The Planar Shape of Rock Joints - 图文

ThePlanarShapeofRockJoints(a)SP1SP2(b)SP1Long axisSP2(c)SP1SP1Long axis Long axisSP2SP2SP1SP1SP2SP2Long axisLong axis Fig.11Threepossiblecasesforwhichtheaveragetracelengthsontwosamplingplanesareaboutequal:ajointsareequidimensional(circular);bjointsarenon-equidimensional(elliptical),withlongaxesinasingleorientation.Thetwosamplingplanes,SP1andSP2,areorientedinsuchawaythatthetracelengthsonthemareaboutequal,andcjointsarenon-equidimensional(elliptical),withlongaxesrandomlyorientated.Thetwosamplingplanes,SP1andSP2,areorientedinsuchawaythattheaveragetracelengthsonthemareaboutequal(modi?edfromZhangetal.2002)1

felT?

l

Zq???????????????????geaT

Mla

l=M

eMaT2àl2

dael aMTe1T

wherelisthelengthofatrace;laisthemeanofmajoraxislengthpa;????????????????????andMisafactorwhichcanbedeterminedby

M?p?????????????????????????tan2bt1

k2tan2bt1e2T

inwhichk=a/bistheaspectratioofthejoint;andbistheanglebetweenthejointmajoraxisandthetraceline(notethatbismeasuredinthejointplane)(seeFig.12).Obvi-ously,bwillchangefordifferentsamplingplanes.Foraspeci?csamplingplane,however,therewillbeonlyonebvalueforajointsetwithadeterministicorientation.Itisnotedthatf(l)inEq.1isthetruetracelengthdistribution.Iff(l)isobtainedfromthemeasuredtracelengthdata,samplingbiasesshouldandcanbeconsidered(PriestandHudson1981;ZhangandEinstein2000).

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MinoraxisLine parallel to traceline and passing b= a/k Majorthrough joint center aaxisβ TracelinelFig.12Parametersusedinthede?nitionofanellipticaljoint(afterZhangetal.2002)BasedonEq.1,Zhangetal.(2002)derivedexpressionsfordeterminingthemeanllandstandarddeviationrloftracelengthsfromthemeanlaandstandarddeviationraofjointsizea,respectivelyforthelognormal,negativeexponentialandGammadistributionofjointsizea(seeTable2)(aisthemajoraxislengthofanellipticaljoint).UsingtheexpressionsinTable2,onecaninvestigatetheeffectofsamplingplaneorientationontracelengths.Figure13showsthevariationofthemeanandstandarddeviationoftracelengthswithb,respectivelyforthelognormal,negativeexponentialandGammadistributionofjointsizea.Forotherdistributionformsofjointsizea,similar?gurescanbeobtained.Sincebistheanglebetweenthetracelineandthejointmajoraxis,itisrelatedtothesamplingplaneorientationrelativetothejoint.Itcanbeseenthat,forallthethreedistributionformsofjointsizea,thereareextensiverangesofsamplingplaneorienta-tions,re?ectedbyb,overwhichboththemeanandstan-darddeviationoftracelengthsshowlittlevariation,especiallyforlargekvalues;thisissodespitetheconsid-erabledifferencebetweenthemaximumandtheminimum,respectively,ofthemeanandstandarddeviationoftracelengths.

TheresultsinFig.13couldwellexplainwhyBridges(1976)andEinsteinetal.(1979)founddifferentmeantracelengthsondifferentlyorientedsamplingplanes,whereasRobertson(1970)andBarton(1977)observedthemtobeapproximatelyequal.Ineachofthesereports,thenumberofdifferentlyorientedsamplingplaneswasverylimitedand,dependingontherelativeorientationsofthesamplingplanes,theauthorscouldobserveeitherapproximatelyequalmeantracelengthsorsigni?cantlydifferentmeantracelengths.Forexample,inBridges(1976)andEinsteinetal.(1979),thetwosamplingplanesmightberespectivelyintheb=0–20°(or160–180°)rangeandtheb=40–140°range,orviceversa.FromFig.13,thiswouldresultinverydifferentmeantracelengths.Ontheotherhand,inRobertson(1970)andBarton(1977),thetwosamplingplanesmightbebothintheb=40–140°range(i.e.,inthe

123

64

Table2Expressionsfordeterminingthemeanllandstandarddeviationrloftracelengthsfromthemeanlaandstandarddeviationraofjointssizea

Distributionformofg(a)Lognormal

llpM?elaT2teraT2??

4la

L.Zhang,H.H.Einstein

(rl)232MelaTteraT

hi2

à3p2M2elaT2elaT2teraT2

48elaT42

h

22

i3

Negative

exponentialGamma

pM2la

pM?elaT2teraT2??

4la

e16àp2TM2

42

elaT2

2

32MelaTteraT

h

elaTt2eraThi2

à3p2M2elaT2teraT2

48elaT22

ih

22

i

Mean trace length, μl, and s. d. of trace length, σl (m) Mean trace length, μl, and s. d. of trace length, σl (m) 9.08.07.06.05.04.014.013.012.011.010.09.08.07.06.05.04.03.02.01.00.00180g(a) is Lognormal withμa= 8.0 m and σa= 4.0 mMean trace lengths. d. of trace lengthg(a) is Negative Exponential withμa=σa= 8.0 mMean trace lengths. d. of trace lengthk= 1k= 1k= 1k= 1k= 2k= 23.02.01.0k= 4k= 4(a)0.00180302106024090270120300k= 8(b)302106024090270120300k= 8150330180360150330180360β(degrees)β (degrees)Mean trace length, μl, and s. d. of trace length, σl (m) 9.08.07.06.05.0g(a) is Gamma withμa= 8.0 m and σa= 4.0 mMean trace lengths. d. of trace lengthk= 1k= 14.03.02.01.00.0k= 2k= 4(c)302106024090270120300k= 80180150330180360β(degrees)Fig.13Variationofmeanandstandarddeviation(s.d.)oftracelengthwithbfordifferentdistributionsofjointsizea:aLognormal,bnegativeexponential,andcgamma123

ThePlanarShapeofRockJoints‘‘?at’’tracelengthpartofFig.13),orrespectivelyinsomebrangesapproximatelysymmetricalaboutb=90°.Itshouldbenotedthatthecommentsabovearesimplyassumptions,becausenoinformationaboutthebvaluescanbefoundintheoriginalpapersorreports.

6DeterminationofJointShapefromTraceDataThediscussionofobservedjointshapeshasshownthatsomejointsareelliptical(possiblycircularbutrarely).Suchjointsoccurinunboundedorweaklyboundedrock.Ontheotherhand,boundedjointsareusuallyrectangular.Todeterminejointshapefromjointtraces,thefollowinggeneraltwo-stepprocesscanbefollowed:

Step1Evaluationofgeologicinformation

Basedongeologicinformationmostlyfromoutcropsorgeneralknowledgeofthearea,evaluatethepossibleshapeoffractures.Informationfromoutcropsmaybelimitedinwhichcaseeducatedassumptionsneedtobemadeifthefracturecanbeapproximatedbyanellipseorarectangle.Jointsnotaffectedbyadjacentgeologicalstructuressuchasbeddingboundariesorpre-existingfracturestendtobeelliptical,butjointsaffectedbyorintersectingsuchgeologicalstructurestendtoberectangles.

Step2Characterizejointsfromjointtraces

Basedonmeasuredtracelengthsondifferentsamplingplanes,jointscanbecharacterizedfollowingtheproce-duresinSects.6.1and6.2,respectively,forellipticalandrectangularjoints.

6.1CharacterizationofEllipticalJoints

Forellipticaljoints,theprocedureofZhangetal.(2002)asoutlinedbelowcanbefollowedtoestimatejointshapeandsize:(a)

Obtainthetracelengthdataondifferentsamplingwindows(outcrops)andobtainthetruetracelengthdistributionbyconsideringthesamplingbiases(PriestandHudson1981;ZhangandEinstein2000).

(b)Assumeamajoraxisorientationandcomputetheb

(theanglebetweenjointmajoraxisandtraceline)valueforeachsamplingwindow(seeFig.12).

(c)Fortheassumedmajoraxisorientation,computelaandraofthejointsize,usingexpressionsderivedfromthegeneralstereologicalrelationship(Eq.1),fromthetracelengthdataofeachsamplingwindow,byassumingdifferentaspectratioskanddifferentdistributionformsofg(a).Theresultsarethenusedtodrawthecurvesrelatingla(andra)tok,respectively,

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Fig.14Variationoflaandrawithaspectratiokforassumedmajoraxisorientationofa0°/0°,b30°/0°,andc60°/0°(afterZhangetal.2002)

fortheassumeddistributionformsofg(a)[see,e.g.,Fig.14forassumedlognormaldistributionofg(a)].(d)Repeatsteps(b)and(c)untilthecurvesrelatingla(andra)tokfordifferentsamplingwindowsintersectinonepoint(see,e.g.,Fig.14b).Themajoraxisorientationforthiscaseistheinferredactualmajoraxisorientation.Thek,laandravaluesattheintersectionpointsarethecorrespondingpossiblecharacteristicsofthejoints.

6.2CharacterizationofRectangularJoints

Forrectangularjoints,oneofthefollowingthreemethodscanbeusedtocharacterizejointshapeandsize.Itisalso

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66possibletocombinetwoorallofthethreemethodsforthecharacterization.

Method1Simplymeasurethejointtracesonexposedbeddingsurfacesandgeometricallyinferrectanglesbasedonsimplegeometry.ThecharacterizationofjointsinisolatedsandstonelayersbyPetitetal.(1994)canbeconsideredanexampleofthismethod.

Method2FollowtheprocedureofZhangetal.(2002)asdescribedinSect.6.1toestimatejointshapeandsize;butthestereologicalrelationshipspeci?callyforrectan-gularjointswillbeused.Forexample,thegeneralstereologicalrelationshipbetweentracelengthdistribu-tionandjointsizedistributionderivedbyWarburton(1980b)forparallelogramscanbeused,arectanglebeingaparallelogramwithrightangles.UsingWarbur-ton’sstereologicalrelationship,theexpressionsfordeterminingthemeanandstandarddeviationofjointsizefromthemeanandstandarddeviationoftracelengthscanbederivedfordifferentdistributionsofjointsize.Intheseexpressions,theorientationofasamplingwindowisde?nedbytheanglebetweenasideoftherectangleandthetraceline.Withthesederivedexpres-sions,theprocedureofZhangetal.(2002)canthenbeusedtoestimatethejointshapeandsize.

Method3TreatrectangularjointsasellipsesatthesameaspectratioandfollowtheprocedureofZhangetal.(2002)asdescribedinSect.6.1toestimatejointshapeandsize.TheappropriatenessofusingellipsestorepresentrectangularjointsisdiscussedinSect.6.3.6.3UsingEllipsestoRepresentRectangularJointsConsideranellipseandarectanglewhichhavethesameareaandthesameaspectratio(seeFig.15),i.e.,

yb/2C (L/2,W/2)0a/2xEllipse and rectangle have the same area and the same aspect ratio, i.e. πab = LW and a/b = L/W = k,which lead to L=(π/2)a and W=(π/2)bFig.15Representingarectanglebyanellipsewiththesameareaandthesameaspectratio123

L.Zhang,H.H.Einstein

pab=4?LWe3Ta=b?L=W?k

e4T

whereaandbarerespectivelythemajorandminoraxislengthoftheellipse;andLandWarerespectivelythelengthandwidthoftherectangle.FromEqs.3and4,LandWcanpberelatedrespectivelytoaL????andbasfollows

pa?0:87ae5p2T

W????p

2b?0:87be6TIfthecentersoftheellipseandrectangleareatthesamelocationandtheirmajoraxesareinthesamedirection,theareaoftherectanglecoveredbytheellipse(theshadedareainFig.15Aab )canpc?2???p???????????beobtainedas

p4àptsinà1??p???p

??

??p???????????4àp??!2

àsinà1

22e7T

Sotheratioofthecoveredareatothetotalareaofthe

rectangleAc2 is

pA????p???????????p4àp

tsinà1??p???p??

àsinà1??p???????????4àp??!tp

222?0:91

e8T

Aratiogreaterthan0.91willresultforjointsboundedbystraightbeddingboundariesbuthavingcurvedendssuchasthoseshowninFigs.5and6.

Onecanthusstatethatsincetheellipsecoversover90%oftherectangle,itisusuallyappropriatetouseellipticaljointstorepresentrectangularjoints.Usinganellipticalapproximationhastheadvantagethatitssizeandshapecanbeobtainedfromtracelengthdata(asproposedbyZhangetal.2002).Thisholdsevenmoreforpolygonaljointswithalargernumberofsides,say[5.Clearly,thereverse,i.e.,multisidedpolygonscanbeusedtorepresentellipsesisalsoapplicable(seeFig.10).Thismaybeoneoftherea-sonswhypolygonsareusedtorepresentjointsindiscretefracturecodes.

7SummaryandConclusions

Abriefliteraturereviewabouttheshapeofrockjointshasbeenconducted.Intheliterature,someactualobservationsofjointshapehavebeenmadebasedoninformationonjointsurface-morphology,whilemanyresearchershaveusedtracelengthobservationstoapproximatelyinferjointshapes.Researchershavealsoconductedexperimentalstudiesontheshapeofjoints.Theliteraturereviewis