Upscaling the flow of generalised Newtonian fluids through anisotropic porous media

The homogenisation method with multiple scale expansions is used to investigate the slow and isothermal flow of generalised Newtonian fluids through anisotropic porous media. From this upscaling it is shown that the first-order macroscopic pressure gradient can be defined as the gradient of a macroscopic viscous dissipation potential, with respect to the first-order volume averaged fluid velocity. The macroscopic dissipation potential is the volume-averaged of local dissipation potential. Using this property, guidelines are proposed to build macroscopic tensorial permeation laws within the framework defined by the theory of anisotropic tensor functions and by using macroscopic isodissipation surfaces. A quantitative numerical study is then performed on a 3D fibrous medium and with a Carreau-Yasuda fluid in order to illustrate the theoretical results deduced from the upscaling.