GENERALIZATION OF THE FORCHHEIMER-EXTENDED DARCY FLOW MODEL TO THE TENSOR PERMEABILITY CASE VIA A VARIATIONAL PRINCIPLE

A convex variational principle is used to obtain a generalization of t he empirical nonlinear one-dimensional Forchheimer-extended Darcy flow equation to the multidimensional and anisotropic (tenser permeability ) case. A modified permeability that is a function of flow velocity (o r pressure gradient) is introduced in order to transform the nonlinear flow equation into a pseudo-linear form. Imposing an incompressibilit y condition on this pseudo-linear equation leads to a flow equation in Euler-Lagrange form which is used to build the corresponding variatio nal principle. It is demonstrated that the variational principle is ba sed on minimizing the power (time rate of doing work) required by the fluid to flow at a certain velocity under a prescribed pressure gradie nt. A consistent generalization of the Forchheimer equation to the ten ser case then follows from the variational principle. The existence an d uniqueness of solutions to the nonlinear flow equations might also b e demonstrated using the variational principle on a case by case basis , once appropriate boundary conditions are chosen.