178 research outputs found
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
Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium
International audienceNeglecting capillary pressure effects in two-phase flow models for porous media may lead to non-physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non-zero capillarity. In this paper, we propose and compare several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi-dimensional case. It has been shown in [Andreianov&Canc'es, Comput. Geosci., 2013, to appear] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one-dimensional case which are used to motivate the construction of multi- dimensional finite volume schemes. We describe and compare a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Canc'es & Hilhorst, HAL preprint no.00675681, 2012) and two IMplicit Pressure - Explicit Saturation (IMPES) schemes with different level of coupling
On interface transmission conditions for conservation laws with discontinuous flux of general shape
International audienceConservation laws of the form with space-discontinuous flux were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of . The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications
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
Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality
We consider the evolution of interfaces in binary mixtures permeating
strongly heterogeneous systems such as porous media. To this end, we first
review available thermodynamic formulations for binary mixtures based on
\emph{general reversible-irreversible couplings} and the associated
mathematical attempts to formulate a \emph{non-equilibrium variational
principle} in which these non-equilibrium couplings can be identified as
minimizers.
Based on this, we investigate two microscopic binary mixture formulations
fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible
fluid formulation without fluid flow; (b) a momentum-driven formulation for
quasi-static and incompressible velocity fields. In both cases we state two
novel, reliably upscaled equations for binary mixtures/multiphase fluids in
strongly heterogeneous systems by systematically taking thermodynamic features
such as free energies into account as well as the system's heterogeneity
defined on the microscale such as geometry and materials (e.g. wetting
properties). In the context of (a), we unravel a \emph{universality} with
respect to the coarsening rate due to its independence of the system's
heterogeneity, i.e. the well-known -behaviour for
homogeneous systems holds also for perforated domains.
Finally, the versatility of phase field equations and their
\emph{thermodynamic foundation} relying on free energies, make the collected
recent developments here highly promising for scientific, engineering and
industrial applications for which we provide an example for lithium batteries
The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions
6 pagesInternational audienceWe consider hyperbolic scalar conservation laws with discontinuous flux function of the type \partial_t u + \partial_x f(x,u) = 0 \text{\;\;with\;\;} f(x,u) = f_L(u) \Char_{\R^-}(x) + f_R(u) \Char_{\R^+}(x). Here are compatible bell-shaped flux functions as appear in numerous applications. It was shown by Adimurthi, S. Mishra, G. D. V. Gowda ({\it J. Hyperbolic Differ. Equ. 2 (4) (2005) 783-837)} and R. Bürger, K. H. Karlsen and J. D. Towers ({\it SIAM J. Numer. Anal. 47~(3) (2009) 1684--1712}) that several notions of solution make sense, according to a choice of the so-called -connection. In this note, we remark that every choice of connection corresponds to a limitation of the flux under the form , first introduced by R. M. Colombo and P. Goatin ({\it J. Differential Equations 234 (2) (2007) 654-675}). Hence we derive a very simple and cheap to compute explicit formula for the Godunov numerical flux across the interface , for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter . A numerical illustration is provided
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
We present a numerical method for approximating the solutions of degenerate
parabolic equations with a formal gradient flow structure. The numerical method
we propose preserves at the discrete level the formal gradient flow structure,
allowing the use of some nonlinear test functions in the analysis. The
existence of a solution to and the convergence of the scheme are proved under
very general assumptions on the continuous problem (nonlinearities, anisotropy,
heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the
efficiency and of the robustness of our approach
Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities
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 non-linear
parabolic equations spatially coupled by non linear 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
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