A COMBINED FINITE VOLUME–NONCONFORMING FINITE ELEMENT SCHEME FOR COMPRESSIBLE TWO PHASE FLOW IN POROUS MEDIA

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ELEMENT SCHEME FOR COMPRESSIBLE TWO PHASE FLOW IN POROUS MEDIA

BILAL SAAD AND MAZEN SAAD

Abstract. We propose and analyze a combined finite volume–nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on the own pressure.

1. Introduction

The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. A large variety of methods have been proposed for the discretization of degenerate parabolic systems modeling the displacement of immiscible incompressible two-phase flows in porous media. We refer to [1] and [23] for the finite difference method.

The finite volume methods have been proved to be well adapted to discretize conservative equations. The cell-centered finite volume scheme has been studied e.g. by [21], [16]

and [4]. Recently, the convergence analysis of a finite volume scheme for a degenerate compressible and immiscible flow in porous media has been studied by Bendahmane et al.

[3] when the densities of each phase depend on the global pressure, and by B. Saad and M.

Saad [26] for the complete system when the density of the each phase depends on the own pressure. In these works, the medium is considered homogeneous, the permeability tensor is proportional to the matrix identity and the mesh is supposed to be admissible in the sense that satisfying the orthogonal property as in [14]. The cell-centered finite volume method with an upwind discretization of the convection term ensures the stability and is extremely robust and have been used in industry because they are cheap, simple to code and robust. However, standard finite volume schemes do not permit to handle anisotropic diffusion on general meshes see e.g. [14].

Various multi-point schemes where the approximation of the flux through an edge involves several scalar unknowns have been proposed, see e.g. Coudi`ere et al. [10], Eymard et al. [15], or Faille [17]. However, such schemes require using more points than the classical 4 points for triangular meshes and 5 points for quadrangular meshes in space dimension two, making the schemes less robust.

On the other hand finite element method allows a very simple discretization of the diffusion term with a full tensor and does not impose any restrictions on the meshes, they were used a lot for the discretization of a degenerate parabolic problems modeling of contaminant transport in porous media. The mixed finite element method by Dawson [11],

Date: October 9, 2013.

Key words and phrases. Finite volume scheme, Finite element method, degenerate problem.

This work was partially supported by GNR MOMAS..

1

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the conforming piecewise linear finite element method has been studied e.g. by Barrett and Knabner [2], Chen and Ewing [8], Nochetto et al. [22], and Rulla et al. [25]. However, it is well-known that numerical instabilities may arise in the convection-dominated case.

To avoid these instabilities, the theoretical analysis of the combined finite volume–finite element method has been carried out for the case of a degenerate parabolic problems with a full diffusion tensors. The combined finite volume–conforming finite element method proposed and studied by Debiez et al. [12] or Feistauer et al. [18] for fluid mechanics equations, are indeed quite efficient.

This ideas is extended by [24] for the degenerate parabolic problems, to the combination of the mixed-hybrid finite element and finite volume methods, to inhomogeneous and anisotropic diffusion–dispersion tensors, to space dimension three, and finally to meshes only satisfying the shape regularity condition. In order to solve this class of equations, Ey- mard et al. [24] discretize the diffusion term by means of piecewise linear nonconforming (Crouzeix–Raviart) finite elements over a triangularization of the space domain, or using the stiffness matrix of the hybridization of the lowest order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a finite volume scheme on a dual mesh, with an upwind discretization of the convection term to ensures the stability, where the dual volumes are constructed around the sides of the original triangularization. The intention of this paper is to extend these ideas to a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media without simplified assumptions on the state law of the density of each phase, to the combination of the nonconforming finite element and finite volume methods, to inhomogeneous and anisotropic permeability tensors, to space dimension three, and finally to meshes only satisfying the shape regularity condition.

Following [24], let us now introduce the combined scheme that we analyze in this paper.

We consider a triangulation of the space domain consisting of simplices (triangles in space dimension two and tetrahedra in space dimension three). We next construct a dual mesh where the dual volumes are associated with the sides (edges or faces). To construct a dual volume, one connects the barycentres of two neighboring simplices through the vertices of their common side.We finally place the unknowns in the barycentres of the sides. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh, hence we obtain the combined scheme. To ensures the stability, the relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique ensures the validity of the discrete maximum principle for the saturation in the case where all transmissibilities are non-negative.

This paper deals with construction and convergence analysis of a combined finite volume–nonconforming finite element for two compressible and immiscible flow in porous media without simplified assumptions on the state law of the density of each phase. The analysis of this model is based on new energy estimates on the velocities of each phase.

Nevertheless, these estimates are degenerate in the sense that they do not permit the control of gradients of pressure of each phase, especially when a phase is not locally present in the domain. The main idea consists to derive from degenerate estimates on pressure of each phase, which not allowed straight bound on pressures, an estimate on global pressure and degenerate capillary term in the whole domain regardless of the presence or the disappearance of the phases. These stabilities results with a priori estimates on the pressures and a function of the saturation that denote capillary terms and some compactness arguments based on the use of the Kolmogorov relative compactness theorem, allow us to derive the convergence of both these approximations to a weak solution of the continuous problem in this paper provided the mesh size tends to zero.

The organization of this paper is as follows. In section 2, we introduce the nonlinear parabolic system modeling the two–compressible and immiscible fluids in a porous media and we state the assumptions on the data and present a weak formulation of the continuous problem. In section 3, we describe the combined finite volume–nonconforming finite element scheme and we present the main theorem of convergence. In section 4, we derive

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three preliminary fundamental lemmas. In fact, we present some properties of this scheme and we will see that we can’t control the discrete gradient of pressure since the mobility of each phase vanishes in the region where the phase is missing. So we are going to use the feature of global pressure. We show that the control of velocities ensures the control of the global pressure and a dissipative term on saturation in the whole domain regardless of the presence or the disappearance of the phases. Section 5 is devoted to a maximum principle on saturation, a priori estimates on the discrete velocities and existence of discrete solutions. In section 6, we derive estimates on difference of time and space translates for the approximate solutions. In section 7 using the Kolmogorov relative compactness theorem, we prove the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous problem.

2. Mathematical formulation of the continuous problem

Let us state the physical model describing the immiscible displacement of two compressible fluids in porous media. We consider the flow of two immiscible fluids in a porous medium. We focus on the water and gas phase, but the considerations below are also valid for a general wetting phase and a non-wetting phase.

The mathematical model is given by the mass balance equation and Darcy’s law for both phases α=l, g. LetT > 0 be the final time fixed, and let be Ω a bounded open subset ofR^d(d≥1). We setQT = (0, T)×Ω, ΣT = (0, T)×∂Ω. The mass conservation of each phase is given inQT

(2.1)

φ(x)∂t(ρα(pα)sα)(t, x) + div(ρα(pα)Vα)(t, x) +ρα(pα)sαfP(t, x) =ρα(pα)s^IαfI(t, x), where φ, ρα and sα are respectively the porosity of the medium, the density of the α phase and the saturation of the αphase. Here the functions f_I and f_P are respectively the injection and production terms. Note that in equation (2.1) the injection term is multiplied by a known saturation s^I_α corresponding to the known injected fluid, whereas the production term is multiplied by the unknown saturation sα corresponding to the produced fluid.

The velocity of each fluidVα is given by the Darcy law:

(2.2) Vα=−Λkrα(sα) µα

∇pα−ρα(pα)g

, α=l, g.

where Λis the permeability tensor of the porous medium, krα the relative permeability of the αphase, µα the constantα-phase’s viscosity, pα theα-phase’s pressure andg is the gravity term. Assuming that the phases occupy the whole pore space, the phase saturations satisfy

(2.3) sl+sg= 1.

The curvature of the contact surface between the two fluids links the jump of pressure of the two phases to the saturation by the capillary pressure law in order to close the system (2.1)-(2.3)

(2.4) pc(sl(t, x)) =pg(t, x)−pl(t, x).

With the arbitrary choice of (2.4) (the jump of pressure is a function ofsl), the application sl7→pc(sl) is non-increasing, (^dp_ds^c

l(sl)<0, for allsl∈[0,1]), and usuallypc(sl= 1) = 0 when the wetting fluid is at its maximum saturation.

2.1. Assumptions and main result. The model is treated without simplified assumptions on the density of each phase, we consider that the density of each phase depends on its corresponding pressure. The main point is to handle a priori estimates on the approximate solution. The studied system represents two kinds of degeneracy: the degeneracy for evolution terms ∂t(ραsα) and the degeneracy for dissipative terms div(ραMα∇pα) when the saturation vanishes. We will see in the section 5 that we can’t control the discrete gradient of pressure since the mobility of each phase vanishes in the region where the phase is missing. So, we are going to use the feature of global pressure to obtain uniform estimates on the discrete gradient of the global pressure and the discrete gradient of the capillary termB(defined on (2.7)) to treat the degeneracy of this system.

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Let us summarize some useful notations in the sequel. We recall the conception of the global pressure as describe in [7]

M(sl)∇p=Ml(sl)∇pl+Mg(sg)∇pg, with theα-phase’s mobilityMαand the total mobility are defined by

Mα(sα) =kr_α(sα)/µα, M(sl) =Ml(sl) +Mg(sg).

This global pressurepcan be written as

p=pg+ ˜p(sl) =pl+ ¯p(sl), (2.5)

or the artificial pressures are denoted by ¯pand ˜pdefined by:

˜ p(sl) =−

Zs_l 0

Ml(z)

M(z)p⁰_c(z)dz andp(sl) = Zs_l

0

Mg(z)

M(z)p⁰_c(z)dz.

(2.6)

We also define the capillary terms by

γ(sl) =−Ml(sl)Mg(sg) M(sl)

dpc

dsl

(sl)≥0, and let us finally define the functionBfrom [0,1] toRby:

B(sl) = Z s_l

0

γ(z)dz=− Z s_l

0

Ml(z)Mg(z) M(z)

dpc

dsl

(z)dz

=− Z sg

0

Ml(z)d¯p dsl

(z)dz= Z s_l

0

Mg(z)d˜p dsl

(z)dz.

(2.7)

Using these notations, we derive the fundamental relationship between the velocities and the global pressure and the capillary term:

(2.8) Ml(sl)∇pl=Ml(sl)∇p+∇B(sl), Mg(sg)∇pg =Mg(sl)∇p− ∇B(sl).

As mentioned above that the mobilities vanish and consequently the control of the gradient of the pressure of each phase is not possible. A main point of the paper is to give sense for the term∇pα,α=l, g. This term is a distribution and it is not enough to give a sense of the velocity. Our approach is based on the control of the velocity of each phase. Thus the gradient of the global pressure and the functionBare bounded which give a rigorous justification for the degenerate problem.

We complete the description of the model (2.1) by introducing boundary conditions and initial conditions. To the system (2.1)–(2.4) we add the following mixed boundary conditions. We consider the boundary∂Ω = Γl∪Γimp, where Γldenotes the water injection boundary and Γimp the impervious one.

(2.9)

( pl(t, x) =pg(t, x) = 0 on (0, T)×Γl, ρlVl·n=ρgVg·n= 0 on (0, T)×Γimp, wherenis the outward normal to Γimp.

The initial conditions are defined on pressures

(2.10) pα(t= 0) =p⁰_αforα=l, gin Ω.

Next we introduce some physically relevant assumptions on the coefficients of the system.

(A1) There is two positive constantsφ0 andφ1 such thatφ0≤φ(x)≤φ1 almost every- wherex∈Ω.

(A2) Λij ∈ L^∞(Ω),|Λij| ≤ ^C_d^Λ a.e. in Ω, 1 ≤i, j≤d, CΛ>0,Λ is a symmetric and there exist a constantcΛ>0 such that

hΛ(x)ξ, ξi ≥cΛ|ξ|²,∀ξ∈R^d.

(A3) The functionsMland Mg belongs toC⁰([0,1],R⁺), Mα(sα = 0) = 0.In addition, there is a positive constantm0>0 such that for allsl∈[0,1],

Ml(sl) +Mg(sg)≥m0.

(A4) (f_P, f_I)∈(L²(QT))²,f_P(t, x),f_I(t, x)≥0 almost everywhere (t, x)∈QT.

(A5) The densityρα is C¹(R), increasing and there exist two positive constants ρm>0 andρM >0 such that 0< ρm≤ρα(pg)≤ρM.

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(A6) The capillary pressure fonctionpc∈ C¹([0,1];R⁺), decreasing and there existspc>0 such that 0< pc≤ |^dp_ds^c

l|.

(A7) The functionγ ∈C¹ [0,1];R⁺

satisfies γ(sl)>0 for 0< sl<1 and γ(sl= 1) = γ(sl= 0) = 0.We assume thatB⁻¹(the inverse ofB(sl) =Rs_l

0 γ(z)dz) is a H¨older¹ function of orderθ, with 0< θ≤1, on [0,B(1)].

The assumptions (A1)–(A7) are classical for porous media. Note that, due to the bound- edness of the capillary pressure function, the functions ˜pand ¯pdefined in (2.6) are bounded on [0,1].

We now give the definition of a weak solution of the problem (2.1)–(2.4).

Definition 1. (Weak solutions). Under assumptions (A1)-(A7) and suppose (p⁰_l, p⁰_g) belong to (L²(Ω))² and0≤s⁰α(x)≤1almost everywhere inΩ, then the pair (pl, pg)is a weak solution of problem (2.1)satisfying :

pα∈L²(QT), 0≤sα(t, x)≤1a.e inQT, (α=l, g), (2.11)

p∈L²(0, T;H¹(Ω)), B(sl)∈L²(0, T;H_Γ¹_l,(Ω)), (2.12)

Mα(sα)∇pα∈(L²(QT))^d, (2.13)

such that for all ϕ, ψ∈C¹([0, T];HΓ¹_l(Ω)) with ϕ(T,·) =ψ(T,·) = 0,

− Z

Q_T

φρl(pl)sl∂tϕdxdt− Z

Ω

φ(x)ρl(p⁰l(x))s⁰l(x)ϕ(0, x)dx +

Z

Q_T

ρl(pl)Ml(sl)Λ∇pl· ∇ϕdxdt− Z

Q_T

ΛMl(sl)ρ²_l(pl)g· ∇ϕdxdt (2.14)

+ Z

Q_T

ρl(pl)slfPϕdxdt= Z

Q_T

ρl(pl)s^IlfIϕdxdt,

− Z

Q_T

φρg(pg)sg∂tψdxdt− Z

Ω

φ(x)ρg(p⁰_g(x))s⁰_g(x)ψ(0, x)dx +

Z

Q_T

Mg(sg)ρg(pg)Λ∇pg· ∇ψdxdt− Z

Q_T

ΛMg(sg)ρ²_g(pg)g· ∇ψdxdt (2.15)

+ Z

Q_T

ρg(pg)sgfPψdxdt= Z

Q_T

ρg(pg)s^IgfIψdxdt.

3. Combined finite volume–nonconforming finite element scheme We will describe the space and time discretizations, define the approximation spaces, and introduce the combined finite volume–nonconforming finite element scheme in this section.

3.1. Space and time discretizations. In order to discretize the problem (2.1), we perform a triangulation Th of the domain Ω, consisting of closed simplices such that Ω =∪K∈T_hKand such that ifK, L∈ Th,K6= L, thenK∩ L is either an empty set or a common face, edge, or vertex ofK and L. We denote byEh the set of all sides, byE_h^int the set of all interior sides, by E_h^ext the set of all exterior sides, and byEK the set of all the sides of an element K∈ Th. We defineh:=max{diam(K), K ∈ Th}and make the following shape regularity assumption on the family of triangulations{Th}h:

There exists a positive constantκT such that

(3.1) min

K∈T_h

|K|

diam(K)^d ≥κT, ∀h >0.

Assumption (3.1) is equivalent to the more common requirement of the existence of a constantθT >0 such that

(3.2) max

K∈T_h

diam(K) DK

≥κT, ∀h >0,

1This means that there exists a positive constantcsuch that for alla, b∈[0,B(1)],one has|B⁻¹(a)−

B⁻¹(b)| ≤c|a−b|^θ.

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whereDK is the diameter of the largest ball inscribed in the simplexK.

K

L E D ×QD

σD

Q×E

σE

σD

σD,E

Figure 1. ^Triangles^{K, L}^{∈ T}hand dual volumesD,E∈ D_hassociated with edgesσD, σE∈ E_h

We also use a dual partitionDhof Ω such that Ω =∪D∈D_hD. There is one dual element D associated with each side σD ∈ Eh. We construct it by connecting the barycentres of every K∈ Ththat contains σD through the vertices of σD. For σD ∈ E_h^ext, the contour ofD is completed by the sideσDitself. We refer to Fig. 1 for the two-dimensional case.

We denote byQDthe barycentre of the sideσD. As for the primal mesh, we setFh,F_hînt, F_hêxtandFDfor the dual mesh sides. We denote byD_hîntthe set of all interior and byD_hêxt the set of all boundary dual volumes. We finally denote byN(D) the set of all adjacent volumes to the volumeD,

N(D) :={E∈ Dh;∃σ∈ Fh^int such thatσ=∂D∩∂E}

and remark that

|K∩D|= |K|

d+ 1, (3.3)

for each K ∈ Th and D ∈ Dh such that σD ∈ EK. For E ∈ N(D), we also set dD,E:=|QE−QD|,σD,E:=∂D∩∂EandKD,Ethe element ofThsuch thatσD,E⊂KD,E. The problem under consideration is time-dependent, hence we also need to discretize the time interval (0, T). The time discretization of (0, T) is given by an integer valueN and by a strictly increasing sequence of real values (tⁿ)n∈[0,N] witht⁰ = 0 and t^N =T. Without restriction, we consider a uniform step time δt=tⁿ−tⁿ⁻¹, forn∈[1, N].

We define the following finite-dimensional spaces:

Xh:={ϕh∈L²(Ω);ϕh|K is linear∀K∈ Th,

ϕhis continuous at the pointsQD, D∈ D_h^int}, Xh⁰:={ϕh∈Xh;ϕh(QD) = 0 ∀D∈ Dh^ext}.

The basis ofXhis spanned by the shape functionsϕD,D∈ Dh, such thatϕD(QE) =δDE, E ∈ Dh,δbeing the Kronecker delta. We recall that the approximations in these spaces are nonconforming since Xh6⊂H¹(Ω). We equipXhwith the seminorm

kuhk²_X

h:= X

K∈T_h

Z

K

|∇uh|²dx, which becomes a norm onX_h⁰.

For a given value uD, D∈ Dh (resp. uⁿ_D, D ∈ Dh, n ∈ [0, N]), we define a constant piecewise function as : u(x) =uDforx∈D (resp. u(t, x) =uⁿDforx∈D,t∈]tⁿ⁻¹, tⁿ].

Next, we define the discret differential operateur :

δD|E(u) =uE−uD, andδD|Eⁿ (u) =uⁿE−uⁿD.

Definition 2. Let the values uⁿ_D, D ∈ Dh, n ∈ {0,1,· · ·, N}. As the approximate solutions of the problem by means of the combined finite volume–nonconforming finite element scheme, we understand:

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(1) A functionuδt,hsuch that uδt,h(x,0) =u⁰h(x)forx∈Ω,

uδt,h(x, t) =uⁿ_h(x) forx∈Ω, t∈(tn−1, tn] n∈ {1,· · ·, N}, (3.4)

whereuⁿ_h=P

D∈D_huⁿ_DϕD; (2) A functionu˜δt,hsuch that

˜

uδt,h(x,0) =u⁰_D forx∈D, D˚ ∈ Dh,

˜

uδt,h(x, t) =uⁿ_D forx∈D, D˚ ∈ Dh, t∈(tn−1, tn] n∈ {1,· · ·, N}.

(3.5)

The functionuδt,his piecewise linear and continuous in the barycentres of the interior sides in space and piecewise constant in time; we will call it anonconforming finite element solution. The function ˜uδt,his given by the values ofuⁿ_Din side barycentres and is piecewise constant on the dual volumes in space and piecewise constant in time; we will call it a finite volume solution.

3.2. The combined scheme. For more clarity and for presentation simplicity, we present the combined scheme for a horizontal field and then we neglect the gravity effect. In remark 1, we indicate how to modify the scheme to include the gravity terms.

Definition 3. (Combined scheme)The fully implicit combined finite volume-nonconforming finite element scheme for the problem (2.1) reads: find the values pⁿ_α,D, D ∈ Dh, n ∈ {1,· · ·, N}, such that

(3.6) p⁰_α,D= 1

|D|

Z

D

p⁰_α(x)dx, s⁰_α,D= 1

|D|

Z

D

s⁰_α(x)dx, for allD∈ Dh^int,

(3.7) |D|φD

ρl(pⁿ_l,D)sⁿ_l,D−ρl(pⁿ⁻¹_l,D )sⁿ⁻¹_l,D

δt − X

E∈N(D)

ρⁿl,D,EMl(sⁿl,D,E) ΛD,EδⁿD|E(pl) +|D|ρl(pⁿl,D)sⁿl,DfP,Dⁿ =|D|ρl(pⁿl,D)(s^Il,D)ⁿfI,Dⁿ ,

(3.8) |D|φD

ρg(pⁿ_g,D)sⁿ_g,D−ρg(pⁿ⁻¹_g,D)sⁿ⁻¹_g,D

δt − X

E∈N(D)

We refer to the matrixΛof the elementsΛD,E,D, E∈ D^int_h , as to the diffusion matrix.

This matrix, the stiffness matrix of the nonconforming finite element method writes in the form

(3.10) ΛD,E:=− X

K∈T_h

(Λ(x)∇ϕE,∇ϕD)_0,K D, E∈ Dh. Notice that the source terms are, for n∈ {1, . . . , N}

f_P,Dⁿ := 1 δt|D|

Z tⁿ tⁿ⁻¹

Z

D

fP(t, x)dxdt, f_I,Dⁿ := 1 δt|D|

Z tⁿ tⁿ⁻¹

Z

D

fI(t, x)dxdt.

The mean value of the density of each phase on interfaces is not classical since it is given as

(3.11) 1

ρⁿ_α,D,E =







1 pⁿ_α,E−pⁿ

α,D

R^pⁿα,E pⁿ_α,D

1

ρα(ζ)dζ ifpⁿ_α,D6=pⁿ_α,E,

1

ρⁿ_α,D otherwise.

This choice is crucial to obtain estimates on discrete pressures.

We denoted

(3.12) Gα(sⁿα,D, sⁿα,E;δD|Eⁿ (pα)) =−Mα(sⁿα,D,E)δD|Eⁿ (pα),

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the numerical fluxes, where Mα(sⁿ_α,D,E) denote the upwind discretization of Mα(sα) on the interfaceσD,E and

sⁿα,D,E =

( sⁿα,Dif (D, E)∈ Eαⁿ, sⁿ_α,Eotherwise, (3.13)

with the setEαⁿis subset ofEhsuch that

Eαⁿ={(D, E)∈ Eh,ΛD,EδⁿD|E(pα) = ΛD,E pⁿα,E−pⁿα,D

≤0}.

(3.14)

Remark 1. To take into account the gravity term, it is enough to modify, for example, the numerical fluxes to be

Gα(sⁿα,D, sⁿα,E;δD|Eⁿ (pα)) =−Mα(sⁿα,D,E)δⁿD|E(pα)+ρα(pⁿα,D,E)

Mα(sⁿα,D)g_D|E⁺ −Mα(sⁿα,E)g⁻_D|E ,

withg_D|E^± = (g·ηD|E)^±. This numerical flux satisfy is consistent, conservative and mono- tone, thus the convergence result remains valid.

In the sequel we shell consider apart the following special case:

All transmissibility’s are non-negative, i.e.

ΛD,E≥0 ∀D∈ Dh^int, E∈ N(D).

(3.15) Since

∇ϕD|K =|σD|

|K|nσ_D, K∈ Th, σD∈ EK

withnσ_D the unit normal vector of the sideσD outward toK, one can immediately see that Assumption (3.15) is satisfied when the diffusion tensor reduces to a scalar function and when the magnitude of the angles betweennσ_D,σD∈ EK, for allK∈ This greater or equal toπ/2.

The main result of this paper is the following theorem.

Theorem 1. There exists an approximate solutions(pⁿ_α,D)n,Dcorresponding to the system (3.7)-(3.8), which converges (up to a subsequence) to a weak solutionpα of (2.1) in the sense of the Definition 1.

4. Preliminary fundamental lemmas

In this section, we will first present several technical lemmas that will be used in our latter analysis to obtain a priori estimate of the solution of the discrete problem.

Lemma 1. For alluh=P

D∈D_huDϕD∈Xh, X

D∈D_h

X

E∈N(D)

ΛD,E(δD|Eu)²≥cΛkuhk²_X

h.

Proof. We have X

D∈D_h

X

E∈N(D)

ΛD,E(δD|Eu)²= X

D∈D_h

X

E∈N(D)

ΛD,E(uE−uD)²

=−2 X

D∈D_h

uD

X

E∈N(D)

ΛD,E(uE−uD)

=−2 X

D∈D_h

uD

X

E∈D_h

ΛD,EuE

= 2 X

K∈T_h

(Λ∇uh,∇uh)0,K≥cΛkuhk²_X

h,

using (3.10) and assumption (A2).

We state this lemma without a proof as well (cf. [24, Lemma 3.1]):

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Lemma 2. For alluh=P

D∈D_huDϕD∈Xh, one has X

σ_D,E∈F_h^int

|σD,E| dD,E

(uE−uD)²≤ d+ 1 2(d−1)κT

kuhk²_X

h. (4.1)

In the continuous case, we have the following relationship between the global pressure, capillary pressure and the pressure of each phase

(4.2) Ml|∇pl|²+Mg|∇pg|²=M|∇p|²+MlMg

M |∇pc|².

This relationship, means that, the control of the velocities ensures the control of the global pressure and the capillary termsB in the whole domain regardless of the presence or the disappearance of the phases.

In the discrete case, these relationship, are not obtained in a straightforward way. This equality is replaced by three discrete inequalities which we state in the following lemma.

Lemma 3. (Total mobility, global pressure, Capillary termB and Dissipative terms)Un- der the assumptions (A1)−(A7) and the notations (2.5). Then for all(D, E) ∈ D and for alln∈[0, N]the following inequalities hold:

(4.3) Ml,D|Eⁿ +Mg,D|Eⁿ ≥m0,

(4.4) m0

δⁿD|E(p)

2

≤Ml,D|Eⁿ

δD|Eⁿ (pl)

2

+Mg,D|Eⁿ

δD|Eⁿ (pg) 2

.

(4.5) (δⁿ_D|E(B(sl)))²≤M_l,D|Eⁿ

δ_D|Eⁿ (pl)2

+M_g,D|Eⁿ

δ_D|Eⁿ (pg)2

,

(4.6) Ml,D|Eⁿ (δD|Eⁿ (¯p(sl)))²≤Ml,D|Eⁿ

δⁿD|E(pl)2

+Mg,D|Eⁿ

δD|Eⁿ (pg)2

, and

(4.7) Mg,D|Eⁿ (δD|Eⁿ (˜p(sl)))²≤Ml,D|Eⁿ

δD|Eⁿ (pl)2

+Mg,D|Eⁿ

δD|Eⁿ (pg)2

.

In [26], the authors prove this lemma on primal mesh satisfying the orthogonal condition. This proof use only two neighbors elements and it is based only on the definition of the global pressure. Thus, we state the above Lemma without proof since the proof made in [26] remains valid on the dual mesh.

5. A priori estimates and existence of the approximate solution We derive new energy estimates on the discrete velocitiesMα(sⁿ_α,D|E)δ_D|Eⁿ (pα). Never- theless, these estimates are degenerate in the sense that they do not permit the control of δⁿ_D|E(pα), especially when a phase is missing. So, the global pressure has a major role in the analysis, we will show that the control of the discrete velocities Mα(sⁿ_α,D|E)δⁿ_D|E(pα) ensures the control of the discrete gradient of the global pressure and the discrete gradient of the capillary termBin the whole domain regardless of the presence or the disappearance of the phases.

The following section gives us some necessary energy estimates to prove the theorem 1.

5.1. The maximum principle. Let us show in the following Lemma that the phase by phase upstream choice yields the L^∞ stability of the scheme which is a basis to the analysis that we are going to perform.

Lemma 4. (Maximum principe). Under assumptions (A1)-(A7). Let(s⁰_α,D)D∈D_h ∈[0,1]

and assume that (pⁿ_α,D)D∈D_h is a solution of the finite volume (3.6)-(3.9). Then, the saturation(sⁿ_α,D)D∈D_h, remains in[0,1]for allD∈ Dh, n∈ {1, . . . , N}.

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Proof. Let us show by induction in nthat for allD∈ Dh, sⁿ_α,D≥0 whereα=l, g. For α=l, the claim is true forn = 0 and for allD ∈ Dh. We argue by induction that for allD∈ Dh, the claim is true up to ordern. We consider the control volumeDsuch that sⁿl,D= min{sⁿl,E}_E∈D

h and we seek thatsⁿl,D≥0.

For the above mentioned purpose, multiply the equation in (3.7) by−(sⁿ_l,D)⁻, we obtain (5.1) − |D|φD

ρl(pⁿl,D)sⁿl,D−ρl(pⁿ⁻¹_l,D )sⁿ⁻¹_l,D δt (sⁿ_l,D)⁻

− X

E∈N(D)

τD|Eρⁿ_l,D,EΛD,EGl(sⁿ_l,D, sⁿ_l,E;δⁿD|E(pl))(sⁿ_l,D)⁻

− |D|ρl(pⁿl,D)sⁿl,DfP,Dⁿ (sⁿl,D)⁻=− |D|ρl(pⁿl,D)(s^Ig,D)ⁿfI,Dⁿ (sⁿl,D)⁻≤0.

The numerical flux Glis nonincreasing with respect tosⁿ_l,E, and consistence, we get Gl(sⁿl,D, sⁿl,E;δD|Eⁿ (pl)) (sⁿl,D)⁻≤Gl(sⁿl,D, sⁿl,D;δⁿD|E(pl)) (sⁿl,D)⁻

=−δⁿD|E(pl)Ml(sⁿ_l,D) (sⁿ_l,D)⁻= 0.

(5.2)

Using the identity sⁿ_l,D = (sⁿ_l,D)⁺−(sⁿ_l,D)⁻, and the mobility Ml extended by zero on ]− ∞,0], thenMl(sⁿ_l,D)(sⁿ_l,D)⁻= 0 and

(5.3) − |D|ρl(pⁿl,D)sⁿl,DfP,Kⁿ (sⁿl,D)⁻ = |D|ρl(pⁿl,D)fP,Kⁿ ((sⁿl,D)⁻)² ≥ 0.

Then, we deduce from (5.1) that

ρl(pⁿl,D)|(sⁿl,D)⁻|²+ρl(pⁿ⁻¹_l,D)sⁿ⁻¹_l,D (sⁿl,D)⁻≤0,

and from the nonnegativity ofsⁿ⁻¹_l,D , we obtain (sⁿ_l,D)⁻ = 0. This implies thatsⁿ_l,D ≥0 and

0≤sⁿl,D≤sⁿl,E for alln∈[0, N−1] andE∈ Dh.

In the same way, we prove sⁿ_g,D≥0.

5.2. Estimations on the pressures. We now give a priori estimates satisfied by the solution valuespⁿ_D,D∈ Dh,{1, . . . , N}.

Proposition 1. Let(pⁿ_l,D, pⁿ_g,D)be a solution of (3.6)-(3.9). Then, there exists a constant C >0, which only depends onMα,Ω,T,p⁰_α,s⁰_α,s^I_α,fP,fI and not onDh, such that the solution of the combined scheme satisfies

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,EMα(sⁿα,D|E)

pⁿα,E−pⁿα,D

2≤C, (5.4)

and

N

X

n=1

δtkphk²_X

h ≤C.

(5.5)

Proof. We define the functionAα(pα) := ρα(pα)gα(pα)−pα,Pc(sl) := Rs_l

0 pc(z)dz and gα(pα) =Rpα

0 1

ρα(z)dz. In the following proof, we denote byCivarious real values which independent onDandn. To prove the estimate (5.4), we multiply(3.7) and (3.8) respectively by gl(pⁿ_l,D), gg(pⁿ_g,D) and adding them, then summing the resulting equation over D∈ Dhandn∈ {1,· · ·, N}. We thus get:

(5.6) E1+E2+E3= 0,

where

E1=

N

X

n=1

X

D∈D_h

|D|φD

(ρl(pⁿl,D)sⁿl,D−ρl(pⁿ⁻¹_l,D)sⁿ⁻¹_l,D )gl(pⁿl,D) +(ρg(pⁿg,D)sⁿg,D−ρg(pⁿ⁻¹_g,D)sⁿ⁻¹_g,D)gg(pⁿg,D)

,

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E2=

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,E

ρⁿl,D,EGl(sⁿl,D, sⁿl,E;δⁿD|E(pl))gl(pⁿl,D) +Gg(sⁿg,D, sⁿg,E;δD|Eⁿ (pg))gg(pⁿg,D)

,

E3=

N

X

n=1

δt X

D∈D_h

|D|

ρl(pⁿ_l,D)sⁿ_l,Df_P,Dⁿ gl(pⁿ_l,D)−ρl(pⁿ_l,D)(s^I_g,D)ⁿf_I,Dⁿ gl(pⁿ_l,D) +ρg(pⁿg,D)sⁿg,DfP,Dⁿ gg(pⁿg,D)−ρg(pⁿg,D)(s^Ig,D)ⁿfI,Dⁿ gg(pⁿg,D)

. To handle the first term of the equality (5.6), let us recall the following inequality : (5.7) ρl(pⁿ_l)sⁿ_l −ρl(pⁿ⁻¹_l )sⁿ⁻¹_l

gl(pⁿ_l) + ρg(pⁿ_g)sⁿ_g −ρg(pⁿ⁻¹_g )sⁿ⁻¹_g gg(pⁿ_g)

≥ Al(pⁿl)sⁿl − Al(pⁿ⁻¹_l )sⁿ⁻¹_l +Ag(pⁿg)sⁿg− Ag(pⁿ⁻¹g )sⁿ⁻¹g − Pc(sⁿl) +Pc(sⁿ⁻¹_l ), using the concavity property ofgα andPc, in [20] the authors prove the above inequality.

So, this yields to (5.8) E1≥ X

D∈D_h

φD|D|

s^N_l,DA(p^N_l,D)−s⁰_l,DA(p⁰_l,D) +s^N_g,DA(p^N_g,D)−s⁰_g,DA(p⁰_g,D)

− X

D∈D_h

φD|D| Pc(s^N_l,D) + X

D∈D_h

φD|D| Pc(s⁰_l,D).

Using the fact that the numerical fluxesGlandGgare conservative, we obtain by discrete integration by parts

E2=1 2

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,E

ρⁿ_l,D,EGl(sⁿ_l,D, sⁿ_l,E;δD|Eⁿ (pl))(gl(pⁿ_l,D)−gl(pⁿ_l,E)) +ρⁿg,D,EGg(sⁿg,D, sⁿg,E;δⁿD|E(pg))(gg(pⁿg,D)−gg(pⁿg,E))

, and due to the correct choice of the density of the phaseαon each interface,

ρⁿ_α,D,E(gα(pⁿ_α,D)−gα(pⁿ_α,E)) =pⁿ_α,D−pⁿ_α,E, (5.9)

we obtain E2= 1

2

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,E

Gl(sⁿl,D, sⁿl,E;δD|Eⁿ (pl))(pⁿl,D−pⁿl,E) +Gg(sⁿg,D, sⁿg,E;δD|Eⁿ (pg))(pⁿg,D−pⁿg,E) . The definition of the upwind fluxes in (3.12) implies

Gl(sⁿl,D, sⁿl,E;δⁿD|E(pl))(pⁿl,D−pⁿl,E) +Gg(sⁿg,D, sⁿg,E;δD|Eⁿ (pg))(pⁿg,D−pⁿg,E)

=Ml(sⁿ_l,D,E)(δⁿD|E(pl))²+Mg(sⁿ_g,D,E)(δⁿD|E(pg))². Then, we obtain the following equality

E2=1 2

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,E

Ml(sⁿl,D,E)(δD|Eⁿ (pl))²+Mg(sⁿg,D,E)(δⁿD|E(pg))²

. (5.10)

In order to estimateE3, using the fact that the densities are bounded and the mapgαis sublinear (a.e.|g(pα)| ≤C|pα|), we have

|E3| ≤C1 N

X

n=1

δt X

D∈D_h

|E3| ≤C1 N

X

n=1

δt X

D∈D_h

|D|(f_P,Dⁿ +f_I,Dⁿ )(2|pⁿD|+|¯pⁿ_D|+|˜pⁿ_D|).

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Hence, by the H¨older inequality, we get that

|E4| ≤C2kfP +fIk_L2(Q_T) N

X

n=1

δtkpⁿhk²_L2(Ω)

¹₂ ,

and, from the discrete Poincar´e–Friedrichs inequality [27], we get

(5.11) |E4| ≤C3

N

X

n=1

δtkpⁿ_hk²_X

h

¹₂ +C4.

The equality (5.6) with the inequalities (5.8), (5.10), (5.11) give (5.4). Then we deduce

(5.5) from (4.4) and Lemma 1.

We now state the following corollary, which is essential for the compactness and limit study.

Corollary 1. From the previous Proposition, we deduce the following estimations:

N

X

n=1

δt B(sⁿ_l,h)

2 X_h≤C, (5.12)

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,EMl,D|Eⁿ (δⁿD|E(¯p(sl)))²≤C, (5.13)

and

N

X

n=1

δt X

D∈D_h

X

E∈N(D)

ΛD,EMg,D|Eⁿ (δⁿD|E(˜p(sl)))²≤C.

(5.14)

Proof. The prove of the estimates (5.12), (5.13) and (5.14) are a direct consequence of the inequality (4.5), (4.6), (4.7), the Lemma 1 and the Proposition 1.

5.3. Existence of the finite volume scheme.

Proposition 2. The problem(3.7)-(3.8)admits at least one solution(pⁿ_l,D, pⁿ_g,D)(D,n)∈D_h×{1,···,N}. The proof is based on a technical assertion to characterize the zeros of a vector field which stated and proved in [13]. This method is used in [3] and [26], so it is easy to adopt their proof in our case, thus we omit it.

6. Compactness properties

In this section we derive estimates on differences of space and time translates of the functionUα,δt,h=φρα(pα,δt,h)sα,δt,hwhich imply that the sequenceφρα(pα,δt,h)sα,δt,his relatively compact inL¹(QT).

The following important relation betweenUδt,hand ˜Uδt,h(see definition 2) is valid:

Lemma 5.

Relation betweenUα,δt,h andU˜α,δt,h

.There holds

Uα,δt,h−U˜α,δt,h

L¹(Q_T)

−→0ash→0.

Proof.

Ul,δt,h−U˜l,δt,h

L¹(Q_T)= Z

Q_T

Ul,δt,h(t, x)−U˜l,δt,h(t, x) dxdt

≤ Z

Q_T

sl,δt,h(t, x)

ρl(pl,δt,h(t, x))−ρl(˜pl,δt,h(t, x)) dxdt +

Z

Q_T

ρl(˜pl,δt,h(t, x))

sl,δt,h(t, x)−˜sl,δt,h(t, x) dxdt

≤T1+T2, whereT1andT2 defined as follows

(6.1) T1=ρM

Z

Q_T

|sl,δt,h(t, x)−s˜l,δt,h(t, x)|dxdt,

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(6.2) T2= Z

Q_T

|ρl(pl,δt,h(t, x))−ρl(˜pl,δt,h(t, x))|dxdt.

To handle the term on saturationT1, we use the fact thatB⁻¹ is a H¨older function, then T1≤ρMC

Z

Q_T

|B(sl,δt,h(t, x))− B(˜sl,δt,h(t, x))|^θdxdt, and by application of the Cauchy-Schwarz inequality, we deduce

T1≤CZ

Q_T

|B(sl,δt,h(t, x))− B(˜sl,δt,h(t, x))|dxdtθ

≤C (T⁰1)^θ, whereT⁰1defined as follows

T₁⁰= Z

Q_T

|B(sl,δt,h(t, x))− B(˜sl,δt,h(t, x))|dxdt.

We have T1⁰ =

N

X

n=1

δt X

K∈T_h

X

σ_D∈E_K

Z

K∩D

|B(sl,δt,h(t, x))− B(˜sl,δt,h(t, x))|dx

=

N

X

n=1

δt X

K∈T_h

X

σ_D∈E_K

Z

K∩D

|B(sl,δt,h(t, x))− B(sl,δt,h(t, QD))|dx

=

N

X

n=1

δt X

K∈T_h

X

σ_D∈E_K

Z

K∩D

|∇B(sl,δt,h(t, x))·(x−QD)|dx

≤

N

X

n=1

δt X

K∈T_h

X

σ_D∈E_K

∇B(sl,δt,h)|K

diam(D)|K∩D|

≤h

N

X

n=1

δt X

K∈T_h

∇B(sl,δt,h)|K

|K|

≤h

N

X

n=1

δtkB(sl,δt,h)k²_X

h+C

!

≤Ch, (6.3)

where we have used the definitions of sl,δt,h and ˜sl,δt,h, the Cauchy-Schwarz inequality and the estimate (5.12), thus

(6.4) T1≤Ch^θ.

To treatT2, we use the fact that the mapρ⁰_lis bounded and the relationship between the gas pressure and the global pressure, namely : pl =p−p¯defined in (2.5), then we have

T2≤max

R

|ρ⁰l| Z

Q_T

|pl,δt,h(t, x)−p˜l,δt,h(t, x)|dxdt

≤max

R

|ρ⁰l| Z

Q_T

|pδt,h(t, x)−p˜δt,h(t, x)|dxdt + max

R

|ρ⁰_l| Z

Q_T

|¯p(sl,δt,h(t, x))−p(˜¯sl,δt,h(t, x))|dxdt, (6.5)

furthermore one can easily show that ¯pis aC¹([0,1];R), it follows, there exists a positive constantC >0 such that

T2≤C Z

Q_T

|pδt,h(t, x)−p˜δt,h(t, x)|dxdt+C Z

Q_T

|sl,δt,h(t, x)−s˜l,δt,h(t, x)|dxdt.

The last term in the previous inequality is proportional toT1, and consequently it remains to show that the term on the global pressure is small with h. In fact, follows (6.3), one