Extending a Gray Lattice Boltzmann Model for Simulating Fluid Flow in Multi-Scale Porous Media

A gray lattice Boltzmann model has previously been developed by the authors of this article to simulate fluid flow in porous media that contain both resolved pores and grains as well as aggregates of unresolved smaller pores and grains. In this model, a single parameter is introduced to prescribe the amount of fluid to be bounced back at each aggregate cell. This model has been shown to recover Darcy-Brinkman flow but with effective viscosity and permeability correlated through the model parameter. In this paper, we prove that the model parameter relates to the fraction of the solid phase of a sub-pore system for a specific set of bounce-back conditions. We introduce an additional parameter to the model, and this enables flow simulation in which cases with variable effective viscosity and permeability can be specified by selecting the two parameters independently. We verify and validate the model for layered channel cases and mathematically analyze fluid momentum and energy losses for the single- and two-parameter models to explain the roles of the parameters in their conservation. We introduce a strategy to upgrade our model to an isotropic version. We discuss the fundamental differences between our model and the Brinkman body-force LBM scheme.