Computational modelling of crackComputational modelling of crackComputational modelling of crackComputational modelling of crack----induced induced induced induced permeability evolution in granite with dilatant permeability evolution in granite with dilatant permeability evolution in granite with dilatant permeability evolution in granite with dilatant

crackscrackscrackscracks

T.J. Massart, A.P.S. Selvadurai

This paper was published in International Journal of Rock Mechanics & Mining Sciences, vol. 70, pp 593-604, 2014

Computational modelling of crack-induced permeability

evolution in granite with dilatant cracks

T.J. Massarta, A.P.S. Selvaduraib

aBuilding, Architecture and Town Planing Department CP 194/2, Université Libre deBruxelles (ULB), Av. F.-D. Roosevelt 50, 1050 Brussels, Belgium

bDepartment of Civil Engineering and Applied Mechanics, McGill University, 817Sherbrooke Street West, Montreal, Canada H3A 2K6

Abstract

A computational homogenization 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 di-

latancy 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 simu-

lations 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.

Keywords: Stress-induced permeability alteration, Computational

homogenisation, Multiscale modelling, Granite failure, Dilatant plasticity,

Triaxial loading, Fluid transport

Published in Int. J. Rock Mech. Min. Sci., Vol 70, pp 593-604, 2014 July 27, 2014

1. Introduction

The isothermal mechanical behaviour of fluid-saturated porous media is

described by Biot’s theory of poroelasticity [1]. This theory has been suc-

cessfully applied to describe many types of geomaterials as illustrated in the

reviews presented in [5, 3, 4, 6, 7]. 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 poten-

tial effect of stress states at the micromechanical level on the permeability of

porous media. Experimental results presented in [8, 9, 10, 11] show substan-

tial 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 utilized to evaluate the effective properties of hetero-

geneous porous media [12, 13]. Computational homogenisation principles

have been employed to evaluate damage-induced permeability alterations

[14]. 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 geo-

materials, inter- and trans-granular cracking, or pore closure on parameters

such as permeability. The effect of damage was investigated in [14] using

2

a phenomenological description of intergranular cracking at the scale of the

constituents. In addition to disregarding transgranular cracking and to main-

tain a purely phenomenological approach, [14] neglects the effect of plastic

dilatancy, which is often considered as one of the main causes of permeability

evolution associated with micro-cracking [15, 16, 17].

Dilatancy is a material property that is difficult to evaluate experimen-

tally, which may explain why it usually receives less attention in routine

engineering applications [18]. Various authors have investigated dilatancy as-

pects and its potential dependence on the confining stress and plastic shearing

[18, 19, 20, 21, 22] .

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 be-

haviour by analysing the effect of frictional cracks on stress-induced anisotropy

[23] and on the propagation of secondary cracks together with local dilatancy

[24]. Subsequent efforts in this stream were focused on characterising compu-

tationally the mechanics of crack-microcrack interactions [25] and the effects

of microcracks interactions on elasticity properties [26]. Other contributions

dealt with the potential use of crack interactions features to model crack co-

alescence [27], and with the validation of the use of the penny-shaped cracks

effective medium theory for irregular fractures, see [28]. Contributions using

such penny-shape cracks approximations were also provided in [29] to inves-

tigate the link between macroscopic dilatancy and multiscale friction; as well

as recently in [30] to incorporate the effect of initial stresses in microcrack-

3

based models. Another sgnificant 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 [31]. This approach was progressively developed,

focusing on volumetric dilatancy, pressure sensitivity and stiffness recovery

[32]; and on the interaction between initial and cracking-induced anisotropy

[33]. The effect of anisotropic damage and unilateral effects on the elastic be-

haviour was analysed in [34]; and subsequent evolutions of the approach were

proposed to account for discrete orientations of cracking in the formulation

of thermodynamical potentials [35, 36].

The coupling between microcracking and fluid transport phenomena was

also investigated using penny-shaped cracks approximations in [37] for per-

meability alterations; as well as for crack propagation [38]. This methodology

was used in [39] to link macrodamage and damage tensors to permeability

evolution using 2D cuts of 3D Representative Volume Elements (RVEs) con-

taining discrete cracks. Non evolving discrete fracture networks (DFN) were

also used in several contributions. Such approaches were used in [40] to

compute permeability tensors from 2D geometrical models for macrosopic

fractures. 2D discrete fracture networks were also used in [41, 42] to com-

pute equivalent permeability with statistical distributions of fracture lengths

and apertures, in [43] with a discussion on the existence of a RVE, and

in [44] to analyse equivalent permeability as a function of discrete fracture

networks properties. The coupling between the mechanical behaviour and

DFN approaches uses discrete elements formulations (DEM) as performed

in [45] to deduce stress-dependent equivalent permeabilities for 2D RVEs

4

incorporating fracture dilation. DEM was also combined with DFNs to as-

sess stress-induced permeability evolution using 2D models in [46, 47], and

to compute bounds on 2D permeability for 3D discontinuity networks using

constant flux and linear pressure boundary conditions on RVEs [48].

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 macrosopic scale. The main objective of this paper is to show that

the homogenisation scheme presented in [14] allows computationally assess-

ing 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 [14], 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 homogenization 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 essen-

tial 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, and based on physically-based mechanical parameters

5

such as cohesion, tensile strength and their corresponding fracture energies,

will be presented in Section 3 . The applied coupling between the local crack-

ing state and the local permeability evolution is also presented in Section 3.

These procedures are used in Section 4 to estimate the variation of perme-

ability 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 prop-

erties

Computational homogenisation techniques were initially used in the field

of mechanics of materials to extract average properties of heterogeneous ma-

terials [49, 50]. 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 ap-

plications of multi-scale techniques have mostly focused on the extraction of

average properties of non-evolving microstructures for uncoupled mechanical

[51, 52] and fluid transport [41, 42, 48] processes in geomaterials . Com-

putational homogenisation was reformulated for allowing nested scale com-

putational schemes in which finite element discretisations are used at both

the macroscopic and fine scales simultaneously, which avoids the use of a

6

priori postulated macroscopic laws [53]. This methodology was subsequently

adapted to model the failure of quasi-brittle materials [54, 55], as well as dif-

fusive phenomena such as thermal conductivity [56, 57]. In the recent study

[14], these techniques were combined to evaluate damage-induced permeabil-

ity evolutions in a geomaterial. The essential features of this approach are

summarized here for completeness. The detailed derivation of the averaging

relationships can be found in [14] 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 [53]. When a macroscopic

strain E is applied to an RVE, the displacement of a point inside the RVE

can be written as

~u (~x) = E.~x+ ~uf (~x) (1)

where ~x is the position vector within the RVE and ~uf is a fluctuation field

caused by the heterogeneity of the material. Assuming that the fluctuation

~uf is periodic, one can show that the macroscopic strain is the volume average

of the fine-scale strain field ε resulting from (1)

E =1

V

∫V

ε dV (2)

Using the Hill-Mandel condition (energy equivalence between the fine-scale

and macroscopic descriptions) written as

Σ : δE =1

V

∫V

σ : δε dV (3)

the macroscopic stress tensor is obtained as the volume average of the mi-

crostructural stress tensor, or equivalently by a summation of RVE tying

7

forces

Σ =1

V

∫V

σdV =1

V

4∑a=1

~f (a)~x(a) (4)

where the summation spans 4 nodes controlling the RVE loading [53].

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 to 4 in Figure 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 forces ~f (a) at the

controlling points, which represent the action of the neighbouring contin-

uum 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 [56]. For fluid flow, the mass conser-

vation equation at the scale of the components has to be solved. Assuming

Darcy flow with a local permeability distribution defined by Km (~x) and with

a fluid of dynamic viscosity µ mass conservation reads

~∇m.(−Km (~x)

µ.~∇pm

)= 0 (5)

A periodic fluctuation pf (~x) of the pressure field is assumed [56, 14] to de-

scribe the pressure variation inside the RVE according to

pm (~x) = pkm + ~∇MpM .

(~x− ~xk

)+ pf (~x) (6)

8

where ~∇MpM 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

~∇MpM =1

V

∫V

~∇mpm dV (7)

In a computational treatment, the periodicity constraint on the pressure

fluctuation field requires that relationships of the form

pSm − pMm = ~∇MpM .(~xS − ~xM

)(8)

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

~∇MpM .~qM =1

V

∫V

~∇mpm.~qm dV (9)

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 [56]

~qM =1

V

∫V

~qm dV (10)

Details of the derivation of relationship (10) starting from the relationship

(9) can be found in [14]. 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 link

between the applied macroscopic pressure gradient terms at the controlling

points and the fluxes developing as a reaction to them. At equilibrium, the

discretized system of equations for the transport problem can be condensed

9

at the controlling points. This allows the fluxes at the controlling point (a)

to be expressed in terms of the applied macroscopic pressure gradients at the

controlling point (b) as

q(a)mn =b=4∑b=1

k(ab)disc (∇jp xj)

(b) (11)

where k(ab)disc results from the condensation of the entire RVE hydraulic stiff-

ness 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 [53]

qMi =1

V

a=4∑a=1

q(a)mnx(a)i (12)

Substituting relation (11) into (12) allows the identification of the macro-

scopic (averaged) permeability KMij as

KMij =µ

V

a=4∑a=1

b=4∑b=1

x(a)i k

(ab)discx

(b)j (13)

All the local permeabilities of the RVE are taken into account in the volume

averaging procedure underlying relationship (13) since the macroscopic flux

is obtained as a volume average of the fine-scale flux.

3. Fine scale modelling of cracking and permeability evolution

3.1. Crack modelling strategy

To model the influence of cracking and to avoid computing-intensive

continuum descriptions, the incorporation of cohesive laws in a priori po-

sitioned interface elements is used here. Such elements will be introduced

10

at the boundaries of all tetrahedral elements in the mesh, see Section 4.1.

This choice is physically similar to discrete orientation-based approaches

[58, 48, 35], with the additional feature that the spatial connectivity of cracks

is explicitly incorporated , similarly to the discrete fracture network approach

[44, 48]. The bulk constituents of the material are modelled with linear elas-

tic tetrahedral finite elements, between which interface elements are inserted

to represent potential microcracks.

In order minimize the number of fine-scale material parameters, the sim-

plest version of a non-associated 3D plastic cohesive model is used, as pre-

sented in [59]. This model links the traction vector ~T across the interface to

the elastic relative displacement vector ~δe

~T = H.~δe (14)

where the matrix H represents the elastic behaviour of the potential cracks.

This matrix is represented as

[H] = diag (kt, kt, kn) (15)

with the elastic parameters defined in terms of the mineral species considered

as

σ =E

hhε︸︷︷︸δen

= knδen; τt =

G

hhγt︸︷︷︸δet

= ktδet ; τs =

G

hhγs︸︷︷︸δes

= ktδes (16)

where E and G are the bulk elasticity properties, and h is an assumed thick-

ness of the cracked zone. This thickness should be chosen small, yet still

avoiding any conditioning problem in the resulting finite element stiffness

matrix. Using a plasticity approach, the plastic relative displacement across

11

the cohesive zone is obtained as

~δp = ~δ − ~δe (17)

and is expressed as the derivative of a plastic potential function with respect

to the stress as {δ̇p}

= λ̇∂g

∂{σ}(18)

in which λ̇ 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. The

choice was made here 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 [60], a Rankine-type yield function is used,

showing that plasticity appears when the normal stress across the cohesive

zone exceeds a threshold value σt

Ftens = σ − σt(κt) (19)

The quasi-brittle failure of the interface is modelled by means of an expo-

nential decay of σt in terms of a plastic parameter κt

σt(κt) = ft exp(−ftGIf

κt) (20)

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

κ̇t =∣∣∣δ̇pn∣∣∣ = λ̇t (21)12

Note that a tension failure criterion is adopted here because of the too 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 accounting for dilatant friction,

under compressive stress states, a non-associated Mohr-Coulomb model is

used as performed in macroscopic approaches for granite in [62, 61, 22]. The

corresponding yield function is written as

Fmc =√τ 2t + τ

2s + σΦ− c (κm) (22)

where τs, τt are shearing stress components across the interface, σ is the

normal stress, and Φ = tan(ϕ) with ϕ the angle of friction.

In order to control the volume variation associated with the frictional

plasticity in cracks, plasticity under compressive-frictional behaviour is gov-

erned by a non-associative flow rule. A plastic potential similar to the form

(22) is used with a dilatancy angle potentially different from the friction

angle. The derivative of the assumed plastic potential is therefore

∂g

∂ {σ}=

{τt√

τ 2t + τ2s

τs√τ 2t + τ

2s

Ψ

}t(23)

where Ψ = tan(ψ), with ψ being the dialatancy angle. The failure behaviour

of the interface under compressive-frictional loading is described by means

of a cohesion softening law given by

c (σ, κm) = c0 e− c0GIIf

κm(24)

13

where c0 is the initial cohesion of the interface, and GIIf is its mode II fracture

energy. The cohesion c is a function of the plastic parameter κm associated

with frictional cracking, which is defined by

κ̇m =

√(δ̇pt

)2+(δ̇ps)2

= λ̇m (25)

Multi-surface plasticity requires a careful treatment of the intersection of

surfaces representing between both yield criteria, as well as the derivations

of the return mapping operators for the stress update, from which the con-

sistent material tangent operator ensues. More details of the derivations of

return mapping algorithm and tangent operators can be found in [59, 63].

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 for granite

recently [64]. More detailed descriptions can be developed to incorporate ad-

ditional features such as frictional softening coupled to cohesional softening,

a confining stress-dependent dilatancy or a confining stress-dependent mode

II fracture energy [65], but were left out of the scope of the present paper.

3.2. Coupling permeability evolution to microcracking

An initial low permeability will be assumed for both the (tetrahedral)

bulk and the interface elements representing potential microcracks. The per-

meability of the bulk elements will remain unchanged during the entire sim-

ulations, focusing on the permeability evolution induced by cracking. The

hydraulic formulation used for the interface elements will be chosen similar

to that presented in [66]. Based on the modelling of the failure of the in-

terfaces and their opening, the local fluid transport properties Km (~x) in (5)

14

are updated. The interface elements forming a connected network, this will

lead to the progressive formation of flow channels. In constrast to the phe-

nomenological damage-permeability coupling used in [14], 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 par-

allel plate model, developed using Stokes’ flow or slow viscous flow between

two parallel plates classically used in rock fractures [67, 68, 69], will be used

to calculate the effective permeability of a fracture [70]:

kloc =e2h12

(26)

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 [71], where

the relationship between the physical and theoretical apertures was dicussed,

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 [72] as

eh = eh0 + f∆em (27)

where ∆em 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

15

interface according to a linear relationship

eh = eh0 + f 〈δpn〉 (28)

where the factor f reflects the roughness of the fracture surfaces, and where

〈.〉 indicates that only positive values of a quantity are considered. In [73]

experimental results indicate a variation of f to be between 0.5 and 1. It is

emphasized that the value of the mechanical plastic normal relative displace-

ment 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 hy-

draulic apertures of potential cracks are updated according to the relationship

(28) and the local permeabilities are consequently updated. 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 [74]. A specific attention

is devoted to the production of a periodic mesh at the boundary of the RVE

in order to satisfy periodic fluctuation fields assumed in the homogenisation

16

procedure. To allow the progressive development of continuous crack paths

under loading, this unstructured mesh is subsequently modified to incorpo-

rate cohesive zone elements (interfaces) between all the tetrahedral elements.

The periodicity relationships therefore need to be adapted to ensure the pos-

sibility 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 accomodating any crack path developed in the material ac-

cording 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 [75]

for Stanstead granite is used as a guide for this purpose, being of a com-

position similar to Lac du Bonnet granite [76]. 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 [75, 77]. The volume fraction of the respective

constituent are V fbiotite = 7%, Vf

quartz = 20% and Vf

feldspath = 73%.

The RVE mesh used in subsequent computations is illustrated in Figure

1b. The spatial arrangement of tetrahedra belonging to each of the phases

(feldspath, biotite and quartz) are also illustrated for completeness in Figure

1c. The procedure described above leads to a mesh containing 14,090 tetra-

hedral elements and 26,580 interface elements, which gives 140,900 nodes

and 422,700 dofs. To efficiently solve algebraic systems of this size, itera-

tive solvers have to be used. For associated plasticity, leading to symmet-

ric systems of equations, the preconditioned conjugate gradient algorithm

17

(PCG) can be used with a preconditioner obtained by means of an incom-

plete Cholesky factorisation of the system matrix. For non-associated cases,

the bi-conjugate gradient algorithm can be used with a preconditioner ob-

tained by a sparse incomplete LU factorisation, see [78].

4.2. Material parameters

Lac du Bonnet granite was used as a test material, as cracking repre-

sents the dominant source of permeability evolution, and experimental data

is available in the literature, both for triaxial tests and permeability measure-

ments. The original set of experimental results for the mechanical response is

given in [11]. The elastic properties used for the bulk materials are as follows:

Ebiotite = 40 GPa, νbiotite = 0.36, Equartz = 100 GPa, νquartz = 0.07,

Efeldspath = 80 GPa, νfledspath = 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 [64]. The elastic parameters

associated with interfaces joining dissimilar bulk phases were selected as the

lowest between these bulk phases.

The fine-scale failure parameters are here chosen based on 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 [62],

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 40◦. Similar values were

found for Barre granite [79] with a cohesion of 50 MPa, and a friction angle

of 35◦. The compressive properties reported in [80] range from 26 MPa to

160 MPa for the (unconfined) compressive strength with compressive fracture

18

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 [81], 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 MPa - 10 MPa for Barre granite

[82] and for Lac du Bonnet granite [83]. The shear fracture energy of Stripa

granite was investigated in [84] 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: h = 0.05 mm, ft = 6 MPa, GIf = 1 N/mm,

c = 26.4 MPa, ϕ = 40◦, GIIf = 10 N/mm, ψ = [20◦, 40◦]. 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 compre-

hensive review of existing experimental results can be found in [18]. In most

sources, upper and lower bounds of the dilatancy effects are considered by

using either an associated flow rule (ψ = ϕ) or by neglecting the microcracks

dilatancy (ψ = 0◦). Recommendations in [85] suggest the use of dilatancy

angles depending on the quality of the rock, ranging from ψ = 0◦ to ψ = ϕ/4,

while the rules of thumb mentioned in [86] approximate dilatancy angles us-

ing ψ = ϕ − 20◦. Consequently, an initial set of simulations was performed

using the upper bound dilatancy associated flow rule, followed by an analy-

19

sis 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 emphasized that apart from the initial stiffness and tensile param-

eters, the fine scale frictional model is restricted to 4 parameters, the same

number as in the macroscopic model discussed in [64]. Note also that the ten-

sile parameters have a low influence for confined stress states as considered

here, and can reasonably be approximated by their macroscopic couterparts.

4.3. Associated dilatant formulation

A first set of simulations was conducted assuming an associated flow rule

(ψ = ϕ = 40◦). In Figure 2, the mechanical response of the RVE under var-

ious confining conditions is compared with experimental results reported in

[64] for Lac du Bonnet granite. The results are compared with those reported

in [64] and [87] based on macroscopic models. The cohesion parameter cho-

sen was based on the mechanical response under a confining stress of σ3 =

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 con-

fining pressure. A fair agreement was obtained at higher confining stresses,

especially in view what appears to be an experimental artefact present in

the experimental results at σ3 = 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 agree-

ment 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

20

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 Figure 3 for a confining stress of σ3 = 10 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 Figure 4 for all three confining stress levels. In this figure, an interface is

considered as a crack when its normal plastic opening exceeds 0.1 µm (com-

pared to the initial hydraulic aperture of 10−4µm . Translated in terms of

local permeability increase using relations (26) and (28) with a coefficient

f = 0.8 and an initial permeability kloc,0 = 10−15mm2, such a plastic open-

ing matches an improved permeability of kloc = 5.10−10mm2, i.e. a local

permeability increase by a factor of 5× 105.

The fluid transport homogenisation is performed assuming a uniform ini-

tial permeability of kloc,0 = 10−15mm2 for all tetrahedral and interface ele-

ments. The tetrahedral elements are considered to have a fixed permeability,

whereas the local permeability evolution in the interfaces is taken into ac-

count in the fluid transport simulations by means of relationships (26) and

(28), with a factor f = 0.8. The resulting axial permeability evolution for the

confining pressure of σ3 = 10 MPa is given in Figure 5a, where experimental

results reported in [11] are included, and where comparisons with modelling

results from [64, 87] are also depicted. As can be seen, the trend of perme-

21

ability 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 experimen-

tal 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 [88, 63].

The effect of the factor f on the obtained permeability evolution is de-

picted in Figure 6, showing that the sensitivity to this parameter remains

limited, and does not affect the global conclusions drawn from the simula-

tions.

Similar results were recently reported by the authors in [14] 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 [64, 89], 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

Figure 5b where the permeability evolution is reported for various confining

stress levels. Note that Figure 5b can be strongly related to the crack density

evolution as defined in Figure 4.

22

4.4. Effect of dilatancy and cohesion variability on permeability alteration

The experimental determination of the dilatancy angle is complex. Ex-

perimental works reported lower values of the dilatancy angle with respect

to the angle of friction for granite, see [18, 22]. 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 σ3 = 10 MPa is first used for this purpose. The influence of the dila-

tancy angle on the average mechanical response is illustrated in Figure 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 Figure 8. Again, the non-associated cases (ψ = 20◦ and

ψ = 0◦) lead to a decrease in the crack density. If this decrease remains mod-

erate for ψ = 20◦, it is much more pronounced for ψ = 0◦. The corresponding

permeability evolution is given in Figure 9 together with the experimental

results reproduced from [11]. Both dilatancy angles ψ = 20◦ and ψ = 40◦

(associated plasticity) can be considered to give results that exhibit the cor-

rect trend in terms of permeability evolution. Conversely, the permeability

evolution obtained for ψ = 0◦ underestimates the permeability evolution 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 Figure 10. The dilatancy angle is chosen as ψ =

23

20◦, a value motivated from [18, 22], in which evolving mobilized dilatancy

angles for granite are reported as a function of the plastic strain. As can be

seen in Figure 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 as 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 are 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 emphasized that the multi-scale procedure used here should

be considered as methodology allowing to assess the macroscopic effect of

fine scale phenomena. The fine scale properties used in 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 com-

position 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-

24

indentation, as advocated in [77]. It is emphasized 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 dif-

ficult to determine experimentally at the scale of individual cracks, hence the

need to introduce additional assumptions, or to analyse the effect of varia-

tions 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, that there is a clear influence of the dilatancy angle, postulated at the

level of microcracks, on the stress-induced permeability evolution. These re-

sults show that the commonly used assumption for non-associated behaviour

(ψ = 0◦) cannot be used in simulations that account for the influence of the

cracking process on the permeability evolution. It is suggested in [18, 85, 86]

that the macroscopic dilatancy angle in rocks should be considerably lower

than the angle of friction. For the selected set of fine scale material pa-

rameters and for the RVE considered in these simulations, a fair agreement

for the mechanical response under a confinement of σ3 = 10 MPa was ob-

tained 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 = 20◦, can also be considered satisfactory, given the potential spatial vari-

ability of the fine scale mechanical properties and the correct trend observed

for the permeability evolution as shown in the simulation in Figure 10. In the

present contribution, as an attempt to use computational homogenisation as

25

a tool for analysis of fine scale phenomena, values for the dilatancy angle

are inspired from the macroscopic values reported in [18, 22]. It is however

emphasized that the fine scale model is restricted to the simplest model pos-

sible, disregarding any of the effects reported in [22] 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 proce-

dure). 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 fine-scale cracks dilatancy with the elastic and strength fine scale param-

eters distributions. Other additional fine scale effects not tested here may

also come into play such as for instance 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 as-

sess 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 multi-

scale computational techniques, as the corresponding permeability evolution

confirms that the approximation of a vanishing dilatancy angle does not yield

26

physically acceptable results. Further research could also be performed to

incorporate more detailed dilatancy models as developed in [88].

The other material parameters used in the interface elements represent-

ing (potential) cracks are also inspired by their macroscopic counterparts.

This choice may be a posteriori further supported by investigating the ef-

fect of their variation. This is illustrated in Figure 11 that presents the

effect of a variation of the angle of friction, keeping the other parameters of

the non-associated 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 [41]. The RVE size was also investigated in

[39] in which a RVE size of 4 times the average crack length was deemed suf-

ficient to obtain sufficient connectivity in Discrete Fracture Networks. Note

also that the RVE size required to obtain a proper estimation of average prop-

erties was shown to depend on the boundary conditions used in the averaging

procedure. Periodic boundary conditions as used in the present contribution

are known to require a lower size than boundary conditions based on uniform

fluxes/tractions or uniform pressure gradients/strains as used in [48] for the

fluid transport case. Finally, it is noted that accounting for the presence

27

of elastic heterogeneities and the cohesion distribution promotes fine scale

localisation of the cracking phenomenon which decreases the importance of

the RVE size issue.

When compared to most existing damage-permeability approaches, the

methodology used here bears similarities with most approaches based on Dis-

crete Fracture Networks (DFNs) [41, 42, 43, 48, 44]. The difference lies in

the fact that the fracture network is obtained as a result of a mechanical

simulation. The approach therefore corresponds to the models using a com-

bination of DEM and DFN [46, 47, 48]. 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 [48].

6. Conclusions

A recently presented periodic computational averaging scheme that cou-

ple mechanical cracking to permeability evolution was used to analyse the

influence of microcracking dilatancy and fine scale 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 constitu-

tive 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 in-

corporating elastic and strength fine scale heterogeneities and on interfacial

dilatant plasticity, the ability of the framework to reproduce stress-induced

28

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 evolu-

tion. Other types of phenomena such as initial crack closure, pore closure,

or gouge generation will require additional fine scale modelling ingredients,

but can be accomodated 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 al-

lows the reproduction of experimentally observed trends by means of compu-

tational homogenisation. In contrast to the damage simulations presented in

[14] 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 non-associative law with

elastic and strength material parameter variability would reproduce the ex-

perimental trends. This conclusion is in agreement with various sources in

29

the literature advocating for dilatancy angles lower than the friction angle

[86].

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 [17] or the

hysterisis in the permeability of limestones [90]. Possible future developments

also relate to the use of real CT scan data of geomaterials [77, 91] to generate

microstructural RVEs, or specific advanced RVE generation techniques [92],

in combination with advanced discretisation techniques [93, 94].

Acknowledgments

The work described in this paper was achieved during the tenure of a

Marie Curie Outgoing Fellowship awarded to the first author by the European

Commission under project ’MULTIROCK’. The support from F.R.S-FNRS

Belgium for intensive computational facilities through grant 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.

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(a)

(b)

(c)

Figure 1: Cubic Representative Volume Element (RVE) of the for the material; (a)

Schematic view of RVE, (b) finite element mesh, (c) distribution of material phases inside

the RVE

44

Figure 2: Stress-Strain response of RVE for various confining pressures, compared to

experimental results reported in [64] (star markers) and to the simulation results reported

in [64] (green dashed line) and in [87] (blue dashed line)

45

Figure 3: Evolution of the normal plastic opening of interfaces for a triaxial test with a

10 MPa confinement.

46

Figure 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

47

(a)

(b)

Figure 5: Cracking-induced permeability evolution: (a) Permeability evolution at a con-

fining pressure of 10 MPa and comparison with experimental results reported in [11] and

the simulation results reported in [64] (green dashed line) and [87] (blue dashed line), (b)

Influence of the confining pressure on stress-induced permeability evolution

48

Figure 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.

Figure 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.

49

Figure 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

Figure 9: Influence of the dilatancy angle on the stress-induced average permeability

evolution of the RVE.

50

Figure 10: Combined effect of fine scale cohesion distribution and non associated dilatant

behaviour (ψ = 20◦)

51

Figure 11: Effect of a variation of the angle of friction on non associated dilatant behaviour

(ψ = 20◦), continuous line: ϕ = 40◦, dashed line: ϕ = 30◦

52