A Perspective on Darcy’s Law across the Scales: From Physical Foundations to Particulate Mechanics

Abstract

This paper puts forward a perspective or opinion that we can demonstrate Darcy's law is valid at any scale where fluid can be modelled/analyzed as a continuum. Darcy's law describes the flow of a fluid through a porous medium by a linear relationship between the flow rate and the pore pressure gradient through the permeability tensor. We show that such a linear relationship can be established at any scale, so long as the permeability tensor is expressed as a function of adequate parameters that describe the pore space geometry, fluid properties and physical phenomena. Analytical models at pore scale provide essential information on the key variables that permeability depends on under different flow regimes. Upscaling techniques based on the Lippman-Schwinger equation, pore network models or Eshelby's homogenization theory make it possible to predict fluid flow beyond the pore scale. One of the key challenges to validate these techniques is to characterize microstructure and measure transport properties at multiple scales. Recent developments in imaging, multi-scale modeling and advanced computing offer new possibilities to address some of these challenges. NOTATION• 𝑎 ′ and 𝑎 ′′ constants in Darcy-Forcheimer law

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