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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
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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

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