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 volumeaveraged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lowerorder terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano 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 volumeaveraged equations of motion gives rise to aclosure 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 ofabrupt changes in the structure of a porous medium is not considered.
This is a preview of subscription content, log in to check access.
Abbreviations
 A _{βσ} :

interfacial area of the βσ interface contained within the macroscopic system, m^{2}
 A _{βe } :

area of entrances and exits for the βphase contained within the macroscopic system, m^{2}
 Aβσ:

interfacial area of the βσ interface contained within the averaging volume, m^{2}
 A ^{*}_{β} σ:

interfacial area of the βσ interface contained within a unit cell, m^{2}
 Aβe :

area of entrances and exits for the βphase 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 βphase (see Figure 2), m
 ℓ_{σ} :

characteristic length scale for the σphase (see Figure 2), m
 n _{βσ} :

unit normal vector pointing from the βphase toward the σphase (n _{βσ}=−n _{σβ})
 n _{βe } :

unit normal vector for the entrances and exits of the βphase contained within a unit cell
 p _{β} :

pressure in the βphase, N/m^{2}
 〈p _{β}〉^{β} :

intrinsic phase average pressure for the βphase, N/m^{2}
 \(\tilde p_\beta \) :

p _{β}−〈p _{β}〉^{β}, spatial deviation of the pressure in the βphase, N/m^{2}
 r _{0} :

radius of the averaging volume and radius of a capillary tube, m
 v _{β} :

velocity vector for the βphase, m/s
 〈v _{β}〉:

phase average velocity vector for the βphase, m/s
 〈v _{β}〉^{β} :

intrinsic phase average velocity vector for the βphase, m/s
 \(\tilde v_\beta \) :

v _{β}−〈v _{β}〉^{β}, spatial deviation of the velocity vector for the βphase, m/s
 V :

averaging volume, m^{3}
 V_{β} :

volume of the βphase contained within the averaging volume, m^{3}
 ∈_{β} :

V_{β}/V, volume fraction of the βphase
 ρ_{β} :

mass density of the βphase, kg/m^{3}
 μ_{β} :

viscosity of the βphase, 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.
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.
Bear, J. and Braester, C., 1972, On the flow of two immiscible fluids in fractured porous media, inProceedings of the First Symposium on Fundamentals of Transport Phenomena in Porous Media, Elsevier, New York, pp. 177–202.
Bear, J., 1972,Dynamics of Fluids in Porous Media, Elsevier, New York.
Biot, M. A., 1962, Mechanics of deformation and acoustic propagation in porous media.J. Appl. Phys. 23, 1482–1498.
Brenner, H., 1968, Personal communication.
Brenner, H., 1980, Dispersion resulting from flow through spatially periodic porous media.Trans. Roy. Soc. (London) 297, 81–133.
Brinkman, H. C., 1947, On the permeability of media consisting of closely packed porous particles.Appl. Sci. Res. A1, 81–86.
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.
Crapiste, G. H., Rotstein, E., and Whitaker, S., 1986, A general closure scheme for the method of volume averaging. To be published inChem. Engng. Sci.
Cushman, J. H., 1983, Multiphase transport equations  I. General equation for macroscopic statistical, local, spacetime homogeneity.Transp. Theory Stat. Phys. 12, 35–71.
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.
Gray, W. G., 1975, A derivation of the equations for multiphase transport.Chem. Engng. Sci. 30, 229–233.
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.
Gray, W. G., 1983, Local volume averaging of multiphase systems using a nonconstant averaging volume.Int. J. Multiphase Flow 9, 755–761.
Greenkorn, R. A., 1984,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, 857–863.
Marle, C. M., 1967, Ecoulements monophasiques en milieu poreuxRev. Inst. Francais du Petrole 22, 1471–1509.
Nozad, I., Carbonell, R. G., and Whitaker, S., 1985 Heat conduction in multiphase systems: I theory and experiment for twophase systems,Chem. Engng. Sci. 40, 843–855.
Nozad, I., Carbonell, R. G., and Whitaker S., 1985, Heat conduction in multiphase systems: II experimental method and results for threephase systems,Chem. Engng. Sci. 40, 857–863.
Nield, D. A., 1983, The boundary correction for the RayleighDarcy problem: limitations of the Brinkman correction,J. Fluid Mech. 128, 37–46.
Raats, P. A. C. and Klute, A., 1968, Transport in soils: the balance of momentum,Soil. Sci. Soc. Amer. Proc. 32, 161–166.
Ross, S. M., 1983, Theoretical model of the boundary condition at a fluidporous interface. AIChE Journal29, 840–845.
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.
Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media.AIChE J. 13, 1066–1071.
Slattery, J. C., 1980,Momentum, Energy and Mass Transfer in Continua, Krieger, Malabar.
Veverka, V., 1981, Theorem for the local volume average of a gradient revised,Chem. Engng. Sci. 36, 833–838.
Whitaker, S., 1967, Diffusion and dispersion in porous media,AIChE J. 13, 420–427.
Whitaker, S., 1969, Advances in the theory of fluid motion in porous media,Ind. Eng. Chem. 12, 14–28.
Whitaker, S., 1983,Fundamental Principles of Heat Transfer, Krieger, Malabar.
Whitaker, S., 1984, Moisture transport mechanisms during the drying of granular porous media,Proceedings Fourth International Drying Symposium, Kyoto.
Author information
Affiliations
Rights and permissions
About this article
Cite this article
Whitaker, S. Flow in porous media I: A theoretical derivation of Darcy's law. Transp Porous Med 1, 3–25 (1986). https://doi.org/10.1007/BF01036523
Received:
Revised:
Issue Date:
Key words
 Volume averaging
 Brinkman
 correction
 closure