development of hubbert’s peak oil theory and analysis of ... · a senior essay presented to the...
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DevelopmentofHubbert’sPeakOilTheoryandAnalysisofitsContinuedValidityforU.S.CrudeOilProduction
TorrenPeeblesProfessorJayAgue&ProfessorMichaelOristaglio
5May,2017
ASeniorEssaypresentedtothefacultyoftheDepartmentofGeologyandGeophysics,YaleUniversity,inpartialfulfillmentoftheBachelor'sDegree.
In presenting this essay in partial fulfillment of the Bachelor’s Degree from theDepartmentofGeologyandGeophysics,YaleUniversity,Iagreethatthedepartmentmaymakecopiesorpostitonthedepartmentalwebsitesothatothersmaybetterunderstand the undergraduate research of the department. I further agree thatextensive copying of this thesis is allowable only for scholarly purposes. It isunderstood,however,thatanycopyingorpublicationofthisthesisforcommercialpurposesorfinancialgainisnotallowedwithoutmywrittenconsent.
TorrenPeebles,5May,2017
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ABSTRACT
InthispaperwereviewthedevelopmentofM.KingHubbert’sbasic
petroleumresourcedepletionmodel,theHubbertcurve,thatbecamethefoundation
forthevarietyofcurvefittingtechniquesthatarestillwidelyusedtoday.Wethen
drawoncurrentliteraturetoassessthestrengthsandweaknessesofsuchmodels,
beforeassessingtheirvalidityforcontinueduseintheUnitedStatesandanalyzing
whetherornotU.S.crudeoilproductionstillfollowsthecurvethatHubbertgained
infamyfor.Specifically,wefocusontheimpactthatadvancesinpetroleum
productiontechnologysuchashydraulicfracturingandEnhancedOilRecovery
(EOR)techniqueshavehadontheU.S.productioncycleandtestderivationsof
Hubbert’soriginalmodelthatmightbetterapproximatetheimpactstheseadvances
havehad.Ourresultsdemonstratethatwhilecurve-fittingtechniquesremainuseful,
theoriginalHubbertcurvenolongerapproximatestheU.S.productioncyclewell.
M.KINGHUBBERT&PEAKOILTHEORY(1956)
ATexasnative,MarionKingHubbertobtainedhisB.S.fromtheUniversityof
Chicagoingeologyandphysicswithamathematicsminorin1926(Narvaez,1989).
HewouldcontinuehisgraduateworkintheUniversity’sDepartmentofGeology,at
thetimerenownedasoneofthenation’sbest,earninghisM.S.in1928,workingona
theoreticalstudyofthermodynamicprocessescontributingtogeologicfaults
(Narvaez,1989).Hubbertgarneredappliedgeophysicsexperienceworkingasan
assistantgeologistfortheAmeradaPetroleumCorporation(Narvaez,1989).After
brieflyreturningtotheUniversityofChicagotopursuehisPh.D.andworkasa
teachingassistant,HubbertwentontoacceptafacultypositionatColumbia
Universityin1931wherehetaughtgeophysicsandcompletedhisPh.D.in1937.
Althoughcolloquiallybestknownforhistheorizationonpeakoil,itwasduringthis
periodthatHubbertpublishedhisseminalpaperTheoryofGroundWaterMotion,
demonstratingthatgroundwaterflowisdeterminedbybothgravityandfluid
pressure.ThepaperprovidedaphysicalinterpretationofDarcy’sLawusinganew
derivationoftheNavier-Stokesequation.Previouslythoughttoremainstaticin
theirreservoirs,thepublicationdrasticallyinfluencedthinkingonhowgaseousand
2
liquidhydrocarbonsweretransportedthroughporousmedia(Priest,2014).During
WorldWarII,afterleavinghispositionatColumbiain1941,Hubbertservedasa
senioranalystfortheUnitedStatesBoardofEconomicWarfare(Narvaez,1989).As
chiefoftheBoard’sMineralResourcesunit,HubbertoversawanalysisoftheAllied
wareffort’sglobalnaturalresources,onethemostcriticalofbeingpetroleum
(Priest,2014).
WorldWarIwasvitaltoestablishing,advancingandstandardizingthe
scienceofestimatingpetroleumreservesandresources.Theadventofmotorized
warfareshiftedtheviewofpetroleumasacommoditysuppliedbyasmallgroupof
entrepreneurialprospectorstoanationalresourcefundamentaltovictoryin
modernwar(forperspective;whenBritainenteredWorldWarIin1914their
militarypossessedonly800motorvehicles,fouryearslateratthewar’send,they
had56,000trucksand36,000cars(ELC,2015).In1916theUnitedStatesGeological
Survey(USGS)establishedtheirOil&GasSection.TheSectionappliedmethodssuch
asdepletioncurvestotrackproductionandreservesremaininginknownfields,and
usedtheirdatatoaidtheU.S.militaryinplanningthewareffortandalsoadvisethe
U.S.governmentonapplyingtaxestopetroleumproduction(Priest,2014).
Throughoutthistimeperiodandthroughthemid1940s,theUnitedStates
accountedforroughly65%ofworldpetroleumproduction(ELC,2015).
Necessarily,thefirstattemptsatquantifyingthenation’sreserves,resources
andproductionratesarosefromtheseanalyses.In1919theUSGSOil&GasSection
estimateddomesticpetroleumreservestobe6.74billionbarrels(Priest,2014).
ChiefgeologistandheadoftheOil&GasSectionDavidWhiteestimatedthesupply
wouldlastthecountry17-18yearsextrapolatingfromcurrentconsumptionrates
(Priest,2014).ArevisedestimateofU.S.petroleumresources,thistimeincluding
newproductionfromfieldsinCalifornia,wasissuedintandemwiththeAmerican
AssociationofPetroleumGeologists(AAPG)in1921(Priest,2014).Withoutseismic
imagingtechnologythataroseinthemid1920s,accurateassessmentofthenation’s
oilfieldswasnearimpossible,andtheseearlyestimatesofdomesticresourcesand
peakproductionprovedwildlylow.
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FollowinghisworkfortheBoardofEconomicWarfare,Hubbertwas
employedasaresearchgeophysicistbyShellPetroleumCompanyfrom1943until
1964,anditwasduringthistimeperiodthathepresentedhisearlyworkonpeak
oil(Narvaez,1989).In1949HubbertpresentedhisEnergyfromFossilFuelsaspart
oftheSymposiumonSourcesofEnergyinWashingtonD.C.,examiningannualworld
productionovertimeforbothcoalandpetroleum.Itisinthispreliminarypaper
thatweobserveseveralaspectsofwhatwouldleadtotheinfamousHubbertcurve
andhispredictionsregardingpeakoil.Hubbert’s1949workfocusedontherateof
increaseofproductionofbothcoalandpetroleum(neglectingnaturalgasforwant
ofworldproductionstatistics),notingthatannualproductionofcoalsince1913had
growngeometricallyatarateof4%peryearsince1913(implyingthatannual
productiondoubledevery17years),whilefrom1860to1929worldcrudeoil
productionhadgrowngeometricallyatarateof9%ayear(implyingthatannual
productiondoubledevery7.5years)(Hubbert,1949).Pairingtheseobservations
withwhathedeemed“oneofthemostdisturbingecologicalinfluencesofrecent
millennia”thatwas“thehumanspeciesproclivityforthecaptureofenergy,
resultinginaprogressiveincreaseinhumanpopulation.”(Hubbert,1949).Hubbert
surmisedthatthattheamountofanyfossilfuelconsumedatanygiventimewould
beproportionaltotheareaunderneathacurvesimilartothosewhichhehad
presented,concluding:
Thuswemayannouncewithcertaintythattheproductioncurveof
anygivenspeciesoffossilfuelwillrise,passthroughoneorseveralmaxima,
andthendeclineasymptoticallytozero.Hence,wholethereisaninfinityof
differentshapesthatsuchacurvemayhave,theyallhavethisincommon:
thattheareaundereachmustbeequaltoorlessthantheamountinitially
present.(Hubbert,1949)
Hubbertwouldwaitanother7yearstospeculateonwhatthiscurvemightlooklike,
althoughhewouldcontinuethepaperbyominouslydiscussinghydropower’sability
tomeetcurrentenergydemandandhumanexistenceonanabsolutetimescale.
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In1956,M.KingHubbertpresentedNuclearEnergyandtheFossilFuels
beforetheSpringMeetingoftheSouthernDistrictDivisionofProductionofthe
AmericanPetroleumInstitute(API).Usingpetroleumproductiondatafromthe
world,thecontinentalUnitedStates(U.S.L48)andTexas,Hubbertcreatedgraphs
foreachinthesamemannerashehaddoneinhis1949publication,plottingrateof
productionovertime.Whileinitialratesofproductioninitiallyincreasedquite
rapidly,ashadobservedforbothcoalandcrudeoilin1949,henotedthatthis
productiongrowthwasquiteobviouslyunsustainableinthatphysicallimits
preventedproductionofafiniteresourcefrombehavingassuchsooverany
prolongedperiod(Hubbert,1956).Tomoreeffectivelyextrapolategrowthcurves
forhisobserveddatasets,Hubbertconcludedthatpetroleumdepletioncouldbe
modeledasfunctionofcumulativeproductionunderthepremisesthat:
(1)Foranyproductioncurveofafiniteresourceoffixedamount,twopointa
onthecurveareknownattheoutset,namelythatatt=0andagainatt=∞.
Theproductionratewillbezerowhenthereferencetimeiszero,andtherate
willagainbezerowhentheresourceisexhausted.
(2)Thesecondconsiderationarisesfromthefundamentaltheoremofthe
integralcalculus;namely,ifthereexistsasinglevaluedfunctiony=f(x),then
𝑦𝑑𝑥 = 𝐴 !!
!
whereAistheareabetweenthecurvey=f(x)andthex-axisfromtheorigin
outtothedistanceisx1.
Inthecaseoftheproductioncurveplottedagainstthetimeonan
arithmeticalscale,wehaveastheordinate
𝑃 = 𝑑𝑄/𝑑𝑡
wheredQisthequantityoftheresourceproducedintimedt.Likewise,from
equation(1)theareaunderthecurveuptoanytimetisgivenby
𝐴 = 𝑃𝑑𝑡 = 𝑑𝑄𝑑𝑡 𝑑𝑡 = 𝑄
!
!
!
!
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whereQisthecumulativeproductionuptothetimet.Likewise,theultimate
productionwillbegivenby
𝑄!"# = 𝑃𝑑𝑡!
!
andwillberepresentedonthegraphofproduction-versus-timeasthetotal
areabeneaththecurve.(Hubbert,1956)
Thegraphicalrepresentationofthesebasicrelationshipsmanifesteditselfin
theformofabell-shapedcurvethatHubbertdidnotdefinemathematicallyinhis
1956publication.Itisspeculatedthathedrewthecurvebyhand,calculatingthe
areabeneathittoarriveathisoriginalultimatereserveestimates(Fig.1).
Fig.1Hubbert’sproposedmathematicalrelationshipencompassingthecompleteproductioncycleof
afiniteresource(Hubbert,1956)
Hubbert’scurveapproachwasbeautifullysimpleinthatitrequiredonlyone
knownvariable,Qmax,tofithiscurvetoadatasetandextrapolateratesof
productionovertime.ThevariableQmax,representingultimateproduction,is
synonymouswithUltimatelyRecoverableResources(URR),theapproximated
cumulativequantityofhydrocarbonthatcanbeeconomicallyextractedfroma
reservoiroveritsproducinglifespan.Inhis1956calculations,HubbertusedaURR
of1250billionbarrelstofithisworldproductioncurve(Fig.2).Hubbertfit
domesticproductioncurvesforbothaconservativeestimateof150billionbarrel
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EURaswellasaslightlymoreoptimistic200billionbarrelEUR(Fig.3),usingan
EURof60billionbarrelsforTexas(Fig.4).
Fig.2Hubbert’sprojectionforultimateworldcrudeoilproduction,assuminga1250billionbarrel
EUR(Hubbert,1956).
Fig.3Hubbert’sprojectionforultimatecrudeoilproductionfromtheLower48UnitedStates,the
bottomcurveassuminga150billionbarrelURRandtheuppercurveassuminga200billionbarrel
EUR(Hubbert,1956).
Fig.4Hubbert’sprojectionforultimatecrudeoilproductionfromthestateofTexas,assuminga60billionbarrelEUR(Hubbert,1956).
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Hubbert’sconceptofamomentofpeakproductionimplicittothesingle
curvehehadappliedtohisdatasetswasnotonethathadbeenconsideredin
previouspetroleumresourceassessments(Priest,2014).Thiswoulddevelopinto
whatbecameknownasthetheoryofpeakoil,whichbynatureofHubbert’sbell-
shapedsymmetricmodeloccurredwhenexactlyhalfoftheultimatereservehad
beenproduced.His1956papercalculatedthatworldpeakproductionofcrudeoil
wouldoccurintheyear2000,withamaximumrateofproductionofroughly12.5
billionbarrelsannually(Hubbert,1956).ForU.S.Lower48EURsof150billion
barrelsand200billionbarrels,Hubbertpredictedpeaksin1965and1970,
respectively,withmaximumpeakproductionratesofroughly2.7billionbarrels
annuallyand3billionbarrelsannually(Hubbert,1956).Hubbertwasintriguedby
thefactthatevenwhendomesticEURwasincreasedby33%from150billion
barrelsto200billionbarrelsdelayedtheoccurrenceofpeakoilbyonly5years
(Hubbert,1956).Notingthattheversatilityandeaseofextractionofliquidand
gaseoushydrocarbonproductshadresultedinarateofproductiondisparateto
theirmagnitude,Hubbertrathersensationallypredictedthatshouldtheworld
continuetousefossilfuelsastheirprimaryenergysource,that“Onthebasisofthe
presentestimatesoftheultimatereservesofpetroleumandnaturalgas,itappears
thattheculminationofworldproductionoftheseproductsshouldoccurwithin
abouthalfacentury,whiletheculminationforpetroleumandnaturalgasinboth
theUnitedStatesandthestateofTexasshouldoccurwithinthenextfewdecades.”
(Hubbert,1956).
Hubbertconsideredpeakproductiontobeacriticaleventinthelifecycleof
anexhaustibleresourceandbelievedthatthispointwouldhavesignificant
implicationsforthemannersinwhichtheywouldbeproducedandconsumed,
exacerbatedbytheincreasingdemandforenergywouldcomewithagrowing
population(Priest,2014).Hewouldtouchbrieflyontheconsequencesalarge
energyimbalanceheldforbothdomesticindustryandnationaldefensepurposes
beforegoingontoassessalternativeenergyoptionsashehaddoneinhis1949
paper,thistimefocusingonenergyfromnuclearsources(Hubbert,1956).AsTyler
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Priestcommentsinhis2014paperHubbert’sPeak:theGreatDebateOvertheEndof
Oil,“WhatwassoshockingaboutHubbert’sprojectionswasthatitofferedaunique
andintuitiveinterpretationofwidelypublisheddatathatoverturnedconventional
wisdom.Thatwisdomheldthatpetroleumresourceswereplentiful,notpoisedfor
decline.”(Priest,2014)Hubbertwaswellawell-respectedpetroleumgeologistand
geophysicist,hisworkheldsignificantweightintheoilindustryandthegravity
evenhispreliminaryworkheldindependentlyestablishedpeakoiltheoryasawell
known,ifcontentious,topicthatwouldbethesubjectofdebateinthefieldof
petroleumresourceassessmentandbeyondfordecadestocome.
HUBBERT’SCONTINUEDWORK
Followinghisinitial1956work,Hubbertwouldcontinuetorefinehis
methodsofanalysis.Mostimportantly,inhisTechniquesofPredictionWith
ApplicationtothePetroleumIndustrypresentedatthe44thAnnualMeetingofthe
AmericanAssociationofPetroleumGeologists(AAPG),heassignedmathematical
functiontothebellcurvehehadfittohistoricaldatainhisearlierwork.Hubbert
assertedthatcumulativeproductionovertimecouldbebestapproximatedbythe
logisticgrowthfunction,withthefirstderivativeofthisfunction,representing
annualrateofproduction,takingthesameformasthecurvehehaddrawnin1956
(Hubbert,1959).
Inthesamepaper,Hubbertproposedthatmagnitudeofultimatereserves
couldbereachedindependentofindustryoracademicexpertspeculationby
examiningtherelationshipbetweencumulativediscoveries,productionandproved
reserves.Atthetimeofhis1956publication,Hubberthaddeclinedtomakehisown
ultimatereservecalculations,insteadusingpreviouslycalculatedindustryestimates
tomakehispreliminaryassertions.Inthe1950s,bestpracticeinestimating
ultimatereserves(usedbyboththepetroleumindustryandtheUSGS)wasa
methodknownasvolumetricyieldanalysis(Priest,2014).Volumetricyieldanalysis
entailedaveragingoilyieldproducedfromaunitvolumeofsedimentinproducing
basinsandassumingthatasofyetundrilled,geologicallysimilarbasinswould
producethesameamountperunitvolumeofsedimenttoextrapolateultimate
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reserves.Thisformofanalysisbothassumedcomparableyieldperunitvolumeof
sedimentwithoutscientificjustificationfordoingso,andhingedonsubjective
measuresofgeologicsimilarity(Priest,2014).Hubbertrealizedthathismodel’s
dependenceonaninherentlyunreliablevariablewasasignificantflaw,anddevised
amathematicalmeansbywhichtoascertainwhatheviewedasamorereliable
estimateindirectly.
Underthepremisethatcumulativediscoveries(QD)foranytimetisthesum
ofoilthathasalreadybeenproduced(QP)andoilthathasbeendiscoveredbutnot
yetproduced,(i.e.reserves,representedbyQR),or:
𝑄! = 𝑄! + 𝑄!
Hubbertalsoascertainedthatthecurveofcumulativeproductionandthecurveof
cumulativediscoverywouldbeofverynearlyidenticalshape,andhavethesame
asymptote,withcumulativeproductiontrailingcumulativediscoverywithalagin
timethatherepresentedwithΔt(7).Withthederivativeofthisequationwith
respecttotime(t):𝑑𝑄!𝑑𝑡 =
𝑑𝑄!𝑑𝑡 +
𝑑𝑄!𝑑𝑡
Notingthatwhenprovedreservesreachtheirmaximum,theirrateofincrease
wouldbeequaltozero,settingtherateofdiscoveryequaltotherateofproduction
atthistimeandimplyingthattheascendingrateofproductioncurvewould
intersecttherateofdiscoverycurvewhichwouldhavebeguntodecrease:
𝑓𝑜𝑟 𝑑𝑄!𝑑𝑡 = 0,
𝑑𝑄!𝑑𝑡 =
𝑑𝑄!𝑑𝑡
Withthesemathematicalfoundationsinplace(graphicalinterpretationsbelowin
Fig.5,6),Hubberthadestablishedwhatwouldcometobeknownasthebasictools
ofmodernHubbertanalysis(Hubbert,1959,Brandt,2006).
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Fig.5(LEFT)Hubbert’stheoreticalsingle-cyclegrowthcurvesforcumulativediscoveries,production,andprovedreserves(Hubbert,1959).Fig.6(RIGHT)Hubbert’stheoreticalrateofdiscoveryandproduction,rateofprovedreserveincrease(Hubbert,1959).
Hubbertwouldgoontoapplythemethodsdescribedinhis1959publication
tosubstantiatehisclaimsinareportsubmittedtotheCommitteeofNatural
Resourcesin1962whileservingaschairofthesubcommitteeonenergy.Using
crudeoilproductiondatafrom1860onwardsandestimatesofU.S.Lower48
provedcrudeoilreservessincefrom1937onwardpublishedannuallybytheAPI,
HubbertcalculatedatimelagbetweenU.S.cumulativeproductionanddiscoveryof
10-11years(bestfitforΔtwas10.5),assertingthatdiscoveryratehadreachedits
peakin1956whileprojectingthatpeakproductionratewouldpeakbetween1966
and1967withultimatecumulativeproductionof175billionbarrels(Hubbert,
1962).Consideringacontingencyallowanceofanadditional50billionbarrels,
bringingthetotalto225billionbarrels,Hubbertclaimedthisadditionwouldonly
pushpeakproductiontotheearly1970s.Hubbert’scontentiouspredictionsfor
peakcrudeoilproductionoftheU.S.Lower48werevalidatedasproductionrates
peakedin1970anddeclinedeveryyearsubsequently.However,whileHubberthad
successfullypredictedthetimingofU.S.peakoil,his1962curveindicatingapeak
productionrateofroughly2.7billionbarrelsannuallyfellmateriallyshortofthe
actual1970productionrateof3.5billionbarrelsperyear(Hubbert,1962,EIA).
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ESTABLISHINGTHEPARAMETERSOFCURVEFITTINGTECHNIQUESANDTHE
UNCERTAINTYTHEYINTRODUCE
Whileitisbeyondthescopeofthispapertorigorouslymathematicallyjustify
theuseofcurvefittingtechniquesversusothermodelsthathavebeendevelopedfor
theanalysisofpetroleumresourceproductionandconsumption,itisimportantto
bothacknowledgetheirlimitationsaswellasestablishsoundparametersforwhen
theirapplicationisbothvalidanduseful.
UltimatelyRecoverableResource(URR)
UltimatelyRecoverableResource(URR)isdefinedbySorrelletal.(2010)as
thesumofcumulativediscoveries,futurereservegrowthatknownfieldsandthe
volumeofoilestimatedtobeeconomicallyrecoverablefromfieldsthathavenotyet
beendiscovered(Sorrelletal.,2010).Inherently,Hubbert’smodelreliesonaccurate
estimationofURR.Becauseultimateproduction(areaundertherateofproduction
curve)delineatesprojectedproductionovertime,ithasalargereffecton
predictionsthemodelmakesthananyothervariable(Brandt,2006).URRestimates
canbeeitherexogenousorendogenous.
Fig.7BreakdownofUltimatelyRecoverableResources(Sorrelletal.,2010)
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EndogenousEstimatesofURR
Asmentionedearlier,HubbertdevisedhisownmethodofcalculatingURRso
thathismodelscouldbeindependentofunreliable(andinsomecasespolitically
motivated)estimates.Whileatthetimethiswasareasonableandnecessary
assumptiontomake,asmethodsofgeologicestimationhavebecomemorecertain
withimprovingtechnologyendogenousestimatesareincreasinglylessreliedupon.
Hubbert’smethod,whichextrapolatespeakproductionrateandtimeofpeak
productionfromtheinflectionpointofthecumulativeproductioncurvehasbeen
showntobeunreliable.AsLaherrere(2000)demonstratesbycomparingGaussian
andaHubbertcurves,theGaussianinflectionpoint(whichoccursearliercompared
tothatoftheHubbertcurve,resultedinapeakrate10%higherandaURR33%
largerthanthatofaHubbertcurvemodelingthesamedataset(Laherrere,2000).
UtilizationoftheinflectionpointincalculatingURRisalsoproblematicinthatit
cannotbeappliedtoacumulativeproductiondatasetthathasyettoreachan
inflectionpoint,limitingthepredictivepowerofthesemodelsearlyinthe
productioncycle.
ExogenousEstimatesofURRandReserveAdditions
ExogenousURRestimatesaremorecommonlyusedinstudiesemploying
curve-fittingtechniquesandactasalimitingexternalvariableindependentofthe
actualcurvefittingprocess(Wang,Feng,2016).Theseestimatesaregenerally
sourcedfromnationalorglobalgeologicauthoritiessuchastheUSGSorEIA.These
estimatespresenttheirownaspectsofuncertaintytocurve-fittingtechniquesin
thathistoricallytheyhaveunderestimatedfutureadditionsduetonewdiscoveries
andreservegrowth(Brandt,2009).Reservegrowthreferstothetendencyof
estimatesforindividualfieldstoincreaseovertimewithouttheadditionofnew
discoveries,andcontributestothemajorityofreserveadditionsinmostregionsof
theworld,andreservegrowthintheUnitesStatescontributedto89%ofproved
reserveadditionsbetween1978and1990(Sorrelletal.,2012).Thiseffectis
attributabletoanumberofdifferentfactors,includingimprovinggeologic
understandingofanarea,improvingtechnologyanddefinitionalfactors(Sorrellet
13
al.,2012).Whiletechnologicalimprovementsinseismicimagingtechnologycan
contributetoreservegrowthinolderfieldswheretheywerenotusedinoriginal
delineationoftheplay,inthefuturetheywilllikelymitigatereservegrowth
attributabletochanginggeologicunderstandingofaplayasnewdiscoverieswillbe
betterunderstoodinthefirstplace.
Improvingtechnologyhasplayedarollinreserveincreasesduetobothother
factorsasithasimprovedtheaccuracyofseismicsurveytechniquesaswellas
cheapenedtheextractionprocess,influencingthedefinitionofwhatcanbereferred
toaseconomicallyextractable.URRestimatesarealsopronetoincreaseovertime
astechnologyimprovesmethodsoffindingandextractinghydrocarbon,and
fluctuateasoilpricedictatesplaysthatareeconomicallyextractable.Whilethisis
potentiallyproblematic,factorsthathaveonlylongtermimpactssuchas
improvementsintechnologywillactmoresignificantlytoslowrateofdeclineafter
peakproductionisreachedandwillhavenegligibleimpactsontimingofthepeak
(Sorrelletal.,2010).
Reservegrowthfunctionshavebeenemployedasameansofpredicting
furtherreservegrowthandrelyontheuseofmeasuredgrowthofsamplefieldsto
maketheirforecasts.TypicalU.S.1Pdatasetsexhibitrapidgrowthimmediately
afterdiscoverywithgrowthslowingovertime,andoldfieldsstillrecordgrowth
after80yeartimeperiods(Sorrelletal.,2012).LikeHubbert’smodel,these
functionsneglecteconomicvariables,butthereasonfortheirgrowthvariesoverthe
productioncycleoftheplay:growthrecentlyafterfirstproductionisheavily
influencedbyadditionstoOriginalOilInPlace(OOIP),withlateradditionsmore
commonlyattributedtochangingrecoveryfactors(Sorrelletal.,2012).
Unfortunately,reservegrowthfunctionshavebeenfoundtobeunreliablewhen
projectingpotentialofawiderangeofareas(McGlade,2012).Calculatedratesof
reservegrowthremaincontentious,butitisclearthatthiseffectintroduces
significantuncertaintytoexogenousestimatesutilizedbycurvefittingtechniques.
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AssumptionsoftheHubbertModelandtheirrelevancytoU.S.L48
Intheory,theUnitedStatespresentanearlyperfectsamplewithwhichto
testthecontinuedviabilityofHubbert’sproposedmodel.AsSorrell&Speirs(2010)
note,extrapolationtechniquesaremostjustifiablyapplicabletogeologically
homogenousareaswhereexplorationhascontinuedrelativelyunimpededover
time:theU.S.L48fitsthiscriteriawell.Additionally,theU.S.haswelldocumented
petroleumproductionanddiscoverydatasetsthrough1859thatarepublicly
availablethroughtheU.S.EnergyInformationAdministration.Wemaintainthat
curvefittingtechniques’utilizationofpubliclyavailableaggregatedata,oftenthe
mostgranularavailable,remainsahugebenefittotheircontinueduse.Untilvery
recently,Hubbert’smodelfitU.S.dataincrediblywell.WhiletheUnitedStates
discoverycurveimitatesthesymmetricnaturethatHubbertdescribesbestofany
majorpetroleumproducingcountry,Sorrell&Speirs(2010)attributethisfittothe
limitedearlydrillingcapabilitiesandexplorationandfirstproductionoccurringin
relativelyminorresourceplays,withmajorbasinsbeingdiscoveredslightlylater.
TheyalsonotethatwhileHubbbert’sassumptionthattheresourcediscoverycycle
trailsproductioncyclesymmetricallyisnotmathematicallywellsupported,U.S.
datasetmimicsthisbehaviorwell(Sorrell&Speirs,2010).
15
Fig.8U.S.provedreserves,cumulativediscoveryandproductionmimicsHubbert’spredictionswell(EIA,2017).
16
Fig.9U.S.crudeoilproductionovertime.ProductionwithouttightoilresemblesHubbert’scurvewell,butexploitationoftightoilresourceshasresultedinasignificantdeviationfromthismodel.
MODERNHUBBERTANALYSIS
AwidevarietyoftechniquesM.KingHubbertdirectlyorindirectly
contributedtothedevelopmentofarestillappliedtodayinvariousforms,withthe
generalframeworkhecreatedforresourceassessmentbroadlyreferredtoas
Hubbertanalysis.Thesemethodsareoftencharacterizedbytheirproductionofa
single-valueestimateobtainedbyextrapolatingoneormultiplecurvesfittohistoric
productionordiscoverydatafromcountry-levelregionswhereonlyaggregatedata
isavailable(Sorrell&Speirs,2010).WefirstgenerateourownsingleHubbertCurve
estimatebeforeassessingtwootherclassesofmodelsthathaveevolvedfromit.
HereweexaminetwoaspectsofHubbert’smodel:1.)Increaseanddecreasein
productionratesapproachingandfollowingpeakproductionareroughly
symmetricalandwellapproximatedusingabellshapedcurve.2.)themonocyclic
natureofresourceproduction.
Fig.10U.S.productionoverlainwitha261GbHubbertcurve,theURRestimategeneratedbyourmodel.
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AsymmetricCurves
Hubbert’sassumptionthatproductionrateoverthelifecycleofaresourceis
symmetrichasbeencriticizedforneglectingtheinteractionofgeologic,technical,
economicandsocialparametersandithasbeenpositedthatitisunreasonableto
expectthatthiswillscenariowillbeapplicabletoawiderangeofproductioncycles
(Wangetal.,2016).Asymmetrycanbeexplainedbythefactthatoperatorsare
motivatedtoincreaseproductionwhileminimizingdeclineratestomaximizepost-
peakprofitviamethodssuchasEnhancedOilRecovery(EOR)(Wangetal.,2016).
EORtechniquesareoftenappliedinplayswithheavyoilthatexhibitpoor
permeabilityandincludechemicalflooding,gasinjection,andthermalrecovery
methodsthatalterthemakeupofthereservoir.Whileexpensivetoapply,theyare
increasinglypopularandhavethepotentialtomorethandoublecurrentaverage
domesticoilrecoveryof30%(DOE,2017).IncreasingapplicationofEORcould
sustainpost-peakproductionofcurrentlyproducedreservesanddampenrateof
decrease.
Reviewingproductiondatafrom139regions,Brandt(2006)foundthat
productionwassignificantlyasymmetricinonedirection:medianrateofincrease
was7.8%wholemedianrateofdeclinewasfoundtobe2.6%(Brandt,2006).
However,Brandt(2006)alsonotesthattheseasymmetricmodelsareespecially
difficulttofittopastproductiondatawhenapeakinproductionisnotyetevident:
Hubbert’smodelascertainsthatdeclinerateismostsimplyapproximatedwhen
assumedtobeequaltothatoftherateofincreaseifthereisnoinformationto
warranttheuseofamorecomplexapproach(Brandt,2006).Herewetestan
asymmetricmodelbasedonaGaussiancurve,describedby:
𝑃 𝑡 = 𝑃!"#𝑒!(!!!!"#$)!/!(!(!)!
wheref(t)isthesigmoidfunctionthatchangesthestandarddeviationinthe
vicinityoft=Tpeakdescribedby:
𝑓 𝑡 = 𝜎!"# −𝜎!"# − 𝜎!"#
1+ 𝑒! !!!!"#$
18
whereP(t)isproductioninyeart,Pmaxismaximumproduction,Tpeakisthe
yearofpeakproduction,𝜎!"# isthestandarddeviationofthepre-peak
productioncurveand𝜎!"# isthestandarddeviationofthepost-peak
productioncurve(Brandt,2006).
Fig.11AsymmetricGaussiancurvefittoU.S.L48productiondata.
Multi-CyclicModels
Curvefittingtechniques,especiallymonocyclicmodelssuchastheHubbert’s,
relyontheassumptionthatfutureexplorationcycleswillnotoccur,oriftheydo
theywillhaveanegligibleimpactonaggregateresourceproductionduetotheir
relativesize(Sorrell&Speirs,2010).Thisisprimarilyanissueincountrieswhere
politicalinstabilityorgeographicinaccessibilityhaslimitedextentorcontinuityof
exploration.Forwell-delineatedregionswheretheexplorationcycleisrelatively
welladvanced(liketheUnitedStates),thisisassumptionappearedvalid.However,
hydraulicfracturinghasdrasticallyimpactedtheU.S.L48productioncycle.Herewe
19
arguethattheadventofhydraulicfracturingintheUnitedStatesnecessitatesthe
useofamulti-cyclicmodelforproductiondata.SincetheEIAbegantrackingtightoil
productionin2000,ithasaccountedfor25%ofallcrudeoilproducedintheU.S.
L48between2000and2016,andaccountedfor51%ofcrudeoilproducedinthe
U.S.L48in2016(EIA,2017).Examinedasanentityseparatefromthatofthepurely
conventionalresourcesHubbertappliedhisanalysisto,cumulativetightoil
productioncorrespondswiththepostulatedlogisticgrowthfunctionextremelywell
(URR48Gbassumed)(EIA,2013)(Fig.12).TightoilintheUnitedStatesisprimarily
sourcedfromtheBakkenshaleformationofNorthDakotaandtheEagleFordshale
formationofsouthernTexas,withBakkenproductionresponsibleforanincreaseof
2.2millionbarrelsofoilperdayin2013(Murray&Henson,2013).
Fig.12U.S.tightoilproductionovertimefitsthelogisticcurveHubbertpostulatedtoapproximate
cumulativeproductionovertimequitewell(EIA,2017).
Laherrere(2000)introducedavariantoftheHubbertcurvegenerallyknown
asmulti-cyclemodels,whichfitanumberofHubbertcurvestoproductiondata.In
20
general,thesemodelswillprovideabetterfittoproductiondata,howeverthiscan
sometimesbearesultof“over-fitting”.Theoretically,eachcurveisrepresentativeof
anew,well-definedresourcethatiseasilydistinguishablefromtheoriginalplay,but
inpracticetheseadditionalcyclescanalsoresultfromimprovementinexploration
andproductiontechnologysuchasEnhancedOilRecoveryTechniques(EOR)or
hydraulicfracturing(Brandt,2009).Hubbert’ssinglecurvemodelfitU.S.L48
productionrelativelywelluntilwidespreaduseofhydraulicfracturing(fracking)
beganintheearly2000’s.ProductionovertimeusingLaherrere’smulti-cycle
modelistypicallydescribedas:
𝑄 𝑡 = 𝑄(𝑡)!
!
!!!
=2𝑄!"#
1+ cosh [𝑏(𝑡 − 𝑡!)
!
!!! !
wherekrepresentsthetotalnumberoflogisticcurvesfittoproductiondata,
andQmaxandtmrepresentpeakproductionandtimeofpeakproduction
respectivelyforeachcycle(Wangetal.,2011).
Laherrere’smulti-cyclecurveshavepreviouslybeenappliedtoregionssuch
asFranceandIllionoiswhoseproductioncyclesfollowsignificantbimodaldiscovery
trend(Brandt,2006,Laherrere,2005).HereapplyingLaherrere’smulti-cyclic
model,weusetwocurves,onerepresentingproductionofU.S.L48conventional
resourcesovertimeandrepresentingU.S.L48productionoftightoilovertime.
Summingthesecurves,wearriveatamulti-cyclicmodelthatcorrespondswith
actualproductionquitewell.Weusedthe219Gbestimateourmodelcalculatedfor
conventionalURRandatightoilURRof48Gbestimatefortightoil(EIA,2013).
21
Fig.13U.S.productiondatafitwithtwocurves,thefirstapproximatingconventionalcrudeoilproductionandthesecondapproximatingtightoilproduction.
22
Fig.14U.S.productiondatafitwiththecurvesummedfromthetwoseparatecurvesdisplayedinFig.13,thisrepresentsLaherrere’sMulti-Cyclicmodel.
Lastly,weattemptedtocreateamulti-cyclicmodelincorporatingan
asymmetriccurvethatmightbetterapproximatetheU.S.productioncyclewhen
tightoilwasmodeledbyaseparatecurve(Curve2).Thismodelnecessarily
incorporatesmorevariablesasitcombinestwodifferentequations,theasymmetric
curvedescribedbyBrandt(2006)andasymmetricHubbertcurvemodelingtightoil
production.Whileitisplausibleandindeedlikelythatthetightoilproductioncycle
mayeventuallybebetterfitbyanasymmetriccurvemodelratherthanasymmetric,
wecannotmodelthisbehaviorasproductionhasyettopeak.
Fig.15U.S.productiondatafitwithanasymmetriccurve(Curve1)approximatingtheproduction
cycleofconventionalresources,withCurve2fittotheproductionoftightoil.
23
Fig.16OurMulti-Cyclic,Asymmetricalmodel(sumofCurve1,Curve2inFig.15)
COMPARINGMODELS
AsBrandt(2006)notes,comparingmodelsofdifferingcomplexity(i.e.
differentnumbersofparameters)usingthesumofsquarederrorsofprediction
(SSE)ortherootmeansquareerror(RMSE)isinadequateasmorecomplexmodels
inherentlyfitbetterwhenmeasuredbySSEduetotheirincreasedflexibility
(Brandt,2006).WereplicateBrandt’s(2006)procedure,usingthecorrected
Akaike’sInformationCriterion(AICC)score,allowingthecomparisonofmodelsof
differingcomplexity.TheAICCformulaisgivenby:
𝐴𝐼𝐶! = 𝑁𝑙𝑛𝑆𝑆𝐸𝑁 + 2𝐾 +
2𝐾 𝐾 + 1𝑁 − 𝐾 − 1
whereNisthenumberofdatapointsintheseries,SSEisthesumofsquarederrors
andKisthenumberofmodelparameters(Motulsky&Christopolus,2004,Brandt,
2006).WhilealowAICCscoreindicatesahighlikelihoodofbest-fit,amodelcannot
bedirectlyacceptedorrejectedbycomparisonAICCasthescoreisbasedon
informationtheoryandnotstatistics.BycalculatingthedifferenceinAICCscoreof
24
twomodelsinquestion(ΔAICC),weestablishtheprobabilitythatonemodelis
‘morecorrect’thantheotherusing:
∆𝐴𝐼𝐶 = 𝐴𝐼𝐶! 𝑏𝑒𝑠𝑡 𝑓𝑖𝑡𝑡𝑖𝑛𝑔 𝑚𝑜𝑑𝑒𝑙 − 𝐴𝐼𝐶! 𝑠𝑒𝑐𝑜𝑛𝑑 𝑏𝑒𝑠𝑡 𝑓𝑖𝑡𝑡𝑖𝑛𝑔 𝑚𝑜𝑑𝑒𝑙
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑒!!.!∆!"#
1+ 𝑒!!.!∆!"#
whereaprobability>99%isconsideredstrongevidenceofbestfit(Motulsky
&Christopolus,2004,Brandt,2006).
Weusedthismethodtocomparethefitofoursingle,multiandassymetric
curvevariants(foroursetofannualproductiondatarangingspanning1859-2016,
N=158):
Model SSE N K AIC
Single Hubbert 14.78 158 2 -370.32
Multi-Cyclic 5.59 158 5 -517.60
Asymmetric Hubbert 10.16 158 5 -423.20
Multi-Cyclic, Asymmetric 2.70 158 7 -627.95
AllvariantsoftheoriginalHubbertmodelscoredbetterthantheoriginal,andall
provedtobesignificantlybetterfits.Comparingthemsequentiallybymagnitudeof
AICscore:
SingleHubbertwiththeAsymmetricHubbert:ΔAIC Probability
-52.88 1.00
TheAsymmetricHubbertwiththeMulti-Cyclic:ΔAIC Probability
-94.3954858 1.00
25
AndfinallytheMutli-CyclicwiththeMutli-Cyclicmodelincorporatingan
asymmetriccurve:ΔAIC Probability
-110.36 1.00
Byanindirectcomparisonofthethreemodelstestedwefindthatthesymmetric
singlecyclemodelHubbertproposedprovidesthepoorestfitforU.S.production
data,whiletheMulti-CyclicAsymmetricmodelprovidesthebestfitevenwhile
accountingforthelargernumberofvariablesitincorporates.
DISCUSSION&CONCLUSION
Asourresultsdemonstrate,technologicaladvanceshavesignificantly
impactedthecontinuedvalidityofHubbert’ssinglecurvemodelfortheU.S.
productioncycle.TheadventofhydraulicfracturingandEORtechniqueshas
introducednewproductioncyclesanddampeddeclineratesinmannersthat
Hubberthadnowayofaccountingfor.ThebestfitofbothMulti-Cyclicmodelsis
unsurprisingconsideringthelarge,welldelineatedresourcehydraulicfracturing
hasmadeexploitable.ThebetterfitofbothAsymmetricmodelsdemonstratesthe
fallibilityofHubbert’sassumedsymmetry.WhileHubbert’ssinglecyclemodel
shouldbeappliedwithcaution,itisimportanttohighlighttheusefulpropertiesall
curvefittingtechniquesshareinthattheycanmakethemostofregionalleveldata
thatisoftenthemostgranularavailable.AsBrandt(2006)notes,Hubbert-like
modelsbasedongoodURRestimateswillnotbeerroneousondecadetimescales
duetothepowerofexponentialgrowth,andasweprovewiththeapplicationMulti
CycliccurvesforU.S.production,modifiedcurvefittingmethodsareusefulwhen
usedwithdiscretion(i.e.avoidingoverfitting).
26
Model P (2100), mmbbl Model Estimated URR, Gb
Single Hubbert 251.36 261.46
Multi-Cyclic 80.88 267.52
Asymmetric Hubbert 610.59 283.10
Multi-Cyclic, Asymmetric 76.42 278.24
AsLaherrere(2005)states“whatgoesupmustcomedown…whatisborn
willdie…constantgrowthhasnofutureinalimitedworld”.TheMulti-Cyclic,
AsymmetricmodelthatfitproductioncycledatabestestimatedaURRof283.10Gb.
CurrentcumulativeU.S.productionamountstomorethan75%ofthatestimate.
Whileourmodelsprovidedawiderangeofestimatesforproductionintheyear
2100,themostoptimistic(AsymmetricHubbert)wasstilllessthan20%ofU.S.
productionin2016.Futureproductioncyclesspurredbyexploitationofother
hardertoaccessorunconventionalresourceswillcomeatthecostofincreasingly
diminishingEROI,anditisimportanttoconsidertheeconomicandpolitical
implicationsthisholdsforanincreasinglyenergydependentnation.
27
28
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