SOCIETYOFPETROLEUMENGINEERSOFAlME 6200NorthCentralExpressway
Dallas,Texas75206 PAPER NUMBERSPE4529 THISISA PREPRINT---
SUBJECTTOCORRECTION TheIsochronalTesti.ngofOiIWells By
M.J.Fetkovich,MemberAIME,PhillipsPetroleumCo. Copyright1973
American In8titute of Mining, Metallurgical, and Petroleum
Engineer8, Inc.
Thispaperwaspreparedforthe48thAnnualFallMeetingoftheSocietyofPetroleum
EngineersofAIME,tobeheldinLasVegas,Nev.,Sept.30-0ct.3,1973.Permissiontocopyis
restrictedtoanabstractofnotmorethan300words.Illustrationsmaynotbecopied.The
abstractshouldcontainconspicuousacknowledgmentofwhereandbywhomthepaperispresented.
PublicationelsewhereafterpublicationintheJOURNALOFPETROLEUMTECHNOLQGYortheSOCIETYOF
PETROLEUMENGINEERSJOURNALisusuallygranteduponrequesttotheEditoroftheappropriate
journalprovidedagreementtogivepropercreditismade.
Discussionofthispaperisinvited.Threecopiesofanydiscussionshouldbesenttothe
SocietyofPetroleumEngineersoffice.Suchdiscussionmaybepresentedattheabovemeetingand,
withthepaper,maybeconsideredforpublicationinoneofthetwoSPEmagazines.
ABSTRACT Thispaperpresentstheresultsand
me.thodeofanalyzingisochronalandflowafter
flowmultipointback-preseuretestsconducted onoil
wells.Testswereconductedin
reser-voirswithpermeabilitiesrangingfrom6MDto >
1000MD.Reservoirsin whichoil well
multipointback-pressuretestswereobtained
rangedfromhighlyundersaturated,tosaturated
atinitialreservoirpressure,toapartially
depletedfieldwithagassaturationexisting
abovethecritical.Eachofthesethree
reservoirfluidstatescanresultindifferent
interpretationmethods.Sack-pressuretests
wereruntopseudo-steadystateinthefield wherethesaturation
wasabovethecritical gassaturation. Inall cases,oil
wellback-pressure curveswerefoundtofollowthesamegeneral
formasthatusedtoexpresstherate-pressure relationshipofagaswell:
JI(- 22)n qo=0PR- Pwf Fromsome40oil wellback-pressuretests
examined,theexponentnwasfoundtolie
between0.568and1.000,verynearthelimits
commonlyacceptedfor"aswellReferencesandillustrationsatendofpaper.
curves.Flowpointalignmenttoestablishan oil well curveonthecustomary
logqvs.log6(p)plotisconsideredtobe
asg08dasthatobtainedongaswellback-preseuretests.
Thispaperdemonstratesthatgaswellsand oil
wellsbehaveverysimilarlyandshouldbe
testedandanalyzedusingthesamebasicflow equations. INTRODUCTION
Multipointback-pressuretestingofgaswells
isanacceptedprocedureforestablishing gas
well'sperformancecurve.Flowafterflowand isochrona12
testingarethetwobasicmethods
commonlyused.Inhighpermeabilityreservoirs,
eithermethodcanbeemployed.Inlowper-meabilityreservoirs,theIsochronal
methodoftestingeliminatesthetransient
effectsthatcanseverelydistorttheresults
obtainedfromaflowafterflowtest.Methods
foranalyzingandcalculatinggaswell
performancecurveshavebeenthesubjectof
numerousinvestigations.Thebulkofthese
investigationshaveexaminednon-Daroyflow
behavior,theprimaryreasonthatmultipoint testsareconducted.
Multipointtestingofoil wellsisnotnow
acurrentpractice.Asearlyas1930,however,
2THEISOCHRONALTESTINGOFOILWELLS SPE4529
T.V.Moore3reportedtheresultsof anoil
wellmultipointtestconductedontheHumble
SmithA-2intheYatesField.Thepurposeof theback-pressuretest
wastodemonstratea methodofestablishingawell ISopenflow potential
withoutproducingthewell wideopen. Theneedforestablishinganaccurate
performancecurveforanoil wellis as
importantasdeterminingoneforagaswell.
Inthesearchfornewoil,theindustryisturn-ingtoremoteareassuchastheArcticand
offshore.Critical questionsofwhetherto develop,andif
so,howtodevelopafield hingeontheabilitytoaccuratelypredicta
wellisdeliverability.Often,becauseof
equipmentlimitations,theratesofproduction
obtainedduringdrillstemtestingaremuchless
thanthoseplannedforfulldevelopment. Thetraditional
methodforpredicting productionratesanddrawdownsforoil wells
hasbeenbasedontheconceptoftheproductiv-ity
index(PI),whichhasbeenusedintheoil
industryformanyyears.Theusualformof theequation
isvalidonlyforsystemsproducinganideal homogeneousliquidobeyingDarcy
ISlaw.This conditionnormallyholdsforoil wellswhenthe oil
isundersaturatedthroughouttheproducing formation.It
haslongbeenrecognizedthat inreservoirsexistingator
belowthebubble-pointpressure,producingwellsdonotfollow
thissimpleequation.Actualfieldtests
indicatethatoilflowratesobtainedat
increasingdrawdownsdeclinemuchfasterthan wouldbepredictedbyEq.1.
EvingerandMuskat4 firstderiveda
theoreticalproductivityindexforsteadystate
radialflowinanattempttoaccountforthe
observednon-linearflowbehaviorof oil wells
andarrivedatthefollowingequation q=7.08kh oIn(::) wheref(p)= k ro
jPe f(p)dp Pwf
o0 (2) CalculationsusingEq.2basedontypical
reservoirandfluidpropertiesindicatedthat
PIatafixedreservoirpressurePe(asdefined
fromEq.1)decreaseswithincreasingdrawdown.
InacomputerstudybyVoge15,results
ba.sedontwo-phaseflowtheorywerepresented
toindicatethatasingleempiricalinflow
performancerelationship(IPR)equationmight
bevalidformostsolution-gasdrivereservoirs.
Hefoundthatasingledimensionless1PRequation
approximatelyheldforseveralhypothetical
solution-gasdrivereservoirsevenwhenusing
awiderangeofoilPVTpropertiesandreservoir
relativepermeabilitycurves.Thefactthathis
studycoveredawiderangeoffluidproperties
andrelativepermeabilitycurvestoobtaina
singlereferencecurve,cannotbeover
emphasized.Vogelthenproposedthathis
equationbeusedtotaketheplaceofthelinear
productivityindexrelationshipforsolution-gasdrivereservoirswhenthereservoirpressure
isator belowthebubble-pointpressure.
Theproposedempiricalreferenceequation (1PR)in
dimensionlessformwasgivenas qo=1_0.200.00 . (3) (q)max )o A
comparisonwasmadeof1PR ISforliquid
flow,gasflow(n=l)andtwo-phaseflow(his
referencecurve)onadimensionlessbasis,
(Fig.1).AsisevidentfromFig.1the
positionofthetwo-phasereferencecurve
relativetoliquidandgasflowindicatesthat oil wellsproducingasif in
asolution-gas drivereservoirshouldactuallymore
likeagaswell,i.e.,(P.a2- Pwf)VB.qo
shouldplotasastraightlineonlog-logpaper withaslope(n)nearunity.
Thispaperpresentstheresultsofmulti-pointback-pressureteststakenatasingle
reservoirpressurelevelTheseresults
showthattheperformancecurveforanoil well
canbeexpressedbyamoregeneralandfamiliar
equationsimilartothatusedforgaswells, q= JI _Pf2)n.(4) ooltw
Reservoirsin whichoil wellmultipoint
back-pressuretestswereobtainedrangedfrom
highlyundersaturated,tosaturatedatinitial
reservoirpressure,toapartiallydepleted
fieldwithagassaturationexistingabovethe
critical(equilibrium)gassaturation.Equation
4wasfoundtobevalidfortestsconductedin
allthreereservoirfluidstates,evenforthe
conditionswhereflowingpressureswerewell
abovethebubble-pointpressure.Permeabilities
ofthereservoirsrangedfrom6to>1000
millidarcys.Flowpointalignmenttoestablish anoil
wellback-pressurecurveonthecustomary log
va.log6(p2)wasfoundtobeasgoodas
thatobtainedongaswellback-pressuretests.
BASICEQUATIONSANDPRESSUREFUNCTIONS
ThebasicflowequationgivenbyEvinger andMuskat4
forsteady-stateflow,applicable toeitheroil orgasflow,is
SPEL.'529M.J.FETKOVICH 3 (2) wheref(p)canbeanyfunctionofpressure.
Usingthetypicalpressurefunctiondepicted inFig.2it
isobviousthatwecanevaluate thetotalintegralintwopartsandwrite + l"
Ib Forflowintheregionwherethepressures
areabovethebubblepointpressureif we assumek=
1(neglectingthepressure dependenOpermeabilitytermforsimplicityof
presentationonly)andtreat(uB)evaluatedat theaveragepressure write
qo=[In:hS] dp (Pe- Ib)] +(6) Exceptfortheadditionofthenecessaryskin
(discussed.laterinthepaper)Eq.6 isidenticaltothatderivedbyHandy21.
Figure2illustratesaplotofllu Basa functionofpressureforanoil
reservoir.Also,drawnonthisfigureisa
dashedlinerepresentingtheeffectofrelative
permeability(k)ondrawdownsbelowthe bubble-pointIt is assumedfor
purposesofdemonstrationthatkI(uB)is linearandits interoeptis
0atrepr3sgure. Thesimplifyingassumptionofthe0intercept for
kro/(uoB)approximatelydefinesVogel'sIPR curveanaexactlydefinesEq.4
whenn=l.)Also, drawnonFig.2isahypotheticalpressure
functionkro/(uoBo)representedasaconstant forall pressures.It
isolearthataconstant valueofkI(uB)overtheentirepressure ro00
drawdownrangerequiredtoaconstant productivityindex(PI).
Figure3illustratesplotsofl/(u B)for
twohighpressuregasreservoirs.gAwas
obtainedfromcalculationsusingthereservoir
gasanalysisandstandardcorrelationsofZand
uasfunctionsofcriticalpressureandtem- CurveB
wasobtaineddirectlyfroma PVTstudy.OnestrikingfeatureofcurveA is
thefactthatit
resemblesthatofanunder-saturatedoilreservoirwithanapparentor
pseudobubble-pointpressurenear2500psia,
thenormalinflectionpointofaZcurve.A
furtherobservationthatcanbemadefrom curvesA
andBisthataregionexistswhere agaswellcanbeconsideredtobehaveasa
liquid,i.e.,l/(uB)isnearlyconstantor onlyslightly withpressureasis
theoaseforthepressurefunctionofan
undersaturatedoilreservoirabovethebubble-pointpressure.
Fortheregionwherethepressure
isaconstant,ornearlyso,wecanimmediately
writeuponintegrationofEq.2thewellknown
steady-statesinglephaseflowequation: _7.0Skh (Pe- Pwf) Ire)'J(uB)
In\- +S rw Notethatthisequationwouldapproximately
holdforgaswellsrepresentedbycurvesA and
BinFig.3overaconsiderablerangeof
pressuredrawdowns.qwillthenbepro-portionalto6pinsteaBof6(p2).This,in
factwasfoundtobethecaseforisochronal
testsconductedontwowellsinareservoir
withfluidpropertiesrepresentedbycurveB.
Nowconsideringtheentirepressurefunction fromPeto0,foreithertheoil
orgascurves, (thedashedlineinFig.2)wenotethatf(p)
canberepresentedapproximatelybytwosepa-ratestraightlinesegments.Theapproximate
flowequationthen,overthetotalpressure
interval,canbewrittenas:(SeeAppendix) q=7.0Skh rIn(:e )+s1Ut!)
LwJPe'pt, "2 +(8) orq= J'(pt, 2 - Pwf2)+ J(Pe- pt,)..(SA)
Fordrawdownsbothaboveandbelowthe
bubble-pointpressure,aback-pressurecurve
plotwillappearastwolinesegments,with
theintersectionyieldinganapproximatevalue
forthereservoirbubble-pointpressure.This
thenoffersanapproachfordetermininga
reservoir'sbubble-pointpressurefroman
isochronaltest.Foranisoohronaltest,a
constantreservoirradiusofinvestigationis
obtainedforeachflow-aninsituconstant volumecell.
Ifthedegreeofundersaturationisslight,
thetwolinesegmentsmaynotbedefinable.
Unstableflowconditionsinthetubingatthe
THEISOCHRONALTESTINGOFOILWEllSSPE4529
lowflowratesnecessarytodefinethesingle-phaseflowconditionsmaypreoludedefiningtwo
straightlines.Further,aswillbedemon-stratedlater,non-Darcyflowoanexisteven
whenallflowingpressuresareabovethebubble-pointpressure.Conceivablythenthiscould
leadtoeventhreelinesegments. Forthecaseofall drawdownsbelowthe
bubble-pointpressureJ(Pe- Ph)isaconstant,
whiletheremainingtermvariesnon-linearly withflowingpressure,Pwf.
Thecompositeeffectresultsinanequation oftheform ( 22)n q= CPe- Pwf
AsdecreasestothepressurePh,1.0and C->J'suchthatfortheoil
wellcase,only thetwo-phaseflowtermremains.Wethusobtain
thebasicequationsuggestedfromVogel's resultsforPe7.08kh[m(Pi)-
m(pwf}] q wherem (p)canalso apressure dependentpermeability, m(p) I
o p kro(S,p)dp uB (13) .(14)
(Theeffectofapressuredependentpermeability
couldreadilybedisplayedinFigs.2and3.)
Equation12or13thenshouldbeappli-cableforanalyzingbothoil
wellandgaswell back-pressuretests _J'(22)1.0 qo- 0Pe- Pwf
(10)RATEANDTIMEDEPENDENTSKIN.s(g.t) A
significantconclusiontobedrawnfromEq.9 isthatagaswelloranoil
wellcanhave2a slopelessthan1.0onalogqvs.log6(p) plot
withoutnon-Darcyflowexisting.The
slope(n)inthiscaseisstrictlyaresultof
theshapeofthewellspressurefunotion.This
possibility,foragaswell,was6recognized andreportedbyRowanandClegg
Eq.10mustbefurthergeneralizedwith
anexponent(n)inlightofresultsobtained
frommultipointback-pressuretestsconducted onoil
wellsforbothsingle-phaseandtwo-phaseflowto q= J' o0 ( 22)n Pe- Pwf
(11) Eq.11isidenticalinformtothegaswell
back-pressureequation.Forconstantrate
transientgasflow,thegaswellback-pressure
equationisusuallyexpressedby7,8. 7.08kh(Pi- Pwf) . q(UB) +s+Dq (12.
Otherthanfortheuniquefluidpropertycases
discussedabove,orapressuredependentper-meabilityeffect,thenon-Daroyflowtermin
Eq.12isrequiredtoobtainanexponent(n) lessthan1.0. Intermsofam(p)
Slopesmuchlessthan1wereconsistently
obtainedfromisochronaltestsconductedon oil
wellsinsaturatedreservoirs.Forunder-saturatedreservoirs,theshapeofthepressure
functionwasshowntobecapableofaccounting
forslopeslessthan1.SinceVogel'swork
basedontwo-phaseflowtheoryindicatedback-pressurecurveslopesshouldbeunityoreven
greater,anearwellboreeffect wassuspected. (Allof
Vogel'sresultsshowthefirstcalculatee IPRcurveafter 0.1%of
originaloil-in-place
isrecovered.Theeffectofinitialgassat-urationbuild-uparoundthewellboremayhavebeenpresentinhisresults.)Handy
studiedtheadverseeffectonPIoftwo-phase
flowinthevicinityofthewellboreforunder-saturatedoils.Muskat12,13presentedasimple
approachtostudytheeffectoftwo-phaseflow
aboutthewellboreforagascondensatewell
thatcouldbeappliedtoasaturatedorunder-saturatedgascondensateoroil
well. s(g,t)FOROONDENSATEWELLS Muskat'sequationtocalculatetherateof
changeofliquidsaturationtakingplaceabout
thewellboreforaproducingcondensatewell is: dSqQ dt=2f1rh!dr dc dp
(15) Saturationisassumedtobuilduponlytothe
limitingequilibriumliquidsaturation;its
radiusthenexpandingwithtime.Forasteady
statepressuredistribution,andsaturation
Sequalto0att=O,wecanobtainanequation SPE4529M.J.FETKOVICH 5
intermsoftheapproximateradiusofthe
equilibriumtwo-phaseflowregion.In engineeringunitsit is r a 2
"'".1135q/ uZyt.(16) h2 kpSclh whereY
isexpressedasreservoircubicfeetof
condensateaccumulationinthereservoirper Msafoffull
wellstreamgasproducedperpsi, Y canbecalculatedusingtheretrograde
liquidvolumedatadeterminedfromPITTstudies.
ThetermSclhistheariticalhydrocarbonliquid
saturationtoreachequilibrium,or mobilliquid
saturation.TheotherpertinentunitsareMsafd cps.,days,ft.andDarcy.
Thedefinitionofskineffect(s)interms
oftheradiusofanalteredzoner(equilib-riumtwo-phaseflowregion),andtRereduced.
permeabilityofthealteredzoneka,canbe expressedas2 s:::(k- ka)In
(ra)(17) 2kr aw SubstitutingEq.16into17weobtain s(q, t) (18)
Equation18definesarateandtimedepen-dentskintermthatcangivetheappearanceof
non-Darcyflow.Theequation,although
approximate,givesasimpleanalyticalexpres-sion
withwhichtoestimatetheeffectsoftwo_
phaseflowinthevicinityofthewellbore.
Thesignificanceofthiseffectinwellshasbeendemonstratedbyothers.14,,16,
17.Eq.18hasbeenusedtosuccessfully
analyzetheresultsobtainedfromisochronal testsoncondensatewells.A
significantpor-tionoftheskin wasattributedtos(q,t).
s(g.t)FOROILWELLS 1819 Inthestudiesof Westetal,Perr1ne
andWeller20,ananalogousbehavioraroundthe
wellborehasbeenshowntoexistinanoil
well.Underconstantrateproductionfor
initiallysaturatedsolution-gasdrive
reservoirs,theirresultsshowthatthegas
saturationquicklybuildsuptotheequilibrium
gassaturation(criticalgas)andremains
constantatitsequilibriumvalue.Its'radius
increaseswithtimeuntilthewellsdrainage
volumeisabovethecriticalgassaturation. (See 4)This
saturationbuild-upin thevicinityofthewellboreiscommonly
referredtoas"gasblockllThecorresronding
oilpermeabilityredUctiohinthisregionis
thereforeconstant,withitsradiusincreasing
withtime.Thisdamagedzonewithinwhichthe
relativepermeabilityhasbeenreducedhasbeen
referredtoasapseudo-skinbyWeller. Utilizing Eq.18
withtheappropreiatevariable
substitution,therateandtimedependentskin s(q,t)foranoil wellis
s(q,t)=2lre,aIn qoXt (19) (k- k)[2j hkScgrw whereX
isexpressedasreservoircubicfeetof
gasevolvedinthereservoirperstocktank barrelofoil producedperpsi,
Xis readilyobtainedfromastandardPVTstudyusing
theliberatedgasdataRLasafunctionof
pressure.ScgistheeqUilibriumorcritical gassaturation,fractionofpore
Other pertinentunitsareSTKBOPD,cps,DAY,FT, DARCYandRESBBL/STKBBL.
Theresultsof Westetal werefirstused
todeterminewhetherEq.19wouldreasonably
predicttheradiusoftheIIpseudo-skinllfor
timesbeforeboundaryeffectsbecameSignificant
Usingthebasicdatagivenintheirpaperand Eq.19acalculatedr=
1.6FTversustheir 1.5FTwasobtainedat2.21days,andra=4.6
versustheir6.0FTat16.8days Eqi.180"- 19areapplicabletoinitially
saturatedandpartiallyundersaturatedreser-voirs.Onceanoil
well'sdrainagevolume exceedstheequilibriumgassaturationEq.19
isnolongerapplicable.Forcondensatewells, Eq.18
willapplyforamuchlongerperiodof
time,atleastuntilrevaporizationbeginsto takeplace.Thenra
willbegintorecede.
Onlyinthecaseofundersaturatedreser-voirs,wecouldassumethatthetwo-phaseregion
isattheequilibriumgassaturationandexists
outtowherethepressureisequaltothebubble
pointpressure.Thissimplerapproach,developec
byHandy2lforwellsproducingfromunder-saturatedreservoirs,leadstothemaximum
reductionofPIwhichcouldbeexpectedfroma
gassaturationbuild-uparoundawellproducing
withaflowingpressurebelowthebubble-point
pressure.Byanalogy,thesameapproachcould
beusedfortreatingundersaturatedgascon-densatewells.
Forcompletenessthen,Eqs.12and13shoulc
bewrittentoincludearateandtimedependent skin,s(q,t).Wewouldthenhave
7.08yill(Pi-Pwr)=In q(UB) +s+s(q,t)+ Dq (20)
6THEISOCHRONALTESTINGOFOILWELLS SPE4529 and - m q (p,)1(I wf
.)14.23kit In2+ i!(uct)irw s+s(q,t)+Dq (21) 22
AfterRamey,wecandefine s'=s+Dq(22) and sIr=S+s(q, t)+Dq (23)
WELLTESTRESULTS Thebasicresultsobtainedfromisochronal
back-pressuretestsandflowafterflowmulti-pointtestsconductedonoil
wellsare summarizedinTables1and2. Reservoirfluidstatesin
whichmultipoint welltestswereobtainedare,inchronological order,
1.Gassaturationexistedthroughoutthe
reservoirabovethecriticalorequilibrium gassaturation.
2.Undersaturatedreservoirwithflowing
pressuresobtainedbothaboveandbelow thebubble-pointpressure.
3.Saturatedreservoirswiththereservoir
pressureatorverynearthebubble-point pressure.
4.Undersaturatedreservoirwithall
flowingpressuresabovethebubble-point pressure.
GASSAWRATIONABOVEEQUILIBRIUM Stabilizedflowafterflowmultipoint
back-pressuretestswereavailableon16wells
producingfromasolution-gasdrivecarbonate
reservoir,FieldA.Reservoirconditionswere
idealfortestingthehypothesisthatqovs. (- 22) PR-
Pwfwouldplotasastraightlineon log-loggraphpaperwithaslope(n)of1.The
reservoirvariablesinthisfieldclosely
approximatedthoseusedbyVogelinhisstudy,
(SeeTable3).Averagegassaturationinthe
reservoiratthetimethetestswereconducted
wasestimatedtobebetween10or12percent.
Producinggas-oilratioswhencomparedtothe
initialsolutiongas-oilratioof684SCF/BBL
indicatesthatthereservoirwaswellabove
abovetheequilibrium(criticalgas)saturation
atthetimethetestswereconducted.Gas-oil
ratiosincreasedonlymoderatelyatincreasing drawdownsformosttests.
Althoughtheunitslopedidpredominate,
fourwellsexhibitedback-pressurecurveslopes muchlessthan1.A
slopelessthan1results inanevenmorerapiddeclineinrateqwith
drawdownthanwouldbepredictedfromVogel's IPRequation.
ThetestonWell6,FieldA(?ig.5) consistedofsevenindividualflows,eachto
apparentstabilization.Thefirstfourflow
rateswereruninanormalincreasingsequence.
Followingthefourthflowat229BOPD,therate
wasreducedto93BOPDthenagainfollowedby
anincreasingsequenceofflows.Allpoints
essentiallyfellonthesameline,indicating
thattransienteffectswerenotthecauseof
thedeviationfromthelinearrelationship
predbtedbytheproductivityindexconcept.
Notethattheflowpointsdefineaperformance
curvewithaslopeof1almosttoitsabsolute
openflowpotential(AOFP).Table1showsthat forall
wellstestedinthisfield,themaximum flowrate
wasveryneartheextrapolatedabsolute
openflowpotential.Intheotherfieldsin
whichmultipointtestswereconducted,equipment
limitationprecludeddefiningtheentirecurve,
requiringagreaterdegreeofextrapolationto AOFP.
WellNo.3,FieldA,(Fig.6)illustrates
themostsignificantresultofthisfirstgroup
oftests.Withanexcellentalignmentoffive
stabilizedflows,theslopeoftheback-pressure
curveis0.648.Theresultsobtainedfromthis
testfirstsuggestedthepossibleexistanc8of
thesamelowerlimitoftheexponent(n)as existsforgaswells(n=
0.500),andafloweffect. WellNo.14,FieldA,(Fig.7)exhibited
themaximumincreaseingas-oilratiowith
increasingdrawdownofallthewellstested.
Evenwiththegas-oilratioincreasingwithrate,
theslopenoftheperformancecurvewas1.0. Inanattempttoutilizetheoil
wellback-pressuretestingmethodtomoreaccuratelypre-dictfulldevelopmentwellperformancefrom
wildcatwelltests,anisochronaltestprogram
wasinitiated.Thefirstknownoil well isochronaltest
wasconductedonApril14,1970
onthePhillipsEkofisk2/4-2Xwell.Surprising
resultswereobtainedfromthesefirsttests.
Twostraightlineswereobtainedwhenalogq vslog plotwasprepared.Figure
8illustratestheresultsobtainedfroma6
hourisochronaltestconductedonzone2.
Handy1s21,workledtotheconclusionthat
thetwostraightlineswerearesultofthe
reservoirbeingundersaturated,withtheinter-sectionpointindicatingtheapparentreservoir
bubble-pointpressure.Usingthefirsttwo SPE4.529M.J.FETKOVICH 7
flowratesandtheconstantPIapproach,an
apparentabsoluteopenflowpotentialof13,000
BOPDisindicated.Thetruepotentialestab-lishedbyextrapolationofdrawdowndatabelow
thebubble-pointpressureis5200BOPD.
Calculatedpermeabilityfrombuild-updata
followingthefirstsinglephaseflowwas6.1
MDwithaskins=O.Forflowsatpressure
drawdownsbelowthebubble-pointpressure,a ratedependentskin
wasTherate dependentskinsextrapolatedtoaskinof 0at
thepointsinglephaseflowended,q 2100 BOPD,asshouldbeexpected,(See9).
Single-phaseskinsof-4arenormallyobtained
fromtestsfollowingacidstimulations.This
favorableresponseusuallyprecludedobtaining
drawdownsbelowthebUbble-pointpressureafter
acidbecauseofequipmentlimitations.Asa
result,noafteracidisochronaltestshave
beenobtainedwhichcoulddemonstratewhether
thenatureoftheperformance'curveissubstan-tially
differentthanthatobtainedbeforeacid.
Isoohronaltestsconductedontwootherzones
inthiswell,beforestimulation,yielded similarresults.
StartingwithEq.8A,weoanoutlinethe
procedureusedtocalculatethebubble-point pressurefromthepre-acidtest
If wethendefine and q(2)= JI o q(l)=JI o 22 (rb- Pwf) (Pe-%)
thenqo=q(2)+q(1) (8A) (24) (26) (Nophysicalsignificanceshouldbe
attachedtoq(l)orq(2)sinceit isobvious
thatforthesteadystateassumptionuponwhich it
wasderived,thetotalqmustbeflowing throughbothregions.)0
Whencombinedtwo-phaseandsinglephase flowareoccuringinawell q(l)=
CONSTANT= J(p-n)(25) o'Pe'Ibe'0 thereforeq(2)= q(measured)- q(1) o
JI(22) o%- Pwf (27)
Withthecorrectvalueofabubble-pointpre8-sure,Pb'aplotofq(2)vs.(%2_PWf2)should
plotastraightlineoneithercartesianora
log-logplot.Onalog-logplot,theslopeis 1.0andtheinterceptJI. o
The1.0slopewasassumedforthetwo-phase
termatthisstageofdevelopmentbecauseofthe
computerresultsobtainedbyVogelandthe resultsobtainedfromtestsin
FieldA.However, thefactthatslopeslessthan1.0areindicated
fromothertestswheretwo-phaseflowexisted
inthereservoir,suggeststhemoregeneralform of Eq.8A.tobe
q=JI(n2_Pf2)n+ J(p-n)..(28) o0'0w0e'o A
trialanderrorcalculationassuming
variousvaluesofPbwasperformeduntila slopeof1was (SeeFig.10).This
resultedinacalculatedbubble-pointpressureof 5874psia.A
bubble-pointpressureof5885 psiawasdeterminedfromaPVTstudyofthe
reservoirfluidobtainedfromthiswell. A simplegraphical
ofthebubble-pointpressurefromtheapparentintersection
pointisprobablyadequatebecauseofthe
unoertaintiesintroducedbyn,theexponentof
thetwo-phaseterm,beingavariable.Once
thetruebubble-pointpressureisdetermined
fromPV'I'data,ncanbedirectlycalculated. SATURATEDRESERVOIRS
Mostofthereservoirsin FieldsCthru H
aresaturatedatinitialreservoirpressure.
Thereservoirsareverysimilarinnatureat
correspondingdepthssincethefieldsarein
closeproximitytoeachother.Allreservoirs
arerelativelycleanTertiarysandstonesranging
indepthoffrom7800to11200feet.Perme-abilitiesdeterminedfrombuild-uptestsranged
from130to2500MDwithnetpaysrangingfrom
20to180feetinthickness.Typicalporosities
are22percentwithwatersaturationsofaround
30percent.Relativepermeabilitymeasurements
exhibitedcriticalgassaturationsrangingfrom 7to13percent.
Humpingeffects,wellborestorage,flat
pressurebuild-upcurvesandtheshortduration
ofthebuild-upsmadethedeterminationof
permeabilitiesdifficultonseveralwells.
Forthosewellsnothavingpermeabilitieslisted
inthetables,itsorderofmagnitudeis reflectedbythewellsA.OFP.A
summaryofall theisochronaltestresultsobtainedappearin Table2.
Thestandardisochronaltestinthesefields
consistedofafourhourflowfollowedbyafour hourshut-in.aflowafterflow
testwasalsoconducted.Increasingand
decreasingsequencesofflowswereperformedon
mostteststocheckreproductability.Because
oftheratherhighpermeabilitiesinthese
reservoirs,flowafterflowtestsoften
duplicatedtheisochronaltestperformance
8THEISOCHRONALTESTINGOFOILWELLSSPE4529
ourve.Performanoecurveslopesobtainedfrom
thesetestsareseentorangefrom0.568to
0.875.Notonesinglewellexhibitedthe1
slopethatwassopredominantinFieldA.
Severalofthewelltestperformanceourves
obtainedininitiallysaturatedreservoirsare showninFigs.11-
21.Ingeneral,flow pointalignmenttoestablishanoil wells
performanoeourveisasgoodasthatobtained
fromgaswellback-pressuretests.Shut-in
pressurerecoverybetweenisochronalflowson
thesetestsissufficienttoestablishtrue
isochronalconditions.Gas-oilratiovariations
areconsideredtobemoreafunctionofsep-aratorpressurethanreservoirdrawdown
pressureeffeots.Themostsignificantobser-vationtobemadefromthesetestsisthatflow
afterflowdatafallonthesameperformance
curveasthatestablishedbyisochronaldata
points.Thelowestpermeabilityofthisgroup of
wellsis130MD.TestresultsforWellNo.
3-C,Field0,(Fig.12andTable4)demonstrate
theflowafterflowandisochronaltestper-formancecurvereproduoabilitybytwoseparate
testsconductedoneweekapart. ThetestonWellNo.5-CinFieldD was
selectedtoapplyEq.20toanalyzethewell
performancedata.Thefourhourisochronalwell
performancecurvewasestablishedbytwo separatetestssix
monthsapart.Nearly100 psireservoirpressuredropoccuredbetween
thesetwotests.Nodetectableshiftinthe
positionofthewell'sperformancecurvewas
noted.WellNo.5-Cwastheonlyoneofthe
saturatedreservoirwellsthathadafully
perforatedinterval,thuseliminatingyet
anothervariable,partialpenetrationeffects.
Further,thepermeabilitycalculatedfrombuild-updatawasconsistentwithmeasuredooreper-meabilitiesforthis
well.Tables5and6 summarizethereservoirandtestdataused
inthecalculationsandtheresultsobtained.
Theratedependentskinterms(q,t),forthis
well,wasfoundtobeinsignifioantateventhe
highestflowrateof2308BOPD.Boths'and
stlwereplottedasafunotionof%.Ineither
case,alinecanbedrawnthrutheplotted pointstoq=
0yeildingaformationskins= O. Non-Darcyflowappearstobesignificantfor
thiswell. Theisochronalperformancecurveobtained
onWellNo.7-e,FieldD,(Fig.16),exhibits
thesteepestslopeofallthetestsconducted
inasaturatedreservoir.Anyoftheflow
rateswouldbereasonableforanormalsingle flowdrillstemtest.A
comparisonofcalculated absoluteopenflowpotential(AOFP)ismade
usingthePImethodandVogel'sIPRmethodfor
eachoftheflowrates.Themaximumerrorin
AOFPisofcourseobtainedwiththelowestflow rate- AOFP=
57,200BOPDPImethod,31,990 BOPDIPRmethodandactualisoohronalAOFP=
7250BOPD.EventhoughtheerrorinAOFP,using
thePIorIPRmethodsisreducedwhendetermined
atthehighestflowrate,theerrorin
evaluatingskinandflowefficiencywillbe increased.
WellNo.8-e,FieldD,(Fig.17andTable7)
demonstratesthechangeinthewellsperformance
curveasaresultofincreasingtheperforated
intervalfrom20Ft.to60Ft.;netpayis182
Ft.Thewellspotentialn9arlydoubledand
theslopeoftheperformancecurveincreased onlyslightly.
UNDERSAWRATEDRESERVOIR(pwrIb) Ofallthetestsconducted,the
mostsurprisingresults 'werethoseobtained
onWellsl-aand2-binFieldG(Figs.22and
24respectively).Withallflowingpressures
wellabovethereservoirbubble-pointpressure,
(single-phaseliquidflow),slopesof0.813
and0.712wereobtainedfromalogqvs.log 1I(p2)plot.
Conolusiveevidenceoftheocourenceof non-Daroyflowinanoil
wellisdemonstrated fromadetailedanalysisoftheisochronal
testdataobtainedonWellNo.1-a.PVT studies ontwobottom-holesamples
andarecombinationofsurfacesamplesbubble-pointpressuresof4495,4756and4785
psia Thelowestflowingpressure obtainedonthistest was5669psiaataflow
rateof2973STKBOPD.Netpayforthiswell
is25feetwithaperforatedintervalof10 feet.
TheisochronalperformancecurveforWell
No.l-a(Fig.22)indicatesaslopenof0.813
withanexcellentalignmentof8separateflow
rates.Threedecreasingsequenoeflowswere
followedbyfivemoreflows.Table 8summarizesthedataobtainedforeachflow
rate.Thefactthatalignmentwasobtained
followingrepeatedflowsandshut-ins,and
flowreversalstendstoindicatethatapressure
dependentpermeabilitywouldnotaccountfor
thenon-linearflowbehaviour23Thenormal hysteresiseffect24,25in a
pressuredependent permeabilitycausedbyrepeatedpressure
reversals,asoocuredduringthisisoohronal
test,shouldnothaveallowedtheflowsto retracetheback-pressurecurve.
Analysesperformedonbuild-upsobtained
afterfouroftheflowsyieldedconsistent
permeabilitiesof222MD.Theskineffect
calculatedfromthesebuild-upanalyseswas
foundtoberatedependent.Whenswas plottedasafunctionofq,askinatq= 0
of+2.2andanon-Darcyflowcoeffioient
Do=.00233BOPD-lwasobtained(Fig.23).
FromReferenoe26,apartialpenetrationskin
sbwascalculatedtobe+2.5,inveryclose SPE4529M.J.FETKOVICH 9
agreementwiththatobtainedto qo=
O.Thereservoirskindamage(s)therefore isconcludedtobeO. A
furtherverificationofnon-Darcyflow
inthiswellwasmadebycheckingtheReynolds number R=?-Vd e)J. A
Reynoldsnumberof8wasobtainedforthewell
underitsflowingconditionof2973BOPDwith
aflashformationvolumefactorof2.70,a
densityof0.48gm/cc,0.22cps.viscosity,and asswningagraindiameterof
0.5mm.According toMuskat12 turbulentflowcanbeexpectedfor
Reynoldsnumbersgreaterthan1. Thenecessityfor conducting multi-rate
tests dnoil wellsforthecorrectevaluation
ofwellperformance,PI,reservoirdamage,flow
efficiencyandpotentialisparticularly
emphasizedbythisexample.Onecanalso
concludethatnon-Darcyflowwouldalsoexist
inthepresenceofagassaturationaroundthe
wellboreandwouldbeevenmoreseverethanis
indicatedforthesinglephaseliquidflows. For
WellNo.1-a,abreakintheperformance
curveshouldoccurforflowsbelowthebubble-pointpressure,withtheabsoluteopenflow
potentialbeingevenlessthanthatindicated bytheextrapolationonFig.22.
CHANGEINPERFORMANCECURVESWITHDEPIETION
Perhapsthebiggestimpedimenttoan earlier
developmentofmultipointtestingof oil
wellswastherealizationthatawell's
performancecurvechangeswithchangingoil
saturationandpressureinacomplexmanner. Standing27
extendedtheutility ofIPR equation(performancecurve)byillustrating
asimplemethodtocorrectaknownIPRcurve
positiontosomefuturepositionasaresult
ofachangeinkThefuturevalueofkin roro uB o0
hisexamplewasobtainedfromaTamermaterial
balancecalculationusingaCorey-type
correlationforkTheinabilitytodefine
arealkcurveroforaspecificwellstill
makesthfgapproachonlyapproximate. It
hasbeenobservedthatinmanymaterial
balancecalculationsforsolution-gasdrive
reservoirs,kisapproximatelylinear with
reservoirAsanapproximationtothe changeinoilpermeabilitywithpressure
depletionwecouldthenwrite (30) or k ro -=% =--PRi (31) wherekrois
withrespecttoki andis defined atavanishingt.p,zerodrawdown.%iisalso
tobeequalto bubble-pointpressure. orlessthanthe Thenkro(PR)plotted
: t; "' .. !1. , "'O' 'G. OR w' PSIGPStG 3403.73394.9 l082.8
3387.23376.5 3391,23062.6 3394.13199.6 3405,53306.5 3402.03385.9
3398.73393,3 2482 1000 STKSOPO ~,-1398 ." '" 4 4 _2 PR= 3420.2 pSia
I AOfP - 7800SOPO n = 0.59610,000 GOOSEP. PRESS. SCF/STKSBLpS'G '"
--,-.--." ,]5 "
'" :I: t: 'T N_ "' 2 1"-2 "-No: ,"-10001000 ..8 88 44
oISOCHRONAL FLOWAFTERFLOW 2 100 24 100 10010.000 Fig.22-
Four-hourisochronalperformancecurveofWellI-a, FieldG,Jan.18.1972.
18 16 14 12 10 'en 8 RESULTS FROMBUILD-UP ANALYSIS FLOW STKqSOPO K
NO.MO ....,. 2i6 2120 ,236 '" 2Z2 5001000 S' + 9.ff 6.65 5.08 '.U
CALCULATEDPARTIALPENETRATIONSKIN. Sb-+2.5(REf.26) 15002000 qo
-STKBOPD 250030003500 8 6 4 2 10,000 8 6 4 2 oISOCHRONAL pSI'
AOFP-I3,300BOPD;I nO.1IZ,/ I' FLOWAFTER FLOW BUBBLEPOINT PRESSURE=
5035 pSI.
Fig.23_Non-Darcyfloweffect,single-phaseliquidflow,Weill-a,FieldG.
1000 8 6 FLOWlJRPwf:!1.< SEP. PREss. PSIG
e PRLPwf2) .,J'ei""=TO 6SBL/DAY/PSI2 4 2 10,000 e e 4 2 1000 e
II 4 2 DRAW DOWNATINITIAL CONDITIONS(PRi=2100PSI ... ) VOGE'Q;IG.7
PSIA(THOUSANDS)BBL/DAYDAY Ii"Oii"BOO --,,- --,-,-1400102
VJ1000J.HO140 o6004050165 Z04410184 OEPLETION MAXAT
:;)(.!a.),-P')EQ.32VOGELFIG.7 oPR\:P"R iRqoqo :t:PSI ...
(THOUSANDS)BeL/DAYBBL/DAY t- """'i'iSO--,,-,- '''' ..... N 'i Q. I
N", IQ. '-' r--. "'I.-IQ. '---" '.50 '50. 12SO""21]9 1607 .,. 34' "
. .. 57 " no .. n =1.000