In this series of works we establish homogenized lattice Boltzmann methods (HLBM) for the simulation of 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. To this end, a matrix of regularly arranged obstacles of equal size is placed into the domain to model fluid flow through structures of different porosities that is governed by the incompressible nonstationary Navier--Stokes equations. Depending on the ratio of geometric parameters in the matrix arrangement, several cases of homogenized equations are obtained. We review the existing methods to homogenize the nonstationary Navier--Stokes equations for specific porosities and interpret connections between the resulting model equations from the perspective of applicability. Consequently, the homogenized Navier--Stokes equations are formulated as targeted partial differential equations which jointly incorporate the derived aspects. Second, we propose a kinetic model, named homogenized Bhatnagar--Gross--Krook Boltzmann equation, which approximates the homogenized nonstationary Navier--Stokes equations. 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 Navier--Stokes equations are consistently approximated by deriving a limit consistent HLBM discretization of the homogenized Bhatnagar--Gross--Krook Boltzmann equation.

翻译：本研究系列建立了用于模拟多孔介质流体流动的均匀化格子玻尔兹曼方法（HLBM）。第一部分的贡献包括两个方面。首先，通过形式化统一多孔介质非定常流体流动的控制方程，构建了目标偏微分方程组。为此，在计算域内放置由规则排列、尺寸相同的障碍物构成的矩阵，以模拟受不可压缩非定常Navier-Stokes方程控制的不同孔隙率结构中的流体流动。根据矩阵排列中几何参数的比值，推导出多种均匀化方程情形。我们回顾了针对特定孔隙率非定常Navier-Stokes方程均匀化的现有方法，并从适用性角度解读所得模型方程之间的关联。由此，均匀化Navier-Stokes方程被表述为统一整合推导要素的目标偏微分方程。其次，我们提出一种名为均匀化Bhatnagar-Gross-Krook玻尔兹曼方程的动力学模型，该模型近似于均匀化非定常Navier-Stokes方程。我们严格证明该动力学模型的零阶矩和一阶矩可在缩放参数的特定阶次下为宏观模型的质量与动量平衡变量提供解。基于当前成果，后续研究（第二部分）将通过推导均匀化Bhatnagar-Gross-Krook玻尔兹曼方程的极限一致性HLBM离散格式，实现对均匀化Navier-Stokes方程的一致近似。