Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. (English) Zbl 1171.76035
Summary: 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 degenerate nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem by 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 for 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.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 76T30 | Three or more component flows |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
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