Hybrid finite volume scheme for two-phase flow in porous media

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Hybrid finite volume scheme for two-phase flow in porous media

Konstantin Brenner

To cite this version:

Konstantin Brenner. Hybrid finite volume scheme for two-phase flow in porous media. 2011. �hal-

00680686�

Hybrid finite volume scheme for two-phase flow in porous media

Konstantin Brenner

^∗

March 6, 2012

Abstract We apply a finite volume method on general meshes for the discretization of an incompressible and immiscible two-phase flow in porous media. The problem is con- sidered in the global pressure formulation. Mathematically, it amounts to solve an elliptic equation for the global pressure, with an anisotropic and heterogeneous permeability ten- sor coupled to a parabolic degenerate convection-diffusion equation for a saturation, again with the same permeability tensor. Extending ideas which we had previously developed for the numerical solution of a degenerate parabolic convection-reaction-diffusion equation we discretize the diffusion terms by means of a hybrid finite volume scheme, while we use a Godunov scheme for the non monotone convection flux. We prove the convergence of the numerical scheme in arbitrary space dimension and we present results of a number of numerical tests in space dimension two.

This work was supported by the GdR MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN), France.

1 Introduction

The two-phase flow in porous media is an important problem arising in many engineering and scientific context as e.g the secondary oil recovery, the basin modeling and the con- taminated sites remediation [9], [7], [13]. In this paper we consider the simplified dead-oil model, that is to say, we assume that there are only two incompressible and immiscible fluids. The problem is described by the mass conservation of each phase together with Darcy-Muskat law

ω∂

t

s

o

− ∇ ·

K kr

o

(s

o

) µ

o

( ∇ p

o

− ρ

o

g)

= k

o

(1.1)

ω∂

_t

s

_w

− ∇ ·

K kr

_w

(s

_w

) µ

w

( ∇ p

_w

− ρ

_w

g)

= k

_w

(1.2)

where ω is the porosity, K absolute permeability, s

w

stands for the wetting phase (water) saturation and s

o

for the non-wetting phase (oil) saturation. The relative permeabilities kr

o

(s

o

) and kr

w

(s

w

) model the effects of coexistence of both phases in porous medium;

ρ

α

and µ

a

denote the densities and the viscosity of phase α ∈ { o, w } , g is the gravity vector. It is also assumed that the porous medium is saturated and that two phases are immiscible

s

o

+ s

w

= 1. (1.3)

∗Laboratoire de Math´ematiques, Universit´e de Paris-Sud 11, F-91405 Orsay Cedex, France

The phase pressures are related by a capillary pressure law

p

o

+ p

w

= π(s). (1.4)

In view of (1.3) we can take as unknown only one saturation, for instance s = s

o

. We denote by λ

_o

(s) = kr

o

(s)

µ

o

and λ

_w

(s) = kr

w

(1 − s) µ

w

the relative mobilities, which are such that λ

o

(0) = λ

w

(1) = 0. In this paper we transform the system (1.1)-(1.4) into the so-called global pressure or fractional flow formulation. The global pressure p which is defined by

p = p

w

+ Z

s

0

λ

o

(a)

λ

_o

(a) + λ

_w

(a) π(a)da

was first introduced in [5] and [13]. We also define the fractional flow f , total mobility λ and capillary diffusion ϕ

λ(s) = λ

o

(s) + λ

w

(s), f (s) = λ

_o

(s)

λ(s) , ϕ(s) = Z

s

0

λ

w

(τ )f (τ)π

^′

(τ )dτ.

The system (1.1)-(1.4) is equivalent to

−∇ · K (λ(s) ∇ p − ξ(s)g) = k

w

+ k

o

in Ω × (0, T ), (1.5) q = − K (λ(s) ∇ p − ξ(s)g) in Ω × (0, T ), (1.6) ω ∂s

∂t + ∇ · (qf (s) + γ(s)Kg) − ∇ · (K ∇ ϕ(s)) = k

o

in Ω × (0, T ), (1.7) where

γ(s) = (ρ

o

− ρ

w

) λ

o

(s)λ

w

(s)

λ

o

(s) + λ

w

(s) and ξ(s) = (λ

o

(s)ρ

o

+ λ

w

(s)ρ

w

) .

The usual assumption λ

o

+ λ

w

≥ λ > 0 implies that the first equation is uniformly elliptic in p, whereas the second one is parabolic degenerate with respect to the saturation s.

The system (1.5)-(1.7) has a remarkable mathematical structure, which permits to obtain a number of energy estimates. Many numerical methods were proposed for solving two- phase problem, such as finite element methods (see e.g. [21], [20], [13], [6], [14], [15], [16], [17]), discontinuous Galerkin (see e.g. [8], [19]) and finite volume methods (see e.g.

[1], [23], [29], [25], [2], [30], [4]). The convergence results for the different finite elements schemes were obtained in [6], [14] and [17]. For finite volume schemes convergence was shown in [1], [29] and [25], where the initial formulation (1.1)-(1.4) was considered. We also would like to mention some interesting convergence results which were obtained in the case of miscible displacement [31], [33], [12].

The heterogeneity and anisotropy of porous media is a numerical challenge even when studying the elliptic problem derived from Darcy’s law for one-phase problem. Many schemes was proposed and analyzed in last decades for its discretization. See [27] and [24]

for more references and for the detailed description and comparison of those numerical

methods. In this paper we propose an implicit fully coupled Hybrid finite volume scheme

for the system (1.5)-(1.7). The discretization of the diffusion terms is based upon a hy-

brid finite volume method [22], which allows the tensor K to be anisotropic and highly

variable in space. Remark that heterogeneity also affects the first order terms K(x)gξ( · )

and qf ( · ) + K(x)gγ( · ), which may be discontinuous with respect to the space variable

x; in that case they require a suitable treatment see e.g. [35], [34], [26]. We apply the

Godunov scheme proposed in [28], which seems to be a natural choice, since the hybrid

(interface) unknowns are used.

Assumptions on the data: ( H

¹

) ϕ ∈ C( R ), ϕ(0) = 0, is a strictly increasing piece- wise continuously differentiable Lipschitz continuous function with a Lipschitz constant L

_ϕ

. We assume that the function ϕ

⁻¹

is H¨older continuous, namely that there exists H

ϕ

> 0 and α ∈ (0, 1] such that | s

1

− s

2

| ≤ H

ϕ

| ϕ(s

1

) − ϕ(s

2

) |

^α

. It is also such that ϕ(s) = s for all s < 0 and ϕ(s) − ϕ(1) = s − 1 for all s > 1;

( H

2

) The functions λ, ξ, γ, f ∈ C([0, 1]) are Lipschitz continuous; for f = λ, ξ, γ, f we denote by L

f

the corresponding Lipschitz constant.

( H

2a

) λ is such that 0 < λ ≤ λ(s);

( H

2b

) ξ and γ are convex functions, moreover γ is such that γ(0) = γ(1) = 0; g is a constant vector from R

^d

;

( H

2c

) f is a nondecreasing function and it satisfies f(0) = 0, f (1) = 1;

( H

3

) The functions λ, ξ, γ, f are constant outside of (0, 1), i.e. for f = λ, ξ, γ, f we assume that f(s) = f(0) for all s < 0 and f(s) = f(1) for all s > 1.

( H

5

) s

0

∈ L

^∞

(Ω); ω ∈ L

^∞

(Ω) and such that 0 < ω ≤ ω(x) ≤ ω for a.e. in x ∈ Ω;

( H

⁶

) k

o

, k

w

∈ L

^∞

(0, T ; L

²

(Ω)) and such that k

o

+ k

w

≥ 0 a.e. in Q

T

.

Remark 1.1 The physical range of values for the saturation is [0, 1], however we extend the definition of all nonlinear functions in (1.5)-(1.7) outside of [0, 1], since the numerical scheme which we study does not preserve neither the maximum principle, nor the posi- tivity of s (nor the bound s ≤ 1). It is worth noting that one can replace ( H

³

) by the assumption that the functions λ, ξ, γ, f are in L

^∞

( R ). In turn the assumption ( H

1

) can also weakened, namely the capillary diffusion ϕ can be extended by an arbitrary strictly increasing function, such that ϕ

⁻¹

is Lipschitz continuous on R \ (0, 1).

Assumptions on the geometry:

( H

^4a

) Ω is a polyhedral open bounded connected subset of R

^d

, with d ∈ N

^∗

, and ∂Ω = Ω \ Ω its boundary.

( H

4b

) K is a piecewise constant function from Ω to M

d

( R ), where M

d

( R ) denotes the set of real d × d matrices. More precisely we assume that there exist a finite family (Ω

i

)

i∈{1,...,I}

of open connected polyhedral in R

^d

, such that Ω = S

i∈{1,...,I}

Ω

i

, Ω

i

T

Ω

j

= ∅ if i 6 = j and such that K(x) |

^Ωi

= K

i

∈ M

^d

( R ). By Γ

i,j

we denote the interface between the sub-domains i, j, Γ

i,j

= Ω

i

T

Ω

j

. We suppose that there exist two positive constants K and K such that for all i ∈ { 1, . . . , I } the eigenvalues of the symmetric positive definite K

i

are included in [K, K].

Remark 1.2 For the sake of simplicity we have assumed that the heterogeneity of the medium is only expressed through the x-dependence of the absolute permeability tensor K = K(x). However it is very simple to extend the analysis to the case where λ, ξ, γ and f depend on the rock type. On the other hand if we suppose that ϕ is discontinuous in space, this may lead to significant difficulties. The analysis of the case where the capillary pressure field (and also ϕ) is discontinuous was carried out in [11] and [10]. It is also worth noting that the partitioning of Ω introduced in H

4b

is only used in order to provide a control on the gravity terms (see Remark 3.2) below). In the case that the gravity effects are neglected, one can consider a fully heterogenous permeability field K.

Let T > 0, we consider the system (1.5)-(1.7) in the domain Q

T

= Ω × (0, T ) together with the initial condition

s( · , 0) = s

0

in Ω (1.8)

and homogeneous Dirichlet boundary conditions

p = 0 and s = 0 on ∂Ω × (0, T ). (1.9)

We now present the definition of a weak solution of the problem (1.5)-(1.9).

Definition 1.1 (Weak solution) A function pair (s, p) is a weak solution of the problem (1.5)-(1.6) if

(i) s ∈ L

^∞

(0, T ; L

²

(Ω));

(ii) ϕ(s) ∈ L

²

(0, T ; H

₀¹

(Ω));

(iii) p ∈ L

^∞

(0, T ; H

₀¹

(Ω));

(iv) for all ψ, χ ∈ L

²

(0, T ; H

₀¹

(Ω)) with ϕ

t

∈ L

^∞

(Q

T

), ϕ( · , T ) = 0, s and p satisfy the integral equalities

− Z

T

0

Z

Ω

sψ

_t

dxdt − Z

Ω

s

₀

ψ( · , 0) dx − Z

T

0

Z

Ω

K(f(s)q + γ(s)g) · ∇ ψ dxdt +

Z

T 0

Z

Ω

K ∇ ϕ(s) · ∇ ψ dxdt = Z

T

0

Z

Ω

k

o

ψ dxdt and

− Z

T

0

Z

Ω

q · ∇ χ dxdt = Z

T

0

Z

Ω

(k

w

+ k

o

)χ dxdt, where q is given by (1.6).

In Section 2 we present the finite volume scheme and some technical lemmas. In Section 3 we provide the a priori estimates and we prove an existence of a discrete solution. We prove the estimates on space and time translates of a discrete solution in Section 4, those estimates allow to establish a strong convergence property for a subsequence of discrete saturation. The convergence result is shown in Section 5. Finally in Section 6 we present a number of numerical results obtained on different two-dimensional meshes.

2 The finite volume scheme

2.1 The main definitions

In order to describe the numerical scheme we introduce below some notations related to the space and time discretization, which follows [22].

Definition 2.1 (Discretization of Ω) Let Ω be a polyhedral open bounded connected subset of R

^d

, with d ∈ N

^∗

, ∂Ω = Ω \ Ω its boundary, and (Ω

_i

)

i∈{1,...,I}

it’s partition in the sense of ( H

⁴

). A discretization of Ω, denoted by D , is defined as the triplet D = ( M , E , P ), where:

1. M is a finite family of non empty connex open disjoint subsets of Ω (the ”control volumes”) such that Ω = S

K∈M

K. For any K ∈ M , let ∂K = K \ K be the boundary of K; we define m(K) > 0 as the measure of K and h

K

as the diameter of K. We also assume that the mesh resolve the structure of the medium, i.e. for all K ∈ M there exist i ∈ { 1, . . . , I } such that K ⊂ Ω

i

.

2. E is a finite family of disjoint subsets of Ω (the ”edges” of the mesh), such that, for all σ ∈ E , σ is a non empty open subset of a hyperplane of R

^d

, whose (d − 1)-dimensional measure m(σ) is strictly positive. We also assume that, for all K ∈ M , there exists a subset E

K

of E such that ∂K = S

σ∈EK

σ. For each σ ∈ E , we set M

σ

= { K ∈ M| σ ∈ E

K

} .

We then assume that, for all σ ∈ E , either M

σ

has exactly one element and then σ ∈ ∂Ω

(the set of these interfaces called boundary interfaces, is denoted by E

ext

) or M

σ

has exactly two elements (the set of these interfaces called interior interfaces, is denoted by E

^int

). For all σ ∈ E , we denote by x

σ

the barycenter of σ. For all K ∈ M and σ ∈ E

^K

, we denote by n

K,σ

the outward normal unit vector.

3. P is a family of points of Ω indexed by M , denoted by P = (x

_K

)

_K∈M

, such that for all K ∈ M , x

K

∈ K ; moreover K is assumed to be x

K

-star-shaped, which means that for all x ∈ K, there holds [x

K

, x] ∈ K. Denoting by d

K,σ

the Euclidean distance between x

_K

and the hyperplane containing σ, one assumes that d

_K,σ

> 0. We denote by D

_K,σ

the cone of vertex x

K

and basis σ.

Next we introduce some extra notations related to the mesh. The size of the discretization D is defined by

h

D

= sup

K∈M

diam(K); (2.1)

moreover we define

θ

D

= max( max

σ∈Eint,{K,L}=Mσ

d

K,σ

d

L,σ

, max

K∈Mσ,σ∈EK

h

K

d

K,σ

). (2.2)

Imposing a uniform bound on θ

D

forces the mesh to be sufficiently regular. Next, we define several discrete spaces, which are going to be used in the sequel.

Definition 2.2 (The hybrid space X

_D

(Ω)) Let D = ( M , E , P ) be a discretization of Ω. We define

X

D

= { ((v

K

)

K∈M

, (v

σ

)

σ∈E

), v

K

∈ R , v

σ

∈ R } ,

X

D,0

= { v ∈ X

D

such that (v

σ

)

σ∈Eext

= 0 } . (2.3) The space X

D

is equipped with the semi-norm | · |

^XD

defined by

| v |

²XD

= X

K∈M

| v |

²XD,K

, where | v |

²XD,K

= X

σ∈EK

m(σ) d

K,σ

(v

σ

− v

K

)

²

for all v ∈ X

D

. (2.4) Note that | · |

XD

is a norm on the space X

D,0

.

Moreover, for each function ψ = ψ(x) regular enough we define its projection P

D

ψ ∈ X

D

on the space X

D

in following way

(P

D

ψ)

K

= ψ(x

K

) for all K ∈ M , (P

D

ψ)

σ

= ψ (x

σ

) for all σ ∈ E .

Definition 2.3 (The discrete flux space Q

^D

(Ω)) Let D = ( M , E , P ) be a discretiza- tion of Ω. We define

Q

D

= { (q

K,σ

)

K∈M,σ∈EK

, q

K,σ

∈ R } . (2.5) Next we introduce the time discretization.

Definition 2.4 (Time discretization) We divide the time interval (0, T ) into N equal time steps of length δt = T /N , where δt is the uniform time step defined by δt = t

_n

− t

_n−1

. Taking into account the time discretization leads us to define of the following discrete spaces

X

D,δt

= X

_D^N

= { (v

ⁿ

)

n∈{1,...,N}

, v

ⁿ

∈ X

D

} and

X

D,δt,0

= X

_D,0^N

= { (v

ⁿ

)

n∈{1,...,N}

, v

ⁿ

∈ X

D,0

} ; moreover we define the following semi-norm on X

D,δt

| v |

²XD,δt

=

N

X

n=1

δt | v |

²XD

. (2.6)

2.2 The numerical scheme

2.2.1 The discrete problem

In this section we present the fully implicit finite volume scheme for the problem (1.5)- (1.9). Let us introduce the discrete saturation (s

ⁿ_K

)

_K∈M

, (s

ⁿ_σ

)

_σ∈E

n∈{1,...,N}

∈ X

D,δt

and the discrete global pressure (p

ⁿ_K

)

_K∈M

, (p

ⁿ_σ

)

_σ∈E

n∈{1,...,N}

∈ X

D,δt

, which are the main discrete unknowns. Moreover let f denote λ, ξ, γ or f we introduce the following notation f

ⁿ_K

= f(s

ⁿ_K

) and f

ⁿ_σ

= f(s

ⁿ_σ

) for all K ∈ M , σ ∈ E and n ∈ { 1, . . . , N } . Let k

_i,Kⁿ

denote the mean value of the source term k

i

(x, t) over a cell K × (t

n−1

, t

n

), i.e.

k

ⁿ_i,K

= 1 m(K )δt

Z

tn

tn−1

Z

K

k

i

(x, t) dxdt with i ∈ { w, n } . (2.7) We denote the porous volume of the element K by ω(K),

ω(K) = Z

K

ω(x) dx.

Next, let Q

ⁿ_K,σ

be an approximation of the total flux through the interface σ Q

ⁿ_K,σ

≈ 1

δt Z

tn

tn−1

Z

σ

K(λ(s) ∇ p − ξ(s)g) · n

K,σ

dνdt (2.8) and let F

_K,σⁿ

be an approximation of the non-wetting phase flux

F

_K,σⁿ

≈ 1 δt

Z

tn

tn−1

Z

σ

(qf (s) + γ (s)Kg − K ∇ ϕ(s)) · n

K,σ

dνdt. (2.9) The numerical fluxes Q

ⁿ_K,σ

and F

_K,σⁿ

have to be constructed as functions of the discrete unknowns. Using the notations (2.7) and (2.8) we discretize the equation (1.5) by

X

σ∈EK

Q

ⁿ_K,σ

= m(K)(k

ⁿ_w,K

+ k

ⁿ_o,K

) for all K ∈ M . (2.10) We also prescribe the continuity of the fluxes

Q

ⁿ_K,σ

+ Q

ⁿ_L,σ

= 0 for all σ ∈ E

^int

with { K, L } = M

^σ

. (2.11) Similarly, the equation (1.7) is discretized by

ω(K) s

ⁿ_K

− s

ⁿ⁻¹_K

δt + X

σ∈EK

F

_K,σⁿ

= m(K)k

_o,Kⁿ

for all K ∈ M , (2.12) and

F

_K,σⁿ

+ F

_L,σⁿ

= 0 for all σ ∈ E

int

with { K, L } = M

σ

. (2.13) The discrete equations (2.16)-(2.13) have to be prescribed at each time step n ∈ { 1, . . . , N } . We prescribe the initial and the boundary conditions for the numerical scheme by setting

s

⁰_K

= 1 m(K)

Z

K

s

0

(x) dx for all K ∈ M (2.14) and

s

ⁿ_σ

= p

ⁿ_σ

= 0 for all σ ∈ E

^ext

. (2.15)

Remark that opposite to the classical two-point flux approximation, the discrete fluxes

Q

ⁿ_K,σ

and F

_K,σⁿ

(which still remain to be constructed) are not a priori continuous across

the element’s interfaces, so that the continuity is prescribed in the scheme by (2.11) and

(2.13).

2.2.2 The discrete weak formulation

Following the ideas of [22] we write the scheme in the variational form.

For each n ∈ { 1, . . . , N } find s

ⁿ

∈ X

D,0

and p

ⁿ

∈ X

D,0

such that for all v

ⁿ

, w

ⁿ

∈ X

D,0

: X

K∈M

X

σ∈EK

(v

_Kⁿ

− v

_σⁿ

)Q

ⁿ_K,σ

= X

K∈M

m(K)v

_Kⁿ

(k

ⁿ_w,K

+ k

ⁿ_o,K

), (2.16) X

K∈M

ω(K )w

_Kⁿ

s

ⁿ_K

− s

ⁿ⁻¹_K

δt + X

K∈M

X

σ∈E_K

(w

ⁿ_K

− w

ⁿ_σ

)F

_K,σⁿ

= X

K∈M

m(K )w

_Kⁿ

k

ⁿ_o,K

, (2.17) s

⁰_K

= 1

m(K) Z

K

s

0

(x) dx. (2.18) In order to complete the scheme we have to define the numerical fluxes Q

ⁿ_K,σ

and F

_K,σⁿ

. Let K

K

denote the mean value of K(x) over a cell K,

K

K

= 1 m(K)

Z

K

K(x) dx (2.19)

and let

g

K,σ

= m(σ)K

K

g · n

K,σ

. (2.20) Note that g

_K,σ

satisfies

X

σ∈E_K

g

K,σ

= 0 for all K ∈ M , (2.21)

but not necessarily

g

K,σ

+ g

L,σ

= 0 with { K, L } = M

^σ

.

The above equality remains true for the interfaces which are ”interior” with respect to some sub-domain Ω

i

, that is to say

g

K,σ

+ g

L,σ

= 0 with { K, L } = M

σ

for all σ 6∈ Γ

i,j

, (2.22) for any i, j. Next, we define Q

ⁿ_K,σ

and F

_K,σⁿ

by

Q

ⁿ_K,σ

= λ

ⁿ_K

F

K,σ

(p

ⁿ

) + G (ξ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

), (2.23) F

_K,σⁿ

= G (Q

ⁿ_K,σ

f( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

) + F

K,σ

(ϕ(s

ⁿ

)), (2.24) where λ

ⁿ_K

= λ

ⁿ_K

for all K ∈ M and n ∈ { 1, . . . , N } , in general. The terms F

^K,σ

( · ) correspond to the diffusive fluxes which are discretized using the SUSHI scheme (Section 2.2.4). The terms G ( · ) stand for the discretization of the convective fluxes, using the Godunov scheme (Section 2.2.3 below, see also [28, p. 3]).

2.2.3 The Godunov scheme and the convection term

Let a, b ∈ R and f ∈ L

^∞

( R ) we define the Godunov flux by G (f; a, b) =

min

_s∈[a,b]

f (s) if a ≤ b,

max

s∈[b,a]

f(s) if b ≤ a. (2.25)

Since it is often useful to write the discrete flux in more explicit form we define S (f ; a, b) =

argmin

_s∈[a,b]

f (s) if a ≤ b,

argmax

_s∈[b,a]

f(s) if b ≤ a,

and we introduce the following notations

ξ

_K,σⁿ

= ξ( S (ξ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

)),

f

_K,σⁿ

= f( S (Q

ⁿ_K,σ

f( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

)), γ

_K,σⁿ

= γ( S (Q

ⁿ_K,σ

f ( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

)).

(2.26)

Using the notations (2.26) we can write the discrete fluxes in the form

Q

ⁿ_K,σ

= λ

ⁿ_K

F

^K,σ

(p

ⁿ

) + ξ

_K,σⁿ

g

K,σ

(2.27) and

F

_K,σⁿ

= Q

ⁿ_K,σ

f

_K,σⁿ

+ γ

_K,σⁿ

g

K,σ

+ F

^K,σ

(ϕ(s

ⁿ

)). (2.28) 2.2.4 The discrete gradient and the diffusion term

In this section we recall a construction of the discrete gradient and of the numerical flux F

^K,σ

( · ) proposed in [22]. Let u ∈ X

D

, for all K ∈ M and σ ∈ E

^K

we define

∇

K,σ

u = ∇

K

u + R

K,σ

u · n

K,σ

, (2.29) where

∇

K

u = 1 m(K)

X

σ∈EK

m(σ)(u

_σ

− u

_K

)n

_K,σ

(2.30) and

R

K,σ

u =

√ d d

K,σ

(u

σ

− u

K

− ∇

^K

u · (x

σ

− x

K

)). (2.31) Note that the stabilizing term R

K,σ

is a second order error term, which vanishes for piecewise linear functions. We define the discrete gradient ∇

D

u as the piecewise constant function equal to ∇

K,σ

u in the cone D

K,σ

with vertex x

K

and basis σ

∇

^D

u(x) |

^x∈DK,σ

= ∇

^K,σ

u.

Let u = (u

ⁿ

)

n∈{1,...,N}

∈ X

D,δt

, taking into account the time discretization, we define the discrete gradient ∇

D,δt

u(x, t) by

∇

^D,δt

u(x, t) |

^t∈(tn−1,tn]

= ∇

^D

u

ⁿ

(x), (2.32) for all x ∈ Ω and all n ∈ { 1, . . . , N } . For an arbitrary u ∈ X

D

the numerical flux F

K,σ

(u) can be defined through the following discrete integration by parts formula

X

σ∈EK

(v

K

− v

σ

) F

^K,σ

(u) = X

σ∈EK

m(D

K,σ

)K

K

∇

^K,σ

u · ∇

^K,σ

v for all v ∈ X

D

, (2.33) which in particular implies that

X

K∈M

X

σ∈EK

(v

K

− v

σ

) F

^K,σ

(u) = Z

Ω

K ∇

^D

u · ∇

^D

v dx for all v ∈ X

D

. (2.34)

The explicit form of F

K,σ

can be obtained by setting v

K

− v

σ

= 1 and v

K

− v

σ^′

= 0 for

all σ

^′

6 = σ. We refer to [22] for more details on construction of F

K,σ

and its practical

implementation. Next we state without proof three results from [22].

Lemma 2.1 (Strong consistency) Let D be a discretization of Ω in sense of Definition 2.1, moreover let θ ≥ θ

D

be given. Then for all ψ ∈ C

²

(Ω), there exist a positive constant C only depending on d, θ and ϕ such that

|∇

^K

P

D

ψ − ∇ ψ(x) | ≤ Ch for all x ∈ K (2.35) and also

k∇

^D

P

D

ψ − ∇ ψ k

(L^∞(Ω))^d

≤ Ch. (2.36) The following lemma, which is the slightly modified version of [22, Lemma 4.1] shows the equivalence between the semi-norm in X

D

and the L

²

-norm of the discrete gradient.

Lemma 2.2 Let D be a discretization of Ω and let θ > θ

D

be given. Then there exists m > 0 and M > 0 only depending on θ and d such that

m | v |

XD,K

≤ k∇

D

v k

L²(K)

≤ M | v |

XD,K

for all K ∈ M for all v ∈ X

D

.

Lemma 2.3 (Discrete Poincar´ e inequality) There exists a positive constant C, in- dependent of the mesh size h

D

such that

k Π

D,δt

u k

L²(Ω)

≤ C | u |

XD

for all u ∈ X

D

. (2.37)

Proof: The result follows from Lemma 5.3 of [22].

The direct consequence of (2.33) and Lemma 2.2 is the lemma below.

Lemma 2.4 Let u, v be an arbitrary couple from X

_D

, then there exist two positive con- stants C

1

, C

2

independent of the mesh size such that

X

σ∈E_K

(v

K

− v

σ

) F

K,σ

(u)

≤ C

1

| u |

XD,K

| v |

XD,K

(2.38)

and X

σ∈E_K

(u

K

− u

σ

) F

K,σ

(u) ≥ C

2

| u |

²XD,K

(2.39) Lemma 2.4 implies the following useful technical result.

Lemma 2.5 Let D = ( M , E , P ) be a discretization of Ω, let q ∈ Q

D

and u ∈ X

D

. Then

X

K∈M

X

σ∈EK

q

_K,σ

F

K,σ

(u)

≤ C

₁

| u |

X^D

X

K∈M

X

σ∈EK

m(σ) d

K,σ

q

_K,σ²

!

¹₂

. (2.40)

Proof: Let K be an element of M , setting v

K

= 0 and v

σ

= f

K,σ

in (2.38) we obtain

X

σ∈E_K

q

K,σ

F

K,σ

(u)

≤ | u |

XD,K

X

σ∈E_K

m(σ) d

K,σ

q

_K,σ²

!

¹₂

. Proceeding the same way for all K ∈ M we get

X

K∈M

X

σ∈EK

q

K,σ

F

^K,σ

(u)

≤ X

K∈M

X

σ∈EK

q

K,σ

F

^K,σ

(u)

≤ C

1

X

K∈M

| u |

^XD,K

X

σ∈EK

m(σ) d

_K,σ

q

_K,σ²

!

¹₂

.

We use the Cauchy-Schwarz inequality to complete the proof.

Lemma 2.6 Let D = ( M , E , P ) be a discretization of Ω, let v ∈ X

D

then there exists a positive constant C independent of the mesh size such that for all n ∈ { 1 . . . N }

X

K∈M

X

σ∈EK

(v

K

− v

σ

) G (Q

ⁿ_K,σ

f ( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

)

≤ C | v |

XD

( | p |

XD

+ 1).

Proof: In view of (2.24), (2.27) and (2.28) we have that X

K∈M

X

σ∈E_K

(v

K

− v

σ

) G (Q

ⁿ_K,σ

f ( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

) = T

1

+ T

2

+ T

3

, where

T

1

= X

K∈M

X

σ∈EK

(v

K

− v

σ

)γ

_K,σⁿ

g

K,σ

T

2

= X

K∈M

X

σ∈EK

(v

K

− v

σ

)ξ

_K,σⁿ

f

_K,σⁿ

g

K,σ

T

₃

= X

K∈M

X

σ∈EK

(v

_K

− v

_σ

)λ

ⁿ_K

F

K,σ

(p

ⁿ

)f

_K,σⁿ

In view of (2.20)

T

1

= X

K∈M

X

σ∈EK

m(σ)(v

K

− v

σ

)γ

_K,σⁿ

K

K

g · n

K,σ

Remark that for all K ∈ M and σ ∈ E

^K

one has m(σ)d

K,σ

= m(D

K,σ

)d, where d is the space dimension. Using Cauchy-Schwarz inequality

(T

1

)

²

≤ X

K∈M

X

σ∈EK

m(σ) d

K,σ

(v

K

− v

σ

)

²

X

K∈M

X

σ∈EK

m(σ)d

K,σ

(γ

_K,σⁿ

)

²

| g |

²

K

²

≤ K

²

| g |

²

m(Ω)d k γ k

²L^∞((0,1))

| v |

²XD

.

(2.41)

In the same way

(T

2

)

²

≤ K

²

| g |

²

m(Ω)d k ξ k

²L^∞((0,1))

k f k

²L^∞((0,1))

| v |

²XD

. (2.42) Applying Lemma 2.5 to the term T

₃

we obtain

(T

3

)

²

≤ | p |

²XD

X

K∈M

X

σ∈E_K

m(σ) d

K,σ

λ

ⁿ_K

(v

K

− v

σ

)f

_K,σⁿ

2

≤ k λ k

²L^∞((0,1))

k f k

²L^∞((0,1))

| p |

²XD

| v |

²XD

.

(2.43)

Gathering (2.41)-(2.43) we complete the proof.

Finally we present the technical lemma which is used in the proof of the a priori estimates.

Lemma 2.7 Let ϕ(s) satisfying the hypothesis ( H

¹

) and let the function Φ be defined by Φ(s) =

Z

s 0

ϕ(τ )dτ. (2.44)

Then,

1 2L

ϕ

(ϕ(s))

²

≤ Φ(s) ≤ L

ϕ

2 s

²

. (2.45)

Proof: The function ϕ is invertible, then setting ξ = ϕ(τ) in (2.44) gives Φ(s) =

Z

ϕ(s) 0

ξdξ

ϕ

^′

(ϕ

⁻¹

(ξ)) ≥ 1 L

ϕ

Z

ϕ(s) 0

ξdξ = 1 2L

ϕ

(ϕ(s))

²

. On the other hand

Φ(s) ≤ L

ϕ

Z

s 0

τ dτ = L

ϕ

2 s

²

.

3 A priory estimates and existence of discrete solu- tion

3.1 A priori estimates

Definition 3.1 (Approximate solution) Let D be a discretization of Ω, N ∈ N

^∗

and δt = T /N > 0. We say that the sequence (s

_D,δt

, p

_D,δt

) = (s

ⁿ

, p

ⁿ

)

n∈{1,...,N}

∈ (X

_D,δt,0

) is an approximate solution of the problem (1.5)-(1.9) if for all n ∈ { 1, . . . , N } , (s

ⁿ

, p

ⁿ

) satisfies (2.16)-(2.18). We also denote by s

D,δt

and p

D,δt

the function pair defined by

s

D,δt

(x, 0) = s

⁰_K

for all x ∈ K, (3.1)

s

D,δt

(x, t) = s

ⁿ_K

for all (x, t) ∈ K × (t

n−1

, t

n

], (3.2) s

D,δt

(x, t) = s

ⁿ_σ

for all (x, t) ∈ σ × (t

n−1

, t

n

] (3.3) and

p

D,δt

(x, t) = p

ⁿ_K

for all (x, t) ∈ K × (t

n−1

, t

n

], (3.4) p

D,δt

(x, t) = p

ⁿ_σ

for all (x, t) ∈ σ × (t

n−1

, t

n

]. (3.5) Remark 3.1 In general the capillary diffusion ϕ(s

D,δt

) is more regular that s

D,δt

and it is easer to work with the function ϕ

D,δt

, thus the element ϕ

D,δt

= ϕ(s

D,δt

) of X

D,δt,0

can be considered as a set of primary discret unknowns. We also define the function ϕ

D,δt

(x, t)

ϕ

_D,δt

(x, t) = ϕ(s

_D,δt

(x, t)) for all (x, t) ∈ Q

_T

. (3.6) Theorem 3.1 (A priori estimate) Let (s

D,δt

, p

D,δt

) be a solution of the discrete prob- lem (2.16)-(2.18). Then there exists a positive constant C independent of h

D

and δt such that

k∇

D,δt

p

_D,δt

k

L^∞(0,T;L²(Ω))

+ k ϕ

_D,δt

k

L^∞(0,T;L²(Ω))

+ k∇

D,δt

ϕ

_D,δt

k

L²(QT)

≤ C. (3.7) Proof: Pressure equation. In order to obtain the estimate on the first term in (3.7) we use p

ⁿ

as a test element in the pressure equation (2.16), which implies

X

K∈M

X

σ∈EK

(p

ⁿ_K

− p

ⁿ_σ

)(λ

ⁿ_K

F

K,σ

(p

ⁿ

) + ξ

_K,σⁿ

g

K,σ

) = X

K∈M

m(K)p

ⁿ_K

(k

_w,Kⁿ

+ k

ⁿ_o,K

). (3.8) Let us first estimate the term T

_ξ

= P

K∈M

P

σ∈EK

(p

ⁿ_K

− p

ⁿ_σ

)ξ

ⁿ_K,σ

g

_K,σ

. In view of (2.20) we deduce from Cauchy-Schwarz inequality that

(T

ξ

)

²

= ( X

K∈M

X

σ∈EK

m(σ)(p

ⁿ_K

− p

ⁿ_σ

)n

K,σ

· ξ

_K,σⁿ

K

K

g)

²

≤ X

K∈M

X

σ∈EK

m(σ) d

K,σ

(p

ⁿ_K

− p

ⁿ_σ

)

²

X

K∈M

X

σ∈EK

m(σ)d

K,σ

( | K

K

g | ξ

_K,σⁿ

)

²

.

Then, in view of the assumptions ( H

2

), ( H

3

) and ( H

4b

), and also thanks to (2.4) and Lemma 2.2 we deduce that

| T

ξ

| ≤ dm(Ω)K | g |k ξ k

L^∞((0,1))

k∇

D,δt

p( · , t

n

) k

L²(Ω)

. (3.9) In view of (2.7) and (3.4) the right-hand side of (3.8) can be written as

X

K∈M

m(K)p

ⁿ_K

(k

ⁿ_w,K

+ k

ⁿ_o,K

) = X

K∈M

1 δt

Z

tn

tn−1

Z

Ω

p

_D,δt

(x, t)(k

_w

(x, t) + k

_o

(x, t)) dxdt Applying Cauchy-Schwarz inequality and the discrete Poincar´e inequality (Lemma 2.3) we obtain

X

K∈M

m(K)p

ⁿ_K

(k

ⁿ_w,K

+ k

ⁿ_o,K

) ≤ k p

D,δt

( · , t

n

) k

L²(Ω)

k k

w

+ k

o

k

L^∞(0,T;L²(Ω))

≤ C k∇

D,δt

p

D,δt

( · , t

n

) k

L²(Ω)

k k

w

+ k

o

k

L^∞(0,T;L²(Ω))

. so that in view of ( H

⁶

),

X

K∈M

m(K)p

ⁿ_K

(k

_w,Kⁿ

+ k

_o,Kⁿ

) ≤ C k∇

D,δt

p

D,δt

( · , t

n

) k

L²(Ω)

. (3.10) Gathering (2.39), (3.8), (3.9), (3.10) we obtain

λ k∇

D,δt

p

D,δt

( · , t

n

) k

²L²(Ω)

≤ X

K∈M

X

σ∈E_K

(p

ⁿ_K

− p

ⁿ_σ

)λ

ⁿ_K

F

K,σ

(p

ⁿ

) ≤ C k∇

D,δt

p

D,δt

( · , t

n

) k

L²(Ω)

, with some positive C; the proof of the estimate on the first term of (3.7) is complete.

The estimate on the other terms of (3.7) can be obtained by setting w

ⁿ

= ϕ(s

ⁿ

) in the saturation equation (2.17) and summing over n ∈ { 1, . . . , m } , which yields

m

X

n=1

X

K∈M

ω(K)ϕ

ⁿ_K

(s

ⁿ_K

− s

ⁿ⁻¹_K

) + δt

m

X

n=1

X

K∈M

X

σ∈EK

(ϕ

ⁿ_K

− ϕ

ⁿ_σ

)F

K,σ

(ϕ

ⁿ

)

=

m

X

n=1

X

K∈M

δtm(K)ϕ

ⁿ_K

k

ⁿ_o,K

. We define the terms

T

t

=

m

X

n=1

X

K∈M

ω(K )ϕ

ⁿ_K

(s

ⁿ_K

− s

ⁿ⁻¹_K

),

T

C

=

m

X

n=1

X

K∈M

X

σ∈EK

δt(ϕ

ⁿ_K

− ϕ

ⁿ_σ

) G (Q

ⁿ_K,σ

f ( · ) + γ( · )g

K,σ

; s

ⁿ_K

, s

ⁿ_σ

).

and

T

D

=

m

X

n=1

X

K∈M

X

σ∈EK

δt(ϕ

ⁿ_K

− ϕ

ⁿ_σ

) F

^K,σ

(ϕ(s

ⁿ

)).

Accumulation term. Remark that using the notations of Lemma 2.7 Φ(s

ⁿ_K

) − Φ(s

ⁿ⁻¹_K

) = ϕ

ⁿ_K

(s

ⁿ_K

− s

ⁿ⁻¹_K

) +

Z

sⁿ_K sⁿ_K⁻¹

(ϕ(τ) − ϕ

ⁿ_K

)dτ. (3.11)

Since the function ϕ is increasing, the second term in the right-hand side of (3.11) is negative, which implies that

T

_t

≥

m

X

n=1

X

K∈M

ω(K )(Φ(s

ⁿ_K

) − Φ(s

ⁿ⁻¹_K

)) = X

K∈M

ω(K)Φ(s

^m_K

) − X

K∈M

ω(K)Φ(s

⁰_K

)).

It follows from (2.45) and the assumption ( H

5

) that T

_t

≥ 1

2L

ϕ

X

K∈M

ω(K)(ϕ

^m_K

)

²

− L

ϕ

2 X

K∈M

ω(K)(s

⁰_K

)

²

≥ ω

2L

ϕ

k ϕ

D,δt

( · , t

m

) k

²L²(Ω)

− ωL

ϕ

2 k s

D,δt

( · , 0) k

²L²(Ω)

. Convection term. In this subsection we use the simplified notation

G

K,σⁿ

(a, b) = G Q

ⁿ_K,σ

f ( · ) + γ( · )g

K,σ

; a, b

for all a, b ∈ R . (3.12) For all K ∈ M , σ ∈ E

^K

and s ∈ R we define the function

G

ⁿ_K,σ

(s) = Z

s

0

G

K,σⁿ

(τ, τ )ϕ

^′

(τ)dτ, (3.13) which is such that

G

ⁿ_K,σ

(s

ⁿ_K

) − G

ⁿ_K,σ

(s

ⁿ_σ

) = (ϕ

_K

− ϕ

_σ

) G

K,σⁿ

(s

_K

, s

_σ

) +

Z

sK

sσ

G

K,σⁿ

(τ, τ ) − G

K,σⁿ

(s

K

, s

σ

)

ϕ

^′

(τ)dτ. (3.14) In view of (2.25) the second term on the right-hand side of (3.14) is negative, which implies the following estimate

T

C

≥

m

X

n=1

X

K∈M

X

σ∈E_K

δt G

ⁿ_K,σ

(s

ⁿ_K

) − G

ⁿ_K,σ

(s

ⁿ_σ

) .

In view of (3.12), (3.13) and (2.25) we have that G

ⁿ_K,σ

(s

ⁿ_K

) − G

ⁿ_K,σ

(s

ⁿ_σ

) = Q

ⁿ_K,σ

Z

sⁿ_K sⁿ_σ

f(τ )ϕ

^′

(τ )dτ + g

K,σ

Z

sⁿ_K sⁿ_σ

γ (τ )ϕ

^′

(τ )dτ.

It follows from (2.16) and the hypotheses ( H

1

), ( H

2c

) that X

K∈M

X

σ∈EK

Q

ⁿ_K,σ

Z

sⁿ_K

sⁿ_σ

f(τ )ϕ

^′

(τ )dτ = X

K∈M

m(K )(k

_w,Kⁿ

+ k

_o,Kⁿ

) Z

sⁿ_K

0

f (τ)ϕ

^′

(τ)dτ ≥ 0, where we have set

v

_Kⁿ

= Z

sⁿ_K

0

f (τ)ϕ

^′

(τ)dτ and v

ⁿ_σ

= Z

sⁿ_σ

0

f(τ )ϕ

^′

(τ)dτ.

Therefore we have that T

_C

≥

N

X

n=1

δt X

K∈M

X

σ∈EK

g

_K,σ

Z

sⁿ_K

sⁿ_σ

γ(τ )ϕ

^′

(τ )dτ

≥

N

X

n=1

δt X

K∈M

Z

sⁿ_K 0

γ(τ)ϕ

^′

(τ)dτ X

σ∈EK

g

K,σ

−

N

X

n=1

δt X

K∈M

X

σ∈EK

g

K,σ

Z

sⁿ_σ 0

γ(τ)ϕ

^′

(τ)dτ