Développement d'un modèle numérique pour l'écoulement triphasique de fluides incompressibles

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Développement d’un modèle numérique pour

l’écoulement triphasique de fluides incompressibles

Lauriane Schneider

To cite this version:

Lauriane Schneider. Développement d’un modèle numérique pour l’écoulement triphasique de flu-ides incompressibles. Sciences de la Terre. Université de Strasbourg, 2015. Français. �NNT : 2015STRAH002�. �tel-01222156�

´

Ecole doctorale des Sciences de la Terre et de l’Environnement

D´

eveloppement d’un mod`

ele

num´

erique pour l’´

ecoulement

triphasique de fluides incompressibles

TH`

ESE

pr´esent´ee et soutenue publiquement le 18 f´evrier 2015 pour l’obtention du

Doctorat de l’universit´

e de Strasbourg

Discipline : Sciences de la Terre et de l’Environnement par

Lauriane Schneider

Rapporteurs : M. Brahim AMAZIANE Maˆıtre de Conf´erences,

Universit´e de Pau et des Pays de l’Adour M. Michel QUINTARD, Directeur de Recherche CNRS,

Institut de M´ecanique des Fluides de Toulouse (IMFT) Examinateurs : M. Philippe ACKERER, Directeur de Recherche CNRS,

Laboratoire d’Hydrologie

et de G´eochimie de Strasbourg (LHyGeS) Mme Danielle HILHORST, Directrice de Recherche CNRS

D´epartement de Math´ematiques Universit´e Paris-Sud

M. Gerhard SCH ¨AFER, Professeur des Universit´es, Universit´e de Strasbourg M. Philippe HELLUY, Professeur des Universit´es, Universit´e de Strasbourg

R

Comme un vol de gerfauts hors du charnier natal Fatigués de porter leurs misères hautaines De Palos de Moguer, routiers et capitaines Partaient, ivres d’un rêve héroïque et brutal. Ils allaient conquérir le fabuleux métal Que Cipango mûrit dans ses mines lointaines Et les vents alizés inclinaient leurs antennes Aux bords mystérieux du monde occidental Chaque soir, espérant des lendemains épiques, L’azur phosphorescent de la mer des Tropiques Enchantait leur sommeil d’un mirage doré Ou penchés à l’avant des blanches caravelles, Ils regardaient monter en un ciel ignoré Du fond de l’océan des étoiles nouvelles

R

R

R

R

Que serais-je sans toi qui vins à ma rencontre Que serais-je sans toi qu’un coeur au bois dormant Que cette heure arrêtée au cadran de la montre Que serais-je sans toi que ce balbutiement. J’ai tout appris de toi sur les choses humaines Et j’ai vu désormais le monde à ta façon

J’ai tout appris de toi comme on boit aux fontaines Comme on lit dans le ciel les étoiles lointaines Comme au passant qui chante on reprend sa chanson J’ai tout appris de toi jusqu’au sens du frisson. J’ai tout appris de toi pour ce qui me concerne Qu’il fait jour à midi, qu’un ciel peut être bleu Que le bonheur n’est pas un quinquet de taverne Tu m’as pris par la main dans cet enfer moderne Où l’homme ne sait plus ce que c’est qu’ètre deux Tu m’as pris par la main comme un amant heureux. Qui parle de bonheur a souvent les yeux tristes N’est-ce pas un sanglot que la déconvenue Une corde brisée aux doigts du guitariste Et pourtant je vous dis que le bonheur existe Ailleurs que dans le rêve, ailleurs que dans les nues. Terre, terre, voici ses rades inconnues.

Que serais-je sans toi qui vins à ma rencontre Que serais-je sans toi qu’un coeur au bois dormant Que cette heure arrêtée au cadran de la montre Que serais-je sans toi que ce balbutiement.

R&D

• •

σ[N · m−1_] α σS−1 σS−2 σ1−2 α σS−1 = σS−2+ σ1−2cos α

Ω ⊂ R3 ∂Ω x (x, y, z) Z(x, y, z) (x, y, z) Z = −z x ∈ O O Ω  O 7→ χ(O, t) = Ot x0 7→ χ(xt=0, t) = xt

✻ ✓ ✓ ✓ ✓ ✴ ✲ ❄ r (x, t) x∈ Ω p(x, t) P a x ∈ Ω P τ τ · = −P · ρ kg · m−3 dµΩ Z O dµΩ= Z O ρ(x, t)dx B(ρ) B(ρ) = ρ(p) ρref µ > 0 P a · s

Ot = χ(O, t) ⊆ Ω d dtm(Ot) = d dt Z Ot ρ(x, t)dx = d dt Z O

ρ(χ(u, t), t)det(χ′)du

= Z

O

 ∂

∂tρ(χ(u, t), t) + ∇(ρ(χ(u, t), t)) · + ρ(χ(u, t), t)∇ ·  det(χ′)du = Z Ot  ∂ ∂tρ(x, t) + ∇(ρ(x, t)) · + ρ(x, t)∇ ·  dx = Z Ot  ∂ ∂tρ(x, t) + ∇ · (ρ(x, t) )  dx d dtm(Ot) = 0 ∂ ∂tρ(x, t) + ∇ · (ρ(x, t) ) = 0 • ∇ · = 0 • ( ) = 1 2(∇ + ∇ t₎

• τ = −p + η (∇ · ) + 2µ ( ) η P a · s τ = −p + 2µ ( ) U ⊂ Ot d dt Z U ρ(x, t) (x, t)dx = Z U ρ(x, t) dx + Z ∂U τ · dΓ = Z U (ρ(x, t) + ∇ · τ )dx d dt Z U ρ(x, t) (x, t)dx = ρd dt Z U (x, t)dx = ρ Z U  ∂ ∂t + ( · ∇)  (x, t)dx ρ∂ ∂t + ρ ( · ∇) = ρ − ∇P + µ∆

mm Ω ⊂ R3 S(x0, r) x0 r γ : Ω → {0, 1} γ(x) =  1 x∈ 0 x∈ ∀x ∈ Ω φ : Ω → [0 : 1] x0 7→ 1 R S(x0,r) x Z S(x0,r) γ(x) x γ rmicro rmacro

φ r φ r_micro r_macro r γj(x, t) =  1 x j 0 ∀x ∈ Ω j Sj(x0, t) = Z V ER γj(x, t) x Z V ER γ(x, t) x S1 S2 S3 0 ≤ Sj ≤ 1 P jSj = 1

Sj,r Sj,r ≤ Sj ≤ 1 − X β6=j Sβ,r ¯ Sj = Sj− Sj,r 1 −P_βSβ,r Sj pc= pnω− pω ≥ 0 pc(Sw) Sω(pc) = Pd pc λ pc ≥ Pd Pd 0, 2 ≤ λ ≤ 0, 3

λ λ Pd pc = Pd(Sω)− 1 λ p c≥ Pd Sω∈ ]0, 1] Sω(pc) = (1 + (αpc)n)m, pc ≥ 0 m = 1−1 n n pc(Sω) = 1 α  S− 1 m ω − 1 1 n , pc≥ 0 h h= krj ρjg µj [P a · m · s] [m2_]

0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Brooks−Corey 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Van Genuchten drainage imbibition Saturation effective ¯Sw Pc Pc Saturation effective ¯Sw λ = 2.89 Pd= 873 P a n = 5.5 j = 0.00077 P a−1 _{λ = 2.29 P} d = 667 P a n = 4 j = 0.0011 P a−1 krw(Sw) = Swa     Z Sw 0 pc(s)−b s Z 1 0 pc(s)−b s     c krnw(Sw) = (1 − Sw)a     Z Sw 0 pc(s)−b s Z 1 0 pc(s)−b s     c a = 2 b = 2 c = 1 krw(Sw) = S 2+3λ λ w krnw(Sw) = (1 − Sw)2(1 − S 2+λ λ w )

a = 1/2 b = 1 c = 2 krw(Sw) = S 1 2 w  1 −  1 − Sm1 w m2 krnw(Sw) = (1 − Sw) 1 2  1 − Sm1 w 2m 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mualem Van Genuchten

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Burdine Brooks Corey k rwimbibition k rwdrainage k rnwimbibition k rnwdrainage Saturation effective ¯Sw k r k r Saturation effective ¯Sw λ = 2.89 Pd = 873 P a n = 5.5 α = 0.00077 P a−1 λ = 2.29 Pd= 667 P a n = 4 α = 0.0011 P a−1 krω = Sω2 krnω= (1 − Sω)2 fj(Sω) = krj(Sω) µj kr1(Sω) µ1 +kr2(Sω) µ2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Flux fractionnaire Saturation en eau S₁ f₁(S₁) f2(S1) G ⊆ Ω σ(x) mG= Z G σ(x)φρ(x, t)dx L(φ, ρ, ) := Z G  ∂(σφρ) ∂t + ∇ · (σρj ) − ρQ  dx = 0 ρQ · [m · s−1] σ = dS L(Sj, ρj, j) := Z G  ∂(σSjφρj) ∂t + ∇ · (σρj j) − ρQα  dx = 0

j = φSj Pj ∂(σφBj(ρj)Sj) ∂t + ∇ · (σρj j) − ρQj = 0 19eme = − ·ρg µ∇h • h = P ρg + z [m] • [m2] • g = 9.81 [m · s−2] • ∇ =           ∂ ∂x ∂ ∂y ∂ ∂z           j = krj

j j = −krjBj(ρj) µj · (∇Pj− ρj(Pj)g∇Z) σ(x) j = −σ krjBj(ρj) µj · (∇Pj− ρj(Pj)g∇Z) ∂(σφBj(ρj)Sj) ∂t − ∇ ·  σkrjBj(ρj) µj · (∇Pj− ρj (Pj)g∇Z)  = 0 dans G. • Sj • Pj • σ j • krj(pj) • fj • dj = Bj(ρj) µj d =X j krjdj • νj = krjdj d • ρj(Pj) ρ =¯ X j νjρj • P12_(S 1, S3) = P1 − P2 • P32_(S 1, S3) = P3 − P2

• j = − krjBj(ρj) µj · (∇Pj− ρj (Pj)g∇Z) j = 1, 2, 3 • ∂(σφBj(ρj)Sj) ∂t + ∇ · (σ j) = 0 j = 1, 2, 3 • X j Sj = 1

           φσ∂ ∂t(Bω(ρω)Sω) − ∇ ·  σkrω µω Bω(ρω) · (∇Pω− ρωg∇Z)  = 0 φσ∂ ∂t(Bnω(ρnω)Snω) − ∇ ·  σkrnω µnω Bnω(ρnω) · (∇Pnω− ρnωg∇Z)  = 0.            φσ∂ ∂t(Bω(ρω)Sω) − ∇ ·  σkrω µω Bω(ρω) · (∇Pω− ρωg∇Z)  = 0 φσ∂ ∂t(Bnω(ρnω)Snω) − ∇ ·  σkrnω µnω Bnω(ρnω) · (∇Pnω− ∇Pc− ρnωg∇Z)  = 0          ∇Pc = dPc dSω∇Sω ou ∇Pc = dPc dSnω ∇Snω

φσ∂ ∂t(Bω(ρω)Sω) − ∇ ·  σkrω µω Bω(ρω) · (∇Pnω− ρωg∇Z)  | {z } Convection + ∇ ·  σkrω µω Bω(ρω)dPc dsω · (∇sω)  | {z } Dif f usion = 0 νj = νj(S1, S3, p) j = 1, 2, 3 Pcg ∇P_cg(S1, S3, p) = ν1(S1, S3, p)∇Pc12(S1, S3) + ν3(S1, S3, p)∇Pc32(S1, S3) +∂P g c ∂p (S1, S3, p)∇P ∂ν1 ∂S1 ∂P_c12 ∂S3 − ∂ν1 ∂S3 ∂P_c12 ∂S1 = ∂ν3 ∂S3 ∂P_c32 ∂S1 − ∂ν3 ∂S1 ∂P_c32 ∂S3

Pcg= 0 (1) oil(2) 0 (3) (s) C T S1 S3 C1 C3 Pcg water gas s1 s3 T j = 1 (j = 3) C ∼ C1 ∪ C3    P (x, t) = P2(x, t) + Pcg(S1(x, t), S3(x, t), p(x, t)) p1≤ p ≤ p2 ∀x ∈ G ∀t > 0 ∂Pcg ∂p (S1, S3, p) < 1 ∀(x, t) ∈ T, pmin≤ p1 ≤ p ≤ p3 ≤ pmax = 3 X j j = − 3 X j dνj · (∇Pj− ρj(pj)g∇Z) P1= Pc12+ P2 P3= Pc32+ P2

ν1∇P1+ ν2∇P2+ ν3∇P3 = ν1∇Pc12+ ν3∇Pc32+ (ν1+ ν2+ ν3) | {z } =1 ∇P2 = ∇Pcg− ∂Pcg ∂p ∇P + ∇P2 = ∇(Pcg+ P2) − ∂Pcg ∂p ∇P = ∇P − ∂P g c ∂p ∇P =  1 −∂P g c ∂p  ∇P                    φσ∂( P3 jBj(ρj)Sj) ∂t + ∇ · (σ ) = 0 = −d .  1 −∂P g c ∂p  ∇P − ¯ρg∇Z  P (x, t) = P2(x, t) + Pcg(S1(x, t), S3(x, t), p(x, t))                      φσ∂(Bj(ρj)Sj) ∂t + ∇ · (σ j) = 0 j = 1, 3 j = −dνj · (∇Pj− ρj(Pj)g∇Z) j = 1, 3 3 X j Sj = 1 j = 1, 3

                                                           φσ∂(Bj(Pj)Sj) ∂t + ∇ · (σ j) = 0 j = 1, 3 j = νj ω − νjd ·  1 − ω ω ρ − ∆ρ¯ j  g∇Z | {z } P artie convective − νjd ·∇(Pcj2− Pcg)  | {z }

P artie dif f usive

∆ρ1 = (ν2(ρ2− ρ1) + ν3(ρ3− ρ2)) ∆ρ3 = (ν2(ρ2− ρ3) + ν1(ρ3− ρ1)) 3 X j=1 Sj = 1 ω =  1 −∂P g c ∂p  Pcg(Snω, p) ω(Snω, p) ∇Pnω+ νω(Snω, p) ∂Pc ∂Snω∇Snω = ω(Snω, p)∇P                            P = Pnω+ Pcg(Snω, p) ∀(Snω, p) ∈ [0, 1[×[pω, pnω] ω(Snω, p) = 1 − ∂Pcg ∂p (Snω, p) ∀(Snω, p) ∈ [0, 1[×[pω, pnω] Pcg(Snω, p) = − Z Snω 0 νω ∂Pc ∂Snω dSnω ∀(Snω, p) ∈ [0, 1[×[pω, pnω] Pc = Pnω− Pω

∂Pcg ∂p                                    φ∂ ∂t(Bnω(ρnω)Snω+ Bω(ρω)Snω) + ∇ · = 0 = −d . [ω∇P − ¯ρg∇Z] P = Pnω+ Pcg Pcg(Sω) = − Z Snω 0 νω ∂Pc ∂Snω dSnω Pc= Pnω− Pω                  φ∂Snω ∂t + ∇ · nω = 0 nω= νnω + νωνnωd(ρnω− ρω) · ∇z − νωνnωd dPc dSnω (Snω) · ∇Snω Sω+ Snω= 1 • Bj = 1 • νj = fj • ρj(p) = ρj • ∂P g c ∂p (S1, p) = 0 • ω = 1

Ω ⊂ R3 Γ = Γe∪ Γs∪ Γl Γe Γs Γl • Γl · = 0 • p = pe · = e Γe Γs

Γe S ≡ 1 Γs ]0, T [ T > 0 Γl Γe Γs                                            φ∂( P3 jBj(Pj)Sj) ∂t + ∇ · ( ) = 0 = −d .  1 −∂P g c ∂p  ∇P − ¯ρg∇Z  P (x, t) = P2(x, t) + Pcg(S1(x, t), S3(x, t), p(x, t)) · = ve Γe P = Ps Γs · = 0 Γl                              φ∂(Bj(Pj)Sj) ∂t + ∇ · ( j) = 0 j = −dνj · (∇Pj− ρj(Pj)g∇Z) 3 X j Sj = 1 (S1,e, S3,e) Γe

                                             ∇ · ( ) = 0 = −ψ · [∇P − ¯ρg∇Z] ψ = d P (x, t) = P2(x, t) + Pcg(S1(x, t), p(x, t)) · = ve Γe · = λ(P − Pe) Γs P = Ps Γs · = 0 Γl                                        φ∂S2 ∂t + ∇ · ( 2) = 0 2 = f2+ f1f2(ρ2− ρ1)gψ · ∇Z − f1f2ddP 12 c dS2 ψ · ∇S2 S1+ S2 = 1 S2,e Γe S2,s Γs PCM(x) ∈ [0 : 1] Pc = PCM(x)Pc(S2)

                                                                     a(S2) = −f1f2 dP_c12 dS2 α = Z S2 1 a(t)dt b0= f1 b1= f1f2dPc12 b2= f1f2d(ρ2− ρ1) γ0 = Z 1 S2 f1 dP_c12 dS2 γ1 = Z 1 S2 f₁′dP 12 c dS2 dS2 γ2 = f1ρ1+ f2ρ2 m > 0                                      φ, ψ ∈ L∞(Ω) = {u ∈ Ω | ∃C ∈ R, |u(x)| ≤ C ∀x ∈ Ω} φ(x) ≥ m, ψ(x) ≥ m Ω j ∈ L2(Ω×]0, 1[) L2(Ω) =u ∈ Ω | ||u||2_L2 = R Ω|u(x)|2dx < ∞ ve∈ L∞(Γe) λ ∈ L∞(Γs) λ ≥ m2 _Ω α β bj γj R R                      d(S2) ≥ m a(S2) = α′(S2) ≥ 0 ∀S2 ∈]0 : 1[ β =qR a(s) b0(S2) ∈]0 : 1[

H1(Ω) = u ∈ Ω | R_Ω|u(x)|2_{+ |∇u(x)|}2_{dx < ∞ ||u||} H1 = kuk2 L2 + k∇uk2_L2
1/2 U =u ∈ H1(Ω), |u|Γe = 0 ; kukU = Z Ω |∇u|2dx ; H = L2(Ω) hu, viφ R Ωφuvdx U = L2(]0 : T [; U ) kuk2U = Z T 0 ku(x, t)k2Udt K = hu ∈ U | |u|Γs ≥ 0i W =  u ∈ U | du dt ∈ L 2_{(]0 : T [; U}′₎  (S2(x, t), P ) ∈ (W × L2(]0 : T [; H1(Ω)) P                                    − Z Ω ∇wdx + λ Z Γs (P − Ps)wdx = Z Γe vewdΓ ∀w H1(Ω) ]0 : 1[ = −d(S2(x, t))ψ · [∇P + γ1b1+ γ2b2] S2(x, t) ∈ W hdS2 dt , u − S2iφ+ Z Ω 2 · ∇(u − S2) ≥ 0 ∀u ∈ K ∀t ∈]0 : T [ S2(x, 0) = S2,0(x)

θ ∈]0 : 1] sup_{0≤ζ≤ξ≤1Rξ} ξ − ζ ζ p a(τ )dτ < ∞ M > 0 SM =u ∈ L2(Ω×]0 : T [)| 0 ≤ S(x, t) ≤ 1 Ω×]0 : T [ γ0(SM) L2(Γ) sup_0≤ζ≤1 bj β(ζ)  < ∞

nc △td_{= n} c× △tc C = △t △xS∈[0,1]max kλ(S)k , λ(S) = φ ∂F ∂S(S) F (S) = fj − fjd · [¯ρ − ∆ρj] g∇Z △ t φ △ xS∈[0,1]max ∂F ∂S ≤ C = 1. ∂u

∂t + (Lconv+ Ldif f)(u) = 0

un+1− un

△td + Ldif f(u) = 0

uk+1− uk

maxkT Ck+1− T Ckk < ǫ1+ ǫ2kT Ck+1k kT Ck+1_{k = max}  max Nf (|T Pk+1|), max Nf (|S₁k+1|), max Nf (|S₃k+1|)  ǫ1, ǫ2 Niter,max Niter,min • Niter> Niter,max • Niter< Niter,min

• Niter,min< Niter< Niter,max

tn T Pn+1 Qn+1 Pn+1 S_{dif f}n+1 Sconvn+1,k nc Sn+1

S0 E PE0 vE0 P_En S_En T P_En T S_En v_En P_En+1,k+1 T P_En+1,k+1 v_En+1,k+1 Niter < Nitermax k T C k Niter< Nitermin ¯ S_En+1,k+1 S_Econv,n+1,k+1 T S_En+1,k+1 N on n = n + 1 k = 0 ∆t × θr k = 0 ∆t × θa k = 0 t = Tf k = k + 1 θr= 0.5 θa= 2 Niter< Nitermin θa= 1

k RTnk(E) = Pk(E)n+ xPk(E)n

RT0(E) j RT20=      u v − 1  ;   u − 1 v  ;   u v      i • ∇ · j = 1 | E | • Z E ∇ · j = 1 • ( j · i) =        1 | ∂E | i = j 0 i 6= j • Z ∂Ej ( j· i)dΓ = δij.            ∇ · = 0 = −d . [∇P − ¯ρg∇Z] P (x, t) = P2(x, t) + Pcg(S1(x, t), S3(x, t), P (x, t)) Λρ¯= ¯ρg PE E SE,j j T Pi T Sj,i j Ei E RT20(E) E = 3 X i=1 QEi i, QEi Ei

u

u

u

v ∈ L2(E)

Z

E

(∇ · E)vdΩ = 0

( E, E) ∈ (RT20(E) × L2(E)) ∀( i, v) ∈ (RT20(E) × L2(E))

             Z E (∇PE − Λρ,E¯ ∇Z) · idΩ = − Z E [dE ]−1 E· idΩ Z E (∇ · E)vdΩ = 0 3 X j=1 QEj Z E [dE ]−1 j· idΩ = Z E Λρ,E¯ ∇Z · idΩ − Z E ∇PE· idΩ = 0 ∀ i ∈ RT20(E) 3 X j=1 QEj 1 dE Z E j −1· idΩ = PE Z E ∇ · idΩ − Z ∂E T PE i· n∂EdΓ + Λρ,E¯ Z E ∇Z · idΩ = PE Z E ∇ · idΩ − 3 X j=1 Z ∂Ej T PEj i· njdΓ + Λρ,E¯ Z E ∇Z · idΩ

Eji = Z E 1 dE j −1_· idΩ GEi = Z E ∇Z · idΩ i 3 X j=1 QEjBEji = PE − T PEi+ Λρ,E¯ GEi BEji ∂E QEj = 3 X i=1 B_E−1 ji(PE− T PEi) + Λρ,E¯ 3 X i=1 B_E−1 jiGEi GEi = Z E ∇Z · idΩ Λρ,E¯ = [fω(S)ρω+ fnω(S)ρnω] g GEi (x1, z1) (x1, z1) (x3, z3)   GE,1 GE,2 GE,3  = −1 6       (z1− z3) + (z2− z3) (z3− z2) + (z1− z2) (z2− z1) + (z3− z1)       S = SE

Z E ∇ · EdΩ = 0 Z ∂E E · EdΓ = 0 3 X i=1 QEi Z ∂E i Ei dΩ = 0 i 3 X i=1 QEi = 0 t ∈ [0, T ] tn= n△t = n T N, ∀n ∈ [|0, n|] 3 X i=1 Qn+1_Ei = 0 tn+1 3 X i,j=1 B−1_Eij(P_En+1− T P_En+1_j ) + Λρ,E¯ 3 X i,j=1 B_E−1 ijGEj = 0 P_En+1 P_En+1= 3 X i,j=1 B_Eij−1T P_En+1 j − Λρ,E¯ 3 X i,j=1 B−1_EijGEj   3 X i,j=1 B_Eij−1  

tn+1 E E′ • Γs TPn+1_Ej = TPD • Γe QEi= QN • TPn+1_Ei = TPn+1_E′_i • Qn+1_Ei + Qn+1_E′_i = 0 ⇔ 3 X j=1 B_Eij−1P_En+1− 3 X j=1 (BEij)−1TPn+1Ej + Λρ,E¯ 3 X j=1 (BEij)−1GEj + 3 X j=1 B_Eij−1P_En+1′ − 3 X j=1 BE′ ij
−1 TPn+1_E′_,j+ Λρ,E¯ 3 X j=1 (BE′ ij)−1GE′ ,j = 0 • αEi = 3 X j=1 (BEij)−1 • αE = 3 X i,j=1 (BEij)−1 tn+1 P_En+1= 3 X i,j=1 B_Eij−1T P_En+1 j − Λρ,E¯ 3 X i,j=1 B_Eij−1GEj αE αEi αE   3 X i,j=1 B−1_EijT P_En+1 j − Λρ,E¯ 3 X i,j=1 B−1_EijGEj  − αEiTPn+1_Ej + Λρ,E¯ αEiGEj +αEi αE′   3 X i,j=1 B_E−1′_ijT P n+1 E′ j − Λρ,E¯ 3 X i,j=1 B_E−1′_ijGE′ j  − α_E′ iTPn+1E′_,j+ Λρ,E¯ αE′iGE′,j = 0

 αEi αE 3 X i,j=1 B_Eij−1 − αEi+ αE′_i αE′ 3 X i,j=1 B−1_E′_ij− αE′i  T P_En+1 j −Λρ,E¯  αEi αE 3 X i,j=1 B_Eij−1 − αEi  GEj − Λρ,E¯ ′  αE ′ i αE′ 3 X i,j=1 B_E−1′_ij − αE′i  G_E′ j = 0 γ_En+1 i = αEi αE Nf (MP)Nf×NfTP n+1,k+1 Nf = FNf − RNf  γ_En+1,k+1 i 3 X i,j=1 (Bn+1,k+1_Eij )−1− αn+1,k+1_Ei  T P_En+1,k+1 j ◮ _MP • mxx mxx = mEii+ mE′_,jj : m_Eii= (γn+1,k+1 Ei − 1)(B n+1,k+1 Eii )−1 • mxy mxy = mEij = 0 i ∈ E j /∈ E

mxy = mEij : mEij = (γ_En+1,k+1_i − 1)(B_Eijn+1,k+1)−1

◮ F_N f fx = −ΛρE  γ_En+1,k i 3 X i,j=1 (B_Eijn+1,k)−1− 3 X j=1 (B_Eijn+1,k)−1  GEj −ΛρE′  γ_En+1,k′ i 3 X i,j=1 (Bn+1,k_E′_ij ) −1₋ 3 X j=1 (B_En+1,k′_ij ) −1  G_E′ j

R_N f i E ΓN rEi RNf  rEi = −QN i ⊂ ΓN rEi = 0

f1f2d (ρ2 − ρ1)g∇Z                            φ∂(S2) ∂t + ∇ · ( 2) = 0 2 = f2+ f1f2d(ρ2− ρ1)g · ∇Z | {z } − f1f2d dP_c12 dS2 · ∇S2 | {z } S1+ S2 = 1                      φ∂(Sl) ∂t + ∇ · ( l) = 0 l = 1, 3 l = −νld ·∇(Pcl2− Pcg) 3 X l=1 Sl= 1 ˜_l= − l,E· ∇Sl l,E = νld · d(P_cl2− Pcg) dSl l = 1, 3 φ∂(Sl) ∂t + ∇ · (˜l) = 0 l = 1, 3

˜_l,E = − [ l,E] · ∇Sl= 3 X j=1 ˜ Ql,Ej˜j ˜i ∈ H(div, E) Z E [ l,E]−1˜l,E· ˜idΩ = Z E ∇Sl· ˜idΩ φ∂ (Sl) ∂t + ∇ · (˜l) = 0 v ∈ L2_(E) φ Z E  ∂Sl,E ∂t + ∇ · (˜l,E)  vdΩ = 0

((˜_1,E, ˜_3,E), (S1,E, S3,E)) ∈ (RT20(E)2 × L2(E)2) ∀(˜i, v) ∈ (RT20(E) ×

L2(E))                                    Z E [ 1,E]−1· ˜1,E · ˜idΩ = Z E ∇S1· ˜idΩ Z E [ 3,E]−1· ˜3,E · ˜idΩ = Z E ∇S3· ˜idΩ φ Z E  ∂S3,E ∂t + ∇ · (˜3,E)  vdΩ = 0 φ Z E  ∂S1,E ∂t + ∇ · (˜1,E)  vdΩ = 0 l,E= νld · d(Pl2 c − Pcg) dSl

σD = νld · d(P_cl2− Pcg) dSl σD(Sl)E = σD ! ₃ X i=1 T Sl,Ei 3 " σD(Sl)E = 3 X i=1 σD(T Sl,Ei) 3 l = 1, 3 3 X j=1 ˜ Ql,Ej Z E [ l,E]−1· ˜j· ˜idΩ = Z E ∇Sl· ˜idΩ 3 X j=1 ˜ Ql,Ej Z E ˜j −1l,E· ˜idΩ = Sl,E Z E ∇ · ˜idΩ − Z ∂E T Sl,Ei˜i· n∂EdΓ ∀˜i ∈ RT 2 0(E). ˜ Bl,Eji = Z E (˜j l,E)−1· ˜idΩ ˜i 3 X j=1 ˜

˜ B ∂Ei ˜ Ql,Ej = 3 X i=1 ˜ B−1_l,E ji(Sl,E− T Sl,Ei). l = 1, 3 φ Z E ∂Sl ∂t + 3 X i=1 ˜ Ql,Ei Z E ∇ · (˜i)vdΩ = 0 v = vE vE φ|E|∂Sl,E ∂t + 3 X i=1 ˜ Ql,Ei = 0 t ∈ [0, T ] tn= n△t = n T N, ∀n ∈ [|0, n|] l = 1, 3 φ|E|S n+1 l,E − Sl,En △t + 3 X i=1 ˜ Qn+1_l,E i = 0 ˜ Qn+1 φ|E|S n+1 l,E − Sl,En △t + 3 X i,j=1 ˜ B_l,E−1 ij(S n+1 l,E − T Sn+1l,Ej) = 0 S_l,En+1 l = 1, 3 S_l,En+1 = φ|E| △t S n l,E− 3 X i,j=1 ˜ B_l,E−1 ijT S n+1 l,Ej φ|E| △t + 3 X i,j=1 ˜ B_l,E−1 ij

tn+1 E E′ • Γe TSn+1_Ej = TSD • Γs Q˜Ei = ˜QN • TSn+1_Ei = TSn+1_E′ i • ˜ Qn+1_l,Ei + ˜Qn+1_l,E′ i = 0 ⇔ 3 X j=1 ˜ B_l,Eij−1 S_j,En+1− 3 X j=1 ˜ B_l,Eij−1 TSn+1_l,E i + 3 X j=1 ˜ B_l,E−1′ ijS n+1 l,E′ − 3 X j=1 ˜ B_l,Eij−1 TSn+1_l,E′ i = 0 l = 1, 3 (MDif f,l)Nf×NfTS n+1,k+1 l,Nf = ˜Fl,Nf − ˜Rl,Nf                                      φ∂(Sl) ∂t + ∇ · ( l) = 0 l = 1, 3 l(S, p) = l = νl − νl∆ρd · g∇Z ∆ρ1 = (ν2(ρ2− ρ1) + ν3(ρ3− ρ2)) ∆ρ3 = (ν2(ρ2− ρ3) + ν1(ρ3− ρ1)) 3 X l=1 Sl = 1 l = 1, 3

Ω φ ∈ P1_(Ω) l Sl( , t)E = 3 X j=1 S_l,jE(t)ϕE_l,j( ) νl νlν2dg νl( , t)E = 3 X j=1 ν_l,jE(t)ϕE_j( ) (νlν2gd)( , t)E = 3 X j=1 ψE_l,j(t)ϕE_j ( ) = 3 X j=1 E j (t)Φj( ) g = ∆ρ · ∇Z = X j GE_j (t)Φj( ) ϕE i ∈ C0∞ φ ∂(S E l ) ∂t ϕ E i  = −∇ · (νl )ϕEi + ∇ · (νlν2dg∆ρ · ∇Z)ϕEi . φ 3 X j=1 # ∂(SE l,j) ∂t ϕ E j ϕEi % = − 3 X j=1 ∇ · (ν_l,jEϕE_j )ϕE_i + 3 X j=1 ∇ · (ψE_l,jϕE_j ∆ρ · ∇Z)ϕE_i .

φ 3 X j=1 # ∂(S_l,jE) ∂t Z E ϕE_j ϕE_i % = − 3 X j=1 Z E (ν_l,jEϕE_j ) · ∇ϕE_i dX | {z } Aj + 3 X j=1 Z ∂E (ν_l,jEϕE_j ϕE_i ) · dΓ | {z } Bj + 3 X j=1 Z E (ψE_l,jϕE_j )∆ρ · ∇Z · ∇ϕEi dX | {z } Cj − 3 X j=1 Z ∂E (ψ_l,jEϕE_j ϕE_i )∆ρ · ∇Z · dΓ | {z } Dj ! φ∂S E l,j ∂t " j

(MConv,l)j,i= Bi,j− Ai,j+ Ci,j− Di,j

∂SE l,j ∂t = 1 φ 3 X i=1

(M−1_Conv,l)i,j(Bi,j− Ai,j+ Ci,j− Di,j)

S_l,jE,n+1 = S_l,jE,n+∆t c φ 3 X i=1

(M−1_Conv,l)i,j(Bi,j− Ai,j+ Ci,j − Di,j)

EJ νl,jE(Sl)

EJ Sl(EJ) = 3 X i=1 SNJ,i 3 ν_l,jE(Sl) · i ν(S) EIEJ νEJ l,j (Sl) =      νEJ l,j(SNJ,1), ν EJ l,j (SNJ,2) · 1≥ 0 νEJ l,j (SNI,1), ν EJ l,j (SNI,2) · 1≤ 0 ψl S_l∗ ψld ψ′_l(S_l∗) = 0 S∗ 2 EJEK

g· 2< 0                                ψEJ l,j(SNJ,1) SNK,1< S∗ SNJ,1< S∗ ψEJ l,j(SNK,1) SNK,1> S∗ SNJ,1> S∗ ψEJ l,j(S∗) SNK,1> S∗ SNJ,1< S∗ min{ψEJ l,j(SNK,1), ψ EJ l,j (SNJ,1)} SNK,1< S∗ SNJ,1> S∗ g· 2≥ 0                                ψEJ l,j (SNJ,1) SNK,1> S∗ SNJ,1> S∗ ψEJ l,j (SNK,1) SNK,1< S∗ SNJ,1< S∗ ψEJ l,j (S∗) SNK,1< S∗ SNJ,1> S∗ min{ψEJ l,j (SNK,1), ψ EJ l,j(SNJ,1)} SNK,1> S∗ SNJ,1< S∗ S∗ S∗ S∗ S∗ S∗ F (S) = f2(qc+ f1(ρ2− ρ1)qg) qc = 10−2 m · s−1 ρ2− ρ1 = 800 kg · m−3

0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.2 0.4 0.6 0.8 1 Fonction flux [m/s] Saturation en eau S₁ q_t/q_g=0.0001 q_t/q_g=100 q_t/q_g=103 q_t/q_g=104 -0.0014 -0.0012 -0.001 -0.0008 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.6 0.65 0.7 0.75 0.8

Derivee de la fonction flux [m/s]

Saturation en eau S₁ q_t/q_g=0.0001 q_t/q_g=100 q_t/q_g=103 q_t/q_g=104 S₁∗ S∗ F (S) = f3(qc+ f1(ρ3− ρ1)qg) qc = 10−5 m · s−1 ρ3− ρ1 = −1000 kg · m−3

-0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.2 0.4 0.6 0.8 1 Fonction flux [m/s] Saturation en eau S₁ q_t/q_g=0.01 q_t/q_g=10 q_t/q_g=103 q_t/q_g=104 -0.0005 0 0.0005 0.001 0.0015 0.002 0.8 0.81 0.82 0.83 0.84 0.85

Derivee de la fonction flux [m/s]

Saturation en eau S₁ q_t/q_g=0.01 q_t/q_g=10 q_t/q_g=103 q_t/q_g=104 S₁∗

u(x, t) : RN× [0, +∞[−→ RN _{A ∈ R}N _{× R}N f (u) = A · u C2_(RN _{× [0, ∞[)}      ∂u ∂t + ∂f (u) ∂x = 0 u : R N _{× [0, +∞[−→ R}N u(x, 0) = u0(x) t > 0 x ∈ R C1(RN × [0, ∞[)

N = 1 L∞_LOC(R × [0, +∞[) Z +∞ 0 Z R u∂tψ + f (u)∂xψ dx dt + Z R u(x, 0)ψ(x, 0) dx = 0, ψ ∈ C1 0(R × [0, +∞[) v          ∂v ∂t + ∂f (v) ∂x = 0 v(x, 0) =  vg(x, t) si x < σ(t) vd(x, t) si x > σ(t) Σ = {σ(t), t} R × R+ vg, vd ∂u ∂t + ∂f (u) ∂x = 0 Σ [v] (t) = lim_t→σ(t)+v_d(t) − lim_t→σ(t)−v_g(t). v t ≥ 0 [f (v)] (t) = dσ dt(t) [v] (t) σ(t) = ζ + st vg(x, t) = vg vd(x, t) = vd f (vg) − f (vd) = s(vg− vd).

C1 ∂v ∂t+ ∂f (v) ∂x = 0 • v v C1 • v s [v] = [f (v)] , s      ∂u ∂t + ∂f (u) ∂x = 0 u(x, 0) = u0(x) u0 ∂tX(x, t) = f′(X(x, t), t) X(x, 0) = x0. du(X(x, t), t) dt = ∂u ∂t(X(x, t), t) + ∂X(x, t) ∂t ∂u ∂x(X(x, t), t) = (∂u ∂t + f ′_{(X(x, t), t)}∂u ∂x)(X(x, t), t) = 0 x0 f′_(u(x 0, 0))

t x 0 ✻ ✲ u0(x) C1(R) ∩ L∞(R) u′0(x) ∈ L∞(R) • f′′(u0(x))u′0(x) ≥ 0, ∀x ∈ R u ∈ C1([0, +∞[×R) • u ∈ C1([0, −1 inf_x∈R{f′′_(u 0(x))u′0(x)} [×R) Tmax= −1 inf_x∈R{f′′_(u 0(x))u′0(x)} u(t, x + tf′(u0(x)) = u0(x)                ∂u ∂t + 1 2 ∂(u2₎ ∂x = 0 u(x, 0) =    1, x < 0 1 − x, 0 ≤ x ≤ 1 0, x > 1

t x 0 ✻ ✲ 1 2(u 2 d+ u2g) = [σ][u] [u] 6= 0 σ = (ud+ ug) 2 t < T = 1 t x 0 u(x) = 1 − x 1 − t σ = 1 2 ✻ ✲ u = 1 u = 0 t = 1 x = 1

     ∂u ∂t + ∂f (u) ∂x = 0 u(x, 0) = u0(x) η ξ : R −→ R ξ′= η′f′ u(x, t) =      ∂u ∂t + ∂f (u) ∂x = 0 u(x, 0) = u0(x) ∈ L∞LOC(R) η ξ ∀ϕ ∈ C₀∞(R × [0, ∞[), ϕ ≥ 0 R  ∂ϕ ∂tη(u) + ∂ϕ ∂xξ(u)  dxdt ≥ 0 u0 ∈ L∞(R)          ∂u ∂t + ∂f (u) ∂x u(x, 0) =  ug(x, t) si x < σ(t) ud(x, t) si x > σ(t) u ∈ L∞(R × [0, T ]). k u(., t) kL∞_(R)≤k u(., 0) k_L∞_(R). u x_t

• u1 u0 u(x, t) =    u0 si x/t ≤ f′(u0) v(x/t) si f′_(u 0) ≤ x/t ≤ f′(u1) u1 si x/t ≥ f′(u1), v(x/t) • u1 u0 u(x, t) =  u0 si x < st u1 si x > st, s = f (u1) − f (u0) u1− u0 S2(x, t)                                        φ∂S2 ∂t + ∇ · (S2) = 0, S2(x, t) ∈ L∞_LOC(R × [0; +∞[) S2(x, 0) =  1 si x < 0 0 si x > 0 (S2) = f2+ f1f2d(ρ2− ρ1)g · ∇Z (S2) ∈ C1([0; 1])

           ∇ · = 0 = −d . [∇P − ¯ρg∇Z] P (x, t) = P2(x, t) = P1(x, t) φ∂S2 ∂t + dF dS2 ∂S2 ∂x = 0, 2= 1 φ dF (S2) dS2 ˜ S2(t, x) dx dt = 2= 1 φ ′_{( ˜S} 2) x(t) = 1 φ Z t 0 ′_{( ˜S} 2)dt S2,f Fobjectif(S2,f) = 1 2φ  d dS2 |S2,f − (S2,f) S2,f 2 = 0 F (S2,d) = 0

v

g

❄

S = 1

S = 0 Ps

❄ Pression (Pa) dans le milieu poreux

0 0.2 0.4 0.6 0.8 1 x (m) -10 -8 -6 -4 -2 0 z (m) 100000 110000 120000 130000 140000 150000 160000 170000 180000 190000 200000

j j

• Sj =

Sj− Sj,r

1 − (S1,r+ S2,r)

• pj(x, t) j Sj(x, t) j pc= p2− p1. ρj [kg · m−3] j ∆ρ = ρ2− ρ1 [kg · m−3] • kr1(S1) = S12, S1∈ [S1,r, 1 − S2,r]; kr2(S2) = S22, S2∈ [S2,r, 1 − S1,r]; • λj = krj(Sj) µj [kg−1· m · s] • d = λ1+ λ2 [kg−1· m · s] • fj = λj λ1+ λ2 [−] • ¯ρ(Sj) =Pj=1,2fjρj [kg · m−3] Ω ⊂ R3 ΓΩ ∂φSj ∂t + ∇ · ( j) = 0 x ∈ Ω, t > 0, j j j = − krj(Sj) µj · (∇pj− ρj g∇z) x ∈ Ω, t > 0 φ z(x) [m] K[m2] µj[kg·m−1·s−1] j

P_cg(S1, p) = Z S1 1 f1(S1)∂pc ∂S1 dS1, p = p2+ Pcg(S1, p), p1≤ p ≤ p2 φ φ∂ ∂t(S1+ S2) − ∇ · ( t) = 0, φ∂S2 ∂t + ∇ · ( 2) = 0, t:= 1+ 2 = −d · (∇p − ¯ρg∇z) . S1+ S2 = 1, pc = p2− p1= 0. p2 = p1 = p t t Γo ⊂ ΓΩ

• ∇ · ( t) = 0, t= −d · [∇p − ¯ρg∇z] , • φ∂S2 ∂t + ∇ · ( 2) = 0, 2 = f2(S2) t+ f1(S2)f2(S2)d(S2)∆ρg∇z = (S2), • S1+ S2 = 1. M = µ1 µ2 [−] , Nc= t µ1 σ12cos(θ) [−] , Nb = a2cos αg△ρ σ12cos θ [−] , t[m · s−1] σ12[kg · s−2] θ α [m]

• Nb ≤ 0 • Nb ≥ 0 N = Nb Nc = a 2_{cos αg△ρ} tµ1 [−] N G = N K a2 = cos αKg△ρ tµ1 [−]. (S2) ∂S2 ∂t + 1 φ∇ · ( (S2)) = 0 S2(x, t = 0) =  1 − S1,r x ∈ Γo S2,r x ∈ Ω (S2) = f2 t.

z z α = 0 ∂S2 ∂t + 1 φ∇ · (S2) = 0, S2(z, t = 0) =  1 − S1,r z ∈ Γo S2,r z ∈ Ω , (S2) · ∇z = F (S2). F′(S2) F′ : [S2,r: 1] → R S2→ t  k′ r2kr1− kr2kr1′ + ∆ρgK tµ1 k2 r1kr2′ + M k2r2kr1′
 µ1µ2d2 . F′ F′(S2) = 2S2(1 − S2) t1 + G (1 − S2)3− M S23
 µ1µ2d2 . F′_(S 2) u : [S2,r: 1] → R u(S2) =1 + G (1 − S2)3− M S23
 . u′(S2) = −3G (1 − S2)2+ M S22
, ∆ρ > 0 ∆ρ > 0 u(S2,r) = 1 + G (1 − S2,r)3− M S2,r3
, u(1 − S1,r) = 1 + G (S1,r)3− M (1 − S1,r)3

t= 10−3m · s−1 ∆ρ = 400kg · m−3 M = 1 − ∆ρ ≥ 0 u(1 − S1,r) ≥ 0 u(S2) F′ RL = F (SR) − F (SL) SR− SL , SL SR F′(SL) > F′(S) > F′(SR) ∀S ∈ [SL, SR]. α G = ∆ρgK tµ1 ≤ 1 M (1 − S1,r)3− (S1,r)3 . ∆ρ > 0 u(1 − S1,r) ≥ 0 G ≤ 1 M. − ∆ρ > 0 u(1 − S1,r) < 0

M = 1 u(1 − S1,r) < 0 G = ∆ρgK tµ1 > 1 M (1 − S1,r)3− (S1,r)3 , G > 1 M. S₂entry 400 kg · m−3 M

t = 10−5m · s−1

∆ρ = 400 kg · m−3 M = 1

tµ1 > maxS2∈[0,1][| ∆ρ | gK] =| ∆ρ | gK. |G| < 1. − ∆ρ < 0 1 + G > 0 ∆ρ < 0 u(0) = 1 + G u(1) = 1 − GM > 0 |G| < 1 S2,r= 0 1 − S1,r M = µ1 µ2 ∆ρ u(S₂entry) = 0.

S_2,front µ₁/µ₂=0.05 -S2,front µ1/µ2=0.1 -S2,front µ1/µ2=0.5 -S2,front µ1/µ2=1 -S2,front µ1/µ2=5 S₂entry, µ₁/µ₂=5 0 200 400 600 800 1000 ∆ρ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−3_{m · s}−1 S_2,front µ1/µ2=0.05 -S_2,front µ1/µ2=0.1 -S_2,front µ1/µ2=0.5 -S_2,front µ₁/µ₂=1 -S_2,front µ₁/µ₂=5 S₂entry, µ₁/µ₂=0.5 S₂entry, µ1/µ2=1 S₂entry, µ1/µ2=5 0 200 400 600 800 1000 ∆ρ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−4m · s−1 S2,front µ1/µ2=0.05 -S_2,front µ₁/µ₂=0.1 -S2,front µ1/µ2=0.5 -S2,front µ1/µ2=1 -S_2,front µ₁/µ₂=5 S2 entry , µ1/µ2=0.05 S₂entry, µ₁/µ₂=0.1 S2 entry , µ1/µ2=0.5 S₂entry, µ₁/µ₂=1 S₂entry, µ₁/µ₂=5 0 200 400 600 800 1000 ∆ρ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−5m · s−1

∂S2 ∂t + 1 φ∇ · ( (S2)) = 0. (S2) = f2 t+ Kf1f2d(ρ2− ρ1)g∇z, S2(x, t = 0) =  1 z ≤ 0 0 z > 0. . f2(S2) t t Kf1(S2)f2(S2)d(S2)△ρg∇z △ρg∇z Ω P1(Ω) Vh=ϕ ∈ L∞(Ω)|ϕ|E ∈ P1(E) . ϕE_i ∈ P1(E) φ Z E ∂S ∂tϕ E i dX = − Z E ∇ · [f2 t]ϕEi dX. − Z E ∇ · [Kf1f2d(ρ2− ρ1)g]ϕEi dX.

φ Z E ∂S ∂tϕ E i dX = + A z }| { Z E f2 t· ∇ϕEi dX − B z }| { Z ∂E f2ϕEi t· dΓE + Z E Kf1f2d(ρ2− ρ1)g∇z · ∇ϕEi dX | {z } C − Z ∂E Kf1f2d(ρ2− ρ1)gϕEi ∇z · dΓ | {z } D . EJ f1(S) f2(S) d(S) EJ S2(EJ) = X n=1,2,3 SNJ,n 3 . f2 t· i EI EJ f2(S) =      f2(SNJ,1), f2(SNJ,2) t· 1 ≥ 0, f2(SNI,1), f2(SNI,2) t· 1 ≤ 0. S∗ f1f2d (f1f2d)′(S∗) = 0.

EJ f1f2d S∗ EJ EK △ρ∇z · 2 ≤ 0 (f1f2d)(SNJ,1) SNK,1< S∗ SNJ,1< S∗, (f1f2d)(SNK,1) SNK,1> S∗ SNJ,1> S∗, (f1f2d)(S∗) SNK,1> S∗ SNJ,1< S∗, min{(f1f2d)(SNK,1), (f1f2d)(SNJ,1)} SNK,1< S∗ SNJ,1> S∗ △ρ∇z · 2 > 0 (f1f2d)(SNJ,1) SNK,1> S∗ SNJ,1> S∗, (f1f2d)(SNK,1) SNK,1< S∗ SNJ,1< S∗, (f1f2d)(S∗) SNK,1< S∗ SNJ,1> S∗, min{(f1f2d)(SNK,1), (f1f2d)(SNJ,1)} SNK,1> S∗ SNJ,1< S∗. (f1f2d) · S∗6= S₂entry C = △t △xS∈[0,1]max kλ(S)k ,

S∗ λ(S) = t φ ∂F ∂S(S) t△ t φ △ xS∈[0,1]max ∂F ∂S ≤ C = 1. △t = 0.5 △x < 10−2 t ρ2= 1456 kg · m−3

[m2_{] 5 × 10}−11 2 × 10−4 α[−] 0 [m · s−2] 9.81 | t| [m · s−1] 10−3 10−3 10−4 10−5 ∆ρ [kg.m−3_] µ1 [P a · s−1] 10−3 10−3 10−3 10−3 Tf S2(x = 0) G [−] 0/0.294/ − 0.294 0.224 2.24 29.4 1 M [−] 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Depth (m) S2

Quasi Analytical Buckley ∆ρ=+456 kg.m-3 Numerical Nc=102 Cells ∆ρ=+456 kg.m-3 Numerical Nc=408 Cells ∆ρ=+456 kg.m-3 Numerical Nc=1832 Cells ∆ρ=+456 kg.m-3 t = 10−3m.s−1 10−3_{m · s}−1 ∆ρ ∆ρ 1 M

0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Depth (m) S2 DGFEM ∆ρ=0 kg.m-3 DGFEM ∆ρ=+600 kg.m-3 DGFEM ∆ρ=-600 kg.m-3 Semi-Analytical ∆ρ=0 kg.m-3 Semi-Analytical ∆ρ=+600 kg.m-3 Semi-Analytical ∆ρ=-600 kg.m-3 t = 10−3m · s−1 M = 1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Depth (m) S₂ DGFEM ∆ρ=+600 kg.m-3 Quasi-Analytical ∆ρ=+600 kg.m-3 Finite Volume Method ∆ρ=+600 kg.m-3

0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Depth (m) S2 DGFEM ∆ρ=+456 kg.m-3 Quasi-Analytical ∆ρ=+456 kg.m-3 Finite Volume Method ∆ρ=+456 kg.m-3

t= 10−3m · s−1 M = 0.05 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 Depth (m) S2 DGFEM ∆ρ=+456 kg.m-3 Quasi-Analytical ∆ρ=+456 kg.m-3 Finite Volume Method ∆ρ=+456 kg.m-3

t = 10−4m · s−1

0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Depth (m) S2 DGFEM ∆ρ=+600 kg.m-3 Quasi-Analytical ∆ρ=+600 kg.m-3 Finite Volume Method ∆ρ=+600 kg.m-3

t = 10−5m · s−1

1e-18 1e-17 1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 0 100 200 300 400 500 600 700 800 900 1000

log(Cumulated Mass Error)

Time (s) DGFEM, Example 1 ∆ρ=+600 kg.m-3 FV, Example 1 ∆ρ=+600 kg.m-3 ∆ρ = +600 kg · m−3 M = 1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 -log(||S Analytic -SNum ||L 1)) -log(∆ x) DGFEM, Example 1 ∆ρ=+600 kg.m-3 0.86x-1.3 FV, Example 1 ∆ρ=+600 kg.m-3 0.79x-2.6 ∆ρ = +600 kg · m−3 M = 1

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 -log(||S Analytic -SNum ||L 1)) -log(∆ x) DGFEM, Example 4 ∆ρ=+600 kg.m-3 0.78x-0.75 FV, Example 4 ∆ρ=+600 kg.m-3 0.765x-0.95 ∆ρ = +600 kg · m−3 M = 1 Ω1 Γ1 Ω2 ⊂ Ω1 K2 = 5 × 10−14 [m2] φ2 = 0.1 K1 = 5 × 10−11 [m2] φ1 = 0.3 Ω1 \ Ω2 S2 = 1 Γ3 ⊂ Γ1 x0 t = 0 Γ1 \ Γ3 t = v Γ3 t Mij = Z R2 S2(x, z)xizjdxdz

¯ x = M10 M00 [m] z =¯ M01 M00 [m]. σxx z σzz x z σ_xx2 = M20 M00 − ¯ x2 [m2] σ_zz2 = M02 M00 − ¯ z2[m2].

α[−] 0 0 | t| [m · s−1] 10−3 10−5 Tf µ1 [P a · s−1] 1.2 × 10−3 1.2 × 10−3 1 M [−] ∆ρ [kg.m−3_] | G | [−] φ1 [−] φ2 [−]

mass center position

0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 0.2 0.4 0.6 0.8 1

mass center position 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 0.2 0.4 0.6 0.8 1

mass center position

0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 0.2 0.4 0.6 0.8 1

mass center position 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 0.2 0.4 0.6 0.8 1 x z z x

mass center position 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 1 2 3 4 5 6 x(m) 0 0.5 1 1.5 2 Depth z (m) 0 0.2 0.4 0.6 0.8 1 z

-0.4 0 0.4 0.8 1.2 0 40000 80000 Depth z (m) Time (s) Example B FV 0 40000 80000 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s) x-x0 (m) Example B FV -0.4 0 0.4 0.8 1.2 0 40000 80000 Depth z (m) Time (s) Example B DGFEM 0 40000 80000 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (s) x-x0 (m) Example B DGFEM -0.4 0 0.4 0.8 1.2 0 4000 8000 Depth z (m) Time (s)

Example A DGFEM on a fine grid 0

4000

8000

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

x-x0 (m)

Example A DGFEM on a fine grid

-0.4 0 0.4 0.8 1.2 0 4000 8000 Depth z (m) Time (s)

Example A FV on a fine grid 0

4000

8000

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

x-x0 (m)

Example A FV on a fine grid

-0.4 0 0.4 0.8 1.2 0 4000 8000 Depth z (m) Time (s)

Example A DGFEM on a coarse grid 0

4000

8000

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

x-x0 (m)

Example A DGFEM on a coarse grid

z z − σ¯ zz x + σ¯ xx

¯

Vh=φ ∈ L∞(Ω)|φ|E ∈ P1(E) ϕE₁(x, y) = 1 − x − y, ϕE₂(x, y) = x, ϕE₃(x, y) = y, ϕE_i (xj, yj) = δij. H( , Ω) ΦE₁(x, y) = (x, y − 1), ΦE₂(x, y) = (x − 1, y) ΦE 3(x, y) = (x, y). SNE j (xj, yj) Vh S2( , t)E = S2E = 3 X j=1 SN_jE(t)ϕE_j ( ). • f1(S2) =Pjf1(SNj)ϕEj • f2(S2) =Pjf2(SNj)ϕEj • d(S2) =P_jd(SNj)ϕEj =X l lΦl( ), = X l lΦl( ) = (ρ2− ρ1)g · ∇z.

S_2,front ∆ρ=50 -S_2,front ∆ρ=200 -S_2,front ∆ρ=456 -S₂* ∆ρ=456 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ₁/µ₂ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−3_{m · s}−1 S_2,front ∆ρ=50 -S_2,front ∆ρ=200 -S_2,front ∆ρ=456 -S₂* ∆ρ=50 -S₂* ∆ρ=200 -S₂* ∆ρ=456 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ₁/µ2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−4_{m · s}−1

S_2,front ∆ρ=50 -S_2,front ∆ρ=200 -S_2,front ∆ρ=456 -S₂* ∆ρ=50 -S₂* ∆ρ=200 -S₂* ∆ρ=456 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ₁/µ2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Oil Saturation S 2 ∆ρ t = 10−5_{m · s}−1

                     φ∂Snω ∂t + ∇ · nω = 0 nω = νnω + νωνnωd(ρnω− ρω) · ∇z − νωνnωd dPc dSnω (Snω) · ∇Snω Sω+ Snω = 1                                ∇ · = 0 = −d . [∇P − ¯ρg∇Z] P = Pnω+ Pcg Pcg(Sω) = − Z Snω 0 νω ∂Pc ∂Snω dSnω Pc= Pnω− Pω

20 C ω nω S2 = 0.0 p3= 1 · 105 P a x = 0 S2= 1 vt= 0 x = 10 ✲ ✻ ✲ ✛

                                     φ∂Snω ∂t + ∇ · (Snω) = 0, Snω(x, t) ∈ L∞_LOC(R×]0; +∞[) Snω(x, 0) = Si,nω (Snω) = −fnωfωd dPc dSnω (Snω) · ∇Snω (Snω) ∈ C1(]0; 1[) = 0) φ 0.3 [−] K 1 ×10−7 _[m2_] ρ3 1.29 [kg/m3] ρ1 1000 [kg/m3] ρ2 1000 [kg/m3] Pd 1 ×103 [P a] λ 2 [−] S1r 0.1 [−] S2r 0.01 [−] S3r 0.01 [−] µ1 1 ×10−3 [P a · s] µ3 1 ×10−5 [P a · s] µ2 1 ×10−3 [P a · s] gz 0 [m/s2] ∆t 0.1/0.5/1 [s]

0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Saturation en eau x(m)

MPTRACES METHODE A σ(TSmoy), ∆t=1

MPTRACES METHODE A σ(TSmoy), ∆t=0.1

MPTRACES METHODE B moy(σ(TS)), ∆t=0.5 Mc Whorter Sunada analytique t=200s

Saturation en eau t=200s 0 0.5 1 1.5 2 2.5 3 3.5 4 x(m) 0 0.2 0.4 0.6 0.8 1 y(m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.5                                                                    φ∂S3 ∂t + ∇ · (S3) = 0, S3(x, 0) =  1 si x < 0 0 si x > 0 S1(x, 0) =  1 si x < 0 0 si x > 0 (S3) = f1f3d dPc dS1 · ∇S3 = 1+ 3 = 0 P (x, t) = P1(x, t) = P3(x, t) S3 = 1 p3 = 1 · 105 P a x = 0 S3 = 0 vt= 0 x = 10 ✲ ✻ ✲ ✛

S1 = 1 p3= 1 · 105 P a x = 0 S1= 0 vt= 0 x = 10 ✲ ✻ ✲ ✛

0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 Saturation en gaz x(m)

Solution numerique MPTRACES t=50s Solution numerique MPTRACES t=100s Solution analytique t=50s Solution analytique t=100s Saturation en gaz t=50s 0 2 4 6 8 10 x(m) 0 0.2 0.4 0.6 0.8 1 y(m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Saturation en gaz t=100s 0 2 4 6 8 10 x(m) 0 0.2 0.4 0.6 0.8 1 y(m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Saturation en eau x(m)

Solution numerique MPTRACES t=100s Solution numerique MPTRACES t=200s Solution analytique t=100s Solution analytique t=200s Saturation en eau t=100s 0 1 2 3 4 5 6 x(m) 0 0.2 0.4 0.6 0.8 1 y(m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Saturation en eau t=200s 0 1 2 3 4 5 6 x(m) 0 0.2 0.4 0.6 0.8 1 y(m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R&D

S3 = 0 P3 = 1 · 105 P a P3= 1.1 · 105 P a x = 0 P3 = 1 · 105 P a S3= 0 P3= 1.1 · 105 P a x = 25 ✲ ✻ S3= 1 P3 = 1 · 105 P a r = −2m r = −2m φ 0.4 [−] K 8 ×10−11 _[m2 ] ρnw 1.29 [kg/m 3 ] ρ1 1000 [kg/m 3 ] ∆ρ −998 [kg/m3 ] Pd 2 ×103 [P a] λ 2 ×100 [−] S1r 0.13 [−] S3r 0.01 [−] µ1 1 ×10−3 [P a · s] µ3 1 ×10−5 [P a · s] gz 9.81 [m/s2] Nf in 1 ×105 [−] ∆t 0.1 [s] Tf 10000 [s] Nmf 200 × 24 [−] Nmm 100 × 12 [−]

                                               φ∂S3 ∂t + ∇ · (S3) = 0 (S3) = f3 − f3f1d dP_c31 dS3 (S3) · ∇S3+ f3f1d∆ρ · ∇z ∇ · = 0 = −d . [∇P − ¯ρg∇Z] P (x, t) = P3(x, t) + Pcg Pcg = − Z S3 0 f1dP 31 c dS3 dS3

Champ de saturation t=0.2s 0 5 10 15 20 25 0 1 2 3 Z (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Champ de saturation t=1h 0 5 10 15 20 25 0 1 2 3 Z (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Champ de saturation t=2h 0 5 10 15 20 25 0 1 2 3 Z (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Champ de saturation t=3h 0 5 10 15 20 25 0 1 2 3 Z (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Champ de saturation t=10h 0 5 10 15 20 25 x (m) 0 1 2 3 Z (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Champ de Pression t=10h 0 5 10 15 20 25 x (m) 0 0.5 1 1.5 2 2.5 3 Z (m) 96000 98000 100000 102000 104000 106000 108000 110000 Champ de Pression t=3h 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 Z (m) 100000 102000 104000 106000 108000 110000 112000 114000 116000 118000 120000 Champ de Pression t=1h 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 Z (m) 95000 100000 105000 110000 115000 120000 125000 Champ de Pression t=0s 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 Z (m) 100000 105000 110000 115000 120000 125000 130000 135000

95000 100000 105000 110000 115000 120000 125000 0 0.5 1 1.5 2 2.5 3

Pression Globale (Pa)

Z (m) Pression Globale 1h Pression Globale 3h Pression Globale 6h Pression Globale 10h -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 5 10 15 20 25 z (m) x(m)

Champ de vecteur vitesse

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 [m/s]

Champ de vitesses t=3h 0 5 10 15 20 25 x (m) -3 -2.5 -2 -1.5 -1 -0.5 0 z (m) 0 5e-05 0.0001 0.00015 0.0002 0.00025 Champ de vitesses t=2h 0 5 10 15 20 25 -3 -2.5 -2 -1.5 -1 -0.5 0 z (m) 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 Champ de vitesses t=1h 0 5 10 15 20 25 -3 -2.5 -2 -1.5 -1 -0.5 0 z (m) 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 [m/s]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 S3 Profondeur Z [m] MPTRACES 1h x=0.1m SIMUSCOPP 1h x=0.1m MPTRACES 1h x=12.5m SIMUSCOPP 1h x=12.5m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 S3 Profondeur Z [m] MPTRACES 2h x=0.1m SIMUSCOPP 2h x=0.1m MPTRACES 2h x=12.5m SIMUSCOPP 2h x=12.5m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 S3 Profondeur Z [m] MPTRACES 3h x=0.1m SIMUSCOPP 3h x=0.1m MPTRACES 3h x=12.5m SIMUSCOPP 3h x=12.5m

Comparaison des champ de vitesses PG-PAir t=1h 0 5 10 15 20 25 x (m) -3 -2.5 -2 -1.5 -1 -0.5 0 z (m) 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 Champ de vitesses de la pression globale t=1h

0 5 10 15 20 25 x (m) -3 -2.5 -2 -1.5 -1 -0.5 0 z (m) 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 [m/s]

∂t  S1 S3  = −∇ · ˆf_ˆ1 f3 

0 Sw ₁ 1 Sg Swi Sgi 1 1 1 krreg w = S∗ w− Swmax 1 − Smax w (1 − krmax w ) + krmaxw Smax w = 1 − Sorw− Sgi krw = 1 krreg o = S∗ o− Somax 1 − Smax o (1 − krmax o ) + krmaxo Smax o = 1 − Swi− Sgi kro = 1 krg= 1 krreg g = S∗ g− S max g 1 − Smax g (1 − krmax g ) + krmaxg Smax g = 1 − Sorg− Swi 1 − Sorg krmax g krmax o krmax w 1 − Sorw S_w∗ S_o∗ S_g∗ ˆ f1 = λ1 λt  1 − N3  (ρd− 1)kr2+ ρd µ2 µ3 kr3  ˆ f3 = λ3 λt  1 − N3  ρd µ2 µ1 kr1+ kr2  ρd = ρ1− ρ3 ρ2− ρ3 N3 = (ρ2− ρ3)K µ2vφ g.                          ∂t  S1 S3  = −∇ · ˆf_ˆ1 f3   S1 S3  =            SL=  S1 S3  L x < 0 SR=  S1 S3  R x > 0

=       ∂ ˆf1 ∂S1 ∂ ˆf1 ∂S3 ∂ ˆf3 ∂S1 ∂ ˆf3 ∂S3       λj = 1 2    ∂ ˆf1 ∂S1 + ∂ ˆf3 ∂S3 ± v u u t ! ∂ ˆf1 ∂S1 − ∂ ˆf3 ∂S3 "2 + 4∂ ˆf1 ∂S3 ∂ ˆf3 ∂S1    T = {(S1, S3) ; S1 ≤ 1 ; S3 ≤ 1} kr23(Sre,3) = S 1 2 2 h 1 − (1 − S n n−1 2 ) n−1 n i2 kr21(Sre,1) =  1 − Sre,1− S3r 1 − S1r− S3r 1 2 h 1 − S n n−1 1 i2(n−1)n krmax₂₁ = kr21(S2r) kr2 = S2kr21(Sre,1)kr23(Sre,3) krmax 21 · (1 − S2) · (1 − S3) Sre,j

kr1(S1) = S 3+2_λ 1 kr3(S2) = (1 − S2)2(1 − S2+ 2 λ 2 ) kr23(S2) = S 3+2 λ 2 kr21(S1) = (1 − S1)2(1 − S 2+_λ2 1 ) kr2 = kr21max  kr21 kr₂₁max + kr1   kr23 krmax₂₃ + kr3  − (kr1+ kr3)  krmax 21 = kr21(S2r) gz = 0 kr1 = 1.09S11.516687− 0.09S14.51668 kr21= 1.95(1 − S1)8.28− 0.95(1 − S1)11.284 kr3 = 0.525S31.02+ 0.475S33.62 kr23= 1.19(1 − S3)2.006− 0.19(1 − S3)2.024 kr2(S1, S3) = (1 − S1− S3) (1 − S1)(1 − S3) kr21kr23

(P3, S1, S3) Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µw = 0, 8 cp µo = 0, 1 cp µg = 0, 05 cp gz= 0 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µw = 0, 8 cp µo= 1 cp µg = 0, 05 cp gz = 0 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µw = 0, 8 cp µo = 5 cp µg= 0, 05 cp gz = 0 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µw = 1, 2 cp µo= 0.5 cp µg= 0, 05 cp gz = 0

(P

, S

, S

)

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gas Water s1 s1 s2 s2 s3 s3 s4 s4 400 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz= 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 s1 s4 s 3 s2 sL,R µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 kr2 400 S =            SL=  0.159 0.25  L x ≤ 0 SR=  0.16 0.249  R x > 0 • SL S1 • S1 S2 • S2 S3 • S3 S4 • S4 SR

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gas Water 400 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp kr2 S =            SL=  0.1599 0.25  L x ≤ 0 SR=  0.16 0.2499  R x > 0,

σ1− S1= τB(S1)

∂S1

∂t τB(S1)

σ1

krdyn(S1) = krstat(σ1) Pcdyn(S1) = Pcstat(σ1)

σ1= S1(t+τB) τB ∂t  σ1 σ3  = ∂t  S1 S3  + ∂t       τ11 τ12 τ31 τ33  | {z } τB(S1,S3) ∂t  S1 S3       ≈ (τ_B)−1  σ1− S1 σ3− S3  + τ_B ∂ 2 ∂t2  S1 S3  ∂2 tSj ≃ 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gas Water 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gas Water 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Gas Water 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 400 µw = 0, 8 cp µo = 1.2 cp µg = 0, 05 cp gz= 0 τ = 0.22 τ = 0.25 τ = 0.3 S =            SL=  0.159 0.25  L x ≤ 0 SR=  0.16 0.249  R x > 0 S =            SL=  0.01 0.8  L x ≤ 0 SR=  0.2 0.2  R x > 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 100000 100100 100200 100300 100400 100500 100600 100700 100800 100900 101000 Oil

Water Static waterStatic oil Dynamic gas pressureStatic gas pressure

400 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 T µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 kr2 τ = 1 400 P3(x, t) =    P3,L= 1.01 × 105P a x ≤ 0 P3,R= 1 × 105P a x > 0 S =            SL=  0.85 0.15  L x ≤ 0 SR=  0.05 0.4  R x > 0, P3(x, t) =    P3,L= 1.01 × 105P a x ≤ 0 P3,R= 1 × 105P a x > 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 100000 101000 102000 103000 104000 105000 106000 107000 108000 109000 110000 Oil Water gas pressure S2 S1 R1 64 µw = 0, 8 cp µo = 1.7 cp µg = 0, 02 cp gz = 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Sw1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Sg 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 sR S2 S1 R1 sL sm s∗ µw = 0, 8 cp µo = 1.7 cp µg = 0, 02 cp gz = 0 kr2 10 ∂ν1 ∂S3 ∂ν3 ∂S1 ≥ M in  0,∂ν1 ∂S1 ∂ν3 ∂S3  ∆2Pcg= 0 T

• • •

•

•

φ r λ = 2.89 Pd = 873 P a n = 5.5 j = 0.00077 P a−1 λ = 2.29 Pd = 667 P a n = 4 j = 0.0011 P a−1 λ = 2.89 Pd = 873 P a n = 5.5 α = 0.00077 P a−1 λ = 2.29 Pd= 667 P a n = 4 α = 0.0011 P a−1 T j = 1 (j = 3) C ∼ C1 ∪ C3 S₂∗ S₁∗ S∗ 1

t= 10−3m · s−1 ∆ρ = 400kg · m−3 M = 1 M = 1 t = 10−5m · s−1 ∆ρ = 400 kg · m−3 M = 1 M = 1 ∆ρ t = 10−3m · s−1 ∆ρ t = 10−4_{m · s}−1 ∆ρ t = 10−5m · s−1 S∗ t = 10−3m.s−1 t= 10−3m · s−1 M = 1 t = 10−3m · s−1 M = 1 t = 10−3m · s−1 M = 0.05 t = 10−4m · s−1 M = 0.5 t = 10−5m · s−1 M = 1 ∆ρ = +600 kg · m−3 M = 1 ∆ρ = +600 kg · m−3 M = 1 ∆ρ = +600 kg · m−3 M = 1

z z − σ¯ zz ¯ x + σxx ¯ x − σxx x + σ¯ xx ∆ρ t = 10−3m · s−1 ∆ρ t = 10−4m · s−1 ∆ρ t = 10−5m · s−1 [m/s] [m/s] [m/s] µw= 0, 8 cp µo= 0, 1 cp µg = 0, 05 cp gz = 0 µw = 0, 8 cp µo = 1 cp µg = 0, 05 cp gz= 0 µw = 0, 8 cp µo = 5 cp µg = 0, 05 cp gz= 0

µw = 1, 2 cp µo= 0.5 cp µg = 0, 05 cp gz = 0 400 µw = 0, 8 cp µo= 1. cp µg = 0, 05 cp gz = 0 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 kr2 400 400 µw= 0, 8 cp µo= 1. cp µg = 0, 05 cp gz = 0 µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp kr2 400 µw = 0, 8 cp µo = 1.2 cp µg = 0, 05 cp gz = 0 τ = 0.22 τ = 0.25 τ = 0.3 400 µw= 0, 8 cp µo= 1. cp µg= 0, 05 cp gz = 0 T µw = 0, 8 cp µo = 1. cp µg = 0, 05 cp gz = 0 kr2 τ = 1 400 64 µw= 0, 8 cp µo= 1.7 cp µg = 0, 02 cp gz= 0 µw = 0, 8 cp µo= 1.7 cp µg= 0, 02 cp gz = 0 kr2 10

                                                           φσ∂(Bj(Pj)Sj) ∂t + ∇ · (σ j) = 0 j = 1, 3 j = νj ω − νjd ·  1 − ω ω ρ − ∆ρ¯ j  g∇Z | {z } P artie convective − νjd ·  ∇(P_cj2− P_cg) | {z }

P artie dif f usive

∆ρ1 = (ν2(ρ2− ρ1) + ν3(ρ3− ρ2)) ∆ρ3 = (ν2(ρ2− ρ3) + ν1(ρ3− ρ1)) 3 X j=1 Sj = 1 ω =  1 −∂P g c ∂p  1 = ν1 + (ν2+ ν3) 1− ν1 2− ν1 3

1= ν1 + d · [ν1ν2(∇P2− ρ2g∇z) + ν1ν3(∇P3− ρ3g∇z) − (ν2+ ν3)ν1(∇P1− ρ1g∇z)] 1 = ν1 + d ·   ν1ν2∇ (P2− P1) | {z } =−P12 c +ν1ν3∇ (P3− P2) | {z } =P32 c −Pc12 + (ν1ν2(ρ1− ρ2) + ν1ν3(ρ1− ρ3)) g∇z    ν1ν2 = ν1(1 − ν1− ν3) 1 = ν1 −d ·   ν1(1 − ν1− ν3)∇Pc12− ν1ν3∇(Pc32− Pc12) + ν1(ν2(ρ2− ρ1) + ν3(ρ3− ρ1)) | {z } ∆ρ1 g∇z    1 = ν1 − ν1d ·         ∇Pc12− (ν1∇Pc12+ ν3∇Pc32) | {z } =∇Pcg− ∂Pcg ∂p ∇p −∆ρ1g∇z         1= ν1 − ν1d ·  ∇(P_c12− P_cg) +∂P g c ∂p ∇p − ∆ρ1g∇z  ∇p =  1 −∂P g c ∂p −1  ¯ ρg∇z − (d )−1_·  1 = ν1 ! 1 +∂P g c ∂p  1 −∂P g c ∂p −1" −ν1d · # ∇(P12 c − Pcg) +∂P g c ∂p  1 −∂P g c ∂p −1  ¯ ρg∇z − (d )−1· − ∆ρ1g∇z %

̟ =  1 −∂P g c ∂p  1= ν1 ̟ − ν1d ·  ∇(Pc12− Pcg) +  1 − ̟ ̟ ρ − ∆ρ¯ 1  g∇z  .  Pcg(Snω, p) ω(Snω, p) ∇Pnω+ νω(Snω, p) ∂Pc ∂Snω∇Snω = ω(Snω, p)∇P                            P = Pnω+ Pcg(Snω, p) ∀(Snω, p) ∈ [0, 1[×[pω, pnω] ω(Snω, p) = 1 − ∂Pcg ∂p (Snω, p) ∀(Snω, p) ∈ [0, 1[×[pω, pnω] Pcg(Snω, p) = − Z Snω 0 νω ∂Pc ∂Snω dSnω ∀(Snω, p) ∈ [0, 1[×[pω, pnω] Pc = Pnω− Pω π(Snω, p), C1(]0, 1] × [pω, pnω]) π(Snω, p) = Pnω ∀(Snω, p) ∇π = ∂π ∂Snω ∇Snω+ ∂π ∂p∇P = ∇Pnω = −νω(Snω, p) ∂Pc ∂Snω ∇Snω+ ω(Snω, p)∇P

         ∂π ∂Snω = −νω(Snω, p) ∂Pc ∂Snω ∂π ∂p = ω(Snω, p) π π(Snω, p) = − Z Snω 0 νω ∂Pc ∂Snω dSnω+ C1 π(0, p) = C1 = Pnω π(Snω, p) = Pnω− Z Snω 0 νω ∂Pc ∂Snω dSnω P = Pnω− Z Snω 0 νω ∂Pc ∂Snω dSnω= Pnω+ Pcg(Snω, p) ω(Snω, p) = 1 − ∂Pcg ∂p                                                                  φ∂Snω ∂t + ∇ · nω = 0, nω= νnω + νωνnωd(ρnω− ρω) · ∇z − νωνnωd dPc dSnω (Snω) · ∇Snω Sω+ Snω= 1 φ∂ ∂t(Bnω(ρnω)Snω) + (Bω(ρω)Snω) + ∇ · = 0, = −d . [ω∇P − ¯ρg∇Z] P = Pnω+ Pcg Pcg(Sω) = − Z Snω 0 νω ∂Pc ∂Snω dSnω Pc = Pnω− Pω

ω = νnω + νω ω− ω nω= νnω + d ·   νωνnω∇ (P2− P1) | {z } =−P12 c +νωνnω(ρnω− ρω)g∇z    nω= νnω + d ·νωνnω∇Pc12+ νωνnω(ρnω− ρω)g∇z 

Vh =  ϕ ∈ L∞(Ω)|ϕ|E ∈ P1(E) (0, 0) s (0, 1) s (1, 0) s !! !!✒ n1 ((x1, z1) = (0, 0), (x2, z2) = (1, 0), (x3, z3) = (0, 1)) • ϕE 1(x, z) = 1 − x − z • ϕE 2(x, z) = x • ϕE 3(x, z) = z ϕE_i (xj, zj) = δij • ΦE 1(x, z) = (x, z − 1) • ΦE₂(x, z) = (x − 1, z)

• ΦE 3(x, z) = (x, z) • ∇ · ϕE 1(x, z) = (−1, −1) • ∇ · ϕE₂(x, z) = (1, 0) • ∇ · ϕE 3(x, z) = (0, 1) ∂Ej • nE₁(x, z) = (0, −1) • nE 2(x, z) = (−1, 0) • nE 3(x, z) = (1, 1)