# Flow in porous media I: A theoretical derivation of Darcy's law

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## Abstract

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate a*no slip* condition at an interface between a porous medium and a bounding solid surface.

The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to a*closure problem*, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem of*abrupt changes* in the structure of a porous medium is not considered.

## Key words

Volume averaging Brinkman correction closure## Nomenclature

## Roman Letters

*A*_{βσ}interfacial area of the β-σ interface contained within the macroscopic system, m

^{2}*A*_{βe}area of entrances and exits for the β-phase contained within the macroscopic system, m

^{2}- Aβσ
interfacial area of the β-σ interface contained within the averaging volume, m

^{2}- A
_{β}^{*}σ interfacial area of the β-σ interface contained within a unit cell, m

^{2}- Aβ
*e* area of entrances and exits for the β-phase contained within a unit cell, m

^{2}**B**second order tensor used to represent the velocity deviation (see Equation (3.30))

**b**vector used to represent the pressure deviation (see Equation (3.31)), m

^{−1}*d*distance between two points at which the pressure is measured, m

**g**gravity vector, m/s

^{2}**K**Darcy's law permeability tensor, m

^{2}*L*characteristic length scale for volume averaged quantities, m

- ℓ
_{β} characteristic length scale for the β-phase (see Figure 2), m

- ℓ
_{σ} characteristic length scale for the σ-phase (see Figure 2), m

**n**_{βσ}unit normal vector pointing from the β-phase toward the σ-phase (

**n**_{βσ}=−**n**_{σβ})**n**_{βe}unit normal vector for the entrances and exits of the β-phase contained within a unit cell

*p*_{β}pressure in the β-phase, N/m

^{2}- 〈
*p*_{β}〉^{β} intrinsic phase average pressure for the β-phase, N/m

^{2}- \(\tilde p_\beta \)
*p*_{β}−〈*p*_{β}〉^{β}, spatial deviation of the pressure in the β-phase, N/m^{2}*r*_{0}radius of the averaging volume and radius of a capillary tube, m

**v**_{β}velocity vector for the β-phase, m/s

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

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

- \(\tilde v_\beta \)
**v**_{β}−〈**v**_{β}〉^{β}, spatial deviation of the velocity vector for the β-phase, m/s*V*averaging volume, m

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

^{3}

## Greek Letters

- ∈
_{β} V

_{β}/*V*, volume fraction of the β-phase- ρ
_{β} mass density of the β-phase, kg/m

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

^{2}- ψ
arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s

- ξ
arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s

^{−1}

## References

- Anderson, T. B. and Jackson, R., 1967, A fluid mechanical description of fluidized beds,
*Ind. Eng. Chem. Fundam.***6**, 527–538.Google Scholar - Baveye, P. and Sposito, G., 1984, The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers.
*Water Resour. Res.***20**, 521–530.Google Scholar - Bear, J. and Braester, C., 1972, On the flow of two immiscible fluids in fractured porous media, in
*Proceedings of the First Symposium on Fundamentals of Transport Phenomena in Porous Media*, Elsevier, New York, pp. 177–202.Google Scholar - Bear, J., 1972,
*Dynamics of Fluids in Porous Media*, Elsevier, New York.Google Scholar - Biot, M. A., 1962, Mechanics of deformation and acoustic propagation in porous media.
*J. Appl. Phys.***23**, 1482–1498.Google Scholar - Brenner, H., 1968, Personal communication.Google Scholar
- Brenner, H., 1980, Dispersion resulting from flow through spatially periodic porous media.
*Trans. Roy. Soc. (London)***297**, 81–133.Google Scholar - Brinkman, H. C., 1947, On the permeability of media consisting of closely packed porous particles.
*Appl. Sci. Res.***A1**, 81–86.Google Scholar - Carbonell, R. G. and Whitaker, S., 1984, Heat and mass transport in porous media, in J. Bear and M. Y. Corapcioglu (eds.),
*Fundamentals of Transport Phenomena in Porous Media*, Martinus Nijhoff, Dordrecht, pp. 121–198.Google Scholar - Crapiste, G. H., Rotstein, E., and Whitaker, S., 1986, A general closure scheme for the method of volume averaging. To be published in
*Chem. Engng. Sci.*Google Scholar - Cushman, J. H., 1983, Multiphase transport equations - I. General equation for macroscopic statistical, local, space-time homogeneity.
*Transp. Theory Stat. Phys.***12**, 35–71.Google Scholar - Cushman, J. H., 1984, On unifying concepts of scale, instrumentation and stochastics in the development of multiphase transport theory.
*Water Resour. Res.***20**, 1668–1672.Google Scholar - Gray, W. G., 1975, A derivation of the equations for multiphase transport.
*Chem. Engng. Sci.***30**, 229–233.Google Scholar - Gray, W. G. and O'Neil, K., 1976, On the general equations for flow in porous media and their reduction to Darcy's law.
*Water Resour. Res.***12**, 148–154.Google Scholar - Gray, W. G., 1983, Local volume averaging of multiphase systems using a nonconstant averaging volume.
*Int. J. Multiphase Flow***9**, 755–761.Google Scholar - Greenkorn, R. A., 1984,
*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**, 857–863.Google Scholar - Marle, C. M., 1967, Ecoulements monophasiques en milieu poreux
*Rev. Inst. Francais du Petrole***22**, 1471–1509.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, I., 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 - Nield, D. A., 1983, The boundary correction for the Rayleigh-Darcy problem: limitations of the Brinkman correction,
*J. Fluid Mech.***128**, 37–46.Google Scholar - Raats, P. A. C. and Klute, A., 1968, Transport in soils: the balance of momentum,
*Soil. Sci. Soc. Amer. Proc.***32**, 161–166.Google Scholar - Ross, S. M., 1983, Theoretical model of the boundary condition at a fluid-porous interface. AIChE Journal
**29**, 840–845.Google Scholar - Ryan, D., Carbonell R. G., and Whitaker, S., 1981,
*A Theory of Diffusion and Reaction in Porous Media*, AIChE Symposium Series, edited by P. Stroeve and W. J. Ward, #202, Vol. 77.Google Scholar - Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media.
*AIChE J.***13**, 1066–1071.Google Scholar - Slattery, J. C., 1980,
*Momentum, Energy and Mass Transfer in Continua*, Krieger, Malabar.Google Scholar - Veverka, V., 1981, Theorem for the local volume average of a gradient revised,
*Chem. Engng. Sci.***36**, 833–838.Google Scholar - Whitaker, S., 1969, Advances in the theory of fluid motion in porous media,
*Ind. Eng. Chem.***12**, 14–28.Google Scholar - Whitaker, S., 1983,
*Fundamental Principles of Heat Transfer*, Krieger, Malabar.Google Scholar - Whitaker, S., 1984, Moisture transport mechanisms during the drying of granular porous media,
*Proceedings Fourth International Drying Symposium, Kyoto.*Google Scholar