# The Forchheimer equation: A theoretical development

- Received:
- Revised:

- 229 Citations
- 2.6k Downloads

## Abstract

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.

### Key words

Forchheimer equation Darcy's law volume averaging closure### Nomenclature

*A*_{βσ}area of the

*β*-σ interface contained with the macroscopic region, m^{2}*A*_{βe}area of the entrances and exits of the

*β*-phase at the boundary of the macroscopic region, m^{2}*A*_{βσ}area of the

*β*-σ interface contained within the averaging volume, m^{2}*A*_{p}surface area of a particle, m

^{2}**b**the vector field that maps μ

_{β}〈v_{β}〉^{β}onto \(\tilde p\)_{β}when inertial effects are negligible, m^{−1}**B**tensor that maps \(\tilde v\)

_{β}onto 〈v_{β}〉^{β}when inertial effects are negligible*d*_{p}6

*V*_{p}/A_{p}, effective particle diameter, m- g
gravitational acceleration, m/s

^{2}**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 μ

_{β}〈v_{β}〉^{β}onto \(\tilde p\)_{β}, m^{−1}**M**tensor that maps \(\tilde v\)

_{β}onto μ_{β}〈v_{β}〈^{β}- n
_{ασ} unit normal vector directed from the

*β*-phase toward the*σ*-phase*p*_{β}total pressure in the

*β*-phase, Pa- 〈
*p*_{β}〉^{β} 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

*r*0radius of the averaging volume, m

*t*time, s

*t*^{*}characteristic process time, s

- v
_{β} velocity in the

*β*-phase, m/s- 〈v
_{β}〉^{β} intrinsic average velocity in the

*β*-phase, m/s- 〈v
_{β}〉 superficial average velocity in the

*β*-phase, m/s- \(\tilde v\)
_{β} v

_{β}− 〈v_{β}〉^{β}, spatial deviation velocity, m/s- ν
local averaging volume, m

^{3}- V
_{β} volume of the

*β*-phase contained within the averaging volume, m^{3}*V*_{p}volume of a particle, m

^{3}**x**position vector locating the centroid of the averaging volume, m

- y
_{β} position vector locating points in the

*β*-phase relative to the centroid of the averaging volume, m

### Greek Symbols

- ε
_{β} V

_{β}/ν, volume fraction of the*β*-phase- ρ
_{β} density of the

*β*-phase, kg/m^{3}- μ
_{β} viscosity of the

*β*-phase, Pa s

## Preview

Unable to display preview. Download preview PDF.

### References

- Barrère, J.: 1990, Modélisation des équations de Stokes et Navier-Stokes en milieux poreux, Thèse de l'Université de Bordeaux I.Google Scholar
- Barrère, J., Olivier, G., and Whitaker, S.: 1992, On the closure problem for Darcy's law,
*Transport in Porous Media***7**, 209–222.Google Scholar - Bensoussan, A., Lions, J. L. and Papanicolaou, G.: 1978,
*Asymptotic Analysis for Periodic Structures*, North-Holland, Amsterdam.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - Darcy, H.: 1856,
*Les fontaines publiques de la Ville de Dijon*, Dalmont, Paris.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - Ene, H.I. and Poliševski, D.: 1987,
*Thermal Flow in Porous Media*, D. Reidel, Dordrecht.Google Scholar - 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.Google Scholar
- 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.CrossRefGoogle Scholar - Forchheimer, P.: 1901,
*Wasserbewegung durch Boden*, Z. Ver. Deutsch. Ing.**45**, 1782–1788.Google Scholar - Gray, W.G.: 1975, A derivation of the equations for multiphase transport,
*Chem. Engng. Sci.***30**, 229–233.Google Scholar - Greenkorn, R. A.: 1983,
*Flow Phenomena in Porous Media: Fundamentals and Applications in Petroleum, Water, and Food Production*, Marcel Dekker, New York.Google Scholar - Howes, F. A. and Whitaker, S.: 1985, The spatial averaging theorem revisited,
*Chem. Engng. Sci.***40**, 1387–1392.CrossRefGoogle Scholar - Joseph, D. D., Nield, D. A. and Papanicolaou, G.: 1982, Nonlinear equation governing flow in saturated porous media,
*Water Resour. Res.***18**, 1049–1052.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Quintard, M. and Whitaker, S.: 1994, Transport in ordered and disordered porous media II: Generalized volume averaging,
*Transport in Porous Media***14**, 179–206.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - Quintard, M. and Whitaker, S.: 1995, Aerosol filtration: An analysis using the method of volume averaging,
*J. Aerosol Sci.***26**, 1227–1255.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - Sanchez-Palencia, E.: 1980,
*Non-homogeneous Media and Vibration Theory*, Lecture Notes in Phys. 127, Springer-Verlag, New York.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - Susskind, H. and Becker, W.: 1967, Pressure drop in geometrically ordered packed beds of spheres,
*AIChE J.***13**, 1155–1163.CrossRefGoogle Scholar - Tennekes, H. and Lumley, J. L.: 1972,
*A First Course in Turbulence*, MIT Press, Cambridge, Massachusetts.Google Scholar - 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.Google Scholar - Whitaker, S.: 1986, Flow in porous media I: A theoretical derivation of Darcy's law,
*Transport in Porous Media***1**, 3–25.Google Scholar - 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.Google Scholar - 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.Google Scholar - Zick, A. A. and Homsy, G. M.: 1982, Stokes' flow through periodic arrays of spheres,
*J. Fluid Mech.***115**, 13–26.Google Scholar