Homogenized lattice Boltzmann methods for fluid flow through porous media -- part I: kinetic model derivation

In this series of studies, we establish homogenized lattice Boltzmann methods (HLBM) for simulating fluid flow through porous media. Our contributions in part I are twofold. First, we assemble the targeted partial differential equation system by formally unifying the governing equations for nonstationary fluid flow in porous media. A matrix of regularly arranged, equally sized obstacles is placed into the domain to model fluid flow through porous structures governed by the incompressible nonstationary Navier--Stokes equations (NSE). Depending on the ratio of geometric parameters in the matrix arrangement, several homogenized equations are obtained. We review existing methods for homogenizing the nonstationary NSE for specific porosities and discuss the applicability of the resulting model equations. Consequently, the homogenized NSE are expressed as targeted partial differential equations that jointly incorporate the derived aspects. Second, we propose a kinetic model, the homogenized Bhatnagar--Gross--Krook Boltzmann equation, which approximates the homogenized nonstationary NSE. We formally prove that the zeroth and first order moments of the kinetic model provide solutions to the mass and momentum balance variables of the macrocopic model up to specific orders in the scaling parameter. Based on the present contributions, in the sequel (part II), the homogenized NSE are consistently approximated by deriving a limit consistent HLBM discretization of the homogenized Bhatnagar--Gross--Krook Boltzmann equation.