Laurent Jeannin

Mikhail Panfilov

Remy Marmier

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 334 (2006) 190–195http://france.elsevier.com/direct/CRAS2B/Averaged model of a cross hydrodynamic–mechanic processin a double porosity mediumMikhail Panfilov a,∗, Remy Marmier b, Laurent Jeannin ba Laboratoire environnement, géomécanique & ouvrages, École nationale supérieure de géologie – INPL, 54501 Vandoeuvre-lès-Nancy, Franceb Institut français du pétrole, 1 and 4, avenue de Bois-Préau, 92852 Rueil-Malmaison cedex, FranceReceived 16 February 2006; accepted 23 February 2006Presented by Charles-Michel MarleAbstractFor the single-phase flow of a compressible liquid in a deformable double porosity medium, the closed homogenized model isobtained with a total splitting between various cross mechanic–hydrodynamic effects. The transfer between matrix and fracturesis completed by the peristaltic effect and the effect of flow arising due to shearing strain. In the equation of deformations, a newstress appears being generated by the cross effects and matrix relaxation. To cite this article: M. Panfilov et al., C. R. Mecanique334 (2006). 2006 Published by Elsevier SAS on behalf of Académie des sciences.RésuméModèle homogénéisé d’un processus hydrodynamique–mécanique couplé en milieu de double porosité. Un modèle ho-mogénéisé de l’écoulement d’un fluide monophasique compressible en milieu déformable double porosité, où les effets couplésmécaniques–hydrodynamiques sont totalement séparés, est obtenu. Le transfert entre la matrice et les fractures est complété par uneffet péristaltique et par un écoulement du aux déformations de cisaillement. De nouvelles contraintes apparaissent suite aux effetscouplés et à la relaxation de la matrice. Pour citer cet article : M. Panfilov et al., C. R. Mecanique 334 (2006). 2006 Published by Elsevier SAS on behalf of Académie des sciences.Keywords: Porous media; Double porosity; Deformation; Transfer; Peristaltic; Darcy; Homogenization; High heterogeneity; Cross effectMots-clés : Milieux poreux ; Double porosité ; Déformation ; Transfert ; Péristaltique ; Darcy ; Homogénéisation ; Forte hétérogénéité ; Effet coupléVersion française abrégéeLe modèle macroscopique d’écoulement en milieu de double porosité est connu dans le cas d’un milieu non dé-formable. Il possède un terme d’échange provoqué par la différence de pression de matrice et de fracture. Dans lecas d’un milieu déformable le modèle macroscopique n’a été obtenu qu’à l’aide de certaines approximations (équi-valence de pressions en matrice et fractures) qui supprimaient la double porosité. La difficulté essentielle d’obtention* Corresponding author.E-mail addresses: mikhail.panfilov@ensg.inpl-nancy.fr (M. Panfilov), remy.marmier@ifp.fr (R. Marmier).1631-0721/$ – see front matter  2006 Published by Elsevier SAS on behalf of Académie des sciences.doi:10.1016/j.crme.2006.02.009M. Panfilov et al. / C. R. Mecanique 334 (2006) 190–195 191du modèle macroscopique est liée à l’inséparabilité des variables d’écoulement et de déformation à l’échelle du pro-blème cellulaire, ce qui amène à l’apparition du comportement macroscopique fortement non local. Dans la présentenote nous avons obtenu le modèle macroscopique pour le cas d’un milieu déformable, en conservant la double poro-sité et la différence de pressions. Le problème original d’écoulement-déformation est décrit par (1)–(3). Le modèlehomogénéisé a la forme (5)–(7), avec des paramètres effectifs définis à l’aide de (8) et de problèmes cellulaires (9).L’approche est basée sur le remplacement de la forte non localité (une mémoire longue) par une non localité faible(une mémoire courte) qui permet de décomposer tous les phénomènes non-locaux sous forme d’une superposition.Ceci a été atteint à l’aide d’un choix entre les paramètres ω et ε définissant le rapport de perméabilités et l’échelle del’hétérogénéité respectivement. La structure des développements asymptotiques a la forme (4).Dans le modèle obtenu, le terme d’échange entre la matrice et les fractures contient deux nouveaux effets : un effetpéristaltique qui signifie l’apparition d’un écoulement en fractures à cause de la compression/détente volumique de lamatrice, tandis que le deuxième effet décrit l’apparition de l’écoulement à cause de la déformation de cisaillement.A son tour, l’équation de déformation montre l’apparition de nouvelles contraintes dues aux effets couplés.1. IntroductionThe flow of a low compressible liquid in an elastic deformable double porosity medium is examined in terms ofthe homogenized behaviour. We apply the two-scale asymptotic homogenization method. The basic properties of adouble porosity medium are a small value of the parameter ω which is the ratio between the matrix and fracturepermeabilities, and a small value of the parameter ε which is the dimensionless heterogeneity scale (a fast oscillatingheterogeneity). The averaged models of flow in a non-deformable medium are well known [1,2]. All they display adifference between the fracture and matrix pressures, a delay in matrix behaviour and a transfer process generated bythe pressure difference.For a double porosity medium, it is usually assumed that ω ∼ ε2, which leads to a non-stationarity of the cell-problem and, as a consequence, to arising of a transfer term in the macroscopic model which is described by anintegro-differential operator meaning the appearance of a strong non-locality (a long-term memory) in the systembehaviour. In a non-deformable medium this non-local operator can be successfully obtained, analyzed and simulated,which is not the case of a deformable medium that occurs to be almost non-homogenizable due to high correlationsbetween the cross effects at the micro-scale. A typical cross effect has a twice nature: the displacement in rocks canbe caused by the pressure gradient (fluid flow), whereas the fluid pressure can be perturbed by a variation in timeof the porosity which is related to the divergence of the rock displacement vector. Due to this the displacement is aspace convolution with the pressure gradient, while the pressure is a space and time convolution with the divergenceof the displacement. The macroscopic model is then expected to represent a coupled system of the integro-differentialequations with implicitly defined space and time kernels. Such a system is useless being more complicated than theoriginal microscale model.The effective mathematical description of such a combined strongly correlated cross process can be obtained onlyin some particular cases. Due to this in various papers devoted to the examined system some approximations arealways introduced in order to obtain constructive results. Basically such an approximation consists in assuming thatthe matrix and fracture pressures are equivalent [4], or the zero approximation for pressure does not depend on thefast variable [3]. Both these hypotheses remove immediately any double porosity.In the present Note we applied the approach developed in Panfilov [5] where both the true double porosity model(with two different pressures) was obtained and the non-locality was reduced to a short-term memory. This case ischaracterized by a softer ratio between ω and ε: ε2 < ω  1. Due to this, we have succeeded in converting the nonlocaloperators into the differential relaxation operators of high order and to split all the cross effects from one other. Theobtained macroscale model is a true double porosity model with two pressures and exchange terms which represent ageneralization of the Barenblatt’s and Warren–Root’s term. We show that various exchange effects are characterizedby a number of various relaxation times. All the effective parameters including the relaxation times are determined ina closed form through the solutions of the cell problems. The obtained cross terms have a clear physical origin whichis discussed in the final part of the Note.192 M. Panfilov et al. / C. R. Mecanique 334 (2006) 190–1952. Problem formulationLet us consider the one-phase weakly compressible flow in a linearly deformable medium consisted of highly per-meable fractures (medium F ) submerged into a tight porous matrix (medium M). The fracture network is connected.Each of the sub-domains Ωα , α = F ,M, is characterized by the rock permeability, Kα , and the porosity φα . The fluidviscosity is assumed to be constant. The medium elastic deformations are described by Lamé’s coefficients, whereasthe Biot parameter is assumed to be equal to one in both sub-domains within the framework of this paper. The lastassumption means that the solid matter of rocks in non-deformable.The medium heterogeneity is periodic. The ratio between the period size and the overall domain size is a smallparameter ε. We assume also that the medium is highly heterogeneous with respect to permeability, so that the matrixand fracture mean permeabilities, KF0 and KM0, are very different from one other: ω = KM0/KF0  1.Let x = (x1, x2, x3) be the space coordinates, τ the time, p the fluid pressure, u the elastic rock displacementvector. The fluid flow is then described by the system of mass balance equations, in which the flow velocity obeys theDarcy law:βF∂p∂τ+ ∂∂xi(∂uFi∂τ)= ∂∂xi(aF∂p∂xi)(1a)βM∂p∂τ+ ∂∂xi(∂uMi∂τ)= ω ∂∂xi(aM∂p∂xi)(1b)while the saturated rock deformations are described by the equilibrium between all the acting stresses (for α = F ,M):∂∂xi(µα(∂uαk∂xi+ ∂uαi∂xk)+ να ∂uαj∂xjδik − æpδik)= 0, k = 1,2,3 (2)The conditions at the matrix-fracture interface Γ imply the continuity for the normal stresses, the fluid pressure,the normal fluid flow rate and the normal displacement:[(µ(∂uk∂xi+ ∂ui∂xk)+ ν ∂uj∂xjδik − æpδik)ni]Γ= 0, [p]Γ = 0 (3a)ωaM∂p∂xini∣∣∣∣Γ M= aF ∂p∂xini∣∣∣∣Γ F, [un]Γ = 0 (3b)where n is the exterior normal vector to the interface Γ , νi and µi are the Lamé coefficients of the medium rocks(dimensionless); aF = KF/KF0 and ωaM = KM/KF0 are the fluid mobility (permeability divided to viscosity);β = φCf where φ is the porosity, while Cf is the fluid compressibility; æ = P 0L/N0U0 is the ratio between thenormal strain in a saturated medium and in a dry medium.System (1)–(3) is formulated in dimensionless form. The following characteristic scales were selected for variableswith dimension: N0: the mean value between all the Lamé coefficients; B0: the mean compressibility between βFand βM; P 0: the maximum pressure drop along the overall medium; KF0: the mean permeability in sub-domain F ;L: the domain size; T = B0L2/KF0: the characteristic time of elastic perturbation propagation along the distance Lthrough medium F ; U0 = P 0B0L: the characteristic elastic deformation caused by the pressure drop P 0.3. Two-scale asymptotic expansionsIn the classic version of the double porosity media parameter ω is usually equal to ε2 which leads to a differencebetween the matrix and the fracture pressures of order one. This leads in turn to arising of an integro-differentialoperator in the macroscale model which is responsible for the transfer between matrix and fracture media [5]. In thecase of a deformable medium, the ratio ω ∼ ε2 does not permit to separate the fast and slow variables both at thescale of the cell problem and of the averaged equation [3]. So the obtained model can not be examined as reallyhomogenized.As shown in [5], the effective averaged model with two various pressures in matrix and fractures can be obtainedif parameter ω is small but larger that ε2. In this case the difference between pressures will be lower than one, but stillM. Panfilov et al. / C. R. Mecanique 334 (2006) 190–195 193significant. We select the ratio between ω and ε in the following way: ε2  ω  1. More strictly, the range betweenparameters is the following: ε2 < ω < ε < ε2/ω  1. The asymptotic expansions for Eqs. (1)–(3) take the form:p(x, y, τ ) = p0(x, τ ) +ε2ωpM01 (x, y, τ ) + εpM10 (x, y, τ ) + ω · · · , y ∈ YMε2ωpF01(x, τ ) + εpF10(x, y, τ ) + εε2ωpF11(x, y, τ ) + · · · , y ∈ YF(4a)u(x, y, τ ) = u0(x, τ ) + ε2ωu01(x, τ ) + εu10(x, y, τ ) + · · · , y ∈ YF , YM (4b)The next steps of the constructive two-scale homogenization technique are as follows: (i) ‘y’ is examined as a newindependent variable; (ii) all the original functions and differential operators formulated in terms of x, τ are replacedby their two-scale extended versions in terms of x, τ, y (∂/∂x → ∂/∂x + ε−1∂/∂y); (iii) the solution to the newtwo-scale problems is searched in the space of y-periodic functions; (iv) Eqs. (4) are substituted into the two-scaleformulations of Eqs. (1)–(3); (v) applying the regular perturbation technique, the obtained flow equations are decom-posed into an infinite system which determines the successive approximations; (vi) the solvability of the obtainedproblems imposes some integral conditions on the coefficients of series (4), according to the Fredholm alternative;(vii) the macroscopic model follows from the solvability condition applied to the zero terms of the asymptotic series.4. Macroscopic modelFor the macroscale pressures in fractures and matrix, PF and PM, and the displacement U we obtain the followingresult with keeping the terms up to ε2/ω:〈β〉∂PF∂τ+ ∂2Uj∂τ∂xj− ∂∂xi(K̂ik∂PF∂xk)= −τϕ ∂2PF∂τ 2− τ θmn∂2Ĝmn∂τ 2− τσ ∂3Uj∂τ 2∂xj(5a)PM = PF + τp ∂PF∂τ+ τGmn∂Ĝmn∂τ+ τu ∂2Uj∂τ∂xj(5b)∂∂xi(µ̂mnki(∂Um∂xn+ ∂Un∂xm)+ ν̂ki ∂Uj∂xj− æPF δik)= −τϕki∂2PF∂τ∂xi+ τ θmnki∂2Ĝmn∂τ∂xi+ τσki∂3Uj∂τ∂xj ∂xk, k = 1,2,3 (5c)where Ĝmn ≡ ∂Um∂xn+ ∂Un∂xmThe effective permeability is determined as:K̂ik =〈aF (y)(∂ψFk∂yi+ δik)〉F(6)The effective Lamé coefficients are defined as in a dry medium:µ̂mnki =〈µ(∂ξkmn∂yi+ ∂ξimn∂yk)+ ν ∂ξjmn∂yjδik + µδkmδik〉(7a)ν̂ki =〈µ(∂ζk∂yi+ ∂ζi∂yk)+ ν ∂ζj∂yjδik + νδik〉(7b)Nine relaxation tensors caused by the double porosity have the form:τp = ε2ω〈ϕ〉M, τGmn =ε2ω〈θmn〉M, τu = ε2ω〈σ 〉M (8a)τϕ = ε2〈βϕ〉M, τ θmn =ε2 〈βθmn〉M, τ σ = ε2〈βσ 〉M (8b)ω ω ω194 M. Panfilov et al. / C. R. Mecanique 334 (2006) 190–195τpki =ε2ω〈µM(∂ϕk∂yi+ ∂ϕi∂yk)+ νM ∂ϕj∂yjδik − ϕδik〉M(8c)τ θmnki =ε2ω〈µM(∂θkmn∂yi+ ∂θimn∂yk)+ νM ∂θjmn∂yjδik − θmnδik〉M(8d)τσki =ε2ω〈µM(∂σ i∂yk+ ∂σ k∂yi)+ νM ∂σ j∂yjδik − σδik〉M(8e)The difference between the macroscale model and the original microscale equations (1) is of order of ε2 + ε3ωwhich tends to zero, as ε,ω → 0.5. Cell problemsThe effective parameters are defined through intermediary functions ψk(y), ϕ(y), θkm(y), σ(y), ξkmn(y), ζk(y),ϕk(y), θkmn(y), σk(y) (k,m,n = 1,2,3), which are the solutions to the following series of cell problems:∂∂yi(µ(∂ξkmn∂yi+ ∂ξimn∂yk)+ ν ∂ξjmn∂yjδik + µδkmδin)= 0, y ∈ Y〈ξkmn〉 = 0, ξkmn is 1-periodic(9a)∂∂yi(µ(∂ζk∂yi+ ∂ζi∂yk)+ ν ∂ζj∂yjδik + νδik)= 0, y ∈ Y〈ζk〉 = 0, ζk is 1-periodic(9b)∂∂yi(aM∂ϕ∂yi)= βM, y ∈ YMϕ|Γ = 0(9c)∂∂yi(aM∂θmn∂yi)= ∂ξjmn∂yj, y ∈ YMθmn|Γ = 0(9d)∂∂yi(aM∂σ∂yi)= 1 + ∂ζj∂yj, y ∈ YMσ |Γ = 0(9e)∂∂yi(µ(∂θkmn∂yi+ ∂θimn∂yk)+ ν ∂θjmn∂yjδik − æθmnδik)= 0, y ∈ YMθkmn|Γ = 0(9f)∂∂yi(µ(∂ϕk∂yi+ ∂ϕi∂yk)+ ν ∂ϕj∂yjδik − æϕδik)= 0, y ∈ YMϕk|Γ = 0(9g)∂∂yi(µ(∂σ k∂yi+ ∂σ i∂yk)+ ν ∂σ j∂yjδik − æσδik)= 0, y ∈ YMσk|Γ = 0(9h)∂∂yi(aF∂ψk∂yi)= −∂aF∂yk, y ∈ YF〈ψk〉F = 0, ψk is 1-periodic(9i)Due to the Fredholm alternative for an elliptic equation, each problem (9a), (9b) and (9i) has a unique periodicsolution defined up to an additive constant which is strictly determined in our case by the condition on the averagevalue. Each system (9c), (9d), (9e), (9f), (9g) and (9h) has a unique solution as a Dirichlet problem for the Laplace-likeequation in a bounded domain YM with piece-wise regular boundary.M. Panfilov et al. / C. R. Mecanique 334 (2006) 190–195 1956. Analysis of the macroscopic modelSystem (4.1) shows the existence of two different macroscopic pressures, PF and PM, which is a typical propertyof a double porosity medium resulting only from the high difference between the matrix and fracture permeabilities.At the same time, the macroscopic displacement U is defined in a uniform way for matrix and fractures.Eq. (5a) can be presented with introducing the transfer function between matrix and fracture, q , for which we canobtain an equivalent formulation:〈β〉F ∂PF∂τ+ ∂2Uj∂τ∂xj− ∂∂xi(K̂ik∂PF∂xk)= qqτp〈β〉M =(PM − PF ) + (τu + τGmm) ∂2Um∂τ∂xm +(τGmn + τGnm)︸ ︷︷ ︸m 	=n∂2Um∂τ∂xn(10)with a summation over indexes m and n in all the terms.The first term is the classic transfer caused by a pressure difference. The second term is a cross deformation-floweffect which implies that a volumetric matrix compaction–extension causes a flow in the fractures. This is a typicalperistaltic effect observed in flexible tubes (blood artery, intestine, rubber hose), when an elastic expansion-dilatationof tube walls produces liquid flow inside the tube. The third term represents a flow in fracture caused by shearingstrains in the matrix.The equation of stresses, (7), can be presented in the following form more convenient to be explained:∂∂xi(Σ(0)ik + Σ(1)ik) = 0, i, k = 1,2,3Σ(0)ik ≡ µ̂mnki(∂Um∂xn+ ∂Un∂xm)+ ν̂ki ∂Uj∂xj− æPF δik,Σ(1)ik ≡ τϕki∂PF∂τ− τ θmnki∂Ĝmn∂τ− τσki∂2Uj∂τ∂xjThe stress Σ(0)ik may be examined as the classic averaged result which corresponds to a moderately heterogeneousmedium without double porosity. The new stress Σ(1)ik is caused by the double porosity, i.e., by the exceeding pressurein matrix, which provokes additional deformations.AcknowledgementsThe research was partially supported by the Institut Français du Pétrole.References[1] G.I. Barenblatt, Yu.P. Zheltov, I.N. Kochina, On main representations in the theory of flow of homogeneous liquids in fractured rocks, Prikl.Mat. Mech. 24 (5) 852–864.[2] J.E. Warren, J. Root, The behaviour of naturally fractured reservoirs, SPE J. 3 (9) 245–255.[3] M.A. Murad, J.H. Cushman, Thermomechanical theories for swelling porous media with microstructure, Internat. J. Engrg. Sci. (2000).[4] Z. Chen, S.L. Lyons, G. Qin, The mechanical behavior of a poroelastic medium saturated with a Newtonian fluid, Internat. J. of Num. An. andMod.[5] M. Panfilov, Averaged Models of Flow in Highly Heterogeneous Media, Kluwer, Dordrecht, 2000.

Averaged model of a cross hydrodynamic–mechanic process in a double porosity medium

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Averaged model of a cross hydrodynamic–mechanic process in a double porosity medium

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