Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities

Clément Cancès

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 5, page 973-1001
  • ISSN: 0764-583X

Abstract

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We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.

How to cite

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Cancès, Clément. "Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 973-1001. <http://eudml.org/doc/250661>.

@article{Cancès2009,
abstract = { We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model. },
author = {Cancès, Clément},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Capillarity discontinuities; degenerate parabolic equation; finite volume scheme; convergence; uniqueness; existence},
language = {eng},
month = {8},
number = {5},
pages = {973-1001},
publisher = {EDP Sciences},
title = {Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities},
url = {http://eudml.org/doc/250661},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Cancès, Clément
TI - Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 5
SP - 973
EP - 1001
AB - We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.
LA - eng
KW - Capillarity discontinuities; degenerate parabolic equation; finite volume scheme; convergence; uniqueness; existence
UR - http://eudml.org/doc/250661
ER -

References

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