# The Forchheimer equation: A theoretical development

## Abstract

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by

$$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$

The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

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## Abbreviations

A βσ :

area of the β-σ interface contained with the macroscopic region, m2

A βe :

area of the entrances and exits of the β-phase at the boundary of the macroscopic region, m2

A βσ :

area of the β-σ interface contained within the averaging volume, m2

A p :

surface area of a particle, m2

b :

the vector field that maps μβ〈vββ onto $$\tilde p$$ β when inertial effects are negligible, m−1

B :

tensor that maps $$\tilde v$$ β onto 〈vββ when inertial effects are negligible

d p :

6V p/Ap, effective particle diameter, m

g:

gravitational acceleration, m/s2

I :

unit tensor

ℓβ:

characteristic length for the β-phase, m

ℓί:

ί = 1,2,3, lattice vectors, m

L :

characteristic length for macroscopic quantities, m

L ρ :

inertial length, m

m:

the vector field that maps μβ〈vββ onto $$\tilde p$$ β, m−1

M :

tensor that maps $$\tilde v$$ β onto μβ〈vββ

nασ :

unit normal vector directed from the β-phase toward the σ-phase

p β :

total pressure in the β-phase, Pa

p ββ :

intrinsic average pressure in the β-phase, Pa

p β〉:

superficial average pressure in the β-phase, Pa

$$\tilde p$$ β :

p β − 〈p β〉 spatial deviation pressure, Pa

r:

position vector, m

r0:

radius of the averaging volume, m

t :

time, s

t * :

characteristic process time, s

vβ :

velocity in the β-phase, m/s

〈vββ :

intrinsic average velocity in the β-phase, m/s

〈vβ〉:

superficial average velocity in the β-phase, m/s

$$\tilde v$$ β :

vβ − 〈vββ, spatial deviation velocity, m/s

ν:

local averaging volume, m3

Vβ :

volume of the β-phase contained within the averaging volume, m3

V p :

volume of a particle, m3

x :

position vector locating the centroid of the averaging volume, m

yβ :

position vector locating points in the β-phase relative to the centroid of the averaging volume, m

εβ :

Vβ/ν, volume fraction of the β-phase

ρβ :

density of the β-phase, kg/m3

μβ :

viscosity of the β-phase, Pa s

## References

• Barrère, J.: 1990, Modélisation des équations de Stokes et Navier-Stokes en milieux poreux, Thèse de l'Université de Bordeaux I.

• Barrère, J., Olivier, G., and Whitaker, S.: 1992, On the closure problem for Darcy's law, Transport in Porous Media 7, 209–222.

• Bensoussan, A., Lions, J. L. and Papanicolaou, G.: 1978, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam.

• Brinkman, H. C. 1947, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A1, 27–34.

• Carbonell, R. G. and Whitaker, S.: 1984, Heat and mass transfer in porous media, pages 123–198 in: J. Bear and M.Y. Corapcioglu (eds), Fundamentals of Transport Phenomena in Porous Media, Martinus Nijhoff, Dordrecht.

• Cioulachtjian, S., Tadrist, L., Occelli, R., et al.: 1992, Experimental analysis of heat transfer with phase change in porous media crossed by a fluid flow, Exp. Thermal Fluid Sci. 5, 533–547.

• Darcy, H.: 1856, Les fontaines publiques de la Ville de Dijon, Dalmont, Paris.

• Du Plessis, J. P.: 1994, Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed, Transport in Porous Media 16, 189–207.

• Edwards, D. A., Shapiro, M., Brenner, H., and Shapira, M.: 1991, Dispersion of inert solutes in spatially periodic, two-dimensional model porous media, Transport in Porous Media 6, 337–358.

• Ene, H.I. and Poliševski, D.: 1987, Thermal Flow in Porous Media, D. Reidel, Dordrecht.

• Eidsath, A.B.: 1981, Flow and dispersion in spatially periodic porous media: A finite element study, MS Thesis, Department of Chemical Engineering, University of California at Davis.

• Eidsath, A. B., Carbonell, R. G., Whitaker, S., and Herrmann, L. R.: 1983, Dispersion in pulsed systems III: Comparison between theory and experiments for packed beds, Chem. Engng. Sci. 38, 1803–1816.

• Ergun, S.: 1952, Fluid flow through packed columns, Chem. Eng. Progr. 48, 88–94.

• Forchheimer, P.: 1901, Wasserbewegung durch Boden, Z. Ver. Deutsch. Ing. 45, 1782–1788.

• Gray, W.G.: 1975, A derivation of the equations for multiphase transport, Chem. Engng. Sci. 30, 229–233.

• Greenkorn, R. A.: 1983, Flow Phenomena in Porous Media: Fundamentals and Applications in Petroleum, Water, and Food Production, Marcel Dekker, New York.

• Howes, F. A. and Whitaker, S.: 1985, The spatial averaging theorem revisited, Chem. Engng. Sci. 40, 1387–1392.

• Joseph, D. D., Nield, D. A. and Papanicolaou, G.: 1982, Nonlinear equation governing flow in saturated porous media, Water Resour. Res. 18, 1049–1052.

• Kannbuk, W. G. and Martin, L. H.: 1933, Proc. Roy. Soc. Lond. A141, 144–153.

• Launder, B. E. and Massey, T. H.: 1978, The numerical prediction of viscous flow and heat transfer in tube banks, J. Heat Trans. 100, 565–571.

• Ma, H. and Ruth, D. W.: 1993, The microscopic analysis of high Forchheimer number flow in porous media, Transport in Porous Media 13, 139–160.

• Macdonald, I. F., El-Sayed, M. S., Mow, K., and Dullien, F. A. L.: 1979, Flow through porous media: The Ergun equation revisited, Ind. Eng. Chem. Fundam. 18, 199–208.

• Martin, J. J., McCabe, W. L., and Mourad, C. C.: 1951, Pressure drop through stacked spheres - effect of orientation, Chem. Engng. Prog. 47, 91–98.

• Mei, C. C. and Auriault, J.-L.: 1991, The effect of weak inertia on flow through a porous medium, J. Fluid Mech. 222, 647–663.

• Nozad, I., Carbonell, R. G., and Whitaker, S.: 1985, Heat conduction in multiphase systems I: Theory and experiment for two-phase systems, Chem. Engng. Sci. 40, 843–855.

• Nozad, L., Carbonell, R. G., and Whitaker, S.: 1985, Heat conduction in multiphase systems II: Experimental method and results for three-phase systems, Chem. Engng. Sci. 40, 857–863.

• Ochoa-Tapia, J. A., Stroeve, P., and Whitaker, S.: 1994, Diffusive transport in two-phase media: Spatially periodic models and Maxwell's theory for isotropic and anisotropic systems, Chem. Engng. Sci. 49, 709–726.

• Ochoa-Tapia, J. A. and Whitaker, S.: 1995, Momentum transfer at the boundary between a porous medium and a homogeneous fluid I: Theoretical development, Int. J. Heat Mass Trans. 38, 2635–2646.

• Ochoa-Tapia, J. A. and Whitaker, S.: 1995, Momentum transfer at the boundary between a porous medium and a homogeneous fluid II: Comparison with experiment, Int. J. Heat Mass Trans. 38, 2647–2655.

• Quintard, M. and Whitaker, S.: 1993, One and two-equation models for transient diffusion processes in two-phase systems, in: Advances in Heat Transfer, Vol. 23, Academic Press, New York, pp. 369–465.

• Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions, Transport in Porous Media 14, 163–177.

• Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media II: Generalized volume averaging, Transport in Porous Media 14, 179–206.

• Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment, Transport in Porous Media 15, 31–49.

• Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media IV: Computer generated porous media, Transport in Porous Media 15, 51–70.

• Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems, Transport in Porous Media 15, 183–196.

• Quintard, M. and Whitaker, S.: 1995, Local thermal equilibrium for transient heat conduction: Theory and comparison with numerical experiments, Int. J. Heat Mass Trans. 38, 2779–2796.

• Quintard, M. and Whitaker, S.: 1995, Aerosol filtration: An analysis using the method of volume averaging, J. Aerosol Sci. 26, 1227–1255.

• Ruth, D. W. and Ma, H.: 1993, Numerical analysis of viscous, incompressible flow in a diverging-converging RUC, Transport in Porous Media 13, 161–177.

• Ruth, D. W. and Ma, H.: 1992, On the derivation of the Forchheimer equation by means of the averaging theorem, Transport in Porous Media 7, 255–264.

• Sahroui, M. and Kaviany, M.: 1994, Slip and no-slip temperature boundary conditions at the interface of porous, plain media: Convection, Int. J. Heat Mass Trans. 37, 1029–1044.

• Sanchez-Palencia, E.: 1980, Non-homogeneous Media and Vibration Theory, Lecture Notes in Phys. 127, Springer-Verlag, New York.

• Sangani, A. S. and Acrivos, A.: 1982, Slow flow past periodic arrays of cylinders with application to heat transfer, Int. J. Multiphase Flow 8, 193–206.

• Snyder, L. J. and Stewart, W. E.: 1966, Velocity and pressure profiles for Newtonian creeping flow in regular packed beds of spheres, AIChE J. 12, 167–173.

• Sorensen, J. P. and Stewart, W. E.: 1974, Computation of forced convection in slow flow through ducts and packed beds II: Velocity profile in a simple array of spheres, Chem. Engng. Sci. 29, 817–819.

• Susskind, H. and Becker, W.: 1967, Pressure drop in geometrically ordered packed beds of spheres, AIChE J. 13, 1155–1163.

• Tennekes, H. and Lumley, J. L.: 1972, A First Course in Turbulence, MIT Press, Cambridge, Massachusetts.

• Whitaker, S.: 1982, Laws of continuum physics for single-phase, single-component systems, in G. Hetsroni (ed.), Handbook of Multiphase Systems, Hemisphere Publ., New York, pp. 1–5 to 1–35.

• Whitaker, S.: 1986, Flow in porous media I: A theoretical derivation of Darcy's law, Transport in Porous Media 1, 3–25.

• Whitaker, S.: 1997, Volume averaging of transport equations, Chap, 1 in J. P. Du Plessis (ed.), Fluid Transport in Porous Media, Computational Mechanics Publications, Southampton, United Kingdom.

• Wodie, J.-C. and Levy, T.: 1991, Correction non linéaire de la loi de Darcy, C.R. Acad. Sci. Paris Série II 312, 157–161.

• Zick, A. A. and Homsy, G. M.: 1982, Stokes' flow through periodic arrays of spheres, J. Fluid Mech. 115, 13–26.

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Whitaker, S. The Forchheimer equation: A theoretical development. Transp Porous Med 25, 27–61 (1996). https://doi.org/10.1007/BF00141261