Abstract
In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- A βσ :
-
interfacial area of theΒ-σ interface contained within the macroscopic system, m2
- A βe :
-
area of entrances and exits for theΒ-phase contained within the macroscopic system, m2
- A βσ :
-
interfacial area of theΒ-σ interface contained within the averaging volume, m2
- A βe :
-
area of entrances and exits for theΒ-phase contained within the averaging volume, m2
- B :
-
second-order tensor used to respresent the velocity deviation
- b :
-
vector used to represent the pressure deviation, m−1
- C :
-
second-order tensor related to the permeability tensor, m−2
- D :
-
second-order tensor used to represent the velocity deviation, m2
- d :
-
vector used to represent the pressure deviation, m
- g :
-
gravity vector, m/s2
- I :
-
unit tensor
- K :
-
εβ C −1,−εβ〈D〉β, Darcy's law permeability tensor, m2
- L :
-
characteristic length scale for volume averaged quantities, m
- ℓβ :
-
characteristic length scale for theβ-phase, m
- l i :
-
i=1, 2, 3, lattice vectors, m
- n βσ :
-
unit normal vector pointing from theΒ-phase toward theσ-phase
- n βe :
-
outwardly directed unit normal vector at the entrances and exits of theΒ-phase
- p β :
-
pressure in theβ-phase, N/m 2
- 〈p β〉β :
-
intrinsic phase average pressure, N/m2
- \(\tilde p_\beta \) :
-
p β−〈p β〉β, spatial deviation of the pressure in theΒ-phase, N/m2
- r :
-
position vector locating points in theΒ-phase, m
- r 0 :
-
radius of the averaging volume, m
- t :
-
time, s
- v β :
-
velocity vector in theβ-phase, m/s
- 〈v β〉β :
-
intrinsic phase average velocity in theβ-phase, m/s
- 〈v β 〉:
-
phase average or Darcy velocity in the \-phase, m/s
- \(\tilde v_\beta \) :
-
v β−〈v β〉β, spatial deviation of the velocity in theΒ-phase m/s
- V :
-
averaging volume, m3
- V β :
-
volume of theΒ-phase contained in the averaging volume, m3
- ε β :
-
V β /V volume fraction of theΒ-phase
- ρ β :
-
mass density of theΒ-phase, kg/m3
- μ β :
-
viscosity of theΒ-phase, Nt/m2
References
Allaire, G., 1989, Prolongement de la pression et homogénéisation des équations de Stokes dans un milieu poreux connexe,C.R. Acad. Sci. Paris 309, Série I, 717–722.
Barrere, J., 1990, Modélisation des équations de Stokes et Navier-Stokes en milieux poreux, Thèse de l'Université de Bordeux I.
Bear, J., 1972,Dynamics of Fluids in Porous Media, Elsevier, New York.
Bensoussan, J., Lions, L., and Papanicolaou, G., 1978,Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam.
Bergelin, O. P., Colburn, A. P., and Hall, H. L., 1950, Heat transfer and pressure drop during viscous flow across unbaffled tube banks, Bulletin No. 2, University of Delaware Experiment Station.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960,Transport Phenomena, John Wiley & Sons, Inc., New York.
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, 121–198, Martinus Nijhoff, Dordrecht.
Crapiste, G. H., Rotstein, E., and Whitaker, S., 1986, A general closure scheme for the method of volume averaging,Chem. Engng. Sci.,41, 227–235.
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.
Eidsath, A., 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.
Eidsath, A. B., 1981, Flow and dispersion in spatially periodic porous media: A finite element study, MS Thesis, Development of Chemical Engineering, University of California, Davis.
Gray, W. G., 1975, A derivation of the equations for multiphase transport,Chem. Engng. Sci. 30, 229–233.
Macdonald, I. F., El-Sayed, M. S., Mow, K., and Dullien, F. A. L., 1979, Flow through porous media; The Ergun equation revisited,Inc. Eng. Chem. Fundam. 18, 199–208.
Nassik, A. Z., 1979, Homogénéisation des équations de Stokes pour un écoulement en milieu poreux périodique, Thèse de l'Université Pierre et Marie Curie, Paris VI.
Quintard, M. and Whitaker, S., 1992, Transport in ordered and disordered porous media I: Darcy's law; II: Some geometrical results, submitted toTransport in Porous Media.
Sanchez-Palencia, E., 1980, Non-homogeneous media and vibration theory,Lecture Notes in Physics 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,A.I.Ch.E. Journal 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.
Tartar, L., 1980, Incompressible fluid flow in a porous medium: Convergence of the homogenization process, Appendix inLecture Notes in Physics 127, Springer-Verlag, New York.
Whitaker, S., 1983,Fundamental Principles of Heat Transfer, R. E. Krieger Pub. Co., Malabar, Florida.
Whitaker, S., 1986a, Flow in porous media I: A theoretical derivation of Darcy's law,Transport in Porous Media 1, 3–25.
Whitaker, S., 1986b, Transport processes with heterogeneous reaction, in S. Whitaker and A. E. Cassano (eds.)Concepts and Design for Chemical Reactors, 1–94, Gordon and Breach, New York.
Whitaker, S., 1989, Heat transfer in catalytic packed bed reactors,Handbook of Heat and Mass Transfer Volume 3, Chapter 10 inCatalysis, Kinetics & Reactor Engineering, edited by N. P. Cheremisinoff, Gulf Publishers, New Jersey.
Zick, A. A. and Homsy, G. M., 1982, Stokes flow through periodic arrays of spheres,J. Fluid Mech. 115, 13–26.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barrere, J., Gipouloux, O. & Whitaker, S. On the closure problem for Darcy's law. Transp Porous Med 7, 209–222 (1992). https://doi.org/10.1007/BF01063960
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01063960