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

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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).

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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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