  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.
