ORIGINAL PAPER

Micromechanical Modeling of Anisotropic Damage-InducedPermeability Variation in Crystalline Rocks

Yifeng Chen • Shaohua Hu •

Chuangbing Zhou • Lanru Jing

Received: 28 April 2013 / Accepted: 19 September 2013

� Springer-Verlag Wien 2013

Abstract This paper presents a study on the initiation and

progress of anisotropic damage and its impact on the per-

meability variation of crystalline rocks of low porosity.

This work was based on an existing micromechanical

model considering the frictional sliding and dilatancy

behaviors of microcracks and the recovery of degraded

stiffness when the microcracks are closed. By virtue of an

analytical ellipsoidal inclusion solution, lower bound esti-

mates were formulated through a rigorous homogenization

procedure for the damage-induced effective permeability

of the microcracks-matrix system, and their predictive

limitations were discussed with superconducting penny-

shaped microcracks, in which the greatest lower bounds

were obtained for each homogenization scheme. On this

basis, an empirical upper bound estimation model was

suggested to account for the influences of anisotropic

damage growth, connectivity, frictional sliding, dilatancy,

and normal stiffness recovery of closed microcracks, as

well as tensile stress-induced microcrack opening on the

permeability variation, with a small number of material

parameters. The developed model was calibrated and val-

idated by a series of existing laboratory triaxial compres-

sion tests with permeability measurements on crystalline

rocks, and applied for characterizing the excavation-

induced damage zone and permeability variation in the

surrounding granitic rock of the TSX tunnel at the Atomic

Energy of Canada Limited’s (AECL) Underground

Research Laboratory (URL) in Canada, with an acceptable

agreement between the predicted and measured data.

Keywords Damage � Permeability �Micromechanics � Crystalline rocks

List of Symbols

a, a0 Radius and initial radius of microcracks

Aa Second-order concentration tensor

corresponding to the dilute, MT, or HS

scheme, a = dil, MT, or HS

c Half opening of microcracks

c0, c1 Parameters representing the initiation and

evolution of damage

Cs, Ss Elastic stiffness tensor and compliance

tensor of rock matrix

d; d0; �d Crack density, initial crack density, and

mean crack density

Es; ms Young’s modulus and Poisson’s ratio of

rock matrix

e Aperture of microcracks

Fd;Fb;Fc Conjugate thermodynamic forces

associated with variables d, b, and c

g, h Yield function and plastic flow potential

function

H0, H1 Elastic parameters

I Second-order identity tensor

ka, ka Effective permeability tensor and isotropic

effective permeability estimated by the

dilute, MT, HS, or Voigt scheme, a = dil,

MT, HS, or Voigt

Y. Chen (&) � S. Hu � C. Zhou

State Key Laboratory of Water Resources and Hydropower

Engineering Science, Wuhan University, Wuhan 430072, China

e-mail: csyfchen@whu.edu.cn

Y. Chen � S. Hu � C. Zhou

Key Laboratory of Rock Mechanics in Hydraulic Structural

Engineering (Ministry of Education), Wuhan University,

Wuhan 430072, China

L. Jing

Royal Institute of Technology, 10044 Stockholm, Sweden

e-mail: lanru@kth.se

123

Rock Mech Rock Eng

DOI 10.1007/s00603-013-0485-5

ks, kc Permeability tensors of rock matrix and

microcracks

ks, kc Isotropic permeability of rock matrix and

that of microcracks

kc0 Initial permeability of microcracks

kn0 Initial normal stiffness of microcracks

m Number of integration points

n Unit normal vector

Pc, Pd Interaction tensors of microcracks

associated with the shape and spatial

distribution

p2, p4 Parameters for brevity of presentation

Ra Parameter for brevity of presentation,

a = dil, MT, or HS

S, S?, S- Surfaces, upper surfaces, and lower

surfaces of microcracks

u?, u- Displacements of upper and lower surfaces

[u], [un], [ut] Displacement discontinuity and its normal

and sliding components on microcrack

surfaces

wi Weighting coefficient at the ith integration

point

N Number of microcracks per unit volume

d Exponent representing the connectivity of

microcracks

b, b0, c Parameters representing the opening,

initial opening, and sliding of microcracks

b1, b2 Compressive component and shear

dilatancy component of be; es; ec Macroscopic strain tensors of cracked

rock, rock matrix, and microcracks

U Energy dissipation function

/c Friction angle of microcracks

‘2 Surface of unit sphere

uc;uc0 Volume fraction and initial volume

fraction of microcracks_kc; _kd Multipliers representing the evolutions of

sliding and damage

r Stress tensor of cracked rock

rc; rcn Local stress tensor and its normal

component on microcrack surfaces

t Unit sliding vector of displacement

discontinuity on microcrack surfaces

x, xd Aspect ratios of microcrack and its

effective environment

X Volume of representative element

W Helmholtz free energy

wc Dilatancy angle of microcracks

fc; fd Geometric parameters of microcrack and

its effective environment

: Inner product of two second-order tensors

(e.g., a:b = aijbij) or double contraction of

adjacent indices of tensors of rank two and

higher (e.g., c:d = cijkldkl)

� Dyadic product of two vectors (e.g.,

a � b = aibj)

�s Symmetric part of the dyadic product of

two vectors (e.g., a�s

b ¼ 12

aibj þ ajbi

� �)

1 Introduction

Damage induced by microcracking in hard crystalline

rocks is an important issue for the safe geological disposal

of radioactive wastes, geothermal energy extraction, com-

pressed air energy storage, and tunnel sealing treatment in

deeply buried underground openings, etc. (see e.g., Kim

et al. 2013). The initiation and accumulation of damage of

host rocks under different loading conditions such as tunnel

excavation, thermal stress, or hydraulic fracturing stimu-

lation may lead to inelastic behavior of the stress–strain

relation and deterioration of strength and stiffness, and may

result in significant variations in groundwater flow and

solute transport properties. The latter is an important issue

for the environmental safety of rock engineering projects.

In recent years, continuous interest has been attracted in

characterizing the damage-induced variation in perme-

ability or thermal conductivity in brittle rocks (Souley et al.

2001; Oda et al. 2002; Shao et al. 2005; Chaki et al. 2008;

Mitchell and Faulkner 2008; Jiang et al. 2010; Bary 2011;

Zhou et al. 2011; Arson and Pereira 2012; Chen et al. 2012;

Pouya and Vu 2012).

Both phenomenological models and micromechanical

models have been developed and applied in modeling the

progressive failure of rocks. The phenomenological models

(Chow and Wang 1987; Ju 1989; Chaboche 1992; Krajci-

novic 1996; Halm and Dragon 1996; Swoboda and Yang

1999; Dragon et al. 2000) use scalar or tensorial internal

variables for representing the degraded state of materials

(Lubarda and Krajcinovic 1993) and formulate the damage

evolution laws in the framework of thermodynamics of

irreversible processes by defining state potential functions.

Some of the complex phenomena induced by microcrack-

ing, such as degradation of moduli, anisotropy effect,

volumetric dilatancy, unilateral effect, hysteretic effect,

frictional sliding over the internal microcrack surfaces, and

normal stiffness recovery of closed microcracks, have been

partially addressed in these evolutionary laws. However,

the damage variables are not quantitatively linked to the

relevant microstructural features in this type of model.

The micromechanical models provide a powerful alter-

native by linking the mechanisms involved in the micro-

structure alteration to the macroscopic behavior observed

Y. Chen et al.

123

in laboratory conditions (Hashin 1988; Nemat-Nasser and

Hori 1993; Pensee et al. 2002; Zhou et al. 2008; Zhu et al.

2008a; Abou-Chakra Guery et al. 2008; Bary 2011). The

performance of a micromechanical model for the macro-

scopic response depends on whether the microstructures

are properly characterized in a quantitative way and how

the interactions between individual inhomogeneities are

modeled by approximation schemes and upscaling proce-

dures. It has been recognized (Shafiro and Kachanov 2000;

Kachanov and Sevostianov 2005; Sevostianov and Ka-

chanov 2008) that the crack density (proportional to the

cube of the crack radius) is a proper parameter for the

quantitative characterization of the effective elasticity for

representing the individual contribution of cracks, while for

the effective permeability, the crack opening (or volume

fraction) is also of importance. In the available upscaling

approaches, the interactions and spatial correlations of

microcracks are considered in different manners. The most

commonly applied estimates of the effective properties are

the dilute scheme, the self-consistent (SC) scheme (Nemat-

Nasser and Hori 1993), the Mori–Tanaka (MT) scheme

(Mori and Tanaka 1973), the differential scheme (Hashin

1988), the generalized Hashin–Shtrikman (HS) variational

structure estimate (Ponte-Castaneda and Willis 1995), and

the interaction direct derivative (IDD) estimate (Zheng and

Du 2001).

The damage-induced variation in permeability in brittle

rocks has been formulated by means of phenomenological

models, with a focus on crack length growth (Souley et al.

2001), crack density distribution (Oda et al. 2002), com-

pressive stress-induced crack opening (Shao et al. 2005), or

change in pore size distribution (Arson and Pereira 2012).

Micromechanical models were also adopted, using the

dilute or SC estimate (Pouya and Vu 2012), the IDD esti-

mate (Zhou et al. 2011), or the volumetric averaging of

local permeability in microcracks (Jiang et al. 2010). For

practical rock mechanics problems, however, the predictive

capability of these damage-induced permeability variation

models is under discussion.

This study focuses on the damage-induced permeability

variation in crystalline rocks of low porosity, in which the

considered porous structure is made of both damage-

induced microcracks and natural pores. The microme-

chanical model proposed by Zhu et al. (2008a) was used,

with an extension to incorporate the recovery of degraded

stiffness of pre-existing microcracks under predominantly

compressive loading (Chen et al. 2012). Various lower

bound estimates (i.e., the dilute, MT, HS, and IDD esti-

mates) were formulated based on a rigorous homogeniza-

tion procedure for the damage-induced permeability of the

microcracks-matrix system. By ascribing infinite perme-

ability to microcracks, the greatest lower bound estimates

were obtained, followed by discussions on the predictive

limitations of the lower bound estimates. It was found that

the limitations resulted from inadequate consideration of

the microcrack connectivity or improper use of micro-

structural parameters in the homogenization schemes. An

empirical upper bound estimation model was then sug-

gested in order to account for the influences of anisotropic

damage growth, connectivity, frictional sliding, dilatancy,

and normal stiffness recovery of closed microcracks, as

well as tensile stress-induced microcrack opening on the

permeability evolution, with a small number of material

parameters. The damage-induced permeability variation

model was formulated in an integral form over the surface

of a unit sphere in order to consider any possible damage

growth pattern at various stress paths and the anisotropic

nature of microcracking. The developed model was cali-

brated and validated by comparing the results with the

laboratory data of triaxial compression tests with perme-

ability measurements on five typical crystalline rocks

(Souley et al. 2001; Mitchell and Faulkner 2008; Ma et al.

2012), and finally applied to characterize the excavation-

induced damage zone (EDZ) and permeability variation in

the surrounding rock of the TSX tunnel at the Atomic

Energy of Canada Limited’s (AECL) Underground

Research Laboratory (URL) in Canada (Martino and

Chandler 2004).

Unless otherwise noted, the conventions in continuum

mechanics are adopted in this paper, i.e., tensile stresses are

positive, while compressive stresses are negative. In the

remainder of this paper, Sect. 2 presents a brief review of

Zhu’s micromechanical damage model, with a minor

extension to consider the compressive deformation of pre-

existing microcracks. The homogenization-based formula-

tion of damage-induced permeability variation is presented

in Sect. 3, in which the predictive capability of different

estimation models is compared. The computational aspects

are discussed in Sect. 4. Validation of the proposed model

against laboratory data and application to the TSX tunnel

are presented in Sects. 5 and 6, followed by conclusions

presented in Sect. 7.

2 A Brief Review of Zhu’s Anisotropic Damage Model

2.1 Damage Variables

Consider a representative element volume (REV) of a

cracked rock, X; in which penny-shaped microcracks with

surface S of unit normal vector n may develop in arbitrary

directions. The volume fraction ucðnÞ of microcracks is

then regarded as a distribution on the surface of a unit

sphere ‘2:

Micromechanical Modeling of Anisotropic

123

ucðnÞ ¼ 4

3pa2cNðnÞ ¼ 4

3pxdðnÞ ð1Þ

where a is the radius of microcracks, c the half opening, Nthe number of microcracks per unit volume, x ¼ c=a the

aspect ratio, and dðnÞ ¼ N a3 the crack density parameter

that has been widely used in the micromechanical damage

modeling (Budiansky and O’Connell 1976).

The displacements of the upper surface S? and lower

surface S- of the microcracks are u? and u-, respectively,

then the displacement discontinuity on the microcrack

surfaces is [u] = u? – u-. The normal component and the

sliding vector of the displacement discontinuity across the

microcrack surfaces are then represented by ½un� ¼ ½u� � nand ½ut� ¼ ½u� � ½un�n; respectively. Two variables, b and c,

are defined to quantify the microcrack opening and sliding

along the surfaces of microcracks of unit normal vector n:

bðnÞ ¼ NZ

Sþ

½un�dS ð2aÞ

cðnÞ ¼ NZ

Sþ

½ut�dS ð2bÞ

The non-negative variable b actually characterizes the

volumetric deformation of the homogenized rock resulting

from opening/closure and growth of existing microcracks

or initiation of new microcracks. If b evolves from its

initial state b0 corresponding to the initial development and

initial opening/closure of the microcracks, Eqs. (2a) and (1)

are linked by the following equation (Chen et al. 2012):

ucðnÞ ¼ bðnÞ ð3Þ

with uc0 ¼ b0 at the initial state. Any possible distribution

pattern of microcracks in space is hence characterized by

the above three internal variables d(n), b(n), and c(n).

2.2 The Free Energy of the Cracked Media

In a cracked rock, both rock matrix and microcracks may

undergo deformation. The macroscopic strain of the

cracked rock, e, is composed of a component from the rock

matrix, es, and another component from the microcracks,

ec. es can be easily computed, given the fact that, in the

inclusion problem, the macroscopic stress field resulting

from the microcracks is self-equilibrated. ec can be

expressed as follows:

ec ¼ 1

4p

I

‘2

bðnÞn� nþ cðnÞ�s

nh i

dS ð4Þ

Suppose, for simplicity, that the stiffness of rock matrix

is uniform in the REV, the macroscopic stress–strain

relation of the cracked medium is then given by:

r ¼ Cs : es ¼ Cs : ðe� ecÞ ð5Þ

where Cs is the fourth-order elastic stiffness tensor of solid

matrix.

Neglecting the interaction of the frictional behaviors of

microcracks, herein it is supposed that the Helmholtz free

energy of Zhu et al. (2008a) applicable to rock materials

weakened by one family of parallel cracks can be

extended to the case of arbitrary distribution of

microcracks:

W ¼ 1

2e� ecð Þ : Cs : e� ecð Þ þ 1

8p

I

‘2

p2b2 þ p4c � c

� �dS

ð6Þ

where the second term on the right hand side represents the

trapped energy from microcracks. If the Mori–Tanaka

homogenization scheme is used, p2 and p4 are expressed as

p2 ¼ H0

dand p4 ¼ H1

d; with H0 ¼ 3Es

16½1�ðvsÞ2� and H1 ¼ H0ð1�vs

2Þ: The parameters Es and vs are, respectively, the elastic

modulus and Poisson’s ratio of rock matrix.

The conjugate thermodynamic forces, Fd, Fb, and Fc,

associated with the internal state variables d, b, and c, are

given by:

FdðnÞ ¼ � oWod¼ 1

2d2ðH0b

2 þ H1c � cÞ ð7aÞ

FbðnÞ ¼ � oWob¼ r : n� n� p2b ð7bÞ

FcðnÞ ¼ � oWoc¼ r � n � ðI � n� nÞ � p4c ð7cÞ

where I is the second-order identity tensor.

The energy dissipation UðnÞ of the damaged micro-

cracks-matrix system is then guaranteed as long as the

constitutive model of the microcracks is dissipative and the

evolution of the damage variable d is irreversible (i.e.,_d� 0) (Chen et al. 2012), written as:

UðnÞ ¼ Fd _d þ Fb _bþ Fc � _c� 0 ð8Þ

2.3 Damage Evolution and Crack Propagation

The normal component of the displacement discontinuity

and the applied normal stress on the microcracks in the

direction n must satisfy the following unilateral condition:

½un� � 0; rcn� 0; ½un� � rc

n ¼ 0; where rcn is the local normal

stress on the microcrack surfaces. The opening–closure

transition condition of the microcracks can be expressed by

removing the normal component of the local stress, rcn¼ 0;

or the corresponding thermodynamic force, Fb = 0 (Zhu

et al. 2008a). Thus, for open cracks (i.e., Fb = 0), the

magnitude of normal deformation of the microcracks is:

Y. Chen et al.

123

b ¼ 1

p2

R : n� n ð9aÞ

The tangential component of the local stress also

vanishes, i.e., Fc = 0, which leads to:

c ¼ 1

p4

R � n � ðI � n� nÞ ð9bÞ

For closed cracks (i.e., Fb B 0), on the other hand, the

normal deformation responds to two mechanisms: the

nonlinear closure of the microcracks due to normal

compression and the opening of the microcracks due to

frictional dilation. This relation can be represented by two

components of b, b1 and b2, representing, respectively, the

above two mechanisms, i.e.:

b ¼ b1 þ b2 ð10Þ

The normal closure of microcracks may result in the

partial recovery of degraded macroscopic deformation

moduli (stiffness) and decrease the effective permeability.

This behavior, therefore, is of particular significance when

the microcracks are under predominantly compressive

loading conditions. By analogy with the normal

deformation of rock joints subjected to normal

compression (Bandis et al. 1983), the first component of

b, b1, is assumed to evolve with a nonlinear hyperbolic

function (Chen et al. 2012):

b1 ¼Fbb0

kn0b0 � Fbð11Þ

where b0 denotes the initial (or maximum) closure of the

microcracks and kn0 is the initial stiffness of the

microcracks.

To determine the sliding vector, c, and the second

component of b, b2, it is assumed that the sliding behavior

of the microcracks obeys a non-associative Mohr–Coulomb

criterion, with the strength criterion defined by:

gðrcÞ ¼ jFcj þ tan /cFb ð12aÞ

where /c is the friction angle of the microcracks, and with

a plastic flow potential given by:

hðrcÞ ¼ jFcj þ tan wcFb ð12bÞ

where wc is the dilatancy angle of the microcracks.

The damage evolution of the microcracks is guided by a

simple yield criterion:

f ðFd; dÞ ¼ Fd � ðc0 þ c1dÞ ð13Þ

where c0 and c1 are two material parameters representing

the initiation and evolution of damage.

According to the normality rules of the sliding and

damage criteria, one has the rates of b2 and the other two

state variables, d and c, defined by:

_c ¼ _kc oh

oFc ¼ _kct ð14aÞ

_b2 ¼ _kc oh

oFb¼ _kc tan wc ð14bÞ

_d ¼ _kd of

oFd¼ _kd ð14cÞ

where t ¼ Fc=jFcj is the sliding vector of the displacement

discontinuity on the microcrack surfaces. The symbols _kc

and _kd are two multipliers for representing the evolutions of

sliding and damage, respectively, which can be determined

by the Kuhn–Tucker complementary conditions of the

sliding and damage criteria, as given by Chen et al. (2012).

2.4 Constitutive Equations for the Cracked Media

Substituting Eq. (4) into Eq. (5), the macroscopic consti-

tutive relation of the homogenized cracked rock with

arbitrary distribution of microcracks in space is expressed

as follows:

e ¼ Ss : rþ 1

4p

I

‘2

bðnÞn� nþ cðnÞ�s

nh i

dS ð15Þ

where Ss is the fourth-order compliance tensor of the solid

matrix.

The damage model involves nine material parameters,

i.e., two elastic coefficients for solid matrix (Es and vs), two

constants for the sliding and dilatancy of microcracks (/c

and wc), two for representing the initiation and evolution of

microcrack damage (c0 and c1), one for representing the

initial normal stiffness of microcracks subjected to com-

pressive pressure (kn0), and two for representing the initial

development of microcracks (b0 and d0). Identification of

the above material parameters by using the laboratory

uniaxial or triaxial test data is critical for applying the

model. In practice, the above model parameters can be

estimated by best fitting the laboratory stress–strain curves.

Through simplifications of the model, however, the number

of parameters can be reduced. If an associative flow rule is

adopted for the microcracks, then wc ¼ /c: Besides, if the

normal closure of microcracks under compressive stress is

negligible, then kn0 !1:The sensitivity of the model with respect to the param-

eters has been examined by Chen et al. (2012), who showed

that Es and vs affect the initial slope of the stress–strain

curve, and c0, c1, and /c condition the initial damage of the

sample and the curvature of the macroscopic stress–strain

curve. Lower values of c0 lead to an anticipated earlier

initiation of microcracks, while lower values of c1 and /c

result in a higher rate of damage growth, larger irreversible

strain, and lower peak strength. The non-associative flow

rule of the microcracking also plays a remarkable role in the

Micromechanical Modeling of Anisotropic

123

overall stress–strain behavior as the peak stress is approa-

ched, especially when lower values of dilatancy angle, wc;

are used. The overall mechanical response is not sensitive to

the initial values of the damage density, d0, in the range

0.001–0.01, and it can be estimated by the initial fracturing/

microcracking status of the samples.

3 Damage-Induced Evolution of Permeability

3.1 Homogenization Schemes for Damage-Induced

Permeability Variation

Starting from the micromechanical damage model pre-

sented in Eq. (15), the effective permeability of a damaged

rock with microcracks distributed in arbitrary directions

can be developed by using different homogenization

methods. By virtue of the analogies between the governing

equations of the steady-state thermal transport and fluid

filtration problems, as well as the dilute solution of thermal

conductivity of a homogeneous rock matrix with ellipsoi-

dal inclusions (Shafiro and Kachanov 2000), the effective

permeability of the microcracks-matrix system can be

represented by the following expressions:

ka ¼ ks þ 1

4p

I

‘2

ucðnÞ kcðnÞ � ks½ � � AaðnÞdS ð16Þ

where ka (a = dil, MT, or HS) is the effective permeability

tensor estimated by the dilute, MT, or HS scheme, ks and

kc(n) the permeability tensors of rock matrix and the

microcracks of unit normal vector n, respectively, and Aa

the second-order concentration tensor corresponding to the

estimation schemes:

AdilðnÞ ¼ fI þ PcðnÞ � ½kcðnÞ � ks�g�1 ð17aÞ

AMTðnÞ ¼ AdilðnÞ

� ð1� ucÞI þ 1

4p

I

‘2

ucðnÞAdilðnÞdS

2

4

3

5

�1

ð17bÞ

AHSðnÞ ¼ AdilðnÞ

� I� 1

4p

I

‘2

ucðnÞ½kcðnÞ � ks�AdilðnÞPdðnÞdS

2

4

3

5

�1

ð17cÞ

where uc is the total volume fraction of the cracks, i.e.,

uc ¼ 14p

H‘2 ucðnÞdS; and Pc and Pd the interaction tensors

of the microcracks of unit normal vector n associated with

the microcrack shape and the spatial distribution,

respectively. If the permeability of both rock matrix and

microcracks is assumed to be isotropic, the interaction

tensors of the ellipsoidal microcracks can be expressed by:

PaðnÞ ¼1a

ksðI � n� nÞ þ 1� 21a

ksn� n ða ¼ c; dÞ ð18Þ

where ks is the isotropic permeability of the rock matrix

and f a geometric parameter related to the aspect ratio x of

the microcracks or the aspect ratio xd of the ellipsoidal

atmosphere (i.e., the area around and locally influenced by

a microcrack) for consideration of the spatial distribution

of microcracks. It can be verified that, for penny-shaped

microcracks (i.e., x! 0), 1! p4x:

Using the expression 14p

H‘2 n� ndS ¼ 1

3I; it can be

inferred that the effective permeability is isotropic for rock

samples of uniform distribution of microcracks (i.e.,

ucðnÞ � uc), and Eq. (16) can be reduced to the following

explicit expression:

ka ¼ ks þ ucðkc � ksÞRa ð19Þ

where ka (a = dil, MT, or HS) is the effective

permeability, kc the isotropic permeability of the

microcracks, and Ra a parameter related to the ratio of kc

to ks and the shapes of the microcracks or the effective

environment in which the microcracks are embedded:

Rdil ¼2

3

ks

ð1� 1cÞks þ 1ckcþ 1

3

ks

21cks þ ð1� 21cÞkcð20aÞ

RMT ¼ Rdil 1� ucð1� RdilÞ½ ��1 ð20bÞ

RHS ¼ Rdil 1� uc 2

3

1dðkc � ksÞks þ 1cðkc � ksÞ þ

1

3

ð1� 21dÞðkc � ksÞks þ ð1� 21cÞðkc � ksÞ

� �� ��1

ð20cÞ

where fd can be computed with the aspect ratio xd of the

effective environment of microcracks by the relationship43pxdð1þ xdÞ2 ¼ 1

dsuggested by Zhou et al. (2011).

Given the fact that the permeability of rock matrix, ks, is

generally much smaller than that of the microcracks, kc,

Eqs. (19) or (16) gives the lower bounds of estimations of

the effective permeability. It is interesting, therefore, to

examine the predictive capability of the models, as kc tends

to infinity and the aspect ratio x tends to zero, since the

greatest lower bound estimations of the effective perme-

ability will be obtained under this condition. In this case,

Eq. (19) is further reduced to:

kdil ¼ 1þ 32

9d

� ks ð21aÞ

kMT ¼ 1þ329

d

1� 163

1cd

!

ks ð21bÞ

kHS ¼ 1þ329

d

1� 329

1dd � 169

1cd 1�21d

1�21c

!

ks ð21cÞ

Y. Chen et al.

123

It should be noted that Eq. (21a) has been given by

Pouya and Vu (2012), where the complementary part of the

flow solution for superconducting penny-shaped

microcracks embedded in an infinite matrix vanishes in

the homogenized estimation, and Eq. (21b) has been given

by Zhou et al. (2011), where the IDD estimate actually

coincides with the Mori–Tanaka estimate. Figure 1 plots

the relationship between the homogenized permeability

and the damage density estimated by Eq. (21) with an

aspect ratio of x = 0.01 for the microcracks. Actually, the

model predictions are almost the same when the aspect

ratio varies from 0.0001 to 0.01. The curves clearly show

that the predicted effective permeability varies almost

linearly with the damage density, and the models can only

predict an increase of effective permeability of one order of

magnitude as the damage density grows from 0 to 3, a

value indicating significant growth of microcracks and

possible failure of rock samples.

These results demonstrate that the above homogeniza-

tion schemes that use the analytical solution for a single

microcrack in an infinite matrix do not reproduce quanti-

tatively the increase of damage-induced permeability of

several orders of magnitude. This shortcoming resulted

from the fact that the effect of the connectivity of the

microcracks is not adequately considered by the homoge-

nization schemes. As an alternative, the upper bound esti-

mations seem more realistic in predicting the permeability

of cracked materials.

3.2 An Empirical Upper-Bound Estimate for Damage-

Induced Permeability Variation

In the Voigt model, the effective permeability is the

average of the permeability values of both rock matrix and

microcracks over their respective volume fractions, and the

resulting estimate constitutes an upper bound for the

effective permeability:

kVoigt ¼ ks þ 1

4p

I

‘2

ucðnÞ kcðnÞ � ks½ �dS ð22Þ

Mathematically, the Voigt estimate, Eq. (22), can be

derived from Eq. (19) by assigning the identity tensor to

the concentration tensor, thus implying that, after

homogenization, the hydraulic gradient in the microcrack

system is exactly the same as the uniform gradient

generated by the conditions on the boundaries of the

REV. This condition is reasonable for relatively more

developed damage states, i.e., higher damage density,

larger crack size, and higher crack connectivity.

Suppose that the flow through the microcracks obeys

Poiseuille’s law, then the permeability of the microcracks

of unit normal vector n is kcðnÞ ¼ e2

12; where e is the

average aperture of the microcracks. Oda et al. (2002) and

Shao et al. (2005) adopted a dimensionless constant, k(0 \ k B 1), as a penalty factor for considering the effect

of the roughness of microcrack walls. In this model,

however, the effect of microcrack roughness is partly

addressed by using other parameters. By using Eqs. (1),

(2a), and (3), e is expressed as follows:

e ¼ 4

3c ¼ b

pa2N ¼abpd

ð23Þ

Assuming that the initial permeability kc0 of the

microcracks (with initial microcrack opening b0) can be

calibrated by laboratory tests, and using Eqs. (23) and (3),

Eq. (22) can be rearranged as follows:

kVoigt ¼ ð1� ucÞks

þ 1

4p

I

‘2

kc0

a

a0

� 2b3

b20

d0

d

� 2

I � n� nð ÞdS ð24aÞ

where a0 is the initial microcrack radius. The above

estimate of effective permeability is related to the radius a

of the microcracks, which, unfortunately, is not an

independent internal variable in the damage model. By

the definition of the damage density d, one has a ¼ dN� �1

3; in

which N is not an independent internal variable either.

However, this expression clearly shows that the radius a of

the microcracks is closely related to the damage density d

and directly influenced by the number N of cracks per unit

volume. As the damage grows, both the length and the

number of microcracks increase, and the microcrack system

tends to be more connected. Following this conceptual

understanding, a simple empirical expression, a / dd; is

established, where d is an exponent to be calibrated by

experimental data. This expression, even empirical, has a

clear physical basis and plays a critical role in considering

the effects of the microstructural alterations (e.g., the

increase of microcrack size and the change of microcrack

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

25

kho

m/k

s

d

HS or IDD MT dilute

Fig. 1 The relationship between the homogenized permeability and

the damage density

Micromechanical Modeling of Anisotropic

123

connectivity) on the permeability variation. Equation (24a)

can then be rewritten as follows:

kVoigt ¼ ð1� ucÞks þ 1

4p

I

‘2

kc0

b3

b20

d

d0

� 2d�2

I � n� nð ÞdS

ð24bÞ

Generally, d is suggested to take values between 0 and 2.

A higher value of d indicates higher persistence and

connectivity of the microcrack system. If the number N of

microcracks per unit volume remains constant in the

damage process, then d ¼ 1=3: On the other hand, if Ngrows proportionally to a, then d ¼ 1=4: Oda et al. (2002)

assumed that the aperture of microcracks increases

proportionally to the increase of the diameter, with the

aspect ratio of microcracks unchanged in the damage

process. Shao et al. (2005) suggested that the increase of

microcrack aperture e is directly proportional to the

increase of microcrack radius a, with de = vda, where vis a proportional coefficient. For these considerations,

however, extra modeling of the effect of microcrack

connectivity is required.

In the proposed formulation, besides the permeability of

rock matrix, ks (that is assumed to be isotropic), only two

parameters, kc0 and d, should be further calibrated with best

fitting of the experimental data. Compared to the works by

Shao et al. (2005) and Jiang et al. (2010), the number of

parameters is reduced by at least three and one, respectively.

4 Computational Aspects

As shown in Eqs. (15) and (24b), the proposed effective

permeability model is formulated in an integral form on the

surface of a unit sphere ‘2: The Gauss-type numerical

integration scheme is available for numerically computing

the integrals:

e ¼ Ss : rþXm

i¼1

wi bn� nþ c�s

n �

ið25aÞ

kVoigt ¼ ð1� ucÞks þXm

i¼1

wi kc0

b3

b20

d

d0

� 2d�2

I � n� nð Þ" #

i

ð25bÞ

where m is the number of integration points and wi the

weighting coefficient. Bazant and Oh (1986) presented

integration schemes for m = 21, 33, 37, or 61 for half of

the symmetric sphere. In practice, the integration scheme

of m = 33 may be a good choice for its reasonable balance

between numerical accuracy and computational cost.

The above model, Eqs. (25), has been integrated in a

computer code, THYME3D, initially developed for cou-

pled deformation/multiphase flow/thermal transport

analysis (Chen et al. 2009), in which linear 3D elements

(i.e., brick, wedge, and tetrahedra) are available. The

damage model is implemented in an incremental form

locally at each Gauss integration point per element by

using a prediction-correction algorithm (Zhu et al. 2008b),

in which microcracks are allowed to develop in any

direction according to the stress state during the loading

process. As the damage grows, the internal variables b and

c in each integration direction are updated using Eqs. (9–

11) and (14), and the macroscopic stress–strain behavior is

then determined. At each loading step, the global system of

equations is solved with the modified Newton–Raphson

method. As the mechanical process converges, the per-

meability is then updated.

It should be noted that this study focuses mainly on the

anisotropic damage-induced permeability variation in

cracked rocks, and the effect of pore water pressure on the

damage and mechanical behavior is not considered.

5 Validation and Discussion of the Proposed Model

The experimental data of triaxial compression tests with

permeability measurements are available for the validation

and calibration of the proposed model, such as the labo-

ratory tests on samples of the Lac du Bonnet granite and

Senones granite (Souley et al. 2001), Beishan granite (Ma

et al. 2012), Westerly granite, and Cerro Cristales grano-

diorite (Mitchell and Faulkner 2008). The experimental test

conditions on the mentioned rock samples are listed in

Table 1.

The elastic properties of the above granitic rocks were

taken directly from Souley et al. (2001), Liu et al. (2007),

and Mitchell and Faulkner (2008); the initial elastic mod-

ulus and Poisson’s ratio of the rocks are about 62–70 GPa

and 0.21–0.28, respectively. The initial porosity of the rock

samples are about 0.3–0.8 %. Some of the triaxial com-

pression test results showed that the closure of the pre-

existing pores and microcracks was not remarkable in the

initial region of the stress–strain curves; however, signifi-

cant permeability decrease was induced in this region. This

fact implies that only a small fraction of the pre-existing

microcracks is sensitive to the initial compression, and the

remaining pores and microcracks can be treated as a

background pore system in the rock matrix. For this reason,

a smaller value of 0.04–0.2 %, rather than the initial

porosity, was used for b0, and the remaining porosity was

regarded as a contribution to the elasticity and permeability

of the rock matrix. The other material parameters were

calibrated by best fitting the stress–strain curves and the

permeability–deviatoric stress curves, as listed in Table 2.

Initial isotropy was assumed for the rock samples, and the

associative flow rule was assumed for modeling sliding and

Y. Chen et al.

123

dilatancy of the microcracks (i.e., /c ¼ wc). By Eqs. (1)

and (3), the values of d0 and b0 imply that the initial aspect

ratio of the microcracks was about 0.01–0.05, and, conse-

quently, the microcracks were initially penny shaped.

Calibration study of the permeability model showed that

the initial permeability of the microcracks in the above

rock samples was on the order of magnitude of 10-18–

10-15 m2, indicating that the initial aperture of the mi-

crocracks was on the order of magnitude of 0.001–0.1 lm.

This estimation is obviously reasonable for the intact

crystalline rocks.

Figures 2, 3, 4, 5, and 6 plot the measured and predicted

stress–strain curves and permeability–deviatoric stress

curves for the selected rocks with (effective) confining

pressures of 5 or 10 MPa (measured data are plotted as dot

points). Note that the experimental data of the stress–strain

curves of the Senones granite are not available in the liter-

ature, and are not plotted in Fig. 3a. One observes from

Figs. 2a, 3a, 4a, 5a, and 6a that the micromechanical damage

model basically characterizes the stress–strain relation of the

granitic rocks, with a clear initial region of microcrack clo-

sure, an elastic region, and a crack initiation and growth

region in the modeled stress–strain curves. Chen et al. (2012)

pointed out that the MT estimate exhibits a strain hardening

behavior and may deviate from the experimental results after

the peak stress. In this study, however, for the purpose of

characterizing the damage-induced permeability variation,

the MT estimate still appears to be a good choice for the

stress–strain behavior, because the estimated spatial distri-

bution of damage density is similar to other estimation

schemes in the pre-peak region.

Figures 2b, 3b, 4b, 5b, and 6b show that the proposed

permeability model captures well the damage-induced

permeability variation in the mechanical loading process,

with a clear decrease of permeability in the initial inelastic

region, a constant permeability in the elastic region, and a

dramatic permeability increase in the crack growth region.

For the five selected rocks, the deviatoric stresses at which

the permeability starts to increase are about 150, 120, 80,

140, and 120 MPa, respectively. This magnitude is influ-

enced by the confining pressure and the initial microstruc-

tures of the rock samples. The deviatoric stress value of the

Beishan granite is much lower than those of other rocks, and

this result is reasonable because significant disturbance was

made to the rock samples taken from borehole BS05 at

300–600 m depth at China’s Beishan preselected area for

high-level radioactive waste disposal (Liu et al. 2007).

Anisotropy of the damage-induced permeability of the

Table 2 Model parameters for five typical crystalline rocks

Rock Lac du Bonnet Senones Beishan Westerly Cerro Cristales

Es (MPa) 6.8 9 104 6.5 9 104 6.2 9 104 7.0 9 104 7.0 9 104

ms 0.21 0.25 0.21 0.25 0.28

/c ¼ wc (�) 46.4 45.0 35.0 43.0 43.0

c0 (MPa) 5 9 10-3 7 9 10-3 0 0 0

c1 (MPa) 1 9 10-2 4 9 10-3 8 9 10-3 6 9 10-3 5 9 10-3

b0 5 9 10-4 5 9 10-4 4 9 10-4 2 9 10-3 5 9 10-4

d0 0.01 0.01 0.01 0.01 0.01

kn0(MPa) 1 9 104 1 9 104 1 9 104 5 9 103 5 9 103

ks (m2) 1 9 10-22 1 9 10-19 8 9 10-20 5 9 10-20 5 9 10-21

kc0 (m2) 1 9 10-17 1 9 10-15 8 9 10-16 2 9 10-17 1 9 10-18

d 0.85 0.85 0.65 0.90 0.90

Table 1 Description of the permeability tests for five typical crystalline rocks

Type of rock Confining pressure (MPa) Pore fluid Control mode Testing method

Lac du Bonnet granite 10 N/A N/A N/A

Senones granite 5 N/A N/A N/A

Beishan granite 10 Water Axial load control to *140 MPa

and then circumferential strain

control at 5 9 10-5 s-1

Classical transient

method

Westerly granite 10 (this is an effective confining

pressure, with pore pressure

&50 MPa and confining

pressure = 60 MPa)

Water Constant axial strain rate at

9 9 10-8 s-1Pore pressure

oscillation

techniqueCerro Cristales granodiorite

Micromechanical Modeling of Anisotropic

123

crystalline rocks is observed in the predicted permeability–

deviatoric stress curves, even though the rock samples are

assumed to be initially isotropic.

Some deviations of the model predictions from the

experimental data occur, but they do not indicate a limi-

tation of the predictive capability of the proposed model,

(a) (b)

Fig. 2 Damage-induced stress–strain behavior and permeability variation for Lac du Bonnet granite under triaxial test condition with 10 MPa

confining pressure: a the stress–strain curves; and b the permeability–deviatoric stress curves

(a) (b)

Fig. 3 Damage-induced stress–strain behavior and permeability variation for Senones granite under triaxial test condition with 5 MPa confining

pressure: a the stress–strain curves; and b the permeability–deviatoric stress curves

(a) (b)

Fig. 4 Damage-induced stress–strain behavior and permeability variation for Beishan granite under triaxial test condition with 10 MPa

confining pressure: a the stress–strain curves; and b the permeability–deviatoric stress curves

Y. Chen et al.

123

given the fact that the experimental results are rather sen-

sitive to test conditions for the low-permeability crystalline

rocks. For example, for the Lac du Bonnet granite (see

Fig. 2), the initial compression of the pre-existing micro-

cracks could hardly be observed in the initial region of the

stress–strain curves, but a dramatic permeability decrease

by two orders of magnitude is shown in the experimental

permeability–deviatoric stress curve. Hence, in numerical

modeling, the main focus is to demonstrate at the micro-

scopic scale the mechanisms for the permeability variation

and to seek a compensated solution for better capturing

both the mechanical and the hydraulic responses. It can be

inferred that the experimental condition on the Beishan

granite (see Fig. 4) might not be well controlled. After its

initial decrease, the measured permeability increases

immediately in a stepwise manner, thus indicating that

microcracks were produced at a rather low stress level, but

the expected nonlinearity and volumetric dilation are not

observed in the stress–strain curves. The stress–strain

curves of the Westerly granite (see Fig. 5) show a

remarkable lateral deformation before unstable microcrack

growth, which may have resulted from the extremely high

pore water pressure (around 50 MPa) prescribed on the

rock sample (Mitchell and Faulkner 2008). As previously

mentioned, the effect of pore pressure on microcracking

and damage was not considered in our model and this

effect possibly leads to the lateral deformation.

It seems that the model in this study is similar to Jiang’s

model (Jiang et al. 2010) in the formulation framework and

in the predictive capability. Both models are based on

Zhu’s anisotropic damage model in the mechanical part,

with some specific extensions, and on the Voigt estimate in

the permeability modeling. The main different features of

the models are discussed below:

1. The proposed model links the volume fraction of

microcracks uc to the internal variable b by Eq. (3).

Compared to Jiang’s model, this relation is more

(a) (b)

Fig. 5 Damage-induced stress–strain behavior and permeability variation for Westerly granite under triaxial test condition with 10 MPa

effective confining pressure: a the stress–strain curves; and b the permeability–deviatoric stress curves

(a) (b)

Fig. 6 Damage-induced stress–strain behavior and permeability variation for Cerro Cristales granodiorite under triaxial test condition with

10 MPa effective confining pressure: a the stress–strain curves; and b the permeability–deviatoric stress curves

Micromechanical Modeling of Anisotropic

123

rigorous given the fact that the crack volume fraction

is another important parameter for characterizing the

effective permeability. Besides, the damage-induced

permeability model is formulated in an integral form

over the surface of a unit sphere, which leads to a more

comprehensive formulation of the model.

2. The recovery of degraded stiffness in the direction

perpendicular to closed microcracks under predomi-

nantly compressive condition is modeled by a nonlin-

ear hyperbolic relation using only one parameter, kn0.

This behavior, however, is modeled with a power

function using two additional parameters in Jiang’s

model. Furthermore, in the proposed model, the total

porosity of the rocks is separated into a volume

fraction of microcracks sensitive to initial compression

and the remaining fraction representing the back-

ground pore system in the matrix. This treatment well

characterizes the influences of the stiffness recovery of

closed microcracks on both the stress–strain behavior

and the induced permeability variation.

3. Instead of an empirical connectivity function which

involves two additional parameters in Jiang’s model,

the proposed model uses only one constant, d, to

consider the effects of microcrack connectivity and

microcrack growth on the permeability evolution.

Although the developed model is also empirical, the

physical meaning is more explicit and easier laboratory

parameterization results because of fewer material

parameters.

6 Simulation Studies of the TSX Tunnel Excavation

and Permeability Test

In this section, the application of the damage-induced

permeability model is presented to demonstrate the damage

growth and permeability changes in the intact surrounding

rocks of a field-scale tunnel. The target of the application is

a test tunnel associated with the tunnel sealing experiment

(TSX) at the AECL’s URL in Canada. The numerical

predictions are validated against the field observations and

measurements of excavation-induced damage and perme-

ability changes made during the TSX tunnel excavation.

The URL is located approximately 120 km NE of

Winnipeg, Manitoba, Canada within the Lac du Bonnet

granite batholith near the western edge of the Canadian

Shield and was developed to study issues related to deep

geologic nuclear fuel disposal, particularly the processes

involved in progressive failure, the development of exca-

vation-induced damage, as well as the thermal and

hydraulic behaviors around underground openings (Marti-

no and Chandler 2004).

6.1 Brief Description of the Experiment

The 40-m-long TSX tunnel (Room 425) was excavated at a

level 420 m below the ground surface, at which the granitic

rock is basically non-fractured. To minimize the excava-

tion-induced damage or disturbance in the surrounding

rock, the tunnel was excavated using a drill-and-blast

technique in the direction of the maximum initial principal

stress. The cross section was elliptical, with approximate

dimensions of 3.5 m in height and 4.375 m in width. The

major axis of the elliptical cross section was aligned to the

intermediate initial principal stress, oriented about 8� with

respect to the horizontal direction, while the minor axis

was subvertical and aligned to the minor initial principal

stress. The magnitudes of the initial principal stresses at the

site are about -55 ± 5 MPa, -48 ± 5 MPa, and

-11 ± 5 MPa, respectively (Souley et al. 2001). Hydraulic

experiments including pulse tests with the SEPPI probe

were performed on eight short radial boreholes (3–4 m in

length) drilled around the tunnel. The EDZ was assessed by

the microvelocity probe (MVP) method for measuring

changes in sonic velocities and the SEPPI method for

measuring changes in transmissivity (or permeability by

dividing the tested values by the thickness of the skin line),

together with microseismic events and borehole camera

surveys, as shown in Fig. 7. In the plot, an inner damage

zone within 0.3 m from the tunnel surface was character-

ized from the remaining outer damage zone by a more

rapid decrease in velocity and more rapid increase in

transmissivity (Martino and Chandler 2004).

6.2 Computational Model

As claimed by Souley et al. (2001), the high anisotropic

stress condition and the comprehensive in situ measure-

ments in the TSX tunnel made it a challenge for numerical

MVP inner damage zone

SEPPI inner damage zone

MVP outer damage zone

SEPPI outer damage zone

2.188

1.750 m

m

Fig. 7 Excavation-induced damage zone around the TSX tunnel

assessed by changes in velocity and transmissivity (Martino and

Chandler 2004)

Y. Chen et al.

123

modeling in the evaluation of change in permeability

induced by microcrack growth. For this purpose, a quarter

of the tunnel was selected for numerical analysis, given the

symmetries of the tunnel geometry with respect to the

in situ stress state. A 3D finite element mesh of 0.2 m in the

tunnel axis and 20 m 9 20 m in the cross section was

generated, with 2,592 brick elements and 5,390 nodes, as

shown in Fig. 8. As a result, the elements immediately

around the tunnel (where the damage is expected to be

more developed) have a size of 0.1 m (radial) 9 0.06 m

(tangential) 9 0.2 m (axial). Such a size of the elements is

close to the size of rock samples in laboratory conditions

(on which the material parameters are calibrated), and the

scale-dependent issue possibly involved in the numerical

modeling is expected to be of secondary importance.

Similar to Souley et al. (2001), the model was applied with

the following initial stress state: r1 = -55 MPa (in the

tunnel axis), r2 = -48 MPa (in the major axis of the

elliptical cross section), and r3 = -12.8 MPa. The ellip-

tical excavation was then modeled by prescribing roller

boundaries on the model surfaces.

The model parameters are listed in Table 3, in which

some of the parameter values were taken directly from the

calibration results under laboratory conditions. Given that

the tested rock samples were taken at a depth of 240 m,

some differences in the characteristics of the granitic rock

around the TSX tunnel could, therefore, be expected

(Souley et al. 2001). Compared to the parameters for the

Lac du Bonnet granite listed in Table 2, lower values of c0,

c1, and /c and higher values of b0 and d were used for

consideration of the influences of the dilute fractures

developed in the surrounding rock (not present in the lab-

oratory rock samples) and the dynamic forces during dril-

ling and blasting (not modeled in this study) on the damage

process. These two effects actually facilitated the initiation

and growth of microcracks, and led to higher connectivity

and persistence of the crack system. For the same reason,

higher values of ks and kc0; of one order of magnitude lar-

ger, were calibrated for the surrounding rock.

6.3 Simulation Results

In the presented damage model, microcracks are allowed to

grow in any direction according to the stress path resulting

from tunnel excavation-induced stress release and redis-

tribution in the surrounding rock. This means that the

damage density d has different values in different direc-

tions at different stress states, and, hence, the accumulated

mean microcrack density �d ¼ 14p

H‘2 ddS provides an overall

view of the damage growth. Figure 9 plots the distributions

of the mean damage density at the cross section and along

profiles P2 and P3 after tunnel excavation, where profiles

P2 and P3 exactly correspond to boreholes MVP03 and

MVP01 drilled for in situ measurements, respectively, in

the directions of initial intermediate and minor principal

stresses. Figure 10 depicts the corresponding distributions

of radial and tangential stresses along profiles P2 and P3,

predicted by the damage model and the elasticity model,

respectively.

One observes from Fig. 9 that the mean damage density

in the surrounding rock grows from its initial value, 0.01, to

a much larger value, 0.51, at the tunnel roof and floor

(along profile P3) and to 0.24 at the side walls (along

profile P2), induced by elliptical excavation. A greater

extent of damage (with the density 2–5 times larger than its

initial value) only appears within 0.5–0.6 m away from the

tunnel surface, and this area can be characterized as the

EDZ, as sketched in Fig. 9a. In the damage zone, the mean

damage density tends to distribute more densely closer to

the tunnel surface, especially around the tunnel roof and

floor. A threshold value of 0.15–0.2 can be reasonably

selected for the mean damage density to characterize the

inner damage zone with greater damage growth, as sket-

ched in Fig. 9a. A comparison between Figs. 9a and 7

shows that this characterization of the inner and outer

damage zones by the mean damage density is rather close

to those measured by using the SEPPI and MVP methods,

especially the SEPPI method, indicating that a relationship

may be established between the internal damage variable

and the sonic velocity or other geophysical measurements,

for better understanding the microscopic mechanisms that

lead to the formation of an EDZ.

The stress redistribution plotted in Fig. 10 explains the

reason for the pattern of damage growth in the vicinity of

Fig. 8 Finite element mesh for modeling the elliptical excavation of

the TSX tunnel. The mesh is generated with 3D brick elements, and

the size along the tunnel axis (i.e., y axis) is 0.2 m

Micromechanical Modeling of Anisotropic

123

the tunnel. At the side walls, the deviatoric stress induced

by tunnel excavation is rather small (i.e., rr � rh ¼4:6MPa), while at the roof and floor, the induced deviatoric

stress shows a concentration up to 102.5 MPa, a magnitude

much closer to an estimation of 100 MPa by Martin (2005).

This deviatoric stress is only slightly lower than the in situ

compressive strength, estimated to be about 120 MPa

(Martin 2005; Rutqvist et al. 2009). The model prediction

is consistent with the in situ measurements by the SEPPI

method and the monitoring of microseismic events, which

indicated a notch-like extension of the inner damage zone

at the top and bottom of the tunnel, with visible fractures

detected by a borehole camera but without observation of

extensive fall-out of rock (Martino and Chandler 2004).

Also shown in Fig. 10 is that the predicted stress by the

damage model deviates visibly from the prediction by

using the elasticity model only in the inner damage zone.

The excavation-induced variations in permeability

along profiles P2 and P3 of the tunnel are plotted in

Fig. 11. The model predictions agree well with the per-

meability change in the damage zone measured by the

in situ SEPPI probes and the anisotropic nature of the

0.02

0.07

0.13

0.13

0.18

0.290.39

0.18

1.75

0 m

2.188 m

Excavation-induced

outer damage zone

0.5 m

0.6

m

0.50

Excavation-induced

inner damage zone

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

P2 (side wall) P3 (roof)

dam

age

dens

ity d

Distance from the tunnel surface (m)

Excavation-induceddamage zone

(b)

Fig. 9 Distribution of the mean density of microcracks �d in the surrounding rock of the TSX tunnel: a the contours of �d at the tunnel cross

section; and b the distribution of �d along profiles P2 and P3

0.0 0.5 1.0 1.5 2.0 2.5 3.0-40

-30

-20

-10

0

Str

ess

(MP

a)

Distance from the tunnel surface (m)

solid line: elasticity modeldot point: damage model

0.0 0.5 1.0 1.5 2.0 2.5 3.0-120

-100

-80

-60

-40

-20

0

Str

ess

(MP

a)

Distance from the tunnel wall (m)

solid line: elasticity modeldot point: damage model

(a) (b)

Fig. 10 Comparison of stress distribution predicted by the damage model and the elasticity model: a along profile P2; and b along profile P3

Table 3 Model parameters for the Lac du Bonnet granite around the TSX tunnel

Es (MPa) ms /c ¼ wc (�) c0 (MPa) c1 (MPa) b0

6.8 9 104 0.21 38.5 0 4 9 10-3 1 9 10-3

d0 kn0 (MPa) ks (m2) kc0 (m2) d

0.01 1 9 104 1 9 10-21 1 9 10-16 1.50

Y. Chen et al.

123

permeability induced by preferential damage growth. The

permeability in the damage zone increases dramatically,

over four orders of magnitude along profile P3 where the

surrounding rock undergoes greater damage, and about

two orders of magnitude along profile P2, where less

damage is induced. The permeability tends to its initial

undisturbed value (1–3 9 10-21 m2) out of the damage

zone (i.e., 0.5–0.6 m away from the excavation surface).

A comparison between Figs. 9b and 11 shows that the

permeability increase evolves in a rather similar pattern to

the mean damage density, with the former considering

more factors, such as the density, closure/opening, con-

nectivity, and persistence of the microcracks. In the

damage zone, the predicted tangential permeability kh is

generally 2–6 times greater than the predicted radial

permeability kr. This anisotropy of permeability is mostly

induced by the stress release on the tunnel wall, where

microcracks tend to preferentially develop along the

direction parallel to the tunnel surface.

Figure 12 plots the spatial distribution of the principal

permeability at the cross section of the tunnel. The plots

further illustrate not only the anisotropic variation in per-

meability induced by excavation, but also the close relation

to the damage growth pattern in the surrounding rock.

Permeability changes dramatically in the roof and floor as a

result of intense damage growth in these regions. The left

and right sides of the tunnel also undergo permeability

change to a certain extent. In the diagonal directions of the

tunnel, the excavation-induced damage seems to be less

developed and smaller permeability change is predicted.

The inner damage zone suffers a permeability increase of

over three orders of magnitude, and the outer damage zone

over one order of magnitude. On the other hand, the cal-

culated minor permeability component is always in the

radial direction of the tunnel, while the major and inter-

mediate permeability components almost have the same

magnitude along the tunnel axis and the tangential direc-

tion of the tunnel. This indicates that the in situ pulse

testing results mainly correspond to the tangential or axial

components of the permeability.

7 Conclusion

In this study, the micromechanical damage mechanisms

induced by mechanical loading or excavation were

0.0 0.5 1.0 1.5 2.0 2.5 3.01E-22

1E-21

1E-20

1E-19

1E-18

1E-17

1E-16

Per

mea

bilit

y (m

2 )

Distance from the tunnel surface (m)

k , predicted

kr, predicted

measured

0.0 0.5 1.0 1.5 2.0 2.5 3.01E-22

1E-21

1E-20

1E-19

1E-18

1E-17

1E-16

Per

mea

bilit

y (m

2 )

Distance from the tunnel wall (m)

k , predicted

kr, predicted

measured

(a) (b)

Fig. 11 Variation of measured and predicted permeability in the surrounding rock of the TSX tunnel: a along profile P2; and b along profile P3

(a) (b) (c)

Fig. 12 Contours of the principal permeability with units m2 at the cross section of the TSX tunnel: a k1; b k2; and c k3

Micromechanical Modeling of Anisotropic

123

formulated for crystalline rocks of low porosity. The

recovery of normal stiffness due to normal closure of com-

pressed microcracks was modeled with an empirical

hyperbolic relation between the normal deformation and the

local normal stress of the microcracks. The sliding and

dilatancy behaviors of the microcracks were described by a

non-associative flow rule. As a major contribution of this

study, lower bound estimates (i.e., the dilute, MT, HS, and

IDD estimates) were formulated, by virtue of an analytical

ellipsoidal inclusion solution, for the damage-induced per-

meability of the microcracks-matrix system, and their pre-

dictive limitations were discussed by ascribing infinite

permeability to microcracks in which the greatest lower

bounds are achieved. On this basis, an empirical upper

bound estimation model was suggested to account for the

influences of damage growth, microcrack connectivity, as

well as microcrack normal stiffness recovery on the per-

meability evolution, with a small number of material

parameters for relating the microstructure alteration to

macroscopic observations. The damage-induced perme-

ability variation model was formulated in an integral form

over the surface of a unit sphere and implemented with the

Gauss-type numerical integration method, for the consid-

eration of any possible damage growth pattern at variable

stress paths and the anisotropic nature of microcracking-

induced permeability changes.

The damage-induced permeability estimation model was

validated against the triaxial compression tests with per-

meability measurements on five typical crystalline rocks of

extremely low porosity. Good agreements were achieved

between the predicted and measured values of both the

stress–strain curves and the permeability–deviatoric stress

curves. The initial microcrack closure region, elastic

region, and crack growth region in a typical stress–strain

relation were well captured by the damage model through

accounting for the microscopic mechanisms of the com-

pressive closure of pre-existing microcracks, damage ini-

tiation and growth, tensile stress-induced opening, and the

sliding and volumetric dilation of closed microcracks. The

damage-induced permeability variation was well modeled,

with a clear permeability decrease in the initial inelastic

region, a constant permeability in the elastic region, and a

dramatic permeability increase in the crack growth region.

Calibration studies also showed that only a small volume

fraction of microcracks was sensitive to initial compres-

sion, while the remaining microcracks or pores contributed

to the mechanical and hydraulic properties of the rock

matrix. This finding is helpful for better understanding the

mechanisms involved in the damage growth, which induces

the initial permeability decrease in cracked rocks.

The proposed model was finally applied to estimate the

damage zone induced by elliptical excavation and the

permeability variation in the surrounding Lac du Bonnet

granite of the TSX tunnel. The mean value of the internal

variable, the damage density d, was selected to charac-

terize the excavation-induced inner and outer damage

zones and resulted in good agreements with those asses-

sed by change in sonic velocities using the MVP or

change in transmissivity using the SEPPI probe, espe-

cially the latter. The model predictions of the excavation-

induced permeability variation agreed well with the per-

meability changes in the vicinity of the tunnel measured

by the in situ SEPPI technique, in about 2–4 orders of

magnitude.

Acknowledgments The authors gratefully appreciate the anony-

mous reviewers for their valuable comments and constructive sug-

gestions in improving this study. The financial support from the

National Basic Research Program of China (No. 2011CB013500) and

the National Natural Science Foundation of China (Nos. 51222903,

51079107, and 51179136) are gratefully acknowledged.

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