The gradient flow structure for incompressible immiscible two-phase flows in porous media

Clément Cancès 1, 2 Thomas Gallouët 3, 4 Léonard Monsaingeon 5
1 RAPSODI - Reliable numerical approximations of dissipative systems
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
4 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : We show that the widely used model governing the motion of two incompressible immiscible fluids in a possibly heterogeneous porous medium has a formal gradient flow structure. More precisely, the fluid composition is governed by the gradient flow of some non-smooth energy. Starting from this energy together with a dissipation potential, we recover the celebrated Darcy-Muskat law and the capillary pressure law governing the flow thanks to the principle of least action. Our interpretation does not require the introduction of any algebraic transformation like, e.g., the global pressure or the Kirchhoff transform, and can be transposed to the case of more phases.
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Submitted on : Wednesday, March 4, 2015 - 4:08:31 PM
Last modification on : Tuesday, May 14, 2019 - 10:10:26 AM
Long-term archiving on : Friday, June 5, 2015 - 11:00:33 AM

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Clément Cancès, Thomas Gallouët, Léonard Monsaingeon. The gradient flow structure for incompressible immiscible two-phase flows in porous media. Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2015, 353, pp.985-989. ⟨hal-01122770⟩

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