HOMOGENIZATION OF IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW IN DOUBLE POROSITY MEDIA

Transcription

1 Electronic Journal o Dierential Equations, Vol (2016), No. 52, pp ISSN: URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu tp ejde.mat.txstate.edu HOMOGENIZATION OF IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW IN DOUBLE POROSITY MEDIA LATIFA AIT MAHIOUT, BRAHIM AMAZIANE, ABDELHAFID MOKRANE, LEONID PANKRATOV Abstract. A double porosity model o multidimensional immiscible compressible two-pase low in ractured reservoirs is derived by te matematical teory o omogenization. Special attention is paid to developing a general approac to incorporating compressibility o bot pases. Te model is written in terms o te pase ormulation, i.e. te saturation o one pase and te pressure o te second pase are primary unknowns. Tis ormulation leads to a coupled system consisting o a doubly nonlinear degenerate parabolic equation or te pressure and a doubly nonlinear degenerate parabolic diusion-convection equation or te saturation, subject to appropriate boundary and initial conditions. Te major diiculties related to tis model are in te doubly nonlinear degenerate structure o te equations, as well as in te coupling in te system. Furtermore, a new nonlinearity appears in te temporal term o te saturation equation. Te aim o tis paper is to extend te results o [9] to tis more general case. Wit te elp o a new compactness result and uniorm a priori bounds or te modulus o continuity wit respect to te space and time variables, we provide a rigorous matematical derivation o te upscaled model by means o te two-scale convergence and te dilatation tecnique. 1. Introduction Te modeling o displacement process involving two immiscible luids is o considerable importance in groundwater ydrology and reservoir engineering suc as petroleum and environmental problems. More recently, modeling multipase low received an increasing attention in connection wit gas migration in a nuclear waste repository and sequestration o CO 2. Furtermore, ractured rock domains corresponding to te so-called Excavation Damaged Zone (EDZ) receives increasing attention in connection wit te beaviour o geological isolation o radioactive waste ater te drilling o te wells or sats, see, e.g., [50]. A issured medium is a structure consisting o a porous and permeable matrix wic is interlaced on a ine scale by a system o igly permeable issures. Te majority o luid transport will occur along low pats troug te issure system, and te relative volume and storage capacity o te porous matrix is muc larger 2010 Matematics Subject Classiication. 35B27, 35K65, 76S05, 76T10. Key words and prases. Compressible immiscible; double porous media; two-pase low; ractured media omogenization; two-scale convergence. c 2016 Texas State University. Submitted February 2, Publised February 18,

2 2 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 tan tat o te issure system. Wen te system o issures is so well developed tat te matrix is broken into individual blocks or cells tat are isolated rom eac oter, tere is consequently no low directly rom cell to cell, but only an excange o luid between eac cell and te surrounding issure system. Tereore te largescale description will ave to incorporate te two dierent low mecanisms. For some permeability ratios and issure widts, te large-scale description is acieved by introducing te so-called double porosity model. It was introduced irst or describing te global beaviour o ractured porous media by Barenblatt et al. [16]. It as been since used in a wide range o engineering specialties related to geoydrology, petroleum reservoir engineering, civil engineering or soil science. For more details on te pysical ormulation o suc problems see, e.g., [17, 49, 51]. During recent decades matematical analysis and numerical simulation o multipase lows in porous media ave been te subject o investigation o many researcers owing to important applications in reservoir simulation. Tere is an extensive literature on tis subject. We will not attempt a literature review ere but will merely mention a ew reerences. Here we restrict ourselves to te matematical analysis o suc models. We reer, or instance, to te books [13, 27, 31, 36, 43, 45, 52] and te reerences terein. Te matematical analysis and te omogenization o te system describing te low o two incompressible immiscible luids in porous media is quite understood. Existence, uniqueness o weak solutions to tese equations, and teir regularity as been sown under various assumptions on pysical data; see or instance [3, 13, 14, 25, 27, 28, 29, 36, 48] and te reerences terein. A recent review o te matematical omogenization metods developed or incompressible immiscible two-pase low in porous media and compressible miscible low in porous media can be viewed in [4, 44, 45]. We reer or instance to [18, 19, 20, 21, 22, 41, 42] or more inormation on te omogenization o incompressible, single pase low troug eterogeneous porous media in te ramework o te geological disposal o radioactive waste. Te double porosity problem was irst studied in [15], and was ten revisited in te matematical literature by many oter autors. Here we restrict oursel to te matematical omogenization metod as described in [45] or low and transport in porous media. For a recent review o te metods developed or low troug double porosity media, we reer or instance to [12, 15, 23, 30, 32, 34, 53] and te reerences terein. However, as reported in [9], te situation is quite dierent or immiscible compressible two-pase low in porous media, were, only recently ew results ave been obtained. In te case o immiscible two-pase lows wit one (or more) compressible luids witout any excange between te pases, some approximate models were studied in [37, 38, 39]. Namely, in [37] certain terms related to te compressibility are neglected, and in [38, 39] te mass densities are assumed not to depend on te pysical pressure, but on Cavent s global pressure. In te articles [26, 40, 46, 47], a more general immiscible compressible two-pase low model in porous media is considered or ields wit a single rock type and [10] treated te case wit several types o rocks. In [4, 11] omogenization results were obtained or water-gas low in porous media using te pase ormulation, i.e. were te pase pressures and te pase saturations are primary unknowns. Let us also mention tat, recently, a dierent approac based on a new global pressure concept was introduced in [5, 7] or modeling immiscible, compressible

3 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 3 two-pase low in porous media witout any simpliying assumptions. Te resulting equations are written in a ractional low ormulation and lead to a coupled system wic consists o a nonlinear parabolic equation (te global pressure equation) and a nonlinear diusion-convection one (te saturation equation). Tis new ormulation is ully equivalent to te original pase equations ormulation, i.e. were te pase pressures and te pase saturations are primary unknowns. For tis model, an existence result is obtained in [8] and omogenization results in [6]. Let us note tat all te aorementioned omogenization works are restricted to te case were te wetting pase (water) is incompressible wile te non-wetting pase (gas) is compressible, contrarily to te present work. In tis paper we extend our previous results obtained in [9] to te more complex case were bot pases are compressible wic is more reasonable in gas reservoir engineering. Te major diiculties related to tis model are in te nonlinear degenerate structure o te equations, as well as in te coupling in te system. In tis case a new nonlinearity appears in te temporal term o te saturation equation. Te compactness result used in [9] is no longer valid. To obtain tese results we elaborated a new approac based on te ideas rom [24, 35] to establis a new compactness result and uniorm a priori bounds or te modulus o continuity wit respect to te space and time variables. In tis paper, we will be concerned wit a degenerate nonlinear system o diusion-convection equations in a periodic domain modeling te low and transport o immiscible compressible luids troug eterogeneous porous media, taking into account capillary and gravity eects. We consider double porosity media, i.e. we consider a porous medium made up o a set o porous blocks wit permeability o order 2 surrounded by a system o connected issures,, is a small parameter wic caracterizes te periodicity o te blocks. Tere are two kinds o degeneracy in te studied system. Te irst one is te classical degeneracy o te capillary diusion term and te second one represents te evolution terms degeneracy. In bot cases te presence o degeneracy weakens te energy estimates and makes a proo o compactness results more involved. Te outline o te rest o te paper is as ollows. In Section 2 we describe te pysical model and ormulate te corresponding matematical problem. We also provide te assumptions on te data and a weak ormulation o te problem in terms o te global pressure and te saturation. Section 3 is devoted to te presentation o some a priori estimates or te solutions o te problem. Tey are essentially based on an energy equality. In Section 4, irstly we construct te extensions o te saturation and te global pressure unctions deined in te issures system and secondly we prove a compactness result adapted to our model. It s based on te compactness criterion o Kolmogorov Riesz Frécet (see, e.g., [24, 35]). Finally, we ormulate te corresponding two scale convergence results. In Section 5 we are dealing wit te dilations o te unctions deined in te matrix part. Firstly, we introduce te notion o te dilation operator and describe its properties. Secondly, we derive te system o equations or te dilated unctions and obtain te corresponding uniorm estimates or tem. Finally, we ormulate te convergence results or te dilated unctions. Te main result o te paper is ormulated in Section 6 and its proo is given in Section 7. Te proo is based on te two-scale convergence and te dilation tecniques.

4 4 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 2. Formulation o te problem Te outline o te section is as ollows. First, in subsection 2.1 we present te model equations wic are valid in ractures and rock matrix. A ractional low ormulation using te notion o te global pressure is discussed in subsection 2.2. Ten in te last subsection 2.3, we give te deinition o a weak solution to our system Microscopic model. We consider a reservoir Ω R d (d = 2, 3) wic is assumed to be a bounded, connected Lipscitz domain wit a periodic microstructure. More precisely, we will scale tis periodic structure by a parameter wic represents te ratio o te cell size to te wole region Ω and we assume tat 0 < 1 is a small parameter tending to zero. Let Y := (0, 1) d be a periodicity cell; we pave R d wit Y. We assume tat Y m is an open set wit piecewise smoot boundary Y m suc tat Y m Y and we reproduce Y m by periodicity, obtaining a periodic open set M in R d. We denote by F te periodic set F := R d \ M, wic is obtained rom te set Y := Y \ Y m. Tus Y = Y m Y Γ m, were Γ m denotes te interace between te two media. Finally, we denote by χ and χ m te caracteristic unctions o te sets F and M. Ten χ m ( x ) is te periodic unction o period Y wic takes te value 1 in te set M, union o te sets obtained rom Y m by translations o vectors n i=1 k i e i, were k i Z and e i, 1 i d, is te canonical basis o R d, and wic takes te value 0 in te set F, complementary in R d o tis union. In oter words, χ m ( x ) is te caracteristic unction o te set M, wile χ ( x ) is te caracteristic unction o F. Now we can deine te subdomains Ω r wit r = or m corresponding to te porous medium wit te index r. We set: Ω m := x Ω : χ m(x) = 1} and Ω := x Ω : χ (x) = 1 }. Ten Ω = Ω m Γ m Ω, were Γ m := Ω Ω m Ω and te subscript m and reer to te matrix and racture, respectively. For te sake o simplicity, we assume tat Ω m Ω =. We also set: Ω T := Ω (0, T ), Ω r,t := Ω r (0, T ), and Γ,T,m := Γ,m (0, T ), (2.1) were T > 0 is ixed. We consider an immiscible compressible two-pase low system in a porous medium wic ills te domain Ω. We ocus ere on te general case were bot pases are compressible, te pases being l and g. Let Φ (x) be te porosity o te reservoir Ω; K (x) be te absolute permeability tensor o Ω; Sl = S l (x, t), Sg = Sg(x, t) be te pase saturations; k r,l = k r,l (Sl ), k r,g = k r,g (Sg) be te relative permeabilities o te pases; p l = p l (x, t), p g = p g(x, t) be te pase pressures; ρ l, ρ g be te pase densities and P c te capillary pressure. In wat ollows, or te sake o presentation simplicity we neglect te source terms, and we denote S = Sl. Te model or te two-pase low is described by

5 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 5 (see, e.g., [27, 31, 43]): Φ (x) Ξ l Φ (x) Ξ g 0 S 1 in Ω T ; div K (x)λ l (S )ρ l (p l) ( p l ρ l (p l) g )} = 0 in Ω T ; div K (x)λ g (S )ρ g (p g) ( p g ρ g (p g) g )} = 0 in Ω T ; P c (S ) = p g p l in Ω T, (2.2) were λ g (S ) = λ g (1 S ); Ξ l := S ρ l (p l ) and Ξ g := (1 S )ρ g (p g); eac unction γ := S, p l, p g, Ξ l, and Ξ g is deined as: γ (x, t) = χ (x)γ (x, t) + χ m(x)γ m(x, t). (2.3) Te velocities o te pases q l, q g are deined by Darcy Muskat s law: ( ) q l := K (x)λ l (Sl ) p l ρ l (p l) g ) q g := K (x) λ g (Sg)( p g ρ g (p g) g wit λ l (S l ) := k r,l µ l (S l ); (2.4) wit λ g (S g) := k r,g µ g (S g) (2.5) wit g, µ l, µ g being te gravity vector and te viscosities, respectively. Now we speciy te boundary and initial conditions. We suppose tat te boundary Ω consists o two parts Γ 1 and Γ 2 suc tat Γ 1 Γ 2 =, Ω = Γ 1 Γ 2. Te boundary conditions are given by Finally, te initial conditions read p g(x, t) = 0 = p l(x, t) on Γ 1 (0, T ); (2.6) q l ν = q g ν = 0 on Γ 2 (0, T ). (2.7) S (x, 0) = S 0 (x) and p g(x, 0) = p 0 g(x) in Ω. (2.8) 2.2. A ractional low ormulation. In te sequel, we use a ormulation obtained ater transormation using te concept o te global pressure introduced in [13, 27]. Te global pressure is deined as ollows: P = p l G l (S ) = p g G g (S ), (2.9) were te unctions G l (S ), G g (S ) are given by: S G g (S λ l (s) ) := G g (0) + 0 λ(s) P c(s)ds, (2.10) G l (S ) = G g (S ) P c (S ), (2.11) wit λ(s) := λ l (s) + λ g (s), te total mobility. Perorming some simple calculations, we obtain te ollowing properties or te global pressure wic will be used in te sequel: Notice tat rom (2.10), (2.13) we obtain λ l (S ) p l + λ g (S ) p g = λ(s ) P, (2.12) G l (S ) = λ g(s ) λ(s ) P c(s ) S. (2.13) λ l (S ) G l (S ) = β(s ) and λ g (S ) G g (S ) = β(s ), (2.14)

6 6 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 were S β(s ) := α(u) du wit α(s) := λ g(s)λ l (s) P 0 λ(s) c(s). (2.15) Furtermore, we ave te important relation were λ g (S ) p g 2 + λ l (S ) p l 2 = λ(s ) P 2 + b(s ) 2, (2.16) b(s) := s 0 a(ξ)dξ wit a(s) := λ g (s)λ l (s) P λ(s) c(s). (2.17) I we use te global pressure and te saturation as new unknown unctions, ten problem (2.2) reads Φ (x) Θ l Φ (x) Θ g 0 S 1 in Ω T ; div K (x) ρ l[ λl (S ) P + β(s ) λ l (S ) ρ l g ]} = 0 in Ω T ; div K (x) ρ g[ λg (S ) P β(s ) λ g (S ) ρ g g ]} = 0 in Ω T, we introduced te notation (2.18) ρ l := ρ l (P + G l (S )) and ρ g := ρ g (P + G g (S )); (2.19) Θ l = Θ l (S, P ɛ ) := S ρ l and Θ g = Θ g (S, P ) := (1 S ) ρ g. (2.20) Te system (2.18) is completed by te ollowing boundary and initial conditions. were Q l and Q g are deined by: P = 0 on Γ 1 (0, T ); (2.21) Q l ν = Q g ν on Γ 2 (0, T ) (2.22) Q l := K (x) ρ l[ λl (S ) P + β(s ) λ l (S ) ρ l g ], Q g := K (x) ρ g[ λg (S ) P β(s ) λ g (S ) ρ g g ]. Finally, te initial conditions read S (x, 0) = S 0 (x) and P (x, 0) = P 0 (x) in Ω. (2.23) 2.3. A weak ormulation o te problem. Let us begin tis section by stating te ollowing assumptions. (A1) Te porosity Φ = Φ(y) is a Y -periodic unction deined by: Φ (x) = χ (x)φ + χ m(x)φ m wit 0 < Φ, Φ m < 1, were Φ and Φ m are constant tat do not depend on. (A2) Te absolute permeability tensor K is given by: K (x) := K χ (x) I + 2 K m χ m(x)i, were I is te unit tensor and K, K m are positive constants tat do not depend on. (A3) Te density ρ k = ρ k (p), (k = l, g) is a monotone C 1 -unction in R suc tat ρ k (p) = ρ min or p p min ; ρ k (p) = ρ max or p p max ; (2.24) ρ min < ρ k (p) < ρ max or p min < p < p max.

7 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 7 ρ min, ρ max, p min, p max are constants suc tat 0 < ρ min < ρ max < + and 0 < p min < p max < +. (A4) Te capillary pressure unction P c C 1 ([0, 1]; R + ). Moreover, P c(s) < 0 in [0, 1] and P c (1) = 0. (A5) Te unctions λ l, λ g belong to te space C([0, 1]; R + ) and satisy te ollowing properties: (i) 0 λ l, λ g 1 in [0, 1]; (ii) λ l (0) = 0 and λ g (1) = 0; (iii) tere is a positive constant L 0 suc tat λ l (s) = λ l (s)+λ g (s) L 0 > 0 in [0, 1]. Moreover, λ l (s) s κ l as s 0 and λ g (s) (1 s) κg as s 1 (κ l, κ g > 0). (A6) Te unction α given by (2.15) is a continuous unction in [0, 1]. Moreover, α(0) = α(1) = 0 and α > 0 in (0, 1). (A7) Te unction β 1, inverse o β deined in (2.15) is a Hölder unction o order θ wit θ (0, 1) on te interval [0, β(1)]. Namely, tere exists a positive constant C β suc tat or all s 1, s 2 [0, β(1)], β 1 (s 1 ) β 1 (s 2 ) C β s 1 s 2 θ. (A8) Te initial data or te global pressure and te saturation deined in (2.23) are suc tat P 0 L 2 (Ω) and 0 S 0 1. Assumptions (A1) (A8) are classical and pysically meaningul or existence results and omogenization problems o two-pase low in porous media. Tey are similar to te assumptions made in [10] tat dealt wit te existence o a weak solution o te studied problem. Next we introduce te Sobolev space H 1 Γ 1 := u H 1 (Ω) : u = 0 on Γ 1 }, wic is a Hilbert space wen it is equipped wit te norm u H 1 Γ 1 (Ω) = u (L 2 (Ω)) d. Deinition 2.1. We say tat te pair o unctions P, S is a weak solution to problem (2.18) (2.23) i (i) 0 S 1 a.e in Ω T. (ii) P L 2 (0, T ; HΓ 1 1 (Ω)). (iii) Te boundary conditions (2.21) (2.22) are satisied. (iv) For any ϕ l, ϕ g C 1 ([0, T ]; HΓ 1 1 (Ω)) satisying ϕ l (T ) = ϕ g (T ) = 0, we ave Φ (x)θ ϕ l l dx dt + Φ (x)θ 0 Ω T lϕ 0 l dx Ω + K (x) ρ l λ l (S ) ( P ρ l g ) } (2.25) + β(s ) ϕ l dx dt = 0; Ω T Φ (x)θ ϕ g g dx dt + Φ (x)θ 0 Ω T gϕ 0 g dx Ω + K (x) ρ g λ g (S ) ( P ρ g g ) } (2.26) β(s ) ϕ l dx dt = 0, Ω T were ρ g and ρ l are deined in (2.19); ϕ0 l = ϕ l(0, x), ϕ 0 g = ϕ g (0, x); Θ 0 l = S 0 ρ l (P 0 + G l (S 0 )) and Θ 0 g = (1 S 0 )ρ g (P 0 + G g (S 0 )).

8 8 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 According to [10], under conditions (A1) (A8), or eac > 0, problem (2.25) (2.26) as at least one weak solution. In wat ollows C, C 1,... denote generic constants tat do not depend on. 3. A priori estimates To obtain te needed uniorm estimates or te solution o problem (2.2) (or te equivalent problem (2.18)), we ollow te coice o te test unctions as in [9]: R l (p l) = p l 0 dξ ρ l (ξ) Ten, as in [9], te ollowing results old. and R g (p g) = p g 0 dξ ρ g (ξ). (3.1) Lemma 3.1. Let p g, p l be a solution to (2.2). Ten we ave te energy equality d Φ (x)ζ (x, t) dx + K (x) λ l (S ) p l [ p l ρ l (p dt l) g ] Ω Ω + λ g (S ) p g [ p g ρ g (p g) g ]} (3.2) dx = 0 in te sense o distributions. Here ζ := S R l (p l) + (1 S )R g (p g) + F (S ), were R k (p) := ρ k (p)r k (p) p, (k = l, g) and F (s) := s 1 P c(u) du. Moreover, ζ 0 in Ω T. Lemma 3.2. Let p g, p l be a solution to (2.2). Ten K (x)λ l (S ) p l L 2 (Ω T ) + K (x)λ g (S ) p g L 2 (Ω T ) C. (3.3) Corollary 3.3. Let p g, p l be a solution to (2.2). Ten λ l (S ) p l, L2 (Ω,T ) + λ g (S ) p g, L2 (Ω,T ) + λ l (Sm) p l,m L2 (Ω m,t ) + λ g (Sm) p g,m L2 (Ω m,t ) C. (3.4) Ten we obtain te ollowing uniorm a priori estimates or te unctions P and β(s ). Lemma 3.4. Let te pair o unctions P, S be a solution to (2.18). Ten Moreover, β(s ) L 2 (Ω,T ) + P L 2 (Ω,T ) + β(sm) L 2 (Ω m,t ) + Pm L2 (Ω m,t ) C. (3.5) P L2 (Ω,T ) + β(s ) L2 (Ω T ) C, (3.6) Pm L2 (Ω m,t ) C. (3.7) Now we pass to te uniorm bounds or te time derivatives o te unctions Θ g and S. In a standard way (see, e.g., [4, 9]) we can prove te ollowing lemma. Lemma 3.5. Let te pair o unctions P, S be a solution to (2.18). Ten or r =, m, t (Φ r Θ l,r)} >0 is uniormly bounded in L 2 (0, T ; H 1 (Ω r)); (3.8) t (Φ r Θ g,r)} >0 is uniormly bounded in L 2 (0, T ; H 1 (Ω r)). (3.9)

9 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 9 4. Convergence o P } >0, S } >0, Θ l, } >0, Θ g, } >0 In tis section, we obtain compactness results tat will be used in passing to te limit as tends to zero in te weak ormulation. Te compactness result used in [9] is no longer valid. To obtain tese results we elaborated a new approac based on te ideas rom [24, 35] to establis a new compactness result and uniorm a priori bounds or te modulus o continuity wit respect to te space and time variables. It is acieved in several steps. First, in subsection 4.1 we extend te unction S rom te subdomain Ω to te wole Ω and obtain uniorm estimates or te extended unction S. Ten in Section 4.2, using te uniorm estimates or te unction P wic ollow rom Lemma 3.4, te deinition o te extension operator, and te corresponding bounds or S, we prove te compactness result or te amilies Θ l, } >0 and Θ g, } >0, were Θ l,, Θ g, are extensions o te unctions Θ l,, Θ g, rom te subdomain Ω to te wole Ω wic will be speciied at te end o subsection 4.1. Finally, in subsection 4.3 we ormulate te two scale convergence wic will be used in te derivation o te omogenized system Extensions o te unctions S, Θ l,, Θ g,. Consider te unction S. To extend S, ollowing te ideas o [23], we use te monotone unction β deined in (2.15). Let us introduce te unction β (x, t) := β(s ) = S 0 α(u) du. (4.1) Ten it ollows rom condition (A6) tat 0 β max s [0,1] α(s) a.e. in Ω,T. Furtermore, rom (3.5) we ave β L 2 (Ω,T ) C. (4.2) Let Π : H 1 (Ω ) H1 (Ω) be te standard extension operator c. [1]. Ten we ave 0 β := Π β max s [0,1] α(s) a.e. in Ω T, β L2 (Ω T ) C. Now we can extend te unction S rom te subdomain Ω denote tis extension by S and deine it as: to te wole Ω. We Tis implies tat S := (β) 1 ( β ). Ω T β( S ) 2 dx dt C, 0 S 1 a.e. in Ω T. (4.3) Finally consider te sequences Θ l, } >0 and Θ g, } >0. We recall tat Θ ( l, := ρ l P + G l (S )) S and Θ g, := ρ g( P + G g (S )) (1 S ). Ten we deine te extension o te unction Θ to te wole Ω by Θ l, := ρ l ( P + G l ( S )) S and Θ g, := ρ g ( P + G g ( S ))(1 S ), (4.4) were P := Π P is te extension o te unction P.

10 10 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/ Compactness o te sequences Θ l, } >0, Θ g, } >0. Te ollowing convergence result is valid. Proposition 4.1. Under our standing assumptions tere exist te unctions L 1, L 2 suc tat Θ l, L 1 strongly in L 2 (Ω T ), (4.5) Θ g, L 2 strongly in L 2 (Ω T ). (4.6) Proo. We will prove te convergence result or te sequence Θ l, } >0, te corresponding result or te sequence Θ g, } >0 can be obtained by similar arguments. Te sceme o te proo is as ollows. We apply te compactness criterion o Kolmogorov-Riesz-Frécet (see, e.g., [24, 35]) in te space L 1 (Ω T ). To tis end we ave to obtain te moduli o continuity wit respect to te space and temporal variables (see Lemmata 4.2, 4.6 below). Finally, te uniorm boundedness o te unction Θ l, imply te desired convergence result (4.5) in te space L2 (Ω T ). We start wit te ollowing result wic can be proved by arguments similar to tose rom Lemma 4.2 in [9]. Lemma 4.2 (Modulus o continuity wit respect to te space variable). Let conditions (A1) (A8) be ulilled. Ten or x suiciently small, T Θ l, (x + x, t) Θ l, (x, t) 2 dx dt C x θ, (4.7) 0 Ω were θ (0, 1) is deined in condition (A7). Now we turn to te derivation o te modulus o continuity wit respect to te temporal variable. To do tis, or any δ > 0, we introduce te unctions S,δ r := min 1 δ, max(δ, S r) } wit r =, m. Let us estimate te norm o te unction S,δ r (r =, m) in L 2 (0, T ; H 1 (Ω r,t )). Lemma 4.3. Under our standing assumptions, S,δ L 2 (0,T ;H 1 (Ω,T )) + Sm,δ L 2 (0,T ;H 1 (Ω m,t )) C δ η, (4.8) were η := (κ l + κ g ) and C is a constant tat does not depend on, δ. Proo. We consider te unction S,δ, te estimate or te unction S,δ m can be obtained in a similar way. It is evident tat S,δ (as well as S ) belongs to te space L 2 (Ω,T ). Now we are going to estimate S,δ. From Lemma 3.4 we know tat β(s ) L 2 (Ω,T ) C. Ten it is clear tat β(s,δ ) L 2 (Ω,T ) β(s ) L2 (Ω,T ) C, (4.9) were C is a constant tat does not depend on, δ. implies te inequalities: Moreover, condition (A5) λ l (S,δ ) C 1δ κ l and λ g (S,δ ) C 2δ κg. (4.10)

11 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 11 Ten taking into account te deinition o te unctions β and α (see (2.15)) and conditions (A4), (A5), rom (4.10), we ave tat Ten rom (4.9), (4.11) α(s,δ ) C 1δ (κ l+κ g) T C β(s,δ ) 2 L 2 (Ω,T ) = 0 T C 2 1δ 2(κ l+κ g) 0 Ω wit C 1 := C 1C 2 2 Ω α 2 (S,δ S,δ 2 dx dt = C 2 1δ 2(κ l+κ g) S,δ 2 L 2 (Ω,T ). Te inequality (4.11) implies tat S,δ 2 L 2 (Ω,T ) C δ 2(κ l+κ g) min P c(s). (4.11) s [0,1] ) S,δ 2 dx dt and inequality (4.8) is proved. Tis completes te proo o Lemma 4.3. (4.12) (4.13) In wat ollows we use a tecnical lemma wic can be proved using te Fubini teorem. Lemma 4.4. For suiciently small, 0 < < T 2 G 1 (t), G 2 (t) it olds: T ( min(t+,t ) G 1 (t) 0 max(t,) and or integrable unctions ) T ( t ) G 2 (τ) dτ dt = G 2 (t) G 1 (τ) dτ dt. t Now, or > 0 and 0 < < T 2, let us introduce te unction min(t+,t ) ϕ,δ, (x, t) := Θ,δ l (x, τ) dτ, max(t,) (4.14) wit v(t) v(t ) v(t) :=, were Θ,δ l (x, t) := ρ l (P + G l (S,δ )) S,δ. (4.15) Te properties o ϕ,δ, are described by te next lemma. Lemma 4.5. Let > 0, 0 < δ < 1, and let > 0 be small enoug. Tere exist a constant C wic does not depend on, δ, and suc tat or te sequence o unctions deined by (4.14) it olds Here ϕ,δ, L 2 (0, T ; H 1 Γ 1 (Ω)); (4.16) ϕ,δ, (x, T ) = 0; (4.17) ϕ,δ, L 2 (Ω T ) C ; (4.18) ϕ,δ, L 2 (Ω,T ) C δ η ; (4.19) ϕ,δ, L 2 (Ω m,t ) C δ η. (4.20) η := (κ l + κ g ). (4.21)

12 12 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 Proo. Te regularity property (4.16) ollows immediately rom Lemma 3.4 and Lemma 4.3. Moreover, taking into account tat S = 1 and P = const on Γ 1 (0, T ) (see Section 2), according to te deinition o te unction S,δ, we ave tat ϕ,δ, = 0 on Γ 1 (0, T ). Result (4.17) ollows directly rom te deinition o te unction ϕ,δ,. In act, ϕ,δ, (x, T ) = min(t +,T ) max(t,) Θ,δ l (x, τ) dτ = T T Θ,δ l (x, τ) dτ = 0. Bound (4.18) also ollows immediately rom te deinition o ϕ,δ, since min(t +, T ) max(t, ) and te unction Θ,δ l is uniormly bounded in L (Ω T ). For bound (4.19), we ave ϕ,δ, 2 L 2 (Ω,T ) [ min(t+,t ) Ω,T max(t,) Θ,δ l, (x, τ) Θ,δ l, (x, τ ) dτ] 2 dx dt := J,δ. (4.22) Let us estimate te rigt-and side o (4.22). Since [min(t +, T ) max(t, )], rom Caucy s inequality, or a.e. (x, t) Ω,T we obtain min(t+,t ) max(t,) 1/2[ min(t+,t ) Θ,δ l, (x, τ) Θ,δ l, (x, τ ) dτ max(t,) Θ,δ l, (x, τ) Θ,δ l, (x, τ ) 2 dτ] 1/2. Tereore, rom tis inequality we obtain T [ min(t+,t ) ] J,δ C Θ,δ l, (x, τ) Θ,δ l, (x, τ ) 2 dτ dx dt. (4.23) 0 Ω max(t,) Now we apply Lemma 4.4 wit G 1 (t) := 1 and G 2 (t) := Θ,δ l, (x, τ) Θ,δ l, (x, τ ) 2 in te rigt and side o (4.23). We ave: T = [ min(t+,t ) 0 max(t,) T Θ,δ l, = T Θ,δ l, Ten, by (4.23), we deduce tat J,δ C 2 Ω C 2[ Ω T T Θ,δ l, C 2 Θ,δ l, 2 L 2 (Ω,T ). Θ,δ l, ] (x, τ) Θ,δ l, (x, τ ) 2 dτ dt (x, t) Θ,δ l, (x, t ) [ 2 t 1 dτ t (x, t) Θ,δ l, (x, t ) 2 dt. (x, t) Θ,δ l, (x, t ) 2 dt dx ] dt T Θ,δ l, (x, t) 2 ] dt dx + Θ,δ l, (x, t ) 2 dt dx Ω (4.24)

13 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 13 It remains to estimate te rigt-and side o (4.24). as ollows: Θ,δ l, To tis end we rewrite Θ,δ l, = ρ l ( P + G l (S,δ ) ) S,δ + ρ l S,δ P + ρ l S,δ G l (S,δ ). (4.25) Ten, rom (4.25), conditions (A3), (A5), and te deinition o te unction G l, we ave Θ,δ l, 2 L 2 (Ω,T ) ρ 2 max S,δ 2 L 2 (Ω,T ) + C [ P 2 L 2 (Ω,T ) + P c S,δ 2 L 2 (Ω,T ) ]. (4.26) To estimate te rigt-and side o (4.26), we make use o condition (A4), bound (4.8), and Lemma 4.3. Taking into account tat δ is suiciently small, we obtain Θ,δ l, 2 L 2 (Ω,T ) C δ 2(κ l+κ g). (4.27) Now, rom (4.22), (4.24), and (4.27), or δ suiciently small, we obtain ϕ,δ, 2 L 2 (Ω,T ) C 2 δ 2(κ l+κ g) (4.28) and te bound (4.19) is proved. Te proo o te bound (4.20) can be done by arguments similar to ones used in te proo o (4.19). Tis completes te proo o Lemma 4.5. Now we are in a position to estimate te modulus o continuity o te unction Θ l,. Lemma 4.6 (Modulus o continuity wit respect to time ). Under our standing assumptions, or all (0, T ) and δ suiciently small, we ave T Θ l, (x, t) Θ l, (x, t ) 2 dx dt C σ wit σ := min 1 2, 1 }, (4.29) 2η Ω were η is deined in (4.21) and C is a constant wic does not depend on and. Proo. Let us insert te unction ϕ,δ, = ϕ,δ, (x, t) in equation (2.25). We ave Φ (x) Θ l Ω T ϕ,δ, dx dt + K (x) ρ l λ l (S ) ( P ρ l g ) } (4.30) + β(s ) ϕ,δ, dx dt = 0. Ω T Taking into account te deinition o te unction ϕ,δ, and condition (A1), or te irst term in te let-and side o (4.30), we ave I1(ϕ,δ, ) : = Φ (x) Θ l Ω T ϕ,δ, dx dt T Θ [ min(t+,t ) ] l, = Φ Θ,δ l, 0 Ω (x, τ) dτ dx dt max(t,) T Θ [ min(t+,t ) ] l,m + Φ m Θ,δ l,m (x, τ) dτ dx dt. 0 Ω m max(t,)

14 14 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 Now Lemma 4.4 implies T I1(ϕ,δ, ) = + Ω T Φ Θ,δ l, (x, t) [ t Ω m T = Φ Ω t Φ m Θ,δ l,m (x, t) [ t Θ l, τ t Θ l, (x, t) Θ l, (x, t ) } Θ l,m τ Θ,δ l, (x, t) Θ,δ l, (x, t )} dx dt T + Φ m Θ l,m (x, t) Θ l,m(x, t ) } Ω m Θ,δ l,m (x, t) Θ,δ l,m (x, t )} dx dt : = J,δ + J,δ m. ], dτ dx dt Considering te integrand in te irst term o (4.31), we ave [ Θ l, (x, t) Θ l, (x, t ) ] [ Θ,δ l, (x, t) Θ,δ l, (x, t )] = Θ l, (x, t) Θ l, (x, t ) 2 J,δ + [ Θ l, (x, t) Θ l, (x, t ) ] [ Θ,δ l, (x, t) Θ l, (x, t) ] ] dτ dx dt [ Θ l, (x, t) Θ l, (x, t ) ] [ Θ,δ l, (x, t ) Θ l, (x, t ) ] : = Θ l, (x, t) Θ l, (x, t ) 2 + T,δ, 1 (x, t) T,δ, 2 (x, t). Ten te irst term in te rigt and side o (4.31) can be rewritten as T = Φ Θ l, (x, t) Θ l, (x, t ) 2 dx dt Ω T + Φ Ω T,δ, 1 (x, t) dx dt Φ T Ω T,δ, 2 (x, t) dx dt. It is easy to see tat Θ,δ l, (x, t) Θ l, (x, t) C δ and Θ,δ l, (x, t ) Θ l, (x, t ) C δ, (4.31) (4.32) (4.33) were C is a constant tat does not depend on, δ,. Let us consider, or example, te irst bound, te second one can be obtained by similar arguments. We ave Θ l, (x, t) Θ,δ l, (x, t) ρl (P + G l (S ))S ρ l (P + G l (S ))S,δ + ρl (P + G l (S ))S,δ ρ max S S,δ ρ max δ + ρ l Gl (S ) G l (S,δ ) ρ max δ + ρ l P c (S ) P c (S,δ ) ρ max δ + ρ l P c S S,δ C δ. ρ l (P + G l (S,δ ))S,δ + ρl (P + G l (S )) ρ l (P + G l (S,δ )) (4.34)

15 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 15 Since te unction Θ l, is uniormly bounded in Ω T, rom (4.33) and (4.34) we ave T = Φ Θ l, (x, t) Θ l, (x, t ) 2 dx dt + j,δ (4.35) J,δ Ω wit j,δ Cδ, were C is a constant tat does not depend on, δ,. In a similar way, one obtains J,δ m T = Φ m Ω m Θ l,m (x, t) Θ l,m(x, t ) 2 dx dt + j,δ m (4.36) wit j,δ m Cδ, were C is a constant tat does not depend on, δ,. Ten we obtain T I1(ϕ,δ, ) Φ Θ l, (x, t) Θ l, (x, t ) 2 dx dt + j,δ + j,δ m. (4.37) Ω Next, we estimate te rigt-and side o (4.30). To tis end we use te notation: K (x) ρ l λ l (S ) ( P ρ l g ) } + β(s ) ϕ,δ, dx dt =: I (ϕ,δ, )+Im(ϕ,δ, ). Ω T We ave I (ϕ,δ, ) + I m(ϕ,δ, ) K ρ l λl (S ) P + β(s ) ρ lλ l (S ) g } L2 (Ω,T ) ϕ,δ, L2 (Ω,T ) + K m ρ l λl (S m) P m + β(s m) ρ lλ l (S m) g } L2 (Ω m,t ) ϕ,δ, L 2 (Ω m,t ). (4.38) Ten or I (ϕ,δ, ), I m(ϕ,δ, ) rom (A3), (A5), te a priori estimates (3.5), (4.19), and (4.20) we obtain I (ϕ,δ, ) + I m (ϕ,δ, ) C1 δ η. (4.39) From (4.30), (4.37), and (4.39), we obtain te bound T Θ l, (x, t) Θ l, (x, t ) 2 dx dt C 1 δ η + C 2 δ. (4.40) Ω Now we set: δ = δ() = 1/2η. Ten rom (4.40) we ave T Θ l, (x, t) Θ l, (x, t ) 2 dx dt C σ wit σ := min 1 2, 1 }. (4.41) 2η Ω Finally, taking into account te act tat te extension is by relection, rom (4.41), we obtain te desired estimate (4.29) and Lemma 4.6 is proved. By te Kolmogorov-Riesz-Frécet compactness criterion, te compactness o te sequences Θ l, } >0, Θ g, } >0 is a consequence o Lemmata 4.2, 4.6 (evidently, wit te corresponding cange o te index l by g or te sequence Θ g, } >0). It assures te ollowing convergence results: Θ l, L 1 strongly in L 1 (Ω T ), (4.42) Θ g, L 2 strongly in L 1 (Ω T ). (4.43)

16 16 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 Now taking into account te L -boundedness o te unctions Θ l,, Θ g, we, inally, arrive to te desired convergence results (4.5), (4.6). Tis completes te proo o Proposition 4.1. Corollary 4.7. Tere exist te unctions P, S L 2 (Ω T ) suc tat (up to a subsequence), we ave P P a.e in Ω T ; (4.44) S S a.e in Ω T. (4.45) Proo. By Proposition 4.1 we conclude tat tere exist te unctions L 1, L 2 suc tat Θ l, L 1 strongly in L 2 (Ω T ) and a.e. in Ω T ; Θ g, L 2 strongly in L 2 (Ω T ) and a.e. in Ω T. Now, we will use te act tat te unction D given by D( S, P ) = ( Θ l, ( S, P ), Θ g, ( S, P )) is a dieomorpism rom [0, 1] R to D([0, 1] R) (or more details see [7]) so it as continuous inverse. Tereore, almost everywere in Ω T convergence o Θ r, ( S, P ) (r l, g}) implies (4.44) and (4.45). Corollary 4.7 is proved Two scale convergence results. In tis section, taking into account te compactness results rom te previous section, we ormulate te convergence results or te sequences P } >0, S } >0, Θ r, } >0 wit r l, g}. In tis paper te omogenization process or te problem is rigorously obtained by using te twoscale approac, see, e.g., [2]. For reader s convenience, let us recall te deinition o te two-scale convergence. Deinition 4.8. A sequence o unctions v } >0 L 2 (Ω T ) two-scale converges to v L 2 (Ω T Y ) i v L 2 (Ω T ) C, and or any test unction ϕ C (Ω T ; C per (Y )) te ollowing relation olds lim 0 v (x, t) ϕ(x, x, t) dx dt = Ω T Ω T Y v(x, y, t) ϕ(x, y, t) dy dx dt. Tis convergence is denoted by v (x, t) 2s v(x, y, t). Now we summarize te convergence results or te sequences P } >0, S } >0, Θ r, } >0 (r l, g}). Tey are given by te ollowing lemma. Lemma 4.9. Tere exist a unction S suc tat 0 S 1 a.e. in Ω T, and unctions P L 2 (0, T ; H 1 (Ω)), P 1, β 1 L 2 (Ω T ; Hper(Y 1 )) suc tat up to a subsequence, S S strongly in L 2 (Ω T ) and a.e. in Ω T ; (4.46) β( S ) β(s) strongly in L 2 (Ω T ); (4.47) P P strongly in L 2 (Ω T ) and a.e. in Ω T ; (4.48) P P weakly in L 2 (0, T ; H 1 (Ω)); (4.49) Θ l, ρ l (P + G l (S))S strongly in L 2 (Ω T ); (4.50) Θ g, ρ g (P + G g (S))(1 S) strongly in L 2 (Ω T ); (4.51)

17 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 17 P (x, t) 2s P (x, t) + yp 1 (x, t, y); (4.52) β( S )(x, t) 2s β(s)(x, t) + yβ 1 (x, t, y). (4.53) Te proo o te above lemma is based on te a priori estimates obtained in Section 3, Proposition 4.1, Corollary 4.7, and two-scale convergence arguments similar to tose in [2] (see also Lemma 4.8 rom [4] or similar arguments). 5. Dilation operator and convergence results It is known tat by te nonlinearities and te strong coupling o te problem, te two scale convergence does not provide an explicit orm or te source terms appearing in te omogenized model, see or instance [23, 32, 53]. To overcome tis diiculty te autors make use o te dilation operator. Here we reer to [15, 23, 32, 53] or te deinition and main properties o te dilation operator. Let us also notice tat te notion o te dilation operator is closely related to te notion o te unolding operator. We reer ere, e.g., to [33] or te deinition and te properties o tis operator. Te outline o te section is as ollows. First, in Section 5.1 we introducete deinition o te dilation operator and describe its main properties. Ten in Section 5.2 we obtain te equations or te dilated saturation and te global pressure unctions and te corresponding uniorm estimates. Finally, in Section 5.3 we consider te convergence results or te dilated unctions Deinition o te dilation operator and its main properties. Deinition 5.1. For a given > 0, we deine a dilation operator D mapping measurable unctions deined in Ω m,t to measurable unctions deined in Ω T Y m by (D ψ(c (x) + y, t), i c (x) + y Ω ψ)(x, y, t) := m; (5.1) 0, oterwise, were c (x) := k i x (Y m + k). Here k Z d denotes te lattice translation point o te -cell domain wic contains x. Te basic properties o te dilation operator are given by te ollowing lemma (see [15, 53]). Lemma 5.2. Let v, w L 2 (0, T ; H 1 (Ω m)). Ten we ave y D v = D ( x v) a.e. in Ω T Y m ; (5.2) D v L2 (Ω T Y m) = v L2 (Ω T ); y D v L 2 (Ω T Y m) = D x v L 2 (Ω m,t ) ; (D v, D w) L 2 (Ω T Y m) = (v, w) L 2 (Ω m,t ) ; (D v, w) L2 (Ω T Y m) = (v, D w) L2 (Ω T Y m). Te ollowing lemma gives te link between te two-scale and te weak convergence (see, e.g., [23]). Lemma 5.3. Let v } >0 be a uniormly bounded sequence in L 2 (Ω m,t ) tat satisies te conditions: (i) D v v 0 weakly in L 2 (Ω T ; L 2 per(y m ));

18 18 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 (ii) 1 m(x)v 2s v L 2 (Ω T ; L 2 per(y m )). Ten v 0 = v a.e. in Ω T Y m. Also we ave te ollowing result (see, e.g., [32, 53]). Lemma 5.4. I v L 2 (Ω m,t ) and 1 m(x)v 2s v L 2 (Ω T ; L 2 per(y m )) ten D v converges to v strongly in L q (Ω T Y m ), were 2s denotes te strong two scale convergence. I v L 2 (Ω T ) is considered as an element o L 2 (Ω T Y m ) constant in y, ten D v converges strongly to v in L 2 (Ω T Y m ) Dilated unctions D S m, D P m and teir properties. Let us introduce te ollowing notation or te dilated unctions deined on te matrix part o te reservoir Ω: p m := D P m, s m := D S m, ϑ m := D β m = β(s m), (5.3) and θ l,m := D Θ l,m = ρ l (p m + G l (s m)) s m, θ g,m := D Θ g,m = ρ g (p m + G g (s m)) (1 s m). (5.4) Te goal o tis section is to derive te equations or te dilated unctions s m, p m and using Lemma 5.2 to obtain te corresponding uniorm estimates. First, we establis te ollowing result. Lemma 5.5. For any > 0 and or a.e. x Ω, te dilated unctions s m, p m deined by (5.3) are te solutions o te system 0 s m 1; Φ m θ l,m div y K m ρ l (s m, p m) [ λ l (s m) p m + ϑ m λ l (s m)ρ l (s m, p m) g ]} = 0; (5.5) Φ m θ g,m div y K m ρ g (s, p m) [ λ g (s m) p m ϑ m λ g (s m)ρ g (s m, p m) g ]} = 0 in L 2 (0, T ; H 1 (Y m )). Here ρ k (s m, p m) := ρ k (p m + G k (s m)) (k = l, g). Te proo o te above can be done by te arguments similar to tose used in te proo o [53, Lemma 6.7]. We complete system (5.5) wit te boundary and initial conditions. Te boundary conditions are p m(x, y, t) = P (y + c (x), t), ϑ m(x, y, t) = β (y + c (x), t) (5.6) in H 1/2 (Γ m ) or (x, t) Ω m,t, were β := β(s ). Te initial conditions are p m(x, y, 0) = D P 0 m(x, y), s m(x, y, 0) = D S 0 m(x, y) in Ω Y m. (5.7) For x outside te matrix part by te deinition (5.1), we set p m(x, y, t) = s m(x, y, t) = 0 in H 1/2 (Γ m ) or x Ω \ Ω m, t (0, T ). (5.8)

19 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 19 Remark 5.6. Tere is no conusion tat in (5.7) we use again te notation D, or te dilatation operator deined or L 2 (Ω)-unctions wic maps tem to te unctions in L 2 (Ω Y m ) by te ormula (D ϕ)(x, y) := ϕ(y + c (x)). (5.9) Remark 5.7. We note tat problem (5.5) wit non-omogeneous Diriclet boundary conditions (5.6) and initial conditions (5.7) corresponds to a amily o problems in Y m (0, T ) parameterized by te slow variable x and depending on it troug te boundary data (5.6). Te ollowing lemma contains te uniorm a priori estimates or te dilated solutions s m, p m o (5.5). Lemma 5.8. Let (p m, s m) be a solution to (5.5)-(5.7). Tere exists a constant C tat does not depend on and suc tat 0 s m 1 a.e. in Ω T Y m ; (5.10) p m L 2 (Ω T ;H 1 per (Ym)) + ϑ m L 2 (Ω T ;H 1 per (Ym)) C; (5.11) t (Φ m θl,m) L 2 (Ω T ;Hper(Y 1 + m)) t(φ m θg,m) L 2 (Ω T ;Hper(Y 1 m)) C; (5.12) were ϑ m := D β m = β(s m). Proo. Statement (5.10) is evident. Te uniorm estimate (5.11) easy ollow rom te uniorm bounds (3.5), (3.7), and Lemma 5.2. Te bound (5.12) ollows rom Lemmata 5.2, 3.5. Te proo is complete Convergence results or te dilated unctions p m, s m, θl,m, θ g,m. In tis section we establis convergence results wic will be used below to obtain te omogenized system. From Lemmata 5.3, 5.8 we get te ollowing convergence results. Lemma 5.9. Let (p m, s m) be a solution to (5.5) (5.7). Ten (up to a subsequence), χ m S m χ m P m 2s s L 2 (Ω T ; L 2 per(y m )), s m s weakly in L 2 (Ω T Y m ); (5.13) 2s p L 2 (Ω T ; L 2 per(y m )), p m p weakly in L 2 (Ω T ; H 1 (Y m )); (5.14) x P m 2s y p L 2 (Ω T ; L 2 per(y m )); (5.15) χ mβ(s m) 2s ϑ ; β(s m) ϑ weakly in L 2 (Ω T ; H 1 (Y m )); x β(s m) 2s y ϑ, were χ m = χ m(x) is te caracteristic unction o te subdomain Ω m. (5.16) Remark Te limit ϑ will be identiied below. More precisely, it will be sown tat ϑ = β(s). It is clear tat te convergence results o Lemma 5.9 above are not suicient or derivation o te equations or te limit unctions s, p. In order to overcome tis diiculty, we introduce te restrictions o te dilated unctions s m, p m wic are deined below. Te key eature o te newly introduced unctions is tat tey possess more compactness properties tan te dilated unctions introduced in (5.3), (5.4). To tis end, ollowing [32], we ix x = x 0 Ω and deine te restrictions o s m, p m to te cell containing te point x 0. Tese unctions are deined in te

20 20 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 domain Y m (0, T ) and are constants in te slow variable x. In order to obtain te uniorm estimates or te restricted unctions (tey are similar to te corresponding estimates or P, S rom Section 3) we make use o te estimates (5.10) (5.11). Ten we proceed as ollows. First, we deine te restrictions o s m, p m to a cell Kx 0 wic is a cube containing x 0 Ω and te lengt. We denote by k(x 0, ) Z d suc tat Kx 0 := ( Y + k(x 0, ) ). Due to te deinition o te dilation operator, te unctions s m, p m are constant in x in Kx 0. Te restricted unctions are given by: s s m,x 0 (y, t) := m, i x Kx 0 ; p p m,x 0, oterwise; 0 (y, t) := m, i x Kx 0 ; (5.17) 0, oterwise. For any > 0, te pair o unctions (s m,x 0, p m,x 0 ) is a solution to (5.5) (5.7) in Y m (0, T ). Now we obtain te uniorm, in, or te unctions s m,x 0, p m,x 0 deined in (5.17), or a.e. x 0 Ω. We ave. Lemma For a.e. x 0 Ω, tere is a constant C = C(x 0 ) wic does not depend on and suc tat p m,x 0 L2 (0,T ;H 1 per (Ym)) + ϑ m,x 0 L2 (0,T ;H 1 per (Ym)) C(x 0 ), (5.18) t (Φ m θl,m,x 0 ) L 2 (0,T ;Hper(Y 1 + m)) t(φ m θg,m,x 0 ) L 2 (0,T ;Hper(Y 1 m)) C; (5.19) were θ l,m,x 0 := ρ l (p m,x 0 + G l (s m,x 0 )) s m,x 0, θ g,m,x 0 := ρ g (p m,x 0 + G g (s m,x 0 )) (1 s m,x 0 ). Next we establis a compactness or te amilies θ l,m,x 0 } >0 and θ g,m,x 0 } >0. We obtain tis compactness result by applying [4, Lemma 4.2]. Proposition Under our standing assumptions, or a.e. x 0 Ω, te amilies θ w,m,x 0 } >0 and θ g,m,x 0 } >0 are compact sets in L 2 (Y m (0, T )). 6. Formulation o te main result We study te asymptotic beavior o te solution to problem (2.2) (troug te equivalent problem (2.18)) as 0. In particular, we are going to sow tat te eective model is 0 S 1 in Ω T ; Φ Ξ l Φ Ξ g div x K ρ l λ l (S) [ P l ρ l g ]} = Q l in Ω T ; div x K ρ g λ g (S) [ P g ρ g g ]} = Q g in Ω T ; P c (S) = P g P l in Ω T. (6.1) Here we use te ollowing notation: S, P l, P g denote te omogenized water saturation, water pressure, and gas pressure, respectively. Φ denotes te eective porosity and is given by Φ := Φ Y Y m, (6.2)

21 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 21 were Y r is te measure o te set Y r, r =, m. K is te omogenized tensor wit te entries K ij deined by: K ij := K [ y ξ i + e i ][ y ξ j + e j ] dy, (6.3) Y m Y were ξ j is a solution o te auxiliary problem Te unctions Ξ l, Ξ g are y ξ j = 0 in Y ; y ξ j ν = e j ν on Γ m ; y ξ j (y) Y -periodic. (6.4) Ξ l := Sρ l (P l ) and Ξ g := (1 S)ρ g (P g ). (6.5) For almost all point x Ω a matrix block Y m R d is suspended topologically. Equations or low in a matrix block are given by 0 s 1 in Ω T Y m ; θ } l Φ m div y K m ρ l (p l )λ l (s) y p l = 0 in Ω T Y m ; θ } g Φ m div y K m ρ g (p g )λ g (s) y p g = 0 in Ω T Y m ; P c (s) = p g p l in Ω T Y m. (6.6) Here we use te ollowing notation: s, p l, p g denote te water saturation, te water and gas pressures in te block Y m, respectively. Te unctions θ l, θ g are deined by θ l := sρ l (p l ) and θ g := (1 s)ρ g (p g ). (6.7) For any x Ω and t > 0, te matrix-racture sources are given by Q l := Φ m θl Y m (x, y, t) dy and Q g := Φ m θg Y m Ym (x, y, t) dy. (6.8) Y m Te boundary conditions or te eective system (6.1) are given by P g (x, t) = P l (x, t) = 0 on Γ 1 (0, T ); K ρ g (P g )λ g (S) ( P g ρ g (P g ) g ) ν = 0 on Γ 2 (0, T ); K ρ l (P l )λ l (S) ( P l ρ l (P l ) g ) ν = 0 on Γ 2 (0, T ). Te interace conditions or te system (6.6) are given by: Finally, te initial conditions read P l (x, t) = p l (x, y, t) on Ω T Γ m ; P g (x, t) = p g (x, y, t) on Ω T Γ m. (6.9) (6.10) S(x, 0) = S 0 (x) and P g (x, 0) = p 0 g(x) in Ω, (6.11) s(x, y, 0) = S 0 (x) and p g (x, y, 0) = p 0 g(x) in Ω Y m. (6.12) Te main result o te paper reads as ollows.

22 22 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 Teorem 6.1. Let assumptions (A1) (A8) be ulilled. Ten te solution o initial problem (2.2) converges (up to a subsequence) in te two-scale sense to a weak solution o omogenized problem (6.1). 7. Proo o Teorem 6.1 Te proo o te omogenization result will be done in several steps. Using te convergence and compactness results rom Section 4 we pass to te limit in equations (2.25), (2.26). Tis is done in Section 7.1. Te passage to te limit in te matrix blocks makes use o te dilation operator deined in Section 5 above. In te passage to te limit in (2.25), (2.26) we see tat te omogenized equations contain, irst o all, te corrector unctions P 1, β 1 deined in Lemma 4.9 as well as non explicit limits L l,m, L g,m (see Section 7.1). Te corrector unctions are ten identiied in Section 7.2. Having obtained te omogenized equations in terms o te global pressure and saturation, we can reormulate tese equations in terms o new unknown unctions tat it is naturally to call te omogenized pase pressures. Tey are deined as ollows: P l := P + G l (S) and P g := P + G g (S). However, in tese equations te limits L l,m, L g,m are still non identiied. To obtain te explicit orm o tese limits we make use o te ideas rom [32]. Tis completes te proo o te main result o te paper Passage to te limit in te equation (2.25), (2.26). We set ϕ l (x, x, t) : = ϕ(x, t) + ζ(x, x, t) = ϕ(x, t) + ζ 1 (x, t) ζ 2 ( x ) (7.1) = ϕ(x, t) + ζ (x, t), were ϕ D(Ω T ), ζ 1 D(Ω T ), ζ 2 C per(y ), and plug te unction ϕ l in (2.25). Tis yields [ ϕ ] Φ χ Ω (x) Θ l, T + ζ dx dt + K χ Ω (x) ρ l, λ l ( S [ ) P ] ρ l, g T + β( S } ) [ ϕ + x ζ + y ζ ] dx dt [ ϕ ] Φ m χ Ω (x) Θ l,m T + ζ dx dt } + 2 K m χ Ω (x) ρ l,m λ l (Sm)( P m ρ l,m g) + β(sm) T (7.2) were [ ϕ + x ζ + y ζ ] dx dt = 0, ρ l, := ρ l ( S, P ) := ρ l ( P + G l ( S )), ρ l,m := ρ l (S m, P m) := ρ l (P m + G l (S m)). (7.3)

23 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 23 Taking into account te convergence results rom Lemmata 4.9, 5.9, we pass to te two-scale limit in (7.2) as 0 and get te irst omogenized equation Y Φ Θ Ω ϕ l dx dt T + K ρ l λ l (S) Ω [ P + y P 1 ρ l g ] + [ ] } β + y β 1 Υ ϕ dy dx dt T Y = Φ m were Ω T Y m L l,m (x, y, t) ϕ (x, t) dy dx dt, (7.4) ρ l := ρ l ( P + Gl (S) ) and Θ l := S ρ l ; (7.5) te unction Υ ϕ is Υ ϕ := [ ϕ + ζ 1 y ζ 2 ], inally, te unction L l,m = L l,m (x, y, t) is deined as a two-scale limit o te sequence Θ l,m } >0 as 0. Namely, Θ l,m 2s L l,m L 2 (Ω T ; L 2 per(y m )). (7.6) In a similar way we pass to te two scale limit in equation (2.26) and get te second omogenized equation Y Φ Θ Ω ϕ g dx dt T + K ρ g λ g (S) Ω [ P + y P 1 ρ g g ] [ ] } β + y β 1 Υ ϕ dy dx dt T Y = Φ m were Ω T Y m L g,m (x, y, t) ϕ (x, t) dy dx dt, (7.7) ρ g := ρ g ( P + Gg (S) ) and Θ g := (1 S) ρ g ; (7.8) te unction L g,m = L g,m (x, y, t) is deined as a two-scale limit o te sequence Θ g,m} >0 as 0. Namely, Θ g,m 2s L g,m L 2 (Ω T ; L 2 per(y m )). (7.9) 7.2. Identiication o te corrector unctions P 1, β 1 and omogenized equations. Step 1. Identiication o P 1. Consider te equations (7.4), (7.7). We set ϕ 0. We get: Taking into account tat ρ l, ρ g are strictly positive and do not depend on y, we obtain λ l (S) [ P + y P 1 ρ l g ] + [ ] } β + y β 1 ζ 2 (y) dy = 0, (7.10) Y Y λ g (S) [ P + y P 1 ρ g g ] [ β + y β 1 ] } ζ 2 (y) dy = 0. (7.11) Now adding (7.10) and (7.11), we obtain λ(s) [ ] [ ] } P + y P 1 λl (S) ρ l + λ g (S) ρ g g y ζ 2 (y) dy = 0. (7.12) Y Taking into account condition (A4), rom (7.12), we obtain } P X(S, P ) g + y P 1 y ζ 2 (y) dy = 0, (7.13) Y

24 24 L. AIT MAHIOUT, B. AMAZIANE, A. MOKRANE, L. PANKRATOV EJDE-2016/52 were X(S, P ) := λ l(s) ρ l + λ g (S)ρ g. (7.14) λ(s) Now we proceed in a standard way (see, e.g., [45]). Let ξ j be a solution o y ξ j = 0 in Y ; y ξ j ν y = e j ν y y ξ j (y) Ten te unction P 1 can be represented as P 1 (x, y, t) = d j=1 on Γ m Y -periodic. (7.15) ( P ) ξ j (y) (x, t) X(S, P )g j. (7.16) x j Step 2. Identiication o β 1. Now we turn to te identiication o te unction β 1. From (7.10) and (7.13), we obtain λ l (S) X(S, P ) g λ l (S) ρ l g + [ ] } β + y β 1 y ζ 2 (y) dy = 0. Y From tis equation we ave [ β+ y β 1 ] y ζ 2 (y) dy = Y Y λ l (S) [ X(S, P ) ρ l ] g } y ζ 2 (y) dy. (7.17) Now, rom (7.17) we get te ollowing equation or te unction β 1 : were Y [ β Z(S, P ) g + y β 1 ] y ζ 2 (y) dy = 0, (7.18) Ten te unction β 1 can be represented as β 1 (x, y, t) = Z(S, P ) := λ l(s)λ g (S)[ ] ρl ρ g. (7.19) λ(s) d j=1 ( β(s) ) ξ j (y) (x, t) Z(S, P ) g j. (7.20) x j Step 3. Homogenized equation or te l pase. Now we are in position to obtain te irst omogenized equation. Coosing ζ 2 = 0, rom (7.4), we obtain Φ Y Θ l div x K λ l (S)ρ l Y [ P + y P 1 X(S, P ) g ] dy [ ] } + K λ l (S)ρ l X(S, P ) ρl g dy Y [ div x K ρ l β + y β 1 Z(S, P ) g Y ] } dy + K ρ l Z(S, P ) g dy Y L l,m = Φ m (x, y, t) dy. Y m (7.21)

25 EJDE-2016/52 IMMISCIBLE COMPRESSIBLE TWO-PHASE FLOW 25 Ten taking into account te deinitions o te unctions X(S, P ) and Z(S, P ), rom (7.21), we ave [ div x K λ l (S)ρ l P + y P 1 X(S, P ) g Y ] } dy Φ Y Θ l div x K Y ρ l [ β + y β 1 Z(S, P ) g ] dy = Φ m Y m L l,m (x, y, t) dy. } (7.22) Now we use te representations or te corrector unctions P 1 and β 1, to obtain Φ Θ l div x K [ ] λ l (S)ρ l P X(S, P ) g + K [ ] } ρ l β(s) Z(S, P ) g (7.23) = Φ m Y m Y m L l,m (x, y, t) dy, were te eective porosity Φ and K, te eective permeability tensor, are given by Φ Y := Φ and K ij := K [ y ξ i + e i ] [ y ξ j + e j ] dy (1 i, j d). Y m Y m Y Since λ l (S)X(S, P ) + Z(S, P ) = λ l (S)ρ l, rom (7.23) we obtain Φ Θ l div x K [ ρ l λl (S) P + β(s) λ l (S)ρ l g ]} = Φ m L l,m (x, y, t) dy. Y m Y m (7.24) Step 4. Homogenized equation or te g pase. In a similar way, coosing ζ 2 = 0, rom te equation (7.7), we derive te second omogenized equation div x Φ Θ g = Φ m Y m Y m K ρ g [ λg (S) P β(s) λ g (S) ρ g g ]} L g,m (x, y, t) dy. (7.25) Let us introduce now te unctions tat is naturally to call te omogenized pase pressures. Namely, we deine P l := P + G l (S) and P g := P + G g (S). (7.26) Ten rom (7.24) (7.26) we get te equations Φ Ξ l div x K ρ l λ l (S) [ P l ρ l g ]} = Φ m Y m Φ Ξ g were div x K ρ g λ g (S) [ P g ρ g g ]} = Φ m Y m Y m Y m Ξ l := Sρ l (P l ) and Ξ g := (1 S)ρ g (P g ). L l,m (x, y, t) dy; (7.27) L g,m (x, y, t) dy, (7.28) At tis stage, te omogenization process is inised by obtaining equations (7.27), (7.28). However, i in addition we want to express te unctions L l,m, L g,m