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Soil losses in rainfed Mediterranean vineyards under climate change scenarios. The effects of drainage terraces.

  • Most vines in the Mediterranean are cultivated on bare soils, due to the scarcity of water. In addition, most traditional soil conservation measures have been eliminated to facilitate the movement of machinery in the fields. In such conditions, high erosion rates are recorded. Given the predicted changes in precipitation and an increasing number of extreme events, an increase in erosion processes is expected. In this study, erosion processes under different climate change scenarios were evaluated as well as the effects of implementing drainage terraces in vineyards. Soil losses were simulated using the WEPP model. The results confirmed the relevance of extreme events on annual soil losses. The WEPP model gave satisfactory results in predicting runoff and soil losses, although the soil losses recorded after some extreme events were under-predicted. The model responded to changes in precipitation and because of that a decrease in precipitation gave rise to a decrease in soil losses. For the scenario in 2050, runoff volumes decreased between 19.1 and 50.1%, while erosion rates decreased between 34 and 56%. However, the expected increase in rainfall intensity may contribute to higher erosion rates than at present. The construction of drainage terraces, perpendicular to the maximum slope, 3 m wide and 30 m between terraces, may lead to an average decrease in soil losses of about 45%.

    Citation: María Concepción Ramos. Soil losses in rainfed Mediterranean vineyards under climate change scenarios. The effects of drainage terraces.[J]. AIMS Agriculture and Food, 2016, 1(2): 124-143. doi: 10.3934/agrfood.2016.2.124

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  • Most vines in the Mediterranean are cultivated on bare soils, due to the scarcity of water. In addition, most traditional soil conservation measures have been eliminated to facilitate the movement of machinery in the fields. In such conditions, high erosion rates are recorded. Given the predicted changes in precipitation and an increasing number of extreme events, an increase in erosion processes is expected. In this study, erosion processes under different climate change scenarios were evaluated as well as the effects of implementing drainage terraces in vineyards. Soil losses were simulated using the WEPP model. The results confirmed the relevance of extreme events on annual soil losses. The WEPP model gave satisfactory results in predicting runoff and soil losses, although the soil losses recorded after some extreme events were under-predicted. The model responded to changes in precipitation and because of that a decrease in precipitation gave rise to a decrease in soil losses. For the scenario in 2050, runoff volumes decreased between 19.1 and 50.1%, while erosion rates decreased between 34 and 56%. However, the expected increase in rainfall intensity may contribute to higher erosion rates than at present. The construction of drainage terraces, perpendicular to the maximum slope, 3 m wide and 30 m between terraces, may lead to an average decrease in soil losses of about 45%.


    The modeling of displacement process involving two immiscible fluids is of considerable importance in groundwater hydrology and reservoir engineering such as petroleum and environmental problems. More recently, modeling multiphase flow received an increasing attention in connection with gas migration in a nuclear waste repository and sequestration of CO2.

    In this paper, we focus our attention on the modeling of immiscible compressible two-phase flow through heterogeneous reservoirs in the framework of the geological disposal of radioactive waste. The long-term safety of the disposal of nuclear waste is an important issue in all countries with a significant nuclear program. One of the solutions envisaged for managing waste produced by nuclear industry is to dispose the radioactive waste in deep geological formations chosen for their ability to delay and to attenuate possible releases of radionuclides in the biosphere. Repositories for the disposal of high-level and long-lived radioactive waste generally rely on a multi-barrier system to isolate the waste from the biosphere. The multibarrier system typically comprises the natural geological barrier provided by the repository host rock and its surroundings and an engineered barrier system, i.e. engineered materials placed within a repository, including the waste form, waste canisters, buffer materials, backfill and seals, for more details see for instance [43]. An important task of the safety assessment process is the handling of heterogeneities of the geological formation.

    In the frame of designing nuclear waste geological repositories, a problem of possible two-phase flow of water and gas, mainly hydrogen, appears, for more details see for instance [43]. Multiple recent studies have established that in such installrations important amounts of gases are expected to be produced in particular due to the corrosion of metallic components used in the repository design, see e.g. [27,42] and the references therein. The French Agency for the Management of Radioactive Waste (Andra) [11] is currently investigating the feasibility of deep geological disposal of radioactive waste in an argillaceous formation. A question related to the long-term performance of the repository concerns the impact of the hydrogen gas generated in the wastes on the pressure and saturation fields in the repository and the host rock.

    During recent decades mathematical analysis and numerical simulation of multiphase flows in porous media have been the subject of investigation of many researchers owing to important applications in reservoir simulation. There is an extensive literature on this subject. We will not attempt a literature review here but will merely mention a few references. Here we restrict ourselves to the mathematical analysis of such models. We refer, for instance, to the books [14,23,26,29,36,38,44] and the references therein. The mathematical analysis and the homogenization of the system describing the flow of two incompressible immiscible fluids in porous media is quite understood. Existence, uniqueness of weak solutions to these equations, and their regularity has been been shown under various assumptions on physical data; see for instance [3,14,15,21,24,25,23,29,41] and the references therein. A recent review of the mathematical homogenization methods developed for incompressible immiscible two-phase flow in porous media and compressible miscible flow in porous media can be viewed in [4,37,38]. We refer for instance to [16,17,18,19,20,34,35] for more information on the homogenization of incompressible, single phase flow through heterogeneous porous media in the framework of the geological disposal of radioactive waste.

    However, as reported in [9], the situation is quite different for immiscible compressible two-phase flow in porous media, where, only recently few results have been obtained. In the case of immiscible two-phase flows with one (or more) compressible fluids without any exchange between the phases, some approximate models were studied in [30,31,32]. Namely, in [30] certain terms related to the compressibility are neglected, and in [31,32] the mass densities are assumed not to depend on the physical pressure, but on Chavent's global pressure. In the articles [22,33,39,40], a more general immiscible compressible two-phase flow model in porous media is considered for fields with a single rock type and [9] treated the case with several types of rocks. In [4,10] homogenization results were obtained for water-gas flow in porous media using the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns.

    Let us also mention that, recently, a new global pressure concept was introduced in [5,7] for modeling immiscible, compressible two-phase flow in porous media without any simplifying assumptions. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation (the global pressure equation) and a nonlinear diffusion-convection one (the saturation equation). This new formulation is fully equivalent to the original phase equations formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. For this model, an existence result is obtained in [8] and homogenization results in [6].

    Let us note that all the aforementioned works are restricted to the case where the gas density is bounded from below and above, contrarily to the present work. This assumption is too restrictive for some realistic problems, such as gas migration through engineered and geological barriers for a deep repository for radioactive waste. In this case the gas obeys the ideal gas law, i.e. the equation of state is given by ϱg(p)def=σp where ϱg is the gas density, pg is the gas pressure and σ is a given constant. Then a new degeneracy appears in the evolution term of the gas pressure equation. In this paper we extend our previous results obtained in [4] to the more complex case of an ideal gas which is more reasonable in gas reservoir engineering. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. To obtain these results we elaborated a new approach based on the ideas from [4] and regularization.

    The rest of the paper is organized as follows. In Section 2 we describe the physical model and formulate the corresponding mathematical problem. We also provide the assumptions on the data.

    The goal of Section 3 is to prove the existence result for the corresponding system of equations. The proof is divided into a number of steps. In subsection 3.1 we consider an auxiliary δ-problem where the gas density admits a positive lower bound. The existence result for the δ-problem is given in subsection 3.2. In subsection 3.3 we obtain a number of a priori estimates for a solution of δ-problem. Then in subsection 3.4 we prove a compactness result adapted to our model. Finally, in subsection 3.5 we pass to the limit in δ-problems and complete the proof of the existence.

    Section 4 is devoted to the corresponding homogenization problem. In subsection 4.1 we introduce the model with a periodic microstructure. We assume that both porosity and absolute permeability tensor are periodic rapidly oscillating functions.

    Then subsection 4.2 we formulate the homogenization result. This result is proved in subsection4.3. The proof is based on the two-scale convergence technique. Our analysis relies essentially on a compactness result [4] which is rather involved due to the degeneracy and the nonlinearity of the system.

    The last section is followed by some concluding remarks.

    We consider an immiscible compressible two-phase flow process in a porous reservoir ΩRd (d=1,2,3) which is a bounded Lipschitz domain. The time interval of interest is (0,T). We focus here on the particular case of water and gas phases, but the consideration below is also valid for a general wetting phase and a non-wetting phase. Let Φ=Φ(x) be the porosity of Ω; K=K(x) be the absolute permeability tensor of Ω; ϱw, ϱg are the densities of water and gas, respectively; Sw=Sw(x,t), Sg=Sg(x,t) are the saturations of water and gas in Ω×(0,T); kr,w=kr,w(Sw), kr,g=kr,g(Sg) are the relative permeabilities of water and gas; pw=pw(x,t), pg=pg(x,t) are the pressures of water and gas in Ω×(0,T).

    In what follows, for the sake of presentation simplicity we neglect the source terms. Then the conservation of mass of each phase is described by (see, e.g., [23,26,36]):

    {Φ(x)t(Swϱw(pw))+div{ϱw(pw)qw}=0inΩT;Φ(x)t(Sgϱg(pg))+div{ϱg(pg)qg}=0inΩT, (1)

    where ΩTdef=Ω×(0,T) with T>0, the velocities of water and gas qw, qg are defined by Darcy-Muskat's law:

    qwdef=K(x)λw(Sw)(pwϱw(pw)g),withλw(Sw)=kr,wμw(Sw); (2)
    qgdef=K(x)λg(Sg)(pgϱg(pg)g),withλg(Sg)=kr,gμg(Sg). (3)

    Here g, μw,μg are the gravity vector and the viscosities of the water and gas, respectively.

    From now on we assume that the density of the water is constant, which for the sake of simplicity will be taken equal to one, i.e. ϱw(pw)=Const=1, and the gas density ϱg obeys the ideal gas law and is given by the following function:

    ϱg(p)=0forp0;ϱg(p)=σpmaxforppmax;ϱg(p)def=σpfor0<p<pmax. (4)

    Here σ, ϱmax, and pmax are positive constants. Note that ϱg is a continuous monotone and non-negative function.

    The model is completed as follows. By the definition of saturations, one has

    Sw+Sg=1withSw,Sg0. (5)

    We set:

    Sdef=Sw. (6)

    Then the curvature of the contact surface between the two fluids links the jump of pressure of two phases to the saturation by the capillary pressure law:

    Pc(S)=pgpwwithPc(s)<0foralls[0,1]andPc(1)=0, (7)

    where Pc(s) denotes the derivative of the function Pc(s).

    Now due to (6) and the assumption on the water density, we rewrite the system (1) as follows:

    {Φ(x)Stdiv{K(x)λw(S)(pwg)}=0inΩT;Φ(x)Θtdiv{K(x)λg(S)ϱg(pg)(pgϱg(pg)g)}=0inΩT;Pc(S)=pgpwinΩT, (8)

    where λg(S):=λg(1S) and

    Θdef=ϱg(pg)(1S). (9)

    The system (8) have to be completed by appropriate boundary and initial conditions.

    Boundary conditions.: We suppose that the boundary Ω consists of two parts Γinj and Γimp such that ΓinjΓimp=, Ω=¯Γinj¯Γimp. The boundary conditions are given by:

    {pg(x,t)=pw(x,t)=0onΓinj×(0,T);qwν=qgν=0onΓimp×(0,T), (10)

    where the velocities qw,qg are defined in (2), (3).

    Initial conditions.: The initial conditions read:

    pw(x,0)=p0w(x)  and pg(x,0)=p0g(x)  inΩ. (11)

    Notice that from (10) and (7) it follows that S=1 on Γinj×(0,T). The initial condition for S is uniquely defined by the equation

    Pc(S0(x))=p0g(x)p0w(x). (12)

    Then according to (9) the initial condition for Θ reads

    Θ0=ϱg(p0g)(1S0). (13)

    Remark 1. It is important to underline that in the earlier works (see, e.g., [4,9,10,30,31,32,33]) it was assumed that the gas density admits a strictly positive lower bound:

    ϱminϱg(p)ϱmaxwith0<ϱmin<ϱmax<+. (14)

    In the sequel, we use a formulation obtained after transformation using the concept of the so called global pressure. In the case of incompressible two-phase flow this concept was introduced for the first time in [12,13]. Following [14,23], see also [26], we first recall the definition of the global pressure. It plays a crucial role, in particular, for compactness results. The idea of introducing the global pressure is as follows. We want to replace the water-gas flow by a flow of a fictive fluid obeying the Darcy law with a non-degenerating coefficient. Namely, we are looking for a pressure P and the coefficient γ(S) such that γ(S)>0 for all S[0,1], and

    λw(S)pw+λg(S)pg=γ(S)P. (15)

    Then the global pressure, P, is defined by:

    pwdef=P+Gw(S)andpgdef=P+Gg(S); (16)

    the functions Gw(s) and Gg(s) will be introduced later on, in (19), (20). Now it is easy to see that

    λw(S)pw+λg(S)pg=λ(S)P+{λg(S)Gg(S)+λw(S)Gw(S)},

    where

    λ(s)def=λw(s)+λg(s) (17)

    We set:

    λg(S)Gg(S)+λw(S)Gw(S)=0. (18)

    Then γ(S)=λ(S). By construction, λ(S)>0 for all S[0,1] (see the condition (A.5) below). Thus the relation (15) is established. Now we specify the functions Gw, Gg. We define Gg as follows:

    Gg(S)def=Gg(0)+S0λw(s)λ(s)Pc(s)ds. (19)

    The functions Gw are then defined by

    Gw(S)def=Gg(S)Pc(S)withGw(S)=λg(S)λ(S)Pc(S)S. (20)

    Notice that from (19), (20) we get:

    λw(s)Gw(s)=α(s)sandλg(s)Gg(s)=α(s)s, (21)

    where

    α(s)def=λg(s)λw(s)λ(s)|Pc(s)|. (22)

    Now we link the capillary pressure and the mobilities. In a standard way (see, e.g., [40] or [9] for more details) we obtain the following identity:

    λg(S)|pg|2+λw(S)|pw|2=λ(S)|P|2+|b(S)|2, (23)

    where

    b(S)def=s0a(ξ)dξwitha(s)def=λg(s)λw(s)λ(s)|Pc(s)|. (24)

    Ii is also convenient to introduce the following function β:

    β(s)def=s0α(ξ)dξ, (25)

    where the function α is defined in (22). Notice that by the definition of the global pressure, (24), (25), and by the boundedness of λw,λg (see the condition (A.5) below) the following relations holds:

    |β(S)|2C|b(S)|2, (26)
    λw(s)pw=λw(s)P+β(s),andλg(s)pg=λg(s)Pβ(s). (27)

    In order to complete this section, let us calculate the value of the global pressure function P on Γinj. In what follows (see condition (A.4) below) we assume that the capillary pressure function Pc satisfies the condition Pc(1)=0. Then from (10)1 we have that the saturation S equals one on Γinj. Now the definition of the global pressure (16) implies that the function P on Γinj is a constant which we denote by P1.

    The main assumptions on the data are as follows:

    (A.1) The porosity ΦL(Ω), and there are positive constants ϕ,ϕ+ such that 0<ϕ<ϕ+ and

    0<ϕΦ(x)ϕ+<1a.e.inΩ. (28)

    (A.2) The tensor K(L(Ω))d×d and there exist constants K,K+ such that 0<K<K+ and

    K|ξ|2(K(x)ξ,ξ)K+|ξ|2forallξRd,a.e.inΩ. (29)

    (A.3) The function ϱg=ϱg(p) is given by (4).

    (A.4) The capillary pressure function Pc(s)C1([0,1];R+). Moreover, Pc(s)<0 in [0,1] and Pc(1)=0.

    (A.5) The functions λw,λg belong to the space C([0,1];R+) and satisfy the following properties:

    (ⅰ) 0λw,λg1 in [0,1];

    (ⅱ) λw(0)=0 and λg(1)=0;

    (ⅲ) there is a positive constant L0 such that λ(s)=λw(s)+λg(s)L0>0 in [0,1].

    (A.6) The function α defined in (22) is an element of C1([0,1];R+). Moreover, α(0)=α(1)=0, and α>0 in (0,1).

    (A.7) The function β1, inverse of β defined in (25) is a Hölder continuous function of order θ with θ(0,1) on the interval [0,β(1)]. There exists a positive constant Cβ such that for all s1,s2[0,β(1)] the following inequality holds:

    |β1(s1)β1(s2)|Cβ|s1s2|θ.

    (A.8) The initial data for the pressures are such that p0g,p0wL2(Ω).

    (A.9) The initial data for the saturation is such that S0L(Ω) and 0S01 a.e. in Ω.

    The assumptions (A.1)-(A.9) are classical for two-phase flow in porous media.

    In order to define a weak solution of the above problem, we introduce the following Sobolev space:

    H1Γinj(Ω)def={uH1(Ω):u=0onΓinj}.

    The space H1Γinj(Ω) is a Hilbert space. The norm in this space is given by

    uH1Γinj(Ω)=u(L2(Ω))d.

    Theorem 3.1. Let assumptions (A.1)-(A.9) be fulfilled. Then there exist functions pg,pw,S such that:

    (Ⅰ) 

    pw,pgL2(ΩT)andλw(S)pw,λg(S)pgL2(ΩT); (30)
    β(S)L2(0,T;H1(Ω))andPP1L2(0,T;H1Γinj(Ω)); (31)
    ΦStL2(0,T;H1(Ω))andΦΘtL2(0,T;H1(Ω)); (32)

    where the function Θ is defined in (9); S=1 on Γinj.

    (Ⅱ) the maximum principle holds:

    0S1a.e.inΩT. (33)

    (Ⅲ) For any φw,φgC1([0,T];H1(Ω)) satisfying φw=φg=0 on Γinj×(0,T) and φw(x,T)=φg(x,T)=0, we have:

    ΩTΦ(x)SφwtdxdtΩΦ(x)S0(x)φw(x,0)dx+ΩTK(x)λw(S)pwφwdxdt//ΩTK(x)λw(S)gφwdxdt=0; (34)
    ΩTΦ(x)ΘφgtdxdtΩΦ(x)Θ0(x)φg(x,0)dx+ (35)
    +ΩTK(x)λg(S)ϱg(pg)pgφgdxdtΩTK(x)λg(S)[ϱg(pg)]2gφgdxdt=0

    with Θ defined in (9), and Pc(S)=pgpw.

    (Ⅳ) The initial conditions are satisfied in a weak sense as follows:

    ψH1Γinj(Ω),ΩΦ(x)S(x,t)ψ(x)dx,ΩΦ(x)Θ(x,t)ψ(x)dxC([0,T]). (36)

    Furthermore, we have

    (ΩΦ(x)Sψdx)(0)=ΩΦ(x)S0ψdx (37)

    and

    (ΩΦ(x)Θψdx)(0)=ΩΦ(x)Θ0ψdx (38)

    with S0 and Θ0 defined in (12) and (13), respectively.

    The proof of Theorem 3.1 is divided into a several steps. It is based on a auxiliary existence result for the system obtained by approximation of the initial degenerate gas density ϱg by a family of functions {ϱδg}δ>0 that admit a positive lower bound. For such kind of system the desired existence result is proved in [9,30,31,32,33]. This result is formulated in subsection 3.1. Using the weak formulation of the regularized problem and the uniform in δ estimates for its solution, we, finally, prove Theorem 3.1.

    In this subsection we approximate the function ϱg by a family of functions {ϱδg}δ>0 that admit a positive lower bound. For each δ>0 we set:

    ϱδg(p)=δfor<pδσ;ϱδg(p)=σpmaxforppmax;ϱδg(p)def=σpforδσ<p<pmax. (39)

    Here σ,ϱmax,pmax are positive constants.

    In addition to (8), consider the following family of problems:

    δproblem:{Φ(x)Sδtdiv{K(x)λw(Sδ)(pδwg)}=0inΩT;Φ(x)Θδtdiv{K(x)λg(Sδ)ϱδg(pδg)(pδgϱδg(pδg)g)}=0inΩT;Pc(Sδ)=pδgpδwinΩT, (40)

    where

    Θδdef=ϱδg(pδg)(1Sδ). (41)

    System (40) have to be completed with the corresponding boundary and initial conditions.

    Boundary conditions.: The boundary conditions read

    {pδg(x,t)=pδw(x,t)=0onΓinj×(0,T);qδwν=qδgν=0onΓimp×(0,T), (42)

    where the velocities qδw,qδg are given by:

    qδwdef=K(x)λw(Sδ)(pδwg)andqδgdef=K(x)λg(Sδ)(pδgϱδg(pδg)g). (43)

    Initial conditions.: The initial conditions read:

    pδw(x,0)=p0w(x)andpδg(x,0)=p0g(x)inΩ. (44)

    The remaining part of the Section is organized as follows. First, in subection 3.2 we recall the existence result for the system (40). Then we obtain the uniform in δ estimates for the solution of δ-problem (40). In subection 3.4 we formulate the compactness and convergence results which we use in the proof of Theorem 3.1.

    The goal of this subsection is to recall the existence result for the δ-problem (40). First, we reformulate the condition (A.3) from subsection 2.2 in order to adapt it to our δ-problem. For this problem the condition (A.3) becomes:

    (A.3δ) The function ϱδg=ϱδg(p) is given by (39).

    Now we are in position to formulate the existence result to δ-problem (40). It reads.

    Theorem 3.2. (see [9,33]) Let assumptions (A.1)-(A.2), (A.3δ), (A.4)-(A.9) be fulfilled. Then, for each δ>0, there exist pδg,pδw,Sδ such that:

    (Ⅰ) 

    pδw,pδgL2(ΩT)andλw(Sδ)pδw,λg(Sδ)pδgL2(ΩT); (45)
    β(Sδ)L2(0,T;H1(Ω))andPδP1L2(0,T;H1Γinj(Ω)); (46)
    ΦSδtL2(0,T;H1(Ω))andΦΘδtL2(0,T;H1(Ω)); (47)

    where the function Θδ is given by (41).

    (Ⅱ) the maximum principle holds:

    0Sδ1a.e.inΩT. (48)

    (Ⅲ) For any φw,φgC1([0,T];H1(Ω)) satisfying φw=φg=0 on Γinj×(0,T) and φw(x,T)=φg(x,T)=0, the following integral identity holds:

    ΩTΦ(x)SδφwtdxdtΩΦ(x)S0(x)φw(x,0)dx+ΩTK(x)λw(Sδ)pδwφwdxdtΩTK(x)λw(Sδ)gφwdxdt=0; (49)
    ΩTΦ(x)ΘδφgtdxdtΩΦ(x)Θδ(x,0)φg(x,0)dx//+ΩTK(x)λg(Sδ)ϱδg(pδg)pδgφgdxdt//ΩTK(x)λg(Sδ)[ϱδg(pδg)]2gφgdxdt=0. (50)

    Here Θδ(x,0)=ϱδg(p0g)(1S0) with the function S0 defined in condition

    (A.9), and Pc(Sδ)=pδgpδw.

    (Ⅳ) The initial conditions are satisfied in a weak sense as follows:

    ψH1Γinj(Ω),ΩΦ(x)Sδ(x,t)ψ(x)dx,ΩΦ(x)Θδ(x,t)ψ(x)dxC([0,T]). (51)

    Furthermore, we have

    (ΩΦ(x)Sδψdx)(0)=ΩΦ(x)S0ψdx (52)

    and

    (ΩΦ(x)Θδψdx)(0)=ΩΦ(x)Θδ(x,0)ψdx. (53)

    We start this subsection by obtaining the energy equality for δ-problem (40). The following result holds.

    Lemma 3.3 (Energy equality for δ-problem Let pδw,pδg,Sδ be a solution to (40). Then

    ddtΩΦ(x)Eδ(pδg(x,t),Sδ(x,t))dx++ΩK(x){λw(Sδ)pδw(pδwg)+λg(Sδ)pδg(pδgϱδg(pδg)g)}dx=0 (54)

    in the sense of distributions. Here

    Eδ(p,S)def=(1S)Rδ(p)ϝ(S),withRδ(p)def=ϱδg(p)Rδg(p)p, (55)

    where

    ϝ(s)def=s0Pc(ξ)dξandRδg(p)def=ppmaxdξϱδg(ξ). (56)

    Notice that in the previous works (see, e.g., [4,9,10,30,31,32,33]), the function Rg was defined by Rg(pg)def=pg0dξϱg(ξ). However, in our case, with such a definition we do not have uniform in δ lower bound for the function Rδ. Thus, we have to modify the definition of Rg, subtracting an appropriate constant C=C(δ). The properties of the functions Rδg,Rδ, and Eδ are given in:

    Lemma 3.4. Let Rδg,Rδ, and Eδ be the functions defined by (55), (56). Then

    (ⅰ) The function Rδ is negative and bounded from below, that is

    CRRδ0withCRdef=min{pmax,minp[0,pmax](p[lnplnpmax]p)}. (57)

    (ⅱ) The function Eδ is bounded from below. Namely,

    EδCRmaxS[0,1]Pc(S). (58)

    Proof of Lemma 3.4. Using the definition of the gas density ϱδg given by (39), it is easy to calculate that

    Rδg(p)def=ppmaxdξϱδg(ξ)={1σ[lnδσlnpmax]+1δ(pδσ)forp(,δσ);1σ[lnplnpmax]forp[δσ,pmax];1ϱmax[ppmax]forp(pmax,+). (59)

    Consider now the function Rδ. Due to (39) and (59), we have:

    Rδ(p)def=ϱδg(p)Rδg(p)p={δσ[lnδσlnpmax]δσforp(,δσ);p[lnplnpmax]pforp[δσ,pmax];pmaxforp(pmax,+). (60)

    The last formula immediately implies (57). Now (58) follows easily from (57) and the estimate:

    Eδ(p,S)=(1S)Rδ(p)ϝ(S)(CR+ϝ(1))CRmaxS[0,1]Pc(S). (61)

    This completes the proof of Lemma 3.4.

    In order to formulate a priori estimates for the solution to δ-problem (40), we remark first that the global pressure Pδ for the problem under consideration can be introduced in a way similar to one used in subsection 2.1 above. Then the desired a priori estimates for the solution of the δ-problem can be easily derived from Lemmata 3.3, 3.4, and the equality (see subsection 2.1 for more details):

    λg(Sδ)|pδg|2+λw(Sδ)|pδw|2=λ(Sδ)|Pδ|2+|b(Sδ)|2, (62)

    where the function b(s) is defined in (24).

    The following result holds.

    Lemma 3.5. Let pδw,pδg,Sδ be a solution to (40), the global pressure Pδ is defined in (16), and the function β(s) is defined in (25). Then

    ΩT{λw(Sδ)|pδw|2+λg(Sδ)|pδg|2}dxdtC; (63)
    ΩT{|Pδ|2+|β(Sδ)|2}dxdtC; (64)
    t(ΦΘδ)L2(0,T;H1(Ω))+t(ΦSδ)L2(0,T;H1(Ω))C. (65)

    Here C does not depend on δ.

    Proof of Lemma 3.5. Integrating (54) over the interval (0,T), we get:

    ΩΦ(x)Eδ(x,T)dx+ΩTK(x){λw(Sδ)pδw(pδwg)+λg(Sδ)pδg(pδgϱδg(pδg)g)}dxdt=ΩΦ(x)Eδ(x,0)dx. (66)

    Let us estimate now the right-hand side of (66) from above. Due to the definition of the function Eδ, (55), and the initial conditions (44) we have that

    Jδdef=ΩΦ(x)Eδ(x,0)dx=ΩΦ(x){(1S0)Rδ(p0g)ϝ(S0)}dx. (67)

    where S0=S0(x) is the initial condition of the saturation function (see condition (A.9) in Section 2.2). Now from condition (A.1) and the maximum principle (48), we easily obtain that

    |Jδ|ϕ+Ω|Rδ(p0g)|dx+ϕ+Ωϝ(S0)dxϕ+|Ω||CR|+ϕ+Ωϝ(S0)dx, (68)

    where |Ω| stands for the measure of the domain Ω. Therefore,

    |Jδ|C0, (69)

    where C0 is a constant which only depends on maxS[0,1]Pc(S), and the constant ϕ+. Combining condition (A.1), (66), (69), and the bound (61) yields:

    ΩTK(x){λw(Sδ)pδw(pδwg)+λg(Sδ)pδg(pδgϱδg(pδg)g)}dxdtC0+ϕ+|Ω|[2|CR|+maxS[0,1]Pc(S)]. (70)

    Applying the Cauchy inequality, from (70), we deduce (63), and consequently (64).

    The uniform estimates (65) can be obtained in the standard way from (40) with the help of (63). Lemma 3.5 is proved.

    In this subsection we recall two compactness results that were obtained in [9].

    Lemma 3.6 (Compactness lemma). Let ΦL(Ω), and assume that there exist positive constants ϕ1,ϕ2 such that 0<ϕ1Φ(x)ϕ2<1 a.e. in Ω. Assume moreover that a family {vδ}δ>0L2(ΩT) satisfies the following properties:

    1. the functions vδ satisfy the ineqiality 0vδC;

    2. there exists a function ϖ such that ϖ(ξ)0 as ξ0, and the following inequality holds true:

    ΩT|vδ(x+Δx,τ)vδ(x,τ)|2dxdτCϖ(|Δx|); (71)

    3. the estimate holds t(Φvδ)L2(0,T;H1(Ω))C.

    Then the family {vδ}δ>0 is a compact set in L2(ΩT).

    This result is a particular case of Lemma 4.2 proved in [4]. We apply the statement of Lemma 3.6 in order to prove the compactness of the sequences {Θδ}δ>0, {Sδ}δ>0. As in [9] we obtain.

    Proposition 1. Let {Θδ}δ>0L2(ΩT) be defined by (41). Then {Θδ}δ>0 is a compact set in the space Lq(ΩT) for all q[1,+).

    Proposition 2. Let {Sδ}δ>0L2(ΩT). Then, for all q[1,+), {Sδ}δ>0 is a compact set in the space Lq(ΩT) for all q[1,+).

    Now from Lemma 3.5 and Propositions 1, 2 we have.

    Lemma 3.7. Up to a subsequence,

    SδSstronglyinL2(ΩT)anda.e.inΩT; (72)
    0S1a.e.inΩT; (73)
    PδPweaklyinL2(0,T;H1Γinj(Ω)); (74)
    β(Sδ)β(S)weaklyinL2(0,T;H1(Ω)); (75)
    Θδdef=ϱδg(pδg)(1Sδ)def=ϱδg(Pδ+Gg(Sδ))(1Sδ)ΘstronglyinL2(ΩT)ΘδΘa.e.inΩT, (76)

    where Θdef=ϱg(P+Gg(S))(1S)def=ϱg(pg)(1S);

    [ϱδg(pδg)]kψ(Sδ)[ϱg(pg)]kψ(S)a.e.inΩT(k=1,2), (77)

    for any ψC([0,1]) such that ψ(0)=0.

    Proof of Lemma 3.7. The convergence (72) follows immediately from Proposition 2 and the limit function S evidently satisfies (73). The relations (74) and (75) are the consequence of (64) from Lemma 3.5. The convergence (76) follows from (72)-(74), the inequality 0Sδ1, and the fact that ϱδg is monotone. In order to justify (76) we first observe that, due to the definitions of the functions ϱδ and ϱ,

    ϱδg(Pδ+Gg(Sδ))(1Sδ)=ϱg(Pδ+Gg(Sδ))(1Sδ)+O(δ)asδ0.

    Then for any vL(ΩT), we have:

    ((ϱg(Pδ+Gg(Sδ))(1Sδ)ϱg(v+Gg(Sδ))(1Sδ)),(Pδv))L2(ΩT)0.

    Denoting ˉΘ the limit of Θδ and passing to the limit, as δ0, in the last inequality, we obtain

    (ˉΘϱg(v+Gg(S))(1S)),(Pv))L2(ΩT)0.

    Choosing v=P+ϰv1 and sending ϰ to zero yields

    (ˉΘϱg(P+Gg(S))(1S)),v1)L2(ΩT)0

    for any v1L2(ΩT). This implies (76).

    Finally, the convergence (77) can be proved by arguments similar to those from Lemma 4.2 in [40]. Lemma 3.7 is proved.

    We begin this subsection by studying the regularity properties of solution to (8).

    Taking into account the lower semi-continuity of the norm, by Lemma 3.5, we obtain:

    ΩT|P|2dxdtlim infδ0ΩT|Pδ|2dxdtC; (78)
    ΩT|β(S)|2dxdtlim infδ0ΩT|β(Sδ)|2dxdtC; (79)
    ΩT|b(S)|2dxdtlim infδ0ΩT|b(Sδ)|2dxdtC; (80)

    Now we set:

    pwdef=P+Gw(S)andpgdef=P+Gg(S). (81)

    We also recall the relation (23):

    λg(S)|pg|2+λw(S)|pw|2=λ(S)|P|2+|b(S)|2.

    Then, taking into account (78), (80), and the last relation we obtain that the functions pw,pg defined in (81) are such that

    ΩT{λg(S)|pg|2+λw(S)|pw|2}dxdt<+. (82)

    Thus properties (30)-(31) are established. The maximum principle (33) follows immediately from (48) and (72). Finally, the interpretation of the initial conditions can be done as in [40] (see also [9]).

    Consider the equation (49), with φwC1([0,T];H1(Ω)) and is such that φw=0 on Γinj×(0,T) and φw(x,T)=0.

    Taking into account (72), one easily gets:

    limδ0ΩTΦ(x)Sδφwtdxdt=ΩTΦ(x)Sφwtdxdt. (83)

    We then recall that

    λw(Sδ)pδw=λw(Sδ)Pδ+β(Sδ). (84)

    Then the third term on the left-hand side of (49) takes the form:

    ΩTK(x)λw(Sδ)pδwφwdxdt=ΩTK(x){λw(Sδ)Pδ+β(Sδ)}φwdxdt.

    Now taking into account the convergence (72), (74), and (75), we obtain that

    limδ0ΩTK(x){λw(Sδ)Pδ+β(Sδ)}φwdxdt=
    =ΩTK(x){λw(S)P+β(S)}φwdxdt.

    Returning now to the water pressure function pw, we finally get:

    limδ0ΩTK(x)λw(Sδ)pδwφwdxdt=ΩTK(x)λw(S)pwφwdxdt. (85)

    Considering (72), one can check that the fourth term of (49) satisfies the relation

    limδ0ΩTK(x)λw(Sδ)gφwdxdt=ΩTK(x)λw(S)gφwdxdt. (86)

    Thus, the saturation equation (34) is derived.

    We turn to (50) with φgC1([0,T];H1(Ω)), φg=0 on Γinj×(0,T), and φg(x,T)=0.

    Taking into account (76), one easily gets:

    limδ0ΩTΦ(x)ϱδg(pδg)(1Sδ)φgtdxdt=ΩTΦ(x)Θ(x,t)φgtdxdt, (87)

    where Θdef=ϱg(pg)(1S) (see (9) above).

    Considering the definition of the functions ϱg and ϱδg (see (4) and (39), respectively) we have

    limδ0ΩΦ(x)ϱδg(p0g)(1S0)φg(x,0)dx=ΩTΦ(x)Θ(x,0)φgtdxdt. (88)

    In order to pass to the limit in the third term of (50) we recall that (see relations (27))

    λg(Sδ)pδg=λg(Sδ)Pδβ(Sδ). (89)

    Then

    ΩTK(x)λg(Sδ)ϱδg(pδg)pδgφgdxdt=
    =ΩTK(x)ϱδg(Pδ+Gg(Sδ)){λg(Sδ)Pδβ(Sδ)}φgdxdt.

    Now taking into account the convergence results (72), (74), and (75) we obtain that

    limδ0ΩTK(x)ϱδg(Pδ+Gg(Sδ)){λg(Sδ)Pδβ(Sδ)}φwdxdt=
    =ΩTK(x)ϱg(P+Gg(S)){λg(S)Pβ(S)}φgdxdt.

    Returning now to the gas pressure function pg, we finally get:

    limδ0ΩTK(x)λg(Sδ)ϱδg(pδg)pδgφwdxdt=ΩTK(x)λg(S)ϱg(pg)pgφgdxdt (90)

    Finally, in view if (77),

    limδ0ΩTK(x)λg(Sδ)[ϱδg(pδg)]2gφgdxdt=ΩTK(x)λg(S)[ϱg(pg)]2gφgdxdt. (91)

    Thus the gas pressure equation (35) is obtained. Theorem 3.1 is proved.

    In this Section we consider the problem describing a reservoir with a periodic microstructure. Then in the model considered in the previous sections one has rapidly oscillating porosity function and absolute permeability tensor. Our goal is to prove the homogenization result for this model. The convergence of the homogenization process is justified by the technique of two-scale convergence [2].

    In this section, we present the mathematical model describing water-gas flow in a periodically heterogeneous porous medium. As above we suppose that the gas density vanishes as the gas pressure is zero. For simplicity, we assume no source/sink terms.

    We consider a bounded Lipschitz domain ΩRd (d=1,2,3) with a periodic microstructure. The microscopic length scale that represents the ratio between the cell size to the size of the whole region Ω, is denoted by ε. We assume that 0<ε1 is a small parameter tending to zero. We denote by Ydef=(0,1)d the periodic cell. Let Φε(x)=Φ(x/ε) be the porosity of Ω; Kε(x)=K(x/ε) be the absolute permeability tensor of Ω; Sεdef=Sεw(x,t), is the saturations of water in Ω×(0,T); pεw=pεw(x,t), pεg=pεg(x,t) are the pressures of water and gas in Ω×(0,T), respectively;

    System (8), in the case of a periodic porous medium, takes the form

    εproblem:{Φε(x)Sεtdiv{Kε(x)λw(Sε)(pεwg)}=0inΩT;Φε(x)Θεtdiv{Kε(x)λg(Sε)ϱg(pεg)(pεgϱg(pεg)g)}=0inΩT;Pc(Sε)=pεgpεwinΩT, (92)

    where

    Θεdef=ϱg(pεg)(1Sε). (93)

    System (92) has to be equipped with appropriate boundary and initial conditions.

    Boundary conditions.: We suppose that the boundary Ω consists of two parts Γinj and Γimp such that ΓinjΓimp=, Ω=¯Γinj¯Γimp. The boundary conditions are given by:

    {pεg(x,t)=pεw(x,t)=0onΓinj×(0,T);qεwν=qεgν=0onΓimp×(0,T), (94)

    where the velocities qεw,qεg are defined as follows:

    qεwdef=Kε(x)λw(Sε)(pεwg),qεgdef=Kε(x)λg(Sε)(pεgϱg(pεg)g). (95)

    Initial conditions.: The initial conditions read:

    pεw(x,0)=p0w(x)andpεg(x,0)=p0g(x)inΩ. (96)

    Let us formulate the main assumptions on the data. First, we replace conditions (A.1), (A.2) from Section 2.2 with the following assumptions:

    (A.1ε) The function Φ=Φ(y) is Y-periodic, ΦL(Y), and there are positive constants ϕ1,ϕ2 such that 0<ϕ1Φ(y)ϕ2<1 a.e. in Y.

    (A.2ε) The tensor K=K(y) is Y-periodic, it belongs to (L(Y))d×d. Moreover, there exist positive constants K,K+ such that

    K|ξ|2(K(x)ξ,ξ)K+|ξ|2forallξRd,a.e.inΩ. (97)

    We also suppose that conditions (A.3)-(A.9) from subsection 2.2 hold true.

    We now provide a weak formulation of problem (92)-(96).

    Definition 4.1. For each ε>0 we say that pεw,pεg,Sε is a weak solution of problem (92), (94), (96) if (30)-(38) are fulfilled for functions pεw,pεg,Sε instead of pw,pg,S.

    Let us recall that for any φw,φgC1([0,T];H1(Ω)) satisfying φw=φg=0 on Γinj×(0,T) and φw(x,T)=φg(x,T)=0, we have:

    ΩTΦε(x)SεφwtdxdtΩΦε(x)S0(x)φw(x,0)dx
    +ΩTKε(x)λw(Sε)pεwφwdxdt
    ΩTKε(x)λw(Sε)gφwdxdt=0 (98)

    and

    ΩTΦε(x)ΘεφgtdxdtΩΦε(x)Θε(x,0)φg(x,0)dx+ (99)
    +ΩTKε(x)λg(Sε)ϱg(pεg)pgφgdxdtΩTKε(x)λg(Sε)[ϱg(pεg)]2gφgdxdt=0,

    where the function Θε is defined in (93).

    Notational convention. In what follows C,C1,.. denote generic constants that do not depend on ε.

    We study the asymptotic behavior of the solution to problem (92), (94), (96) as ε0. In particular, we are going to show that the effective model reads:

    {0S1inΩT;ΦStdivx{Kλw(S)[Pwg]}=0inΩT;ΦΘtdivx{Kϱg(Pg)λg(S)[Pgϱg(Pg)g]}=0inΩT;Pc(S)=PgPwinΩT, (100)

    where S, Pw, Pg denote the homogenized water saturation, water pressure, and gas pressure, respectively. Φ denotes the mean value of the function Φ over the cell Y. K is the homogenized tensor with the entries Kij defined by:

    Kijdef=YK(y)[yξi+ei][yξj+ej]dy, (101)

    where the function ξj is a Y-periodic solution of the following local problem:

    {divy(K(y)[yξj+ej])=0inY,yξj(y)Y-periodic (102)

    with ej being the j-th coordinate vector.

    The function Θ=Θ(x,t) is given by: Θdef=(1S)ϱg(Pg).

    Remark 2. The homogenized system (100) generalizes the result obtained earlier in [4] in two ways. First, this system allows the gas density to degenerate. In addition, this system is written in terms of the homogenized phase pressures Pw,Pg and not in terms of the homogenized global pressure and water saturation as was done in [4] (see (3.1)).

    System (100) has to be completed with the following boundary and initial conditions.

    Boundary conditions.: The boundary conditions are given by:

    {Pg(x,t)=Pw(x,t)=0onΓinj×(0,T);qwν=qgν=0onΓimp×(0,T), (103)

    where the velocities qεw,qεg are defined as follows:

    qwdef=Kλw(S)(Pwg)andqgdef=Kλg(S)(Pgϱg(Pg)g). (104)

    Initial conditions.: The initial conditions read:

    Pw(x,0)=p0w(x)andPg(x,0)=p0g(x)inΩ. (105)

    The rigorous justification of the homogenization process relies on the two-scale convergence approach, see, e.g., [2]. For the reader's convenience, we recall the definition of the two-scale convergence.

    Definition 4.2. A sequence of functions {vε}ε>0L2(ΩT) two-scale converges to vL2(ΩT×Y) if vεL2(ΩT)C, and for any test function φC(ΩT;C#(Y)) the following relation holds:

    limε0ΩTvε(x,t)φ(x,xε,t)dxdt=ΩT×Yv(x,y,t)φ(x,y,t)dydxdt.

    This convergence is denoted by vε(x,t)2sv(x,y,t).

    The homogenization result reads.

    Theorem 4.3. Let assumptions (A.1ε), (A.2ε), (A.3)-(A.9) be fulfilled. Then a solution of problem (92) converges (up to a subsequence) to a weak solution of the homogenized problem (100).

    The proof is divided into a number of steps.

    4.3.1. A priori estimates for solutions to problem (92)

    In this section we derive the a priori estimates for problem (92). For any ε>0, we consider the following ε,δ-problem:

    {Φε(x)Sε,δtdiv{Kε(x)λw(Sε,δ)(pε,δwg)}=0inΩT;Φε(x)Θε,δtdiv{Kε(x)λg(Sε,δ)ϱδg(pε,δg)(pε,δgϱδg(pε,δg)g)}=0inΩT;Pc(Sε,δ)=pε,δgpε,δwinΩT, (106)

    where the family of functions {ϱδg}δ>0 is defined in (39) and

    Θε,δdef=ϱδg(pε,δg)(1Sε,δ). (107)

    The (ε,δ)-problem is completed by the boundary and initial conditions (94) and (96), respectively.

    The energy equality for problem (106) can be obtained as in Section 3.3.

    Lemma 4.4 (Energy equality for (ε,δ)-problem. Let pε,δw,pε,δg,Sε,δ be a solution to (106). Then

    ddtΩΦε(x)Eε,δ(x,t)dx++ΩKε(x){λw(Sε,δ)pε,δw(pε,δwg)+λg(Sε,δ)pε,δg(pε,δgϱδg(pε,δg)g)}dx=0 (108)

    in the sense of distributions. Here

    Eε,δdef=(1Sε,δ)Rδ(pε,δg)ϝ(Sδ),withRδ(p)def=ϱδg(p)Rδg(p)p, (109)

    where the functions ϝ(s) and Rδ(p) are defined by (56).

    Then following the lines of Section 3.3 one can prove the following statement which is similar to that of Lemma 3.5.

    Lemma 4.5. Let pε,δw,pε,δg be a solution to (106), the global pressure Pε,δ be defined in (16), and the function β(s) be defined in (25). Then

    ΩT{λw(Sε,δ)|pε,δw|2+λg(Sε,δ)|pε,δg|2}dxC; (110)
    ΩT{|Pε,δ|2+|β(Sε,δ)|2}dxC. (111)

    Here C does not depend on ε,δ.

    Now, as in Section 3.5.1, we conclude that

    ΩT|Pε|2dxdtlimδ0ΩT|Pε,δ|2dxdtC; (112)
    ΩT|β(Sε)|2dxdtlimδ0ΩT|β(Sε,δ)|2dxdtC; (113)
    ΩT|b(Sε)|2dxdtlimδ0ΩT|b(Sε,δ)|2dxdtC; (114)
    ΩT{λg(Sε)|pεg|2+λw(Sε)|pεw|2}dxdtC, (115)

    where C is a constant that does not depend on ε,δ.

    The uniform estimates for the time derivatives of the functions ΦεΘε and ΦεSε can be derived from (92) using (115). These estimates read:

    t(ΦεΘε)L2(0,T;H1(Ω))+t(ΦεSε)L2(0,T;H1(Ω))C, (116)

    where C is a constant that does not depend on ε.

    4.3.2. Compactness and convergence results

    First, we recall the following compactness result established in~[4].

    Lemma 4.6 (Compactness lemma). Let ΦL(Y), and assume that there are positive constants ϕ1,ϕ2 such that 0<ϕ1Φ(y)ϕ2<1 a.e. in Y. Assume moreover that a family {vε}ε>0L2(ΩT) satisfies the following properties:

    1. vεL(ΩT), and 0vεC;

    2. there exists a function ϖ such that ϖ(ξ)0 as ξ0, and

    ΩT|vε(x+Δx,τ)vε(x,τ)|2dxdτCϖ(|Δx|);

    3. t(Φεvε)L2(0,T;H1(Ω))C.

    Then the family {vε}ε>0 is a compact set in L2(ΩT).

    Remark 3. In the formulation of the above compactness lemma the periodicity of Φ can be replaced with the assumption that Φε1 weakly in L2(Ω), as ε0.

    Now we turn to the compactness result for the family {Θε}ε>0.

    Proposition 3. Under our standing assumptions, the set {Θε}ε>0 is compact in the space Lq(ΩT) for all q[1,+).

    A similar result holds for the set {Sε}ε>0.

    Proposition 4. Under our standing assumptions, the set {Sε}ε>0 is compact in the space Lq(ΩT) for all q[1,+).

    Summarizing the above statements yields.

    Lemma 4.7 There exist a function S with 0S1 a.e. in ΩT and a function PL2(0,T;H1(Ω)) such that up to a subsequence:

    Sε(x,t)S(x,t)stronglyinLq(ΩT) 1q<+; (117)
    Pε(x,t)P(x,t)weaklyinL2(0,T;H1(Ω)); (118)
    β(Sε)β(S)stronglyinLq(ΩT) 1q<+; (119)
    ΘεΘdef=(1S)ϱg(Pg)stronglyinL2(ΩT). (120)

    The Proof of Lemma 4.7 relies on the arguments similar to those used in the proof of Lemma 4.8 in [4].

    4.3.3. Passage to the limit in equations (98), (99)

    In this subsection we apply the method of a cut-off function introduced in [10].

    It is easy to justify the passage to the two-scale limit in the temporal terms using the convergence results (117) and (120) from Lemma 4.7 as it was done, for example, in [4]. Namely, let φ0D(ΩT). The first two terms in (98) become:

    JεSdef=ΩTΦε(x)Sε(x,t)ψt(x,t)dxdt. (121)

    Now we pass to the limit on the right-hand side of (121). Taking into account (117), we have that

    \lim\limits_{{\varepsilon }\to0} {\mathcal{J}}^{\varepsilon }_S = -\langle \Phi \rangle\, \int\limits_{\Omega_T} S(x, t) \frac{\partial \psi}{\partial t}(x, t)\,dx\, dt. (122)

    For any \eta > 0, we introduce the family of functions \{S^{{\varepsilon },\eta}\} defined by:

    S^{{\varepsilon },\eta} \overset{def}{\mathop =} \min\left\{(1 - \eta),\, \max(\eta, S^{{\varepsilon }})\right\}.

    These functions satisfy the estimate:

    \|S^{{\varepsilon },\eta} \|_{L^2(0,T;H^1(\Omega))}\leqslant C(\eta),

    where C(\eta) \to + \infty as \eta \to 0. Therefore,

    S^{\,\eta} \overset{def}{\mathop =} \min\bigg\{(1 - \eta),\, \max(\eta, S)\bigg\}\in L^2(0,T;H^1(\Omega)) {\rm for\,\, any}\,\, \eta > 0.

    Now, taking into account (117), (118), for a subsequence,

    \nabla \left[{\mathsf P}^{{\varepsilon }} + {\mathsf G}_{w}\left(S^{{\varepsilon },\eta}\right)\right] \stackrel {2s} \rightharpoonup \nabla_x\left[{\mathsf P} + {\mathsf G}_{w}\left(S^{\eta}\right)\right] + \nabla_y {\mathsf V}^\eta_w(x, t, y) (123)

    with {\mathsf V}^\eta_w \in L^2(\Omega_T; H^1_\#(Y)). We set:

    \varphi^{\varepsilon }_w(x, t) \overset{def}{\mathop =} {\varepsilon }\,\varphi(x, t)\,{\mathsf Z}(S^{{\varepsilon }})\, \zeta\left(\frac{x}{{\varepsilon }}\right) (124)

    with {\mathsf Z}(s) being a smooth function equal to zero for s\not\in(\eta, 1 - \eta); \zeta(y) is smooth periodic, and \varphi is a smooth function with a compact support in \Omega_T. Using \varphi^{\varepsilon }_w as a test function in (98) and considering the global pressure definition, we get:

    \int\limits_{\Omega_T}K^{\varepsilon }(x)\lambda_w(S^{{\varepsilon }}) \left[\nabla {\mathsf p}_w^{\varepsilon } - \vec g\right] \nabla\zeta \Big(\frac{x}{{\varepsilon }}\Big)\, \varphi(x, t)\, {\mathsf Z}(S^{{\varepsilon }}) \,dx \,dt = O({\varepsilon }). (125)

    We pass to the two-scale limit in (125). Taking into account (117), (118), and (123), we obtain:

    \int\limits_{\Omega_T\times Y} K(y) \lambda_w(S) \bigg[\nabla {\mathsf P} + \nabla {\mathsf G}_{w}(S) + \nabla_y {\mathsf V}^\eta_w(x, t, y) - \vec g \bigg]\, \nabla\zeta(y)\, {\mathsf Z}(S)\, \varphi(x, t)\,dy\, dx\, dt = 0. (126)

    Therefore,

    {\mathsf V}^\eta_w = \xi(y)\, \big(\nabla_x {\mathsf P} + \nabla_x{\mathsf G}_{w}(S) - \vec g\big) (127)

    for all (x,t)\in\Omega_T such that S \in (\eta, 1 - \eta). Here \xi \in \mathbb{R}^d is a vector with the components \xi_j that are the solutions of the auxiliary problem (102).

    Since \eta is an arbitrary positive number, representation (127) is valid for all (x,t) such that S \in (0,1). In particular, {\mathsf V}^\eta_\ell does not depend on \eta: {\mathsf V}^\eta_w = {\mathsf V}_w. This leads to the following equation:

    \int\limits_{Y} K(y)\,\lambda_w(S)\, \bigg\{ \big[\nabla {\mathsf p}_w - \vec{g}\big] + \nabla_y {\mathsf V}_w \bigg\} \cdot \nabla_y \zeta_2(y) \, dy = 0 {\rm for\, all}\,\, \zeta_2 \in C^\infty_\#(Y). (128)

    Finally, with the help of our a priori estimates we deduce in a standard way that

    K^{\varepsilon }\lambda_w(S^{\varepsilon }) \Big[ \nabla{\mathsf p}_w^{\varepsilon } - \vec g\,\Big]\stackrel {2s} \rightharpoonup K(y)\lambda_w(S) \Big[\mathbb{I} + \nabla_y\xi(y)\Big]\Big(\nabla_x {\mathsf P} + \nabla_x{\mathsf G}_{w}(S) - \vec g\Big), (129)

    where \mathbb{I} is the unit matrix. This allows us, with the help of (122), to obtain the weak formulation of the homogenized saturation equation (100)_2.

    The derivation of the weak formulation for the homogenized gas pressure equation can be done in a similar way. This completes the proof of Theorem 4.3.

    We have presented new results for immiscible compressible two-phase flow in porous media. More precisely, we give a week formulation and an existence result for a degenerate system modeling water-gas flow through a porous medium. The water is assumed to be incompressible and the gas phase is supposed compressible and obeying the ideal gas law leading to a new degeneracy in the evolution term of the pressure equation. Furthermore, a homogenization result for the corresponding system is established in the case of a single rock-type model. The extension to a porous medium made of several types of rocks, i.e. the porosity, the absolute permeability, the capillary and relative permeabilities curves are different in each type of porous media, is straightforward. Let us also mention that this homogenization result has been used successfully in [1] to simulate numerically a benchmark test proposed in the framework of the European Project FORGE: Fate Of Repository Gases [28]. The study still needs to be improved in several areas such as the cases of unbounded capillary pressure and double porosity media. These more complicated cases appear in the applications. Further work on these important issues is in progress.

    Most of the work on this paper was done when L. Pankratov and A. Piatnitski were visiting the Applied Mathematics Laboratory of the University of Pau & CNRS. They are grateful for the invitations and the hospitality. The work of L. Pankratov has been partially supported by the RScF, project No. 15-11-00015. This work was partially supported by the Carnot Institute, ISIFoR project (Institute for the sustainable engineering of fossil resources). The supports are gratefully acknowledged.

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