Alain Bourgeat

Brahim Amaziane

Leonid Pankratov

Mariya Goncharenko

Comptes Rendus Mécanique (CRM)

affectederrger’apparitionméabilitéous avonsbilités.mectifsC. R. Mecanique 332 (2004) 79–84Characterization of the flow for a single fluidin an excavation damaged zoneBrahim Amazianea, Alain Bourgeatb, Mariya Goncharenkoc, Leonid Pankratovca Université de Pau, LMA-FRE 2570 CNRS, avenue de l’Université, 64000 Pau, Franceb Université Lyon 1 – ISTIL, équipe MCS, 15, bld. Latarjet, 69622 Villeurbanne cedex, Francec Dépt. de mathématiques, B. Verkin institut des basses températures, 47, av. Lénine, 61103, Kharkov, UkraineReceived 16 July 2003; accepted after revision 21 November 2003Presented by Évariste Sanchez-PalenciaAbstractThe aim of this Note is to quantify the change of characteristics of the media of an Excavated Damaged Zone (EDZ)by several fractures. For this, we consider Darcy flow through matrix blocks and fractures with permeability of ordε2δθand 1 respectively.ε is the size of a typical porous block,δ representing the relative size of the fracture andθ is a parametecharacterising the permeability ratio. We derive the global behavior from the limit asε and δ tend to zero. The resultinhomogenized equation is of dual-porosity type forθ = 2, but it is a simple-porosity model with effective coefficients forθ > 2,and there is no flow at the macroscopic level when 0< θ < 2. To cite this article: B. Amaziane et al., C. R. Mecanique 332(2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméCaractérisation de l’écoulement d’un fluide monophasique en zone endommagée.Le but de cette Note est de quantifiles changements dans les paramètres de l’écoulement au sein d’un milieu poreux lorsque celui-ci est endommagé par lde fissures en grand nombre. Pour cela, on considère l’écoulement d’un fluide régi par la loi de Darcy avec une perde l’ordre deε2δθ dans les matrices et d’ordre un dans le réseau de fissures. Pour décrire les diverses situations ncaractérisé respectivement parε, δ et θ la taille des blocs, l’épaisseur relative des fractures et le rapport des perméaOn étudie alors le comportement asymptotique de ce problème lorsqueε et δ tendent vers zéro. On montre que le problèhomogénéisé est un modèle à double porosité pourθ = 2, un modèle à simple porosité avec des coefficients effelorsqueθ > 2 mais qu’il n’y a pas d’écoulement pour le modèle globalement équivalent avec 0< θ < 2. Pour citer cetarticle : B. Amaziane et al., C. R. Mecanique 332 (2004). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords:Porous media; EDZ; Double porosity modelMots-clés :Milieu poreux ; EDZ ; Milieu poreux fracturé ; Modèle de double porositéE-mail addresses:brahim.amaziane@univ-pau.fr (B. Amaziane), bourgeat@mcs.univ-lyon1.fr (A. Bourgeat),goncharenko@ilt.kharkov.ua (M. Goncharenko), pankratov@ilt.kharkov.ua (L. Pankratov).1631-0721/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2003.11.00680 B. Amaziane et al. / C. R. Mecanique 332 (2004) 79–84d underns havingrt of thege fluidifferentachievedhaviorcialitiesacturedention inafts.porousand wasl of thisDZ, bythere set toused inwill leade that theected andlocksdizationichin [10],del.e thereifferentre size1. IntroductionWe assume a geological situation where originally the host rock was a low permeability porous rock, antectonic stresses, these rocks became fissured. The resulting porous system is then made of two regiovery different properties (see, e.g., [1,2]). The region constituted by the fissure set represents a small patotal volume, but it is highly permeable, whereas, on the contrary, the porous block has a very high storacapacity with a low permeability. Therefore, the large-scale description will have to incorporate the two dflow mechanisms. For some permeability ratios and some fissure width, the large-scale description isby introducing the so-called double porosity model. It was introduced first for describing the global beof fractured porous media by Barenblatt et al. [1] and it is since used in a wide range of engineering sperelated to geohydrology,petroleum reservoir engineering, civil engineering or soil science. More recently, frrock domains corresponding to the so-called Excavation Damaged Zone (EDZ) received an increasing attconnection with the behavior of geological isolation of radioactive waste after the drilling of the wells or shThe usual double porosity model assumes that the width of the fractures containing highly permeablemedia is of the same order as the block size. The related homogenization problem was studied in [3],then revisited in the mathematical literature by many other authors (see, e.g., [4–8], and [9]). The main goaNote is to describe all the different situations, according to the different type and size of fissuring in the Emeans of two parameters, the relative size of the fissures,δ, and the permeability ratio. An example is given atend. In order to have explicit formulae and a better understanding of the physics, we will assume the fissube described by a thin, periodic network of intersecting plane fractures. The periodicity assumption wasorder to get more ‘tractable’ physical models, but the situation where the fractures are randomly spacedqualitatively to similar results, as soon as the associated random field is ergodic and stationary. We assumdomainΩ is made of a set of porous blocks calledMε,δ with permeability of orderε2δθ , whereε is the normalizedmicroscopic length scale of a typical porous block (0< ε  1), δ is the relative fracture width andθ > 0 is aparameter characterizing the permeability ratio; these porous blocks are surrounded by a system of connthin fissuresFε,δ with permeability of order 1. The fissure thickness is much smaller than the size of the band, therefore, the measure of the setFε,δ = Ω \ Mε,δ tends to zero asε → 0 andδ → 0. This model is describeby the following linear parabolic equation:Φε,δ(x)∂tuε,δ − div(aε,δ(x)∇uε,δ) = f ε,δ(x) (1)whereaε,δ is the permeability of the medium,Φε,δ is the porosity of the medium, andf ε,δ is the source density.In this Note, in order to describe the various situations occurring in the EDZ, we study the homogenproblem for (1) whenε andδ tend to zero. In the caseθ = 2, the macroscopic model is nonlocal in time, whcorresponds to a memory phenomenon. In contrast with the usual double porosity model we obtain, asan additional linear source term. Forθ > 2, the macroscopic behavior is given back by a single-porosity moWhen 0< θ < 2, the solution of (1) converges in a suitable topology to zero, which means that in this casis no flow at the macroscopic level; the flow is trapped. Similar questions, with different parameters and dscope, were considered in [5,7] and [10]. In [10], this type of microstructure, but with a fixed relative fissuδ = εα/2, α > 0, was modeled with only one parameterε. In [7] the simpler caseθ = 0 was considered.2. Problem statement and the main resultWe consider the following initial boundary value problem for the functionuε,δ :ΩT → R:Φε,δ(x)∂tuε,δ − div(aε,δ(x)∇uε,δ) = f ε,δ(x) in ΩTuε,δ(t, x) = 0 on(0, T ) × ∂Ωuε,δ(0, x) = 0 in Ω(2)B. Amaziane et al. / C. R. Mecanique 332 (2004) 79–84 81rintettheing to thectivewhereΩ is a bounded domain inR3 with a smooth boundary∂Ω, T > 0 is given, andΩT denotes the cylindeΩT = (0, T ) × Ω . Let Y = (0,1)3 be a periodic cell. We assume thatMδ is an open cube centered at the po(12,12,12) with length equal to 1− δ and we reproduceMδ by periodicity, obtaining a periodic open setMδ in R3.We denote byFδ the periodic setFδ = R3 \Mδ , which is obtained from the setFδ = Y \ Mδ.The functionsaε,δ :R3 → R andΦε,δ :R3 → R are defined byaε,δ(x) = aε2δθ · 1δM(xε)+ 1 · 1δF(xε), Φε,δ(x) = ΦM · 1δM(xε)+ ΦF · 1δF(xε)(3)with 0< a,θ,ΦM,ΦF < +∞, where1δF ( xε ) and1δM(xε ) denote the characteristic periodic functions of the setsFδandMδ defined in the following way:1δM(xε) is the periodic function of periodεY which takes the value 1 in the sMε,δ, union of the sets obtained fromεMδ by translations of vectorsε∑ni=1 ki ei , whereki ∈ Z andei, 1  i  3,is the canonical basis ofR3, and which takes the value 0 in the setFε,δ, complementary inR3 of this union. Inother words,1δM(xε) is the characteristic function of the setMε,δ, while 1δF (xε) is the characteristic function ofFε,δ.The functionaε,δ therefore takes the value 1 on the setFε,δ which is of asymtotically small measure (of orderδ),while it takes small valuesaε2δθ on the setMε,δ (whereε andδ take small values, as we will assume below).Notice that the measure ofFδ , is calculated as follows:∣∣Fδ∣∣ = 1− (1− δ)3 = 3δ − 3δ2 + δ3 ∼ 3δ (4)asδ → 0. Then the measure of the setFε,δ is such that limδ→0 limε→0 |Fε,δ| = 0.We make the following assumption on the source termf ε,δ :Ω → R:(H1) f ε,δ ∈ L2(Ω) such thatf ε,δ(x) ≡ 0 for x ∈ Mε,δ;(H2) f ε,δ Lδ-converges to a functionf ∈ L2(Ω), according to Definition 2.1, below.It is already known (see, e.g., [11]) that for anyε, δ > 0 there exists a unique solutionuε,δ of problem (2)belonging toC([0, T ];H 1(Ω)) and that theε and δ limits are commuting [6]. We then choose to studyasymptotic behavior of the solutionsuε,δ first asε → 0 and then asδ → 0.Due to the vanishing measure of the fissure, we should define the convergence of sequences accordsingularity of the fissure measure. For this, inspired by [12,6,13] and [14] we define:Definition 2.1.A sequence{vε,δ} ⊂ L2(F ε,δ) is said toLδ-converge to a functionv ∈ L2(Ω) iflimδ→0 limε→01|Fε,δ|∥∥vε,δ − v∥∥2L2(F ε,δ)= 0Models corresponding to the various situations are given by the following convergence results:Theorem 2.2.Let θ = 2 in (3), then for anyt ∈ ]0, T [, under assumptions(H1), (H2), uε,δ solution of (2) Lδ-con-verges tou∗ solution of a global model with an additional linear source term and the fracture porosity as effeporosity:ΦFu∗t −23u∗ = f (x) − 2√aΦM√πt∫0u∗t (x, τ )√t − τ dτ in ΩTu∗(t, x) = 0 on(0, T ) × ∂Ωu∗(0, x) = 0 in Ω(5)82 B. Amaziane et al. / C. R. Mecanique 332 (2004) 79–84d inmss],ceTheorem 2.3.If θ > 2 in (3), then for anyt ∈ ]0, T [, under assumptions(H1), (H2), uε,δ solution of (2) Lδ-con-verges tou∗ solution of a simple porosity model with effective constant porosity and permeability:ΦFu∗t −23u∗ = f (x) in ΩTu∗(t, x) = 0 on(0, T ) × ∂Ωu∗(0, x) = 0 in Ω(6)Theorem 2.4.If θ < 2 in (3), then for anyt ∈ ]0, T [, under assumptions(H1), (H2), uε,δ solution of (2) Lδ-con-verges to0. Thus, there is no flow through the effective porous medium at the global scale.3. Sketch of the proof of Theorem 2.2Knowing from [6] that the limit as(ε, δ) → 0 does not depend on the order, Theorem 2.2 will be provetwo main steps. On the first step fixingδ we pass to the limit asε → 0; we obtain then a boundary value probleconsidered in the whole domainΩ but with coefficients depending on the parameterδ. On the second step we pato the limit asδ → 0 and obtain, finally, the macroscopic model (5).Step1. Let us first fixδ.In the following, forΩε any subdomain of the domainΩ , we will say that the the sequence{vε} ⊂ L2(Ωε)converges in the spaceL2(Ωε) to a functionv ∈ L2(Ω) if lim ε→0 ‖vε − v‖L2(Ωε) = 0 (see, e.g., [15], p. 14, or [4p. 1247).The convergence of the homogenization process of problem (2) for fixedδ > 0 whenε → 0 will be given byTheorem 3.1 (see, e.g., [4]), and for this, we consideruε,δλ :Ω → the Laplace transform ofuε,δ solution of Eq. (2)and study then the corresponding initial boundary value problem:{Φε,δ(x)uε,δλ − div(aε,δ(x)∇uε,δλ) = λ−1f ε,δ(x) in Ωuε,δλ (x) = 0 on∂Ω(7)λ > 0. It is known that, for anyε > 0, there exists a unique solutionuε,δλ ∈ H 10 (Ω) of problem (7). The asymptotibehavior of the functionsuε,δλ asε → 0 is given by the following theorem.Theorem 3.1.Let the conditions of Theorem2.2 be fulfilled. Thenuε,δλ solution of (7), converges in the spacL2(F ε,δ) to uδλ = uδλ(x) solution ofλ∣∣Fδ∣∣ΦFuδλ − divx(Aδ∇xuδλ) + bδλuδλ = λ−1∣∣Fδ∣∣f (x) in Ωuδλ(x) = 0 on∂ΩλΦMUδλ − aδ2yUδλ = 0 in MδUδλ(y) = 1 on∂Mδ(8)whereAδ = {aδij } is the homogenized permeability tensor defined by:aδij =∫Fδ(ei + ∇ywi) · (ej + ∇ywj )dy (9)wi being the unique solution inH 1# (Fδ) \ R of−wi = 0 in Fδ(ei + ∇ywi) · ν = 0 on∂Mδy → wi(x, y) Y-periodic(10)B. Amaziane et al. / C. R. Mecanique 332 (2004) 79–84 83ntm [17],ce.4non-rderand wherebδλ is a positive constant given by:bδλ = λΦM∫MδUδλ(y)dy (11)Remark 1. Theorem 3.1 can be reformulated as follows. Letũε,δλ be an extension ofuε,δλ from the setFε,δ toΩ which exists (see [16]). Then we have that under conditions of Theorem 2.2 the sequence{ ˜ε,δλ } converges inL2(Ω) to uδλ solution of (8).Step2. Now we pass to the limit asδ tends to 0. First we study the asymptotic behavior ofbδλ in (11) asδ → 0.Applying the ideas of [10] one can show thatbδλ = 6δ√aλΦM ·(1+ o(1)), asδ → 0 (12)Following now the arguments of [15] and using (4) one easily shows that the sequence{ δλ} onverges inL2(Ω)to a functionu∗λ, the solution of{λΦFu∗λ − 23u∗λ + 2√aλΦMu∗λ = λ−1f (x) in Ωu∗λ(x) = 0 on∂Ω(13)Let us show now thatuε,δλ , solution of (7),Lδ-converge tou∗λ, solution of (13). In fact, we have1|Fε,δ|∥∥uε,δλ − u∗λ∥∥2L2(F ε,δ)  C(∥∥ũε,δλ − uδλ∥∥2L2(Ω) + ∥∥uδλ − u∗λ∥∥2L2(Ω)) (14)whereũε,δλ is the extension ofuε,δλ from the setFε,δ to Ω defined in Remark 1 andC is a constant independeof ε, δ. Now theLδ-convergence ofuε,δλ to u∗λ easy follows from Remark 1 and the strong convergence inL2(Ω)of the sequence{uδλ} to u∗λ.Therefore, the following result is proved.Theorem 3.2.Let θ = 2. Then under assumptions(H1), (H2) and (3), uε,δλ solution of (7) Lδ-converges tou∗λsolution of (13).Now we are in position to complete the proof of Theorem 2.2. Using the arguments similar to ones frowe prove that the inverse Laplace transform ofu∗λ, denotedu∗, is solution of (5) and we are in position to deduthe statement of Theorem 2.2 from Theorem 3.2 . This completes the proof of Theorem 2.2.4. Sketch of the proofs of Theorems 2.3 and 2.4The proofs of Theorems 2.2 and 2.3 are exactly the same, except thatbδλ/δ tends to 0 whenθ > 2 andδ → 0.The idea of the proof of Theorem 2.4 is the following. The termbδλ/δ tends to+∞ whenθ < 2 andδ → 0.Since the energy of the system is finite,uε,δλ solution of (7) shouldLδ-converge to 0. Then the proof of Theorem 2may be completed in a similar way as the proof of Theorem 2.2.5. ConclusionA typical example of the situation described by the above theory is the following. We start with an initiallydamaged porous media, with no source or sink inside the considered zone, with effective permeability of oε4,84 B. Amaziane et al. / C. R. Mecanique 332 (2004) 79–84neernetrder ofreffectiverezone,,of thekratovtefully, J. Appl.Elsevier,, SIAM(1999)2) 75–92.in press.matical2 (2001)rtial7) (2000)ew York,l periodici.e., permeability and porosity of orderεk = ε12 andεm = ε8, respectively. Assume, the damaging of this zocorresponds to a net of 10 connected fissures per square meter (ε = 1/10) each of which has thickness of ord1/100= ε2, with effective permeability of order (εαδ−θ = ε) and with a source term of flow inside the fissureof orderε. The above theorems applied to this situation give three different behaviors, according to the othe fissure net porosityδθ . If the fissure porosity is of orderε5 (θ = 2.5), then, according to Theorem 2.3, thewill be some flow through the damaged zone, which then behaves effectively like a porous media with epermeability of order O(1), exactly= (2/3)/φf , and with source flow of densityε. On the other hand, if the fissuporosity is of orderε3 (θ = 1.5), then, according to Theorem 2.4, there will be no flow through the damagedwhich then behaves effectively like the non-damaged porous media. In between (θ = 2), according to Theorem 2.2the damaged zone will then behave effectively according to a simplified ‘double porosity’ model as in (5).AcknowledgementsThis paper was completed when L. Pankratov was visiting the Applied Mathematics LaboratoryUniversity of Pau. He is grateful for the invitation and the hospitality. The work of B. Amaziane and L. Panhas been partially supported from the CNRS ANDRA BRGM CEA GdR 2439 whose support is graacknowledged.References[1] G.I. Barenblatt, Yu.P. Zheltov, I.N. Kochina, Basic consepts in the theory of seepage of homogeneous liquids in fissured rocksMath. Mech. 24 (1960) 1286–1303.[2] T.D. Van Golf-Racht, Fundamentals of Fractured Reservoir Engineering, in: Development in Petroleum Science, vol. 12,Amsterdam, 1982.[3] T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theoryJ. Appl. Math. 21 (1990) 823–826.[4] A. Bourgeat, M. Goncharenko, M. Panfilov, L. Pankratov, A general double porosity model, C. R. Acad. Sci. Paris, Sér. IIb 3271245–1250.[5] A. Bourgeat, Overall behaviour of fractured porous media versus fractures’ size and permeability ratio, Contemp. Math. 295 (200[6] A. Bourgeat, G.A. Chechkin, A.L. Piatnitski, Singular double porosity model, Appl. Anal. 82 (2003) 103–116.[7] A. Bourgeat, L. Pankratov, M. Panfilov, Study of the double porosity model versus the fissures thickness, Asymptotic Anal. (2003)[8] U. Hornung (Ed.), Homogenization and Porous Media, Interdisciplinary Appl. Math., vol. 6, Springer-Verlag, New York, 1997.[9] G.V. Sandrakov, Homogenization of parabolic equations with contrasting coefficients, Izv. Math. 63 (5) (1999) 1015–1061;English translation from Izv. Ross. Akad. Nauk Ser. Mat. 63 (4) (1999) 179–224.[10] L. Pankratov, V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, Mat. Sb. 194 (1) (2003) 121–146.[11] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, American MatheSociety, Providence, RI, 1968.[12] G. Bouchitté, I. Fragala, Homogenization of thin structures by two-scale method with respect to measure, SIAM J. Math. Anal. 31198–1226.[13] G.P. Panasenko, Homogenization of lattice-type domains:L-convergence, in: D. Cioranescu, J.-L. Lions (Eds.), Nonlinear PaDifferential Equations, Vol. XIII, in: Res. Notes Math., vol. 220, Longman, 1997, pp. 259–280.[14] V.V. Zhikov, On one extension and application of the method of two-scale convergence, Russian Acad. Sci. Sb. Math. 191 (31–72.[15] D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, in: Appl. Math. Sci., vol. 136, Springer-Verlag, N1999.[16] E. Acerbi, V. Chiadò Piat, G. Dal Maso, D. Percival, An extension theorem from connected sets and homogenization in generadomains, Nonlinear Anal. 18 (1992) 481–496.[17] L. Pankratov, Homogenization of parabolic equations in domains of degenerating measure, Appl. Anal. 81 (2002) 753–769.

Characterization of the flow for a single fluid in an excavation damaged zone

https://comptes-rendus.academie-sciences.fr/mecanique/item/10.1016/j.crme.2003.11.006.pdf

Characterization of the flow for a single fluid in an excavation damaged zone

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mécanique (CRM)