A micro–macro approach of permeability evolution in rocks excavation damaged zones

Computers and Geotechnics 49 (2013) 245–252

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A micro–macro approach of permeability evolution in rocks excavation damaged zones S. Levasseur a,⇑, F. Collin a, R. Charlier a, D. Kondo b a b

Université de Liége (ULg), Chemin des chevreuils, 1 4000 Liége 1, Belgium Institut d’Alembert, UMR 7190 CNRS, Université Pierre et Marie Curie (UPMC), Paris, France

a r t i c l e

i n f o

Article history: Received 31 January 2012 Received in revised form 11 October 2012 Accepted 4 December 2012 Available online 16 January 2013 Keywords: Anisotropic damage Induced permeability Initial stresses Homogenization Micromechanics Geomaterials

a b s t r a c t Excavation damaged zone, with significant irreversible deformations and nonnegligible changes in flow and transport properties generally occurs in indurated clay around underground structures. The stress perturbation around the excavation could lead to a significant increase of the permeability physically due to diffuse and/or localized microcracks growth in the material. In the present study, we investigate microcracks-induced damage processes together with the subsequent modification in permeability. The proposed approach is based on a homogenization-based upper bound extended to the context of microcracked media in presence of initial stress. Application of this approach is done on a borehole excavation problem related to the Selfrac in situ experiments on Opalinus Clay. Although, the model fails to quantitatively account for the in situ permeability change (which may also originated from existing macrofractures), its prediction shows a significant evolution of the material permeability around the borehole. This is in qualitative agreement with available data. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Excavation of underground structures in rock masses usually generates an Excavation Damaged Zone (EDZ) in which geotechnical and hydro-geological properties are deteriorated. This degradation process, which results from microcracks growth, is generally accompanied by significant changes in flow and permeability properties. In fact, damage influences permeability that changes the original pore pressure distribution, which affects in turn mechanical response of materials via poromechanical coupling. In the context of radioactive waste disposal, as potential host rocks must be characterized by a very low hydraulic conductivity, occurrence of EDZs plays a nonnegligible role and has various implications for the long term performance of underground repository. Consequently, there is still a need to deeply investigate the link between mechanical deformation process and the variations of permeability properties. The general hydro-mechanical behavior of host rocks like overconsolidated clay mainly depends on the stress history and the elastic and strength properties. It is also affected by load-induced microcracks which generate not only a reduction of elastic stiffness but also evolutions of permeability [31]. Modeling of such complex behavior is generally performed by using purely macroscopic approaches. In these models, evolution of the permeability within the EDZ is often associated with discontinuities, represented by ⇑ Corresponding author. E-mail address: [email protected] (S. Levasseur). 0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2012.12.001

strain localization or damage. When rock is damaged, the creation of a crack network is assumed to constitute preferential flow paths, depending of crack interactions. As a consequence, the rock hydraulic conductivity increases and become heterogeneous and anisotropic with crack aperture [2,22,27,33,35]. On one hand, the issue can be to propose models providing relationships between rock mass permeability and crack aperture evolution through stress or strain state during loading (see for instance [8,18,21,26]). On the other hand, it can be to directly relating permeability evolution with induced damage as proposed by [14,32,34]. These models are mainly based on a semi-empirical and engineering-oriented approaches and are expressed in the macroscopic framework to be easily implemented for engineering application. Unfortunately, permeability evolution is not well related to the crack growth in multiple orientations. As recent developments in homogenization of microcracked materials provide more physical models for the description of damage induced anisotropy, it becomes now possible to assess the performance of the appropriate upscaling methods in some geomechanics problems. In particular, due to the striking link between damage and permeability [7], damage-induced anisotropy as well as cracks closure effects may be carefully taken into account [12,13,15,19,23,37]. This paper proposes to answer the question of damage-induced permeability micromechanical definition through an application to the indurated Opalinus Clay of Mont Terri Underground Research Laboratory in Switzerland [3].

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Nomenclature

X Xs Xr

ur ks

ls dr

wr cr ar Keq

W

r

Fd

fr r Rðd Þ

z x nr q Q gradp rP I r(z)

R

r0

representative elementary volume (rev) located at macroscopic point. Its boundary is @ X domain occupied by the solid matrix domain occupied by the inclusion or of the rth family of cracks volume fraction of the rth family of cracks elastic bulk modulus of the isotropic solid phase elastic shear modulus of the isotropic solid phase crack density parameter of the rth cracks family (indice 0 refers to initial state); d denotes the set of these parameters for all cracks families crack aspect ratio of the rth cracks family (indice 0 refers to initial state) crack opening of the rth cracks family (indice 0 refers to initial state); dcr denotes the crack opening increment crack radius of the rth cracks family (indice 0 refers to initial state) Von-Mises-like equivalent permeability potential of the microcracked material energy release rate; thermodynamic force associated to the rth cracks family damage yield function of the rth cracks family resistance to damage of the rth cracks family; it is chosen as an affine function defined by two constants h0 (initial damage threshold) and g (damage hardening) position vector at the microscopic scale position vector at the macroscopic scale normal vector to the rth cracks family local rate of flow macroscopic rate of flow local gradient of pressure macroscopic gradient of pressure second order unit tensor microscopic Cauchy stress tensor at point z macroscopic Cauchy stress tensor at point x initial uniform Cauchy stress tensor in the solid

For this, we briefly present in Section 2 the micromechanical modeling of elastic materials weakened by a system of arbitrarily oriented microcracks. This model accounts for the spatial distribution of microcracks. Moreover, unilateral effects due to microcracks closure, as well as initial stresses (which are crucial in geotechnical applications) are considered. In Section 3, an estimate of the overall permeability of microcracked materials is provided by taking advantage of recent micromechanical developments. The identification of model parameters from data available for the Opalinus Clay is presented in Section 4. Finally, in order to asses the validity of the proposed procedure, we present an application to a borehole excavation process related to the in situ experiment Selfrac. 2. Micro–Macro modeling of microcracks-induced damage with account of initial stresses Consider a representative elementary volume (rev, X – see Fig. 1) constituted of a solid matrix s (occupying a domain Xs) and an arbitrary system of inhomogeneous inclusions or microcracks; each inclusion/microcrack family is denoted r and occupies a domain Xr. The matrix and the inclusions/microcracks are assumed elastic whereas the initial stress field r0 is considered uniform. The rev is subjected to uniform strain boundary conditions:

@X : n ¼ E  z

ð1Þ

rp e(z)

heterogeneous prestress tensor field microscopic strain tensor at point z E macroscopic strain tensor at point x Ks macroscopic permeability tensor at point x in solid matrix Kr macroscopic permeability tensor at point x in rth crack (K rt tangential component, K rn normal component) Khom homogenized permeability tensor K hom PCW estimate of homogenized permeability tensor pcw D macroscopic second order tensor of the approximate anisotropic damage model C macroscopic second order tensor of the approximate crack opening P0 second order Hill tensor second order Hill-type tensor describing the spatial disPd tribution of cracks ar gradient of pressure localization tensor of phase r aresh dilute estimate of gradient of pressure localization tensor of phase r arpcw PCW estimate of gradient of pressure localization tensor of phase r Ar strain localization tensor of phase r Arpcw PCW estimate of the strain localization tensor of phase r I fourth order symmetric unit tensor J ¼ 13 I  I and K ¼ I  J fourth order projectors unit tensor elastic stiffness tensor of the solid Cs Cr elastic stiffness tensor of inclusion or of the rth family of cracks Chom homogenized elastic stiffness tensor Chom PCW estimate of the homogenized elastic stiffness tenpcw sor Chom homogenized elastic stiffness tensor t Eshelby tensor of rth family of cracks Sr Sd Eshelby-type tensor describing the spatial distribution of cracks

in which the quantity z denotes the vector position, n the displacement vector, and E the macroscopic strain tensor. Following [19], the homogenization problem with initial stresses of a microcracked medium is recalled and extended by considering the Ponte-Castaneda and Willis (PCW) bound [30] to take into account microcracks interaction and effects of spatial distribution of cracks. To this end, it is convenient to formulate the local constitutive equations in the heterogeneous medium with prestress as:

ðz 2 XÞ rðzÞ ¼ CðzÞ : eðzÞ þ rp ðzÞ

ð2Þ

where r(z) denotes the local stress tensor, CðzÞ represents the heterogeneous stiffness tensor and rp(z) corresponds to a prestress tensor such as:

 CðzÞ ¼

Cs

in ðXs Þ

Cr

in ðXr Þ

rp ðzÞ ¼



r0 in ðXs Þ 0

in ðXr Þ

ð3Þ

in which Xs and Xr represent the domain occupied by the matrix and by cracks, respectively. A direct application of the standard Levin theorem (see for instance [20] or [17]) delivers the thermodynamic potential which reads (see [19]):

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Fig. 1. Rev of microcracked material and details of a penny-shaped crack for which the crack aspect ratio is wr = cr/ar (according to [36]).

!     N X 1 1 1 s r r : Chom : E W ¼ E : C þ r0 : I  u A : E ¼ E þ r0 : Cs 2 2 r¼1 ð4Þ r

where u represents the volume fraction of the r family of microcracks, related to the corresponding microcracks density dr by: ur ¼ 43p wr dr , in which wr is the microcracks aspect ratio (following damage definition of [6]). For a family of penny-shaped cracks, microcracks are approximated by a flat ellipsoid, characterized by r its unit normal vector (orientation) nr and the aspect ratio wr ¼ ac r , r r with a the radius of the circular crack and c the half-length of the small axis (see Fig. 1). According to the PCW bound for open microcracks ðCr ¼ 0Þ,1 the homogenized stiffness tensor can be put in the form: s Chom pcw ¼ C :

I

N 4p X r wr d Arpcw 3 r¼1

!

"

N 4p X j wj d ðI  Sj þ Sd Þ : ðI  Sj Þ1 3 j¼1

#1

ð6Þ It must be recalled that this localization tensor accounts for the Eshelby tensor Sr of the rth crack family and for the spatial distributions of microcracks through the corresponding Eshelby tensor Sd . Expression (5) reads then:

2

Chom pcw

N 4p X r ¼ C : 4I  d Tr : 3 r¼1 s

!1 3 N 4p X j j j 5 Iþ d T : ðI  S þ Sd Þ 3 j¼1 ð7Þ

r

r

r

Fd ¼ 

  @W 1 @Chom s1 : E þ ¼  r : C 0 r r : E 2 @d @d

ð9Þ

One considers a damage yield function corresponding to each crack family in the form: r

r

r

r

f r ðFd ; d Þ ¼ Fd  Rðd Þ 6 0

ð10Þ

with

ð5Þ

The localization tensor Arpcw , which relates the microscopic strain tensor to the macroscopic one, E, reads:

Arpcw ¼ ðI  Sr Þ1 : I þ

in which ks and ls refer to the classical elastic parameter of material [13] The formulation of a microcracks-induced damage model requires a damage function and a damage evolution law. To this end, we take advantage of the thermodynamics approach classically applied to irreversible processes occurring in solids materials. r Then, after defining the damage energy release rate Fd :

r

r

Rðd Þ ¼ h0 ð1 þ gd Þ

ð11Þ

the local resistance to the damage propagation inspired from those reported in literature dealing with fracture mechanics (see Ref. [28]). This linear function of damage has been adopted following the proposition of Marigo [24] in the context of a purely macroscopic isotropic damage model. h0 corresponds to the initial damage threshold value, g represents damage hardening. Both have been assumed independent of crack family. By combining the first state law (together with Eq. (7)) with the damage evolution equation obtained by assuming normality rule for damage and consistency condition f_ r ¼ 0, one obtains the incremental form of the homogenized constitutive law (see Ref. [19]): : E_ with R_ ¼ Chom t

h i h i 1 1 hom hom N ðE þ r0 : Cs Þ : @C@dr  ðE þ r0 : Cs Þ : @C@dr X r   2 hom Chom ¼ Chom  H t 1 : @ Cr2 : E h0 g þ 12 E þ r0 : Cs r¼1 @d

ð12Þ

r 1

where T ¼ limwr !0 w ðI  S Þ . For sake of simplicity, a spherical spatial distribution of microcracks is physically assumed in the following. Consequently, Sd for a spherical distribution of microcracks is given by the isotropic tensor: s

3k Sd ¼ aJ þ bK with a ¼ s 3k þ 4ls

s

6ðk þ 2ls Þ and b ¼ s 5ð3k þ 4ls Þ ð8Þ

and

( r

H ¼

if f r < 0 or if f r ¼ 0 and f_ r < 0 1 if f r ¼ 0 and f_ r ¼ 0

0

ð13Þ

3. Coupling between microcracking and permeability 3.1. Permeability of microcracked materials

1 The computation of closed microcracks can be performed by following the approach already introduced by [29,13].

We follow here an homogenization approach of permeability introduced by Dormieux and Kondo [11] and recently ex-

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tended by Barthélémy [1] in the context of opened microcracks. The local equations of the heterogeneous permeability problem related to the existence of microcracks can be put in the form:

Eq. (19) can be recast in the form:

8 div q ¼ 0 ðXÞ > > > > < q ¼  Ks  grad p ðXs Þ l



1 in which aresh ¼ I þ Pr0 ðKr  Ks Þ represents the localization tensor in the dilute (Eshelby) approximation given by:

ð14Þ

r > > q ¼  Kl  grad p ðXr Þ > > : ð@ XÞ p¼r Px

Kr ¼ K rt ðI  nr  nr Þ þ K rn nr  nr

ð15Þ

For this heterogeneous Darcy problem, it is convenient to introduce an appropriate localization relation between the local gradient of pressure and the macroscopic one:

grad pðzÞ ¼ aðzÞ  rP

ð16Þ

It follows that the macroscopic relation:

Q ¼ hqiX ¼ 

Khom

ð17Þ

in which the macroscopic permeability, which presents analogy with the mechanical problem, is defined by:

Khom ¼ hK  aiX ¼ Ks þ

N  4p X r wr d Kr  Ks  ar 3 r¼1

ð18Þ

Note that the permeability tensor of the solid matrix is taken isotropic: Ks = KsI. For the tangential component K rt of the crack perme2

ability, assuming a Poiseuille flow, one obtains K rt ¼ cr =12 in which cr = arwr characterizes the crack opening, wr being the crack aspect ratio and ar the crack radius. Note also that the latter is  r 1=3 , in which ar0 is the initial linked to the damage by: ar ¼ ar0 ddr 0

r

crack radius and d0 is the initial damage value as proposed by [6]. Computation of the cracks opening is performed by considering the strain localization relation [10]:

_  nr Þ cr ¼ ar wr ¼ cr0 þ dcr ¼ cr0 ð1 þ nr  e_r  nr Þ ¼ cr0 ð1 þ nr  ðAr : EÞ with cr0 the initial cracks opening and dcr the increment of cracks opening. It remains now to determine the localization tensor ar, required for the computation of Khom. Adapting the methodology proposed by [11] and by [1] (both in the context of self consistent scheme), it follows that the PCW-type localization tensor reads:



arpcw ¼ I þ Pr0 ðKr  Ks Þ (

 Iþ

N X

h

h





i

)1

uj I þ Pj0  Pd  ðKj  Ks Þ  ajesh

ð22Þ

1 1   ðI  nr  nr Þ þ  r  nr  nr r 1 þ p4Kwsr K rt  K s 1 þ 22Kpw Kn  Ks s ð23Þ

This leads to: s Khom pcw ¼ K þ

N  4p X r wr d Kr  Ks  aresh 3 r¼1

N h    i 4p X j  Iþ wj d ajesh  I þ Pj0  Pd  Kj  Ks 3 j¼1

!1

ð24Þ 3.2. Numerical results and comparison with self-consistent scheme

 rP

l

N X j¼1

aresh ¼

in which, Ks represents the permeability tensor of the solid matrix and Kr that of microcracks; the latter involves normal and tangential components, that is

8z 2 X;

(

arpcw ¼ aresh  I þ

In order to asses the predictive capabilities of the present model, Eq. (24) together with 20, 21 and 23 is applied to the case of an uniform distribution of cracks orientation and a constant cracks aperture. In this case, the effective permeability tensor, Khom is isotropic: Khom = KhomI. For low permeable matrix (with a nonzero permeability), we have compared the PCW bound and the self-consistent scheme proposed by [11]. For the comparison purpose, an aspect ratio fixed at w = 103 and an initial permeability value of the matrix equal to Ks = 1020 m2 (this is of the order of the value expected for clay rocks) are chosen. Both PCW and self-consistent models (SC model) predict a significant variation of the overall permeability Khom with the damage level (see Fig. 2). It must be pointed out that for this configuration, in which the matrix permeability is chosen very low comparatively to the microcracks one, the predictions of the PCW s model suggest very small dependence with the ratio KK t . In contrast, the self consistent estimates predict significant differences between the permeability evolution with damage for the different ras s tios KK t . Moreover, for relatively high values of KK t , the SC model leads to effective permeabilities less than that given by the PCW model. s While, for relatively low values of KK t , the SC model provides values lower than the PCW predictions for relatively low damage and greater ones when damage is over a critical value about 0.4. It is also interesting to compare these effective permeabilities Khom to Kt (see Fig. 3). For this comparison for which Ks is still fixed

1 



i h



uj I þ Pj0  Pd  ðKj  Ks Þ  I  Pj0 Kj  Ks

i1

)1

j¼1

ð19Þ P0r

is the Hill tensor for the r-crack family in the context of the heterogeneous permeability. It is expressed as:

Pr0 ¼

nr  nr p þ wr ðI  3nr  nr Þ Ks 4K s

ð20Þ

Pd corresponds to the Hill-type tensor describing the spatial distribution of cracks; assuming a spherical distribution of cracks (see [13]), one has:

Pd ¼

1 I 3K s

ð21Þ

Fig. 2. Estimates of permeability evolution with damage for an initial value of matrix permeability Ks = 1020 m2 and constant aspect ratio w = 103: case of an uniform isotropic cracks orientation distribution.

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Fig. 3. Estimates of permeability evolution with damage for an initial value of matrix permeability Ks = 1020 m2 and constant aspect ratio w = 103: case of an uniform cracks orientation distribution.

Fig. 4. Results of calibration of the micro–macromodel on a Opalinus Clay triaxial compression test.

s

at 1020 m2, both models provide a dependence with KK t . At the difference of the SC model which exhibits similar behaviors for low s values of KK t with percolation-like effects, the PCW model predicts s different evolutions of permeability for different ratios KK t . For damage values under SC model percolation threshold value, both two models provide quite similar evolutions of permeability whatever s is KK t . But, when damage level is higher than the percolation threshs old, both models diverge. For very low values of KK t (from 106 to 5 10 in Fig. 3), the self consistent estimates provide permeability evolutions higher than that predicted by PCW model. On contrary, s for higher values of KK t (i.e. 104 in Fig. 3), the SC model predicts permeability evolutions lower than the one given by PCW. One can notice that in the context of radioactive waste disposal, intact permeability of host rocks is very small one can expect a s small KK t ratio. In this context, for small damage levels SC model and PCW would provide equivalent estimates of the effective permeability. Difference between these models could be observed only for damage values higher than percolation threshold. 4. Applications In this section, applications of the proposed model are in the context of radioactive waste disposal. As potential host rocks must be characterized by a very low hydraulic conductivity, occurrence of EDZ plays a nonnegligible role and has various implications for the long term performance of underground repository. Since EDZ take their origin from the load-induced microcracks and their growth process, the above micro–macro model appears to be particularly suitable for their analysis. In particular, we aim at linking the elastic stiffness reduction and the permeability increase to the state of induced microcracks. To this end, we perform such analysis in the case of Opalinus Clay, which is an overconsolidated claystone involved in Mont Terri Underground Research Laboratory (URL) in the Jura Mountains of the north-western Switzerland. The PCW micromechanical model is calibrated on a triaxial compression test and then we asses the validity of the proposed procedure by modeling a borehole excavation process inspired from the in situ Selfrac experiment (cf. [3]). 4.1. Micromechanical model calibration on triaxial test For the calibration of the model, a 15 MPa confining pressure triaxial compression test has been considered (see [16] for the description of this test). The values of elastic parameters, generally chosen for this material are: Es = 10 GPa and m = 0.24 (see [4,25]). Identification of the model is then reduced on the parameters characterizing the local resistance to damage propagation: h0 and g (cf.

Fig. 5. Results of calibration of the micro–macromodel on a Opalinus Clay triaxial compression test: corresponding permeability and damage evolutions.

Eq. (10)). Fig. 4 shows the verification of the PCW bound-based model with the identified values h0 = 100 J/m2 and g = 1.5  103. It shows that the model reproduced the softening regime observed on the experimental curve. The consequence in term of damage and permeability is presented in Fig. 5. Damage variable corresponds to the sum of damage variables on each r-family of microcracks:

d¼

N X 1 r¼1

N

r

d

ð25Þ

Permeability variable is expressed as a Von Mises-like equivalent permeability tensor, in 2D:

K eq ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2  2  2 u hom hom þ K hom þ K hom t K1  K2 1 2 2

ð26Þ

in which K hom and K hom are the eigenvalues of Khom permeability 1 2 tensor. It must be noticed that damage and permeability evolutions follow the same trends. For the effective permeability, damage induces a modification about one order of magnitude: an increase of effective permeability from 2.0  1020 m2 to 2.4  1019 m2). This observation, which is in agreement with experiments carried out by [9], can be explained by the fact that, although a damage growth is noted, the microcracks average opening does not significantly evolve (stay quite constant around the fixed initial opening c0 = 106 m).

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Fig. 6. Selfrac experiment design.

4.2. Borehole excavation 4.2.1. Studied geometrical model inspired from Selfrac experiment The Selfrac project aimed at understanding and quantifying the EDZ in clay in order to assess its impact on the performances of radioactive waste geological repositories [3,18]. The general concept of the experiment performed in Mont Terri URL, illustrated in Fig. 6, was to combine dilatometer tests and numerous hydraulic tests with multi-packer system. The later serves to evaluate the influence of excavation and bentonite swelling pressure (simulated by dilatometer loading) on the axial transmissivity of the Excavation Damaged Zone (EDZ). As this experiment has been very well instrumented, its general configuration is used to asses the ability of the proposed model to simulate borehole excavation. However as first step of modeling, a plane strain approach is adopted by considering a borehole cross section. Rocks behavior and initial stresses are assumed isotropic. The material parameter obtained during previous calibration are assumed; isotropic initial stresses are equal to r0 = 5.6  I MPa. Because of symmetries, only a quarter of the domain is modeled in the finite element code Lagamine. Borehole radius is equal to 5 cm, domain size is equal to 6m  6m (meshed with 3992 quadratic elements). By this modeling, the goal is to predict the extension of the EDZ resulting from the excavation process (isotropic loading) and its impact on permeability. The obtained results will be qualitatively compared to observations in Opalinus Clay (see

[4,5]). It is convenient to point out that the global geometry of the borehole and the hydromechanical loadings involved in Selfrac experiment, as well as the well known mechanical anisotropy of Opalinus Clay behavior, are not considered in the present study and will be taken into account in a further study for quantitative comparison between the micro–macro model and in situ measurements. 4.2.2. Modeling results Fig. 7 presents the damage field after borehole excavation. On one hand, the distribution of the average value of damage (cracks density parameter defined in Eq. (25)) around the borehole is shown in order to characterize the EDZ; on the other hand, crosses representing the principal directions of damage in EDZ are presented, corresponding to the principal directions of the following tensor:

D¼

N X 1 r¼1

N

r

d ðnr  nr Þ

ð27Þ

Moreover the distribution of the average permeability Keq (cf. Eq. (26)) is provided in Fig. 8, together with the representation of the principal directions of microcracks opening tensor in EDZ:

C¼

N X 1 r¼1

N

cr ðnr  nr Þ

Fig. 7. Damage field around borehole after excavation and associated crosses of damage.

ð28Þ

S. Levasseur et al. / Computers and Geotechnics 49 (2013) 245–252

251

Fig. 8. Equivalent permeability field around borehole after excavation and associated crosses of microcracks openings.

we propose a modification of the micro–macro model in order to account of this ‘‘macrofracture-like’’ phenomenon. 4.3. Discussion

Fig. 9. Damage and permeability evolutions in a radial section around borehole excavation (with Eq. (24)).

It can be observed that the principal directions in these two figures coincide; this is due to the fact that the most opened microcracks are that corresponding to the most developed. It is observed that the most important microcracks are normal to the radial direction, suggesting that microcracks are parallel to borehole wall. To complete the analysis, damage and permeability evolutions on a radial section around the borehole is proposed in Fig. 9. From the above results, one can notice that the predicted EDZ has a size equal to the borehole radius (’5 cm) and damage strongly decreases from the borehole wall to the limit of the EDZ. These observations seem realistic and correspond to fractures occurring during unloading and commonly observed in Mont Terri URL [4]. Despite the interest of the above numerical results, because of very small damage variations due to the high rigidity of Opalinus Clay, the level of in situ stresses and to the main characteristics of the coupling between microcracking and permeability defined in Section 3, the modification of permeability appears to be very small and largely underestimates the reported values from Selfrac experiment: the in situ evolution of permeability in the EDZ has been evaluated about 5 orders of magnitude higher than the initial permeability [3]. This is probably due to the contribution of microcracks coalescence zones which can occur at some highly developed microcracking state. In the following subSection 4.3,

We have noticed from the detail of our simulations that for some microcracks, crack aspect ratio (whose initial values are wr = 103) significatively evolve with the mechanical loading through damage and strain rate variations as detailed in Section 3.1. In particular, they do not comply with the assumption of infinitesimal aspect ratio. For instance in the previous modeling, 20% of the microcracks have an aspect ratio as 0.1 < wr < 2. For these cracks, the PCW bound type approach of microcracked media is not well suited. Following an idea introduced by [1], we model this types of cracks as crosscutting fractures generating some percolation-like threshold despite the small damage level reached. Fluid flow is then directly related to effective pressure gradient and the contribution to Khom in (18) can be identified as:

Kr  haiXr ¼ Krt ¼ K rt ðI  nr  nr Þ

ð29Þ

This implies haiXr ¼ ðI  nr  nr Þ and then:

Khom ¼ Ks þ

N X 4p r¼1

3

r

wr d K rt ðI  nr  nr Þ

ð30Þ

Fig. 10. Damage and permeability evolutions in a radial section around borehole excavation (with Eq. (30)).

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With this new approach, the effective permeability can be reevaluated as shown in Fig. 10. Due to the direct relation between Khom, wr, dr and K rt , it appears that a significant evolution of permeability (several orders of magnitude) can be obtained by mean of the modified relation (30), the damage state evolution being the same as before. 5. Conclusion The present study has been devoted to a micro–macro approach of microcracking and its consequence on permeability evolution around boreholes. To this end, we have proposed a promising homogenization based formulation which takes advantage of Ponte Castaneda and Willis bound and account for initial stresses. Application of this model on a hardened clay such as Opalinus Clay shows that it describes well the material behavior and the induced damage. Compared to experimental in situ available data in the context of radioactive waste repository, the predicted damaged zone is qualitatively relevant to the EDZ shape observed around excavations reported in [5]. Moreover, permeability evolution qualitatively follows the good trends. However, the amplitude of the predicted damage distribution does not permit to reproduce the magnitude of the permeability increase observed in Selfrac in situ experiments after excavation. This has motivated further modification in which crosscutting fractures are considered. It is shown that such modification allows to reproduce permeabilities evolution similar to that noted in the in situ experiments. Acknowledgements The authors thank the F.R.S.–FNRS, the national funds of scientific research in Belgium, for their financial support in the FRFC project. They are also grateful to anonymous reviewer whose suggestions and comments have allowed to clarify some points of this study and to improve its presentation. References [1] Barthélémy J-F. Effective permeability of media with a dense network of long and microfractures. Transport Porous Med 2009;76:153–78. [2] Barton N, Bandis S, Bakhtar K. Strength, deformation, conductivity coupling of rock joints. Int J Rock Mech Min Sci 1985;22(2):121–40. [3] Bernier F, Li XL, Bastiaens W, Ortiz L, Van Geet M, Wouters L, et al. Fractures and self-healing within the excavation disturbed zone in clays (selfrac). In: Final report, 5th EURATOM Framework Programme, 1998–2002; 2007. [4] Bossart P, Meier PM, Moeri A, Trick Th, Mayor JC. Geological and hydraulic characterization of the excavation disturbed zone in the opalinus clay of the mont terri rock laboratory. Eng Geol 2002;66:19–38. [5] Bossart P, Trick Th, Meier PM, Mayor JC. Structural and hydrogeological characterisation of the excavation disturbed zone in opalinus clay (Mont Terri Project, Switzerland). Appl Clay Sci 2004;26:429–48. [6] Budiansky B, O’Connell R. Elastic moduli of a cracked solid. Int J Solids Struct 1976;12:81–97. [7] Chatzigeorgiou G, Picandet V, Khelidj A, Pijaudier-Cabot G. Coupling between progressive damage and permeability of concrete: analysis with a discrete model. Int J Numer Anal Meth Geomech 2005;29:1005–18. [8] Chen YF, Zhou CB, Sheng Y. Formulation of strain dependant hydraulic conductivity for fractures rock mass. Int J Rock Mech Min Sci 2007;44:981–96.

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