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.

翻译：本研究系列建立了用于模拟多孔介质流体流动的均质化格子玻尔兹曼方法（HLBM）。我们在第一部分中的贡献有两方面：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。将规则排列、尺寸相等的障碍物矩阵置于计算域中，以模拟由不可压缩非定常Navier-Stokes方程（NSE）控制的多孔结构流体流动。根据矩阵排列中几何参数的比值，推导出多个均质化方程。我们回顾了针对特定孔隙率非定常NSE均质化的现有方法，并讨论了所得模型方程的适用性。由此，均质化NSE被表述为联合整合了推导要素的目标偏导数方程组。其次，我们提出了一种动力学模型——均质化Bhatnagar-Gross-Krook玻尔兹曼方程，该方程逼近均质化非定常NSE。我们形式化证明了该动力学模型的零阶和一阶矩在缩放参数的特定阶次上提供了宏观模型质量与动量平衡变量的解。基于当前贡献，在后续工作（第二部分）中，将通过推导均质化Bhatnagar-Gross-Krook玻尔兹曼方程的极限一致HLBM离散格式，实现对均质化NSE的一致性逼近。