Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution

We consider an immiscible two-phase flow in a heterogeneous one-dimensional porous medium. We suppose particularly that the capillary pressure field is discontinuous with respect to the space variable. The dependence of the capillary pressure with respect to the oil saturation is supposed to be weak, at least for saturations which are not too close to 0 or 1. We study the asymptotic behavior when the capillary pressure tends to a function which does not depend on the saturation. In this paper, we show that if the capillary forces at the spacial discontinuities are oriented in the same direction that the gravity forces, or if the two phases move in the same direction, then the saturation profile with capillary diffusion converges toward the unique optimal entropy solution to the hyperbolic scalar conservation law with discontinuous flux functions

A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rocks media

International audienceWe discuss the convergence of the upstream phase-by-phase scheme (or upstream mobility scheme) towards the vanishing capillarity solution for immiscible incompressible two-phase flows in porous media made of several rock types. Troubles in the convergence where recently pointed out in [S. Mishra & J. Jaffré, Comput. Geosci., 2010] and [S. Tveit & I. Aavatsmark, Comput. Geosci, 2012]. In this paper, we clarify the notion of vanishing capillarity solution, stressing the fact that the physically relevant notion of solution differs from the one inferred from the results of [E. F. Kaasschieter, Comput. Geosci., 1999]. In particular, we point out that the vanishing capillarity solution de- pends on the formally neglected capillary pressure curves, as it was recently proven in by the authors [B. Andreianov & C. Canc'es, Comput. Geosci., 2013]. Then, we propose a numerical procedure based on the hybridization of the interfaces that converges towards the vanishing capillarity solution. Numerical illustrations are provided

An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field

International audienceWe consider the system of equations governing an incompressible immiscible two-phase flow within an heterogeneous porous medium made of two different rock types. Since the capillary pressure funciton depends on the rock type, the capillary pressure field might be discontinuous at the interface between the rocks. We prove the existence of a solution for such a flow by passing to the limit in regularizations of the problem

Dimensional reduction of a fractured medium for a two-phase flow

We consider a porous medium containing a single fracture, and identify the aperture to length ratio as the small parameter ɛ with the fracture permeability and the fracture porosity scaled as exponents of ɛ. We consider a two-phase flow where the flow is governed by the mass balance and the Darcy law. Using formal asymptotic approach, we derive a catalogue of reduced models as the vanishing limit of ɛ. Our derivation provides new models in a hybrid-dimensional setting as well as models which exhibit two-scale behaviour. Several numerical examples confirm the theoretical derivations and provide additional insight.publishedVersio

A gravity current model with capillary trapping for oil migration in multilayer geological basins

We propose a reduced model accounting capillary trapping to simulate oil migration in geological basins made of several rock types. Our model is derived from Darcy type models thanks to Dupuit approximation and a vertical integration in each geological layer. We propose a time-implicit finite volume scheme which is shown to be unconditionally stable and to admit discrete solutions. Numerical outcomes are then provided in order to illustrate the behavior of our reduced model