Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-18T01:13:20.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalization of the Forchheimer-extended Darcy flow model to the tensor premeability case via a variational principle

Published online by Cambridge University Press:  26 April 2006

P. M. Knupp  and
J. L. Lage
P. M. Knupp
Affiliation:
Ecodynamics Research Associates, PO Box 9229, Albuquerque, NM 87119, USA
J. L. Lage
Affiliation:
Mechanical Engineering Department, Southern Methodist University, Dallas, TX 75275-0337, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A convex variational principle is used to obtain a generalization of the empirical nonlinear one-dimensional Forchheimer-extended Darcy flow equation to the multidimensional and anisotropic (tensor permeability) case. A modified permeability that is a function of flow velocity (or pressure gradient) is introduced in order to transform the nonlinear flow equation into a pseudo-linear form. Imposing an incompressibility condition on this pseudo-linear equation leads to a flow equation in Euler–Lagrange form which is used to build the corresponding variational principle. It is demonstrated that the variational principle is based on minimizing the power (time rate of doing work) required by the fluid to flow at a certain velocity under a prescribed pressure gradient. A consistent generalization of the Forchheimer equation to the tensor case then follows from the variational principle. The existence and uniqueness of solutions to the nonlinear flow equations might also be demonstrated using the variational principle on a case by case basis, once appropriate boundary conditions are chosen.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 299 , 25 September 1995 , pp. 97 - 104
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bachmat, Y. 1965 Basic transport coefficients as aquifer characteristics. IASH Symp. on the Hydrology of Fractured Rocks, Dubrovnik, vol. 1, pp. 63–75.
Bear, J. & Bachmat, Y. 1990 Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer.
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.Google Scholar
Dacorogna, B. 1989 Direct Methods in the Calculus of Variations. Springer.
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Victor Dalmond.
Dupuit, J. 1863 Estudes Theoriques et Pratiques sur le Mouvement des Eaux. Dunod.
Forchheimer, P. 1901 Wasserbewegung durch Boden. VDIZ. 45, 1782–1788.Google Scholar
Givler, R. C. & Altobelli, S. A. 1994 A determination of the effective viscosity for the Brinkman—Forchheimer flow model. J. Fluid Mech. 258, 355–370.Google Scholar
Hsu, C. T. & Cheng, P. 1990 Thermal dispersion in a porous medium. Intl J. Heat Mass Transfer 33, 1587–1597.Google Scholar
Joseph, D. D., Nield, D. A. & Papanicolaou, G. 1982 Nonlinear equation governing flow in a saturated porous medium. Water Resources Res. 18, 1049–1052.Google Scholar
Kaviany, M. 1991 Principle of Heat Transfer in Porous Media. Springer.
MacDonald, I. F., El-Sayed, M. S., Mow, K. & Dullien, F. A. L. 1979 Flow through porous media: the Ergun equation revisited. Indust. Chem. Fundam. 18, 199–208.Google Scholar
Stanek, V. & Szekely, J. 1974 Three-dimensional flow of fluids through nonuniform packed beds. AIChE J. 20, 974–980.Google Scholar
Ward, J. C. 1964 Turbulent flow in porous media.. J. Hydraul. Div. ASCE 90 (HY5), 1–12.Google Scholar
Whitaker, S. 1969 Advances in theory of fluid motion in porous media. Indust. Engng Chem. 61, 14–28.Google Scholar