International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
Contents lists available at ScienceDirect
International Journal of
Rock Mechanics & Mining Sciences
journal homepage: www.elsevier.com/locate/ijrmms
Computational modelling of crack-induced permeability evolution in
granite with dilatant cracks
T.J. Massart a,n, A.P.S. Selvadurai b
a
b
Building, Architecture and Town Planing Department CP 194/2, Université Libre de Bruxelles (ULB), Av. F.-D. Roosevelt 50, 1050 Brussels, Belgium
Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, Canada H3A 2K6
art ic l e i nf o
a b s t r a c t
Article history:
Received 7 August 2012
Received in revised form
21 April 2014
Accepted 21 June 2014
A computational homogenisation technique is used to investigate the role of fine-scale dilatancy on the
stress-induced permeability evolution in a granitic material. A representative volume element incorporating the heterogeneous fabric of the material is combined with a fine-scale interfacial decohesion
model to account for microcracking. A material model that incorporates dilatancy is used to assess
the influence of dilatant processes at the fine scale on the averaged mechanical behaviour and on
permeability evolution, based on the evolving opening of microcracks. The influence of the stress states
on the evolution of the spatially averaged permeability obtained from simulations is examined and
compared with experimental results available in the literature. It is shown that the dilatancy-dependent
permeability evolution can be successfully modelled by the averaging approach.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
Stress-induced permeability alteration
Computational homogenisation
Multiscale modelling
Dilatant plasticity
Triaxial loading
Fluid transport
1. Introduction
The isothermal mechanical behaviour of fluid-saturated porous
media is described by Biot's theory of poroelasticity [1]. This
theory has been successfully applied to describe many types of
geomaterials as illustrated in the reviews presented in [2–6].
It accounts for the coupling between the porous fabric and the
fluid saturating the pore space, often with the implicit assumption
that the material properties such as porosity and permeability
remain unchanged during the coupled interaction. However, it is
known that such properties can evolve as a result of mechanical,
thermal or chemical actions. A number of experimental contributions have identified the potential effect of stress states at the
micromechanical level on the permeability of porous media.
Experimental results presented in [7–10] show substantial evolution of permeability in granite and limestone samples associated
with microcracking under deviatoric stresses. Such permeability
alterations in turn may strongly affect the poromechanical processes that control pore fluid pressure dissipation.
In addition to these experimental procedures, computational
approaches have recently been utilised to evaluate the effective
properties of heterogeneous porous media [11,12]. Computational
n
Corresponding author: Tel.: þ32 2 650 27 42; fax: þ 32 2 650 27 89.
E-mail address: [email protected] (T.J. Massart).
URL: http://batir.ulb.ac.be (T.J. Massart).
http://dx.doi.org/10.1016/j.ijrmms.2014.06.006
1365-1609/& 2014 Elsevier Ltd. All rights reserved.
homogenisation principles have been employed to evaluate
damage-induced permeability alterations [13]. The versatility of
this approach is that it enables the incorporation of a variety of
micromechanical constitutive laws to investigate the potential
influence of various features or phenomena, such as the local
fabric of geomaterials, inter- and trans-granular cracking, or pore
closure on parameters such as permeability. The effect of damage
was investigated in [13] using a phenomenological description of
intergranular cracking at the scale of the constituents. In addition
to disregarding transgranular cracking and to maintain a purely
phenomenological approach, Ref. [13] neglects the effect of plastic
dilatancy, which is often considered to be one of the main causes
of permeability evolution associated with micro-cracking [14–16].
Dilatancy is a material property that is difficult to evaluate
experimentally, which may explain why it usually receives less
attention in routine engineering applications [17]. Various authors
have investigated dilatancy aspects and its potential dependence
on the confining stress and plastic shearing [17–21].
Regarding the effects of microcracking on the average response
of rocks, a number of important contributions were published
based on approaches of phenomenological and micromechanical
nature. The works by Kachanov and co-workers investigated the
influence of microcracks on the material behaviour by analysing
the effect of frictional cracks on stress-induced anisotropy [22] and
on the propagation of secondary cracks together with local
dilatancy [23]. Subsequent efforts along these lines focused on
characterising computationally the mechanics of crack–microcrack
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interactions [24] and the effects of microcrack interaction on
elasticity properties [25]. Other contributions dealt with the
potential use of crack interaction features to model crack coalescence [26], and with the validation of the use of the penny-shaped
crack effective medium theory for irregular fractures, see [27].
Contributions using such penny-shaped crack approximations
were also provided in [28] to investigate the link between
macroscopic dilatancy and multiscale friction and, recently, in
[29] to incorporate the effect of initial stresses in microcrack-based
models. Another significant stream of research for models incorporating cracking effects was proposed by Dragon and co-workers
initially based on a micromechanically motivated decomposition
in the frame of thermodynamics of irreversible processes, see [30].
This approach was progressively developed, focusing on volumetric dilatancy, pressure sensitivity and stiffness recovery [31];
and on the interaction between initial and cracking-induced
anisotropy [32]. The effect of anisotropic damage and unilateral
effects on the elastic behaviour was analysed in [33]; subsequent
evolutions of the approach were proposed to account for discrete
orientations of cracking in the formulation of thermodynamical
potentials [34,35].
The coupling between microcracking and fluid transport phenomena was also investigated using penny-shaped crack approximations in [36] for permeability alterations; as well as for crack
propagation [37]. This methodology was used in [38] to link
microdamage and macrodamage tensors to permeability evolution
using 2D cuts of 3D representative volume elements (RVEs)
containing discrete cracks. Non-evolving discrete fracture networks (DFN) were also used in several contributions. Such
approaches were used in [39] to compute permeability tensors
from 2D geometrical models for macroscopic fractures. 2D discrete
fracture networks were also used in [40,41] to compute equivalent
permeability with statistical distributions of fracture lengths and
apertures, in [42] with a discussion on the existence of a RVE, and
in [43] to analyse equivalent permeability as a function of discrete
fracture networks properties. The coupling between the mechanical behaviour and DFN approaches uses discrete element formulations (DEM) as performed in [44] to deduce stress-dependent
equivalent permeabilities for 2D RVEs incorporating fracture dilation. DEM was also combined with DFNs to assess stress-induced
permeability evolution using 2D models in [45,46], and to compute bounds on 2D permeability for 3D discontinuity networks
using constant flux and linear pressure boundary conditions on
RVEs [47].
In view of these contributions, there is an interest for an
approach that allows investigating the link between the dilatancy
parameters postulated at the scale of evolving microcracks and the
permeability evolution in rocks at the macroscopic scale. The main
objective of this paper is to show that the homogenisation scheme
presented in [13] allows for the computational assessment of the
dilatancy-dependent permeability evolution in a granitic material,
based on the simplest possible dilatant crack formulation at the
scale of the heterogenous fabric of the material. Unlike the
phenomenological approach used in [13], the coupling between
cracking and local permeability evolution will be directly based on
the opening of microcracks caused by the externally applied stress
state and the local dilatancy. This effort should be considered in a
broader context in which computational homogenisation is used
as a generic methodology to assess the effect of specific fine scale
phenomena and microstructure evolution on average properties,
thereby paving the way for the incorporation of other evolving fine
scale features, such as crack closure or pore collapse.
The paper is structured as follows: Section 2 will briefly outline
the essential aspects of the averaging scheme used to extract
the averaged mechanical and transport properties from representative volume elements (RVEs) of an heterogeneous material.
The dilatant plastic interface formulation used in the simulations,
which is based on the mechanical parameters such as cohesion,
tensile strength and their corresponding fracture energies, will be
presented in Section 3 . The coupling applied between the local
cracking state and the local permeability evolution is also presented in Section 3. These procedures are used in Section 4 to
estimate the variation of permeability of a granitic material with
the confining pressure and the deviatoric stresses applied in
triaxial testing. The results obtained are compared with experimental data on both the mechanical response and the permeability evolution, leading to a detailed assessment of the influence
of the fine scale parameters. The results are discussed in Section 5,
and concluding remarks are given in Section 6.
2. Computational homogenisation of mechanical and
transport properties
Computational homogenisation techniques were initially used
in the field of mechanics of materials to extract average properties
of heterogeneous materials [48,49]. They allow the identification
of the macroscopic constitutive parameters or the investigation of
the behaviour of heterogeneous materials by using scale transitions applied to RVEs, based on averaging theorems and laws
postulated for constituents of the microstructure. Geomechanical
applications of multi-scale techniques have mostly focused on the
extraction of average properties of non-evolving microstructures for
uncoupled mechanical [50,51] and fluid transport [40,41,47] processes in geomaterials. Computational homogenisation was reformulated to allow nested scale computational schemes in which
finite element discretisations are used at both the macroscopic and
fine scales simultaneously, which avoids the use of a priori
postulated macroscopic laws [52]. This methodology was subsequently adapted to model the failure of quasi-brittle materials
[53,54], as well as diffusive phenomena such as thermal conductivity [55,56]. In the recent study [13], these techniques were
combined to evaluate damage-induced permeability evolutions in
a geomaterial. The essential features of this approach are summarised here for completeness. The detailed derivation of the
averaging relationships can be found in [13] and references
therein.
The homogeneous equivalent mechanical properties of a material with a heterogeneous microstructure can be deduced by
applying a loading to an RVE containing the main microstructural
features of the material, and solving the corresponding equilibrium problem [52]. When a macroscopic strain E is applied to an
RVE, the displacement of a point inside the RVE can be written as
!!
! ! !
u ð x Þ¼E x þ u fð x Þ
ð1Þ
!
!
where x is the position vector within the RVE and u f is a
fluctuation field caused by the heterogeneity of the material.
!
Assuming that the fluctuation u f is periodic, one can show that
the macroscopic strain is the volume average of the fine-scale
strain field ε resulting from (1)
Z
1
E¼
ε dV
ð2Þ
V V
Using the Hill–Mandel condition (energy equivalence between the
fine-scale and macroscopic descriptions) written as
Z
1
Σ : δE ¼
σ : δε dV
ð3Þ
V V
the macroscopic stress tensor is obtained as the volume average of
the microstructural stress tensor, or equivalently by a summation
T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
of RVE tying forces
Z
1
1 4 !ðaÞ ðaÞ
Σ¼
σ dV ¼ ∑ f !
x
V V
V a¼1
ð4Þ
where the summation spans four nodes controlling the RVE
loading [52].
Any type of material behaviour can be postulated at the fine
scale, and the periodicity of the microfluctuation field can be
enforced by homogeneous linear connections between corresponding nodes of the boundary of the RVE, provided identical
meshes are used on its opposing faces. Four controlling points
(denoted 1–4 in Fig. 1a) are used to apply the macroscopic stress or
deformation tensors on the RVE. The RVE equilibrium problem
under the macroscopic stress loading is then solved by imposing
!ðaÞ
forces f
at the controlling points, which represent the action of
the neighbouring continuum on the RVE. The displacements of the
controlling points, energetically conjugated to the imposed controlling forces, can be used to extract the macroscopic strain.
Likewise, the homogenised permeability for a given local
permeability distribution within the RVE can be evaluated using
the upscaling scheme developed for heat conduction given in [55].
For fluid flow, the mass conservation equation at the scale of the
595
components has to be solved. Assuming Darcy flow with a local
!
permeability distribution defined by Km ð x Þ and with a fluid of
dynamic viscosity μ mass conservation reads
!
!
Km ð x Þ !
!
∇ m: ∇ pm ¼ 0
ð5Þ
μ
!
A periodic fluctuation pf ð x Þ of the pressure field is assumed
[13,55] to describe the pressure variation inside the RVE according
to
!
!
! !k
!
pm ð x Þ ¼ pkm þ ∇ M pM ð x x Þ þ pf ð x Þ
ð6Þ
!
where ∇ M pM is the macroscopic pressure gradient to be applied in
an average sense on the RVE, and where pkm is the pressure of an
arbitrary point in the RVE. An averaging relation for the pressure
gradient is required as
Z
!
1 !
∇ M pM ¼
∇ m pm dV
ð7Þ
V V
In a computational treatment, the periodicity constraint on the
pressure fluctuation field requires that relationships of the form
!
!S !M
Þ
pSm pM
m ¼ ∇ M pM ð x x
ð8Þ
be satisfied between points (M) and (S) at opposite surfaces of the
boundary, which are related by a periodicity condition. Imposing
consistency between scales of the product of the pressure gradient
by the flux
Z
1 !
!
!
!
∇ M pM q M ¼
∇ m pm q m dV
ð9Þ
V V
combined with pressure gradient averaging (7), it can be shown
that the macroscopic flux is obtained as the RVE average of the
fine-scale fluxes [55]
Z
1 !
!
qM¼
q dV
ð10Þ
V V m
Details of the derivation of relationship (10) starting from the
relationship (9) can be found in [13]. The linear constraints (8) can
be enforced through controlling points to apply the macroscopic
pressure gradients to the RVE. The averaged permeability tensor of
the RVE can be identified from the relationship between the
applied macroscopic pressure gradient terms at the controlling
points and the fluxes developing as a reaction to them. At
equilibrium, the discretised system of equations for the transport
problem can be condensed at the controlling points. This allows
the fluxes at the controlling point ðaÞ to be expressed in terms of
the applied macroscopic pressure gradients at the controlling
point ðbÞ as
b¼4
ðabÞ
ðbÞ
qðaÞ
mn ¼ ∑ kdisc ð∇j pxj Þ
ð11Þ
b¼1
ðabÞ
where kdisc results from condensing the entire RVE hydraulic
stiffness values. Manipulating the expression of the macroscopic
flux and using periodicity, the averaged flux is obtained from the
‘reaction’ fluxes at the controlling points as [52]
qMi ¼
1 a ¼ 4 ðaÞ ðaÞ
∑ q x
V a ¼ 1 mn i
ð12Þ
Substituting relation (11) into (12) allows the identification of the
macroscopic (averaged) permeability KMij as
KMij ¼
Fig. 1. Cubic representative volume element (RVE) of the material; (a) schematic
view of RVE, (b) finite element mesh, and (c) distribution of material phases inside
the RVE.
μa¼4b¼4
ðabÞ
∑ ∑ xðaÞ k xðbÞ
V a ¼ 1 b ¼ 1 i disc j
ð13Þ
All the local permeabilities of the RVE are taken into account in the
volume averaging procedure underlying relationship (13) since the
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T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
macroscopic flux is obtained as a volume average of the finescale flux.
where ft is the tensile strength of the interface, and GIf is its mode I
fracture energy. The plastic parameter κt is chosen according to a
strain softening assumption:
κ_ t ¼ jδ_ n j ¼ λ_ t
p
3. Fine scale modelling of cracking and permeability evolution
3.1. Crack modelling strategy
To model the influence of cracking and to avoid computingintensive continuum descriptions, the incorporation of cohesive
laws in a priori positioned interface elements is used here. Such
elements will be introduced at the boundaries of all tetrahedral
elements in the mesh, see Section 4.1. This choice is physically
similar to discrete orientation-based approaches [34,47,57], with
the additional feature that the spatial connectivity of cracks is
explicitly incorporated, similar to the discrete fracture network
approaches [43,47]. The bulk constituents of the material are
modelled with linear elastic tetrahedral finite elements, between
which interface elements are inserted to represent potential
microcracks.
In order minimise the number of fine-scale material parameters, the simplest version of a non-associated 3D plastic
cohesive model is used, as presented in [58]. This model links
!
the traction vector T- across the interface to the elastic relative
e
displacement vector δ
e
-
T ¼Hδ
ð14Þ
where the matrix H represents the elastic behaviour of the
potential cracks. This matrix is represented as
½H ¼ diagðkt ; kt ; kn Þ
ð15Þ
with the elastic parameters defined in terms of the mineral species
considered as
E
h
e
hε ¼ kn δn ;
σ ¼ |{z}
δen
G
h |{z}
τt ¼ hγ t ¼ kt δet ;
δet
G
h |{z}
τs ¼ hγ s ¼ kt δes
ð16Þ
-
-
-
ð17Þ
and is expressed as the derivative of a plastic potential function
with respect to the stress as
n po
∂g
ð18Þ
δ_ ¼ λ_
∂fσ g
in which λ_ is a plastic multiplier and g is the plastic potential.
When considering quasi-brittle failure of geomaterials, different failure modes have to be considered to account for the various
loading modes. Here the choice was made to select the simplest
set of constitutive laws accounting for such failure modes. For
tension failure, an associated flow rule is selected. Similarly to the
choice made in [59], a Rankine-type yield function is used,
showing that plasticity appears when the normal stress across
the cohesive zone exceeds a threshold value σt
F tens ¼ σ σ t ðκ t Þ
Note that a tension failure criterion is adopted here because of the
high tensile strength predicted by a single frictional criterion (see
below). Under the confined stress states considered in this contribution, this tensile criterion only becomes active in points
where tensile stresses appear as a result of the heterogeneous
nature of the material.
To select the simplest possible model that accounts for dilatant
friction, under compressive stress states, a non-associated Mohr–
Coulomb model is used as performed in macroscopic approaches
for granite in [21,60,61]. The corresponding yield function is
written as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F mc ¼ τ2t þ τ2s þ σΦ cðκ m Þ
ð22Þ
where τs, τt are shearing stress components across the interface, σ
is the normal stress, and Φ ¼ tan ðφÞ with φ being the angle of
friction.
In order to control the volume variation associated with
frictional plasticity in cracks, plasticity under compressivefrictional behaviour is governed by a non-associative flow rule.
A plastic potential similar to the form (22) is used with a dilatancy
angle potentially different from the angle of friction. The derivative
of the assumed plastic potential is therefore
(
)t
τt
τs
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ψ
∂g
¼
ð23Þ
2
2
2
2
τt þ τs
τt þ τs
∂fσ g
where Ψ ¼ tan ðψ Þ, with ψ being the dilatancy angle. The failure
behaviour of the interface under compressive-frictional loading is
described by means of a cohesion softening law given by
!
c0
δes
where E and G are the bulk elasticity properties, and h is an
assumed thickness of the cracked zone. This thickness should be
chosen small, yet avoiding any conditioning problem in
the resulting finite element stiffness matrix. Using a plasticity
approach, the plastic relative displacement across the cohesive
zone is obtained as
δp ¼ δ δe
ð21Þ
ð19Þ
The quasi-brittle failure of the interface is modelled by means of
an exponential decay of σt in terms of a plastic parameter κt
!
f
σ t ðκ t Þ ¼ f t exp tI κ t
ð20Þ
Gf
cðσ ; κ m Þ ¼ c0 e
GIIf
κm
ð24Þ
GIIf
is its mode
where c0 is the initial cohesion of the interface, and
II fracture energy. The cohesion c is a function of the plastic
parameter κm associated with frictional cracking, which is defined
by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p
κ_ m ¼ ðδ_ t Þ2 þ ðδ_ s Þ2 ¼ λ_ m
ð25Þ
Multi-surface plasticity requires a careful treatment of the
intersection of surfaces corresponding to both yield criteria, as
well as the derivations of the return mapping operators for the
stress update, from which the consistent material tangent operator
ensues. More details about the derivations of the return mapping
algorithm and tangent operators can be found in [58,62]. Note that
this-fine scale constitutive setting was purposely kept simple to
limit the number of material parameters, and that the number of
strength parameters used here is the same as in macroscopic
models used recently for granite [63]. More detailed descriptions
can be developed to incorporate additional features such as
frictional softening coupled to cohesional softening, a confining
stress-dependent dilatancy or a confining stress-dependent
mode II fracture energy [64], but were left out of the scope of
the present paper.
3.2. Coupling permeability evolution to microcracking
An initially low permeability is assumed for both the (tetrahedral) bulk and the interface elements representing potential
microcracks. The permeability of the bulk elements remains
unchanged during the entire simulations, focusing on the
T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
permeability evolution induced by cracking. The hydraulic formulation used for the interface elements is chosen similar to that
presented in [65]. Based on the modelling of the failure of the
interfaces and their opening, the local fluid transport properties
!
Km ð x Þ in (5) are updated. The interface elements forming a
connected network lead to the progressive formation of flow
channels. In contrast to the phenomenological damagepermeability coupling used in [13], the normal plastic opening of
interfaces can be used here to couple cracking and permeability at
any point within the RVE. Still in the same spirit of selecting the
simplest approximation to couple microcracking and permeability
evolution, the parallel plate model, developed using Stokes' flow
or slow viscous flow between two parallel plates classically used in
rock fractures [66–68], will be used to calculate the effective
permeability of a fracture [69]:
kloc ¼
e2h
12
ð26Þ
where eh is the hydraulic aperture of the crack. Fractures are,
however, different from ideal parallel plates, and their hydraulic
apertures are not equal to their mechanical aperture. These
aspects were investigated in [70], where the relationship between
the physical and theoretical apertures was discussed, together
with the influence of roughness on fluid flow. Starting from an
initial permeability, an initial value of hydraulic aperture eh0 for
potential cracks may be deduced from (26). Using this initial
hydraulic aperture, a linear relationship between the hydraulic
and mechanical apertures of a fracture was proposed in [71] as
eh ¼ eh0 þ f Δem
ð27Þ
where Δem is the variation in the mechanical aperture of the
fracture, and f is a proportionality factor. Here, the variation of the
hydraulic aperture will be related to the mechanical plastic normal
relative displacement of the interface according to a linear
relationship:
eh ¼ eh0 þ f 〈δn 〉
p
ð28Þ
where the factor f reflects the roughness of the fracture surfaces,
and where 〈:〉 indicates that only positive values of a quantity are
considered. In [72] experimental results indicate a variation of
f between 0.5 and 1. It is emphasised that the value of the
mechanical plastic normal relative displacement δpn depends on
the response of the interface under general stress states, including
the tangential stress inducing a plastic uplift through the dilatant
response. The effect of the magnitude of this dilatancy and of the
parameter f will be assessed in Section 4.
Based on the results of the mechanical simulation of an RVE
using the computational homogenisation procedure described in
Section 2, the hydraulic apertures of potential cracks are updated
according to the relationship (28) and the local permeabilities are
consequently adapted. The resulting averaged axial permeability of
the RVE can then be computed using the transport homogenisation relationships (5)–(13).
4. Permeability evolution in a granitic rock
4.1. Representative volume element and computational procedure
The RVE used in the present paper is a cubic RVE with
dimensions 10 10 10 mm3. This cube is discretised using
an unstructured mesh of tetrahedral elements with quadratic
shape functions [73]. A specific attention is devoted to the
production of a periodic mesh at the boundary of the RVE in
order to enforce periodic fluctuation fields assumed in the homogenisation procedure. To allow the progressive development of
597
continuous crack paths under loading, this unstructured mesh is
subsequently modified to incorporate cohesive zone elements
(interfaces) between all the tetrahedral elements. The periodicity
relationships therefore need to be adapted to ensure the possibility of crack extension between opposite faces of the RVE connected by periodicity. The interconnection of all these interface
elements results in a mesh capable of accommodating any crack
path developed in the material according to the evolution of its
microstructure caused by cracking. In order to trigger the development of a non-homogeneous deformation state in the RVE and
to initiate physically admissible cracking, the elastic heterogeneity
of the material is taken into account. The microstructural data
used in [74] for Stanstead granite is used as a guide for this
purpose, being of a composition similar to Lac du Bonnet granite
used here as a test material [75]. Accordingly, the tetrahedral
elements are split into three phases, representing biotite, quartz
and feldspar. A random spatial selection of tetrahedra is organised
to obtain the prescribed volume fractions reported in [74,76]. The
volume fraction of the respective constituents is V fbiotite ¼ 7%,
V fquartz ¼ 20% and V ffeldspar ¼ 73%.
The RVE mesh used in subsequent computations is illustrated
in Fig. 1b. The spatial arrangement of tetrahedra belonging to each
of the phases (feldspar, biotite and quartz) are also illustrated for
completeness in Fig. 1c. The procedure described above leads to a
mesh containing 14,090 tetrahedral elements and 26,580 interface
elements, which gives 140,900 nodes and 422,700 dofs. To
efficiently solve algebraic systems of this size, iterative solvers
have to be used. For associated plasticity, leading to symmetric
systems of equations, the preconditioned conjugate gradient
algorithm (PCG) can be used with a preconditioner obtained by
means of an incomplete Cholesky factorisation of the system
matrix. For non-associated cases, the bi-conjugate gradient algorithm can be used with a preconditioner obtained by a sparse
incomplete LU factorisation, see [77].
4.2. Material parameters
Lac du Bonnet granite was used as a test material, as cracking
represents the dominant source of its permeability evolution, and
experimental data is available in the literature, both for triaxial
tests and permeability measurements. The original set of experimental results for the mechanical response is given in [10].
The elastic properties used for the bulk materials are as follows:
Ebiotite ¼ 40 GPa, νbiotite ¼ 0:36, Equartz ¼ 100 GPa, νquartz ¼ 0:07,
Efeldspar ¼ 80 GPa, νfeldspar ¼ 0:32. Note that these Young's moduli
are in the range of values for these species and were selected to
match the elastic slopes reported in experimental results [63]. The
elastic parameters associated with interfaces joining dissimilar
bulk phases were selected as the lowest among these bulk phases.
The fine-scale failure parameters chosen here are based on
the macroscopic values for granite. The macroscopic values of the
cohesion and the angle of friction in Lac du Bonnet granite were
measured and discussed in [61], showing that friction and cohesion are not activated simultaneously during the fracture process.
From these tests, an average cohesion is of the order of 30 MPa,
whereas the angle of friction φ is around 401. Similar values were
found for Barre granite [78] with a cohesion of 50 MPa, and a
friction angle of 351. The compressive properties reported in [79]
range from 26 MPa to 160 MPa for the (unconfined) compressive
strength with compressive fracture energies ranging from 2.3 N/
mm to 10 N/mm for the pre-peak component and from 11 N/mm
to 45 N/mm for the post-peak response. In [80], the tensile
strength of granite ranged from 2 MPa to 8 MPa, with a Mode-I
fracture energy between 0.1 N/mm and 0.3 N/mm, while the
reported tensile strengths were in the range of 6–10 MPa for Barre
granite [81] and for Lac du Bonnet granite [82]. The shear fracture
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T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
energy of Stripa granite was investigated in [83] with values
between 5 N/mm and 50 N/mm. Based on these sources, the
reference values of the mechanical parameters of the potential
cracks used in subsequent computations were selected in the
ranges mentioned as follows: h¼0.05 mm, f t ¼ 6 MPa, GIf ¼
1 N=mm, c ¼ 26:4 MPa, φ ¼ 401, GIIf ¼ 10 N=mm, ψ ¼ ½201; 401.
Note that in the absence of detailed experimental data, the effect
of variability in the strength properties of the interfaces was
assessed, and is discussed in the sequel.
The dilatancy angle is a material property that is difficult to
evaluate in rock joints since it depends on both the material
density and the stress state. These experimental procedures either
use volume change or evaluate the dilatancy indirectly through its
influence on other properties. A comprehensive review of existing
experimental results can be found in [17]. In most sources, upper
and lower bounds of the dilatancy effects are considered by using
either an associated flow rule (ψ ¼ φ) or by neglecting the microcrack dilatancy (ψ ¼01). Recommendations in [84] suggest the use
of dilatancy angles depending on the quality of the rock, ranging
from ψ ¼01 to ψ ¼ φ/4, while the rules of thumb mentioned in [85]
approximate dilatancy angles using ψ ¼ φ 201. Consequently, an
initial set of simulations was performed using the upper bound
dilatancy associated flow rule, followed by an analysis of the
effects of dilatancy variations. These simulations are presented
for both the mechanical response and the evaluation of the
ensuing permeability evolutions.
It is emphasised that apart from the initial stiffness and tensile
parameters, the fine scale frictional model is restricted to four
parameters, the same number as in the macroscopic model discussed
in [63]. Note also that the tensile parameters have a low influence for
confined stress states as considered here, and can reasonably be
approximated by their macroscopic counterparts.
4.3. Associated dilatant formulation
A first set of simulations was conducted assuming an associated
flow rule (ψ ¼ φ ¼ 401). In Fig. 2, the mechanical response of the
RVE under various confining conditions is compared with experimental results reported in [63] for Lac du Bonnet granite. The
results are compared with those reported in [63,86] based on
macroscopic models. The cohesion parameter chosen was based
on the mechanical response under a confining stress of σ3 ¼
10 MPa, and the other parameters chosen were in the ranges
reported in the literature; as can be seen, a good agreement was
obtained for this confining pressure. A fair agreement was
obtained at higher confining stresses, especially in view of what
appears to be an experimental artefact present in the experimental
results at σ3 ¼ 20 MPa for the transverse strain response. Note
that for this confining pressure, the upper part of the deviatoric
stress vs. transverse strain curve is almost parallel to the experimental curve and the RVE response shows a good agreement for
the axial response. The agreement obtained for a confining stress
of 40 MPa is slightly less satisfactory, which might be due to a
slight overestimation of the friction angle, or to an evolution of the
fine scale dilatancy angle with compressive confinement, not
taken into account by the simple fine-scale dilatant model
used here.
Based on the fine-scale state of the RVE, the cracking state is
illustrated in Fig. 3 for a confining stress of σ3 ¼10 MPa, in which
the interfaces are represented with the value of their current
normal plastic opening at increasing load levels. This figure shows
the progressive interface opening as a result of the increase of the
deviatoric stress.
The cracking state can also be quantified more globally by
representing the crack density information as a function of the
stress level; this is shown in Fig. 4 for all three confining stress
Fig. 2. Stress–strain response of RVE for various confining pressures, compared to
experimental results reported in [63] (star markers) and to the simulation results
reported in [63] (green dashed line) and in [86] (blue dashed line). (For
interpretation of the references to color in this figure caption, the reader is referred
to the web version of this paper.)
levels. In this figure, an interface is considered as a crack when its
normal plastic opening exceeds 0.1 μm (compared to the initial
hydraulic aperture of 10 4 μm). Translated in terms of local permeability increase using relations (26) and (28) with a coefficient f¼ 0.8
and an initial permeability kloc;0 ¼ 10 15 mm2 , such a plastic opening
matches an improved permeability of kloc ¼ 5 10 10 mm2 , i.e. a local
permeability increase by a factor of 5 105.
T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
599
Fig. 3. Evolution of the normal plastic opening of interfaces for a triaxial test with a 10 MPa confinement.
Fig. 4. Stress-induced evolution of the crack density inside the RVE for associated
plastic flow. The density relates to the surface of interfaces on which at least a point
presents a plastic normal opening larger than 10 4 mm.
The fluid transport homogenisation is performed assuming a
uniform initial permeability of kloc;0 ¼ 10 15 mm2 for all tetrahedral
and interface elements. The tetrahedral elements are considered to
have a fixed permeability, whereas the local permeability evolution in
the interfaces is taken into account in the fluid transport simulations
by means of relationships (26) and (28), with a factor f¼0.8. The
resulting axial permeability evolution for the confining pressure of
σ 3 ¼ 10 MPa is given in Fig. 5a, where experimental results reported
in [10] are included, and where comparisons with modelling results
from [63,86] are also depicted. As can be seen, the trend of permeability increase by several orders of magnitude is correctly reproduced,
even though the sharp transition at a deviatoric stress of approximately 200 MPa is not present in the simulations. It should also be
noted that the experimental results show a slower permeability
increase for deviatoric stresses above 250 MPa. This could likely be
due to either gouge production in the cracks or due to a decrease of
the dilatancy angle; fine-scale modelling could be extended in future
developments to account for these aspects [62,87].
The effect of the factor f on the obtained permeability evolution
is depicted in Fig. 6, showing that the sensitivity to this parameter
remains limited, and does not affect the global conclusions drawn
from the simulations.
Similar results were recently reported by the authors in [13]
based on a mechanical damage description (stiffness degradation)
that disregards any dilatancy effect. In that case, the local permeability alterations were related to the local damage state, which is
a purely phenomenological quantity. In the present model, the
local permeability evolution is directly deduced from the crack
opening appearing in the simulations, leading to a more physically
motivated description.
Finally, the confinement-dependent evolution of the permeability, also reported in [63,88], is captured by the homogenisation
approach. The slower cracking development with a deviatoric
stress increase for highly confined RVEs results in a slower
permeability increase as well. This is visible in Fig. 5b where the
permeability evolution is reported for various confining stress
levels. Note that Fig. 5b can be strongly related to the crack density
evolution as defined in Fig. 4.
4.4. Effect of dilatancy and cohesion variability on permeability
alteration
The experimental determination of the dilatancy angle is
complex. Experimental works reported lower values of the dilatancy angle with respect to the angle of friction for granite, see
[17,21]. The effect of the dilatancy angle postulated at the fine
scale (i.e. at the scale of microcracks within the RVE) is therefore
now investigated.
The homogenised mechanical response of the RVE for a confining stress of σ 3 ¼ 10 MPa is first used for this purpose. The
influence of the dilatancy angle on the average mechanical
response is illustrated in Fig. 7 for unchanged values of other
parameters. As expected, the radial (transverse) strain is strongly
reduced in the non-linear range when the angle of dilatancy is
decreased. The corresponding density of cracks with an opening
above 0.1 μm is reported in Fig. 8. Again, the non-associated cases
(ψ ¼201 and ψ ¼01) lead to a decrease in the crack density. If this
decrease remains moderate for ψ ¼201, it is much more
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Fig. 7. Influence of dilatancy on the average mechanical response of the RVE
subjected to a deviatoric triaxial stress state at a confining stress of 10 MPa.
Fig. 5. Cracking-induced permeability evolution: (a) permeability evolution at a
confining pressure of 10 MPa and comparison with experimental results reported
in [10] and the simulation results reported in [63] (green dashed line) and [86]
(blue dashed line); (b) influence of the confining pressure on stress-induced
permeability evolution.
Fig. 8. Influence of the dilatancy angle on the stress-induced evolution of the crack
density inside the RVE. The density relates to the surface of interfaces on which at
least a point presents a plastic normal opening larger than 10 4 mm.
Fig. 6. Influence of the factor f on the obtained macroscopic permeability evolution
for the RVE subjected to a deviatoric triaxial stress state at a confining stress of
10 MPa.
pronounced for ψ ¼01. The corresponding permeability evolution
is given in Fig. 9 together with the experimental results reproduced from [10]. Both dilatancy angles ψ ¼201 and ψ ¼401
Fig. 9. Influence of the dilatancy angle on the stress-induced average permeability
evolution of the RVE.
T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
601
Fig. 10. Combined effect of fine scale cohesion distribution and non-associated dilatant behaviour (ψ¼ 201).
(associated plasticity) can be considered to give results that exhibit
the correct trend in terms of permeability evolution. Conversely,
the permeability evolution obtained for ψ ¼01 underestimates the
experimental data by several orders of magnitude.
A second set of mechanical simulations at different confining
stress levels is performed with a non-associated behaviour
coupled to a spatially variable cohesion of the interface elements.
A lognormal distribution is used for this purpose as illustrated in
Fig. 10. The dilatancy angle is chosen as ψ ¼ 201, a value motivated
from [17,21], in which evolving mobilised dilatancy angles for
granite are reported as a function of the plastic strain. As can be
seen in Fig. 10, the agreement of with experimental results is
unchanged. This is due to the introduction of the fine-scale
cohesion variability that induces lower stress levels for a given
transverse strain, while a decrease of the dilatancy angle has an
inverse effect. The macroscopically observed dilatancy is thus
influenced in opposite ways by the postulated fine-scale dilatancy
and the fine-scale strength heterogeneity.
5. Discussion
In view of the results presented in Section 4, a number of
additional comments can be given concerning the parameters
used in the simulations. The first set of parameters required is
related to the fine-scale properties of the individual phases and
interfaces. It could be argued that the experimental determination
of some of these properties remains a challenge. However, it
should be emphasised that the multi-scale procedure used here
should be considered as a methodology to assess the macroscopic
effect of fine scale phenomena. The fine-scale properties associated with the models defined in Section 3 and used in simulations in Section 4 should be split into two categories.
The elastic fine-scale parameters are obtained based on the
mineral composition of the granitic rock under investigation. In
the present case, these elastic moduli were taken as the average of
the range of values experimentally obtained for these minerals.
When not available, these local elastic properties can be obtained
using local characterisation devices, such as micro-/nano-indentation, as advocated in [76]. It is emphasised that a proper account
for the elastic heterogeneity of the material properties is crucial to
capture the progressive non-uniform cracking of the material.
The properties corresponding to fine-scale failure parameters
are more difficult to determine experimentally at the scale of
individual cracks, hence the need to introduce additional assumptions, or to analyse the effect of variations of those. In the context
of coupling between the mechanical behaviour and the permeability evolution, the dilatancy angle has to be considered in
addition to the usual failure properties. From results given in
Section 4.4, there is a clear influence of the dilatancy angle,
postulated at the level of microcracks, on the stress-induced
permeability evolution. These results show that the commonly
used assumption for non-associated behaviour (ψ ¼01) cannot be
used in simulations that account for the influence of the
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T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
cracking process on the permeability evolution. It is suggested in
[17,84,85] that the macroscopic dilatancy angle in rocks should be
considerably lower than the angle of friction. For the selected set of
fine-scale material parameters and for the RVE considered in these
simulations, a fair agreement for the mechanical response under a
confinement of σ 3 ¼ 10 MPa was obtained for the simulation that
uses an associated flow rule. However, this fact should not lead to the
conclusion that the associated case should be selected. The results
based on the non-associated flow rule, for an angle ψ ¼201, can also be
considered satisfactory, given the potential spatial variability of the
fine scale mechanical properties and the correct trend observed for the
permeability evolution as shown in the simulation in Fig. 10. In the
present contribution, in an attempt to use computational homogenisation as a tool for analysis of fine-scale phenomena, values for the
dilatancy angle are inspired from the macroscopic values reported in
[17,21]. It is however emphasised that the fine-scale model is
restricted to the simplest model possible, disregarding any of the
effects reported in [21] concerning the variation of the dilatancy angle
with the level of plastic straining and of confinement. Some other
detailed aspects were not taken into account that may influence the
transition between the fine-scale dilatancy angle and the macroscopically observed effect, such as the clustering of mineral species within
the RVE (their distribution is purely random in the current RVE
generation procedure). The results in Section 4 show that the effect
of a decrease of the fine scale dilatancy angle on the stress–strain
response (decrease of macroscopic dilatancy) can be ‘compensated’ by
the introduction of a spatial variability of the fine scale strength
properties (in this case the cohesion). The latter indeed leads to an
increase of the macroscopically observed dilatancy. The macroscopically observed dilatancy in the simulations is thus a convolution of the
dilatancy of the fine-scale cracks with the elastic and strength finescale parameter distributions. Other additional fine-scale effects not
tested here may also come into play such as the non-convexity of
grain shapes. A clear advantage of the methodology proposed here is
that each of these fine scale features can be tested individually in
future developments to assess the magnitude of its macroscopic scale
effects. The results presented do, however, clearly point to the need to
account for dilatancy at the scale of microcracking in order to predict
permeability evolution by means of multiscale computational techniques, as the corresponding permeability evolution confirms that the
approximation of a vanishing dilatancy angle does not yield physically
acceptable results. Further research could also be performed to
incorporate more detailed dilatancy models as developed in [87].
The other material parameters used in the interface elements
representing (potential) cracks are also inspired by their macroscopic counterparts. This choice may be a posteriori further
supported by investigating the effect of their variation. This is
illustrated in Fig. 11 that presents the effect of a variation of the
angle of friction, keeping the other parameters of the nonassociated model unchanged. This figure clearly illustrates a better
agreement with the experimental curves for the original angle of
friction used in Section 4.
As in any averaging procedure, the size of the RVE may have an
influence on the obtained results, as its size should be large
enough to account for all possible fine-scale interactions. In the
present case, the crucial feature is that the RVE should be large
enough to account for the elastic heterogeneities and to contain a
sufficient number of (interacting) cracks. A dependency of the RVE
size on the distribution of crack lengths was illustrated for the case
of non-evolving crack networks in [40]. The RVE size was also
investigated in [38] in which a RVE size of four times the average
crack length was deemed sufficient to obtain sufficient connectivity in discrete fracture networks. Note also that the RVE size
required to obtain a proper estimation of average properties was
shown to depend on the boundary conditions used in the averaging procedure. Periodic boundary conditions as used in the
Fig. 11. Effect of a variation of the angle of friction on non-associated dilatant
behaviour (ψ¼ 201), continuous line: φ¼ 401, and dashed line: φ¼ 301.
present contribution are known to require a lower size than
boundary conditions based on uniform fluxes/tractions or linear
pressure gradients/ uniform displacements as used in [47] for the
fluid transport case. Finally, it is noted that accounting for the
presence of elastic heterogeneities and the cohesion distribution
promotes fine scale localisation of the cracking phenomenon
which decreases the importance of the RVE size.
When compared to most existing damage-permeability
approaches, the methodology used here bears similarities with
most approaches based on discrete fracture networks (DFNs)
[40–43,47]. The difference lies in the fact that the fracture network
T.J. Massart, A.P.S. Selvadurai / International Journal of Rock Mechanics & Mining Sciences 70 (2014) 593–604
is obtained here as a result of a mechanical simulation. The
approach is therefore similar to the models using a combination
of DEM and DFN [45–47]. The averaging scheme is however more
naturally coupled with a finite element approach, and in spite of
being more complex, the periodic boundary conditions are known
to deliver better estimates than before [47].
603
author by the European Commission under project ‘MULTIROCK’.
The support from F.R.S.-FNRS Belgium for intensive computational
facilities through number 1.5.032.09.F is also acknowledged. The
first author is grateful for the facilities provided by the Environmental Geomechanics Group, Department of Civil Engineering and
Applied Mechanics, McGill University during the Marie Curie
fellowship.
6. Conclusions
A recently presented periodic computational averaging scheme
that couples mechanical cracking to permeability evolution was
used to analyse the influence of microcracking dilatancy and finescale heterogeneities on the overall response of a granitic material.
Based on the general nature of this averaging framework, which can
incorporate any type of microscale constitutive behaviour, dilatant
plasticity interface laws were used for this purpose. Periodic RVEs
were constructed to allow for cracking and flow channels to develop
in a general manner, by introducing a cohesive zone between all the
elements of an unstructured tetrahedral mesh. Based on these RVEs
incorporating elastic and strength fine-scale heterogeneities and on
interfacial dilatant plasticity, the ability of the framework to reproduce stress-induced macroscopic permeability evolution was
demonstrated. A proper fit of the macroscopic mechanical response
and of the related permeability evolution of Lac du Bonnet granite
was obtained using fine scale material parameters in the range of
acceptable values available in the literature. It is noted that cracking
was considered here as the dominant source of permeability
evolution. Other types of phenomena such as initial crack closure,
pore closure, or gouge generation will require additional fine-scale
modelling ingredients, but can be accommodated without change by
the averaging procedure.
In particular, the following conclusions can be drawn from the
presented results.
A physically motivated coupling between the micro-cracking
evolution within a granitic material and its stress-induced permeability evolution allows the reproduction of experimentally
observed trends by means of computational homogenisation.
In contrast to the damage simulations presented in [13] and based
on a phenomenological coupling, the approach presented here
relies on local permeability evolution driven by a more physically
motivated quantity, i.e. crack opening.
Based on the experimental information from the literature used in
this paper, a vanishing dilatancy angle does not allow the reproduction
of both the average mechanical response and the permeability
evolution of granite samples subjected to triaxial tests.
Even though a proper fit is obtained with a flow rule of the
associated type, the results suggest that the combination of a nonassociative law with elastic and strength material parameter
variability would reproduce the experimental trends. This conclusion is in agreement with various sources in the literature
advocating for dilatancy angles lower than the friction angle [85].
The results presented in this contribution call for further extensions
of the fine scale modelling to investigate additional physics, such as
the permeability reduction in sandstones linked to microcracks or
pore closure [16] or the hysteresis in the permeability of limestones
[89]. Possible future developments also relate to the use of real CT scan
data of geomaterials [76,90] to generate microstructural RVEs, or
specific advanced RVE generation techniques [91], in combination
with advanced discretisation techniques [92,93].
Acknowledgments
The work described in this paper was achieved during the
tenure of a Marie Curie Outgoing Fellowship awarded to the first
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