Numerical Calculation of the Equation of Flow in Porous Media: The Lattice Gas Approach

Abstract

Lattice gasses are models of gasses where the particles move in discretized space and time. A lattice gas model is defined by a lattice and a set of rules defining particle movements. The hydrodynamical equations of the gas are then found as successive terms in a perturbation expansion of the lattice Boltzmann equations. A lattice gas has been introduced in a previous publication, where it was shown that the lowest order term in the above-named expansion is the equation of one-phase flow in a homogeneous porous medium with two independent permeability components. The model assumes that the lattice gas density, as characterized by a density scale, is small. This paper presents a generalization to inhomogeneous media with three independent permeability components. The main contribution of the paper is, however, a calculation of the flow equation of the lattice gas to the next order in the perturbation expansion of the Boltzmann equations, showing that correction terms proportional to the gas density appear in the permeability coefficients. A numerical example is given, in a case where the exact solution is known. The numerical results contain errors due to statistical fluctuations and deviations due to the correction terms mentioned above. For the particular example, the relative deviations are shown to be in the neighborhood of 5% for a 600 × 600 lattice and a density scale of about 0.1.