Rock mechanics characterisation of the rock mass – summary ...
Rock mechanics characterisation of the rock mass ...
Transcript of 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
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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.
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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.
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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
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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.
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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.
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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.
Mean Standard deviation
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
Deformation modulus, GPa
Data from 3DEC simulationsAdapted distribution
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
Data from 3DEC simulationsAdapted distribution
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, °
Data from 3DEC simulationsAdapted distribution
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
Data from 3DEC simulationsAdapted distribution
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
Data from 3DEC simulationsAdapted distribution
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.
Mean Standard deviation
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
Deformation modulus, GPa
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
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%
30 35 40 45 50 55 60 65
Deformation modulus, GPa
Data from 3DEC simulationsAdapted distribution
�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
Data from 3DEC simulationsAdapted distribution
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, °
Data from 3DEC simulationsAdapted distribution
�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
Data from 3DEC simulationsAdapted distribution
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
Uniaxial compressive strength, MPa
Data from 3DEC simulations
Adapted distribution
�2
Table 4‑2. DFN geometry‑induced variability in rock mass properties of Rock Domain A, paralleltoσ2.
Mean Standard deviation
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
Deformation modulus, GPa
Data from 3DEC simulationsAdapted distribution
��
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
Deformation modulus, GPa
Data from 3DEC simulationsAdapted distribution
�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
Data from 3DEC simulationsAdapted distribution
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, °
Data from 3DEC simulationsAdapted distribution
��
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
Data from 3DEC simulationsAdapted distribution
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
Uniaxial compressive strength, MPa
Data from 3DEC simulationsAdapted distribution
��
Table 4‑3. DFN geometry‑induced variability in rock mass properties of Rock Domain B.
Mean Standard deviation
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.
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
Uniaxial compressive strength1)
72 MPa/13.4 MPa 45–105 MPa 65 MPa/14.6 MPa 32–113 MPa
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 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.
Parameter for the rock mass (20×20×20 m scale)
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)
72 MPa/13.4 MPa 45–105 MPa 72 MPa/13.4 MPa 45–105 MPa
Deformation Modulus 59 GPa/8 GPa2) 62 GPa/5 GPa3)
36–82 GPa 45–75 GPa
59 GPa/8 GPa2) 62 GPa/5 GPa3)
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.
DFN realisation Poisson’s ratio,νm
Deformation modulus, Em, GPa
Vertical stress at failure,σvf, MPa
1 0.34 53.79 123.502 0.34 48.83 70.593 0.33 52.78 88.684 0.30 53.88 82.085 0.30 56.19 151.967 0.32 55.27 116.378 0.37 39.05 69.879 0.28 57.72 99.0010 0.28 62.03 109.7911 0.37 43.17 57.4712 0.33 53.62 131.7013 0.35 50.97 75.9014 0.33 57.27 79.6117 0.31 60.66 97.7818 0.32 54.20 124.4719 0.33 56.21 125.4120 0.29 61.48 52.44Mean 0.32 53.95 97.45Standard dev. 0.03 6.01 28.56Min. 0.28 39.05 52.44Max. 0.37 62.03 151.96
��
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
Uniaxial compressive strength, MPa
1 46.18 14.89 74.042 44.87 5.04 24.263 42.63 10.33 47.114 44.19 7.87 37.275 43.83 23.00 107.947 44.36 14.98 71.198 38.64 8.48 35.279 38.26 15.75 64.9810 46.63 11.77 59.1811 34.04 7.74 29.1312 41.29 20.97 92.6613 45.34 5.85 28.4814 46.20 6.05 30.1217 47.20 8.95 45.6818 49.46 12.15 65.8119 48.50 13.20 69.6920 38.99 4.13 17.30Mean 43.56 11.24 52.95Standard dev. 4.12 5.41 25.54Min. 34.04 4.13 17.30Max. 49.46 23.00 107.94
��
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.
DFN realisation Poisson’s ratio,νm
Deformation modulus, Em, GPa
Vertical stress at failure,σvf, MPa
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.
DFN realisation Poisson’s ratio,νm
Deformation modulus, Em, GPa
Vertical stress at failure,σvf, MPa
1 0.35 53.51 64.762 0.35 48.71 69.913 0.35 50.07 85.364 0.30 48.46 45.875 0.35 52.17 93.027 0.36 45.77 77.088 0.37 40.73 64.069 0.36 38.73 58.3210 0.28 52.04 24.1111 0.42 34.50 71.9512 0.31 57.56 76.6213 0.34 55.24 68.3614 0.39 37.68 47.6315 0.33 52.77 82.4016 0.33 54.01 100.9517 0.39 39.23 75.3918 0.31 58.28 76.3119 0.39 42.18 37.7020 0.42 33.46 54.54Mean 0.35 47.11 67.07Standard dev. 0.04 7.88 19.08Min. 0.28 33.46 24.11Max. 0.42 58.28 100.95
�8
Table B‑3. Friction angle, cohesion and uniaxial compressive strength for all DFN realisations,traceplanesparalleltoσh, Rock Domain A.
DFN realisation Friction angle,φm
Cohesion, cm, MPa
Uniaxial compressive strength, MPa
1 43.11 10.01 46.152 49.36 8.22 44.393 48.50 11.55 60.984 49.67 3.66 19.925 45.74 14.61 71.867 42.68 12.89 58.858 45.44 8.85 43.229 45.61 7.61 37.3010 33.74 3.17 11.8611 48.64 8.94 47.3812 49.24 9.52 51.2513 49.58 7.83 42.5314 43.32 6.22 28.8315 54.21 7.89 48.8416 45.70 16.25 79.8317 43.47 12.14 56.4518 49.72 9.22 50.2919 46.71 3.07 15.4720 40.61 8.73 37.98Mean 46.05 8.97 44.92Standard dev. 4.45 3.56 17.61Min. 33.74 3.07 11.86Max. 54.21 16.25 79.83
�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.
DFN realisation Poisson’s ratio,νm
Deformation modulus, Em, GPa
Vertical stress at failure,σvf, MPa
1 0.27 64.00 140.882 0.26 60.19 291.303 0.26 60.02 210.564 0.29 68.50 320.405 0.27 66.53 196.136 0.27 61.29 175.267 0.27 64.57 216.938 0.26 64.65 288.889 0.28 67.06 341.0610 0.28 62.03 284.8211 0.27 67.42 196.1212 0.26 63.09 199.4113 0.26 63.64 244.2414 0.28 58.56 181.3015 0.27 61.08 193.2316 0.26 61.68 177.7717 0.26 66.30 236.7118 0.28 61.99 217.8120 0.27 63.10 225.08Mean 0.27 63.46 228.31Standard dev. 0.01 2.78 53.89Min. 0.26 58.56 140.88Max. 0.29 68.50 341.06
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.
DFN realisation Poisson’s ratio,νm
Deformation modulus, Em, GPa
Vertical stress at failure,σvf, MPa
1 0.27 64.67 44.282 0.27 57.38 113.313 0.25 64.01 81.774 0.31 62.61 123.415 0.35 48.35 71.776 0.25 59.54 62.617 0.31 57.09 87.678 0.28 60.26 134.619 0.30 61.28 145.6810 0.31 55.54 139.2811 0.29 59.33 74.7212 0.27 63.00 90.2513 0.32 53.74 96.9314 0.27 62.03 58.8315 0.35 44.20 73.6916 0.32 47.92 66.2017 0.29 59.59 106.6318 0.35 44.19 89.2920 0.32 51.93 95.71Mean 0.30 56.67 92.45Standard dev. 0.03 6.54 28.58Min. 0.25 44.19 44.28Max. 0.35 64.67 145.68
�0
Table C‑3. Friction angle, cohesion and uniaxial compressive strength for all DFN realisations,traceplanesparalleltoσH, Rock Domain B.
DFN realisation Friction angle,φm
Cohesion, cm, MPa
Uniaxial compressive strength, 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