Boris Andreianov
email: boris.andreianov@univ-fcomte.fr

Clément Cancès
email: cances@ann.jussieu.fr

Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium

Keywords: scalar conservation laws with discontinuous flux functions, discontinuous capillarity; two-phase flows in heterogeneous porous media, finite volume schemes AMS classification (2010) 35L02, 35L65, 65M08, 76S05

For the hyperbolic conservation laws with discontinuous flux function there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley-Leverett equations, each notion of solution is uniquely determined by the choice of a "connection", which is the unique stationary solution that takes the form of an undercompressive shock at the interface. To select the appropriate connection, following Kaasschieter (Comput. Geosci., 3(1): 1999) we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Netw. Heterog. Media, 5(3):635-647, 2010) that the "optimal" connection and the "barrier" connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the "optimal" entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter's results which is sometimes encountered in the literature).

We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley-Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.

Introduction

The Buckley-Leverett equation is a scalar conservation law

∂ t s + ∂ x f (x, s) = 0
with a particular form of the flux function f (x, •); the dependence in x describes the medium heterogeneities, and the whole equation serves as a model for two-phase immiscible flow in one-dimensional medium with neglected capillarity effects. The details of the models (with and without capillarity) are recalled in the sequel. When the dependence of f on x is regular, the notion of Kruzhkov entropy solution [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] has been recognized as the appropriate one; in particular, it is known that, whatever be the form of the capillary pressure curves, the "vanishing capillarity limit" yields the Kruzhkov solution (e.g., in the autonomous case, one can deduce this convergence result from the approach of [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]; for the general case, the result of [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] can be used). The situation is much more delicate when the medium consists of two or more geological layers with radically different physical properties and a sharp transition between the layers; mathematically, this means that x → f (x, •) presents discontinuities. Several works were devoted to the study of such discontinuousflux Buckley-Leverett model ; let us mention Gimse and Risebro [START_REF] Gimse | Riemann problems with a discontinuous flux function[END_REF][START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF], Kaasschieter [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], Adimurthi et al. [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF][START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], Bürger et al. [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF] (see also [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF]), and the works [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF][START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF] of the second author. These works were mainly considering the model problem with interface located at x = 0 and piecewise constant in 1 1 x>0 ; this will also be our framework in this paper.

x flux f (x, •) = f L (•)1 1 x<0 + f R (•)
In particular, Adimurthi, Mishra and Veerappa Gowda in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] have pointed out the fact that infinitely many notions of solution, all of them equally consistent from the mathematical point of view, may coexist for the discontinuous-flux Buckley-Leverett equation; this fact was illustrated numerically in [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF]. The so-called "optimal entropy solutions" (here and in the sequel, we follow the terminology of [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF][START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF]) were recognized as the vanishing capillarity limits (with discontinuous capillarity π(x, •) = π L (•)1 1 x<0 + π R (•)1 1 x>0 ) in some physical situations: see [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF][START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF]. Let us highlight the fact that in many physical situations, the "optimal entropy solutions" are not appropriate. Indeed, in [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF] it was shown that the so-called "barrier entropy solutions" appear, in another physical range of parameters. Roughly speaking, the optimal entropy solutions correspond to the maximization of the flux of one phase across the interface while the barrier entropy solutions correspond to the situation where the flux of this same phase across the interface is minimized (cf. [START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF]). As shown in [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF], the occurrence of the barrier entropy solution can be linked to the oil trapping phenomenon. In this paper we show, both theoretically and numerically, that all intermediate notions of entropy solutions, described by Adimurthi, Mishra and Veerappa Gowda in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] and by Bürger, Karlsen and Towers in [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], do appear as vanishing capillarity limits for some choice of nonlinearities (see Theorem 5 and the subsequent comment). More importantly, we indicate a simple procedure that permits to identify the adequate notion of solution, given the graphs of the flux functions f L,R and of the capillarity functions π L,R . In Section 2.3 we make clear the relation of the conclusions of our work to the conclusions of the pioneering work [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] of Kaasschieter that are sometimes misinterpreted in the recent literature.

While the starting point of our analysis is exactly the same as in the work [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], we exploit the theoretical framework of the paper [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] of Karlsen, Risebro and the first author (see also Bürger et al. [20]) in order to avoid the lengthy analysis of vanishing capillarity profiles corresponding to different initial Riemann data. Namely, from the facts established in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and those assessed in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF][START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF], we deduce that only one vanishing capillarity profile should be constructed explicitly. The choice of the profile follows a simple geometrical rule (see Fig. 1 and Proposition 3). The main result of the paper, i.e. Theorem 5, combines elements of the general approach to scalar conservation laws with discontinuous flux (see [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]) with some recent results on two-phase flows in two rocks' media (see [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]).

The paper is organized as follows. In Section 1, we recall the parabolic model for two-rocks' porous medium, and the notions of bounded-flux and mild solutions as introduced in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]. The key point here is the so-called Kato inequality, which is a localized L 1 contraction principle satisfied by two mild solutions. In Section 1.3, we point out a particular mild solution; this is a viscosity profile connecting some states (s π L , s π R ) defined from transmission conditions across the interface. This profile gives rise to the particular stationary solution c(x) = s π L 1 1 x<0 + s π R 1 1 x>0 for the hyperbolic Buckley-Leverett model in two-rocks' medium described in Section 2. Namely, c(•) can be obtained as a vanishing capillarity limit, therefore it must be considered as an admissible solution for the hyperbolic model. Using this fact and the general structure of entropy solutions to our hyperbolic model, in Theorem 5 we eventually identify the vanishing capillarity limits as the G (s π L ,s π R ) -entropy solutions in the sense of [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. Finally, in Section 3 we illustrate numerically the above theoretical results. For solving the hyperbolic model obtained as the vanishing capillarity limit, we use a simple finite volume Godunov scheme designed in [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] to approximate the discontinuous-flux Buckley-Leverett equation in a way compatible with the more precise parabolic model with capillarity. In order to illustrate the efficiency of the procedure, we compare the results provided by this Godunov scheme with those provided by the scheme (analyzed in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]) that approximates the parabolic problem. In particular, we observe a remarkable computational gain in considering the simplified model, as well as a good concordance in the numerical results.

1 Parabolic model for two-phase flow in two-rocks' medium This section is devoted to the parabolic model of two-phase flow with discontinuous capillary pressure in one space dimension. Following the previous work of the second author [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF][START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF] (see also [START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF][START_REF] Buzzi | Interface conditions for degenerate two-phase flow equations in one space dimension[END_REF]), the frame of multivalued capillary pressures is introduced in order to give a extended sense to the continuity of the capillary pressure at the medium's discontinuity. We will use the notions of bounded-flux and mild solutions that have been proved to be well-suited for this problem in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]. This model will be re-scaled, letting a scaling parameter appear in front of the capillary diffusion. Letting the capillarity parameter ǫ tend to zero will be the main purpose of this paper, and especially of Section 2.2.

Immiscible two-phase flows with discontinuous capillary pressure

We consider a one-dimensional porous medium made of two different rocks Ω L = (-∞, 0) and Ω R = (0, +∞), separated by an interface Γ = {x = 0}. The medium is assumed to be vertical, but we use the subscripts L ("Left") for the lower rock, and R ("Right") for the upper rock in order to comply with the notation used in the context of conservation laws with discontinuous flux. Two immiscible and incompressible phases a, b are flowing within this medium. Writing the volume balance of each phase in Ω i yields

φ i ∂ t s α + ∂ x v α = 0 (α ∈ {a, b}, i ∈ {L, R}), (1)
where s α ∈ [0, 1] denotes the saturation of the phase α and φ i ∈ (0, 1) denotes the porosity of the rock Ω i . The filtration speed v α of the phase α is prescribed by the Darcy-Muskat law (see e.g. [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF])

v α = -K i kr α,i (s α ) µ α (∂ x p α -ρ α g) (α ∈ {a, b}, i ∈ {L, R}), (2)
where K i is the intrinsic permeability of Ω i , µ α , p α and ρ α are respectively the viscosity, the pressure and the density of the phase α, and g is the gravity. Whenever ρ a = ρ b , the presence of gravity induces the buoyancy force. The relative permeability kr α,i of the phase α in Ω i is supposed to be Lipschitz continuous, increasing on [0, 1] and such that kr α,i (0) = 0. The pore volume is supposed to be fully saturated by the fluid, i.e.

s a + s b = 1, (3)
while the phase pressures are supposed to be linked by the capillary pressure relation

p a -p b = π i (s a ), (i ∈ {L, R}), (4)
where the functions π i are increasing. As noticed by H. W. Alt et al. [START_REF] Alt | On nonstationary flow through porous media[END_REF], the natural topology for the phase pressure p α stems from the estimate

i Ωi kr α,i (s α ) (∂ x p α ) 2 dx ≤ C. (5)
Therefore, if s α = 0 (and thus kr α,i (s α ) = 0), no control is provided by (5) on the pressure p α . As suggested in [START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF] (see also [START_REF] Brenner | A convergent finite volume scheme for two-phase flows in porous media with discontinuous capillary pressure field[END_REF][START_REF] Brenner | Convergence of finite volume approximation for immiscible two-phase flows in porous media with discontinuous capillary pressure field in several dimensions[END_REF]), we extend the pressure in the following multivalued way

p a (x, t) = [-∞, p b (x, t) + π i (0)] if x ∈ Ω i and s a (x, t) = 0, ( 6a
)
p b (x, t) = [-∞, p a (x, t)) -π i (1)] if x ∈ Ω i and s a (x, t) = 1, (6b)
for i = L, R. Therefore, as it was already the case in [START_REF] Buzzi | Interface conditions for degenerate two-phase flow equations in one space dimension[END_REF][START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF], the capillary pressure function has to be extended into the maximal monotone graph πi from [0, 1] to [-∞, +∞] defined by

πi (s) =      π i (s) if s ∈ (0, 1), [-∞, π i (0)] if s = 0, [π i (1), +∞] if s = 1. (7)
At the interface Γ, we require the balance of the phase fluxes, i.e. (formally)

v α (0 -, t) = v α (0 + , t) (α ∈ {a, b}), (8)
and the continuity of the extended phase pressures, i.e.

p α (0 -, t) ∩ p α (0 + , t) = ∅. (9)
Here and at the sequel, the values at x = 0 ± denote the one-sided traces of different quantities, in some sense that has to be made precise in each case. Now, summing (1) for α = a, b we find that ∂ x (v a + v b ) = 0. Thanks to [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF], we can claim that the total flow rate q := v a + v b only depends on time. For the sake of simplicity, we assume that q is constant in time. However, our results can be generalized to the case of time dependent q by means of an adaptation of the tools developed in [START_REF] Andreianov | On vanishing viscosity approximation of conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF][START_REF] Cancès | Two-phase Flows Involving Discontinuities on the Capillary Pressure In Proceeding of the conference FVCA[END_REF]. Without loss of generality, we assume that q ≥ 0 and that the buoyancy coefficient (ρ a -ρ b )g is nonnegative (these conditions can be enforced by changing x by -x and by exchanging the role of a and b). The equation ( 1) for the phase a can now be rewritten under the form

φ i ∂ t s a + ∂ x (f i (s a ) -λ i (s a )∂ x π i (s a )) = 0, (10)
where, for i = L, R,

λ i (s) = K i kr a,i (s)kr b,i (1 -s) µ b kr a,i (s) + µ a kr b,i (1 -s) , f i (s) = q kr a,i (s) kr a,i (s) + µa µ b kr b,i (1 -s) + (ρ a -ρ b )gλ i (s). (11)
Since we assumed that kr a,i (s), kr b,i (s) are zero if and only if s = 0, the functions λ i verify λ i (0) = λ i (1) = 0 and λ i (s) > 0 if s ∈ (0, 1), while the functions f i are such that f i (0) = 0 and f i (1) = q. For classical choices of relative permeabilities kr a,i and kr b,i (see e.g. [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF]), the flux functions f i , i = L, R, are bell-shaped in the sense (A1) below.

For the sake of readability, we remove the index a in s a ; thus s stands for the saturation of the phase a. Denoting by ϕ i the Kirchhoff's transform function defined by

ϕ i (s) = s 0 λ i (z)π ′ i (z)dz,
we convert equation [START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF], valid in Ω i , into

φ i ∂ t s + ∂ x (f i (s) -∂ x ϕ i (s)) = 0. ( 12
)
Thus equation ( 8) becomes

lim x→0 - (f L (s) -∂ x ϕ L (s)) = lim x→0 + (f R (s) -∂ x ϕ R (s)) ; (13)
the precise sense of equality (13) will be specified later. Notice that traces at x = 0 ± of ϕ i (s) exist whenever

ϕ i (s(t, •)) ∈ H 1 (Ω i ).
Since each ϕ i admits a continuous inverse function, also the one-sided traces of s on Γ exist in the strong L 1 (0, T ) sense. Denote by s L , s R the traces on Γ from Ω L and Ω R respectively; it has been shown in [START_REF] Buzzi | Interface conditions for degenerate two-phase flow equations in one space dimension[END_REF][START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF] that relation (9) implies

πL (s L ) ∩ πR (s R ) = ∅. (14)
Note that in this paper, buoyancy is taken into account, and, as it will be stressed in the sequel, it plays a major role in the following study. Indeed, it makes the flux f i defined by [START_REF] Baiti | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data[END_REF] bell-shaped in the sense of assumption (A1) below. In the case where the gravity was neglected, existence of traveling wave solutions to problem ( 12)-( 14) was investigated in [START_REF] Van Duijn | The effect of capillary forces on immiscible two-phase flows in heterogeneous porous media[END_REF], while existence and uniqueness of (regular) weak solutions was

shown in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF][START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF]. The effective equations in a stratified porous medium were formally derived in [START_REF] Van Duijn | Effective equations for two-phase flow with trapping on the micro scale[END_REF], and rigorously recovered in [START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF]. Numerical schemes were proposed in [START_REF] Ersland | Numerical methods for flows in a porous medium with internal boundary[END_REF][START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF][START_REF] Hoteit | Numerical modeling of two-phase flow in heterogeneous permeable media with different capillary pressure[END_REF][START_REF] Ern | Discontinuous galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures[END_REF] and analyzed in [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF]. To our knowledge, the only results available concerning the analysis of problem ( 12)-( 14) in presence of gravity are [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] for the traveling waves and [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] for the existence and uniqueness of the solutions, existence being proved by establishing the convergence of a suitable finite volume scheme. Multi-dimensional extensions have been recently performed [START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF][START_REF] Brenner | A convergent finite volume scheme for two-phase flows in porous media with discontinuous capillary pressure field[END_REF][START_REF] Brenner | Convergence of finite volume approximation for immiscible two-phase flows in porous media with discontinuous capillary pressure field in several dimensions[END_REF]. Due to the large dimensions of the sedimentary basins, and since the time scale involved in the migration of hydrocarbons is also large, it is natural to rescale the variables by choosing x := x/ǫ, t := t/ǫ for some small positive ǫ. The problem ( 12)-( 14), completed with the initial condition (15d), thus turns into

φ i ∂ t s ǫ + ∂ x (f i (s ǫ ) -ǫ∂ x ϕ i (s ǫ )) = 0 in Ω i × (0, ∞), (15a)
lim x→0 -(f L (s ǫ ) -ǫ∂ x ϕ L (s ǫ )) = lim x→0 + (f R (s ǫ ) -ǫ∂ x ϕ R (s ǫ )) in (0, T ), (15b) πL (s ǫ L ) ∩ πR (s ǫ R ) = ∅ in (0, T ), (15c)
s ǫ |t=0 = s 0 in R. (15d)
Here, as usual, i = L, R and s ǫ L , s ǫ R denote the traces of s ǫ at x = 0 -and x = 0 + , respectively. The flux transmission property (15b) should be understood in the weak sense, e.g., according to the theory of [START_REF] Chen | Divergence-measure fields and hyperbolic conservation laws[END_REF].

Let us now make precise the assumptions on the data required for our analysis. It is worth noting that all of them are fulfilled by the model commonly used in oil-engineering (see [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF][START_REF] Aziz | Petroleum Reservoir Simulation[END_REF]).

(A1) The flux functions f i belong to Lip([0, 1]) and satisfy f i (0) = 0, f i (1) = q ≥ 0. Moreover, f i is a so-called bell-shaped function, i.e. there exists s i ∈ (0, 1] such that f ′ i (s)(s is) > 0 for a.e. s ∈ (0, 1).

(A2) The capillary pressure functions π i belong to Lip loc ((0, 1)) ∩ L 1 ((0, 1)) and they are strictly increasing on (0, 1).

(A3) The Kirchhoff transforms ϕ i belong to Lip([0, 1]) and they are strictly increasing on [0, 1].

(A4) s 0 is measurable with 0 ≤ s 0 ≤ 1.
Hereabove, i = L, R; and by Lip and Lip loc we denote the spaces of Lipschitz and locally Lipschitz continuous functions, respectively.

1.2 Bounded-flux solutions and mild solutions of [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] This section is devoted to a brief summary of the state of the art for the mathematical analysis of the system (15) for fixed ǫ. Some existence and uniqueness results can be found in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF][START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF], but we focus here on the frame developed in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]. In particular, up to our knowledge, the result of [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] is the only one that allows to take non-monotone flux function f i (as it is the case when buoyancy is taken into account). The mathematical analysis of the system (15) for fixed ǫ is carried out in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF][START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF][START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] in cases where the gravity (and thus the buoyancy) is neglected, and in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] in presence of buoyancy. Let us recall the framework of bounded-flux solutions introduced in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] for this problem.

Definition 1 (bounded-flux solution) A function s ǫ ∈ L ∞ (R × R + ; [0, 1]) is said to be a bounded flux solution of problem (15) with initial datum s 0 if ∂ x ϕ i (s ǫ ) ∈ L ∞ (Ω i × R + ), if πL (s ǫ L (t)) ∩ πR (s ǫ R (t)) = ∅ for a.e. t ∈ R + , and if, for all ψ ∈ C ∞ c (R × R + ), R×R+ φ i s ǫ ∂ t ψ + R φ i s 0 ψ(•, 0) + i∈{L,R} Ωi×R+ (f i (s ǫ ) -ǫ∂ x ϕ i (s ǫ )) ∂ x ψ = 0. ( 16
)
From now on, we use the Kruzhkov and semi-Kruzhkov (or Serre) entropy fluxes

Φ i (a, b) = sign(a -b)(f i (a) -f i (b)), Φ ± i (a, b) = sign ± (a -b)(f i (a) -f i (b)), (17)
where sign + (a) = 1 if a > 0 and 0 otherwise, and sign -(a) = -sign + (-a). In the sequel, for a ∈ R, we denote by a + (resp. a -) the positive (resp. negative) part of a, i.e. a ± = sign ± (a)a.

Proposition 1 Let s ǫ , šǫ be two bounded-flux solutions of (15) in the sense of Definition 1 corresponding to initial data s 0 , š0 respectively. Then for all ψ ∈ C ∞ c (R × R + ; R+), the following Kato inequality holds:

i∈{L,R} Ωi×R+ φ i (s ǫ -šǫ ) ± ∂ t ψ + i∈{L,R} Ωi φ i (s 0 -š0 ) ± ψ(•, 0) + i∈{L,R} Ωi×R+ Φ ± i (s ǫ , šǫ ) -ǫ∂ x (ϕ i (s ǫ ) -ϕ i (š ǫ )) ± ∂ x ψ ≥ 0. ( 18
)
Corollary 2 For all initial datum s 0 satisfying (A4) there exists at most one bounded-flux solution s ǫ to [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF].

The proof of Proposition 1 and Corollary 2 is a straightforward generalization of the results of [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]Section 4] at least in the case of L 1 data with values in [0, 1], the L 1 assumption being used to ensure that ∂ x ϕ L,R (s) → 0 as x → ±∞. For the general case, let us point out that the L 1 assumption is bypassed, e.g., by exploiting the Kato inequality in the way of Maliki and Touré [START_REF] Maliki | Uniqueness of entropy solutions for nonlinear degenerate parabolic problems[END_REF]. Thus (A4) is a sufficient assumption in Corollary 2.

Still in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF], the existence of a bounded-flux solution is proven thanks to the convergence of a Finite Volume scheme under the assumption (A5) that the initial flux is bounded. Putting this existence result together with the uniqueness result exposed in Corollary 2 yields to following theorem.

Theorem 1 Assume that (A1)-(A4) hold. In addition, let the initial datum be regular in the sense

(A5) for i = L, R, assume ∂ x ϕ i (s 0 ) ∈ L ∞ (Ω i ).
Furthermore, assume that the initial data are connected; namely, denoting by s 0,i the trace of s 0 on Γ from Ω i , we suppose that πL (s 0,L ) ∩ πR (s 0,R ) = ∅.

Then there exists a unique bounded-flux solution s ǫ of problem (15) corresponding to s 0 . Furthermore, s ǫ belongs to C(R + ; L 1 loc (R)). Moreover, if š0 also satisfies (A4) and (A5), if s 0 -š0 ∈ L 1 (R) and if we denote by šǫ the unique bounded-flux solution corresponding to š0 , then for all t ≥ 0 we have

i∈{L,R} Ωi φ i (s ǫ (•, t) -šǫ (•, t)) ± ≤ i∈{L,R} Ωi φ i (s 0 (x) -š0 (x)) ± . (19)
Upon generalizing the notion of solution by a closure procedure, the above existence and uniqueness framework can be extended to initial data that only satisfy (A4), but not (A5). This approach is a slightly improved variant of the technique exploited in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF].

Definition 2 (mild solution) A function s ǫ ∈ L ∞ (R×R + ; [0, 1]) is said to be a mild solution if for i = L, R, ∂ x ϕ i (s ǫ ) ∈ L 2 loc ( Ωi × R + ), if πL (s ǫ L (t)) ∩ πR (s ǫ R (t)) = ∅ for a.e. t ∈ R + ,
and if there exists a sequence (s ν,ǫ ) ν of bounded flux solutions tending towards

s ǫ in L 1 loc (R × R + ).
Theorem 2 Assume that (A1)-(A4) hold, then there exists a unique mild solution s ǫ of (15) corresponding to s 0 . Furthermore, s ǫ belongs to C(R + ; L 1 loc (R)). Moreover, if šǫ is a mild solution corresponding to an initial datum š0 then the Kato inequality (18) holds.

Proof: Let us start with the case of a compactly supported initial datum. In this case, smoothing s 0 and modifying it near the origin as proposed in [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF], we can approximate s 0 in L 1 (R) by a sequence (s ν 0 ) ν∈N of initial data that are regular in the sense (A5). Denoting by s ν,ǫ the unique bounded-flux solution corresponding to the initial data s ν 0 , we see from [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF] that the sequence (s ν,ǫ ) ν is a Cauchy sequence in C(R + ; L 1 (R)) Therefore, it admits a unique limit value s ǫ .

Let us show that s ǫ is a mild solution. Let K be an arbitrary bounded interval of R, let T > 0, and let

χ i : Ω i → [0, 1] be a smooth function with compact support such that χ i ≡ 1 on Ω i . Choosing formally (x, t) → π(x, s ν,ǫ (x, t))1 1 (0,T ) (t)χ i (x)
as test function in the weak formulation ( 16) on s ν,ǫ (this point is thoroughly justified, by means of two steps of regularization of the problem in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] -see also [START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF] for the multidimensional case) provides that

ǫ T 0 i∈{L,R} Ki (∂ x ϕ i (s ν,ǫ )) 2 ≤ C, (20)
where K i = K∩Ω i and where C does not depend on ν nor on ǫ (but on K).

Then ϕ i (s ν,ǫ ) is uniformly bounded in L 2 ((0, T ); H 1 (K i )) with respect to ν. Since ϕ i (s ν,ǫ ) converges strongly towards ϕ i (s ǫ ) in C([0, T ]; L 2 (K i ))
, by interpolation, it also converges strongly in L 2 ((0, T ); H s (K i )) (to the same limit) as soon as s < 1. Hence, we infer the strong convergence of the one-sided traces ϕ i (s ν,ǫ i ) on the interface in the L 2 (0, T ) sense towards ϕ i (s ǫ i ). The functions ϕ -1 i being invertible, we deduce that s ν,ǫ i tends to

s ǫ i . Since the set (s L , s R ) ∈ [0, 1] 2 | πL (s L ) ∩ πR (s R ) = ∅ is closed, one recovers at the limit ν → ∞ the property πL (s ǫ L ) ∩ πR (s ǫ R ) = ∅ a.
e. in (0, T ), and then a.e. in R + since T has been chosen arbitrary. This ends the existence proof for compactly supported data.

Next, given a general initial datum s 0 , we can approximate it by a monotone sequence (s m 0 ) m∈N by setting s m 0 := s 0 1 1 |x|<m . Using the comparison principle contained in [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF], we see that the corresponding sequence (s m,ǫ ) of mild solutions is non-decreasing, then it converges to some limit s ǫ a.e. on R × [0, T ]. It follows that s ǫ is itself a mild solution.

Finally, mild solutions being constructed as L 1 loc limits of bounded-flux solutions, the Kato inequality ( 18) remains true because it is stable by L 1 loc convergence. A mild solution is a weak solution, i.e. it satisfies ( 16); therefore, we deduce from [START_REF] Cancès | On the time continuity of entropy solutions[END_REF] that

s ǫ belongs to C([0, T ]; L 1 loc (Ω i )). Since s ǫ ∈ L ∞ (R × R + ), one obtains that s ǫ ∈ C([0, T ]; L 1 loc (R)).
As a consequence of the Kato inequality, the comparison and L 1 -contraction property [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF] remains valid for mild solution instead of bounded-flux solution. Last but not least, all the equations of system ( 15) are still fulfilled, in the distributional sense or in the appropriate trace sense, by the mild solutions, ensuring that they are effective solutions to the problem. To sum up, Theorem 2 sets up a well-posedness framework for [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF], for all ǫ > 0.

Remark 1 Let us explain the terminology used in this section. The denomination bounded-flux solutions has been introduced in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]; the name is due to the regularity property

∂ x ϕ i (s ǫ ) ∈ L ∞ (Ω i × R + ).
Note that this non-trivial property is derived thanks to a maximum principle for the flux

F ǫ i (x, t) = f ǫ i (s ǫ ) -ǫ∂ x ϕ i (s ǫ ) (recall that ǫ > 0 is fixed).
Further, the denomination mild solution is the usual term used in the theory of nonlinear semigroups generated by accretive operators: is denotes the solution obtained by means of implicit semi-discrete in time approximation. Solution in the sense of Definition 2 being the limit of bounded flux solutions, it is indeed a mild solution in the latter sense. This fact can be inferred from the arguments of [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF], where the solutions provided by a fully discrete time-implicit finite volume approximation are shown to converge towards the mild solution.

Looking for a stationary profile solution

Clearly, ǫ in (15) can be seen as a vanishing capillarity parameter. In order to understand the limit problem, as ǫ → 0, in this paragraph we point out an evident stationary profile U : R → [0, 1] such that for all ǫ, U (x/ǫ) yields a bounded-flux solution to problem [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. In the simplest case, U is constant on each side from zero; in the other case, U is constant on one side only.

Given π L,R and f L,R , we define two curves P and U in the unit square [0, 1] × [0, 1] (see Fig. 1). Recall that we have extended π L,R to maximal monotone graphs πL,R from [0, 1] to R, thus extending the domain of π -1 L,R to whole R (the inverse of a maximal monotone graph is a maximal monotone graph). Define the set

P := (s L , s R ) ∈ [0, 1] 2 | πL (s L ) ∩ πR (s R ) = ∅ , (21)
then the curve P is the maximal monotone graph from [0, 1] to [0, 1] defined as the composition π-1 R • πL of two maximal monotone graphs. The curve U is implicitly given by

U := (s L , s R ) | f L (s L ) = f R (s R ), s L ≥ sL and s R ≤ sR . ( 22
)
Due to assumption (A1), U is the graph of a strictly decreasing function on an interval that we denote [s opt L , s bar L ]. More specifically, the extremity (s opt L , s opt R ) of the curve U lies inside (0, 1) 2 ; we have either s opt L = sL or s opt R = sR according to the order of the values max [0,1] f L and max [0,1] f R . The other extremity (s bar L , s bar R ) lies either on the part {1} × [0, s opt R ] or on the part [s opt L , 1] × {0} of the boundary of the unit square, according to the sign of the total flux q.

0 s L s π R s bar R 1 1 s opt L s R s opt R s π L U D P Figure 1:
The maximal monotone graph P (in red) is defined by ( 21) from the capillary pressure function π L,R , and the decreasing curve U (in green) is defined by ( 22) from the convective flux functions f L,R . The vanishing capillarity limit is fully characterized by their intersection, as stated in the "selection rule" at the end of the section. The segment D (horizontal, in the case s opt L = sL ) is used in the proof of Prop. 3. Proposition 3 (i) Assume that U ∩P = ∅, and denote by (s π L , s π R ) its unique element. Then c(x) = c ǫ (x) := s π L 1 1 x<0 +s π R 1 1 x>0 is a bounded-flux solution of (15) for every ǫ > 0, it is therefore a vanishing capillarity limit. (ii) Assume that U ∩ P = ∅. Then c(x) := s opt L 1 1 x<0 + s opt R 1 1 x>0 is a vanishing capillarity limit, i.e., there exists a sequence c ǫ of stationary bounded-flux solutions of (15) that converges to c(•) in L 1 loc (R), as ǫ → 0. In the above statement, saying that a function is a solution of (15) we do not specify the initial condition. Proof: (i) It is enough to check that the function c(•) fits the definition of a bounded-flux solution. Indeed, it is constant on each side of the interface, so that the equation is verified pointwise away from {x = 0}. Next, the capillary pressures are connected in the sense π

L (s π L ) ∩ π R (s π R ) = ∅ because (s π L , s π R ) ∈ P. Finally, because (s π L , s π R ) ∈ U, we have (f L (c) -ǫ∂ x c)| x=0 -= f L (s π L ) = f R (s π R ) = (f R (c) -ǫ∂ x c)| x=0 + .
(ii) The proof in this case is similar to the proof of [24, Proposition 2.9], in which a particular choice of P was done. We consider separately two cases: either s opt L = sL , or s opt R = sR . In the first case we complement U by the horizontal segment D := [0, s opt L ] × {s opt R } (see Fig. 1); in the second case we complement U by the vertical segment

D := {s opt L } × [s opt R , 1].
In each of the cases, there is an intersection point ( s π L , s π R ) of the maximal monotone graph P with the union U ∪ D which is a maximal anti-monotone graph. Since U ∩ P = ∅ by assumption, the point ( s π L , s π R ) belongs to D. Consider the first case: we have

s π R = s opt R , s π L < s opt L , and f L (•) -f L (s opt L ) ≤ 0 on [0, 1].
We construct the solution of the following Cauchy problem for the ordinary differential equation:

λ L (U (ξ)) π L (U (ξ)) ′ = f L (U (ξ)) -f L (s opt L ), ξ ∈ (-∞, 0] U (0) = s π L . (23)
Existence of a local solution is clear from the Cauchy-Peano theorem, and it is easily seen that the solution is non-increasing and it can be continued to a global on (-∞, 0] solution satisfying lim ξ→-∞ U (ξ) = s opt L .

Set c ǫ (x) := U (x/ǫ)1 1 x<0 + s opt R 1 1 x>0 ; as in (i), we check that this function is a bounded-flux solution of (15) for every ǫ > 0. Indeed, differentiating [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] in the weak sense and recalling the definition of ϕ L we see that equation (15a) is satisfied pointwise for x = 0. The capillary pressures are connected at {x = 0} because ( s π L , s opt R ) ∈ P; and the fluxes are connected at {x = 0} because

(f L (c) -ǫ∂ x c)| x=0 -= f L (s opt L ) = f R (s opt R ) = (f R (c) -ǫ∂ x c)| x=0 + . The limit of s ǫ (•) being c(x) := s opt L 1 1 x<0 + s opt R 1 1 x>0
, this ends the proof for this case. In the second case, we have

s π L = s opt L , s opt R < s π R , and f R (•) -f R (s opt R ) ≤ 0 on [0, 1]
. Analogously to the first case, we construct a profile c ǫ (x) := s opt L 1 1 x<0 + U (x/ǫ)1 1 x>0 . Here U (•) is a non-increasing function with lim ξ→+∞ U (ξ) = s opt R ; it solves the ODE problem analogous to [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] but posed on [0, +∞), with f R , s π R ,s opt R replacing f L , s π L ,s opt L , respectively. With the above proposition in hand, we highlight the following Selection Rule: We set (s π L , s π R ) to be the intersection point of U and P if the two curves cross (see Fig. 1), and we set it to be (s opt L , s opt R ) if U and P do not cross.

2 Buckley-Leverett equation in two-rocks' medium

Taking the limit ǫ → 0 in the problem (15) provides formally that the limit s of s ǫ satisfies the hyperbolic scalar conservation law with discontinuous flux function

φ(x)∂ t s + ∂ x f (x, s) = 0, (24)
that is known to have several mathematically consistent notions of solution (see [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]). In Section 2.1, we recall some elements of the theory on the scalar conservation laws with discontinuous flux functions detailed in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], that will be of great interest to identify the notion of solution that describes the vanishing capillarity limit.

The formal discontinuous-flux model, connections, entropy solutions

Buckley-Leverett equation in two-rocks' medium is a particular case of conservation law with discontinuous flux. When the interface between the media is located at {x = 0}, this general problem takes the form

∂ t (φ L 1 1 x<0 + φ R 1 1 x>0 ) s + ∂ x f L (s)1 1 x<0 + f R (s)1 1 x>0 = 0. ( 25
)
Remark 2 In the case φ L = φ R , problem [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF] has been much studied in the literature (see the references in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]). Let us stress that the introduction of constant coefficients φ L and φ R does not change the properties of problem: namely, the definitions and results stated below can be reduced to those of [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and the other references upon introducing the new unknown u(x, t) := (φ L 1 1 x<0 + φ R 1 1 x>0 ) s(x, t) and the new fluxes g L,R :

u → f L,R (u/φ L,R ).
The notion of L 1 -dissipative germ (L 1 D germ, for short) has been formulated in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] in order to describe the different semigroups of entropy solutions satisfying the L 1 contraction principle. For fluxes f L,R satisfying (A1), ( 25) can be seen as the formal limit, as ǫ → 0, of [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. We interpret this idea by saying that an admissible solution s to [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF], in the Buckley-Leverett context, should be a vanishing capillarity limit, i.e., a limit of some sequence (s ǫ ) ǫ→0 of solutions of [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. Due to Theorems 1,2, it is clear that the vanishing capillarity limits do satisfy the L 1 contraction principle; thus the setting of [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] is suitable for our needs.

Let us give the definitions underlying the theory of problem [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF].

Definition 3 (admissibility germs; complete, maximal and definite germs)

• Any set G of couples (s L , s R ) ∈ [0, 1] 2 satisfying the Rankine-Hugoniot relation ∀(s L , s R ) ∈ G f L (s L ) = f R (s R ) ( 26
)
and the L 1 -dissipativity relation

∀(s L , s R ), (z L , z R ) ∈ G Φ L (s L , z L ) ≥ Φ R (s R , z R ), ( 27
)
where Φ L,R are defined in [START_REF] Brenner | A convergent finite volume scheme for two-phase flows in porous media with discontinuous capillary pressure field[END_REF], is called an L 1 D admissibility germ (a germ, for short) associated with the couple of fluxes

(f L , f R ) defined on [0, 1].
• A germ G is called complete if all Riemann problem at x = 0 for (25) admits a self-similar solution s such that (s L , s R ) ∈ G, where s L , resp. s R , is the limit of s(t, •) as x → 0 -, resp.as x → 0 + .

• We say that G ′ is an extension of a germ G if G ⊂ G ′ and G ′ still satisfies the L1 -dissipativity property in [START_REF] Cancès | On the time continuity of entropy solutions[END_REF] and the Rankine-Hugoniot condition in [START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF].

• A germ G is called maximal, if it does not admit a nontrivial extension.

• A germ G is called definite, it it admits only one maximal extension.

In relation with definite and maximal germs, consider one more definition.

Definition 4 (dual of a germ) Let G be an L 1 D-admissibility germ. The dual of G is the set

G * := (z L , z R ) ∈ [0, 1] 2 f L (z L ) = f R (z R )
and for all

(s L , s R ) ∈ G, Φ L (s L , z L ) ≥ Φ R (s R , z R ) . (28)
It is shown in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] that, if G is a definite germ, then its dual G * is the unique maximal extension of G.

We are in a position to define different notions of entropy solution.For simplicity, consider a finite time horizon T > 0.

Definition 5 Given a couple of continuous functions (f L , f R ) defined on [0, 1] and a definite germ G associated with this couple, we say that s ∈ L ∞ (R × (0, T ); [0, 1]) is a G-entropy solution of (25) if the Kruzhkov entropy inequalities hold away from the interface {x = 0}:

∀κ ∈ [0, 1] ∂ t φ L,R |s -κ| -∂ x Φ L,R (s, κ) ≤ 0 in D ′ (Ω L,R × (0, T )), (29)
and and for a.e. t ∈ (0, T ), one has s L (t) , s R (t) ∈ G * , where s L (•) (the trace as x → 0 -) and s R (•) (the trace as x → 0 + ) are the interface traces of s in the strong L 1 (0, T ) sense. We say that s is a G-entropy solution of the Cauchy problem with s(•, 0) = s 0 if the initial condition s 0 is assumed in the sense of strong L 1 loc initial trace. Notice that under assumption (A1), the traces s L,R and s(•, 0) do exist ( [START_REF] Vasseur | Strong traces of multidimensional scalar conservation laws[END_REF][START_REF] Yu | Existence of strong traces for generalized solutions of multidimensional scalar conservation laws[END_REF][START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF][START_REF] Cancès | On the time continuity of entropy solutions[END_REF]).

Remark 3 According to the results of [START_REF] Vasseur | Strong traces of multidimensional scalar conservation laws[END_REF], [START_REF] Yu | Existence of strong traces for generalized solutions of multidimensional scalar conservation laws[END_REF] and [START_REF] Cancès | On the time continuity of entropy solutions[END_REF], it is not a restriction to assume that, up to a re-definition of s(t, •) on a set of zero measure of t ∈ [0, T ], a G-entropy solution of (25) belongs to

C([0, T ]; L 1 loc (R)).
The following result is contained in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] (see in particular [7, Theorem 6.4])

Theorem 3 (Well-posedness for G-entropy solutions) Assume (A1) holds, and G is a definite germ of which the dual G * is complete. Then for all measurable initial datum s 0 with values in [0, 1] there exists a unique G-entropy solution to problem [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF]. Moreover, the finite volume scheme for [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF] with Godunov flux converges to the corresponding G-entropy solution, for all initial datum.

Remark 4 It is required in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]Theorem 6.4] that f L,R be defined on R. Nevertheless, let us point out that in our case, solutions with [0, 1]-valued initial data always take values in [0, 1]. Indeed, assumptions (A1) contain the compatibility conditions

f L (0) = f R (0), f L (1) = f R (1)
. Moreover, it is easily seen that (0, 0) and (1, 1) belong to G * , whatever be the germ G; therefore 0 and 1 are constant G-entropy solutions. This ensures, in particular, that approximate solutions constructed by the Godunov scheme lie in between zero and one.

Under assumptions (A1), it is easy to classify all possible L 1 D admissibility germs. According to the analysis of [7, Section 4.8] 1 , each maximal germ is complete, and it is entirely determined by a definite germ which is a singleton. Such singletons are called connections in the below definition.

Definition 6 (Adimurthi et al. [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], Bürger et al. [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF])

For f L,R satisfying (A1), a couple (A, B) ∈ [0, 1] 2 is said to be a connection if A ∈ [s L , 1], B ∈ [0, sR ] and f L (A) = f R (B).
Being a connection means that u(t, x) := A1 1 x<0 +B1 1 x>0 is a stationary weak solution of (25) that represents an undercompressive shock: the (strict) Lax condition fails from both sides from the jump.

Notice that the set U of all connections (see Fig. 1) is given by [START_REF] Cancès | Two-phase Flows Involving Discontinuities on the Capillary Pressure In Proceeding of the conference FVCA[END_REF]. Let us describe its extremities. We define the optimal connection (A opt , B opt ) by (A opt , B opt ) ∈ U, with either A opt = sL or B opt = sR .

= f L (A bar ) = f R (B bar ), while F opt = min{max [0,1] f L , max [0,1] f R } = f L (A opt ) = f R (B opt ).
Reciprocally, the connection at level F ∈ [ F bar , F opt ] is denoted by (A F , B F ). Such a connection is indeed unique, since f L,R are strictly monotone on [0, sL,R ] and on [s L,R , 1].

Further, set O := G * (A opt ,B opt ) (see Fig. 2b). From the bell-shapedness assumption in (A1) one easily sees that O \ {(A opt , B opt )} is the set of all couples (a, b)

∈ [0, 1] 2 \ U such that f L (a) = f R (b).
In contrast to under-compressive states (A, B) ∈ U, every couple (a, b) ∈ O will be called an over-compressive state (note that (A opt , B opt ) ∈ U ∩ O is both under-and over-compressive). We have Proposition 4 (see Section 4.8 in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], see also [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF]) For every connection (A, B) ∈ U, the singleton G (A,B) := {(A, B)} is a definite germ; its dual is given by

G * (A,B) = {(A, B)} ∪ O F(A,B) , where O F(A,B) := (z L , z R ) ∈ O s.t. f L (z L ) = f R (z R ) ≤ F(A,B) . (30)
Moreover, every maximal germ contains one and only one connection (A, B) ∈ U, therefore it can be represented under the form [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF].

f L,R (s) sL = s opt s s bar R B s opt R f R (s) f L (s) F opt F bar = q s bar L = 1 0 A F(A,B) (a
(s L , s R ) ∈ [0, 1] 2 | (f L (s L ) > F(A,B) )&(f R (s R ) > F(A,B) ) . So, the maximal germ G * (A,B
) is made of the union of singleton {(A, B)} and of the subset O F(A,B) of O which is outside of the grey rectangle.

Remark 5

The point of view developed in our note [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] is that, at least for the purpose of interpretation of the solutions' behavior and for their numerical approximation, it is convenient to characterize different notions of G-entropy solution by the connection level F rather than by the corresponding connection (A F , B F ). Indeed, as one can see from the representation (30), the possible trace couples (s L , s R ) of G (A F ,B F ) -entropy solutions obey the constraint f L,R (s L,R ) ≤ F . In particular, the only free parameter required to construct the Godunov scheme for problem [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF] with fluxes (A1) is the connection level F (see [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] and Section 3.2 below for details).

Finally, we recall an equivalent characterization of G (A,B) -entropy solutions with the help of adapted entropy inequalities introduced by Baiti and Jenssen [START_REF] Baiti | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data[END_REF] and Audusse and Perthame [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF].

Theorem 4 (see [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], see also [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF]) Given a connection (A, B) ∈ U, a function s ∈ L ∞ (R × (0, T )) is a G (A,B) -entropy solution of (25) with fluxes (A1) if and only if it satisfies, away from the interface, the Kruzhkov entropy inequalities (29) and moreover, given c(x) = A1 1 x<0 +B1 1 x>0 , it satisfies the global adapted entropy inequality

∂ t φ(x)|s -c(x)| -∂ x Φ(x; s, c(x)) ≤ 0 in D ′ (R × (0, T )). ( 31
)
Here

φ(x) = φ L 1 1 x<0 + φ R 1 1 x>0 ; similarly, Φ(x; s, c) = Φ L (s, c)1 1 x<0 + Φ R (s, c)1 1 x>0 .

Identifying the vanishing capillarity solutions

We have now introduced enough material to be able to carry out the proof of our main result. The following theorem permits to characterize the semigroup of vanishing capillary limits by identifying it to the appropriate G (A,B) -entropy solutions' semigroup; we see that the underlying connection (A, B) only depends on the nonlinearities present in the problem.

Theorem 5 (Main result) Assume we are given nonlinearities f L,R and π L,R satisfying (A1),(A2),(A3). Let (s π L , s π R ) ∈ U be the connection obtained according to the Selection Rule of Section 1.3, i.e., it is either the intersection point of the curves U and P (see Fig. 1) or the optimal connection (s opt L , s opt R ) when U ∩P = ∅. Let s ǫ be the unique mild solution of problem (15), and let s denote the unique G (s π L ,s π R ) -entropy solution of the discontinuous-flux Buckley-Leverett equation (25) corresponding to the same initial datum s 0 , then

s ǫ → s in L 1 loc (R × R + ).
In particular, any solution of (25) obtained as vanishing capillarity limit obeys the flux limitation constraint at the interface:

f L (s(t, 0 -)) = f R (s(t, 0 + )) ≤ F π where F π = f L,R (s π L,R ) is the corresponding connection level.
Let us point out that, choosing π L,R appropriately, we can make appear any given point of the curve U as the intersection point (s π L , s π R ). Thus it follows from Theorem 5 that, given a notion of G (A,B) -entropy solution, this notion corresponds to some choice of vanishing capillarity profiles π L,R . Proof: The proof combines the results of Propositions 1, 3 and characterization [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF] of G (A,B) -entropy solutions.

To start with, fix some (not labelled) sequence ǫ decreasing to zero. According to Theorem 2, for all ǫ > 0 fixed, the problem ( 15) is well posed in the setting of mild solutions, i.e there exists a unique corresponding mild solution s ǫ . Moreover, the Kato inequality [START_REF] Bürger | Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[END_REF] holds for all couple of solutions s ǫ , šǫ corresponding to initial data s 0 , š0 . Assume for a moment that there exists s ∈ L ∞ (R × R + ; [0, 1]) such that up to a subsequence, s ǫ tends to s in L 1 loc (R × R + ) as ǫ tends to 0.

First, write the Kato inequality ( 18) for a solution s ǫ of problem [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] and for the capillarity profile c ǫ constructed in the proof of Proposition 3. Using the convergence

s ǫ → s, c ǫ → c as ǫ → 0, c(x) = c π L 1 1 x<0 + c π R 1 1 x>0
, we can pass to the limit in this inequality. We inherit the "hyperbolic Kato inequality" i∈{L,R} Ωi×R+

φ i |s -c(x)|∂ t ψ + Φ i (s, c(x))∂ x ψ + i∈{L,R} Ωi φ i |s 0 -c(x)|ψ(•, 0) ≥ 0 for all ψ ∈ D(R × [0, T )), ψ ≥ 0.
Restricting the choice of test functions to D(R × (0, T )), we find the global adapted entropy inequality [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF] with (A, B) = (s π L , s π R ). Second, it follows from classical arguments (see e.g. [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF]) that s is a Kruzhkov entropy solution away from the interface, in the sense [START_REF] Cancès | An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field[END_REF]. Moreover, it assumes the initial datum s 0 , hence s is the (unique) G (s π L ,s π R ) -entropy solution corresponding to datum s 0 . Now, applying [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] to subsequences of (s ǫ ) ǫ , from the uniqueness of the accumulation point we deduce that lim ǫ→0 s ǫ exists. Thus, provided [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] is justified, we prove that the vanishing capillarity limit exists and it coincides with the unique G (s π L ,s π R ) -entropy solution. This ends the proof of the theorem, except for the justification of [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF].

If we assume that f L,R are genuinely nonlinear on every interval, then according to the well-known compactification results of [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF][START_REF] Yu | On sequences of measure valued solutions for a first order quasilinear equation (Russian)[END_REF][START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] we can extract an L 1 loc convergent subsequence of s ǫ . In the general case, we can use the framework of G-entropy-process solutions in the way of [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]. Indeed, extracting a nonlinear weakly- * convergent subsequence of (s ǫ ) ǫ , due to the existence of G-entropy solutions (see Theorem 3) we can prove that the G (s π L ,s π R ) -entropy-process solution coincides with the unique G (s π L ,s π R ) -entropy solution for the same initial datum. Let us point out that the proof is not straightforward, because one global adapted entropy inequality (as in Theorem 4) is not sufficient in this argument (see [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF] for the case f L ≡ f R ).

Remark 6 Another way to prove [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] is to restrict our attention to a dense set of initial data s 0 , and to derive additional estimates on the solution, like a BV estimate on a Temple function [START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF], or, using a variant of the technique of Bürger, García, Karlsen and Towers [START_REF] Bürger | Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], one can derive a BV loc estimate on the solution with small capillarity s ǫ . This latter point is detailed in Appendix A.1. [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] Our work builds on the idea of Kaasschieter [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] that the physically admissible solutions of the Buckley-Leverett equation with discontinuous flux should be seen as vanishing capillarity limits. In [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], the author analyzes solutions of the general Riemann problem; here, due to the tools borrowed from [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF], we reduce the analysis to a study of one particular stationary solution.

Comparison of our conclusions with those of Kaasschieter

Then we are able to observe the following important fact. Although the analysis of [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] is fully correct under a seemingly non-restrictive assumption of "genericity", it follows from our analysis that the assumption made by Kaasschieter is truly restrictive. Namely, in [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] the case where a solution u(t, x) = U x t to the Riemann problem fulfulls simultaneously the constraints

f L (U (0 -)) = f R (U (0 + )) and π L (U (0 -)) = π R (U (0 + )) ( 33
)
is eluded because it is considered as "merely coincidental". Our analysis shows, in an indirect way, that this case is realized for many Riemann problems. Namely, whenever Fπ < F opt and whenever the values s ± at ±∞ are such that the flux given by min (f L (min(s -, sL )), f R (max(s + , sL )))

exceeds the value Fπ , the situation (33) does happen (see formula [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] in the the next section), moreover,

f ′ L (U (0 -)) ≤ 0 and f ′ R (U (0 + )) ≥ 0. ( 34
)
Therefore, our conclusion differs drastically from the one of [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF]. Indeed, the conclusion of [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] should sound as follows: "the appropriate entropy solution is the optimal entropy solution, except may be when ( 33)- [START_REF] Eymard | Finite Volume Methods. Handbook of Numerical Analysis[END_REF] happen". Yet the case (33) is not merely coincidental, and it cannot be seen as exceptional. Thus one should recast the conclusion of the Kaasschieter work [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] as follows: "the appropriate solution notion is G (s π L ,s π R ) -entropy solution, in particular, the optimal entropy solution occurs whenever ( 33)-( 34) is impossible".

Numerical approximation of the flow in two-rocks' medium

The goal of this section is, first of all, to provide numerical evidence for convergence of s ǫ towards the appropriate entropy solution s (recall that the notion of solution strongly depends on the capillarity profiles π L,R , see Section 2, and secondly, to discuss about "time saved versus accuracy lost" by solving the simpler problem (25) instead of solving the finer problem [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. To do so, we introduce two numerical schemes: the first one, used to discretize the parabolic problem [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF], was proved to be convergent by the second author in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]; the second one, introduced by the authors in [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF], is the exact Godunov scheme adapted to the connection (s π L , s π R ), and is based on the notion of flux limitation ( [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]) discussed in Section 2.1.

A finite volume scheme for the parabolic model

First, we have to compute the mild solutions s ǫ of the degenerate parabolic problem. This is done by means of the fully implicit finite volume scheme studied in [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF].

For ∆x > 0, we denote by x j+1/2 j∈Z = {(j + 1/2)∆x | j ∈ Z} the set of the "cell centers" and by (x j ) j∈Z = {j∆x | j ∈ Z} the sets of the "edges". Given ∆t > 0, we use (t n ) n = {n∆t | n ∈ N} for time steps.

For s 0 ∈ L ∞ (R; [0, 1]), the initial data is discretized as follows:

s ǫ,0 j+1/2 = 1 ∆x xj+1 xj s 0 (x)dx. (35)
The implicit scheme is then given by ∀j ∈ Z, ∀n ∈ N,

φ j s ǫ,n+1 j+1/2 -s ǫ,n j+1/2 ∆t ∆x + F ǫ,n+1 j+1 -F ǫ,n+1 j = 0, (36)
where the fluxes F ǫ,n+1 j have to be made explicit. Let j ∈ Z \ {0}; for ψ standing for one of the symbols φ, f, ϕ, π, s, we denote a space dependent function which is constant in Ω L,R as follows:

ψ j := ψ(•, x j ) = ψ L if j < 0 ψ R if j > 0.
Now we introduce the exact Riemann solver for the convection within Ω L,R . For (a, b) ∈ [0, 1] 2 and j ∈ Z \ {0}, we set

G j (a, b) = min s∈[a,b] f j (s) if a ≤ b, max s∈[a,b] f j (s) if a ≥ b.
Note that G j (a, a) = f j (a), that G j is Lipschitz continuous w.r.t. both variables, and that G j is nondecreasing w.r.t to its first argument and non-increasing w.r.t. the second. It is well known that for bellshaped fluxes, G j can be computed by the formula

G j (a, b) = min (f j (min(a, s j )), f j (max(b, s j ))) ; (37)
let us recall that sL,R = arg max f L,R (see Assumption (A1)). For j = 0 (i.e. in the case where the edge j is not at the interface), one defines

F ǫ,n+1 j = G j (s ǫ,n+1 j-1/2 , s ǫ,n+1 j+1/2 ) -ǫ ϕ j (s ǫ,n+1 j+1/2 ) -ϕ j (s ǫ,n+1 j-1/2 ) ∆x . (38)
It remains to define the flux F ǫ,n+1 0 across the interface so that everything be defined in [START_REF] Ern | Discontinuous galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures[END_REF]. To do so, following [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF], we introduce additional unknowns s ǫ,n+1 0,L , s ǫ,n+1 0,R that solve the following nonlinear system πL (s ǫ,n+1 0,L ) ∩ πR (s ǫ,n+1

0,R ) = ∅, (39a)
F ǫ,n+1 0 := G L (s ǫ,n+1 -1/2 , s ǫ,n+1 0,L ) -ǫ ϕ L (s ǫ,n+1 0,L ) -ϕ L (s ǫ,n+1 -1/2 ) ∆x/2 (39b) = G R (s ǫ,n+1 0,R , s ǫ,n+1 1/2 ) -ǫ ϕ R (s ǫ,n+1 1/2 ) -ϕ L (s ǫ,n+1 0,R ) ∆x/2 . ( 39c
)
It is proven in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] that for all (s ǫ,n+1 -1/2 , s ǫ,n+1 1/2 ), the system (39) admits a unique solution (s ǫ,n+1 0,L , s ǫ,n+1 0,R ), hence the flux F ǫ,n+1 0 is well defined. The results of the paper [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] can be summarized as follows.

Proposition 5 Let ǫ > 0 be fixed and let s 0 ∈ L ∞ (R; [0, 1]), then 1. the scheme (36), [START_REF] Gimse | Riemann problems with a discontinuous flux function[END_REF],(39) admits a unique solution s ǫ,n+1 j+1/2 j∈Z,n∈N ; 2. if we define the approximate solution s ǫ h almost everywhere on R + × R by

s ǫ h (x, t) = s ǫ,n+1 j+1/2 if (x, t) ∈ (x j , x j+1 ) × (t n , t n+1 ), then s ǫ h ∈ L ∞ (R × R + ; [0, 1]) converges in L 1 loc (R × R + )
towards the unique mild solution of the problem as ∆x, ∆t → 0.

A finite volume scheme for the hyperbolic model

The scheme introduced in previous section is asymptotic preserving, in the sense that choosing ǫ = 0, and obtaining therefore an approximate solution s 0 h (the solution to the scheme in the case ǫ = 0 is once again unique), one can show that s 0 h tends to the vanishing capillarity limit described in Theorem 5. This point is made explicit in Appendix A.2. Nevertheless, to produce numerical results for the hyperbolic problem [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF], we use the Godunov scheme under the form explained in our note [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] (see also [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF]). Namely, we have shown in [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF]Theorem 3.1] that in order to obtain the Godunov scheme for approximation of G (A,B) -entropy solutions of [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping[END_REF] with fluxes (A1), it is enough to take the scheme of Adimurthi et al. [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF] known for the optimal connection (A opt , B opt ) and to limit the flux at the interface to the maximum value F(A,B) . More precisely, in our case, the explicit Godunov scheme for computing the unique G (s π L ,s π R ) -entropy solution can be rewritten as

φ j s n+1 j+1/2 -s n j+1/2 ∆t ∆x + F n j+1 -F n j = 0, (40)
where the fluxes F n j are given by

F n j = G j (s n j-1/2 , s n j+1/2 ) if j = 0, (41)
F n 0 = min F π , f L (min(s n -1/2 , sL )), f R (max(s n 1/2 , sL )) . (42)
In the previous formula [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], the exact Riemann solver G j was defined by (37) while, in formula ( 42), the quantity

F π = f L,R (s π L,R
) is the connection level corresponding to the connection chosen using the selection rule of Section 1.3.

We now state a convergence result which is a consequence of the fact that the scheme prescribed by ( 40)-( 42) is monotone and preserves

G (s π L ,s π R ) (it even preserves G * (s π L ,s π R )
since the scheme is the Godunov one). Recall that it has been stated in Theorem 3 that the Godunov scheme is convergent. We refer to [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] for further explanations.

Proposition 6 Define s

h : R × R + → R by s h (x, t) = s n+1 j+1/2 if (x, t) ∈ (x j , x j+1 ) × (t n , t n+1
), then, if we denote by L f a Lispchitz constant of both f L,R , and if there exists ζ ∈ (0, 1) such that

∆t ≤ (1 -ζ)∆x L f , (43)
then s h ∈ L ∞ (R × R + ; [0, 1]
). Moreover, under the CFL condition [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF], when ∆x (and thus also ∆t) tends to zero the discrete solution s h converges in L 1 loc (R × R + ) towards the unique G (s π L ,s π R ) -entropy solution of the problem.

Numerical illustrations of convergence

We now give numerical evidence of convergence of the mild solution s ǫ of the parabolic problem towards the G (s π L ,s π R ) -entropy solution by comparing their respective approximations s ǫ h and s h .

The test cases

Concerning the design of the test cases, we have chosen a particularly simple configuration. The capillary pressure functions π L,R are defined by

π L,R (s) = P L,R -ln(1 -s), (44)
where the quantities P L,R , called entry pressures, play an important role in the selection of the correct solution notion (cf. Section 1.3) and will vary from one case to another. Note that in the case where P R ≥ P L , the set P defined in Section 1.3 by ( 21) has the particular simple expression P = s, max 0, 1 + (s -1)e PR-PL , s ∈ [0, 1] . Numerical values of the parameters. The only parameter we let vary between the two test cases is the entry pressure P R . In the first case, which leads to the optimal connection, we choose P R = 0.5. In the second case, we chose P R = 2, so that the selection rule presented in Section 1.3 provides another solution, despite the fact that formally, the equation remains the same. The physical parameters and functions used in the simulations are collected in the following tables. Concerning the scaling parameter ǫ, several values has been used in order to illustrate the convergence of s ǫ towards s (see Fig. 10). All the numerical tests have been performed for the initial data u 0 ≡ 0.5.

total flow rate q = 0; gravity g = -9.81; intrinsic permeabilities K L = 10 -2 , K R = 5.10

The optimal connection

In the case where P R = 0.5, the connection diagram (Fig. 4) is such that P ∩ U = ∅. Therefore, the selection rule of Section 1.3 and Theorem 5 claim that the good notion of solution for the vanishing capillarity limit is the G (s opt L ,s opt R ) -entropy solution. The numerical approximation of the optimal entropy solution obtained via the Godunov scheme described in Section 3.2 is presented in Fig. 5a. It appears to be in good accordance with the solution for a small value of ǫ given by the implicit scheme described in Section 3.1, and represented on Fig. 5b. In particular, the wave starting from the interface with negative speed has the expected amplitude and the expected speed. As already noticed on Fig. 5, we see on Fig. 6 that the shocks of the hyperbolic solution are smoothed by adding some capillary diffusion. Let us also point out that the one-sided traces s L , s R of the hyperbolic solution on the interface {x = 0} do not satisfy πL (s L ) ∩ πR (s R ) = ∅. Therefore, these traces are not suitable for the parabolic approximation. We can see on both figures (particularly on Fig. 6a) that a boundary layer is present on the right hand side from the interface. 7a, where it clearly appears that P ∩ U = ∅. As previously, we denote by (s π L , s π R ) the connection belonging to P ∩ U. Following the selection rule of Section 1.3 and Theorem 5, the appropriate notion of entropy solution for the vanishing capillarity limit is then the G (s π L ,s π R ) -entropy solution. As a consequence, the interface flux is limited to the maximal value F π (formally, the limitation is equal to F opt in the case where the optimal entropy solution is selected). Here again, the approximate solution s ǫ h for small capillarity (ǫ = 10 -3 ) is really close to the vanishing capillarity solution (the G (s π L ,s π R ) -entropy solution). On Fig. 8, one can see that the shocks (for the hyperbolic solution) are smoothed in presence of capillary diffusion. Three waves are generated by the medium discontinuity: one wave with negative speed joining s 0 = 0.5 to s π L ≃ 0.87, one wave with zero speed joining s π L to s π R ≃ 0.11, and one wave with positive speed joining s π R to s 0 = 0.5. Note that since πL (s π L ) ∩ πR (s π R ) = ∅, then there is no boundary layer for s ǫ as x → 0. at each time step, making the solution expensive to compute. For example, computing the approximate solution s ǫ h presented on Fig. 5b requires 2182.31 s of CPU time with Scilab, while the computation of the approximate solution s h presented on Fig. 5a only requires 3.185 s of CPU time, the speed-up ration being hence of about 685. Moreover, since it is explicit, the computation of s h requires less memory than the one needed to obtain s ǫ h ; this allows to solve the hyperbolic problem on a finer mesh. Concerning the convergence speed, we first illustrate in Fig. 10 the convergence of s ǫ towards s by plotting log s ǫ hs h L 1 (0,T ;L 1 (-1,1)) as a function of ǫ. In accordance with the theory (see e.g. [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF][START_REF] Serre | Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems[END_REF]), Fig. 10 lets us think that for all T > 0, one has

T 0 R |s ǫ (x, t) -s(x, t)|dxdt ≤ Cǫ 1/2 . (45)
Figure 10: log s ǫ h -s h L 1 (0,T ;L 1 (-1,1)) as a function of ǫ (in blue) and a straight line with slope -1/2 (dashed green). We recover numerically the order of convergence that was expected from [START_REF] Yu | On sequences of measure valued solutions for a first order quasilinear equation (Russian)[END_REF]. Note that the slope of the blue curve is damaged when ǫ is too large. This phenomenon is due to the fact that the solution is computed on the finite domain x ∈ (-1, 1). When the diffusion is large, the boundary conditions affects the numerical solution. The convergence rate is also damaged for small ǫ. This comes from the fact that the numerical error become comparable to the modeling error s ǫs (this effect is particularly visible since the convection is discretized in an implicit way in the scheme presented in Section 3.1 and in an explicit way in the Godunov scheme presented in Section 3.2.

We now look at the convergence rate of the Godunov scheme. To our knowledge, no uniform bound on the total variation of s h has been proved in the case f L = f R (see [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF] for the case f L ≡ f R ). Yet the particularly simple configuration we are dealing with (a Riemann problem) ensures the existence of a variation bound.

Carrying out a proof similar to the one performed in [START_REF] Cancès | Error estimate for Godunov approximation of locally constrained conservation laws[END_REF] provides an error estimate of type

T 0 R |s h (x, t) -s(x, t)|dxdt ≤ C∆x 1/2
(recall that ∆t ≤ C∆x thanks to ( 43)) This estimate is optimal in the case where f R or f L is linear. In the framework of the test case presented in Section 3.3.1, the flux functions are genuinely nonlinear (see Fig. 3b).

As it is usual in this case, a convergence of order 1 is observed numerically: see Fig. 11.

Conclusion

The goal of this paper was to investigate the limit, as ǫ → 0, of the system [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF]. This study is close to the one performed by E. Kaasschieter [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], but here, we have taken advantage of the recent developments in the theory of the scalar conservation laws with discontinuous flux function (see [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and references therein) to avoid difficult calculations, and eventually achieved a full classification of possible physical situations. We have identified the correct interface coupling in the discontinuous-flux Buckley-Leverett model in terms of the profiles of the flux functions and capillary pressure functions on two sides from the interface. In particular, we clarified the conclusions of the work [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] by proving that "optimal entropy solution" is not always the right notion of solution in the Buckley-Leverett context. Finally, we constructed an adequate numerical method and gave strong evidences on its efficiency.

A Appendix A.1 The BV loc technique

The goal of this appendix is to prove the relation ( 32) by adapting to the continuous case, and under the additional assumption that ϕ L,R ∈ W 2,∞ ([0, 1]), a technique developed in [START_REF] Bürger | Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF]. Because of the finite speed of propagation and the L 1 loc contraction property for G-entropy solutions, completely analogous to the classical estimate of [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], it is enough to prove [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] for an L 1 loc -dense subset of initial data. Indeed, a limit of vanishing viscosity limits is still a vanishing viscosity limit.

Thus we pick s 0 ∈ C ∞ 0 (R) and such that s 0 ≡ 0 on some interval around zero (this is a way to ensure a smooth transition across the interface {x = 0}). We extend the corresponding solution s ǫ of (15) continuously by s 0 for t ≤ 0; notice that for t < 0, the so extended function s ǫ satisfies

∂ t (φ L 1 1 x<0 + φ R 1 1 x>0 )s ǫ + ∂ x f L (s ǫ )1 1 x<0 + f R (s ǫ )1 1 x>0 = ǫ∂ x ∂ x ϕ L (s ǫ )1 1 x<0 + ∂ x ϕ R (s ǫ )1 1 x>0 + r(x) where r : x → ∂ x f L (s 0 ) -ǫ∂ x ϕ L (s 0 ))1 1 x<0 + f R (s 0 ) -ǫ∂ x ϕ R (s 0 ))1 1 x>0 (46)
is an L ∞ (R)∩L 1 (R) function, by the assumptions on s 0 and because f L,R , ϕ L,R were assumed regular enough. Therefore the so extended function s ǫ is an entire solution (i.e., a solution defined for t ∈ R) of problem [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] with the additional source term r(x)1 1 t<0 . Now, the key fact is that we can control the L 1 time translates of s ǫ by a linear modulus of continuity, because solutions of (15) with a source term verify the L 1 contraction principle completely analogous to [START_REF] Bürger | On conservation laws with discontinuous flux[END_REF]:

i∈{L,R} Ωi φ i |s ǫ (t) -s ǫ (t -τ )| ≤ i∈{L,R} Ωi φ i |s ǫ (0) -s ǫ (-τ )| + t 0 R |r 1 1 s<0 -r 1 1 s-τ <0 | ds = τ r L 1 .
Therefore s ǫ ∈ BV (0, T ; L 1 (R)), with a uniform in ǫ bound. Then we can use the idea of [START_REF] Bürger | Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[END_REF]Lemma 4.2] and [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF]Lemma 5.4]: for a > 0, using the mean-value theorem for each ǫ > 0 we can find a contour (0, T ) × {a ǫ } with 0 < a ǫ < a such that TotVar a ǫ along these contours is uniformly bounded by C a . The variation of s 0 is also bounded, therefore in the same way as in the classical estimate of Bardos, LeRoux and Nédélec [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] for the Dirichlet problem for viscous conservation law (with boundary datum given by the values of s ǫ on our contour), we get the bound TotVar s ǫ | {(t,x) | t∈(0,T ), x≥a} ≤ C a ,

with C that only depends on s 0 and on the Lipschitz constant of f L,R and of ϕ ′ L,R . Analogous estimate holds for the variation on the set {(t, x) | t ∈ (0, T ), x ≤ a}. With the Cantor diagonal argument, we deduce compactness of (s ǫ ) ǫ in L 1 loc ((0, T ) × R + ) and thus justify [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF].

A.2 An asymptotic preserving scheme

As a consequence of Proposition 5 and Theorem 5, we have lim ǫ→0 lim ∆t,∆x→0

s ǫ h = s in L 1 loc (R × R + ),
where s ǫ h is the solution to the scheme ( 35)- [START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF]. In order to justify the comparison of the numerical solutions s ǫ h and s h on Figures 5,6 First of all, we need to identify which scheme governs lim ǫ→0 s ǫ h .

Lemma 7 Let s ǫ h be the solution of (35)- [START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF], then s 0 h := lim ǫ→0 s ǫ h (in the L 1 loc sense) is a solution of the scheme (35)- [START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF] where ǫ has been set to 0.

Proof: First of all, since, for all compact subset K of R × R + , the restriction of s ǫ h to K lies in a finite dimensional space, the L 1 loc convergence means the convergence of each s ǫ,n j+1/2 (j ∈ Z, n ∈ N) towards some s 0,n j+1/2 . Assume that this holds for n ∈ N (this is true for n = 0), let us show it for n + 1.

Since, for all ǫ > 0, s ǫ,n+1 j+1/2 ∈ [0, 1], then, up to a subsequence, s ǫ,n+1 j+1/2 tends to some s 0,n+1 j+1/2 ∈ [0, 1], and, by a diagonal extraction process, one can assume that this convergence occurs for all j ∈ Z. Up to an new subsequence, one can assume that s ǫ,n+1 L,R tends to s 0,n+1

L,R
as well as ǫ tends to 0. Note that since the set P in ( 21) is closed, (s 0,n+1 L , s 0,n+1 R ) ∈ P.

For j = 0, the flux F ǫ,n+1 j := G j (s ǫ,n+1 j-1/2 , s ǫ,n+1 j+1/2 )ǫ ϕj (s ǫ,n+1 j+1/2 )-ϕj(s ǫ,n+1 j-1/2 ) ∆x satisfies lim ǫ→0 F ǫ,n+1 j = G j (s 0,n+1 j-1/2 , s 0,n+1 j+1/2 ) := F 0,n+1 j .

Similarly, it follows from the formulas the one where F opt = f R (s R ). Let us treat the first situation in detail: see Fig. 1, but imagine now that P crosses D and not U. We see that there exists a value s L < s opt L = sL such that (s L , s opt R ) ∈ P. Moreover, since s L < sL = s π, * L ≤ u L , we do have G L (u L , s L ) = f L (s opt L ). The latter value coincides with f R (s opt R ), which also equals G R (s opt R , u R ) because u R ≤ s opt R . We arrive to the desired equality G L (u L , s L ) = G R (s opt R , u R ) = F opt with (s L , s opt R ) ∈ P: again, we see that formulas [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] and ( 50) yield the same value.

(ii) Assume that u L ≤ s π,⋆ L and that u R ≤ u R,⋆ L , so that formula [START_REF] Serre | Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems[END_REF] provides that the flux at the interface should be given by F 0 (u L , u R ) = f L (u L ). Let us find a convenient choice of (s L , s R ) solution to [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] so that F 0 0 (u L , u R ) = F 0 (u L , u R ). The fact that G L (u L , s L ) = G R (s R , u R ) = f L (u L ) implies, because of Assumption (A1), that s L can be chosen arbitrarily in [0, u ⋆ L ], while s R has to be equal to u R L . Note that (u ⋆ L , u R L ) ∈ U, and that u ⋆ L ≥ s π L , u R L ≤ s π R . It can thus be seen on Fig. 1 that [0, u ⋆ L ] × {u R L } ∩ P = ∅. Choosing (s L , s R ) at this last intersection in [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] ensures that formulas [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] and (50) yield the same value for the flux F 0 (u L , u R ).

(iii) The last case is then u R ≥ s π,⋆ R and u L ≥ u L,⋆ R , so that the flux given by (50) turns to be equal to f R (u R ). From similar argument as in the previous case, we deduce from G L (u L , s L ) = G R (s R , u R ) that s R can be chosen arbitrary in [u ⋆ R , 1] while the condition s L = u L R is enforced. Here again, the segment {u L R } × [u ⋆ R , 1] has a non-empty intersection with P. Choosing (s L , s R ) ∈ {u L R } × [u ⋆ R , 1] ∩ P ensures that, in this case again, the values given by the formulas ( 48) and (50) coincide.

The above case by case study is illustrated by Fig. 12b.

As a direct consequence of formula (50) and of [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF], taking ǫ = 0 in the scheme defined by ( 35)-( 39) yields the implicit Godunov scheme corresponding to the notion of G (s π L ,s π R ) -entropy solution. From the monotonicity of the scheme, we deduce that the discrete solution s 0 h is unique (e.g. [START_REF] Eymard | Finite Volume Methods. Handbook of Numerical Analysis[END_REF][START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]). The analysis carried out in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] for the explicit Godunov scheme can be straightforwardly adapted to the implicit case. Corollary 10 Let s 0 h be the unique approximate solution provided by the scheme (35)- [START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF] in the case ǫ = 0, then lim ∆x,∆t→0

s 0 h = s in L 1 loc (R × R + ),
where s is the unique G (s π L ,s π R ) -entropy solution to the hyperbolic Buckley-Leverett equation in two-rocks' medium.

  and the barrier connection (A bar , B bar ) by (A bar , B bar ) ∈ U, with either A bar = 1 or B bar = 0. The common value F = f L (A) = f R (B) is called the connection level and denoted by F(A,B) ; when (A, B) runs over U, F(A,B) fills the interval [ F bar , F opt ]; here F bar = max{0, q}

)Figure 2 :

 2 Figure 2: On Figure 2a, the two flux functions f L,R have been plotted. Given a value F(A,B) ∈ [ F bar , F opt ], we construct the unique corresponding connection (A, B) ∈ U. On Figure 2b, we have plotted the corresponding sets O (green solid line) and U (red dashed line). For a given flux limitation F(A,B) , the grey rectangle represents the open set(s L , s R ) ∈ [0, 1] 2 | (f L (s L ) > F(A,B) )&(f R (s R ) > F(A,B) ) . So, the maximal germ G * (A,B) is made of the union of singleton {(A, B)} and of the subset O F(A,B) of O which is outside of the grey rectangle.

Figure 3 :

 3 Figure 3: The capillary pressure curves (Fig. 3a) defined by (44)-here P L = 0 (blue) and P R = 2 (green)-satisfy lim s→1 π L,R (s) = +∞. Therefore, the maximal extension πL,R of π L,R is obtained by adding {0} × [-∞, P L,R ) and {1} × {+∞} to the graph {(s, π L,R (s) | s ∈ [0, 1)}. For the particularly simple choice of parameters and functions done in the simulations, the flux functions f L,R are proportional one to the other. We have represented on Fig. 3b the optimal connection that is relevant for the case presented in Section 3.3.2, but not in the one presented in Section 3.3.3.

  -3 ; porosity φ L = φ R = 1; entry pressures P L = 0, P R = 0.5 in Section 3.3.2, P R = 2 in Section 3.3.3; viscosities µ a = 10 -3 , µ b = 3.10 -3 ; densities ρ a = 0.87, ρ b = 1; relative permeabilities kr a,i (s) = s , kr b,i (s) = (1s); time step ∆t = 2.5 * 10 -3 , space step ∆x = 10 -2 .

Figure 4 :

 4 Figure 4: The connection diagram in a case where the intersection of P (in blue) and U (in green) is empty; according to the Selection Rule of Section 1.3, it leads to considering the optimal entropy solution.

Figure 5 ( 6 3. 3 . 3

 5633 Figure 5

Figure 7 :

 7 Figure 7: The connection diagram in a case where the intersection of P (in blue) and U (in green) is non-empty is presented on Fig. (7a). This results in a flux limitation, in the sense that at the interface, the flux of the hyperbolic solution may not exceed the value F π = f L,R (s π L,R ), where (s π L , s π R ) is the intersection point of U and P.

Figure 8 3. 3 . 4

 834 Figure 8

( a )

 a Figure 9

Figure 11 :

 11 Figure 11: Illustration of the convergence of order 1 in the genuinely nonlinear case. The blue curve correspond to the plot of s hs ref L 1 as a function of ∆x, where s ref is a reference solution computed with a small ∆x = 10 -3 . The green dashed line has a slope equal to 1.

  ,8,9,10, in this appendix we aim to prove that lim ∆x,∆t→0 lim ǫ→0s ǫ h = s in L 1 loc (R × R + ).

F ǫ,n+1 0 = 2 =)F 0

 020 G L (s ǫ,n+1 -1/2 , s ǫ,n+1 L )ǫ ϕ L (s ǫ,n+1 L )ϕ L (s ǫ,n+1 -1/2 ) ∆x/G R (s ǫ,n+1 R , s ǫ,n+1 1/2 )ǫ ϕ R (s ǫ,n+1 1/2 )ϕ R (s ǫ,n+1 R = f L (u L ) F 0 (u L , u R ) = F π F 0 (u L , u R ) = f R (u R) the case by case study's zones

Figure 12

 12 Figure 12

While this analysis has been carried out under the assumption q = 0, the general case is completely analogous

Acknowledgement The first author thanks John D. Towers for the discussion that was at the origin of this work. The work of the first author was partially supported by the French ANR project CoToCoLa. The work of the second author was partially supported by GNR MoMaS, CNRS-2439 (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN).

and from the property (s 0,n+1 L , s 0,n+1 R ) ∈ P, that πL (s 0,n+1

The following lemma ensures that the transmission conditions system (47) yields a flux that is well defined.

admits a least one solution (s L , s R ) ∈ P; moreover, the value F 0 0 (u L , u R ) is defined uniquely by [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF].

Proof: The set P can be naturally parametrized by p ∈ R as follows:

Therefore, finding (s L , s R ) solution of (48) reduces to finding p ∈ R such that

where the left-hand side Ψ L is non-increasing while the right-hand side Ψ R is non-decreasing. In addition,

due to the consistency and the monotonicity properties of the numerical fluxes G L,R (•, •). As a consequence, there exists at least one value of p and a unique value of Ψ L,R (p) such that (49) holds.

In the following proposition, we identify the flux given by [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] with the Godunov flux at the interface, whose explicit formula was derived in [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF].

given by the nonlinear system (48) is equal to the Godunov flux

where

is the connection selected in Section 1.3. Proof: We perform the proof by a case by case study relying on the resolution of the Riemann problem. First, we need to introduce some notation. To start with, we extend the graphs f i by setting f i (0) = (-∞, 0] and f i (1) = (-∞, q]. In this case, for all F < F opt the sets of level F for f L , R have exactly two elements, one on each side from sL,R . Then we denote by u ⋆ i the unique value of [0, 1], called conjugate of u i , such that

) lies on the decreasing branch of O. We denote by (s π L , s π R ) the connection defined by the Selection Rule at the end of Section 1.3, and by s π,⋆ L,R , the conjugate values of s π L,R .

(i) Assume first that u L ≥ s π,⋆ L and u R ≤ s π,⋆ R , then, thanks to Assumption (A1), the Godunov flux given by formula (50) provides F 0 (u L , u R ) = F π . Reciprocally, assume firstly that P ∩ U = ∅, so that (s π L , s π R ) ∈ P. From Assumption (A1) on the flux functions, we deduce that G L (u L , s π L ) = G R (s π R , u R ) = F π . Thus formulas [START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF] and ( 50) yield the same value. The remaining case is when P ∩ U = ∅, thus F π = F opt . There are two symmetric situations: the one where F opt = f L (s L ) and