Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

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Existence of a solution for two phase flow in porous media: The case that the porosity depends on the

pressure

Fatima-Zahra Daim, Robert Eymard, Danielle Hilhorst

To cite this version:

Fatima-Zahra Daim, Robert Eymard, Danielle Hilhorst. Existence of a solution for two phase flow in

porous media: The case that the porosity depends on the pressure. Journal of Mathematical Analysis

and Applications, Elsevier, 2007, 326 (1), pp.332-351. �10.1016/j.jmaa.2006.02.082�. �hal-00693093�

Existence of a solution for two phase flow in porous media: The case that the porosity

depends on the pressure

aLaboratoire de Mathématique (UMR 8628), Université de Paris-Sud, 91405 Orsay cedex, France bLaboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université de Marne-La-Vallée,

77454 Marne La Vallée cedex, France

Inthispaperweprovetheexistenceofasolutionofacoupledsysteminvolvingatwophaseincom- pressibleflowinthegroundandthemechanicaldeformationoftheporousmediumwheretheporosityisa function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation.Inordertoshowtheexistenceofaweaksolution,weconsiderasequenceofrelateduniformlypar- abolicproblemsandapplytheSchauderfixedpointtheoremtoshowthattheypossessaclassicalsolution.

WethenprovetherelativecompactnessofsequencesofsolutionsbymeansoftheFréchet–Kolmogorov theorem;thisyieldstheconvergenceofasubsequencetoaweaksolutionoftheparabolicsystem.

Keywords:Porous medium; Subsidence model; Nonlinear parabolic degenerate equations; Schauder fixed point theorem; Fréchet–Kolmogorov theorem

1. Introduction

In this article we prove the existence of a solution for a system involving a two phase in- compressible flow in the ground and the mechanical deformation of a porous medium which is

F.Z. Daïm

, R. Eymard

, D.Hilhorst

subject to the weakening water phenomena: this means that the medium absorbs the water which is already present or artificially injected to develop a draining process. The most well-known geophysical example is the Ekofisk deposit in the North Sea, where the rock is essentially made of chalk. The mathematical model which we present in Section 2 is a system of two coupled equations where the unknown functions are the saturationS and the pressure pw of the water phase. In general, the coupling between the two phase flow and the mechanical deformations is taken into account by means of Biot’s law; however Biot’s law is not sufficient to describe the coupling in a sensitive porous medium in weakening water. Instead the model which we study involves the dependence of the porosity on the global pressure.

We rewrite the model in Section 2, where the saturationSand the global pressurepare the two unknown functions. We present two equivalent forms which both involve a parabolic equation which degenerates forS=1: the first one allows to apply the maximum principle to the saturation Swhereas the second one permits to obtain uniform estimates on some quantities depending on the saturationS. We indicate the precise hypotheses and present a definition of a weak solution of the problem. We consider a sequence(P)of regularized problems. In Section 3 prove that they possess a classical solution(S, p)by means of the Schauder fixed point theorem. We present a priori estimates uniform in the parameterfor the pair of functions(p, S). In Section 4, we apply the Fréchet–Kolmogorov theorem to prove the relative compactness of the sequences and show the convergence of subsequences to a weak solution of the problem.

As related work let us mention the book of Gagneux and Madaune-Tort [5] about the mathe- matical analysis of petrol problems. Further Chen [2] considered a coupled system of equations modelling two phase flow in porous medium where the porosity only depends on space. He proved the existence of a weak solution which he obtained as a limit of solutions of associated discrete time problems. In [2], Chen proved the uniqueness of the solution under the hypothesis that the global pressure is a continuous Lipschitz function. Finally let us cite [4] where Eymard, Herbin and Michel deduced the existence of solutions of a general class of problems modelling two phase incompressible flow from the convergence proof of a finite volume method.

2. The mathematical model

We consider a system of nonlinear equations coupling two phase flow in the ground and the mechanical deformation of the porous media; the porosityϕ=ϕ(p)depends on the global pressurepintroduced in [1]. The unknowns are the saturationSand the pressurep_wof the water phase. Problem(P)is given by the coupled system

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∂

∂t

ϕ(p)S

=kdiv

ρ_wkr_wo(S) μ_w ∇p_w +α

f_w(S^∗)(p−p)⁺−f_w(S)(p−p)⁻ ,

∂

∂t

ϕ(p)(1−S)

=kdiv

ρ_okr_ow(S)

μo ∇(p_w+p_c) +α

fo(S^∗)(p−p)⁺−fo(S)(p−p)⁻ ,

(2.1)

inQ_T =Ω×(0, T], together with the boundary and initial conditions

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∇p_w·n=0 on∂Ω×(0, T],

∇(p_w+p_c)·n=0 on∂Ω×(0, T],

∇S.n=0 on∂Ω×(0, T], S(x,0)=S₀(x) inΩ,

p(x,0)=p₀(x) inΩ,

(2.2)

where

• a⁺=max(0, a)anda⁻=max(0,−a),

• pc=pc(S)is the capillary pressure, which is decreasing function,

• krij =krij(S)is the mobility of the phasei in the presence of the phase j, where i, j∈ {w, o}andi=j. The functionskrwo andkrow are respectively increasing and decreasing functions. Typical expressions in the case of an incompressible nonmiscible flow are given by

kr_wo(S)=S^a, kr_ow(S)=(1−S)^a, witha∈ {1,2,3}, (2.3)

• ρ_i andμ_i are respectively the density and the viscosity of the phasei,

• S^∗is a constant such thatS^∗∈ [0,1],

• p=p(x, t )is the global pressure given by

p=p_w+ S

0

k_o(u)

M(u)p_c(u) du, (2.4)

where

k_i(S)=ρ_ikr_ij(S)

μ_i withi, j∈ {w, o}, M(S)=k_w(S)+k_o(S).

Fori∈ {w, o}, we define the functionf_iby f_i(S)= k_i(S)

M(S). (2.5)

We remark that 0fi(S)1 andfw+fo=1.

2.1. Equivalent form of Problem(P)

We formulate below a problem equivalent to Problem(P)with the saturationSand the global pressurepas unknown functions. We add up the two equations of system (2.1) to obtain:

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p). (2.6)

We rewrite the first equation of system (2.1) as

∂

∂t

ϕ(p)S

=kψ (S)+kdiv

f_w(S)M(S)∇p +α

fw(S^∗)(p−p)⁺−fw(S)(p−p)⁻

, (2.7)

where

ψ (S)= S

0

−k_o(u)k_w(u)

M(u) p_c(u) du (2.8)

is a continuously differentiable function on[0,1]. We denote byκ its Lipschitz constant. This yields the first problem equivalent to Problem(P)with the unknown functions(S, p)namely,

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⎨

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⎪⎪

⎪⎩

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p),

∂

∂t

ϕ(p)S

=kψ (S)+kdiv

fw(S)M(S)∇p +α

f_w(S^∗)(p−p)⁺−f_w(S)(p−p)⁻ ,

(2.9)

inQ_T, with the boundary and initial conditions

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⎪⎩

∇p·n=0 on∂Ω×(0, T],

∇ψ (S)·n=0 on∂Ω×(0, T], S(x,0)=S₀(x) inΩ,

p(x,0)=p₀(x) inΩ.

(2.10)

Since it is difficult to apply the maximum principle to the second equation of (2.9) in order to show that 0S1 we write a second formulation which allows us to obtain these estimates forS. We multiply the first equation of (2.9) byf_w(S)and subtract it from the second equation of (2.9). We deduce from a formal computation that:

S−f_w(S)∂

∂tϕ(p)+ϕ(p)∂

∂tS=kψ (S)+kM(S)∇p.∇f_w(S) +α

f_w(S^∗)−f_w(S)

(p−p)⁺, (2.11)

where the functionψis given by (2.8). Consequently, a second form of Problem(P)with the unknown functions(S, p)can be written as

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⎩

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p),

∂

∂tS= k

ϕ(p)ψ (S)+kM(S)

ϕ(p) ∇p.∇fw(S)+ α ϕ(p)

fw(S^∗)−fw(S)

(p−p)⁺

−(S−fw(S)) ϕ(p)

∂

∂tϕ(p),

(2.12)

with again the boundary and initial conditions (2.10).

2.2. Hypothesis

We suppose that the following hypotheses are satisfied:

(H0) Ωis a bounded domain ofRⁿ,∂Ω∈C^5+β,withβa given constant in(0,1);

(H1) Mis a differentiable function in[0,1]given by M(S)=ρ_w

μ_oS^a+ ρ_o

μ_o(1−S)^a, (2.13)

so that there exist positive constantsM_∗andM^∗such that

0< M_∗M(S)M^∗ in[0,1]; (2.14)

(H2) α=α(x)is a continuous function onΩ and we defineα^∗=sup_Ωα;

(H3) p=p(x)0 is a continuous function onΩ and we also definep_∗=inf_Ωpandp^∗= sup_Ωp,so that 0p_∗p^∗;

(H4) ϕ is a continuously differentiable, strictly increasing function on [p_∗, p^∗], such that ϕ(p_∗) >0;

(H5) ψ∈C¹([0,1])satisfies

C_∗S^m−1ψ(S)C^∗S^m−1 for 0S1,withm∈N^∗, and the constantain (2.13) satisfies

amax

1,m−1

2 ; (2.15)

(H6) f_wis a continuous differentiable and increasing function on[0,1]such that

0fw1, fw(0)=0 and fw(1)=1; (2.16)

(H7) (p0, S0)∈(L²(Ω))²satisfies

p_∗p₀p^∗ a.e.Ω, (2.17)

and

0S₀1 a.e.Ω. (2.18)

2.3. Notations

Further we recall the standard definitions of Hölder spaces (cf., [6]). Letm∈Nand 0< β <1.

Letf be a continuous function defined inQ_T such that all its derivatives of the form _∂t^∂rr⁺∂x^sfs, with 2r+sm, are continuous. We define the norm|f|^m_Q⁺_T^β,m+β2 by

|f|^m_Q⁺_T^β,m+2β = m j=0

f^{(j )}+ f^(m_QT^+β), (2.19)

such that

f^{(j )}=

2r+s=j

maxQ_T

∂^r+sf

∂t^r∂x^s ,

f^(m_QT^+β)=

2r+s=m

∂^r+sf

∂t^r∂x^s (β)

x,QT

+

2r+s=m

∂^r+sf

∂t^r∂x^s (^β₂)

t,QT

,

f^(β)_x,QT = sup

(x₁,t ),(x₂,t )∈Q_T

|f (x1, t )−f (x2, t )|

|x1−x2|^β , f⁽_t,Q^β2)_T = sup

(x,t₁),(x,t₂)∈Q_T

|f (x, t1)−f (x, t2)|

|t₁−t₂|^β2 .

We denote byC^m+β,m+2β(Q_T)the functional space of functionsf such that|f|^m_Q⁺_T^β,m+β2 <+∞: it is a Banach space with the norm|.|^m_Q⁺_T^β,m+2β.

2.4. Weak solutions of Problem(P)

In general, a solution of the degenerate parabolic Problem(P)is not smooth. Hence, we have to define weak solutions.

Definition 2.4.1. We say that (p, S)∈(L^∞(Q_T))² is a weak solution of Problem (P), if it satisfies:

(i) p∈L²(0, T;H¹(Ω))andψ (S)∈L²(0, T;H¹(Ω)); (ii) the integral equations

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⎩

Q_T

ϕ(p)∂tv dxdt=

Q_T

kM(S)∇p.∇v−α(p−p)v dxdt

−

Ω

ϕ(p0)v(0) dx,

Q_T

ϕ(p)S∂tv dxdt=

Q_T

−kψ (S)v+kfw(S)M(S)∇p.∇v

−α

fw(S^∗)(p−p)⁺−fw(S)(p−p)⁻ v

dxdt

−

Ω

ϕ(p0)S0v(0) dx,

(2.20)

for allvin V=

w∈C^2,1(QT), w(., T )=0 inΩ, ∂w

∂n =0 on∂Ω× [0, T]

.

3. A sequence of approximate problems(P)

In order to prove the existence of a solution of Problem (P), we introduce a sequence of regularized problems(P). Letbe a real constant small enough, we define the sequences:

• {ϕ}>0∈C^∞(R)such that

ϕ −−−→→0 ϕ uniformly in the interval[p_∗, p^∗], ϕ(p_∗)ϕ(p)ϕ(p^∗)+1,

0< cϕ(p) for allp∈R, ϕ(p) sup

[p_∗,p^∗]ϕ for allp∈ [p_∗, p^∗], (3.1) wherecis a constant depending on;

• α∈C₀^∞(Ω)such that

α −−−→→0 α inL²(Ω)and a.e. inΩ, αα^∗; (3.2)

• p∈C₀^∞(Ω)such that

p _→

−−−→0 p inL²(Ω)and a.e. inΩ, p_∗pp^∗; (3.3)

• S_0,∈C₀^∞(Ω)such that S_0,

→0

−−−→S₀ inL²(Ω)and a.e. inΩ,0S_0,1; (3.4)

• p0,∈C₀^∞(Ω)such that p_0,

→0

−−−→p₀ inL²(Ω)and a.e. inΩ, p_∗p_0,p^∗; (3.5)

• ψ∈C^∞(R)such that

ψ _→

−−−→0 ψ inC¹ [0,1]

,

C_∗S^m−1+λ_∗ψ(S)C^∗S^m−1+λ^∗ in[0,1], (3.6) whereλ_∗andλ^∗are strictly positive constants depending onψ.

For allr∈R, we define the functions a⁺(r)=r⁺ and a⁻(r)=r⁻

and sequences of functions{a⁺}>0and{a⁻}>0inC^∞(R)such that

a⁻(r)=a⁺(r)−r, (3.7)

and

• a⁺ →a⁺anda⁻→a⁻uniformly on each compact set ofR;

• 0a⁺ 1,−1a⁻ 0,a⁺0 anda⁻0.

Next we consider the sequence of regularized problems(P):

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⎪⎪

⎪⎪

⎩

∂

∂tS= k

ϕ(p)ψ(S)+kM(S)

ϕ(p)∇p.∇f_w(S) + α

ϕ(p)

f_w(S^∗)−f_w(S)

a⁺(p−p)−(S−f_w(S)) ϕ(p)

∂

∂tϕ(p), inQ_T,

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p), inQT, (3.8)

with the following boundary and initial conditions:

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⎪⎪

⎨

⎪⎪

⎪⎩

∇S·n=0 on∂Ω×(0, T],

∇p·n=0 on∂Ω×(0, T], S(x,0)=S_0,(x) x∈Ω,

p(x,0)=p_0,(x) x∈Ω.

(3.9)

Theorem 3.1(Classical solution of the approximate problem(P)). For all >0, there exists a classical solution(p, S)∈(C^5+β,5+2β(Q_T))²of the approximate problem(P). This solution satisfies

p_∗pp^∗ and 0S1.

Proof. We define the closed convex set K:=

w∈C^5+β,5+2β(Q_T), p_∗wp^∗inQ_T, w(x,0)=p_0,(x)inΩ,

∇w·n=0 on∂Ω×(0, T]

. (3.10)

For an arbitrary elementpofK, we consider the problem(P^S)

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⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂

∂tS= k

ϕ(p)ψ(S)+kM(S)

ϕ(p)∇p.∇f_w(S) + α

ϕ(p)

fw(S^∗)−fw(S)

a⁺(p−p)

−(S−f_w(S)) ϕ(p)

∂

∂tϕ(p) inΩ×(0, T],

∇S·n=0 on∂Ω×(0, T],

S(x,0)=S_0,(x) inΩ.

(3.11)

The problem(P^S)is uniform parabolic so that it possesses a unique classical solution S in C^2+β,1+β2(Q_T) [6, Theorem 7.4, Chapter 5, p. 491]. Applying the comparison theorem [7, Lemma 3.6, Chapter 2, p. 61], we obtain

Lemma 3.1.The functionSis such that

0S1 inQ_T. (3.12)

Since the coefficients of the differential operator in problem(P^S)are smooth, one can show thatS∈C^5+β,5+2β(Q_T)and that

|S|⁵_Q⁺_T^β,5+2β B¹|p|⁵_Q⁺_T^β,5+2β +B², (3.13) where the positive constantsB¹andB²only depend onandS_0,. Indeed the idea is to apply four times [6, Theorem 5.3, Chapter 4, p. 320] to the problem (3.11) which we consider as linear.

We successively show thatS∈C^{i+2+β,i+2+β2} (Q_T)fori=0,1,2,3. We remark that to obtain the desired regularity, we have to check compatibility conditions of orderi=0,1,2. Here, the condition of orderiis given by

∇∂ⁱS

∂tⁱ

t=0

·n=0 on∂Ω. (3.14)

This condition is satisfied for all i ∈ {0,1,2} thanks to the properties of the sequences {S_0,},{p_0,},{p},{α} and to the fact that the functions{p} belong to the set K. Hence, we obtain (3.13). LetS be the solution of problem (3.11); we consider the problem(P^p)given by

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⎩

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p) inΩ×(0, T],

∇p·n=0 on∂Ω×(0, T],

p(x,0)=p_0,(x) inΩ.

(3.15)

The parabolic equation in problem(P^p)is uniform parabolic and we deduce from [6, Theo- rem 7.4, Chapter 5, p. 491] that it possesses a unique classical solutionpˆ ∈C^2+β,1+β2(Q_T).

Applying the comparison theorem [7, Lemma 3.6, Chapter 2, p. 61], we obtain the following bounds

Lemma 3.2.The functionpˆis such that

p_∗pˆ(x, t )p^∗ inQ_T. (3.16)

Since the coefficients of the partial differential equation in Problem(P^p)are smooth, we apply five times [6, Theorem 5.3, Chapter 4, p. 320] to problem (3.15) to deduce that pˆ∈ C^6+β,6+2β(Q_T)and that

| ˆp|⁶_Q⁺_T^β,6+2β M¹|S|⁵_Q⁺_T^β,5+2β +M², (3.17) where the positive constantsM¹ andM² only depend on, p_0,. To do that, we successively show thatpˆ∈C^{i+2+β,i+22+β}(Q_T)for i=0, . . . ,4. We have to verify the compatibility condi- tions of orderi=0, . . . ,4, which reduce to

∇∂ⁱpˆ

∂tⁱ

t=0

·n=0 on∂Ω. (3.18)

These conditions are satisfied thanks to the properties of sequences{S_0,}, {p_0,},{p}, {α} and{S}. Thus, we have defined a mapF:p→ ˆp fromK into itself. In order to apply the Schauder fixed point theorem, we have to show that the mapFis compact and continuous.

Compactness: We define the closed bounded set ofK, BC=

w∈K, |w|⁵_Q⁺_T^β,5+2β C

.

We show below thatF(BC)is compact inK. Letpbe arbitrary inBCand letS the solution of (3.11). Since|p|⁵_Q⁺_T^β,5+2β Cand in view of (3.13), we obtain

|S|⁵_Q⁺_T^β,5+β2 M(C), (3.19)

whereM(C) is a positive constant depending onC. Letpˆ=F(p)be the solution of problem (3.15). From (3.17), we get

| ˆp|⁶_Q⁺_T^β,6+2β C(C), (3.20)

whereC(C) is a positive constant depending onC. Hence, we have proved that F(BC)⊂

w∈C^6+β,6+2β(Q_T), |w|^6+β,6+2β(Q_T)C(C) ,

so thatF(BC)is bounded inC^6+β,6+2β(Q_T)and since the embedding fromC^6+β,6+2β(Q_T)into C^5+β,5+2β(Q_T)is compact, we deduce thatF(BC)is compact inK and finally the mapF is compact.

Continuity: Let{p^δ}δ>0⊂Kbe such that p^δ

δ→0

−−−→p∈K inC^5+β,5+2β(Q_T). (3.21)

There exist a sequence of functions{S^δ}δ>0and a functionSinC^5+β,5+2β(Q_T)satisfying

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⎪⎪

⎪⎪

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⎪⎪

⎩

∂

∂tS^δ= k ϕ(p^δ)ψ

S^δ

+kM(S^δ)

ϕ(p^δ)∇p^δ.∇fw

S^δ

+ α ϕ(p^δ)

fw(S^∗)−fw

S^δ a⁺

p−p^δ

−(S^δ−f_w(S^δ)) ϕ(p^δ)

∂

∂tϕ

p^δ

inΩ× [0, T],

∇S^δ·n=0 on∂Ω× [0, T],

S^δ(x,0)=S0,(x) inΩ,

(3.22)

and ⎧

⎪⎪

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⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

∂

∂tS= k

ϕ(p)ψ(S)+kM(S)

ϕ(p) ∇p.∇f_w(S) + α

ϕ(p)

fw(S^∗)−fw(S)

a⁺(p−p)

−(S−f_w(S)) ϕ(p)

∂

∂tϕ(p) inΩ× [0, T],

∇S·n=0 on∂Ω× [0, T],

S(x,0)=S_0,(x) inΩ.

(3.23)

Since the sequence{p^δ}δ>0is bounded, in view of (3.13) and of the compactness of the embed- ding fromC^5+β,5+β2 (QT)intoC^4+β,4+β2 (QT), there exist a functionS∈C^5+β,5+β2 (QT)and a subsequence{S^δn}n0 such thatS^δn−−−→δn→0 S inC^4+β,4+β2 (QT). Letδn tend to 0 in problem (3.22); by the uniqueness of the solution of problem (3.23), we deduce thatScoincides withS. Let(pˆ^δn)_n>0be the sequence of solutions of the problems

⎧⎪

⎪⎨

⎪⎪

⎩

∂

∂tϕ ˆ p^δn

=kdiv M

S^δn

∇ ˆp^δn +α

p− ˆp^δn

inΩ× [0, T],

∇p^δn·n=0 in∂Ω× [0, T],

p^δn(x,0)=p_0,(x) inΩ.

(3.24)

In view of (3.17) and of the compactness of the embedding from C^6+β,6+2β(Q_T) into C^5+β,5+2β(Q_T), there exist a functionθ∈C^6+β,6+2β(Q_T)and a subsequence(pˆ^δn)_n>0, which we also denote by(pˆ^δn)_n>0, such that

p^δn _δ

n→0

−−−→θ inC^5+β,5+β2 (Q_T).

Lettingδ_ntend to 0 in problem (3.24), we deduce thatθ satisfies the problem

⎧⎪

⎪⎨

⎪⎪

⎩

∂

∂tϕ(θ)=kdiv

M(S)∇θ

+α(p−θ) inΩ× [0, T],

∇θ·n=0 on∂Ω× [0, T],

θ(x,0)=p_0,(x) inΩ.

(3.25)

On the other hand, the functionpˆ=F(p)satisfies problem (3.25) as well. By the uniqueness of the solution [6, Theorem 7.4, Chapter 5, p. 491] we deduce that

θ=F(p).

We conclude that the application F:p˜→ ˆp is continuous and compact for the topology of C^5+β,5+2β(QT)from the convex closed setKinto itself. Therefore it follows from the Schauder fixed point theorem that there exists a pair of functions(S, p)satisfying Problem(P).

We formally showed above the correspondence between two formulations of Problem(P);

the regularity of the coefficients in (3.8)–(3.9) and (3.26)–(3.9) and the fact that a⁺(r)−a⁻(r)=r, for allr∈R,

allow us to obtain the following result.

Lemma 3.3.The regularized Problem(P)is equivalent to the problem

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂

∂tϕ(p)=kdiv

M(S)∇p

+α(p−p), inQ_T,

∂

∂t

ϕ(p)S

=kψ(S)+kdiv

fw(S)M(S)∇p

+α

f_w(S^∗)a⁺(p−p)−f_w(S)a⁻(p−p)

, inQ_T,

(3.26)

with the boundary and initial conditions given in(3.9).

Lemma 3.4(A priori estimates forp). Let(p, S)be a solution of the Problem(P);there exists a positive constantCindependent ofsuch that

(i) pL²(0,T;H¹(Ω))C, (ii) ϕ(p)

L²(0,T;H¹(Ω))C, (iii)

∂

∂tϕ(p)

L²(0,T;(H¹(Ω)))

C. (3.27)

Proof. (i) We multiply the second equation of problem (3.8) by p and integrate onQ_T. We obtain:

Q_T

p

∂

∂tϕ(p) dxdt+

Q_T

kM(S)(∇p)²dxdt=

Q_T

α(p−p)pdxdt.

We define the function Φ(s)=

s

0

rϕ(r) dr.

We have that

Q_T

p∂

∂tϕ(p) dxdt=

Q_T

∂

∂tΦ(p) dxdt=

Ω

Φ

p(., T )

−Φ(p_0,) dx.

Hence

Ω

Φ

p(., T )

dx+kM_∗

QT

(∇p)²dxdt

QT

α(p−p)pdxdt+

Ω

Φ(p_0,) dx,

which implies kM_∗

Q_T

(∇p)²dxdtα^∗T|Ω|p^∗2+ |Ω|p^∗2 sup

[p_∗,p^∗]ϕ(p).

(ii) In view of (3.1) we immediately deduce (ii) from (i).

(iii) Letφ∈L²(0, T;H¹(Ω)), we have

T

0

∂

∂tϕ(p), φ k

QT

M(S)∇p.∇φdxdt+

QT

α(p−p)φdxdt

kM^∗∇p_L²_(QT)∇φ_L²_(QT)+α^∗p^∗|Q_T|φ_L²_(QT) CφL²(0,T;H¹(Ω)),

which completes the proof of (iii). 2

Lemma 3.5(A priori estimates forS). Let(p, S)be a solution of Problem(P);there exists a positive constantCindependent ofsuch that

(i)

Q_T

∇G

S(x, t )²dxdtC,

(ii)

QT

∇ψ

S(x, t )²dxdtC,

(iii)

QT

∇K ϕ

p(x, t )

S(x, t )²dxdtC, (iv)

T

0

Ω_ξ

K ϕ

p(x+ξ, t )

S(x+ξ, t )

−K ϕ

p(x, t )

S(x, t )2

dxdtC|ξ|²,

(v)

T−τ

0

Ω

K ϕ

p(x, t+τ )

S(x, t+τ )

−K ϕ

p(x, t )

S(x, t )2

dxdtCτ,

where G(S)=

S

0

C_∗r^m−1+λ_∗ dr, K(r)= 1

mr^m, (3.28)

and

Ω_ξ:= {x∈Ω|x+rξ∈Ωfor all0r1}.

Proof. (i) We multiply the second equation of (3.26) byϕ(p)S and integrate onQT. This yields

I0+I1+I2+I3+I4+I5=0, (3.29)

where I0=

QT

1 2

∂

∂t

ϕ(p)S2

dxdt,

I1=k

QT

ϕ(p)ψ(S)(∇S)²dxdt,

I2=k

QT

Sψ(S)∇S∇ϕ(p) dxdt,

I3=kρ_w μ_w

QT

S^a+1∇p∇ϕ(p) dxdt,

I4=kρ_w μ_w

QT

ϕ(p)S^a∇S∇pdxdt,

I5= −

Q_T

α

f_w(S^∗)a⁺(p−p)−f_w(S)a⁻(p−p)

ϕ(p)Sdxdt.

We have to estimate each of the quantitiesI0,I1I2,I3,I4andI5. We have that I0=

Ω

1 2

ϕ

p(x, T )

S(x, T )2

dx−

Ω

1 2

ϕ p_0,(x)

S_,0(x)2

dx

−1

2|Ω|ϕ(p^∗). (3.30)

ForI1, we have I1=k

QT

ϕ(p)

ψ(S)∇S2

dxdt

kϕ(p_∗)

Q_T

C_∗S^m−1+λ_∗

∇S 2

dxdt

kϕ(p_∗)

Q_T

∇G(S)2

dxdt. (3.31)