Role of heterogeneities in jointing (fracturing) of geological media: Numerical analysis of fracture mechanisms

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To cite this version:

Alexandre Chemenda, Julien Ambre, Junchao Chen, Jinyang Fan, Jean-Pierre Petit, et al.. Role of

heterogeneities in jointing (fracturing) of geological media: Numerical analysis of fracture mechanisms.

Comptes Rendus Géoscience, Elsevier Masson, 2018, 350 (8), pp.452-463. �10.1016/j.crte.2018.06.009�.

�hal-01983594�

media:

Numerical

analysis

of

fracture

mechanisms

Alexandre

I.

Chemenda

a,

*

,

Julien

Ambre

a

,

Junchao

Chen

a,b

,

Jinyang

Fan

b

,

Jean-Pierre

Petit

c

,

Deyi

Jiang

b

a

UniversityofCoˆted’Azur,CNRS,OCA,IRD,Ge´oazur,250,rueAlbert-Einstein,SophiaAntipolis,06560Valbonne,France b

StateKeyLaboratoryfortheCoalMineDisasterDynamicsandControls,ChongqingUniversity,400044Chongqing,China c

UniversityofMontpellier,Ge´osciencesMontpellier,placeEuge`ne-Bataillon,34095Montpelliercedex05,France

1. Introduction

Therearetwoprincipaltypesoffracturesingeological media and notably in sedimentary basins (reservoirs), joints and shear fractures.The principal difference bet-ween them is that shear fractures accommodate shear displacements,whereasalongjointsonlynormaltojoints displacements arepossible.Another difference is in the orientationof fractures compared tothe acting (during fracturing)stressdirections.Jointsareparalleltothemost compressivestress

s

1andnormaltotheleastcompressive

orthemosttensilestress

s

3.Jointsthusmakeangle

c

=08

with

s

1 and typically are organized in regular parallel

(tabular)setsororthogonalnetworks.Shearfracturesare oblique toboth

s

1 and

s

3 andtypically formconjugate

networks,orientedto

s

1atangles

c

=108–308.Allfracture

typesareparalleltotheintermediateprincipalstress

s

2.

1.1. Theoreticalbackgroundoffractureformation

The mechanism of fracture formation remains not completelyclear.Thisisparticularlytrueforjoints,which isreflectedindifferentnamesusedforthisfracturetype: tensile, extension, opening mode or cleavage fractures. Most commonly, they are believed to form (or more exactly,topropagatefromapreexistingflaw)accordingto Mode-Imechanism(Fig.1)stemmingfromLinearElastic FractureMechanics(LEFM),(e.g.,Lawn,1993). According-ly, shear fractures are believed to be Mode-II fractures (Fig.1).Fromatheoreticalpointofview,LEFMisapplicable

ARTICLE INFO Articlehistory: Received29June2018

Acceptedafterrevision29June2018 Availableonline6September2018 HandledbyVincentCourtillot Keywords: Fracture Damage Joints Finite-differencemodeling Plumose Plasticitytheory AMScodes:65,74,86 ABSTRACT

Fracture process is investigated using finite-difference simulations with a new constitutivemodel.Itisshownthatbothgeometryandfracturemechanismitselfdepend onthepreexistingheterogeneitiesthatarestressconcentrators.Inthebrittleregime(low pressure,P),Mode-Ifracturespropagatenormaltotheleaststresss3fromtheimposed

weakzones.AthighP,sheardeformationbandsareformedobliquetos3.Atintermediate

valuesofP,thefractureprocessinvolvesbothshearbandingandtensilecrackingand resultsintheinitiationandpropagationofpuredilationbands.Thepropagatingbandtip undulates,reactingonthefailuremechanismchanges,butitsglobalorientationisnormal tos3.Thes3-normalfracturesarejoints.Therearethustwotypesofjointsresultingfrom

Mode-Icrackinganddilationbanding,respectively.Theobtainednumericalresultsarein goodagreementwithandexplaintheresultsfromprevioussimilarexperimentalstudy.

C 2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.Thisisan

openaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/ by-nc-nd/4.0/).

* Correspondingauthor.

E-mailaddress:chem@geoazur.unice.fr(A.I.Chemenda).

https://doi.org/10.1016/j.crte.2018.06.009

1631-0713/C _{2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense}

only when thesize ofthe near tipzone (processzone) undergoinginelastic(irreversible)deformationordamage issmallcomparedtoothergeometricdimensionsofthe problem (fracture length, size of thedeformingobject). Thisassumptioncanholdonlyatarelativelylowpressure P(orstresslevel),whendeformationandfractureoccurin apurelybrittleregime.However,thisiscertainlynotthe caseintheexperimentaltestsatarelativelyhighP,when damage(revealedbyacousticemissions)involves practi-callytheentiredeformingspecimenpriortothefracture (e.g.,Fortinetal.,2006).Inthis‘‘ductile’’regime,thefailure occursvialocalizationoftheinitiallydistributeddamage (inelasticdeformation)withinanarrowbandthatthencan becomeafracture.Thetheoreticalframeworkfor address-ing the mechanical instability resulting in deformation localizationconstitutesthebifurcationtheoryappliedto elasto-plasticity models (Chemenda, 2007; Issen and Rudnicki, 2000; Rudnicki and Rice, 1975; Vardoulakis andSulem,1995).Thistheoryiscompletelydifferentfrom LEFM (which does not explicitly take into account the inelasticdeformation),andexplainsverywellthe forma-tion ofvarioustypes ofsheardeformation bandsatthe origin of shear fractures. However, the prediction of dilation banding (which can be at the origin of joints, see below)does nogostraightforward fromthis theory whenrealisticmaterialparameters(notablythedilatancy factor)areadopted.

1.2. Insightsfromexperimentalstudies

Thetransitionfromverybrittleto‘‘ductile’’fracturing withincreasingpressurehasbeenobservedinnumerous experimental tests (e.g., Be´suelle et al., 2000; Fortin et al., 2006;Wong et al.,1997)including the conven-tional (axisymmetric) extension tests on specimens made of Granular Rock Analogue Material (GRAM), which behavesas real rocks,butis muchweakerthan hard rocks (Nguyen et al., 2011). The dog-bone and cylindrical specimens are first loaded hydrostatically intheseteststoagivenconfiningpressurePthatisequal to

s

1 and

s

2, andthenunloaded in theaxialdirection

that becomes

s

3, P being maintained constant. With

increasingP, specimen fracture occursat progressively smaller j

s

3j (which is negative),but the failure plane

orientation does not change until

s

3 reaches

approxi-mately zero (Fig. 2a, b): this plane is parallel to

s

1

(

c

08), Fig. 2b, which corresponds to jointing. With further Pincrease,

c

increases(Fig. 2b),which corres-pondstoshearfracturing.

Althoughtheorientationoffailureplanesinthestress spaceinGRAMexperimentsis thesame(

c

08)within thePrangeofjointsformation,the‘‘ductility’’ofthefailure processclearlyincreaseswithP,whichisexpressedinthe morphologyoftheformingfracturesurfaces(Chemenda etal.,2011a).AtlowP,thesesurfacesarerathersmooth, whileathighP,theyarerough,withtherugositiesbeing organizedinniceplumosefeatures(Fig.2c)sotypicalfor naturaljoints(e.g.,Bahat,1991;PollardandAydin,1988; Savalli and Engelder, 2005; Woodworth, 1896). The damage/failure process occurs under these pressures within a band of a finite thickness comparable to the plumoserelief.Thisprocessisthereforeratherductileand isnotlimitedtotheopeningofa‘‘zero’’-thicknessplane, whichwouldbethecase duringMode-Ifracturing.SEM observations of the failure zones forming at high P in GRAM specimens reveal a several grains-thick irregular wavybandofdamagedmaterialwithincreasedporosity (Chemenda et al., 2011a). This

s

3-normal band can be

called a pure dilation band. The conclusion made by

Chemendaetal.(2011a,b)isthatthejointingmechanism changes from Mode-I cracking to dilatancy/dilation bandingwithincreasingP.Withfurtherpressureincrease, failureoccursviatheformationofshear-dilatant deforma-tionbands.

Thedilationbandingclearlyinvolvesductile,butalso tensiledamage/failuremechanisms.Thisisthereasonwhy the dilatancy banding could not be correctly predicted theoreticallyfromdeformationbifurcationanalysiswithin anelasto-plasticityframeworkasmentionedabove.Inthis paper,weinvestigatethisprocessnumericallyby consid-ering both brittle-tensile (BT) and shear-ductile (SD) failure/damagemechanisms.

2. Constitutiveformulation

The adequate constitutive formulation is the most importantingredientintheoreticalanalysisandnumerical modelingofinelasticdeformation(damage)andfailureof geomaterials.Rockmechanicsexperimentalstudieswith differentloadingconfigurations(considerablyaccelerated recently(e.g.,Ingrahamet al.,2013; Li etal., 2018;Ma etal.,2017),clearlyshowtheimportanceoftheloading type (controlled by the Lode angle), which affects the mechanicalresponseofrocks.Anewconstitutivemodel has been proposed based on these experimental data (ChemendaandMas,2016).Themodelincludesa three-invariantyieldfunction

F¼

t



t

eð

s

Þ 1 2 1þcosð3

u

ÞþRð

s

Þ 1 n_ð1cosð3

_u

_ÞÞ h i  n (1) andtherelatedplasticpotential(nistheconstantscalar thatcantake valuesfrom 0.1to1).The bifurcation analysiswithintheframeworkofthisconstitutivemodel ledtothepredictionsofrockfailureregimesthatarein goodagreementwiththeexperimentaldata,andthatare differentfromthepredictionsmadebasedontheclassical 2-invariant theories like that of Drucker–Prager (ChemendaandMas,2016).Here

t

isvonMises’sstress; R¼

t

c=

t

e;

t

c and

t

earefunctionsofthemean stress

s

definedfromtheconventionalcompressionandextension tests,respectively;

u

istheLodeangle.Thisconstitutive model is rather complicated and has not been imple-mented into a numerical code as yet. For this reason, we use here much simpler, but also 3-invariant Coulomb-typetheory.Theyieldfunctionistakeninthe form

F¼A

s

1

m

c

s

3kc (2)

Itdoesnotinclude

s

2,whichisasimplificationofreality

thatcanbejustifiedbytheexperimentalfactthattherock propertiesdependmuchstrongeron

s

1and

s

3thanon

s

2

(e.g.,Ingrahametal.,2013; Maetal.,2017).Theplastic potentialfunctionreads

G¼A

s

1

m

b

s

3 (3)

where

m

c,kc,

m

barethefriction,cohesionanddilatancy

parameters(notcoefficients);Atakesvalue0or1forthe brittle-tensile(BT)andshearductile(SD)failure/damage mechanisms,respectively.Theinitialvaluesof

m

candkcas

wellasofthetensilestrength

s

t(

m

c0,k0cand

s

0t)aredefined from theGRAM experimental data(from the simplified

s

3(

s

1)curveinFig.3a)tobe

m

0 c¼1;A¼0; kc¼

s

t if

s

3¼

s

t

m

0 c¼2; A¼1; k0c¼0:65MPa if

s

3>

s

t; (4) ThemuchhigherbrittlenessoftheBT(thanthatofthe SD) mechanism is taken into account in the following softeningrules:

s

t¼

s

0t 1

e

t

e

0 t   ;kc¼k0 c 1

e

e

0   ;

m

c¼

m

0cþ

m

end c 

m

0 c  

_e

e

0 (5) where

s

0 t =0.07MPa;

e

0t=0.5103 and

e

0=1102 (

e

0

t <<

e

0);

m

endc =1.7,

e

t¼Rd

e

p₃ and

e

t¼

e

tþRd

e

p₁ (the superscript‘‘p’’standsforplasticorinelastic).Notethat

s

1

in (5) evolves only with the accumulated inelastic extensionstrain,whichisasumofstrainsresultingfrom BTandSDdeformations.Thisisthewayhowbothfailure mechanismsare coupled.According tothis formulation, thetensilestrengthdecreasesevenduringSDdeformation alongwiththecohesiondecrease,meaningthatthesetwo parametersarerelated,bothcharacterizingthe pressure-independentpartofthestrength.Accordingtotheresults oftheprocessingofalargeexperimentaldataset(Masand Chemenda, 2015), the dilatancy parameter

b

c evolves

(reduces)with

e

inthesamewayas

m

c,seeEq.(5).

Theelasticpropertiesofthemodelbothbeforeandafter reaching the yield condition are described by Hooke’s equations with Poisson’s ratio

n

=0.25 and Young’s modulus E=6.7108_{Pa, characterizing GRAM material}

(Nguyenetal.,2011).

The formulated constitutive model has been imple-mentedintothefinite-differencedynamictime-matching explicit code Flac3D (Itasca, 2012) used for numerical modelinginthiswork.Numeroussimulationswerecarried outinquasi-static,large-strainmodewithdifferentmodel parameters.Belowwereportrepresentativeresults.

Fig.2. Resultsfromconventional(axisymmetric)extensiontestsofGranularRockAnalogueMaterial,GRAM(fromChemendaetal.,2011a).(a)Minimum nominalstresss3atfracturing/rupturevsconfiningpressureors1.(b)Anglecbetweentheruptureplaneands1directionvss1atfailure.(c)Surfaceofthe fracture(openedrupturezone)formedats1=s2=0.6MPa.Theinitiationpoint(zone)oftheplumosemorphologyisduetothestressconcentrationcaused bytherigidsupportoftheLVDTgageattachedtothespecimen(tothejacket)atthatlocation.

3. Setupofnumericalmodeling

Themodelingsetup(Fig.3b)istwo-dimensionalinthe sensethatthemodelthicknessinthe

s

2-directionisonly

onenumericalzone(or0.25mm),butthestrainrateinthis directionis notzero(seebelow).Themodelisfirst pre-stressedtotheinitialstresses

s

1=

s

2andtoasufficiently

large

s

3forthe stress state tobeabovethe initialyield

envelopeinFig.3a(thiscorrespondstothepurelyelastic mechanicalstate).Then,

s

3isprogressivelydecreasedby

applyingvelocitiesV3tothe

s

3-normalmodelboundaries,

whilemaintainingtheinitialvaluesof

s

1and

s

2byapplying

velocities V1 and V2 as shown in Fig. 3b. In reality,

maintainedandmonitoredareaveragesovereach bound-ary or nominal stresses as it is typically done in the experimentaltests.SinceV2isnotzero,themodelsarenot

plane-strain. 4. Results

We reporthere theresultscorrespondingtothefour pointsshownontheyieldenvelope

s

3(

s

1)inFig.3a.The

firstpoint(point1,

s

1=0.3MPa;

s

3=

s

0t =0.07MPa)is locatedinthemiddleoftheBTsegmentoftheenvelope. Point2(

s

1=0.54MPa;

s

3=0.055MPa)islocatedonthe

SDsegmentclosetothetransitiontotheBTsegment.Point 3 (

s

1=0.6MPa;

s

3=0.025MPa)is on theSD segment

close to

s

3=0 (

s

3 < 0), and point 4 (

s

1=0.66MPa;

s

3=5103Pa)isalsoontheSDsegmentwith

s

3>0.

4.1. Point1inFig.3a,

s

1=

s

2=0.3MPa

4.1.1. Homogeneousmodel

Inthiscase,thetensilefailureoccursexactlyat(along) oneofthe

s

3-normalendsofthemodelduetotheend

effect. Although very small, this effect exists in the

numerical models, and occurs where the extension is appliedtothemodel,i.e.atthe

s

3-normalboundaries.

4.1.2. ‘‘Homogeneouslyheterogeneous’’modelwithsmall variationsofthePoisson’sratio

n

TheGaussdistributionof

n

isappliedinthismodelwith ameanvalueof0.25andastandarddeviationof0.02.This leads to a non-uniform stress distribution during the loading, which eliminates the boundary (end) affects because the stress perturbations within the model are largerthanthose(verysmall)causedbytheseeffects.

Fig.4showstheevolutionoffracturing,whichoccurs exclusivelyaccordingtotheBT mechanism asexpected (Fig.4e);theSDmechanismwasnotactivatedatall.The formedtensilefracturesconstituteagenerallynormalto

s

3but ratherirregularset.Thisis becausethemodelis

almosthomogeneousandthereforefracturingisinitiated simultaneouslyindifferentpointsresultinginstressdrops and perturbations of the stress field. Therefore, the initiated fracturespropagatein theheterogeneousfield, which explains their irregular geometry. To avoid such multiple stress perturbations, in the following model a smallweakzoneisintroduced.

4.1.3. Modelwithweakzone

ThiszoneisshowninFig.3bandcanberecognizedin

Fig.5a.Thestrengthparametersinthiszonearereduced, asindicatedinthecaptionofFig.5;theotherparameters arethesameasintherestofthemodel.Onecanseethat thefracturingoccursinthiscaseinacompletelydifferent waycomparedtothepreviousmodelinFig.4.Thereis onlyonefracturethatisinitiatedattheweakzoneand propagatesstrictlynormalto

s

3.Propagationoccursdue

to the tensile stress concentration at the fracture tip, whereitreaches

s

t(Fig.6b).ThisMode-I-type

propaga-tionoccursexclusivelyduetotheBTfailuremechanism,

Fig.3.(a)Initialyieldsurface/envelopeorfunctionusedinthenumericalsimulations(simplifiedplotinFig.2a).Points1to4correspondtothestressstates appliedinthenumericalmodels.(b)Setupofnumericalmodeling.Themodelswerepre-stressedtotheinitialstressess1ands2(s2=s1)correspondingto points1to4inFig.3a,andtoasufficientlylarges3forthemodeltoremaininapurelyelasticstate.Thens3wasprogressivelydecreased(byapplying velocitiesV3tothes3-normalboundaries),whilemaintainingtheinitialvaluesofs1ands2byapplyingvelocitiesV1andV2asshowninthisfigure.The modelsizeis510cm(200400numericalzones)inthes1ands3directions,andonenumericalzone,ins2direction.Totakeintoaccountinthemodels heterogeneitiestypicalforrealrocks,twoscalepropertyheterogeneitieswereintroducedindifferentmodels(indicatedinthecaptionsofthefigures showingthecorrespondingnumericalresults):(1)GaussdistributioninthenumericalzonesofthePoisson’sratiowithameanvalueof0.25andstandard deviationof0.02;(2)aweakzoneofalargerscale(includingseveralnumericalzones)showninthisfigure.Thestrengthinthiszoneisindicatedinthe captionsofthefigurespresentingthenumericalresults.

SDmechanismbeingoperatedonlywithintheweakzone (Figs.6c,d).

4.2. Point2inFig.3a,

s

1=

s

2=0.54MPa

4.2.1. Homogeneousmodel

A dense network of conjugate shear bands is first initiatedthroughouttheentiremodel(Fig.7a).Then,most initialbandsdie,whiletheremainingonesaccommodate

growinginelasticdeformation.Thisprocessresultsinafew principal (well-developed)shear bands(Fig.7c). Oneof themthenistransferredintothree-segmentfracturewhen theBDmechanismcomesinplay(Fig.7d,e).

4.2.2. Homogeneouslyheterogeneousmodel

Beforefracturing,thismodelundergoesdistributedBT, butmostlySDdistributeddamage(inelasticdeformation),

Fig. 8a. Then, an oblique fracture is initiated and

Fig.5.Evolutionoftheep

patterninthemodelwithGaussianndistributionandaweakzonewiththereducedinitialstrengthparameters:st=0.03MPa; kc=0.1MPa.Theinitialstressesandboundaryconditionsarethesameasinthepreviousmodel.

Fig.4.(a–d)Evolutionoftheinelasticstrain,ep_{,patternduringverticalunloading(s}

3reduction)ofthemodelatconstants2=s1=0.3MPa,correspondingto point1inFig.3a.Poisson’srationvariesinthenumericalzonesaccordingtoGaussdistribution.(e)Mechanicalstate(elasticandelastic-plastic),damage/ failureorinelasticdeformationmechanism(tensileorshear)anditshistoryforthedeformationstagein(d).Differentcolorsinthisimagecorrespondto differentdamage/failuremechanismsandtheirsuccession.Forexample,‘‘tensile-n(nmeansnow),shear-p,tensile-p(pmeanspast)’’indicatesthatthe correspondingzonesundergotensiledamageatthestageshown,buttheyalreadyunderwentshearandtensiledamageduringthepreviousdeformation stages(inthepast).Zonesindicatedas‘‘elastic’’didnotundergoinelasticstrainingatall.

propagatesduetobothBTandSDmechanisms(Fig.8b–d). Theobliquefracturingprocessisfollowedbyanalmost

s

3

-normalone(Fig.7d). 4.2.3. Modelwithweakzone

The introduction of the weak zone again changes radically thefractureprocessandgeometry(Fig.9).The fracturepropagatingfromtheweakzonedoesnotfollow exactly a linear trajectory in this case. Its tip slightly undulates during propagation, resulting in globally

s

3

-normal macro fracture orientation, whose formation involves both BT and SD mechanisms(Fig. 9). One can recognizeboththeprocess(bluedots)anddamage(green dots)zonesonthemapsofdamagedistributioninFig.9

(thesecondrowinthisfigure). 4.3. Point3inFig.3a,

s

1=

s

2=0.6MPa

4.3.1. Homogeneousmodel

Thedeformationpatterninthiscaseissimilartothatin

Fig.7,whichisexplainedbythefactthat,inbothcases,the imposedstressstatescorrespondtotheSDdomain. 4.3.2. Modelwithweakzone(Fig.10)

TheresultissimilartothatinFig.9,buttheprocessand damagezonesrelatedtothefracture(ormoreexactlyto thedeformationband)propagationaremuchlarger.They developexclusivelythroughanSDmechanismandinvolve almosttheentiremodel(thelast rowinFig.10). Atthe band/fracture tip,

s

3 istensile. Itsmaximummagnitude

reaches around0.06MPa, which is somewhat less than

s

0

t=0.07MPa. The nominal stress (which would be

measuredatthemodel(sample)endintheexperimental test)is

s

3=0.0011MPa,whichismuchlessinmagnitude

than

s

0 t.

Within the band, both BT and SD mechanisms are active.Asinthepreviouscase(Fig.9),thefracture/bandtip undulatesduring propagation. Thistime, thereasonfor this is clear from the octahedral strain rate ˙

g

and

s

t

patternsinFig.10.Duringgenerally

s

3-normalbandtip

propagation,theinitiationofobliqueshearbandsoccurs. Thefracturetiptendstofollowthese(oblique)directions alternately,keepinggeneraltrajectorynormalto

s

3.Inthe

nextmodel,itistheobliquedirectionthatisdefinitively followedbythedeformationband.

4.4. Point4inFig.3b,

s

1=

s

2=0.66MPa

4.4.1. Modelwithweakzone

Thesameweakzoneasinthepreviousmodelleadsin thiscasetotheinitiationofthe

s

3-normalextensionband

(that can hardly be recognized in Fig. 11) and of two conjugateshear bands (Fig.11a), thelatter propagating muchfasterthantheformer.Thenoneshearbanddiesand theotheronecutstheentiremodelandbecomesashear fracture(Fig.11e).Onecanseealargeprocesszoneinfront ofthepropagatingband(thesecondrowinFigs.11b–d). 5. Concludingdiscussion

Thereportedresultsshowthatnotonlythegeometrical attributes of fracture in geomaterials, but the fracture mechanism itself strongly depend on the presence of heterogeneities. Joints,which arebelievedtobeMode-I

Fig.6.Theinelasticstrainep

(a),theleaststresss3(b),andthemechanicalstate(c)patternsatthesameevolutionarystageofthemodelinFig.5;(d) correspondstothelaststageoffracturepropagationinFig.5e.

fractures, are supposed to form exclusively when the tensilestress

s

3reachesthetensilestrength

s

t(ormore

exactly,whenthestressintensityfactorreachesacritical value). Thisconceptworks fairlywellfor purelytensile fractures forming according to the brittle-tensile (BT) deformation/fracture mechanism. This mechanism is possible at a relatively low far-field stress level (or pressure P), when the inelastic process zone near the fracturetipisverysmallasinFigs.5and6.However,when P is high enough (comparedto

s

0

t), joints can form at j

s

3j<<

s

0t (Figs.8and9).Here

s

3isnotthestressatthe

micro-(granular)scale,butthe‘‘far-field’’stressappliedto theboundariesoftheobject(specimenorstructure).This corresponds to the nominal stress

s

3, which can be

measuredexperimentallyinthelaboratory.Atthe micro-scale,j

s

3jistensile,butlowerthan

s

0t.Purelytensile,

s

3

-normal fractures cannot propagate in this case. The damage (inelastic deformation) involves a wide zone (almost the entire model)emanating from the fracture front(Figs.10b–d).Thefailureofthematerialoccursinthis zoneaccordingtoashearductile(SD)mechanism,leading totheinitiationofobliqueshearbands.Thefracturetip tendstofollow theseobliquedirectionsalternately, but keeps general

s

3-normal orientation because of the

involvementoftheBTmechanisminthefailureprocess. Asaresult,thefracturezoneisnotlinear.Itisthickerand more heterogeneous than at lower P.This zone corres-pondstothedilation bands(embryonicjoints)obtained

Fig.7.Evolutionoftheep

experimentallyinChemendaetal.(2011a).Thewallsof theopenedbandcannotbesmooth(asinthecaseofthe ‘‘ideal’’Mode-Ifracture)andshouldbeartraces (irregular-ities) reflecting complex brittle-ductile failure process.

These traces are likely at the origin of a plumose morphologyofthewallsofnatural(aswellas experimen-tal,Fig.2c)joints,whichisusuallyinterpretedasaresultof mixedmode (IandIII, Fig.1)fracturingin thesenseof

Fig.8.Evolutionoftheep

andmechanicalstatepatternsinthemodelwithaGaussiandistributionofn.Theinitialstressesandboundaryconditionsarethe sameasinthepreviousmodel(thestressstatecorrespondstopoint2inFig.3a).

Linear Elastic Fracture Mechanics, LEFM, (Pollard and Aydin,1988).Bothopeningandshearareindeedpresentin thedescribeddilatancybanding(Fig.10),butthisprocess iswellbeyondthelimitsofapplicabilityofLEFM,which doesnottakeintoaccountexplicitlyinelasticdeformation. Noteagainthatthisfracturemodeispossibleonlyinthe presenceofstressconcentrators,whichcanberelatedto theheterogeneitiesofthematerial(caseanalyzedinthis work)typicalforrocksortostructuralheterogeneitiesalso widelypresentinsedimentarybasins.These heterogenei-tiesandtherelatedfractureinitiationpointsaretypicalfor naturaljointsandwereextensivelydiscussedingeological literature(e.g.,McConaughyandEngelder,2001;Pollard andAydin,1988).InFig.2c,theinitiationpoint(zone)is

located at theplace where theLVDT gage supportwas attachedtothesample(seecaptiontothisfigure).

Thenumericalmodelsthuspredictacomplicationof thefractureprocessandrougheningofthejointsurfaces (walls)withincreasingP.Thisisintotalagreementwith the experimental results reported in Chemenda et al. (2011a).Ifthestresslevelisstillhigher,theincipientshear bands (that do not evolve beyond the initiation stage duringthedescribeddilatancybanding)areresolvedinto macroshearbandsandthenshearfractures(Fig.11).The angle

c

betweenthesefracturesand

s

1isratherhighin

thisfigurebecausetheslopeoftheyieldenvelopeadopted intheconstitutivemodelchangesabruptlyattheBT-to-SD transition(Fig.3a).Inreality,theslopechangeisgradual

Fig.9.Evolutionoftheep

andmechanicalstatepatternsinthemodelwithaGaussiandistributionofnandaweakzonewithfollowingstrengthparameters:

st=0.03MPa;kc=0.3MPa.Theinitialstressesandboundaryconditionsarethesameasinthetwopreviousmodels(thestressstatecorrespondstopoint 2inFig.3a).

Fig.10.Evolutionofep

,oftheoctahedralstrainrate ˙g(incrementofthetotaloctahedralstrainduringonecalculationstep),ofthetensilestrengthst,andof themechanicalstatepatternsinthemodelwithGaussiandistributionofnandaweakzonewithst=0.03MPa,kc=0.3MPa.Theinitialstressescorrespond topoint3inFig.3a(s2=s1=0.6MPa).Onlycentralsegmentsoftheimageswithepandg˙patternsareshown,asbeyondthesesegments,strainsandstrain ratesarenegligible.

(Fig. 2a), which shouldbe taken intoaccount in future models. The impact of structural heterogeneities and notablythoserelatedtomultilayernatureofthe sedimen-tarybasins(McConaughyandEngelder,2001)shouldbe addressedaswell.

Acknowledgements

Thispaperisinvitedintheframeoftheprizesawarded by the ‘Acade´mie des sciences’ in 2017. It has been reviewed and approved by Jean-Paul Poirier and by VincentCourtillot

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