Transport in Porous Media

, 81:35 | Cite as

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

  • Matthew Balhoff
  • Andro Mikelić
  • Mary F. Wheeler


In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.


Forchheimer Inertial effects Homogenization Cubic law 

List of symbols


Physical velocity [L/t]


Pressure [F/L2]


Characteristic velocity [L/t]


Viscosity [F/L2t]


Density [M/L3]


Characteristic length M


Characteristic pressure [F/L2]


Pressure drop


Reynolds number


Froude’s number




Unit cell


Solid part of the unit cell




Pore space (the fluid part of Ω)


Ratio between the pore size and the reservoir size L




Dimensionless forcing term


Dimensionless physical velocity


Dimensionless pressure


Dimensionless permeability


  1. Ahmad, N.: Physical properties of porous medium affecting laminar and turbulent flow of water. Ph. D. Thesis, Colorado State University, Fort Collins (1967)Google Scholar
  2. Allaire G.: Homogenization of the Stokes flow in connected porous medium. Asymptotic Anal. 3, 203–222 (1989)Google Scholar
  3. Balhoff, M., Wheeler, M.F.: A predictive pore-scale model for non-Darcy flow in anisotropic porous media. SPE 110838, presented at the Annual Technical Conference and Exhibition, Anaheim, 11–14 November 2007Google Scholar
  4. Barree, R.D., Conway, M.W.: Beyond beta factors: A complete model for Darcy, Forchheimer and Trans-Forccheimer flow in porous media. Paper SPE 89325, presented at the SPE Annual Technical Conference and Exhibition held in Houston, Texas, USA, 26–29 September 2004Google Scholar
  5. Barree, R.D., Conway, M.W.: Reply to discussion of ‘Beyond beta factors: a complete model for Darcy, Forchheimer, and Trans-Forchheimer flow in porous media’. J. Pet. Tech. 73–74 (2005)Google Scholar
  6. Barenblatt G.I., Entov V.M., Ryzhik V.M.: Theory of Fluid Flows Through Natural Rocks. Kluwer, Dordrecht (1990)Google Scholar
  7. Bear J.: Hydraulics of Groundwater. McGraw-Hill, Jerusalem (1979)Google Scholar
  8. Blake F.C.: The resistance of packing to fluid flow. Trans. Am. Inst. Chem. Engrs. 14, 415–421 (1922)Google Scholar
  9. Bourgeat A., Marušić-Paloka E.: Nonlinear effects for flow in periodically constricted channel caused by high injection rate. Math. Models Methods Appl. Sci. 8(3), 379–405 (1998)CrossRefGoogle Scholar
  10. Bourgeat A., Marušić- Paloka E., Mikelić A.: Weak non-linear corrections for Darcy’s Law. M3 AS Math. Models Methods Appl. Sci. 6(8), 1143–1155 (1996)CrossRefGoogle Scholar
  11. Brownell L.E., Dombrowski H.S., Dickey C.A.: Pressure drop through porous media. Chem. Eng. Prog. 43, 537–548 (1947)Google Scholar
  12. Chen Z., Lyons S.L., Qin G.: Derivation of the Forchheimer law via homogenization. Transp. Porous Med. 44, 325–335 (2001)CrossRefGoogle Scholar
  13. Couland O., Morel P., Caltagirone J.P.: Numerical modelling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech. 190, 393–407 (1988)CrossRefGoogle Scholar
  14. Cvetkovic V.D.: A continuum approach to high velocity flow in porous media. Transp. Porous Med. 1, 63–97 (1986)CrossRefGoogle Scholar
  15. Deiber J.A., Schowalter W.R.: Flow through tubes with sinusoidal axial variations in diameter. AICHE J. 25, 638–645 (1979)CrossRefGoogle Scholar
  16. Deiber J.A., Peirotti M., Bortolozzi R.A., Durelli R.J.: Flow of Newtonian fluids through sinusoidally constricted tubes; numerical and experimental results. Chem. Eng. Commun. 117, 241–262 (1992)CrossRefGoogle Scholar
  17. Edwards, D.A., Shapiro, M., Bar-Yoseph, P., Shapira, M.: The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders. Phys. Fluids A. 2 (1990)Google Scholar
  18. Fancher G.H., Lewis J.A.: Flow of simple fluids through porous materials. Ind. Eng. Chem. 25, 1139–1147 (1933)CrossRefGoogle Scholar
  19. Firdaouss M., Guermond J.L., Le Quéré P.: Nonlinear corrections to Darcy’s law at low Reynolds numbers. J. Fluid Mech. 343, 331–350 (1997)CrossRefGoogle Scholar
  20. Forchheimer P.: Wasserbewegung durch Boden. Zeits V. Deutsch Ing. 45, 1782–1788 (1901)Google Scholar
  21. Forchheimer P.: Hydraulik, 3rd edn. Teubner, Leipzig (1930)Google Scholar
  22. Hassanizadeh S.M., Gray W.G.: High velocity flow in porous media. Transp. Porous Med. 2, 521–531 (1987)CrossRefGoogle Scholar
  23. Huang, H., Ayoub, J.: Applicability of the Forchheimer equation for non-Darcy flow in porous media. SPE 102715, presented at the 2006 SPE Annual Technical Conference and Exhibition, San Antonio, 24–27 September 2006Google Scholar
  24. Kim, B.Y.K.: The resistance to flow in simple and complex porous media whose matrices are composed of spheres. M.Sc. thesis, University of Hawaii at Manoa (1985)Google Scholar
  25. Koch D.L., Ladd A.J.C.: The first effects on fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–241 (2001)Google Scholar
  26. Lindquist E.: On the flow of water through porous soils. Premier Congrès des Grands Barrages, Stockholm 5, 81–101 (1933)Google Scholar
  27. Lions J.L.: Some Methods in the Mathematical Analysis of Systems and Their Control. Gordon and Breach, New York (1981)Google Scholar
  28. Lions, J.L.: Some problems connected with Navier–Stokes equations. Lectures at the IV Latin-American School of Mathematics, Lima, 1978, in Lions, Jacques-Louis (Euvres choisies de Jacques-Louis Lions. Vol. II. (French)) [Selected works of Jacques-Louis Lions. Vol. II] Contrôle. Homogénéisation [Control. Homogenization]. Edited by Alain Bensoussan, Philippe G. Ciarlet, Roland Glowinski, Roger Temam, François Murat and Jean-Pierre Puel, and with a preface by Bensoussan. EDP Sciences, Les Ulis; Société de Mathématiques Appliquées et Industrielles, pp. xvi+864. Paris, (2003)Google Scholar
  29. Marušić–Paloka E., Mikelić A.: The derivation of a non-linear filtration law including the inertia effects via homogenization. Nonlinear Anal. Theory Methods Appl. 42, 97–137 (2000)CrossRefGoogle Scholar
  30. Mei C.C., Auriault J.-L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991)CrossRefGoogle Scholar
  31. Mikelić, A.: Homogenization theory and applications to filtration through porous media. In: Filtration in Porous Media and Industrial Applications, Lecture Notes Centro Internazionale Matematico Estivo (C.I.M.E.) Series, Lecture Notes in Mathematics, vol. 1734, pp. 127–214. Springer (2000)Google Scholar
  32. Mikelić A.: On the justification of the Reynolds equation, describing isentropic compressible flows through a tiny pore. Ann. Univ. Ferrara 53, 95–106 (2007)CrossRefGoogle Scholar
  33. Mobasheri, F., Todd, D.K.: Investigation of the hydraulics of flow near recharge wells. Water Resources Center Contribution No. 72, University of California Berkeley (1963)Google Scholar
  34. Rasoloarijaona M., Auriault J.-L.: Non-linear seepage flow through a rigid porous medium. Eur. J. Mech. B Fluids 13(2), 177–195 (1994)Google Scholar
  35. Rojas S., Koplik J.: Nonlinear flow in porous media. Phys. Rev. E. 58(4), 4776–4782 (1988)CrossRefGoogle Scholar
  36. Ruth D.W., Ma H.: On the derivation of the Forchheimer equation by means of the averaging theorem. Transp. Porous Med. 7, 255–264 (1992)CrossRefGoogle Scholar
  37. Ruth D.W., Ma H.: The microscopic analysis of high Forchheimer number flow in porous media. Transp. Porous Med. 13, 139–160 (1993)CrossRefGoogle Scholar
  38. Sanchez-Palencia E.: Non-Homogeneous Media and Vibration Theory, Springer Lecture Notes in Physics 127. Springer-Verlag, Berlin (1980)Google Scholar
  39. Scheidegger, A.E.: Hydrodynamics in Porous Media. In: Flügge, S. Encyclopedia of Physics, vol.VIII/2 (Fluid Dynamics II), pp. 625–663. Springer-Verlag, Berlin (1963)Google Scholar
  40. Skjetne E., Auriault J.L.: New insights on steady, non-linear flow in porous media. Eur. J. Mech. B Fluids 18(1), 131–145 (1999)CrossRefGoogle Scholar
  41. Stanley H.E., Andrade J.S.: Physics of the cigarette filter: fluid flow through structures of randomly-packed obstacles. Physica A 295, 17–30 (2001)CrossRefGoogle Scholar
  42. Sunada, D.K.: Laminar and turbulent flow of water through homogeneous porous media. PhD dissertation, University of California at Berkeley (1965)Google Scholar
  43. van Batenburg, D., Milton-Taylor, D.: Discussion of SPE 89325, ‘Beyond beta factors: a complete model for Darcy, Forchheimer, and Trans-Forchheimer flow in porous media’. J. Pet. Tech. 72–73 (2005)Google Scholar
  44. Whitaker S.: The Forchheimer equation: a theoretical development. Transp. Porous Med. 25, 27–61 (1996)CrossRefGoogle Scholar
  45. Wodié, J.-C., Levy, T.: Correction non linéaire de la loi de Darcy. C.R. Acad. Sci. Paris t.312, Série II, 157–161 (1991)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Matthew Balhoff
    • 1
  • Andro Mikelić
    • 2
    • 3
  • Mary F. Wheeler
    • 4
  1. 1.Petroleum and Geosystems EngineeringThe University of Texas at AustinAustinUSA
  2. 2.Université de LyonLyonFrance
  3. 3.Institut Camille Jordan, UFR MathématiquesUniversité Lyon 1Lyon Cedex 07France
  4. 4.Institute for Computational and Engineering ScienceThe University of Texas at AustinAustinUSA

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