Abstract
A theoretical analysis, based on the search for a normal dissipation potential, is performed in order to generalize the empirical non-Darcy one-dimensional flow models to 3-D flows through anisotropic porous media. In an abstract framework, it is proven that a large number of heuristic non-linear equations governing the multidimensional flow through isotropic porous media can be derived starting from a potential strictly related to the mechanical power dissipated by the fluid. Such a formulation allows to define, for the tensor permeability case, a wide class of filtration models according to the Onsager's generalized theory of dissipative mechanical systems. A consistent generalization to anisotropic permeability case of polynomial flow models is proposed. Both primal and dual mixed variational formulations associated to the proposed quadratic and incomplete cubic flow models are introduced and discussed.
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References
Bachmat, Y.: 1965, Basic transport coefficients as aquifer characteristics, I.A.S.H. Symposium Hydrology of Fractured Rocks, Dubrovnik.
Cvetković, V. D.: 1986, A continuum approach to high velocity flow in porous medium, Transport in Porous Media 1, 63-97.
Darcy, H.: 1856, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmond, Paris.
Dupuit, J.: 1863, Ètudes Théoriques et Pratiques sur le Mouvement des Eaux, Dunod, Paris.
Fand, R. M., Kim, B. Y. K., Lam, A. C. C. and Phan, R. T.: 1987, Resistance of the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres, Trans. ASME, J. Fluids Engng. 109, 268-274.
Firdaouss, M., Guermond, J. and Le Quere, P.: 1997, Nonlinear correction to Darcy's law at low Reynolds number, J. Fluid Mech. 343, 331-350.
Forchheimer, P.: 1901a, Wasserbewegung durch Boden, Zeitscrift des Vereines Deutscher Ingenieure 49, 1736-1741.
Forchheimer, P.: 1901b, Wasserbewegung durch Boden, Zeitscrift des Vereines Deutscher Ingenieure 50, 1781-1788.
Hsu, C. T. and Cheng, P.: 1990, Thermal dispersion in a porous medium, Int. J. Heat Mass Transfer 33, 1587-1597.
Inoue, M. and Nakayama, A.: 1998, Numerical modelling of non-Newtonian fluids flow in a porous medium using a three-dimensional periodic array, Trans. ASME, J. Fluids Engng. 120, 131-135.
Kececioglu, I. and Rubinski, Y.: 1989, A continuum model for the propagation of discrete phasechange fronts in porous media in the presence of coupled heat flow, fluid flow and species transport processes, Int. J. Heat Mass Transfer 32, 1111-1130.
Kececioglu, I. and Jiang, Y.: 1994, Flow through porous media of packed spheres saturated with water, Trans. ASME, J. Fluids Engng. 1116, 164-170.
Knupp, P. M. and Lage, J. L.: 1995, Generalization of the Forchheimer-extended Darcy model to the tensor permeability case via a variational principle, J. Fluid Mech. 299, 97-104.
Lage, J. L.: 1998, The fundamental theory of flow through permeable media from Darcy to turbulence, in: D. B. Ingham and I. Pop (eds), Transport Phenomena in Porous Media, Elsevier Science, Oxford.
McCorquodale, J. A.: 1970, Variational approach to non-Darcy flows, J. Hydraul. Div. ASCE 96(HY11), 2265-2278.
Mei, C. C. and Auriault, J. L.: 1991, The effect of weak inertia on flow through porous media, J. Fluid Mech. 222, 647-663.
Moreau, J. J.: 1970,‘ Sur les lois de frottment de plasticitç et de viscosit’, C. R. Acad. Sc. Paris 271, 608-611.
Muskat, M.: 1946, The Flow of Homogeneous Fluids through Porous Media, J. Edwards (ed.), Ann Arbor, Michigan.
Nakayama, A. and Shenoy, A. V.: 1993, Non-Darcy forced convective heat transfer in a channel embedded in a non-Newtonian inelastic fluid-saturated porous medium, Canadian J. Chem. Engng. 71, 168-173.
Nield, D. A. and Bejan, A.: 1999, Convection in Porous Media, Springer-Verlag, New York, 2nd edn.
Nield, D. A.: 2000, Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium, Transport in Porous Media 41, 349-357.
Oden, J. T. and Reddy, J. N.: 1976, Variational Methods in Theoretical Mechanics, Springer.
Onsager, L.: 1942, Theories and problems of liquid diffusion, N. Y. Ac. Sci. 78, 622-645.
Rasoloarijaona, M. and Auriault, J.-L.: 1994, Nonlinear seepage flow through porous media, European J. Mech. B/Fluids 13, 177-195.
Russo Spena, A.: 1968, Moti di Filtrazione, Invited Lecture, Proc. of XI Nat. Conf. of Hydr., pp. 131-138 (in Italian).
Russo Spena, F. and Vacca, A.: 2001, A minmax formulation of nonlinear seepage flow problem, J. Inf. Opt. Sci., in press.
Ruth, D. W. and Ma, H.: 1992, On the derivation of the Forchheimer equation by means of the averaging theorem, Transport in Porous Media 7, 255-264.
Strang, G.: 1988, A framework for equilibrium equations, SIAM Rev. 30, 283-297.
Wang, X., Thauvin, F. and Mohanty, K. K.: 1999, Non-Darcy flow through anisotropic porous media, Chem. Engng. Sci. 54, 1859-1869.
Ward, J. C.: 1964, Turbulent flow in porous media, J. Hydraul. Div. ASCE 90(HY5): 1-12.
Whitaker, S.: 1996, The Forchheimer equation: A theoretical development, Transport in Porous Media 25, 27-61.
Zeidler, E.: 1985, Nonlinear Functional Analysis and its Applications, Vol. III, Springer-Verlag International.
Ziegler, H.: 1962, A Possible Generalization of Onsager's Theory, Advances in Solid Mechanics, Ac. Press, N.Y.
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Spena, F.R., Vacca, A. A Potential Formulation of Non-Linear Models of Flow through Anisotropic Porous Media. Transport in Porous Media 45, 405–421 (2001). https://doi.org/10.1023/A:1012044015534
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DOI: https://doi.org/10.1023/A:1012044015534