17 Uniformly-fracturedaquifers,double-porosityconcept dt1

0, / 1." 7\.L

.-.. 'r::._.>.-.,,1

Li-::::~v/

17.1 Introduction

Fracturesinarockformationstronglyinfluencethefluidflowin thatformation-Con-ventionalwell-flowequations,developedprimarilyfor homogeneousaquifers,there-foredo notadequatelydescribetheflow in fracturedrocks.An exceptionoccursinhardrocksof verylow permeabilityif thefracturesarenumerousenoughandareevenlydistributedthroughouttherock;thenthefluidflowwill onlyoccurthroughthefracturesandwillbesimilartothatinanunconsolidatedhomogeneousaquifer,

A complicatingfactorin analyzingpumpingtestsin fracturedrockis thefracturepattern,whichisseldomknownprecisely-Theanalysisisther~o.J~_l1!a!terofidentify-inganunknownsystem(Section2.9).Systemidentific'ittlonrelieson~o'defS~'wn6secharacterIsticsareassumedto representthecharacteristicsof theactualsystem.Wemustthereforesearchfor awell-definedtheoreticalmodelto simulatethebehaviourof theactualsystemandtoproduce,ascloselyaspossible,itsobservedresponse.

In recentyears,manytheoreticalmodelshavebeendeveloped,allofthemassumingsimplifiedregularfracturesystemsthatbreaktherockmassintoblocksof equaldi-mensions(Figure17.1).Thesemodelsusuallyallowconventionaltype-curvematchingproceduresto beused.But,becausethemechanismof fluidflowin fracturedrocksis complex,themodelsarecomplextoo,comprising,astheydo,severalparametersor a combinationof parameters.Consequently,fewof theassociatedwellfunctionshavebeentabulated,so,fortheothermodels,onefirsthastocalculateasetoffunctionvalues.Thismakessuchmodelslessattractivefor ourpurpose.

A B c

matrix

fracture

t matrixfracture

Figure 17,1 Fractured rock formationsA: A naturally fracturedrock formationB: Warren-Root's idealizedthree-dimensional,orthogonal fracturesystemC: Idealized horizontal fracturesystem

..

249

.,-

- '--- !

Evenmoreseriousis theon-goingdebateaboutfractureflow,whichindicatesthatthetheoryof fluidflowin fracturedmediais lesswell-establishedthanthatin porousmedia.In reviewingtheliteratureonthesubject,Streltsova-Adams(1978)states:'Pub-lishedworkonwelltestsin fracturedreservoirsclearlyindicatesthelackof aunifiedapproach,whichhasledtocontradictoryresultsin analyzingthedrawdownbehav-iour'.And Gringarten(1982),in hisreview,states:'A carefulinspectionof thepub-lishedanalyticalsolutionsindicatesthattheyareessentiallyidentical.Apparentdiffer-encescomeonlyfromthedefinitionof thevariousparametersusedin thederivation'.Indeed,in theliterature,thereis anenormousoverlapof equations.In thischapter,therefore,wepresentsomepracticalmethodsthatdo not requirelengthytablesoffunctionvaluesandwhich,whenusedin combination,allowacompleteanalysisofthedatatobemade.

Themethodswepresentareallbasedonthedouble-porositytheorydevelopedini-tiallyby Barenblattetal. (1960).This conceptregardsa fracturedrock formationasconsistingof twomedia:thefracturesandthematrixblocks,bothof themhavingtheirowncharacteristicproperties.Twocoexistingporositiesandhydraulicconducti-vitiesarethusrecognized:thoseofprimaryporosityandIowpermeabilItyIII thematnx6TockS;-andthoseofi()'; storagecapacityandhighpermeabilityin thefractures.Thisconceptmakesit possibletoexplaintheflowmechanismasa re-equalizationof thepressuredifferentialin thefracturesandblocksbytheflowof fluidfromtheblocksinto thefractures.No variationin headwithinthematrixblocksis assumed.Thisso-calledinterporosityflowisin pseudo-steadystate.Theflowthroughthefracturestothewellisradialandinanunsteadystate.

Theassumptionofpseudo-steady-stateinterporosityflowdoesnothaveafirmtheo-reticaljustification.Transientblock-to-fractureflow wasthereforeconsideredbyBoultonandStreltsova(1977),Najurieta(1980),andMoench(1984).FromMoench'swork,it is apparentthattheassumptionof pseudo-steady-stateinterporosityflowis onlyjustifiedif thefacesof thematrixblocksarecoatedbysomemineraldeposit(astheyoftenare).Onlythenwill therebelittlevariationin headwithintheblocks.Thepseudo-steady-statesolutionisthusaspecialcaseof Moench'ssolutionof tran-sientinterporosityflow.

Themethodsin thischapterareall basedonthefollowinggeneralassumptionsandconditions:- Theaquiferisconfinedandof infinitearealextent;- The thicknessof theaquiferis uniformovertheareathatwill beinfluencedby

thetest;- Thewellfullypenetratesafracture;- Thewellispumpedataconstantrate;- Prior to pumping,thepiezometricsurfaceis horizontalovertheareathatwill be

influencedbythetest;- Theflowtowardsthewellisinanunsteadystate.

Thefirstmethodin thischapter,in Section17.2,is theBourdet-Gringartenmethodanditsapproximation,whichis moreuniversallyapplicablethanothermethods;itusesdrawdowndatafromobservationwells.Next,in Section17.3,wepresenttheKazemietal.method;it isanextensionof themethodoriginallydevelopedbyWarren

250

andRoot(1963)forapumpedwell;theKazemietal.methodusesdatafromobserva-tionwells.Finally,in Section17.4,wepresenttheoriginalWarrenandRootmethodforapumpedwell.

17.2 Bourdet-Gringarten'scurve-fittingmethod(observationwells)

BourdetandGringarten(1980)statethat,inafracturedaquiferofthedouble-porositytype(Figure17.1B),thedrawdownresponsetopumpingasobservedin observationwellscanbeexpressedas

s= Q F(4rcTf U*,A,CO)

(17.1)

whereTft

u* = (Sf + ~Sm)r2

2Km= rJ.r Kf

(17.2)

A (17.3)

SfCO =-Sf+~Srn

f = ofthefracturesm = of thematrixblocksT = jTf(x)Tf(y)= effectivetransmissivity(m2/d)S = storativity(dimensionless)K = hydraulicconductivity(m/d)A = interporosityflowcoefficient(dimensionless)rJ. = shapefactor,parametercharacteristicof thegeometryof thefractures

andaquifermatrixof a fracturedaquiferof thedouble-porositytype(dimension:reciprocalarea)

~ = factor;for early-timeanalysisit equalszeroandfor late-timeanalysisit equals1/3(orthogonalsystem)or 1(stratatype)

x,y = relativetotheprincipalaxesofpermeability

(17.4)

To avoidconfusion,notethatourdefinitionoftheparameterAdiffersfromthedefini-tionofAcommonlyusedin thepetroleumliterature;A = (r/rw)2Aoil.Notealsothatfora fracturesystemasshowninFigure17.1B, rJ.= 4n(n+2)/F,wherenisthenumberofanormalsetoffractures(1,2,or3)andI isacharacteristicdimensionof a matrix block. For a systemof horizontal slab blocks (n=1)as shown in Figure17.1C, rJ.= 12/h~,wherehmis thethicknessof a matrixblock.Typicalvaluesof Aandcofall withintherangesof 10-3(rw/rf to 10-9(rw/r)2for A and 10-1to 10-4forco(Serraetal.1983).For smallvaluesofpumpingtime,Equation17.1reducesto

- ~ W(u)s - 4rcTf (17.5)251

where(Sr+ ~Srn)r2u=

4 Trt (17.6)

Equation17.5isidenticaltotheTheisequation.It describesonlythedrawdownbehav-iourinthefracturesystem(~equalszero).For largevaluesofpumpingtime,Equation17.1alsoreducestotheTheisequation,whichnowdescribesthedrawdownbehaviourin thecombinedfractureandblocksystemWequals1/3or 1).

Accordingto thepseudo-steady-stateinterporosityflow concept,thedrawdownbecomesconstantatintermediatepumpingtimeswhenthereisatransitionfromfrac-tureflowtoflowfromfracturesandmatrixblocks.Thedrawdownatwhichthetransi-tionoccursisequalto

s = Q K (fie2nT 0 A)r (17.7)

whereKo(x)isthemodifiedBesselfunctionof thesecondkindandofzeroorder.BourdetandGringarten(1980)showedthat,for Avalueslessthan0.01,Equation

17.7reducesto

- 2.30Q1 1.26s - 4nTr og A

Thedrawdownatwhichthetransitionoccursis independentof early-andlate-timedrawdownbehavioursandissolelyafunctionofA.

BourdetandGringarten(1980)presentedtypecurvesof F(u*,A,w)versusu* fordifferentvaluesofAandw(Figure17.2).Thesetypecurvesareobtainedasasuperposi-tionofTheissolutionslabelledinWvalues,withasetofcurvesrepresentingthebehav-iourduringthetransitionalperiodanddependinguponA.

As canbeseenfromFigure17.2,thehorizontalsegmentdoesnotappearin thetypecurvesathighvaluesofw.For highWvalues,thetypecurvesonlyhaveaninflec-tionpoint.NumerouscombinationsofWandAvaluesarepossible,eachpairyieldingdifferenttypecurves.But, insteadof presentingextensivetablesof functionvaluesrequiredto preparethesemanydifferenttypecurves,wepresentasimplifiedmethod.It is basedonmatchingboththeearly-andlate-timedatawiththeTheistypecurve,whichyieldsvaluesofTrandSf,andTrandSr+ Srn,respectively.Fromthesteady-statedrawdownat intermediatetimes,a valueof A canbeestimatedfromEquation17.7or 17.8.

(17.8)

TheBourdet-Gringartenmethodcanbeusedif, inadditiontothegeneralassumptionsandconditionslistedin Section17.1,thefollowingassumptionsandconditionsaresatisfied:- Theaquiferisofthedouble-porositytypeandconsistsofhomogeneousandisotro-

picblocksor strataof primaryporosity(theaquifermatrix),separatedfromeachothereitherbyanorthogonalsystemofcontinuousuniformfracturesorbyequally-spacedhorizontalfractures;

- Anyinfinitesimalvolumeof theaquifercontainssufficientportionsof boththeaquifermatrixandthefracturesystem;

252

108

6

Figure 17.2 Type curvesfor the function F(u*,A,m) (after Bourdet and Gringarten 19~O)

- The aquifermatrixhasa lower permeabilityand a higherstorativitythan thefrac-turesystem;

- The flow from the aquifer matrix into the fractures(i.e. the interporosityflow) isin apseudo-steadystate;

- Theflowtothewellisentirelythroughthefractures,andisradialandinanunsteadystate;

- Thematrixblocksandthefracturesarecompressible;- A < 1.78.

BourdetandGringarten(1980)showedthatthedouble-porositybehaviourofa frac-turedaquiferonlyoccursin a restrictedareaaroundthepumpedwell.Outsidethatarea(i.e.forAvaluesgreaterthan1.78),thedrawdownbehaviouristhatofanequiva-lentunconsolidated,homogeneous,isotropicconfinedaquifer,representingboththefractureandtheblockflow.

Procedure17.1- Preparea typecurveof theTheiswellfunctiononlog-logpaperbyplottingvalues

ofW(u)versusIju, usingAnnex3.1;- On anothersheetof log-logpaperof thesamescale,plotthedrawdownsobserved

inanobservationwellversusthecorrespondingtimet;- Superimposethedataploton thetypecurveandadjustuntila positionis found

wheremostof theplottedpointsrepresentingtheearly-timedrawdownsfall onthetypecurve;

253

- ChooseamatchpointA andnotethevaluesof thecoordinatesof thismatchpoint,W(u),lju, s,andt;

- SubstitutethevaluesofW(u),s,andQintoEquation17.5andcalculateTr;- Substitutethevaluesof Iju, Tr,t,andr intoEquation17.6andcalculateSr(J3= 0);- Ifthedataplotexhibitsahorizontalstraight-linesegmentoronlyaninflectionpoint,

notethevalueof thestabilizeddrawdownor thatof thedrawdownattheinflectionpoint.SubstitutethisvalueintoEquation17.7or 17.8andcalculateA;

- Nowsuperimposethelate-timedrawdowndataplotonthetypecurveandadjustuntilapositionisfoundwheremostof theplottedpointsfallonthetypecurve;

- ChooseamatchpointB andnotethevaluesof thecoordinatesof thismatchpoint,W(u),Iju, s,andt;

- SubstitutethevaluesofW(u),s,andQintoEquation17.5andcalculateTr;- Substitutethevaluesof Iju, Tr,t,andr intoEquation17.6andcalculateSr+Srn

(13= Ij3 or I).

Remarks

- For relativelysmallvaluesof (J),matchingthelate-timedrawdownswiththeTheistypecurvemaynot bepossibleandtheanalysiswill onlyyieldvaluesof Tr andSf;

- For highvaluesof A (i.e.for largevaluesof r), thedrawdownin anobservationwellnolongerreflectstheaquifer'sdouble-porositycharacterandtheanalysiswillonlyyieldvaluesofTrandSr+ Srn;

- Gringarten(1982)pointedoutthattheBourdet-Gringarten'stypecurvesareidenti-cal to thetime-drawdowncurvesfor anunconsolidatedunconfinedaquiferwithdelayedyieldaspresentedby Boulton(1963).(SeealsoChapter5.)If onehasnodetailedknowledgeof theaquifer'shydrogeology,thismayleadtoa misinterpre-tationofthepumpingtestdata.

17.3 Kazemietal.'sstraight-linemethod(observationwells)

Kazemietal.(1969)showedthatthedrawdownequationsdevelopedbyWarrenandRoot(1963)forapumpedwellcanalsobeusedforobservationwells.Theirextensionof theapproximationof theWarren-Rootsolutionis,in fact,alsoanapproximationof thegeneralsolutionofBourdetandGringarten(1980).It canbeexpressedby

s = 4~r F(u*,A,(J)

where A * A *

F(u*,A,(J)= 2.3Iog(2.25u*)+ Ei-((J)(1~)-Ei(-(1~(J)) (17.9)

Equation17.9isvalidforu* valuesgreaterthan100,inanalogywithJacob'sapproxi-mationof theTheissolution(Chapter3).

A semi-logplotof thefunctionF(u*,A,(J)versusu* (for fixedvaluesof Aand(J)will revealtwoparallelstraightlinesconnectedbya transitionalcurve(Figure17.3).Consequently,thecorrespondingsversust plotwill theoreticallyshowthesamepat-tern(Figure17.4).

(17.1)

254

F(u:"",,)24

22

14

20

18

16

12

106 108 1010u'

Figure 17.3 Semi-logplot ollhe lunctlon I( u*,A,m)versusu* for fixed valuesof Aand m

+

//

//

//

-.- logt

Figure17.4Semi-logtime-drawdownplot for an observationwellin a fracturedrockformationof thedouble-porositytype

255

6

4

2

0100 102 104

'!'

For earlypumpingtimes,Equations 17.1and 17.9reduceto

- 2.30QI 2.25Trts - 4 T og S 2n r rr(17.10)

Equation17.9is identicalto Jacob'sstraight-lineequation(Equation3.7).Thewaterflowingto thewellduringearlypumpingtimesis derivedsolelyfromthefracturesystem(~= 0).For latepumpingtimes,Equations17.1and17.9reduceto

- 2.30Q1 2.25Trts - 4nTrog(Sr+~Sm)r2

Equation17.11is alsoidenticalto Jacob'sequation.Thedrawdownresponse,how-ever,is nowequivalentto theresponseof anunconsolidatedhomogeneousisotropicaquiferwhosetransmissivityequalsthetransmissivityof thefracturesystem,andwhosestorativityequalsthearithmeticsumof thestorativityof thefracturesystemandthatof theaquifermatrix.Hence,thewaterflowingto thewellatlatepumpingtimescomesfromboththefracturesystemandtheaquifermatrix.

Kazemietal.'smethodisbasedontheoccurrenceof thetwoparallelstraightlinesin thesemi-logdataplot.Whethertheselinesappearin sucha plotdependssolelyonthevaluesofAandm.AccordingtoMavorandCincoLey(1979),Equation17.10,describingtheearly-timestraightline,canbeusedif

(17.11)

u* ::;m(1-ro)3.6A (17.12)

andEquation17.11,describingthelate-timestraightline,canbeusedif

I-mu* ? 1.3A ? 100 (17.13)

If thetwoparallelstraightlinesoccurin a semi-logdataplot,thevalueof mcanbederivedfromtheverticaldisplacementof thetwo lines,L1s"andtheslopeof theselines,L1s(Figure17.4).

m = 10-L1sv/L1s (17.14)

AccordingtoMavorandCincoLey(1979),thevalueofmcanalsobeestimatedfromthehorizontaldisplacementofthetwoparallelstraightlines(Figure17.4)

m= t,/t2 (17.15)

Followingtheprocedureof theJacobmethodonbothstraightlinesin Figure17.4,wecandeterminevaluesofTr,Sf,andSrn.UsingEquation17.7or 17.8,wecanestimatethevalueofAfromtheconstantdrawdownatintermediatetimes.

Kazemietal.'smethodcanbeusedif, in additionto theassumptionsandconditionsunderlyingtheBourdet-Gringartenmethod,theconditionthatthevalueofu* islargerthan100issatisfied.

Accordingto Van Golf-Racht(1982),theconditionu* > 100is veryrestrictiveandcanbereplacedbyu* > 100m,if A 1,orbyu* > 100-1/A,if m 1.

256

Procedure17.2- Onasheetofsemi-logpaper,plotsversust (tonlogarithmicscale);- Drawastraightlinethroughtheearly-timepointsandanotherthroughthelate-time

points;thetwolinesshouldplotasparallellines;- Determinetheslopeof thelines(i.e.thedrawdowndifferenceAsperlogcycleof

time);- SubstitutethevaluesofAsandQ intoTf =2.30Q/4rcAs,andcalculateTf;

Extend.theearly-timestraightlineuntil it interceptsthetimeaxiswheres = 0,anddeterminet,;

- SubstitutethevaluesofTr, tt,andr intoSf= 2.25Trt1/r2,andcalculateSf;- Extendthelate-timestraightlineuntilit interceptsthetimeaxiswheres = 0,and

determinet2;- Substitutethevaluesof Tf, t2,r, and~into Sf +~Srn =2.25Tft2/r2,and calculate

Sf + Srn;- Calculatetheseparatevaluesof SfandSrn.

Remarks

The two parallel straight lines can only be obtainedat low levalues(i.e. le < 10-2).At higherlevalues,only thelate-timestraightline,representingthefractureandblockflow, will appear,providedof coursethatthepumpingtimeislong enough.The analy-sisthenyieldsvaluesofTf andSf+Srn.

To obtainseparatevaluesof Sfand Srnwhenonly onestraightline is present,Proce-dure 17.3canbeapplied.

Procedure17.3- Follow Procedure17.2to obtainvaluesof Tr andSf fromthefirst straightline,

or if it isnotpresent,valuesofTf andSr+Srnfrom thesecondstraightline;- Determinethecentreof thetransitionperiodofconstantdrawdownanddetermine

1/2As,;- CalculatethevalueofcousingEquation17.14;- Substitutingthevaluesofcoand~intoEquation17.4,determinethevalueof Srn

if Sfis known, or viceversa.

RemarkTo estimatethecentreofthetransitionperiodwithconstantdrawdown,theprecedingandfollowingcurved-linesegmentsshouldbepresentin thetime-drawdownplot.

17.4 Warren-Raat'sstraight-linemethod(pumpedwell)

AsKazemietal.'sstraight-linemethodforobservationwellsisanextensionofWarren-Root'sstraight-linemethodfor a pumpedwell,wecanuseEquations17.7to 17.15toanalyzethedrawdowninapumpedwellifwereplacethedistanceoftheobservationwelltothepumpedwell,r,withtheeffectiveradiusof thepumpedwell,rw.

FollowingProcedure17.2on bothstraightlinesin thesemi-logplotof Swversust,wecandetermineTf, Sf,andSrn,providedthattherearenowelllosses(i.e.noskin)andthatwell-borestorageeffectsarenegligible.

257

,AccordingtoMavorandCincoLey(1979),well-borestorageeffectsbecomeneglig-iblewhen

u* > C' (60+ 3.5skin) (17.16)

where,atearlypumpingtimesC' = C/2nSfr~(dimensionless)C = well-borestorageconstant= ratioof changein volumeof waterin the

wellandthecorrespondingdrawdown(m2)

For a water-levelchangein a perfectwell(i.e.no welllosses),whichis pumpingahomogeneousconfinedaquifer,thedimensionlesscoefficientC' isrelatedtothedimen-sionlessClasdefinedby Papadopulos(1967)(seeSection11.1.1)bytherelationship(Ramey1982)

C' = 1/2Cl

Whenwell-borestorageeffectsarenotnegligible,thelimitingconditionfor applyingEquation17.10,asexpressedbyEquation17.12,shouldbereplacedby

. w(1-w)C'(60 + 3.5skm) < u* < ~ (17.17)

Theearly-timestraightlinemaythusbeobscuredby storageeffectsin thewellandin thefracturesintersectingthewell.But,withProcedure17.3,a completeanalysisisthenstillpossible.

RemarksWelllosses(skin)donotinfluencethecalculationofTf andw.If the linearwell lossesarenot negligible,Equation17.8becomes(BourdetandGringarten1980)

2.30Q1 1.26Sw= 4nTf og lee-2skin (17.18)

From theconstantdrawdown Swand the calculatedvalueof Tf, the valueof lee-2skincan be determined.If the well lossesare known or negligible,the value of lecan beestimated.

Example17.1For thisexample,weusethetime-drawdowndatafromPumpingTest3conductedonWellUE-25b#1in thefracturedTertiaryvolcanicrocksof theNevadaTestSile.US.A., aspublishedbyMoench(1984).

Thewell(rw= 0.11m; totaldepth1219m)wasdrilledthroughthicksequencesof fracturedandfaultednon-weldedtodenselyweldedrhyoliticashflowandbeddedtuffsto a depthbelowthewatertable,whichwasstruckat470m belowthegroundsurface.Fivemajorzonesof waterentryoccurredovera depthintervalof 400m.Thedistancebetweenthesezoneswasroughly100m.Coresamplesrevealedthatmostof thefracturesdip steeplyandarecoatedwithdepositsof silica,manganese,andironoxides,andcalcite.Thewater-producingzones,however,hadmineral-filledlow-anglefractures,asobservedincoresamplestakenat612mbelowthegroundsurface.

258

Thewellwaspumpedataconstantrateof35.8l/sfornearly3days.Table17.1showsthetime-drawdowndataofthewell.

LikeMoench,weassumethatthefracturedaquiferisunconfinedandof thestratatype(i.e.~ = 1).Figure17.5showsthelog-logdrawdownplotof thepumpedwellandFigure17.6thesemi-logdrawdownplot.Thesefiguresclearlyrevealthedoubleporosityof theaquiferbecausetheyshowtheearly-time,intermediate-time,andlate-timesegmentscharacteristicofdouble-porositymedia.At earlypumpingtimes,how-ever,well-borestorageaffectsthetime-drawdownrelationshipof thewell.In alog-logplotofdrawdownversustime,well-borestorageisusuallyreflectedbyastraightlineof slopeunity.Consequently,thetwoparallelstraightlinesof theWarrenandRootmodeldo notappearin Figure17.6.Only thelate-timedataplotasa straightline.

swim)102

86

10'86

4

100

10-2 2 4 6 810-1 2 4681002 4 6810' 2 4681022 4681032 4 68104,Im;n)

Figure 17.5 Time-drawdown log-log plot of data from the pumped well UE-25b# I at the Nevada TestSite,D.S.A. (after Moench 1984)

Sw Im)12

10

4 6 810-2 2 4 6 810-1 2 4 6 8100 2 4 6810' 2 4681022 4681032 468104,(m;nl010-3 2

Figure 17.6 Time-drawdown semi-logplot of data from thc pumped well UL-25b # I at the 0

Well-boreskineffectsareunlikely,becauseair wasusedwhenthewellwasbeingdrilled,themajorwater-producingzoneswerenotscreened,andpriorto testingthewellwasthoroughlydeveloped.

To analyzethedrawdownin thiswell,wefollowProcedure17.3.FromFigure17.6,wedeterminetheslopeof thelate-timestraightline,whichis L1s= 1.70m.Wethencalculatethefracturetransmissivityfrom

T = 2.30Q= 2.3x 3093.12= 333m2fdf 4nL1s 4 x 3.14x 1.70

Table 17.1 Drawdown data from pumpedwell UE-25b # 1,test3 (afterMoench 1984)

t Sw t Sw(min) (m) (min) (m)

0.05 2.513 30.0 8.840.1 3.769 35.0 8.840.15 4.583 40.0 8.860.2 4.858 50.0 8.860.25 5.003 60.0 8.900.3 5.119 70.0 8.910.35 5.230 80.0 8.920.4 5.390 90.0 8.930.45 5.542 100.0 8.950.5 5.690 120.0 8.970.6 5.990 140.0 8.980.7 6.19 160.0 8.990.8 6.42 180.0 9.000.9 6.59 200.0 9.021.0 6.74 240.0 9.041.2 6.96 300.0 9.071.4 7.17 400.0 9.111.6 7.33 500.0 9.141.8 7.45 600.0 9.172.0 7.56 700.0 9.182.5 7.76 800.0 9.213.0 7.93 900.0 9.253.5 8.03 1000.0 9.304.0 8.12 1200.0 9.445.0 8.24 1400.0 9.556.0 8.32 1600.0 9.647.0 8.41 1800.0 9.748.0 8.46 2000.0 9.789.0 8.54

/ 2200.0 9.8010.0 8.62 2400.0 9.8412.0 8.67 2600.0 9.9314.0 8.70 2800.0 10.0316.0 8.74 3000.0 10.0818.0 8.76 3500.0 10.2620.0 8.77 4000.0 10.3025.0 8.81 4200.0 10.41

Extending the straightline until it interceptsthe time axis wheres = 0 yields t2 -3.4 X IQ-3min. The overall storativity is then calculated from

260

S S = 2.25Tf t2= 2.25x 333x 3.4x 10-3= 0 15I.+ rn r2w 1440(0.11)2 .

Thesemi-logplotof timeversusdrawdownshowsthatthecentreof thetransitionperiodis at t ~ 75minutes.At t = 75minutes,1/2L1sy= 1.65m.SubstitutingtheappropriatevaluesintoEquation17.14yields

m = 1O-,",5y/,",5= 10-2x 1.65/1.70= 0.011

SubstitutingtheappropriatevaluesintoEquation17.4yields

Sf=m(Sf+ Srn)=0.011X0.146=0.0016and

Srn= 0.15

Thishighvalueof Srnis anorderof magnitudenormallyassociatedwiththespecificyieldofunconfinedaquifers.Moench(1984),however,offersanexplanationfor sucha highvaluefor thestorativityof thefracturedvolcanicrock,namelythatit maybeduetothepresenceofhighlycompressiblemicrofissureswithinthematrixblocks.Weconsiderthisaplausibleexplanation,becausethereislittlereasontoassumehomo-geneousmatrixblocks,asinFigure17.1C.

Wemustnowchecktheconditionthatu* > 100,whichunderliestheWarren-Rootmethod.SubstitutingtheappropriatevaluesintoEquation17.2,weobtain

100(Sf + Srn)r2w- 100x 1440X 0.15(0.11)2- 08.

t > Tf - 333 - . mm

Hencethisconditionissatisfied.Next,wemustchecktheconditionstatedin Equation17.13.For this,weneedthe

value00...Theconstantdrawdownduringintermediatetimesistakenas8.9m.UsingEquation17.8,weobtain

le= 1.26/10(4x 3.14x 333x 8.9)/(2.3x 309312)= 7.3 X 10-6

Substitutingtheappropriatevaluesinto Equation 17.13gives

t> (I-m) Sf+ Srn)r~= 1440(1-0.011)0.15(0.11)2= 818min1.3leTf 1.3 x 7.3 X 10-6X 333

Theconditionforthesecondstraight-linerelationshipisalsosatisfied.Finally,wemustcheckour assumptionthatthefirst straight-linerelationshipis

obscuredbywell-borestorageeffects.UsingC' = 1/20:andassumingre= rwgivesusC' = 1/2Sf.TakingthisC' valueandusingEquation17.16,weget

t 60r~ - 1440x 60 (0.11? - 16.

> 2 Tf - 2 x 333 -. mm

So, accordingto Equation17.16,afterapproximately1.6min,thedrawdowndataareno longerinfluencedby well-borestorageeffects.A checkof Figure17.6showsusthattheearly-timestraight-linerelationshipwouldhaveoccurredbeforethenandisthusobscuredbywell-borestorageeffects.

261

p.249 Uniformly fractured aquifers, double porosity concept, Introductionp.250p.251 Bourdet-Gringarten's curve-fitting method (observation wells)p.252p.253p.254 Kazemi et al.'s straight line method (observation wells)p.255p.256p.257 Warren-Root's straight line method (pumped well)p.258p.259p.260p.261