Volume average of Navier-Stokes equations in soils and its application

Abstract

Seepage flow in soils is inherently governed by microscopic water flow occurring at the pore scale; however, in practical and engineering analyses, the primary objective is often to understand macroscopic seepage behavior. In this paper, volume averaging is introduced as a theoretical approach to link microscopic flow phenomena with macroscopic behavior, and its concept and physical meaning are explained. After presenting the definitions of volume averaging and intrinsic phase averaging and organizing the relationships between averaged quantities and their spatial and temporal derivatives, the Navier-Stokes equations are volumeaveraged for saturated soils. This procedure yields governing equations for the Darcy velocity and the averaged pore water pressure, through which the relationship between microscopic flow and Darcy’s law is clarified. Furthermore, using a flow problem spanning a fluid region and a porous medium as an example, it is shown that the volume-averaged equations enable a continuous treatment of flow across different domains.