Jean-Louis Auriault

Jolanta Lewandowska

Comptes Rendus Mécanique (CRM)

highlye stronglyrformedroscopicretentioneffectiveteinclusionsRichards.méthode-linéaire àme deson raide.C. R. Mecanique 332 (2004) 91–96Modelling of unsaturated water flow in soilswith highly permeable inclusionsJolanta Lewandowskaa, Jean-Louis Auriaultba Laboratoire LTHE, UJF, INPG, CNRS, IRD, UMR 5524, BP 53X, 38041 Grenoble cedex, Franceb Laboratoire 3S, UJF, INPG, CNRS UMR 5521, BP 53X, 38041 Grenoble cedex, FranceReceived 25 September 2003; accepted after revision 21 October 2003Presented by Évariste Sanchez-PalenciaAbstractIn this paper the mathematical macroscopic modeling of unsaturated water flow in a porous medium (soil) withpermeable porous inclusions is presented. It is supposed that water flow in each sub domain can be described by thnon-linear Richards’ equation. Gravity effects are considered. The upscaling process of this stiff problem is peusing the homogenization method of periodic structures with asymptotic expansions. The resulting non-linear macdescription is a one equation model, revealing the local equilibrium of the capillary pressure head. The effective watercapacity was found to be the volume average of the water retention capacities of the two porous sub-domains. Theconductivity tensor is obtained from a linear and non-stiff boundary value problem at the heterogeneity scale.To cite this article:J. Lewandowska, J.-L. Auriault, C. R. Mecanique 332 (2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméModélisation de l’écoulement dans les sols non saturés avec inclusions très perméables. Nous étudions dans cette Nole modèle mathématique macroscopique de l’écoulement non-saturé d’eau dans un milieu poreux contenant desporeuses très conductrices. L’écoulement est décrit dans chaque milieu par l’équation fortement non-linéaire deLa gravité est prise en considération. La macroscopisation de ce problème raide est conduite au moyen de lad’homogénéisation de structures périodiques. Le modèle résultant à l’échelle macroscopique est un modèle nonune équation qui traduit l’équilibre local de la pression capillaire. La rétention d’eau effective est la moyenne de volurétentions locales. Le tenseur effectif de conductivité est obtenu à partir de la solution d’un problème local linéaire nPour citer cet article : J. Lewandowska, J.-L. Auriault, C. R. Mecanique 332 (2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Granular media; Soils; Porous mediaMots-clés : Milieux granulaires ; Sols ; Milieux poreuxE-mail addresses: jolanta.lewandowska@hmg.inpg.fr (J. Lewandowska), jean-louis.auriault@hmg.inpg.fr (J.-L. Auriault).1631-0721/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2003.10.01292 J. Lewandowska, J.-L. Auriault / C. R. Mecanique 332 (2004) 91–96saturés isoléespropriétéssions).lencia [4],ulementndescriptiondans leevitaire :ionnellepressionmaineeion (21)ation nouseffectivee.s dans lan.highlyracticalcceleratefound.. In theivity inVersion française abrégéeL’objectif de ce travail est le développement d’un modèle mathématique décrivant l’écoulement nondans un sol hétérogène. Nous considérons un milieu poreux périodique (sol) contenant des inclusiontrès perméables. Nous supposons donc qu’il est composé de deux sous-domaines poreux dont leshydrodynamiques sont très contrastées. On noteΩ la période,Ω1 et Ω2 les deux domaines poreux, etΓ leurinterface. Le domaineΩ2 est connexe et le milieu 2 est beaucoup moins perméable que le milieu 1 (incluLe modèle macroscopique est obtenu par la méthode des développements asymptotiques (Sanchez-PaBensoussan et al. [5]). Le point de départ de notre analyse est l’équation de Richards’ [3] qui décrit l’écod’eau dans un sol non saturé pour les domaines 1 et 2, Éq. (2). Dans cette équationC est la capacité de rétentiod’eau etK la conductivité hydraulique. Sur l’interfaceΓ on suppose la continuité des pressions capillairesh1 eth2et des flux d’eau, Éq. (3). Les rétentions d’eauθ1(h) etθ2(h) et les conductivitésK1(h) etK2(h) sont fortement nonlinéaires et sont supposées connues. Pour estimer les ordres des grandeurs des différents termes de lalocale, nous avons choisi comme temps caractéristique le temps d’observation de l’écoulement de l’eaumilieux 2,T = C2cL2/K2c, où l’indicec indique une propriété caractéristique etL est la longueur caractéristiqumacroscopique. Le contraste entre les paramètres est le suivant :K2c/K1c =O(ε) etC2c/C1c =O(1), oùε = l/L(1) et l est la taille de la période. Enfin, le terme capillaire est supposé dominant par rapport au terme grahc/l = O(ε). En utilisant ces estimations, la description microscopique est donnée sous forme adimenspar (6)–(8). C’est un problème raide fortement non linéaire.Premier ordre d’approximation h(0)Après introduction du développement asymptotique (9), nous obtenons au premier ordre un champ decapillaire macroscopique de la formeh(0)1 = h(0)2 = h(0)(x, t∗).Deuxième ordre d’approximation h(1)L’identification des termes à l’ordre suivant conduit à (13), (14), (16), dont la solution est (15) pour le do1 et (17) pour le domaine 2. Le problème local définissant le champ vectorielχ (y), linéaire et non-raide, est lsystème (18), qui présente certaines similitudes avec le problème de filtration décrit dans Levy [1].Description macroscopiqueAfin de trouver la description macroscopique, nous considérons les Éqs. (19) et (20) avec la conditsur l’interface, ainsi que la condition de périodicité. On intègre l’Éq. (19) surΩ1 et l’Éq. (20) surΩ2. Aprèsquelques transformations, la somme membres à membres des deux équations résultant de cette opérdonne l’équation recherchée (24). Les paramètres effectifs rentrant dans cette description sont la capacitéde rétention d’eauCeff(h(0)) (23), et la conductivité effectiveKeff(h(0)) (25). L’Éq. (24) est fortement non linéairNous pouvons vérifier que cette description est valable dans le cas de plusieurs inclusions présentepériode et/ou dans le casK2c/K1c = O(εp), p > 1. Cette équation reste aussi valable pourK2c/K1c = O(1) etC1c/C2c =O(εp), p  1. EnfinK2c/K1c =O(1), C2c/C1c =O(1) conduit au cas classique d’homogénéisatio1. IntroductionTo our knowledge, the mathematical problem of unsaturated water flow in a porous medium (soil) withpermeable inclusions has not received much attention in the literature. However, this is a problem of pinterest in soil hydrology of vadose zone. The presence of highly permeable inclusions can considerably athe water and pollutant transfers. In the literature two mathematically alike though linear problems can beThe first one by Levy [1] addresses the saturated flow through non deformable porous fissured rockssecond paper, Moctezuma-Berthier et al. [2], the effect of porosity distribution on the electrical conductreconstructed bimodal porous media is investigated.J. Lewandowska, J.-L. Auriault / C. R. Mecanique 332 (2004) 91–96 93ablethe stiffoscopicalidity.mediumth,uctingchards’ure duringn for therethehatispondingly. Timeroscopictiff) andIn this paper the modelling of water flow in soils of double porosity (soil matrix with highly permeinclusions) in the vadose zone is presented. Due to the non-linearity of the hydrodynamic parameters,problem into consideration becomes strongly non-linear. The modeling includes the derivation of the macrdescription and the effective hydrodynamic parameters, together with the definition of the domain of its vThe effective conductivity is obtained from a non-stiff linear boundary value problem.2. General assumptionsWe consider a porous medium (soil) with highly permeable porous inclusions. Let us assume that thisis periodic and there exists the following separation of scalesε = lL 1 (1)wherel is the period length (the characteristic microscopic length) andL is the characteristic macroscopic leng(e.g., the dimension of the macroscopic domain). We noteΩ the period,Ω1 andΩ2 the two porous domainsrespectively, andΓ the interface between them. For the moment, the periodΩ contains a single inclusionΩ1. Wealso assume that the domainΩ2 is continuously connected and that the porous medium 2 is much less condthan the porous medium 1.The unsaturated water flow in a homogenous rigid soil is often described by the strongly non-linear Riequation [3] which assumes that the air pressure in the soil is constant and equal to the atmospheric pressthe whole flow process. Under isothermal conditions this equation is formulated in each porous subdomaicapillary pressureh[L] being the water pressure head relative to the atmospheric pressure (h  0 ) as follows:Cα(hα)∂hα∂t− divX(Kα(hα)gradX(hα + X3)) = 0, α = 1,2, in Ωα (2)whereC(h) = dθ/dh is the soil water retention capacity [L−1], θ(h) is the volumetric water content [·], K(h) is thehydraulic conductivity positive tensor [LT−1], X = (X1,X2,X3) denotes the physical spatial variable [L], whethe axisX3 is positively oriented upwards andt [T] is the time. This equation is strongly non-linear because ofnon-linear functionsC(h) andK(h).At the interfaceΓ , the capillary pressure head and the water flux are continuous:h1 = h2, K1(h1)gradX(h1 + X3)N = K2(h2)gradX(h2 + X3)N, onΓ (3)whereN is a unit vector normal toΓ . In Eqs. (2) and (3) theX3 term is related to the gravity. We assume tthe water retention curvesθα(h), as well as the conductivity curvesKα(h), α = 1,2, are known and that hystereseffects are negligible.3. A priori estimation and dimensionless micro-descriptionLet us introduce dimensionless variables as follows:hα = hαch∗α, Kα = KαcK∗α, Cα = CαcC∗α, α = 1,2, t = T t∗ (4)where the subscriptc denotes a characteristic quantity (constant) and the asterisk denotes the corresdimensionless variable. We introduce also two dimensionless space variablesy = X/l and x = X/L, themicroscopic dimensionless space variable and the macroscopic dimensionless space variable, respectiveTis the characteristic time of observation, chosen to be the time of the water flow in the medium 2 at the macscaleL, T = C2cL2/K2c.We assume that medium 2 is much less conductive than medium 1 (therefore the problem (2), (3) is sthat the two water retention capacities are of the same order of magnitude:K2cK1c=O(ε), C2cC1c=O(1) (5)94 J. Lewandowska, J.-L. Auriault / C. R. Mecanique 332 (2004) 91–96ionlessanchez-iault [6].emFinally, we notice that the ratio of characteristic length of the periodl t the characteristic capillary pressure headhcis of the orderO(ε) in a period. It means that the capillary term is dominating over the gravity one.By taking into account the above estimation, the local problem (2), (3) can be put in the following dimensform:ε3C∗1∂h∗1∂t− divy(K∗1 grady(h∗1 + εy3)) = 0, in Ω1 (6)ε2C∗2∂h∗2∂t− divy(K∗2 grady(h∗2 + εy3)) = 0, in Ω2 (7)h∗1 = h∗2, K∗1 grady(h∗1 + εy3)N = εK∗2 grady(h∗2 + εy3)N, onΓ (8)4. UpscalingIn this study, the classical method of homogenization by formal asymptotic expansions was applied (SPalencia [4], Bensoussan et al. [5]). The procedure adopted in this paper was presented in details by AurThe dimensionless capillary pressure headh∗ is looked for in the form of the following expansionh∗ = h(0)(y,x, t∗) + εh(1)(y,x, t∗) + +ε2h(2)(y,x, t∗) + · · · , (9)which results in similar expansions forK andC, with K(0) = K(h(0)) andC(0) = C(h(0)).4.1. First order variable h(0)After introducing the above expasions in Eqs. (6)–(8) and extracting like powers ofε, we obtain for the domainΩ1:∂∂yi(K(0)1ij∂h(0)1∂yj)= 0, in Ω1, K(0)1ij∂h(0)1∂yjNi = 0, onΓ (10)which yieldsh(0)1 = h(0)1 (x, t∗), after noting thatK(0) = K(h(0)) is a positive tensor. The boundary value problin Ω2 is in the form∂∂yi(K(0)2ij∂h(0)2∂yj)= 0, in Ω2, h(0)2 = h(0)1 , onΓ (11)whereh(0)2 is y-periodic. We finally obtain:h(0)2 = h(0)1 = h(0)(x, t∗) (12)4.2. Second order variable h(1)After using Eq. (12), we obtain the following boundary value problem forh(1)1 :∂∂yi[K(0)1ij(∂h(0)∂xj+ I3j)+ K(0)1ij∂h(1)1∂yj]= 0, in Ω1 (13)[K(0)1ij(∂h(0)∂xj+ I3j)+ K(0)1ij∂h(1)1∂yj]Ni = 0, onΓ (14)J. Lewandowska, J.-L. Auriault / C. R. Mecanique 332 (2004) 91–96 95arylations,nceAfter multiplying the two members of Eq. (13) byyi(∂h(0)/∂xi + I3i ), integrating by parts and using the boundcondition (14) and the positiveness ofK(0)1 , we obtain:∂h(0)∂xi+ I3i + ∂h(1)1∂yi= 0, h(1)1 = −yi (∂h(0)∂xi+ I3i ) + h̄(1)(x, t∗) (15)whereh̄(1) is an arbitrary function. Forh(1)2 we obtain:∂∂yi[K(0)2ij(∂h(0)∂xj+ I3j + ∂h(1)2∂yj)]= 0, in Ω2, h(1)2 = h(1)1 , onΓ (16)whereh(1)2 is y-periodic. That yields:h(1)2 = −χi(∂h(0)∂xi+ I3i)+ h̄(1)(x, t∗) (17)The boundary value problem for the vectorial functionχ is a linear non-stiff problem:∂∂yi[K(0)2ij(−∂χk∂yj+ Ijk)]= 0, in Ω2, χk = yk, onΓ, k = 1,2,3 (18)whereχ is y-periodic.4.3. Macroscopic modellingIn order to determine the macroscopic model we extract from the asymptotic expansions the following rewhere the results (12) and (15) are accounted for:∂∂yi[K(0)1ij(∂h(2)1∂xj+ ∂h(3)1∂yj)+ K(1)1ij(∂h(1)1∂xj+ ∂h(2)1∂yj)+ K(2)1ij(∂h(0)∂xj+ ∂h(1)1∂yj+ I3j)]+ ∂∂xi[K(0)1ij(∂h(1)1∂xj+ ∂h(2)1∂yj)]= C(0)1∂h(0)∂t∗, in Ω1 (19)∂∂yi[K(0)2ij(∂h(1)2∂xj+ ∂h(2)2∂yj)+ K(1)2ij(∂h(0)∂xj+ ∂h(1)2∂yj+ I3j)]+ ∂∂xi[K(0)2ij(∂h(0)∂xj+ ∂h(1)2∂yj+ I3j)]= C(0)2∂h(0)∂t∗, in Ω2 (20)[K(0)1ij(∂h(2)1∂xj+ ∂h(3)1∂yj)+ K(1)1ij(∂h(1)1∂xj+ ∂h(2)1∂yj)+ K(2)1ij(∂h(0)∂xj+ ∂h(1)1∂yj+ I3j)]Ni=[K(0)2ij(∂h(1)2∂xj+ ∂h(2)2∂yj)+ K(1)2ij(∂h(0)∂xj+ ∂h(1)2∂yj+ I3j)]Ni, onΓ (21)Now, integrate Eqs. (19) and (20) over the domainsΩ1 andΩ2, respectively, apply to both integrals the divergetheorem, and use the periodicity condition and the boundary condition (21). After dividing by|Ω | we get:Ceff∂h(0)∂t∗= 1|Ω |∂∂xi[∫Ω1K(0)1ij(∂h(1)1∂xj+ ∂h(2)1∂yj)dΩ +∫Ω2K(0)2ij(∂h(0)∂xj+ ∂h(1)2∂yj+ I3j)dΩ](22)96 J. Lewandowska, J.-L. Auriault / C. R. Mecanique 332 (2004) 91–96w in) can beelusionsar, sinceed as thelusionsetentiond) whenlysis inlurces en(2002)n, 1980.sorption,where the effective water retention capacityCeff is defined by:Ceff = 1|Ω |∫Ω1C(0)1 dΩ +1|Ω |∫Ω2C(0)2 dΩ (23)Eq. (22) is independent of the micro-space variabley. It represents a macroscopic equivalent model of the flothe heterogeneous porous medium. After some analytical investigations, the macroscopic description (22put in the simplified formCeff∂h(0)∂t∗= ∂∂xi[Keffij(∂h(0)∂xj+ I3j)](24)where the effective permeability tensorKeff is obtained from the solutionχ of the linear non-stiff boundary valuproblem (18), by the surface integral:Keffij =〈K(0)2ik(−∂χj∂yk+ Ikj)〉Σi, 〈·〉Σi =1|Σi |∫Σi·dΓ (25)whereΣk is the surfaceyk = l∗k = lk/ l of the period andlk is the dimension of the period in the directionek . It canbe demonstrated that the tensorKeff is positive symmetric.5. Concluding remarksAt the macroscopic scale, the unsaturated water flow in double porosity soils with highly permeable incis described by a one equation model with two effective parameters. The macroscopic model is non-lineboth its effective parameters depend on the capillary pressure. The effective conductivity can be calculatsolution of a linear non-stiff problem by using numerical methods. The presence of very permeable incincreases the effective conductivity and influences the macroscopic behaviour through the modified rcapacity.The above analysis can be shown to apply in case when the period contains several inclusions or (anK2c/K1c = O(εp), p > 1, where it yields similar results. The case whenK2c/K1c = O(1), C1c/C2c = O(εp),p  1, is also described by the above macroscopic model. It can be obtained by following the anaLewandowska et al. [7]. Finally, note that the caseK2c/K1c = O(1), C2c/C1c = O(1) yields the classicahomogenization case.AcknowledgementsThe financial support from the French PNRH program “Transferts complexes en milieu poreux et ressoeau” is greatly acknowledged.References[1] T. Levy, Filtration in a porous fissured rock: influence of the fissures connexity, Eur. J. Mech. B Fluids 4 (1990) 309–327.[2] A. Moctezuma-Berthier, O. Vizika, P. Adler, Macroscopic conductivity of vugular porous media, Transport in Porous Media 49313–332.[3] L.A. Richards, Capillary conduction of liquids through porous medium, Physics 1 (1931) 318–333.[4] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, in: Lecture Note in Phys., vol. 127, Springer-Verlag, Berli[5] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, 1987.[6] J.-L. Auriault, Heterogeneous medium. Is an equivalent description possible?, Int. J. Engrg. Sci. 29 (7) (1991) 785–795.[7] J. Lewandowska, J.-L. Auriault, S. Empereur, P. Royer, Solute diffusion in fractured porous media with memory effects due to adC. R. Mécanique 330 (2002) 879–884.

Modelling of unsaturated water flow in soils with highly permeable inclusions

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Modelling of unsaturated water flow in soils with highly permeable inclusions

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