caa a22 4500 463162676 CHVBK 20180406164752.0 cr unu---uuuuu 170326e20071101xx s 000 0 eng 10.1007/s11242-006-9096-x doi (NATIONALLICENCE)springer-10.1007/s11242-006-9096-x Upscaling Forchheimer law [Elektronische Daten] [Jean-Louis Auriault, Christian Geindreau, Laurent Orgéas] We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy. Springer Science+Business Media, Inc., 2007 Flow in porous media nationallicence High velocity flow nationallicence Forchheimer law nationallicence Heterogeneous porous media nationallicence Upscaling nationallicence Roman letters nationallicence G : Macroscopic gradient of pressure nationallicence K : Permeability nationallicence K F : Forchheimer equivalent permeability nationallicence K eff : Effective permeability nationallicence l : Characteristic size of the pores nationallicence L : Macroscopic characteristic length nationallicence n : Medium 1 volume fraction nationallicence n : Unit normal vector to Γ nationallicence p : Pressure nationallicence $${\mathcal{R}}$$ : Dimensionless numbers nationallicence v : Fluid velocity nationallicence V : Periodic cell nationallicence V 1, V 2 : Medium 1 and 2 volumes in the periodic cell nationallicence x : Dimensionless macroscopic space variable nationallicence X : Dimensional space variable nationallicence y : Dimensionless microscopic space variable nationallicence Greek letters nationallicence β : Inertial coefficient nationallicence β eff : Effective inertial coefficient nationallicence $${\varepsilon}$$ : Small parameter of separation of scale nationallicence Γ : Media 1 and 2 interface nationallicence ψ : Potential nationallicence μ : Viscosity nationallicence ρ : Fluid density nationallicence Auriault Jean-Louis Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut Geindreau Christian Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut Orgéas Laurent Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut Transport in Porous Media Springer Netherlands 70/2(2007-11-01), 213-229 0169-3913 70:2<213 2007 70 11242 https://doi.org/10.1007/s11242-006-9096-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9096-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Auriault Jean-Louis Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut NATIONALLICENCE 700 1- Geindreau Christian Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut NATIONALLICENCE 700 1- Orgéas Laurent Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53, 38041, Grenoble Cedex, France aut NATIONALLICENCE 773 0- Transport in Porous Media Springer Netherlands 70/2(2007-11-01), 213-229 0169-3913 70:2<213 2007 70 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer