International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921

Contents lists available at ScienceDirect

International Journal ofRock Mechanics & Mining Sciences

1365-16

doi:10.1

� Corr

F-54501

E-m

journal homepage: www.elsevier.com/locate/ijrmms

A composite sphere assemblage model for porous oolitic rocks

N.B. Nguyen a, A. Giraud a,b,�, D. Grgic a

a LAEGO-ENSG, BP40, Vandoeuvre-l�es-Nancy Cedex F-54501, Franceb LPMM - FRE CNRS 3236, Ile du Saulcy, 57045 Metz cedex 01, France

a r t i c l e i n f o

Article history:

Received 6 May 2010

Received in revised form

14 April 2011

Accepted 15 May 2011Available online 15 June 2011

Keywords:

Poroelasticity

Micromechanics

Multistep homogenization

Effective properties

Four phase model

Limestone

Composite sphere assemblage model

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ijrmms.2011.05.003

esponding author at: LAEGO-ENSG, BP40, V

, France. Tel./fax: þ33 3 83 59 63 87.

ail address: Albert.Giraud@ensg.inpl-nancy.fr

a b s t r a c t

The present work is devoted to the determination of linear effective poroelastic properties of porous

rocks characterized by an assemblage of grains (oolites) coated by a matrix. In the framework of Hashin

composite sphere assemblage (CSA [1]) models, a two-step homogenization method is developed.

Three porous materials are considered and assembled into a CSA: oolite (o) coated by an interfacial

transition zone ITZ (i) and a matrix (m). The overall porosity is supposed connected and decomposed

into oolite porosity, interphase porosity and matrix porosity. The three constitutive porous materials

(o, i, m) are homogenized at the first step and a sensitive study is performed on the homogenization

scheme by comparing self consistent SC, differential self consistent DSC, generalized self consistent GSC,

iterative self consistent ISC and Mori Tanaka schemes. At the second step, the simple three-phase model

[2] is firstly recalled and general relations for effective linear poroelastic coefficients are presented in

the four-phase model case [3]. Mathematical analogy between linear thermoelasticity and linear

poroelasticity is used to establish theoretical results in the context of CSA.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

This paper treats the estimation of the effective linear poroelasticproperties of isotropic porous rock materials or geomaterials whosemicrostructure is composed of nearly spherical grains surroundedby a matrix. Both grains and matrix are porous materials andthe corresponding porosities are supposed connected. The volumefraction of the grains may be very significant compared to thematrix (as an example, the volume fraction of the grains maybe approximately comprised in the range 0.2—0.8). Such kind ofmaterials are encountered in geomechanics: as an example cementbased materials such as concretes and some rocks, such as oolitic

rocks (limestones, iron ore, etc.), may be characterized by largefraction of grains coated by a continuous matrix. In oolitic lime-stones, the nearly spherical grains (the oolites) are porous andsurrounded by a porous calcareous matrix, and grains are stifferthan matrix.

In the limiting case of low or moderate volume fractions of grains,a matrix-inclusion homogenization method based on Eshelbyapproach with appropriate scheme may be used to estimate effectiveporoelastic properties (see among others [4–7]). For intermediateand large volume fraction of grains, and for such microstructures, analternative approach is given by the composite sphere assemblage

ll rights reserved.

andoeuvre-l�es-Nancy Cedex

(A. Giraud).

(CSA) models introduced by Hashin [1]. This homogenization methodis powerful and has been extensively developed by many authors fortwo-phase, three-phase and N-phase composite spheres (referred asmulti-coated spheres). In the context of geomaterials, micromecha-nical based model for porous rock-like composites may be found inRefs. [8–12]. Many applications of CSA models to cement basedmaterials and concrete have been presented. Among many others see[13,14,3,15,16] and recently [17]. Differential effective mediumtheory has been developed in the context of sphere assemblagemodels in [18].

As the analysis is focused on estimation of linear poroelasticproperties, the well-known correspondence between the equationsof static linear poroelasticity and the equations of linear thermo-elasticity including entropy will be used in this paper to deriveeffective linear poroelastic properties. The correspondence is ofpractical use in determining the effective parameters in an inhomo-geneous poroelastic medium using known results from the literatureon the effective thermal expansion coefficient and the effective heatcapacity of a disordered thermoelastic continuum as it has beenshown in early papers of Berryman and coauthors (see Refs. [19–21]),see also [22–24]. From this correspondence, it is interesting to notethat latent heat tensor and effective heat capacity at constant strainin linear thermoelasticity respectively play the role of Biot tensor andinverse of solid Biot modulus in linear poroelasticity. In this paper,this correspondence will be used in the context of composite sphere

assemblage (CSA) models which have been extensively studiedin linear thermoelasticity. Presented theoretical results for linearporoelastic composite spheres are deduced from corresponding

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921910

results obtained in similar thermoelastic microstructures, in parti-cular three-phase model, four-phase model or N-phase model compo-site sphere models. See among many others [25–27,2,28–35].

One of the most important problem in composite materialscomposed of grains surrounded by a matrix with a strong contrastof stiffness between them, such as concrete materials and alsorock–composite materials, is the interphase zone or interfacial

transition zone (ITZ) between grain and matrix. See [36] for amodel for composites in which a thin layer existed outside of eachinclusion. See [29,3] for extension to a finite number of concentricshells around each inclusion, with the moduli uniform withineach shell. See Refs. [37–39] for composite sphere model takinginto account inhomogeneous interfacial transition zone. In thispaper, only perfect interface is considered but debonding andimperfect interfaces is an alternative approach and an importantsubject of research actually. Among others see [40–42]. See also[16] for a nonlinear analysis of the impact of the ITZ on cohesionof cement based materials. In this paper, ITZ poroelastic proper-ties will be identified through an inverse approach taking intoaccount the macroscopic experimental results and microstruc-tural informations such as volume fractions of constituents, andsolid constituent properties.

It may be noticed that recent papers [43,44] present a linearporoelastic model in the particular case of porous materialscontaining two populations of voids (pores and cracks), with ascale separation between them. The micromechanical modelpresented in these two latter references is consistent and generaland includes linear and nonlinear analyses. The paper is organizedas follows. In Sections 2 and 3, a two-step homogenization modelis presented, with derivation of theoretical results. More pre-cisely, the pores are homogenized, both in grains, interface andmatrix, at the first step which represents the transition from themicroscopic scale to the mesoscopic scale. A sensitive study isperformed on the homogenization method for the first step.The three poroelastic materials obtained from the first step,which constitute a doubly coated composite sphere (three-phasecomposite sphere) are homogenized at the second step by usingthe four-phase model [3]. In Section 4, numerical results arepresented for representative oolitic limestone porous rocks.Model and macroscopic experimental data are compared and asensitive study is performed on the key parameters of the model(in particular parameters defining ITZ).

1.1. Notations and definitions

Hereafter we define some notations and recall some resultswhich are needed later (see among others [45,46]).

Barred letters A, C, D, Q refer to fourth-order tensors, twice

underlined letters e , s , i refer to second-order tensors, underlined

letters z, x refer to first order tensors. Einstein’s summation

convention over repeated indices is used unless otherwise indi-cated. �, : and :: respectively represent tensor product, contracted

products on two and four indices. i , I, J and K respectively

represent the second-order identity tensor, the fourth-ordersymmetric identity tensor, spherical and deviatoric tensors

(dij denotes Kronecker delta symbol, dij ¼ 1 if i¼ j, dij ¼ 0 otherwise)

a � b ¼ aibjei� e

j,a : b ¼ aijbji

A : B¼ Aijkl Blkmnei� e

j� e

m� e

n

i ¼ dijei� e

j

Iijkl ¼12 ðdikdjlþdildjkÞ, Jijkl ¼

13dijdkl, Kijkl ¼ Iijkl�Jijkl ð1Þ

The brackets refer to an intrinsic average, /aS represents the

intrinsic average of a over the RVE denoted O, /aSa represents

the intrinsic average of a over the a phase

/aS¼1

O

ZO

a dO, /aSa¼

1

Oa

ZOa

a dO ð2Þ

2. Two phase composite spheres and three-phase model

This section is devoted to the presentation of a two-stephomogenization scheme for linear poroelastic properties ofporous rocks. The microstructure of the investigated material ischaracterized by grains (referred as oolites in the paper, index o)surrounded by a matrix m. Both oolites and matrix are porousmaterials and the respective parts (matrix and oolite porosities)of the porosity are supposed connected.

1.

The oolites and matrix containing microporosities is homogenizedat the first step. As in [43], the first step will be referred as thetransition from the microscopic to the mesoscopic scale. Homo-genization method at the first step is based upon Eshelby tensorapproach and a sensitive study is performed at the first step bycomparing homogenization schemes: Mori Tanaka, self consis-tent and differential self consistent schemes. The result of thisfirst step is two isotropic homogenized linear poroelastic materi-als: porous oolite and porous matrix, with isotropic linear por-oelastic properties denoted by kI

j , mIj , bI

j , NIj (j¼ o,mÞ.

2.

The oolites and surrounding matrix are homogenized at thesecond step in the framework of Hashin composite sphere

assemblage (CSA) models: grains (oolites) are supposed sphericaland the surrounding matrix is modelled by a spherical shellcoating the grains (Fig. 1). The generalized self consistent scheme

(GSCS), or three-phase introduced by [27,2], is used at the secondstep to obtain effective properties of the porous composite.

It is worth to notice that the assumption of spherical shape forgrains particle is not always necessary to work in the frameworkof Hashin composite sphere assemblage, providing the distribu-tion of the centres of inclusions be random. In the particular caseof the oolitic limestones studied in this paper, microstructuralobservations show that oolite are randomly distributed and ofovoidal shape. The average oolitic ovoidal shape may be quiteaccurately approximated by the simpler spherical shape.

As similar two-step scheme developed in the context of porousmatrix-inclusions rock composite such as argillaceous rocks andargillites has been presented in [47,7]. Multi-step homogenizationmethods for porous media have been presented in [48]. See also[43,44] for a poroelastic analysis of a porous materials containingtwo different populations of voids.

2.1. Volume fractions and constituents

At the macroscopic scale, one considers a two-phase compo-site material composed of two isotropic poroelastic media: oolites(o) and matrix (m) surrounding the oolites

O¼OoþOm, fo ¼Oo

O, fm ¼

Om

O, foþ fm ¼ 1 ð3Þ

At the mesoscopic scale the porosity is homogenized and the totalporosity is decomposed into two parts, oolite porosity Oop andmatrix porosity Omp supposed connected to each other

Op ¼OopþOmp

fop ¼Oop

O, fmp ¼

Omp

O, fp ¼

Op

O¼ fopþ fmp ð4Þ

Fig. 1. A two-step model for porous oolitic rocks three phase sphere assemblage

model (two-phase, or coated composite sphere, oolite o, matrix m).

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921 911

At the first step one defines the oolite and matrix porosities as

f Iop ¼

Oop

Oo¼

fop

fo, f I

mp ¼Omp

Om¼

fmp

fmð5Þ

The solid particles constituting oolites o and surrounding matrixm are both supposed isotropic (j¼o, m), elastic stiffness tensor Cj

and compliance tensor Sj write

Cj ¼ 3kjJþ2mjK, Sj ¼1

3kjJþ

1

2mj

K ð6Þ

where kj and mj and represent the bulk modulus and shearcoefficient of constituent j.

2.2. First transition from the microscopic to the mesoscopic scale

(step I)

The first step represents the transition from the microscopicscale to the mesoscopic scale. Oolite pores and matrix pores arehomogenized and the result of the first step is two porous materials:porous oolite and porous matrix. Oolite solid and matrix solid areboth supposed isotropic. Due to the relative uncertainty on thebehaviour of constituents at the mesoscopic scale, a sensitive studyhas been performed and five homogenization schemes have beenused and compared for the first homogenization step: Mori Tanaka(MT), self consistent (SC), differential self consistent (DSC) scheme,iterative self consistent (ISC) scheme and generalized self consistent(GSC) scheme. Only the particular case of spherical pores is inves-tigated in this paper but similar formula could be easily obtained forrandomly oriented oblate spheroidal pores (see [49,50,47]). In thissection, the linear poroelastic behaviour is first recalled. The well-known homogenization results for linear poroelastic properties atmesoscopic scale corresponding to the three investigated schemes,under hypothesis of isotropic solid constituent and spherical poresand/or solid particles will be given as they will be used as input forthe second homogenization step. Homogenized linear poroelasticconstitutive behaviour at the mesoscopic scale may be written as(see finding in Biot’s papers [51,52], and among others [53,54])

S ¼CIj : E�B I

jP

f�f0 ¼ B I

j: Eþ

P

NIj

ð7Þ

S , E , P and f respectively represent Cauchy stress tensor, deforma-

tion tensor, pore pressure and porosity. In this paper, for the sake ofsimplicity one considers zero initial stress and zero initial pore

pressure. CIj and B I

jrespectively denote effective stiffness and Biot

tensors, NIj represents solid Biot modulus of material j (j¼o, m) at

the mesoscopic scale. Micro–macro compatibility equations give(see [55–57,20])

B I

j¼ i : ðI�Sj : C

IjÞ,

1

NIj

¼ ðB I

j�f I

jpi Þ : Sj : i ð8Þ

In the particular case of isotropic homogeneous solid constituent j

and isotropic solid skeleton, relations (7) rewrite

S ¼ ð3kIjJþ2mI

jKÞ : E�bIjPi

f�f0 ¼ bIji : Eþ

P

NIj

ð9Þ

and relations (8) between skeleton and matrix properties write

B I

j¼ bI

ji , bIj ¼ 1�

kIj

kj,

1

NIj

¼bI

j�f Ijp

kjð10Þ

It must be emphasized that Biot relations ((8)–(10)) are only valid inhomogeneous solid case (microhomogeneity hypothesis, see [58]).Mori Tanaka homogenization scheme corresponds, in the particularcase of spherical pores, to the Hashin Shtrikman upper bound (HSþ ,see [59] and among others, reference books [45–60]) and corre-sponding isotropic poroelastic coefficients write

kMTj

kj¼ð1�f I

jpÞð1�ajÞ

1�ajð1�f IjpÞ

,mMT

j

mj

¼ð1�f I

jpÞð1�bjÞ

1�bjð1�f IjpÞ

bMTj ¼

f Ijp

1�ajð1�f IjpÞ

,1

NMTj

¼ajð1�f I

jpÞfIjp

kjð1�ajð1�f IjpÞÞ

ð11Þ

with

aj ¼3kj

3kjþ4mj

, bj ¼2ðkjþ2mjÞ

5ð3kjþ4mjÞð12Þ

It may be recalled that Biot coefficient bIj only depends on the

Poisson ratio nj and on the relative porosity f Ijp (recalling

kj ¼ Ej=3=ð1�2njÞ and mj ¼ Ej=2=ð1þnjÞ)

bMTj ¼

3f Ijpð1�njÞ

2ð1�2njÞþ f Ijpð1þnjÞ

ð13Þ

and expression (13) is consistent with the limit nj-12 ðkj-1) which

corresponds to incompressible solid material

limnj-

12

bMTj ¼ 1 ð14Þ

The self consistent approximation gives two uncoupled implicit

nonlinear equations for bulk modulus kSCj and shear coefficient mSC

j

(see among many others [9], relations (12) are used with kSCj and mSC

j

instead of kj and mj)

kSCj

kj¼

1�f Ijp

1þaSCj ðkj�kSC

j Þ=kSCj

,mSC

j

mj

¼1�f I

jp

1þbSCj ðmj�mSC

j Þ=mSCj

ð15Þ

A differential self consistent approximation (DSC, see [61]), [62] hasbeen taken into account. In the case of spherical pores and solidgrains, differential equations can be integrated to give the two

explicit nonlinear equations to solve for kDSCj and mDSC

j (relations

10.55 and 10.56 in [9])

mDSCj

mj

¼ ð1�f IjpÞ

21þkjðmDSC

j =mjÞ3=5

1þkj

!1=3

kDSCj

kj¼mDSC

j

mj

1þ2kj

1þ2kjðmDSCj =mjÞ

3=5

!

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921912

kj ¼1�5nj

2ð1þnjÞ¼�

3kj�4mj

6kjð16Þ

An iterative self consistent approach (ISC, see [63]), has been alsoused and compared to the previous schemes. It is similar tothe differential self consistent approach and to the differentialeffective method theory (DEMT) presented in [18]. The iterativehomogenization method allows to take into account various familiesof heterogeneities according to their geometrical or mechanicalproperties and to progressively introduce them in a volume ofmatrix. This is a flexible method, easy to implement, and welladapted to complex microstructures. In the investigated isotropiccase, two nonlinear equations need to be solved for the bulk

modulus kSCðnÞj and shear coefficient mSC

ðnÞj at each increment

kðnþ1Þj

kðnÞj

¼1�f ðnÞIjp

1þaðnþ1Þj ðkðnÞj �kðnþ1Þ

j Þ=kðnþ1Þj

mðnþ1Þj

mðnÞj

¼1�f ðnÞIjp

1þbðnþ1Þj ðmðnÞj �m

ðnþ1Þj Þ=mðnþ1Þ

j

aðnþ1Þj ¼

3 kðnþ1Þj

3 kðnþ1Þj þ4 mðnþ1Þ

j

, bðnþ1Þj ¼

2ðkðnþ1Þj þ2 mðnþ1Þ

j Þ

5ð3 kðnþ1Þj þ4 mðnþ1Þ

j Þð17Þ

with

kð0Þj ¼ kj, mð0Þj ¼ mj, f ð0ÞIjp ¼ 0, Df Ijp ¼

f Ijp

N

f ðnÞIjp ¼Df I

jp

1�f Ijpþ

Pni ¼ 1 Df I

jp

ð18Þ

0.2 0.4 0.6 0.8

f jpI

0.2

0.4

0.6

0.8

1.0

kjIkj

Step I : SC ISC DSC GSC MT HS

MT HSGSCDSCISC 200SC

Fig. 2. Step I: reduced moduli (kIj=kj and mI

j=m

0.0 0.2 0.4 0.6 0.8

f jpI

0.2

0.4

0.6

0.8

1.0

jkI

kj

Step I : ISC

ISC 200ISC 10ISC 5ISC 3SC

Fig. 3. Step I: reduced modu

The iterative process ((17)–(18)) is performed from n¼1 to the finalstate n¼N. Comparisons of the MT, SC, DSC, and ISC estimates forbulk modulus and shear coefficient are presented in Fig. 2. Asexpected, MT approximate which exactly corresponds to HashinShtrikman upper bound, provides upper estimates for bulk modulusand shear coefficient whereas lower estimates are given by SC

scheme. GSC scheme exactly corresponds to MT approximate andHashin Shtrikman upper bound for bulk modulus and it providesintermediate values between MT and DSC estimates for shearcoefficient. It must be emphasized that DSC, GSC (and MT schemesin this particular case) can be used for the whole range of porositywith no percolation threshold. ISC estimate converges towards DSC

estimate for a sufficiently large number of increments (N¼200 inFig. 2). ISC scheme may be used as an intermediate scheme, betweenSC and DSC. Estimates for different values of increments (N¼3, 5, 10,200) are presented in Fig. 3. It may be noticed that, as materialconstituting oolite and matrix may have different microstructures,different homogenization could be used for each material, at the firststep. Self consistent estimate vanishes for the level of porosity

f Ijp ¼ 1=2 which is classically interpreted as a percolation threshold

of the pores. The relevancy of the Mori Tanaka (MT) estimates fortwo-phase composite materials depends on the pore shape and onthe porosity (see [50]). The validity range of MT estimate is higherfor spherical pores than flat pores, and it may be used approximately

in the range f Ijpr0:320:4. DSC or ISC approximates may be used,

instead of MT and SC estimates, for high values of porosity (0:4r f Ijp).

Schematically:

�

jIj

��

j), D

jIj

��

li (k

higher porosities may be observed in the matrix phase m, andMT scheme could be used only for oolites with low ormoderate porosity,

0.2 0.4 0.6 0.8

f jpI

0.2

0.4

0.6

0.8

1.0

Step I : SC ISC DSC GSCS MT HS

MT HSGSCDSCISC 200SC

SC and ISC (N¼200) are superposed.

0.2 0.4 0.6 0.8

f jpI

0.2

0.4

0.6

0.8

1.0Step I : ISC

ISC 200ISC 10ISC 5ISC 3SC

Ij=kj and mI

j=mj), ISC.

Fig. 4. A two-step model for porous oolitic rocks four-phase sphere assemblage

model (three phase, or doubly coated composite sphere, oolite o, interphase i,

matrix m).

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921 913

�

percolation threshold of the pores of the SC scheme may bereached in the matrix (depending on the level of porosity) andDSC or ISC schemes need to be used for highly porous matrix, � microstructure of oolites and matrix seems closer to a

polycrystalline disordonned type microstructure than amatrix-inclusion type microstructure, so MT scheme is certainlythe less relevant scheme for investigated materials (in parti-cular for matrix phase) (Fig. 4).

2.3. Second transition from the mesoscopic scale to the macroscopic

scale (step II)

In the second transition, from the mesoscopic scale to themacroscopic scale, the microstructure of the medium is describedby a composite sphere assemblage of two-phase compositespheres (or coated spheres). It corresponds to the three-phasemodel introduced by [2] and used by many authors (see amongothers [30,15]). As results for elastic properties of such sphereassemblage are well known, some results for bulk modulus andshear coefficient will be briefly recalled. Analysis presented in thispaper is centred on poroelastic couplings, and then only thesimplest loading case is studied: strain or stress boundary condi-tions are spherical and all quantities respect spherical symmetry(functions of radius only). Assembled materials (oolites andmatrix) are poroelastic and oolite porosity is supposed connectedwith matrix porosity. Homogenized linear poroelastic constitutivebehaviour at the macroscopic scale may be written as

S ¼CIIhom : E�B II

homP

ðf�f0ÞII¼ B II

hom: Eþ

P

NIIhom

ð19Þ

CIIhom and B II

homrespectively denote effective macroscopic stiffness

and Biot tensors, NIIhom represents effective solid Biot modulus (see

[5]). It is worth to notice that, as linear poroelastic theory andlinear thermoelastic theory are similar, exact and known resultsobtained in isotropic thermoelasticity may be adapted and usedin isotropic poroelasticity thanks to proper equivalence betweenvariables and properties. As an example, in isotropic case, the Biotcoefficient bII

hom plays the role of the latent heat 3aIIhomkII

hom inthermoelasticity, and the inverse of solid Biot modulus 1=NII

hom

plays the role of specific heat at zero deformation, porosity being(only mathematically !) analogous to entropy. So results estab-lished in linear thermoelasticity by many authors, including[25,26,29,32,33], may be used. In this study, thermoelastic resultshave been systematically used to check and validate presented

results for poroelasticity. Effective bulk modulus kIIhom, Biot

coefficient bIIhom, and effective solid Biot modulus NII

hom write

kIIhom ¼ kI

mþð3kI

mþ4mImÞðk

Io�kI

mÞfo

4mImþ3kI

ofmþ3kImfo

ð20Þ

bIIhom ¼ bI

mþð3kI

mþ4mImÞðb

Io�bI

mÞfo

4mImþ3kI

ofmþ3kImfo

ð21Þ

1

NIIhom

¼fo

NIo

þ1�fo

NIm

þ3ðbI

m�bIoÞ

2ð1�foÞfo

3foðkIm�kI

oÞþ3kIoþ4mI

m

ð22Þ

Effective Biot modulus MIIhom writes

1

MIIhom

¼1

NIIhom

þfp

k‘ð23Þ

where k‘ is the bulk modulus of liquid water. The effective bulkmodulus of a sphere assemblage of a two-phase sphere compo-sites (relation 20) is well known and may be found in [1,27].Relation (20) is consistent with relation (10) given in [3]

kIIhom ¼ kI

mþfo

1=ðkIo�kI

mÞþ3ð1�foÞ=ð3kImþ4mI

mÞð24Þ

Relation (21) is consistent with relation (18) given in [15].Analogously, relation (22) for solid Biot modulus NII

hom is consis-tent with relation (3.29) given in [33] (where 1=N is replaced byce, specific heat at zero deformation and b is replaced by latentheat s¼ 3ak). The effective shear modulus of the sphere assem-blage is given in many references including [1,27] and it will notbe recalled in this paper, mainly centred on poroelastic couplingsand hydrostatic loadings (relations 4.12–4.16 of the book [27]have been used to calculate the effective shear coefficient mII

hom).

3. Three phase composite spheres and four-phase model

In this section, the previous model, coated spheres or two-phase composite spheres assembled in the generalized self con-sistent three-phase model is generalized by the introduction of aninterphase between oolite and matrix (Fig. 4). The analysis is thenfocused on the interfacial transition zone (ITZ) previously cited inintroduction and extensively studied in literature. The assump-tion of perfect interface for (ITZ) will be followed in this paper.The new model is the Hashin CSA model of three-phase compositespheres (doubly coated spheres), it is referred as the four-phasemodel (see [3]) or generalized self consistent model (GSCM). As thebasic equations have been given for the three-phase model, onlythe new supplementary added terms will be detailed in whatfollows.

3.1. Volume fractions and constituents

At the macroscopic scale, one considers a three-phase compo-site material composed of three isotropic poroelastic media:oolites (o), interphase (i) surrounding oolite and matrix (m)

O¼OoþOiþOm

fo ¼Oo

O, fi ¼

Oi

O, fm ¼

Om

O, foþ fiþ fm ¼ 1 ð25Þ

At the mesoscopic scale the porosity is homogenized and the totalporosity is decomposed into three parts, oolite porosity Oop,interphase porosity Oip and matrix porosity Omp

Op ¼OopþOipþOmp

fop ¼Oop

O, fip ¼

Oip

O, fmp ¼

Omp

O, fp ¼

Op

O¼ fopþ fipþ fmp ð26Þ

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921914

At the first step one defines the oolite, interphase and matrixrelative porosities as (j¼o, i, m)

f Ijp ¼

Ojp

Oj¼

fjp

fjð27Þ

3.2. Two step homogenization model for the four-phase model

The first homogenization step, the transition between micro-scopic scale to mesoscopic scale, is identical to the one presentedin previous case of the three step model. The only difference is theadding of a new layer, the interphase i. Relations presented inSection 2.2 applies for (j¼o, i , m). In the second transition, fromthe mesoscopic scale to the macroscopic scale, the microstructureof the medium is described by a composite sphere assemblage ofthree-phase composite spheres (doubly coated spheres). As pre-viously mentioned it corresponds to the four-phase model intro-duced by [2,64,3] and used by many authors. It may be noticedthat the assembled materials (oolites, interphase and matrix) areporoelastic materials. The oolite porosity, the interphase porosityand the matrix porosity are also supposed connected but parti-cular situations of pore disconnection could be easily considered.As described in [3,15] compact expressions for the bulk modulusmay be obtained by using relation (24). The following procedureis a particular case of the general N-phase homogenizationprocedure introduced in [29] (phases o–i–m correspond to phases1–2–3 in N-phase model). The two-phase composite sphere o–i isfirst homogenized (note that it is important to consider therelative volume fractions of constituents o-i inside the two-phasecomposite sphere o–i)

kIIoi ¼ kI

iþfo=ðfoþ fiÞ

1=ðkIo�kI

iÞþ3ð1�fo=ðfoþ fiÞÞ=ð3kIiþ4mI

iÞð28Þ

and secondly assembled with the matrix m to provide theeffective bulk modulus of the four-phase model

kIIhom ¼ kI

mþfoþ fi

1=ðkIIoi�kI

mÞþ3ð1�fo�fiÞ=ð3kImþ4mI

mÞð29Þ

Results for bulk modulus of composite spheres (28)–(29) are wellknown since a long time, see among others [1,27,2,18,29] for thegeneralization to N phases. As presented in [15], compact expres-sions of Biot coefficient may be deduced by successive homo-genizations of the multicoated sphere. Thermoelastic resultsobtained in the general case of N-phase model (see [29,32]) couldhave been directly used for bulk modulus, Biot coefficient andsolid Biot modulus at the second homogenization step. By usingrelation (21) established for the two-phase composite poroelasticsphere with the relative volume fraction of constituents o–i, thehomogenized Biot coefficient of the two-phase composite sphereo–i (oolite–interphase, 0rrrB) writes

bIIoi ¼ bI

iþð3kI

iþ4mIiÞðb

Io�bI

iÞfo=ðfoþ fiÞ

4mIiþ3kI

oð1�fo=ðfoþ fiÞÞþ3kIi fo=ðfoþ fiÞ

ð30Þ

and the homogenized Biot coefficient of the three-phase compo-site sphere is obtained by assembling the third phase m and byusing relations (30)–(28) containing the pre-homogenized Biotcoefficient bII

oi and bulk modulus kIIoi of the coated sphere o–i. The

compact expression of Biot coefficient may be written as

bIIhom ¼ bI

mþð3kI

mþ4mImÞðb

IIoi�bI

mÞðfoþ fiÞ

4mImþ3kII

oið1�fo�fiÞþ3kImðfoþ fiÞ

ð31Þ

A similar reasoning provides leads to a compact expression ofsolid Biot modulus

1

NIIoi

¼fo=ðfoþ fiÞ

NIo

þ1�fo=ðfoþ fiÞ

NIi

þ3ðbI

i�bIoÞ

2ð1�fo=ðfoþ fiÞÞfo=ðfoþ fiÞ

3foðkIi�kI

oÞ=ðfoþ fiÞþ3kIoþ4mI

i

ð32Þ

and then

1

NIIhom

¼foþ fi

NIIoi

þ1�fo�fi

NIm

þ3ðbI

m�bIIoiÞ

2ð1�fo�fiÞðfoþ fiÞ

3ðfoþ fiÞðkIm�kII

oiÞþ3kIIoiþ4mI

m

ð33Þ

Effective Biot modulus MIIhom of the four-phase model is obtained

thanks to relations (33) and (23). The shear coefficient of the four-phase model mII

hom has been independently calculated by using themethod presented in [3] and the N-phase method presented in[29] and the two methods (both computed with Mathematica)perfectly agree.

4. Numerical results and discussion

4.1. Microstructure of a porous oolitic limestone rock

Microstructure of oolitic limestone rocks is described in[65,66] and it is illustrated in Figs. 5 and 6. The geologicalformation of most oolitic limestone rocks can be schematicallysummarized in three steps [67]: formation of concentric oolites orooids constituted from micro-crystallized calcite (micrite), sedi-mentation of oolitic sand and crystallization of a palissadic calciteore around the oolites; this sedimentation has also a marineorigin, and cementation of the oolites by precipitation of crystal-line calcite of sparite type. SEM observations are presented inFig. 6 and low magnification images show oolites cemented bydifferent types of calcites: large syntaxial crystals of severalhundreds of micrometers, small equant calcites with an averagesize around 10 mm. Those two types of calcite crystals, equantcalcite and syntaxial calcite will be homogenized with porosityand isotropized to constitute the matrix material (m). Oolites aremade of concentric layers of microcalcite (see 6). As shown inFig. 6, oolites are complex multicoated porous spheres withvariable porosities: oolite porosity increase from the centre(nucleus) to the outer boundary (see also [68,67]). The outerlayer of the oolite, referred as macroporosity, is a highly poroustransition zone between inner oolite and surrounding matrix. Thisouter layer will be identified, in the model as the interfacialtransition zone (ITZ), or interface I.

4.2. Properties of constituents and volume fractions

Oolitic porous limestone rocks studied in this paper are nearlymono mineral, composed of 98% of calcite. Pure calcite is ananisotropic material (see [69]). As calcite grains composing bothoolites and matrix are randomly oriented, the elastic behaviour ofcalcite is supposed to be isotropic (see [9]) and correspondingisotropic properties are calculated by using the isotropisationmethod introduced by [70]

Cisca ¼ 3kis

ca Jþ2misca K

kisca ¼

Cca :: J

3� 76:0 GPa, mis

ca ¼Cca :: K

10� 36:8 GPa ð34Þ

The pure calcite is used as a reference value for the solids grainsconstituting oolites (ko, mo), interphase (ki, mi) and matrix(km, mmÞ:

kj ¼ wjkisca, mj ¼ wjmis

ca, j¼ o,i,m ð35Þ

At the present stage, coefficients wj are unknown and needs to becharacterized for the three constitutive materials (j¼o, i, m).Elastic properties of pure calcite certainly strongly overestimateelastic properties of oolite and matrix but we will consider equal

Fig. 6. Low magnification SEM (scanning electron microscopy) images of oolitic Lavoux limestone (see [78]).

Fig. 5. Microstructure of oolitic limestone rock (see [78]): Petrographic observation on polished thin section of Lavoux limestone by optical microscopy. Syn. cal.: syntaxial

calcite, Equant cal.: equant calcite, Ech.: test of echinoderm, Dis.: dissolution.

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921 915

coefficients

wo ¼ wm ¼ 1 ð36Þ

Coefficients wm and wo need to be experimentally justified, thanksto tests such as micro or nano-indentation for example. Thedevelopment of these test, for limestone rocks is actually underprogress and it will be taken into account in further works. Itmust be emphasized that the relative hardness between ooliteand matrix is an open question because recently [67] supposed,on the basis of indentation tests on limestone rocks, that hardnessof oolite is smaller than the hardness of the matrix (it wouldimply woowm in the present model). The presented model takeinto account the volumetric effect of the porosities f I

j (f Ioo f I

m inmost limestone rocks), and also, potentially, the microstructureby choosing different homogenization methods for the differentphases at the first homogenization step. The coefficient wi whichcharacterizes the degradation of elastic properties in ITZ (wio1)will be estimated from experimental data on various limestonerocks (see [71,72]).

The connectivity between oolite porosity and matrix porosityis also an open question and its response depends on the precisetype of limestone. In [67], authors supposed that oolite porosityand matrix porosity are occluded and the connected porosity issupposed to be only located around the central oolite core, withshould correspond to ITZ in the proposed model. In the presentmodel it would imply fop-0, fmp-0, fip-fp, occluded porositybeing homogenized at an initial step (step 0, which should need tobe performed before step I), to provide the solid constituents in thesense of a poroelastic theory: ko, mo, and km, mm would be obtainedthanks to relations presented in Section 2.2, replacing relativeporosities f I

jp by corresponding relative occluded porosities, forboth oolites and matrix. A sensitive study is performed on volumefraction of oolites fo and ITZ fi, and on relative fractions of

porosities associated to phases o, i, m

fop ¼ Zofp, fip ¼ Zifp, fmp ¼ Zmfp ð37Þ

fo and fi being fixed, one deduces the volume fraction of matrix

fm ¼ 1�fo�fi, Zm ¼ 1�Zo�Zi ð38Þ

On the basis of comparison between macroscopic poroelastic dataobtained by [72] and model, the volume fraction of ITZ has beenchosen linearly proportional to the porosity

fi ¼ gifp ð39Þ

The relative porosity f Iip of ITZ has been chosen higher than

relative porosities of oolites (f Iop) and matrix f I

mp (f IipZ f I

op andf IipZ f I

mp, 8fp). This assumption is performed without precise dataon the microstructure of ITZ. Microstructural observations showthat a peak of porosity is concentrated near the outer boundary ofoolites. In the presented four-phase model, it corresponds tohighly porous ITZ. By considering relations ((37)–(39)), it maybe deduced that the relative porosity of ITZ has been assumedconstant

f Ii ¼

fip

fi¼Zi

gi

, Ziogi ð40Þ

It should be emphasized that microstructures of limestone rocksmay differ very strongly depending on the type of limestone andthe particular geological history. As a consequence volume frac-tion parameters fo, fi, fm, and porosity parameters fp, Zj may vary ina large extent. A common reasonable assumption seems to be, forthe studied limestones

ZooZiþZm

f Iopr f I

ip and f Impr f I

ip, 8fp ð41Þ

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921916

Representative effective linear poroelastic experimental data ofvarious oolitic limestones are given in Table 1 (see [73,74],Lion04thesis, [75,76]). On the basis of experimental data andmicrostructural observations, one proposes a reference data setfor limestone rock similar to Lavoux limestone

fo ¼ 0:7, 0r fpr0:3

gi ¼ 0:9375, Zo ¼ 0:4, Zi ¼ 0:045, f Ii ¼ 0:6

0.05 0.10 0.15 0.20 0.25 0.30

fp

20

40

60

kG

Pa

Four Phase step II DSC step I

0.05 0.10 0.15 0.20 0.25 0.30

fp

0.2

0.4

0.6

0.8

b

Four Phase step II DSC step I

0.05 0.10 0.15

fp

0.05

0.10

0.15

1M

Four Phase step II

Fig. 7. Effective poroelastic properties:

Table 1Linear poroelastic properties of porous oolitic limestones (experimental data).

Limestone fp khom (GPa) khoms (GPa) bhom

Villeperdue 0.064 36 61 0.41

Tonnerre 0.13 19.3 41.4 0.53

Salem 0.13 21.78 72.6 0.7

Lavoux 0.219 13.8 54.9 0.80

Chauvigny 0.174 16.3 52.6 0.69

Anstrude 0.199 15.5 63.25 0.82

wo ¼ wm ¼ 1, wi ¼ 1=5 ð42Þ

4.3. Numerical results and discussion

The differential self consistent (DSC) estimate, or equivalentiterative self consistent ISC (with NISC¼200) scheme, has beenchosen for the first homogenization step, for the three constitu-tive materials o, i, m. Effective poroelastic properties obtainedwith reference data set are presented in Fig. 7. Results arepromising in the sense that global experimental trends arecorrectly captured by the model. A sensitivity analysis to gi andwi parameters, which characterize ITZ, is respectively presented inFigs. 8 and 9. As expected, gi (with related parameter Zi) and wi arekey parameters which controls the effective macroscopic para-meters. Without ITZ, it is clear that the assumption of pure calciteelastic coefficients for solids grains constituting oolites andmatrix (wo ¼ wm ¼ 1) would strongly overestimate effective elasticparameters. In the present proposed model, and, at the presentstage, the correction is performed in the ITZ only. The poroelastic

0.05 0.10 0.15 0.20 0.25 0.30

fp

5

10

15

20

25

30

35G

Pa�

Four Phase step II DSC step I

0.05 0.10 0.15 0.20 0.25 0.30

fp

0.002

0.004

0.006

0.008

0.010

1N

Four Phase step II DSC step I

0.20 0.25 0.30

DSC step I

four Phase and DSC, reference case.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP fP

0

10

20

30

40

50

60

70

kGPa

DSC �i 0.6 i i 1 50

expexp� i 0.375� i 0.15� i 0.075� i 0.015

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20

25

30

35

GPa

�

DSC i 0.6 � i i 1 50

expexp� i 0.375� i 0.15� i 0.075� i 0.015

0.00 0.05 0.10 0.15 0.20 0.25 0.30fPfP

0.0

0.2

0.4

0.6

0.8

1.0

b

DSC i 0.6 � i i 1 50

exp� i 0.375� i 0.15� i 0.075� i 0.015

0.00 0.05 0.10 0.15 0.20 0.25 0.300.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1N

GPa

1

DSC i 0.6 � i i 1 50

Fig. 8. Effective poroelastic properties: parameter gi.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0

10

20

30

40

50

60

70

kGPa

DSC i 0.6 i i 0.075

expexpi 1 2500i 1 500i 1 50i 1 5

0.00 0.05 0.10 0.15 0.20 0.25 0.30fP

0

5

10

15

20

25

30

35

GPa

�

DSC i 0.6 i i 0.075

expexpi 1 2500i 1 500i 1 50i 1 5

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0.0

0.2

0.4

0.6

0.8

1.0

b

DSC i 0.6 i i 0.075

expi 1 2500i 1 500i 1 50i 1 5

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1N

GPa

1

DSC i 0.6 i i 0.075

Fig. 9. Effective poroelastic properties: parameter wi .

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921 917

parameters obtained at the mesoscopic scale (step I), arecertainly overestimated for oolite and matrix and experimentalcharacterization at the mesoscopic scale will be necessary to

quantify, more precisely, difference of poroelastic propertiesbetween oolites and matrix. Compared to parameters gi and wi,the effect of the volume fraction of oolite fo seems to be low or

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP fP

0

10

20

30

40

50

60

70k

GP

a

DSC i 0.6 i i 0.075 i 1 50

expexpfo 0.7fo 0.6fo 0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20

25

30

35

GP

a�

DSC i 0.6 i i 0.075 i 1 50

expexpfo 0.7fo 0.6fo 0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP fP

0.0

0.2

0.4

0.6

0.8

1.0

b

DSC i 0.6 i i 0.075 i 1 50

exp

fo 0.7

fo 0.6

fo 0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.300.000

0.002

0.004

0.006

0.008

0.010

0.012

1N

GP

a1

DSC i 0.6 i i 0.075 i 1 50

exp

fo 0.7

fo 0.6

fo 0.5

Fig. 10. Effective poroelastic properties: parameter fo.

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921918

moderate (se Fig. 10) but it is mainly due to the fact that it isnecessary to consider small variations of fo around the referencedata set (42) in order to avoid inconsistency in data. As anexample, volume fraction of oolite fo and relative oolite porosityfop (and then parameter Zo) are certainly correlated in the generalcase. Maintaining Zo ¼ 0:4, with fo¼0.8 and other referenceparameters (42) would conduce to a matrix relative porosityf Ipm � 0:94 which is inconsistent with assumption (41).

A comparison of the homogenization schemes (step I), SC, ISC

(N¼3), DCS, and MT, is presented in (11). As the self consistentscheme is considered, this comparison is performed with amodified data set: Zi ¼ 0:48gi, fo¼0.6 and others parameters beingequal to the corresponding in reference data set ((42), withgi ¼ 0:075). Relative porosities of ITZ f I

ip ¼ 0:48 and matrixf Impr0:45 (maximal value for fp¼0.3) are then lower than the

porosity threshold of the SC scheme (fp¼1/2). Due to the veryhigh relative porosity of ITZ, f I

ip ¼ 0:48, very close to the porositythreshold of the SC scheme, SC scheme is significantly differentfrom the three other investigated schemes (see Fig. 11). It may benoticed that maximal value of relative matrix porosity f I

mp is alsovery close to the porosity threshold so the particular casepresented in Fig. 11 corresponds to a limit of applicability of SC

scheme. The effective drained kIIhom bulk modulus is presented in

Fig. 12 in the particular case of the reference data set, and for ISC

and MT homogenization schemes (SC scheme for step I is out ofrange for ITZ).

In the particular case of Lavoux limestone, a sensitive studyhas been performed on parameters gi and wi, and comparisonbetween model and experimental data is presented in Table 2. Itmay be noticed that different set of parameters gi2wi correctly fitthe data. An increase in ITZ volume fraction gi is, roughly,compensated by an increase of the degrading coefficient wi. Inthe particular case presented in Table 2, it seems that correctestimates are obtained for a similar values of ratio gi=wi (equals to

3.75–4.12–4.69 in the three cases). Estimates are more accuratefor drained bulk modulus, Biot coefficient and shear coefficientthan for Biot modulus (slightly underestimated by the model). Itis very important to have the most complete set of macroscopicdata to assess the relevancy of a micromechanical model. Evolu-tions of drained bulk modulus are presented in Figs. 12–14. It maybe noticed that effective drained bulk modulus is not significantlysensitive to parameters gi, wi, with the particular investigatedvalues of ratio gi=wi.

It may be emphasized that the aim of the numerical study isnot to fit very accurately experimental data but to show thepotentiality of the micromechanical model to recover generalexperimental trends. A more accurate modelling needs moreprecise microstructural informations such as properties ofconstituents at the mesoscopic scale for instance. For a givenlimestone rock, a precise microstrucural characterization com-bined with micro and nano intendation tests on various consti-tuents is necessary to improve the micromechanical approach. Inparticular properties of oolite and matrix need to be precised(parameters wo and wm) because they are certainly stronglyoverestimated in the present approach.

5. Closing remarks

In this study, a two-step homogenization model has been devel-oped in the framework of the Hashin composite sphere assemblage

(CSA) model and it is proposed to estimate linear effective poroelasticproperties of composite rocks whose microstructure is characterizedby porous grains (such as oolites) embedded in a porous matrix. Aporous interfacial transition zone ITZ coats the grains and insuretransition between grains and matrix. The pores, divided intosupposed connected oolite-pores, ITZ-pores and matrix pores, areintroduced and homogenized in the first homogenization step. At the

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0

10

20

30

40

50

60

70

kG

Pa

fo 0.6 i 0.48 i i 0.075 i 1 50

MT

DSCISC 3SC

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0

5

10

15

20

25

30

35

kG

Pa

fo 0.6 i 0.48 i i 0.075 i 1 50

MT

DSCISC 3SC

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0.0

0.2

0.4

0.6

0.8

1.0

b

fo 0.6 i 0.48 i i 0.075 i 1 50

MT

DSCISC 3SC

0.00 0.05 0.10 0.15 0.20 0.25 0.30

fP

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

1N

GP

a1

fo 0.6 i 0.48 i i 0.075 i 1 50

MTDSCISC 3SC

Fig. 11. Effective poroelastic properties (step II): comparisons of homogenization schemes (step I): SC, ISC, DSC, MT.

Table 2Effective poroelastic properties of Lavoux limestone: fp ¼0.22, fo¼0.7.

ISC(200) ISC(200) ISC(200) Experiment

kIIhom (GPa) 13.7 13.5 13.5 13.8

mIIhom (GPa) 9.6 9.5 9.5 9.0

MIIhom (GPa) 8.39 8.67 8.73 9.69

bIIhom

0.76 0.75 0.75 0.80

gi 0.075 0.4125 0.9375

Zi=gi 0.6 0.6 0.6

wi 1/50 1/10 1/5

0.05 0.10 0.15 0.20 0.25 0.30

fP

0

10

20

30

40

kG

Pa

Fully drained Step II

MT HSISC 200ISC 10ISC 5ISC 3

Fig. 12. Effective poroelastic properties (step II): drained kIIhom bulk modulus,

ISC, MT.

0.05 0.10 0.15 0.20 0.25 0.30

fP

0

10

20

30

40

kG

Pa

i 0.4125� i 1 10 Fully Drained Step II

MT HSISC 200ISC 10ISC 5ISC 3

Fig. 13. Effective poroelastic properties (step II): drained kIIhom bulk modulus, ISC,

MT, gi ¼ 0:4125, wi ¼ 1=10.

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921 919

second homogenization step, the three different porous constituents,oolite, ITZ and matrix are assembled in a CSA model referred as a four-

phase model [3]. For its simplicity, the three-phase model whichneglects ITZ has been first recalled but the analysis is centred onthe effect of ITZ which plays a crucial role in such materials.

The novelty of the analysis mainly concerns the development ofsuch micromechanical model, including a two-step analysis, in thecontext of linear poroelasticity and porous rocks such as limestones.From a theoretical point of view it may be noticed that all necessaryresults for linear poroelasticity are available in the huge body ofliterature devoted to CSA models in thermoelasticity. Mathematicalanalogy between linear poroelasticity and linear thermoelasticity

0.05 0.10 0.15 0.20 0.25 0.30

fP

0

10

20

30

40

kG

Pa

� i 0.9375 i 1 5 Fully Drained Step II

MT HS

ISC 200

ISC 10

ISC 5

ISC 3

Fig. 14. Effective poroelastic properties (step II): drained kIIhom bulk modulus, ISC,

MT, gi ¼ 0:9375, wi ¼ 1=5.

N.B. Nguyen et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 909–921920

has been used in this paper to obtain effective linear poroelasticcoefficients.

The two-step homogenization model has been applied, in thispaper, to estimate effective linear poroelastic properties of alimestone rock and the analysis is focused on the impact of theITZ (interfacial transition zone). Linear isotropic poroelastic proper-ties are estimated by comparisons between theoretical effectiveresults and experimentally characterized properties at the labora-tory scale (typically on samples with sizes comprised between1 and 10 cm). Some presented results seems promising:

�

the impact of porosity, oolite porosity and matrix porosity, onporoelastic properties is correctly captured by the model, � comparison model—experimental data underlines the impor-

tance of ITZ properties,

� for the considered material, the differential self consistent

scheme (DSC) and similar iterative self consistent scheme (ISC)for the first step, seem more relevant than the two otherhomogenization schemes (Mori Tanaka and self consistent(SC)). As the oolite and matrix are also constituted by anassemblage of grains and pores, the Mori Tanaka associated tomatrix-inclusion microstructure is not relevant, but the SC

scheme associated to perfectly disordonned microstructureoverestimates the decrease of elastic properties with porosity.MT and SC are also limited in porosity range.

Homogenization models used in this paper, such as three-phase andfour-phase CSA models do not directly account to packing and sizeeffects. Recent approach referred as ‘morphologically representative

pattern-based approach’ (MRP) in [77] seems promising because itincludes, in the context of CSA models, packing and size effects, Inmaterials such as the studied limestone rock, the size effect iscertainly important. The different behaviours between oolites andsurrounding matrix is correlated to the size of the respectiveconstituting particles and associated pores (large size of matrixparticles compared to oolite solid particle size). Concerning applica-tion, the respective properties of constituents, oolites, matrix (andeventually interface) needs to be experimentally characterized. Microand nano-indentation tests are actually under development. Suchkind of tests will be extremely important to evaluate the relevancy ofthe micromechanical approach because it will provide a secondexperimental reference at the step one of the model, referred as themesoscopic scale.

Nonlinear effects are also under study but it is beyond the scopeof this paper centred on coupled linear poroelastic behaviour.

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