The DFLU flux for systems of conservation laws

Abstract

Abstract:The DFLU numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space. We show how this flux can be used to solve certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering. Furthermore, these results are extended to the case where the flux function is discontinuous in the space variable. Such a situation arises for example while dealing with oil reservoirs which are heterogeneous. Numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility, Lax-Friedrichs and Force schemes. Key-words:Finite volumes, finite differences, Riemann solvers, system of conservation laws, flow in porous media, polymer flooding. Le flux DFLU pour les systèmes de lois de conservationRésumé : Le flux numérique DFLU a été introduit pour résoudre des lois de conservation scalaires hyperboliques dont la fonction de flux est discontinues en espace. Nous montrons comment ce flux peut être utilisé pour résoudre une certaine classe de systèmes de lois de conservation tels que les systèmes modélisant l'injection de polymères en ingéniérie de réservoirs pétroliers. En outre, ces résultats s'étendent au cas de fonctions de flux discontinus par rapport à la variable d'espace. Une telle situation apparaît par exemple quand on considère des réservoirs pétroliers qui sont hétérogènes. Des expériences numériques sont présentées pour illustrer l'efficacité de ce nouveau schéma comparé à d'autres shémas standard tels que les schémas Mobilités Amont, Lax-Friedrichs et Force.

Errors in the Upstream Mobility Scheme for Counter-Current Two-Phase Flow With Discontinuous Permeabilities

“…In the following, the application of polymer flooding presented in Section 2.2.2 is discussed in this context. The theory presented herein follows the framework of [51] and [4]. hence the eigenvectors are not linearly independent and the problem is non-strictly hyperbolic.…”

“…Thus, the exact solution of the Riemann problem is more difficult to construct. General Riemann problems for this system are considered in [4,84]. Following [4], the main theory of the related solution strategy is provided below.…”

“…For the special application of polymer flooding, the corresponding Riemann problem was solved in Section 3.4.2 and the corresponding Godunov flux approximation can be defined in a straightforward manner, see e.g., [4]. In our experience, not unexpectedly, the first-order Godunov scheme has the least numerical dissipation, also for the polymer system.…”

“…The general conservative finite-volume scheme (4.4) for the system of equations governing polymer flooding (3.21) is written as The modified Godunov flux approximation that was presented for a spatial conservation law with discontinuous flux function in [1], was in [4] extended to the polymer system. The polymer concentration c(x,t) in f (s, c) is treats as known function which may be discontinuous at the space discretization points.…”

“…In [4] a discussion on total variation bounds and a convergence analysis for the DFLU scheme applied to the polymer system is provided. Note that the saturation s do not need to be of total variation bounded since f (s, c) and c(x,t) is discontinuous.…”