the numerical simulation of a lid-driven cavity flow for different fundamental parameters, e.g., Darcy number, inclination of the principal permeability direction, and permeability ratio. By increasing grid size, the LBM is able to decrease the compressibility effect, and reduces the deviation of the maximum values of the stream function between the LBM and the FDM. On the same grid size, the LBM takes less time than the conventional FDM to get the steady solutions. This paper leads to the conclusion that the LBM can simulate incompressible flow in anisotropic porous media at the representative elementary volume scale.