rock mechanics characterisation of the rock mass

Rock mechanics characterisation of the rock mass – theoretical approach

Preliminary site description Simpevarp subarea version 1.2

Anders Fredriksson, Isabelle Olofsson

Golder Associates AB

October 2006

R-05-87

Svensk Kärnbränslehantering ABSwedish Nuclear Fueland Waste Management CoBox 5864SE-102 40 Stockholm Sweden Tel 08-459 84 00 +46 8 459 84 00Fax 08-661 57 19 +46 8 661 57 19

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Rock mechanics characterisation of the rock mass – theoretical approach

Preliminary site description Simpevarp subarea version 1.2

Anders Fredriksson, Isabelle Olofsson

Golder Associates AB

October 2006

ISSN 1402-3091

SKB Rapport R-05-87

This report concerns a study which was conducted for SKB. The conclusions and viewpoints presented in the report are those of the author(s) and do not necessarily coincide with those of the client.

A pdf version of this document can be downloaded from www.skb.se

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Symbols and abbreviations

ci Cohesionofintactrock[MPa]cf Peakcohesionoffracture[MPa]cm Peakcohesionoftherockmass,Mohr-Coulomb[MPa]Ei Young’smodulusoftheintactrock[GPa]Em Young’smodulusoftherockmass[GPa]Kn Jointnormalstiffnessatexpectednormalstress[MPa/m]Ks Jointshearstiffnessatexpectednormalstress[MPa/m]kr ExponentinPowerLawsizedistributionTi Tensilestrengthofintactrock[MPa]UCSi Uniaxialcompressivestrengthofintactrock[MPa]X0 MinimumradiusinPowerLawsizedistribution

φi Internalfrictionangleofintactrock[°]φf Internalfrictionangleoffracture,Mohe-Coulomb[°]φm Internalfrictionangleofrockmass[°]νi Poisson’sratiooftheintactrockνm Poisson’sratiooftherockmassσ1 Maximumprincipalinsitustress[MPa]σ2 Intermediateprincipalinsitustress[MPa]σ� Minimumprincipalinsitustress[MPa]σa Levelofhorizontalconfiningstressforsimulations[MPa]σb Levelofhorizontalconfiningstressforsimulations[MPa]σH Maximumhorizontalinsitustress[MPa]σh Minimumhorizontalinsitustress[MPa]σvf Verticalstressatfailure[MPa]

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Abstract

ThepresentReportsummarisesthetheoreticalapproachtoestimatethemechanicalpropertiesoftherockmassinrelationtothePreliminarySiteDescriptiveModelling,Simpevarpsubarea,version1.2.

ThetheoreticalapproachisbasedonthegeometricalDFN-description(DiscreteFractureNetwork)ofthefracturesystemintherockmassandontheresultsofmechanicaltestingofintactrockandonrockfracturesfromthesite.

Toestimatethemechanicalpropertiesoftherockmassaloadtestonarockblockwithfracturesissimulatedwiththenumericalcode�DEC.ThelocationandsizeofthefracturesaregivenbyDFN-realisations.Therockblockisloadedinplainstraincondition.Fromdecalculatedrelationshipbetweenstressesanddeformationsthemechanicalpropertiesoftherockmassaredetermined.

Theinfluenceofthegeometricalpropertiesofthefracturesystemonthemechanicalpropertiesoftherockmassisanalysedbyloading20blocksbasedondifferentDFN-realisations.Thematerialpropertiesoftheintactrockandthefracturesarekeptconstant.Thepropertiesaresetequaltothemeanvalueofeachmeasuredmaterialproperty.

Theinfluenceofthevariationofthemechanicalpropertiesoftheintactrockandvariationofthemechanicalpropertiesofthefracturesareestimatedbyanalysingnumericalloadtestsononespecificblock(oneDFN-realisation)withcombinationsofpropertiesforintactrockandfractures.Eachparameterisvariedfromitslowestvaluestoitshighestvalueswhiletherestoftheparametersareheldconstant,equaltothemeanvalue.Theresultingdistributionisexpressedasavariationaroundthevaluedeterminedwithmeanvaluesonallparameters.

ToestimatetheresultingdistributionofthemechanicalpropertiesoftherockmassaMonteCarlosimulationisperformedbygeneratingvaluesfromthetwodistributions,causedbyfracturenetworkvariationandpropertyvariation,independentofeachother.Thetwovaluesareaddedandthestatisticalpropertiesoftheresultingdistributionaredetermined.

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Sammanfattning

Dennarapportsammanfattardetteoretiskaangreppssättetattuppskattabergmassansmekaniskaegenskaperisambandmeddenplatsbeskrivandemodellenversion1.2förSimpevarp

DetteoretiskaangreppssättetbaserasdelspådengeometriskaDFN-beskrivningen(DiscreteFractureNetwork)avbergmassansspricksystemochdelsmekaniskalaboratorietesterutfördapåintaktbergochpåbergsprickorfrånplatsen.

Förattuppskattabergmassansmekaniskaegenskaperutförsettnumerisktbelastningsförsökpåettbergblockidennumeriskakoden�DEC.LägeochstorlekpåsprickornaiblocketbaseraspåDFN-realiseringar.Blocketbelastasunderplanttöjningstillstånd.

Inverkanavspricksystemetsgeometriskautformningbestämsgenomattanalyseraca20stDFN-realiseringarmedkonstantaegenskaperhosdetintaktabergetochhossprickorna.Egenskapernaharsattslikameddeuppmättamedelvärdenaförrespektiveegenskap.

InverkanavvariationhosdetintaktabergetsochsprickornasmekaniskaegenskaperbestämsgenomattförenDFN-realiseringutföraanalysermedkombinationeravegenskaper.Varjeparametervarierasmellandesslägstaochhögstavärdemedanövrigaparametrarhållskonstanta.Denresulterandefördelningenuttryckssomvariationkringdetvärdesombestämtsmedmedelvärdepåallaegenskaper.

FöratterhålladenresulterandefördelningenpåbergmassansegenskapergörsMonte-Carlosimuleringardärettvärdeslumpasframurdebestämdafördelningarnaöverspricksystemetsgeometriskainverkanochinverkanavvariationavdelkomponenternasegenskaper.Detvåvärdenaadderasföratterhålladenresulterandefördelningenhosbergmassansmekaniskaegenskaper.

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Contents

1 Introduction 9

2 Indata 112.1 Intactrock 112.2 Fractures 11

2.2.1 Geometryoffractures 112.2.2 Mechanicalpropertiesoffractures 14

2.� Insitustresses 1�

3 Set-upofthemodel 1��.1 Descriptionofthenumericalsimulations 1��.2 Assumptions 20�.� Indatatothenumericalsimulations 20

4 Simulations 2�4.1 Descriptionoftheprocedure 2�4.2 DFNgeometry-inducedrockmassvariability 2�

4.2.1 SimulationsparalleltoσH,RockDomainA 2�4.2.2 Simulationsparalleltoσh,RockDomainA 284.2.3 SimulationsparalleltoσH,RockDomainB �24.2.4 SummaryofDFNgeometry-inducedrockmassvariability ��

4.� Materialpropertyinfluenceonrockmassparameters ��4.4 Monte-Carlosimulations 40

4.4.1 Adjustingboundaries 444.4.2 Combinedresults 4�

5 Discussion 49

6 Conclusions �1

7 References ��

AppendixA ��AppendixB ��AppendixC �9

9

1 Introduction

Thisworkreportsresultsfromoneofthefourrockmechanicsactivitiesthathavebeenrecognisedwithintheproject“SimpevarpArea–SiteDescriptiveModelduringtheinitialSiteInvestigationstageversion1.2”.ThisactivityaimstodeterminetheundisturbedmechanicalpropertiesoftherockmassinthelocalmodelareaforSimpevarp1.2.Theseparameterswillbeusedforthepreliminarydesignandtoevaluatethesuitabilityofthesite.

Theapproachusedinthisactivityisbasedonnumericalsimulationswiththeuseofthe�DECsoftware/�DEC200�/.ThemethodologyhasbeendevelopedinthepurposeoftheSiteInvestigationsandisbuiltuponthreedifferentmodels:theDFNmodelwhichisusedtosimulatethefracturenetworkintherockmass,the�DECmechanicalmodelwhichisusedtocalculatetherockmassmechanicalproperties,andtheGoldSimmodelwhichisthetoolforestimationofcombinedvariabilities.

Themodellingprocedureisdescribedindetailin/OlofssonandFredriksson200�/.

TheDFNmodel,theinsitustressesaswellasthemechanicalpropertiesofintactrockandfracturesconstitutetheinputdatathatarenecessarytobuildthe�DECmodel,andaredescribedinChapter2.Thentheset-upofthe�DECmodelandtheprocedureusedfornumericalsimulationsaredescribedinChapter�.Theresultsobtainedfromsimulationsin�DECandGoldSimarereviewedandanalysedinChapter4,andthesummarytablesofmechanicalpropertiesoftherockmassarepresented.Chapters�and�presentashortdiscussionandconclusionsofthestudy.

11

2 Indata

2.1 Intact rockInordertodeterminewhatintactrockparametersshouldbeassignedtothematerialinaspecificrockdomain,themainandsubordinaterocktypesweregivenwithanestimationoftheiroccurrenceineachrockdomain,Table2-1.RegardingthecompositionoftherockdomainsandtherocktypesthathavebeentestedvaluesareavailableforrockdomainAandB,calledrespectivelyRDAandRDB.

Laboratorytestdataareavailableonlyfortworocktypes,thequartzmonzonitetomonzodioriteandthefine-graineddioritoid,/LanaroandFredriksson200�/.

2.2 Fractures2.2.1 Geometry of fracturesTheparametersfortheDFNmodelusedinthisstudy(Simpevarpversion1.2)weredeliveredandpresentedattheendofJune2004.Thestatisticalparametersaredescribedin/LaPointeandHermanson200�/.

Onealternativewasdevelopedwhichisbasedonsixsub-verticalsetsoffracturesandonesub-horizontalsetoffractures.Threeofthesub-verticalsets(NNE-NE,EW-WNWandNW-NNW)aredefinedasregionalandtheircharacterisation(orientation,sizedistributionandintensity)isbasedoninformationfromoutcropsandlineaments.Theotherthreesub-verticalsets(BGNE,BGNSandBGNW)areconsideredtorepresentthebackgroundfracturingintherockmassandtheircharacterisationisbasedonoutcropdata.

Sub-horizontalfractures(SubHZ)arealsoconsideredtobelongtothebackgroundfracturingoftherockmassbuttheircharacterisationisbasedonlyonboreholedata.

TheparametersfortheDFNmodelhavebeenstudiedandusedforgeneratingthe�Dfracturenetworkrequiredforsetting-upthenumericalmechanicalmodel.TheparametersintheDFNmodelarepresentedbelow.

Table 2‑1. Rock types identified in the different rock domains (from /Appendix 6, SKB 2005/).

Rock domain Main rock type % Subordinate rock types %

RDA Ävrö granite 75.8–84.7 Fine- to medium- grained granite 0.8–21.5Fine-grained dioritoid 9–17

Fine-grained mafic rock 3–4.9RDB Fine-grained dioritoid 90.6–94.2 Fine- to medium- grained granite 0.9–6.7

Quartz monodiorite 0–3.5RDC Quartz monzodiorite 51.5–73.9 Fine-grained dioritoid 6.5

Ävrö granite 22.9–34.1 Fine- to medium- grained granite 1.8–4.2Granite 2

RDD Quartz monzodiorite – Fine- to medium- grained granite –Pegmatite –Fine-grained mafic rock –

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Orientation

Themeantrendandplungetogetherwithdispersionaregivenforeachsetdisregardingifthefracturesareopen,partlyopenorclosed(definitionaccordingtoBOREMAPmapping).InTable2-2theparametersfortheorientationofsub-verticalfracturesetsaregivenandinTable2-�theparametersfortheorientationofthesub-horizontalfractureset.TheparametersfororientationofthefracturesetsareequalinallrockdomainsA,B,CandD.

Size distribution

ThesizedistributionsusedaretheonesprovidedintheDFNmodel,Simpevarpversion1.2.Table2-4containsdataforthesub-verticalsetsandTable2-�forthesub-horizontalset.Fornumericalreasonsin�DEConlyfractureswitharadiuslargerthan1mweregenerated.TheparametersforsizedistributionforthefracturesetsareequalinallrockdomainsA,B,CandD.

Table 2‑2. Orientation of the sub‑vertical fracture sets, from /LaPointe and Hermanson 2005/.

OrientationSet name Mean pole trend/

plunge/dispersion1)Model/K‑S2) Relative % of total

population of sub‑vertical fractures

NNE-NE 118.0/1.9/17.3 Fisher Not significant

18.99%

EW-WNW 17.1/7.3/11.2 Fisher Not significant

17.75%

NW-NNW 73.1/4.7/13.7 Fisher Not significant

22.50%

BGNE 326.3/5.5 K1:17.65 K2:18.14

Bivariate Fisher 0.041/45.4%

18.60%

BGNS 96.8/3.8/20.32 Fisher not significant

15.44%

BGNW 22.1/2.4 K1:5.36 K2: 6.66

Bivariate Fisher 0.051/61.3%

6.71%

1) k for univariate distribution, k1 and k2 for bivariate distribution.2) Distribution model/Statistics for the Kolmogorov-Smirnov Goodness-of-fit test.

Table 2‑3. Orientation of the sub‑horizontal fracture set, from /LaPointe and Hermanson 2005/.

OrientationSet name Mean pole trend/

plunge/dispersionModel/K‑S Relative % of total

population of sub‑horizontal fractures

SubHZ 33/86/15 Selection by visual inspection, dispersion 15 degrees.

100%

1�

Table 2‑4. Size distribution for the sub‑vertical fracture sets, from /LaPointe and Hermanson 2005/.

SizeSet name Model Minimum

size (X0) (m)kr (parent population) or Std. deviation

Comments (used data etc)

NNE-NE Powerlaw 0.36 2.58 (mass, median) Estimated from outcrop data and lineaments.

EW-WNW Powerlaw 0.36 2.8 (mass, median) Estimated from outcrop data and lineaments

NW-NNW Powerlaw 0.49 2.87 (mass, median) Estimated from outcrop data and lineaments

BGNE Log-normal 0.48 0.55 Estimated from outcrop data. Univariate Fisher also significant at 43.9% (K = 16.9)

BGNS Log-normal 0.67 0.82 Estimated from outcrop dataBGNW Log-normal 0.45 1.00 Estimated from outcrop data. Weakly-

developed set; Bivariate normal also significant at 18.8%

Table 2‑5. Size distribution for the sub‑horizontal fracture set.

SizeModel Minimum size (X0)

or mean radius (m)kr (parent population) or Std. deviation

Comments (used data etc)

Lognormal 0.57 1.86 Estimated from borehole data (size from outcrop). Size model not well known (small data sample)

Intensity

Fractureintensitycanbequantifiedbyseveralmeasures,includingthenumberoffracturesperunitlength(P10),thenumberoffracturesperunitarea(P20),theamountoftracelengthperunitarea(P21),andtheamountoffracturesurfaceareaperunitvolumeofrock(P�2).TheparameterP�2isoftenthemostusefulwaytodescribefractureintensityinastochasticDFNmodel,asitisscale-anddirectionally-independent.

ThetablesprovidedfortheDFNmodelv1.2/LaPointeandHermanson200�/presentintensi-tiesforsub-verticalsetsandsub-horizontalsetsforrockdomainsA,BandC(NoinformationareprovidedforrockdomainD).Accordingtothesetablestheproportion(expressedinP�2)ofsub-horizontalfracturesintherockmassis�0–��%.Neverthelesstheproportionofsub-horizontalfracturesintherockmassisestimatedfromboreholestobebetween12and20%(respectivelyweightedandunweightedplotsoffractures).Henceduetoinconsistencyofdatatheintensitiesofsub-verticalsetswerere-calculatedtakingintoaccounttheirrelativeproportionintherockmass.ThevaluesofP�2arespecifictorockdomain,andforeachrockdomaintherelativeP�2foreachfracturesetwascalculated,seeTable2-�forrockdomainAandTable2-�forrockdomainB.

14

Table 2‑6. P32 for all fracture sets in the rock domain A (RDA).

All fractures Open fractures Sealed fracturesP32 total 3.02 0.97 2.06% horizontal 12% 20% 12% 20% 12% 20%

NNE-NE 0.50 0.46 0.16 0.15 0.34 0.31EW-WNW 0.47 0.43 0.15 0.14 0.32 0.29NW-NNW 0.60 0.54 0.19 0.17 0.41 0.37BGNE 0.49 0.45 0.16 0.14 0.34 0.31BGNS 0.41 0.37 0.13 0.12 0.28 0.25BGNW 0.18 0.16 0.06 0.05 0.12 0.11SubHZ 0.36 0.60 0.12 0.19 0.25 0.41

Table 2‑7. P32 for all fracture sets in the rock domain B (RDB).

All fractures Open fractures Sealed fracturesP32 total 7.66 1.42 6.24% horizontal 12% 20% 12% 20% 12% 20%

NNE-NE 1.28 1.16 0.24 0.22 1.04 0.95EW-WNW 1.20 1.09 0.22 0.20 0.97 0.89NW-NNW 1.52 1.38 0.28 0.26 1.24 1.12BGNE 1.25 1.14 0.23 0.21 1.02 0.93BGNS 1.04 0.95 0.19 0.18 0.85 0.77BGNW 0.45 0.41 0.08 0.08 0.37 0.33SubHZ 0.92 1.53 0.17 0.28 0.75 1.25

TheP�2giveninTable2-�andTable2-�representsthemeanfractureintensityofthefracturenetworkinthegivenrockdomains.Thefractureintensityactuallyvariesinsidetherockdomainsbutthisisneitherdescribednoranalysedinthisreport.

2.2.2 Mechanical properties of fracturesLaboratorynormalloadtestsupto10MPaandsheartestsatthedifferentnormalstresslevels,0.�,�and20MPahavebeenperformedonfracturesfromboreholeKSH01A,KSH02AandKAV01.Thelaboratorytestsareevaluatedandtheresultsgivenby/LanaroandFredriksson200�/.

Thedatawasstatisticalanalysed.Atruncatednormaldistributedwaschosenbyexpertjudgementtodescribethemodel.ThepreliminarymechanicalpropertiesoffracturesthatwereusedatthisstageispresentedinTable2-8intermsofmean,spanandrangeofpotentialvaluesforeachparameter.Thecohesionisexpressedasafunctionofthefrictionangleasthetwoparametersarecorrelated.

1�

Table 2‑8. Summary of mechanical properties of fractures evaluated from laboratory tests /Lanaro and Fredriksson 2005/.

Parameter for single fractures (small scale).

All fracture set 1) Truncated normal distribution mean/standard deviation;

Min trunc. – max trunc.

Normal stiffness 100/32 MPa/mm 49–179 MPa/mmShear stiffness 29/11 MPa/mm 10–49 MPa/mm

Peak friction angle, φ 32°/4° 24°–40°Cohesion 2) cmean = 2.35–0.058 · φ/0.25 MPa cmin = cmean – 0.37 MPa

cmax = cmean + 0.69 MPa

1) In later versions there may be different parameters for different sets.2) The cohesion is dependent on the friction angle. The friction angle given in °.

2.3 In situ stressesTwodifferentstressdomainsweredefinedinSimpevarp/HakamiandMin200�/.Thestateofstresswasestimatedforeachdomainasafunctionofdepthandtheseestimationswereusedtoselecttheconfiningstresslevelsforthenumericalloadingtestsrepresentingtheconditionsatrepositorydepth,�00m.ThesevaluesaregiveninTable2-9.Thestressesdivergeinmagnitudebetweenthetwodifferentstressdomainsbuttheirorientationissimilar.ForbothlithologicaldomainsAandBonlythestressdomainIwasconsideredfordirectloadingtesttoenabledirectcomparisonsofrockmassproperties.

Table 2‑9. In situ stress magnitude and orientation for both stress domains at 500 m depth.

Stress domain I Stress domain IIσ1 σ2 σ3 σ1 σ2 σ3

Mean magnitude, MPa 32 14 9.5 16 9 5.5Mean strike, ° 132 90 42 132 90 42Mean dip, ° 0 90 0 0 90 0

1�

3 Set‑up of the model

3.1 Description of the numerical simulationsTheparameterspresentedinSection2.2.1wereusedtogeneratethe�Dfracturenetworkusedforextractionoffracturedatainto�DEC.

Thefracturenetworksweregeneratedfortworockdomains,RDAandRDB,basedonthedifferentfractureintensityinthetworockdomains.TwodifferentsetsofparameterswereusedforP�2dependingontheestimatedrelativeproportionofsub-horizontalfracturesintherockmass.

Foreachrockdomain20realisationsofthesamefracturenetwork(i.e.withallinputparametersequivalent)aresimulatedforthe“basecase”(i.e.20%ofsub-horizontalfracturesintherockmass).

Onlyopenfractures(includingpartlyopenfractures)weregeneratedintheDFNmodel.Basedontheresultsoflaboratorytests,theassumptionthatsealedfracturesdonotsignificantlyinfluencethemechanicalbehaviouroftherockmasswasmade.

Whenthe�Dfracturenetworksaregenerated2Dverticalsamplingplanesorientedparalleltothehorizontalinsitustresses(σHandσh)areextracted.Thetracedataontheseplanesareusedforinputin�DEC.Theidentificationofeachfracturesetismaintainedthroughouttheprocessallowingassigningdifferentmechanicalpropertiestothedifferentfracturesets.

InFigure�-1anexampleofgeneratedfracturetracesinaverticalplainisshown.InFigure�-2thecorresponding�DECmodelisshown,andinFigure �-�thecontactpointsalongeachfractureinthe�DECmodelareillustrated.

Figure 3‑1. Example of fracture traces in a vertical plan. Fracture traces from different fracture sets have different colours.

x

y

-10 -5 0 5 10-10

-5

0

5

10

18

Figure 3‑2. 3DEC model generated from the fracture traces shown in Figure 3-1.

x

y

- 10 -5 0 5 10-10

-5

0

5

10

54321

matID

Figure 3‑3. Contact points along fractures in the 3DEC model.

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Theresultintheformofverticalstress-verticalstrainandhorizontalstrain–verticalstraincurvesfromonesimulationwith�DECisshowninFigure �-4.

Thedeformationmodulus,Em,andPoisson’sratio,νm,oftherockmassareevaluatedfromstress-verticalstrainandhorizontalstrain–verticalstraincurves.Thestrengthparametersoftherockmass,uniaxialstrength,UCSm,cohesion,cm,andfriction,φm,areevaluatedfromsimulationswithdifferentconfiningstress.Thefollowingequationsareused:

φm=arcsin(k–1⁄ k+1) (�.1)

UCSm=σ vfb+k·σb (�.2)

cm=UCSm·(1–sinφm)⁄2·cosφm (�.�)

wherek = (σ vfa – σ vfb)⁄(σ a – σ b)andσvfa,σvfb,σaandσbaretheprincipalverticalstressesat

failureattwoconfiningstresslevelsaandb.

Distributionsofthefourrockmassparameters(Em,νm,cm,andφm)areestimatedatablockscaleof20m,usingthesoftware�DECfortherockmechanicalmodelingpartandGoldSimforsubsequentMonte-Carlosimulations.

Theprocedureisinmoredetaildescribedin/OlofssonandFredriksson200�/.

Theuncertaintyofamodelcanbeseparatedintoconceptualuncertainty,datauncertaintyandspatialvariability.Theconceptualuncertaintyoriginatesfromanincompleteunderstandingoftheprincipalstructureoftheanalyzedsystemanditsinteractingprocesses.Thisuncertaintyisnotfurtherdiscussed.

Datauncertaintyconcernstheuncertaintyinparametervaluesbeingusedinamodel;itmaybecausedbymeasuringerrors,interpretationerrorsoruncertaintyinextrapolationofspatiallyvariableparameters.

Figure 3‑4. Example of stress- strain curves.

0

50

100

150

200

250

300

350

400

0.000 0.002 0.004 0.006 0.008 0.010

vertical strain

vert

ical

str

ess,

MPa

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

horiz

onta

l str

ain

vertical strain - vertical stresshorizontal strain- vertical strain

20

Spatialvariabilityconcernsthevariationinspaceofaparametervalue;althoughthisisnotstrictlyanuncertainty,incombinationwithpracticallimitationsinrockcharacterization,itconstitutesanindirectsourcefordatauncertainty.Hence,inthefollowing,nodistinctionismadetowhatextenttheestimatedrockmassparameterdistributionsrelatetospatialvariabilityand/ordatauncertainty.

Inthecaseofthepresentdata,stochasticmaterialpropertiesofintactrockandoffracturesareapproximatedbyempirical,truncated,normaldistributionsthataredefinedbytheirmean,standarddeviation,minimumandmaximumvalues(Table�-1).Likewise,theDFNgeometryisgivenasstochasticdistributions.

Ideally,rockmasspropertydistributionscouldbeestimatedbyiterative�DECsimulationsinvolvingnumerousstochasticDFNrealizations,wheretheDFNgeometryandmaterialpropertyparametersareallowedtotakeonanyvaluefromtheirdefinedinputdistributions.However,suchadirectapproachbecomesimpracticalduetoitscomputationaldemandandlimitationsinparameterdescriptionsin�DEC.

Instead,asimplerstochasticapproachisused.Here,�DECisonlyusedtoestimatetheDFNgeometry-inducedvariabilityandtheinfluenceinputmaterialparameters(intactrockandfractures)haveonrockmassproperties.ThecombinedeffectofDFNgeometry-inducedvariabilityandthematerialproperty-inducedvariabilityisestimatedbyMonte-CarlosimulationsusingasimpleGoldSimmodel.

Theprocedureformanagementofuncertaintyisdescribedinthemethodologyreport/OlofssonandFredriksson200�/.

3.2 AssumptionsThekeyconceptusedhereisthattherockmassvariabilitydependingonthegeometryofthefracturenetwork(DFN-model)canbeeevaluatedindependentofthevariabilityfromthevariationofmechanicalpropertiesofthefracturesandtheintactrocki.etheyareindependentofanother.Thevariabilitycanbeevaluatedseparatelyandthetotalvariabilitycanbeestimatedbysuperimposingtheeffectsofthetwocomponents.

Sealedfracturesarenotexplicitlysimulatedandtestsamplescontainingsealedfracturesaretreatedas“intactrock”samples.

3.3 Indata to the numerical simulationsFromthelaboratorywehaveuniaxialandtiaxialloadtests.Foreachrocktypemoreuniaxialloadtestareperformedthantriaxialtests.Thereforetheuniaxialtestsgiveabetterbasistoestimatethevariationinstrength,UCSiandtypeofdistributionthanthetriaxialtests.Fromthetriaxialtestsitispossibletoestimateofthevariationofthefrictionangle,φioftheintactrock.Therelationshipbetween,φi,ciandUCSiis

i

iii

cUCSφφ

sin1cos2

−⋅⋅

= (�.4)

Knowingthedistributionoftheuniaxialstrength,UCSi,andthedistributionofthefrictionangle,φi,thedistributionofthecohesion,ci,(Equation�.4)canbedeterminedbyGoldsimsimulations,assumingthatthecohesion,ci,andthefrictionangle,φi,arenotcorrelated.

21

Howevertotestwhethertheassumptionofuncorrelatedciandφiisactuallyreasonable,triaxialtestdataweresimulatedfromafirstestimateofthedistributionsofciandφi.These“simulatedtriaxialtestdata”werethencomparedtothe“realtriaxialtestdata”(Figure�-�aandb).Ascanbeseen,thegivenciandφiproduceatoonarrowrangeforRDAandatoowiderangeforRDB,ifcomparingsimulatedandmeasuredUCSi-values.Also,thegivenlowerlimitofUCSiforRDBishigherthantherealdataindicates.Toconclude,thegiveninputparametersdefineanover-determinedsystem.

Inordertoadjustciandφi,soastobettermatchthetri-anduniaxialmeasureddata,ciisinsteadcalculatedfromUCSiandφi(whichalsoareassumeduncorrelated),using(Equation�.�).TheUCSilimitsofRDBareredefinedaccordingtouniaxialloadingtestmeasurements.AscanbeseeninFigure�-�aandb,thenew“simulatedtriaxialtests”matchtherealdatabetter.

( )1 sin2cos

rr

r

UCSc

φφ

−= (�.�)

Figure 3‑5a and b. Probability distributions of simulated triaxial test data from given distributions of cr and φr (assumed non-correlated). Pink boxes are real intact rock data and white boxes refer to sampled including sealed fractures. Red lines indicate given limits of UCS.

Figure 3‑6a and b. Probability distributions of simulated triaxial test data from given distributions of UCSi and φi (assumed non-correlated). Pink boxes are real intact rock data and white boxes refer to sampled including sealed fractures. Red lines indicate modified limits of UCSi.

22

ThenewcalculatedcidistributionsaresummarizedinTable�-1.Asaconsequence,ciandφibecomecorrelatedandthecorrelationcoefficientis–0.�2�forRDAand–0.241forRDB.Aswillbediscussedlater,therockmassUCSofRDBdependsstronglyonci,anditslargespan(14–59),whichisadirectconsequenceofthelargerangeofφi,isfoundunrealistic.Instead,φiin(Equation�.�)isalwayschosensuchthattheprevioustruncationlimitsofci(20–42)stillapplyforRDB.TheincreasedrangeforciinRDAhasaminorimpactonrockmassUCS.

StatisticaldistributionsofinputparametersareshowninTable�-2.Thefracturepropertiesareassumedtobeequalforbothrockdomains.Thecohesionandfrictionanglesforintactrockofbothdomains,ciandφi,areassumedtobeindependent(non-correlated).Statisticsoftheuniaxialtensilestrengths,Ti,arealsogiven.

Table 3‑1. Cohesion for intact rock.

Mean Standard deviation

Min Max

RDA ci (MPa) 22 3.2 14 29RDB ci

(1) (MPa) 32.5 5.4 (14) (59)

ci(2) (MPa) 32.5 5.4 20 42

(1) Strictly applying (Equation 3.2).(2) Applying the truncation limits of ci (20–42).

Table 3‑2. Input parameter and distributions for intact rock and fracture properties.


Min Max

Intact rock, RDA Ei (GPa) 80 10 70 90νi (–) 0.27 0.05 0.18 0.33

φi (°) 60 3 57 62ci (MPa) 22 3.2 14 29Ti (MPa) 17 4 12 24

Intact rock, RDB Ei (GPa) 85 10 70 110νi (–) 0.26 0.03 0.19 0.31φi (°) 55 6 35 60ci (MPa) 32.5 5.4 20 42Ti (MPa) 20 2 14 24

Fractures Kn (MPa/mm) 100 32 49 179Ks (MPa/mm) 29 11 10 49φf (°) 32 4 24 40cf (MPa) 2.35–0.058×φf 0.25 cf mean–0.37 cf mean+0.69

2�

4 Simulations

4.1 Description of the procedureThedistributionsofrockmasspropertiesbeingestimatedhere(Em,νm,cm,andφm)areassumedtoconsistoftwomaincomponents:a)anintrinsicvariabilitycomponentcausedbyitsstochasticDFNgeometryandb)acomponentinducedbystochasticmaterialpropertiesoffractures(Kn,Ks,φf,andcf)andthoseofintactrock(Ei,vi,φi,ci,andTi).Further,thesetwocomponentsareassumedtobeindependent,suchthatthetotalrockmasspropertydistributionscanbeestimatedbysuperimposingtheDFNgeometry-basedandthematerialproperty-relatedvariabilitycomponents.Theprocedureoutlinecanbesummarizedasfollows:

1. ThevariabilitycomponentcausedbystochasticfracturenetworkgeometryisevaluatedformultipleDFNrealizations;theseareallassignedmeanmaterial-propertyvalues.

2. Theinfluencethateachindividualmaterialpropertyhasontherockmasspropertiesisestimatedforonespecific“average”realization;itisdonebyexaminingtheeffectonrockmassparametersaseachmaterialpropertyisassigneditsminimumandmaximumparametervalues,whileallothermaterialpropertiesaresettotheirmeanvalues.

�. Next,theeffectthatvariablematerialpropertieshaveontherockmassisthenestimatedinastochasticframework;materialparametersaresampledfromtheirempiricaldistributions(Table�-1)andappliedtotherelationshipsobtainedinstep2,toprovideestimatesoftheirimpactonrockmasspropertyvariability.

4. Finally,theDFNgeometry-inducedandthematerialproperty-relatedcomponentsaresuperimposedtoestimatethetotalrangesofrockmassparameterdistributions.

4.2 DFN geometry‑induced rock mass variability4.2.1 SimulationsparalleltoσH, Rock Domain AThefirstvariabilitycomponent,relatingtovariabilityarisingfromthestochasticfracturenetworkgeometryalone,isevaluatedby�DECmodelingofDFNrealizationswithfracturetracesinaplaneparalleltoσ1subjecttotwoconfiningstresses:�2MPaand8MPa(seeSection2.�).32MPaisequivalenttoσ1instressdomainI,and8MPaisselectedas2�%ofthisvalue.Themeanmaterialpropertyvalues(Table�-2)areassignedbothtofracturesandtotheintactrock.Outof20generatedDFN-realizations�DECcouldgeneratezonedivisionfor1�ofthem,�realizationshadtoberejected.

Thenumericalmodelswereloadedwithaconstantvelocityintheverticaldirectionwhilethehorizontalconfiningstress,σarespectiveσb,waskeptconstantduringtheloadingtest.Thedeformationmodulus,E,Poisson’sratio,ν,andtheverticalstressatfailure,σvf,wereevaluatedatbothconfiningstresslevelstoprovideanestimationofφmandcm.

Theevaluatedrockmassparametersatconfingstress�2MPaand8MPaarepresentedinAppendixA.InFigure4-1,Figure4-2,Figure4-�andFigure4-4thedistributionsfordeformationmodulusandPoisson’sratioareillustrated.

24

Figure 4‑1. Distribution of deformation modulus at high confining stress level (32.0 MPa), Rock Domain A, trace plane parallel to σH.

Figure 4‑2. Distribution of Poisson’s ratio at high confining stress level, (32 MPa) Rock Domain A, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

57 58 59 60 61 62 63 64 65 66 67 68 69 70

Deformation modulus, GPa

Data from 3DEC simulations

Adapted distribution

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31

Poisson´s ratio

Data from 3DEC simulationsAdapted distribution

2�

Figure 4‑3. Distribution of Deformation modulus at low confining stress level, (8.0 MPa) Rock Domain A, trace plane parallel to σH.

Figure 4‑4. Distribution of Poisson’s ratio at low confining stress level, (8,0 MPa) Rock Domain A, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

35 40 45 50 55 60 65



0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.26 0.28 0.3 0.32 0.34 0.36 0.38

Poisson´s ratio


2�

TheevaluatedcohesionandfrictionangleoftherockmassforeachsimulationarepresentedinAppendixA.Theseparameterswereevaluatedbyfittingastraightlinebetweentheverticalstressatfailureatbothstresslevels.Theuniaxialcompressivestrengthoftherockmasshasbeencalculatedfromtheevaluatedcohesionandfrictionangle.Thedistributionsoffrictionangle,cohesionanduniaxialcompressivestrengthareshowninFigure4-�,Figure4-�andFigure4-�.

Figure 4‑5. Distribution of friction angle, rock mass in Rock Domain A, trace plane parallel to σH.

Figure 4‑6. Distribution of cohesion, rock mass in Rock Domain A, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

30 35 40 45 50 55

Friction angle, °


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

2 4 6 8 10 12 14 16 18 20 22 24 26

Cohesion, MPa


2�

Somerealizationsgiveaverylowvaluefortheuniaxialstrengthoftherockmass.Ifyouexaminetheserealizationsindetailyouseethatusuallyatleastonefracturecutsofacorneroftheblockandslidingoccursalongthisfracture.OneexampleisillustratedinFigure4-8,whereonefracturecutofthelowerrightcornerofthemodel.Theresultsoftheserealizationsareomittedwhenthefinaldistributionsforφmandcmarecalculated.ThefinalobtaineddistributionsofEm�2MPa,νm�2MPa,Em8MPa,νm8MPa,φmandcmaresummarizedinTable4-1forrockdomainA.Thedistributionsofparametersthataregiveninthistableonlyaccountfortheinfluenceofvariationinthefracturepatternontherockmassproperties(asinputmechanicalparametersareconstant).

Figure 4‑7. Distribution of the uniaxial strength of the rock mass in Rock Domain A, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 20 40 60 80 100 120

Uniaxial compressive strength, MPa


Figure 4‑8. Fracture traces for realisation nr 8. mat_IDi refers to the fracture sets.

x

y

-10 -5 0 5 10-10

-5

0

5

10

mat_ID1mat_ID2mat_ID3mat_ID4mat_ID5mat_ID6mat_ID7

28

Table 4‑1. DFN geometry‑induced variability in rock mass properties of Rock Domain A, paralleltoσ1.


Min Max

Em 32 MPa (GPa) 65.5 2.2 59.6 69νm 32 MPa (–) 0.27 0.01 0.25 0.3Em 8 MPa (GPa) 54 6 39.1 62νm 8 MPa (–) 0.32 0.03 0.28 0.37φm (°) 44.83 3.45 38.26 49.46cm (MPa) 41.30–0.5954 × φm 3.96 cm mean–5.58 cm mean+7.8

4.2.2 Simulationsparalleltoσh, Rock Domain ADFNrealizationsparalleltoσhwerealsogeneratedandloadedin�DECfortwoconfiningpressures:14MPaand3.5MPa(seeSection2.3).14MPaisequivalenttoσ2instressdomainI,and�.�MPais2�%ofthisvalue1.Themeanmaterialpropertyvalues(Table�-2)areassignedbothtofracturesandtotheintactrock.Outof20generatedDFN-realizations�DECcouldgeneratezonedivisionfor19ofthem,1realizationhadtoberejected.TheevaluatedrockmassparametersarepresentedinAppendixB.

Theevaluatedrockmassparametersanddistributionsat14MPaarepresentedinFigure4-9andFigure4-10,andtheparametersanddistributionsevaluatedat�.�MPainFigure4-11andFigure4-12.ThecohesionandfrictionangleoftherockmassandthedistributionsarepresentedinFigure4-1�andFigure4-14.Theseparameterswereevaluatedbyfittingastraightlinebetweentheverticalstressatfailureatbothstresslevels.Theuniaxialcompressivestrengthoftherockmasshasbeencalculatedfromtheevaluatedcohesionandfrictionangle,seeFigure4-1�.

1 AccordingtothestressmodelpresentedinSection2.�theminimumhorizontalstressσhinSimpevarpcorrespondstoσ�.Howeverthemodellingontheverticaltraceplanesextractedparalleltoσhwereloadedatconfiningstressescorrespondingtoσ2in-situstresses.

Figure 4‑9. Distribution of deformation modulus at high stress level (14.0 MPa), Rock Domain A, trace plane parallel to σh.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

50 55 60 65 70 75


Data from 3DEC simulatiosAdapted distribution

29

Figure 4‑10. Distribution of Poisson’s ratio at high stress level, (14.0 MPa) Rock Domain A, trace plane parallel to σh.

Figure 4‑11. Distribution of deformation modulus at low stress level (3.5 MPa), Rock Domain A, trace plane parallel to σh.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34

Poisson´s ratio



0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

30 35 40 45 50 55 60 65



�0

Figure 4‑12. Distribution of Poisson’s ratio at low stress level, (3.5 MPa) Rock Domain A, trace plane parallel to σh.

Figure 4‑13. Distribution of friction angle, rock mass in Rock Domain A, trace plane parallel to σh.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45

Poisson´s ratio


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

30 35 40 45 50 55 60

Friction angle, °


�1

Figure 4‑14. Distribution of cohesion, rock mass in Rock Domain A, trace plane parallel to σh.

Figure 4‑15. Distribution of the uniaxial strength of the rock mass in Rock Domain A, trace plane parallel to σh.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

2 4 6 8 10 12 14 16 18

Cohesion, MPa


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 10 20 30 40 50 60 70 80 90




�2

Table 4‑2. DFN geometry‑induced variability in rock mass properties of Rock Domain A, paralleltoσ2.


Min Max

Em 14 MPa (GPa) 62.9 4.1 53.6 69.2νm 14 MPa (–) 0.28 0.01 0.26 0.31

Em 3.5 MPa (GPa) 47.1 7.9 33.5 58.3νm 3.5 MPa (–) 0.35 0.04 0.28 0.42φm (°) 46 4.4 33.7 54.2cm (MPa) 9 3.6 3.1 16.2

ThefinalobtaineddistributionsofEm14MPa,νm14MPa,Em�.�MPa,νm�.�MPa,φmandcmaresummarizedinTable4-2forrockdomainA.

4.2.3 SimulationsparalleltoσH, Rock Domain BDFNrealizationsparalleltoσHwerealsogeneratedforRockDomainBandloadedin�DECfortwoconfiningpressures:32MPaand8MPa(seeSection2.3).32MPaisequivalenttoσ1instressdomainI,and8MPais2�%ofthisvalue.Themeanmaterialpropertyvalues(Table�-2)areassignedbothtofracturesandtotheintactrock.Outof20generatedDFN-realizations�DECcouldgeneratezonedivisionfor19ofthem,1realizationhadtoberejected.TheevaluatedrockmassparametersforeachrealizationarepresentedinAppendixC.

Theevaluatedrockmassparametersat�2MPaarepresentedinFigure4-1�andFigure4-1�,andtheparametersevaluatedat8MPainFigure4-18andFigure4-19.ThecohesionandfrictionangleoftherockmassarepresentedinFigure4-20andFigure4-21.Theseparameterswereevaluatedbyfittingastraightlinebetweentheverticalstressatfailureatbothstresslevels.Theuniaxialcompressivestrengthoftherockmasshasbeencalculatedfromtheevaluatedcohesionandfrictionangle,seeFigure4-22.

Figure 4‑16. Distribution of deformation modulus at high stress level (32.0 MPa), Rock Domain B, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

56 58 60 62 64 66 68 70 72



��

Figure 4‑17. Distribution of Poisson’s ratio at high stress level (32.0 MPa), rock Domain B, trace plane parallel to σH.

Figure 4‑18. Distribution of deformation modulus at low stress level (8.0 MPa), Rock Domain B, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3

Poisson´s ratio

Data from 3DEC simulstionsAdapted distribution

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

40 45 50 55 60 65 70



�4

Figure 4‑19. Distribution of Poisson’s ratio at low stress level, (8.0 MPa) Rock Domain B, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37

Poisson´s ratio


Figure 4‑20. Distribution of friction angle, rock mass in Rock Domain B, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

35 37 39 41 43 45 47 49 51 53 55

Friction angle, °


��

Figure 4‑21. Distribution of cohesion, rock mass in Rock Domain B, trace plane parallel to σH.

Figure 4‑22. Distribution of the uniaxial strength of the rock mass in Rock Domain B, trace plane parallel to σH.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

2 4 6 8 10 12 14 16 18 20

Cohesion, MPa


0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 10 20 30 40 50 60 70 80 90 100



��

Table 4‑3. DFN geometry‑induced variability in rock mass properties of Rock Domain B.


Min Max

Em 32 MPa (GPa) 63.46 2.78 58.56 68.50νm 32 MPa (–) 0.27 0.01 0.26 0.29

Em 8 MPa (GPa) 56.67 6.54 44.19 64.67νm 8 MPa (–) 0.30 0.03 0.25 0.35φm (°) 44.93 3.57 39.76 51.52cm (MPa) 0.3349×φm–4.16 5.4 cm mean–6.7 cm mean+11.7

Somerealizationsgiveaverylowvaluefortheuniaxialstrengthoftherockmass.ThesameproblemasdescribedinSection4.2.1andillustratedinFigure4-8isthesourceoftheselowvalues.Theresultsoftheserealizationsareomittedwhenthefinaldistributionsforφmandcmarecalculated.ThefinalobtaineddistributionsofEm�2MPa,νm�2MPa,Em8MPa,νm8MPa,φmandcmaresummarizedinTable4-�forrockdomainB.Thedistributionsofparametersthataregiveninthistableonlyaccountfortheinfluenceofvariationinthefracturepatternontherockmassproperties(asinputmechanicalparametersareconstant).

4.2.4 Summary of DFN geometry‑induced rock mass variabilityTheresultsfromallthe�DECsimulationsonDFN-realizationsforRockDomainAandBareplottedinFigure4-2�andFigure4-24.Theseillustraterespectivelythevariationoftherockmassdeformationmoduluswithconfiningstressandthemajorandminorstressatfailureatthedifferentstresslevels.ThedifferencebetweenRockDomainAandBisnotsignificant.ThespreadismaybealittlelargerinRockDomainB.

Figure 4‑23. Variation of the deformation modulus as a function of confining stress.

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

0 5 10 15 20 25 30 35

Confining stress, MPa

Def

orm

atio

n m

odul

us, G

Pa

RDA // sig H

RDA // sig h

RDB // sig H

Logg. (all)

��

BasedontheseobservationsthesimulationsforDFN-realizationsparalleltoσ2inRockDomainBhavebeenomittedassimilarresultsasforrockdomainAareexpected.

Figure4-2�illustratesadependencyofthedeformationmoduluswithconfiningstress.Whatevertherockdomainandtheorientationofthetraceplanesthedeformationmodulusincreaseswithconfiningstressuptoaconstantvaluereachesaboveabout1�MPaconfiningstress.HoweverthedeformationmodulusdoesnotappeartobedependentontotheorientationoftheDFNtraceplanesorientations,whichcanbeexplainedbyanalmostisotropicDFNmodel.

ThereforewithconsiderationtotimeconstraintstheinfluenceofvariationofinputparametershasonlybeenanalyzedforthetraceplanesparalleltoσHinrockdomainsAandB.

4.3 Material property influence on rock mass parametersNext,theinputmaterialpropertyinfluenceonrockmassparametersisestimatedasindependentcomponents.

ThematerialpropertyinfluenceonrockmassparametershasbeenevaluatedforsimulationsparalleltoσHinrockdomainAandB.SimilarlytoSection4.1,thetwoconfiningpressures�2MPaand8MPaareusedtoestimatetheinfluencethatindividualmaterialparametershaveonEm�2MPa,νm�2MPa,Em8MPa,νm8MPa,φmandcm.Thisisdonebyperforming�DEC-simulationsonaspecific“average”realization(hererealization14forthetworockdomains),whereeachmaterialproperty,one-by-one,isassigneditsminimumanditsmaximumvalue,whileallothermaterialpropertiesaresettotheirmeanvalues(Table�-1).Relationshipscanthenbeestablishedbetweenvariationsinallinputmaterialparametersandtheirrespectiveimpactonrockmassproperties.

Figure 4‑24. Major and minor principle stresses at failure in the rock mass.

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

0 5 10 15 20 25 30 35

Minor stress, MPa

Maj

or s

tres

s, M

PaRDA // sig 1

RDA // sig 2

RDB // sig 1

�8

Asastartallrelationshipsbetweeninputmaterialpropertiesandrockmasspropertiesareassumedlinearandindependent,i.e.canbeapproximatedbyseparateproportionalityconstantskXi,Ym,whereXiisaninputproperty(intactrockorfractures)andYmisaresultingrockmassparameter.Therockmasspropertiesareevaluatedforthreedifferentvaluesofeachinputproperty:itsminimum,mean,andmaximumvalue.Thus,twoproportionalityconstantscanbeachieved,oneforcaseswhentheinputpropertyislessthanitsmean(Equation4.1)andonewhentheinputpropertyislargerthanitsmean(Equation4.2):

(4.1)

and

(4.2)

whereXi,min,Xi,0,andXi,maxaretheminimum,meanandmaximuminputparametervalues,respectively,andYmandYm,0aretheresultingrockmassparameterscalculatedwith�DEC.

Asanexample,theinfluencethatthedeformationmodulusofintactrock,Ei,hasonthedefor-mationmodulusofrockmass,Em,isshowninFigure4-2�.�DECsimulationswithEisettoitsminimum,meanandmaximumvalues(allotherparameterssettotheirmeanvalues),providethreecorrespondingvaluesofEm.TwoproportionalityconstantskEi,Em(Ei<Ei,0)andkEi,Em

(Ei>Ei,0)arethenevaluated;thesearefoundtobeinthiscase0.�81and0.�48,respectively.

TheinfluencesofallinputparametersonrockmasspropertiesaresummarizedinTable4-4andTable4-�,andascanbenoted,someproportionalityconstantsmaychangesigndependingonifitsinputparametervalueisaboveorbelowitsmean.Notethat,sincetheunitsofthevariousproportionalityconstantsaremixed,adirectcomparisonoftheirrelativemagnitudesmaybemisleading.

Figure 4‑25. Evaluation of the influence the deformation modulus of intact rock, Ei, has on the deformation modulus of rock mass, Em, in Rock Domain A for confinement 32 MPa.

Em = Em,0

+ 0.548 (Er - Er,0 )(for Er > Er,0 )

58

60

62

64

66

68

70

72

65 75 85 95

Deformation modulus intact rock, Er (GPa)

Def

orm

atio

n m

odul

us ro

ck m

ass,

Em

(G

Pa)

3DEC resultsLinear fit, for Er > Er, 0Linear fit, for Er < Er, 0

Em,0

Er,0Er,min Er,max

Em = Em,0

+ 0.581 (Er - Er,0 )

(for Er < Er,0 )

�9

Table 4‑4. Dependency of rock mass parameters on input parameters set above the mean value, proportionality constant kXi,Ym.

Confinement 32 MPa Confinement 8 MPa Rock mass stengthEm 32 MPa (GPa) νm 32 MPa (–) Em 8 MPa (GPa) νm 8 MPa (–) cm (MPa) φm (°)

RDA, intact rock

Ei (GPa) 0.55 0 1.26 –0.01 0.24 –0.12νi (–) –21.42 0.68 28.87 0.3 29.83 –5.65ci (MPa) 0.01 0 0.52 –4.E–03 0.69 –0.31Ti (MPa) 5.E–03 –3.E–06 0.01 2.E–05 0.42 –0.21φi (°) 0.01 –6.E–06 0.03 1.E–04 1. –0.56

RDA, Fractures

Kn (MPa/mm) 0.01 2.E–04 0.09 –7.E–04 0.02 –0.01Ks (MPa/mm) 0.09 –4.E–04 0.3 –4.E–03 0.19 –0.16φf (°) 0.12 –6.E–04 0.6 –5.E–03 0.08 0.37cf (MPa) 0.13 –3.E–03 0.2 –0.02 1.03 –0.01

RDB, intact rock

Ei (GPa) 0.62 0 0.36 3.E–04 0.07 0.01νi (–) –0.86 0.66 165.33 –0.12 58.4 –21.8ci (MPa) 3.E–03 6.E–06 3.E–04 –2.E–06 0.83 –0.24Ti (MPa) 0.01 2.E–05 0.E+00 0.E+00 0.29 –0.16φi (°) 0.01 2.E–07 –3.E–04 8.E–07 –0.43 0.61

RDB, Fractures

Kn (MPa/mm) 0.04 3.E–04 0.02 3.E–04 –0.01 –0.01Ks (MPa/mm) 0.1 –5.E–04 0.13 2.E–05 0.04 –0.02φf (°) 0.03 –2.E–04 0.34 –2.E–03 –0.44 0.73cf (MPa) 0.25 –3.E–04 0.21 –8.E–04 1.45 –0.29

Table 4‑5. Dependency of rock mass parameters on input parameters set below the mean value, proportionality constant kXi,Ym.

Confinement 32 MPa Confinement 8 MPa Rock mass stengthEm 32 MPa (GPa) νm 32 MPa (–) Em 8 MPa (GPa) νm 8 MPa (–) cm (MPa) φm (°)

RDA, intact rock

Ei (GPa) 0.58 3.E–04 0.66 2.E–03 –0.18 0.19νi (–) –1.46 0.83 11.94 0.13 4.8 1.23ci (MPa) 0.01 0.E+00 –0.78 0.01 –0.19 0.18Ti (MPa) –0.01 4.E–06 –0.01 –2.E–05 –0.54 0.27φi (°) –0.02 –4.E–06 –0.03 4.E–07 –0.72 0.53

RDA, Fractures

Kn (MPa/mm) 0.07 6.E–04 0.05 5.E–04 –0.04 0.03Ks (MPa/mm) 0.6 –2.E–03 –0.22 1.E–03 –0.03 0.05φf (°) 0.5 –3.E–03 0.73 –4.E–03 –0.62 0.85cf (MPa) 0.3 –6.E–04 4.03 –0.01 –3.08 0.4

RDB, intact rock

Ei (GPa) 0.63 5.E–05 0.52 –6.E–05 0.2 –0.14νi (–) 2.77 0.88 –83.05 1.54 9.71 6.29ci (MPa) –4.E–03 –2.E–05 4.E–03 –2.E–05 0.26 0.04Ti (MPa) –0.01 –1.E–05 3.E–03 –1.E–05 –0.54 0.25φi (°) 1.E–03 –2.E–05 5.E–03 –3.E–05 0.18 0.18

RDB, Fractures

Kn (MPa/mm) 0.14 6.E–04 0.13 6.E–04 –0.02 –0.01Ks (MPa/mm) 0.46 –2.E–03 –0.04 –2.E–04 0. 0.03φf (°) –0.16 –3.E–04 0.81 –4.E–03 –0.01 0.22cf (MPa) –0.21 –0.01 2.46 –0.01 –3.35 3.88

40

4.4 Monte‑Carlo simulationsThetotalrangeofrockmassparametervariabilityinEm�2MPa,νm�2MPa,Em8MPa,νm8MPa,φmandcmisfinallyestimatedusingaMonte-CarlobasedGoldSimmodel.Thisisdonebycombiningthetwofollowingdistributions:

1. OnedistributionwhichaccountsonlyforthevariationofthefracturepatternbymeansofDFNrealisationsrunin�DEC,seeSection4.2.1and4.2.�,and

2. Onedistributionwhichaccountsforthevariationoftheinputmechanicalparametersinthe�DECsimulations,seeSection4.�.Thisdistributionwasobtainedfrom�DECsimulationsononeDFNrealisation.TheinfluenceofthevariationoftheinputmechanicalparametersisassumedtobesimilarforallDFNrealisations.

Theprocedureforsimulationsisthefollowing:

• Onerandomvalueisextractedfromthedistributionwhichdescribestheinfluenceofthevariationofthefracturepattern.

• Arandomvalueextractedfromthedistributionaccountingforthevariationofinputmechanicalparametersisaddedtotheprecedentvalue.

• 100,000randomvaluesareproducedfrombothdistributionsandtheresultingpropertiesarestatisticallyanalysed.ThedistributionoftherockmasspropertiesisillustratedinFigure4-28toFigure4-28forrockdomainsAandB.

Figure 4‑26. Probability density function of Deformation modulus is Rock Domain A and B.

41

Figure 4‑27. Probability density function of Poisson’s ratio in Rock Domain A and B.

TheobtaineddistributionsofUCSm,cmandφmarealsoshownasprobabilitydistributionsofsimulatedtriaxialloadingtestsinFigure4-29.Ascanbeseen,thelowerlimitofUCSmis��MPaforRDAand�MPaforRDB.

ThecovariancematricesinTable4-�andTable4-�indicatethatEmdependsstronglyonthedeformationmodulusoftheintactrock,Er,andonthefourfractureproperties;moststronglyonfractureshearstiffness,Ks.Similarly,thePoisson’sratiooftherockmass,νm,dependsstronglyonPoisson’sratioofintactrock,νr,andalsoonthefourfractureproperties.Thefrictionangleoftherockmass,φm,ispositivelycorrelatedtothefrictionangleofintactrock,φr,andthatoffractures,φf,whileitisnegativelycorrelatedtocrandcf.Theoppositeholdsforthecohesionoftherockmasscm.However,theparameterTrseemstobeoflittlesignificancetoanyoftheexaminedrockmassparameters.Theuniaxialcompressivestrengthoftherockmass,UCSm,isstronglycorrelatedtothecohesionoftheintactrockinRDB,whilethiscorrelationismuchweakerforRDA.ThiscanbeexplainedbythelargerfractureintensityinRDB,whichimpliesthatUCSmislargelydeterminedbyDFNgeometry-inducedvariability(patternandintensity)andhencethecorrelationstoinputpropertyparametersaresuppressed.

42

Figure 4‑28. Probability density function of the rock mass mechanical properties in Rock Domain A and B (accounting for variation in fracture pattern and input parameters)

4�

Table 4‑6. Correlation coefficient matrix between rock mass parameters and input parameters, RDA.

Confinement 32 MPa Confinement 8 MPaEm νm Em νm cm φm UCSm

Ei 0.59 0.01 0.64 –0.26 0.02 0.05 0.05νi –0.08 0.82 0.08 0.18 0.14 –0.01 0.16ci 0.01 0.00 0.05 –0.09 0.34 –0.18 0.30φi 0.00 0.00 –0.01 0.04 –0.15 0.13 –0.11Ti 0.00 0.01 0.00 0.00 0.02 –0.01 0.02Kn 0.18 0.29 0.24 –0.10 –0.04 0.02 –0.03Ks 0.60 –0.34 0.04 –0.32 0.16 –0.14 0.11cf –0.14 0.10 –0.18 0.19 0.16 –0.40 –0.04φf 0.22 –0.18 0.32 –0.40 –0.26 0.60 0.04

Table 4‑7. Correlation coefficient matrix between rock mass parameters and input parameters, RDB.

Confinement 32 MPa Confinement 8 MPaEm νm Em νm cm φm UCSm

Ei 0.78 –0.01 0.51 0.04 0.20 –0.10 0.19νi –0.01 0.74 0.09 0.56 0.18 –0.04 0.18ci 0.01 0.00 –0.01 0.01 0.72 –0.28 0.67φi 0.00 0.00 0.01 0.00 –0.19 0.36 –0.08Ti 0.01 0.01 0.00 0.01 –0.06 0.03 –0.06Kn 0.33 0.43 0.25 0.30 –0.06 –0.08 –0.09Ks 0.37 –0.42 0.05 –0.04 0.04 0.01 0.04cf 0.03 0.01 –0.19 0.19 0.13 –0.28 0.06φf –0.03 –0.04 0.31 –0.32 –0.22 0.49 –0.08

Figure 4‑29. Probability distributions of simulated triaxial test for rock mass of RDA and RDB. Pink boxes are results from 3DEC modelling of different DFN realizations at confining stress levels 8 MPa and 32 MPa. Red boxes refer to the realization that was used to evaluate the influence that various input parameters have on rock mass properties.

44

4.4.1 Adjusting boundariesTheresultsintheprevioussectionrelyontheassumptionthatallsourcesofvariabilityontherockmassparametersarelinearandindependent.Inordertoexaminethisassumption,theinputparametercombinationsthatyieldthemaximumandminimumUCSmvalueswereexaminedforbothdomains.Theseextremeparametercombinationsweremodeledwith�DECandwerealsousedinGoldSim.ThevaluesobtainedaresummarizedandcomparedinTable4-8.Ascanbeseen,theresultsofMonte-Carlosimulationsindicatethatthe“best”parametercombinationincreasesUCSmby�4.�MPaforRDA,whilethe“worst”combinationdecreasesUCSmby2.�MPa.Quitecontradictory,the“best”combinationdecreasesby�.�MPa,asevaluatedby�DEC.ForRDB,Monte-Carlosimulationsand�DECmodelingseemtoprovidemoreconsistentresults,althoughtherangeofvariationissmallerforthe�DECvalues.

Aconclusionisthattheassumptionoflinearindependencyexaggeratestheimpactthatvariableinputparametershaveonrockmasscompressivestrength,atleastforthespecificextremecombinationsthathavebeen“validated”with3DEC.ThevaluesofφmandUCSmthatwereobtainedfortheextremecombinationsarealsoshownascompressivestrengthsforatriaxialloadingexperimentinFigure4-�0.

InordertoremovethisexaggerationoftheMonte-Carlosimulations,thesewerere-runsuchthatthepredictedimpactonφmandUCSmwererescaledaccordingtothe“maximum”and“minimum”limitsdeterminedby�DEC.ThevaluespresentedasProbabilityDensityFunctionareillustratedinFigure4-�1.

ThevaluesobtainedforUCSmareplottedasresultsofsimulatedtriaxialloadingtests.Figure4-�2illustratesthevariationrelatedtothefracturepatternandFigure4-��therelationimpededtothevariationofinputparameters.

Table 4‑8. Input parameter combinations that yield maximum influence on UCSm.

Ei νi ci φi Ti kn ks cf φf ∆UCSm, GoldSim

∆UCSm, 3DEC

RDA

Best max max max min max min max high max

90 0.33 29 57 24 49 49 0.69 40 74.7 –5.5Worst avg min avg avg avg avg avg avg avg

80 0.18 20 60 17 100 29 0.50 32 –2.6 –1.28

RDBBest max max high avg low min max high max

110 0.31 40 55 24 49 49 0.73 40 68.5 29.4Worst min min min max min max min min max

70 0.19 20 60 20 179 10 0.065 40 –58.3 –3.1

Max, min and avg refer to the minimum, maximum and average value of the input parameter.

4�

Figure 4‑30. φm and UCSm for extreme cases of input parameter combinations, shown as a triaxial test for rock mass, rock domains A and B.

Figure 4‑31. Probability density function of cohesion, friction angle and unixial compressive strength of the rock mass, Rock Domains A and B.

RDA

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

Maj

or p

rinci

pal s

tres

s, σ1

[MPa

]

3DEC AvgGoldsim best3DEC bestGoldsim worst3DEC worst

RDB

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

Minor principal stress, σ3 [MPa]Minor principal stress, σ3 [MPa]

Maj

or p

rinci

pal s

tres

s, σ

1 [M

Pa]

3DEC AvgGoldsim best3DEC bestGoldsim worst3DEC worst

4�

Figure 4‑32. Probability distributions of simulated triaxial test for rock mass of RDA and RDB. Pink boxes are results from 3DEC modelling of different DFN realizations at confining stress levels 8 MPa and 32 MPa. Red boxes refer to the realization that was used to evaluate the influence that various input parameters have on rock mass properties.

Figure 4‑33. Probability distributions of simulated triaxial test for rock mass of RDA and RDB. Red boxes are results from 3DEC modelling on one DFN realization at confing stress levels 8 and 32 MPa, when one input property at a time was set to its maximum or minimum value, while all other parameters were kept at their respective mean values.

4�

4.4.2 Combined resultsThedistributionofthepredictedrockmassmechanicalpropertiesisgiveninTable4-9forrockdomainAandB.Thesevaluesaccountforboththeinfluenceofthefracturepatternandtheinfluenceofthevariationofinputpropertyparameters.

Table 4‑9. Distribution of the predicted rock mass mechanical properties, rock domain A and B.

Parameter for the rock mass (20×20×20 m scale)

Rock Domain A Truncated normal distribution mean/standard dev.

Rock Domain A Min trunc. – max trunc

Rock Domain B Truncated normal distribution mean/standard dev.

Rock Domain B Min trunc. – max trunc

Deformation Modulus 59 GPa/8 GPa2) 62 GPa/5 GPa3)

36–82 GPa 45–75 GPa

57 GPa/7 GPa2) 62 GPa/7 GPa3)

36–76 GPa 42–82 GPa

Poisson’s ratio 0.25/0.042) 0.27/0.033)

0.11–0.36 0.17–0.32

0.28/0.042) 0.27/0.033)

0.15–0.38 0.19–0.35

Tensile strength 0 MPa 0 MPa

Before adjusting maximum impact of extreme input parameter combination to 3DEC resultsUniaxial compressive strength1)

99 MPa/15.3 MPa 60–143 MPa 70 MPa/21 MPa 14–133 MPa

Mohr-Coulomb, φm 40°/3.8° 28°–49° 44°/3.9° 35°–57°Mohr-Coulomb, cm

4) 23.3/4.2 (–0.5421) 12–36 14.6/4.7 (–0.3729) 2.6–28

After adjusting maximum impact of extreme input parameter combination to 3DEC resultsUniaxial compressive strength1)

72 MPa/13.4 MPa 45–105 MPa 65 MPa/14.6 MPa 32–113 MPa

Mohr-Coulomb, φ 41°/3.1° 32°–49° 45°/3.5° 36°–56° Mohr-Coulomb, c4) 16.3/3.2 (–0.3156) 10–24 13.3/3.0 (–0.1911) 6–22

1) This description parameter is not a standard parameter, it refers to the strength of a block of 30 m size with low confinement at boundaries.2) For confining stress, 8 MPa and lower. 3) For confining stress, 32 MPa.4) The cohesion is correlated to the friction angle. The friction angle given in o and the correlation coefficient is specified within brackets.

49

5 Discussion

Assumptionoflinearindependencycouldbetestedwithadditional�DECsimulations,wheremorethanonepropertyisvariedatatime.TheboundaryadjustmentinSection4.4.1isundertakenfortheparametercombinationsthatproducetheextremecasesofimpactonUCSm.Thesecombinationsarethemselvesdeterminedunderassumptionoflinearindependency,andconsequentlydonotguaranteethatother,moreextreme,combinationsdoesnotexist.Incominganalysesitmightbemoreappropriatetorunmoresimulationswith�DECfordifferentparam-etercombinationstogetthematerialpropertyinfluenceontherockmassparameters.

StochasticvariabilityinfracturepropertiesamongfracturesinDFNrealizationshavenotbeenexamined,becauseoflimitationsin�DEC.Instead,allfractureswithinaDFNrealizationhavebeenassignedthesamevalues:eithertheirminimum,meanormaximumparametervalues,whichseemsunrealistic.Itisalsodifficulttotellwhetherthissimplificationexaggeratesorunderestimatesthefractureinputparametervariabilityimpactonrockmassproperties.Howeversometestswereconductedduringthedevelopmentofthemodelingstrategyontheinfluenceoffractureparametersfordifferentfracturesets.Theresultsarepresentedin/OlofssonandFredriksson200�/.

TheDFN-inducedvariabilitycomponentisonlyevaluatedforalimitednumberofrealizations(n≤17forRDAandn≤19forRDB).

TheinfluencesofinputparametersonrockmasspropertieshaveonlybeenexaminedforoneDFNrealizationofRDAandoneforRDB.Theinfluenceinrockmasspropertiesfrommaterialpropertiesofintactrockandfracturesshouldbetestedusingafewotherrealizationsinordertoevaluatetheirpotentialsimilarityinbehavior.

Themostimportantlimitationinthedescriptionofvariabilityisthattheanalysespresentedinthisreportarebasedonlyonthemeanvaluesofthefractureintensity.NovariabilityofthefractureintensityinsidearockdomainwastestedalthoughthefractureintensityinSimpevarpisshowntovaryquitesignificantly.Thereforetheinfluenceoffractureintensityshouldbeanalyzedindetailincomingmodelingstages.

�1

6 Conclusions

Therockmassmechanicalpropertieshavebeendeterminedbymeansofnumericalmodelling.Themodellingiscarriedoutin�DECandtheblockmodelisbuiltusingthefracturenetworkdescribedbythesitespecificDFNmodel.

Thedatauncertaintyandvariabilityisstudiedintwosteps,firstbyanalysingtheinfluenceofthefracturepattern,andthenbystudyingtheinfluenceofthevariationoftheinputparameters.TheircombinedeffectisanalysedbymeansofMonte-Carlosimulations.

Therockmasspropertiesweredeterminedforeachrockdomain.Therockdomainsarecharacterisedbytheirstructureandlithologiesandassuchthefracturenetworkmightbedifferent.HoweverthegeologicaldescriptionofthefourrockdomainsillustratesthatrocktypesandfracturecharacteristicsinrockdomainsCandDdonotsignificantlydifferfromthepropertiesobservedinrockdomainA.HenceonlyrockdomainsAandBwereanalysedinthisstudy(andtheestimatedrockmassmechanicalpropertiesofrockdomainsAandDarederivedfromthoseestimatedforrockdomainA).

Table�-1andTable�-2presentthedistributionofthepredictedrockmassmechanicalpropertiesforthefourrockdomainsidentifiedinSimpevarp.

Table 6‑1. Predicted rock mechanical properties for the mass, rock domain A and B.


Rock Domain A Truncated normal distribution mean/standard dev.

Rock Domain A Min trunc. – max trunc

Rock Domain B Truncated normal distribution mean/standard dev.

Rock Domain B Min trunc. – max trunc

Uniaxial compressive strength1)


Deformation Modulus 59 GPa/8 Gpa2) 62 Gpa/5 GPa3)

36–82 GPa 45–75 GPa


36–76 GPa 42–82 GPa

Poisson’s ratio 0.25/0.042) 0.27/0.033)

0.11–0.36 0.17–0.32

0.28/0.042) 0.27/0.033)

0.15–0.38 0.19–0.35

Tensile strength 0 MPa 0 MPaMohr–Coulomb, φm 41°/3.1° 32°–49° 45°/3.5° 36°–56° Mohr–Coulomb, cm

4) 16.3/3.2 (–0.3156)4) 10–24 13.3/3.0 (–0.1911)4) 6–22

1) This desription parameter is not a standard parameter, it refers to the strength of a block of 20 m size with low confinement at boundaries.2) For confining stress, 8 MPa and lower. 3) For confining stress, 32 MPa.4) The cohesion and the friction angle are correlated. The correlation coefficient is specified within brackets.

�2

Table 6‑2. Predicted rock mechanical properties for the mass, rock domain C and D.


Rock Domain C Truncated normal distribution mean/standard dev.

Rock Domain C Min trunc. – max trunc

Rock Domain D Truncated normal distribution mean/standard dev.

Rock Domain D Min trunc. – max trunk

Uniaxial compressive strength1)


Deformation Modulus 59 GPa/8 GPa2) 62 GPa/5 GPa3)

36–82 GPa 45–75 GPa


36–82 GPa 45–75 GPa

Poisson’s ratio 0.25/0.042) 0.27/0.033)

0.11–0.36 0.17–0.32

0.25/0.042) 0.27/0.033)

0.11–0.36 0.17–0.32

Tensile strength 0 MPa 0 MPaMohr-Coulomb, φm 41°/3.1° 32°–49° 41°/3.1° 32°–49°Mohr-Coulomb, cm

4) 16.3/3.2 (–0.3156)4) 10–24 16.3/3.2 (–0.3156)4) 10–24

1) This desription parameter is not a standard parameter, it refers to the strength of a block of 20 m size with low confinement at boundaries.2) For confining stress, 8 MPa and lower. 3) For confining stress, 32 MPa.4) The cohesion and the friction angle are correlated. The correlation coefficient is specified within brackets.

��

7 References

3DEC,2003.�DimensionalDistinctElementCode,User’sGuide.ItascaconsultinggroupInc.,Minneapolis.

HakamiE,MinK-B,2005.Modellingofthestateofstress.Preliminarysitedescription,Simpevarpsubarea–version1.2.SKBR-0�-19,SvenskKärnbränslehanteringAB.

LanaroF,FredrikssonA,2005.Rockmechanicscharacterisationoftherockmass–Sumamryofprimarydata.Preliminarysitedescription,Simpevarpsubarea–version1.2.SKBR-0�-21,SvenskKärnbränslehanteringAB.

LaPointePR,HermansonJ,2005.Statisticalmodelforfracturesanddeformationzones,Simpevarp1.2.Inprogress,SKBR-0�-28,SvenskKärnbränslehanteringAB.

OlofssonI,FredrikssonF,2005.StrategyforanumericalRockMechanicsSiteDescriptiveModel.Furtherdevelopmentofthetheoretical/numericalapproach.SKBR-0�-4�,SvenskKärnbränslehanteringAB.

SKB,2005.PreliminarySiteDescription.Simpevarpsubarea–version1.2.SKBR-0�-08,SvenskKärnbränslehanteringAB.

��

Appendix A

Table A‑1. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσH, high stress level (32.0 MPa), Rock Domain A.

DFN realisation Poisson’s ratio,νm

Deformation modulus, Em, GPa

Vertical stress at failure,σvf, MPa

1 0.28 67.82 271.892 0.27 64.43 209.583 0.26 65.93 213.384 0.26 66.10 216.525 0.27 65.11 284.057 0.29 63.67 251.908 0.27 64.10 173.669 0.25 59.59 201.0810 0.27 66.91 261.5911 0.27 67.35 142.5112 0.27 69.01 248.8413 0.30 66.90 218.1414 0.28 65.71 228.0917 0.27 66.72 254.0618 0.28 63.18 300.4519 0.28 65.96 292.5820 0.27 64.77 157.86Mean 0.27 65.49 230.95Standard dev. 0.01 2.16 45.99Min. 0.25 59.59 142.51Max. 0.30 69.01 300.45

Table A‑2. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσH, low stress level (8.0 MPa), Rock Domain A.





��

Table A‑3. Friction angle, cohesion and uniaxial compressive strength for all DFN realisations,traceplanesparalleltoσH, Rock Domanin A.

DFN realisation Friction angle,φm

Cohesion, cm, MPa



��

Appendix B

Table B‑1. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσh, high stress level (14.0 MPa), Rock Domain A.




1 0.28 69.19 120.592 0.27 65.45 146.483 0.30 60.76 158.494 0.28 60.78 123.725 0.29 64.58 156.507 0.28 61.56 131.778 0.28 59.28 126.599 0.26 57.72 121.4010 0.28 57.30 60.8411 0.28 66.79 145.6612 0.26 67.10 152.7113 0.28 66.48 145.8514 0.26 65.65 104.0315 0.28 61.23 183.1016 0.29 61.45 164.3017 0.31 53.64 132.2018 0.27 63.18 154.3719 0.30 64.34 104.3720 0.27 68.08 104.19Mean 0.28 62.87 133.54Standard dev. 0.01 4.11 28.04Min. 0.26 53.64 60.84Max. 0.31 69.19 183.10

Table B‑2. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσh, low stress level (3.5 MPa), Rock Domain A.





�8

Table B‑3. Friction angle, cohesion and uniaxial compressive strength for all DFN realisations,traceplanesparalleltoσh, Rock Domain A.


Cohesion, cm, MPa



�9

Appendix C

Table C‑1. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσH, high stress level (32.0 MPa), Rock Domain B.





Table C‑2. Poisson’s ratio, deformation modulus and vertical stress at failure for all DFN realisations,traceplanesparalleltoσH, low stress level (8.0 MPa), Rock Domain B.





�0

Table C‑3. Friction angle, cohesion and uniaxial compressive strength for all DFN realisations,traceplanesparalleltoσH, Rock Domain B.


Cohesion, cm, MPa


1 37.01 3.01 12.072 49.67 9.91 53.983 43.30 8.38 38.844 51.52 10.08 57.755 42.57 6.66 30.326 40.45 5.78 25.057 43.38 9.60 44.588 46.95 16.40 83.189 51.37 14.12 80.5610 45.80 18.43 90.7611 42.06 7.62 34.2512 39.76 12.63 53.8613 46.04 9.65 47.8214 42.25 3.98 18.0015 41.73 7.58 33.8416 40.24 6.73 29.0017 43.51 13.59 63.2818 43.26 10.04 46.4520 43.40 11.33 52.59Mean 43.91 9.76 47.17Standard dev. 3.86 4.00 21.62Min. 37.01 3.01 12.07Max 51.52 18.43 90.76

rock mechanics characterisation of the rock mass

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