On the developments of Darcy's law to include inertial and slip effects

Abstract The empirical Darcy law describing flow in porous media, whose convincing theoretical justification was proposed almost 130 years after its original publication in 1856, has however been extended to account for particular flow conditions. This article reviews historical developments aimed at including inertial and slip effects (respectively, when the Reynolds and Knudsen numbers are not exceedingly small compared to unity). Despite the early empirical extensions to include inertia and slip effects, it is striking to observe that clear formal derivations of physical models to account for these effects were reported only recently.

[1]  H. E. Stanley,et al.  Inertial Effects on Fluid Flow through Disordered Porous Media , 1999 .

[2]  Flow of low pressure gas through dual-porosity media , 2007 .

[3]  J. Auriault,et al.  Nonlinear seepage flow through a rigid porous medium , 1994 .

[4]  L. Klinkenberg.  The Permeability Of Porous Media To Liquids And Gases , 2012 .

[5]  J. Auriault,et al.  High-Velocity Laminar and Turbulent Flow in Porous Media , 1999 .

[6]  Michel Quintard,et al.  Ecoulement monophasique en milieu poreux: effet des hétérogénéités locales , 1987 .

[7]  Didier Lasseux,et al.  An improved macroscale model for gas slip flow in porous media , 2016, Journal of Fluid Mechanics.

[8]  J.-L. Auriault,et al.  Homogenization of Wall-Slip Gas Flow Through Porous Media , 1999 .

[9]  F. J. Uribe,et al.  Beyond the Navier-Stokes equations: Burnett hydrodynamics , 2008 .

[10]  M Muskat,et al.  THE FLOW OF HOMOGENEOUS FLUIDS THROUGH POROUS MEDIA: ANALOGIES WITH OTHER PHYSICAL PROBLEMS , 1937 .

[11]  F. Dullien.  Porous Media: Fluid Transport and Pore Structure , 1979 .

[12]  Stephen Whitaker,et al.  The equations of motion in porous media , 1966 .

[13]  F. Dullien,et al.  Flow rate‐pressure gradient measurements in periodically nonuniform capillary tubes , 1973 .

[14]  B. Goyeau,et al.  A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media , 2014 .

[15]  Hiroshi Adzumi.  Studies on the Flow of Gaseous Mixtures through Capillaries. II. The Molecular Flow of Gaseous Mixtures , 1937 .

[16]  James Clerk Maxwell.  Illustrations of the Dynamical Theory of Gases , 1860 .

[17]  J. Auriault,et al.  New insights on steady, non-linear flow in porous media , 1999 .

[18]  S. Ergun.  Fluid flow through packed columns , 1952 .

[19]  M. Knudsen.  Die Gesetze der Molekularstrmung und der inneren Reibungsstrmung der Gase durch Rhren , 1909 .

[20]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[21]  J. Geertsma.  Estimating the Coefficient of Inertial Resistance in Fluid Flow Through Porous Media , 1974 .

[22]  R. Jackson,et al.  Transport in porous catalysts , 1977 .

[23]  Anthony J. C. Ladd,et al.  Moderate Reynolds number flows through periodic and random arrays of aligned cylinders , 1996, Journal of Fluid Mechanics.

[24]  R. G. Deissler.  An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases , 1964 .

[25]  Hiroshi Adzumi.  Studies on the Flow of Gaseous Mixtures through Capillaries. III. The Flow of Gaseous Mixtures at Medium Pressures , 1937 .

[26]  W. Hall.  An analytical derivation of the Darcy equation , 1956 .

[27]  Sheng Shen,et al.  A kinetic-theory based first order slip boundary condition for gas flow , 2007 .

[28]  Garcia-Colin L.s.,et al.  Navier‐Stokes方程式を越えて:Burnett流体力学 , 2008 .

[29]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[30]  J. Gudmundsson,et al.  High-velocity flow in a rough fracture , 1999, Journal of Fluid Mechanics.

[31]  J. Auriault.  Nonsaturated deformable porous media: Quasistatics , 1987 .

[32]  R. V. Edwards,et al.  A New Look at Porous Media Fluid Mechanics — Darcy to Turbulent , 1984 .

[33]  G. Chauveteau,et al.  Régimes d'écoulement en milieu poreux et limite de la loi de Darcy , 1967 .

[34]  Didier Lasseux,et al.  On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media , 2011 .

[35]  M. Muskat,et al.  FLOW OF GAS THROUGH POROUS MATERIALS , 1931 .

[36]  P. Carman,et al.  Flow of gases through porous media , 1956 .

[37]  D. Lasseux,et al.  From steady to unsteady laminar flow in model porous structures: an investigation of the first Hopf bifurcation , 2016 .

[38]  Chahid Kamel Ghaddar,et al.  On the permeability of unidirectional fibrous media: A parallel computational approach , 1995 .

[39]  J. Maxwell,et al.  On Stresses in Rarified Gases Arising from Inequalities of Temperature , 2022 .

[40]  W Steckelmacher,et al.  Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems , 1986 .

[41]  Chiang C. Mei,et al.  The effect of weak inertia on flow through a porous medium , 1991, Journal of Fluid Mechanics.

[42]  S. Whitaker.  The method of volume averaging , 1998 .

[43]  S. Whitaker.  The Forchheimer equation: A theoretical development , 1996 .

[44]  S. Lorenzani.  Higher order slip according to the linearized Boltzmann equation with general boundary conditions , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[45]  R. R. Kairi,et al.  Effect of viscous dissipation on natural convection heat and mass transfer from vertical cone in a non-Newtonian fluid saturated non-Darcy porous medium , 2011, Appl. Math. Comput..

[46]  S. Irmay.  On the theoretical derivation of Darcy and Forchheimer formulas , 1958 .

[47]  Hiroshi Adzumi.  Studies on the Flow of Gaseous Mixtures through Capillaries. I The Viscosity of Binary Gaseous Mixtures , 1937 .

[48]  J.-C. Wodie,et al.  Correction non linéaire de la loi de Darcy , 1991 .

[49]  William G. Gray,et al.  High velocity flow in porous media , 1987 .

[50]  Zhangxin Chen,et al.  Derivation of the Forchheimer Law via Homogenization , 2001 .

[51]  P. Painlevé,et al.  Mémoire sur la transformation des équations de la Dynamique , 1894 .

[52]  T. Giorgi.  Derivation of the Forchheimer Law Via Matched Asymptotic Expansions , 1997 .

[53]  H. Brinkman.  A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles , 1949 .

[54]  Liu,et al.  Boundary condition for fluid flow: Curved or rough surfaces. , 1990, Physical review letters.