Modelling EROEI and net energy in the exploitation of non renewable resources

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ContentslistsavailableatScienceDirect

Ecological

Modelling

j o u r n al hom ep age : w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l

Modelling

EROEI

and

net

energy

in

the

exploitation

of

non

renewable

resources

Ugo

Bardi

a,∗

_{, Alessandro}

_Lavacchi

b

_{, Leigh}

_Yaxley

c

a_Dipartimento_di_Chimica_–_Università_di_Firenze,_Via_della_Lastruccia_3,_Sesto_Fiorentino_[Fi],_Italy b_ICCOM-CNR,_Polo_scientiﬁco_di_Sesto_Fiorentino,₅₀₀₁₉_Firenze_Italy

c_Society_for_Petroleum_Engineers,₁₀₇₇₇_Westheimer_Rd.,_Suite_1075,_Houston,_United_States

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received3February2011

Receivedinrevisedform15May2011 Accepted17May2011

Available online 7 July 2011 Keywords: Hubbertmodel Naturalresources Lotka–Volterra EROEI Netenergy

a

b

s

t

r

a

c

t

Recently,BardiandLavacchi(2009)showedthatasimplesystemofcoupleddifferentialequationscan beusedforaquantitativedescriptionoftheexploitationofnonrenewableresourcesinafreemarket economy.Thepresentpaperexamineshowthemodeldescribesthebehaviorofthesystemintermsof energyreturnforenergyinvested(EROEI)andnetenergy(energyreturnedminusenergyexpended).We showthatthemodelgeneratesabehaviorofthesefactorscomparabletotheresultsobtainedbyother methods,forinstanceforthecaseofcrudeoilproductionintheUS.

1. Introduction

Theexploitationofaﬁniteresourcemustnecessarilyfollowa

productioncyclethatstartswithzeroandendswithzero.In

sev-eralhistoricalcases,andinparticularforfossilfuels,ithasbeen

observedthattheshapeoftheproductioncurveisrelatively

sim-ple:itis“bellshaped”withasinglemaximumwhenapproximately

halfoftheresourcehasbeenproduced.Thismodelwasproposed

fortheﬁrsttimebyHubbert(1962)asanempiricaldescriptionof

theproductionofcrude oilintheUS48 lowerstates.Thesame

behaviorisoftenobservedforseveralcasesofmineralresources

(BardiandPagani,2008)andforslowlyrenewableresources,such

aswhaleoil(Bardi,2007).

Areasonableexplanationforthedeclineofproductionafterthe

peakisrelatedtothedecliningenergyreturnforenergyinvested

(EROEIorEROI)(Halletal.,2008;Murphy,2009).Producerswill

normally tend to exploit ﬁrst the“easy” resources, those with

highEROEI and mustprogressivelymove tolower EROEI ones,

withincreasingcostsofextraction.Withlowerproﬁts,companies

involvedinextractionﬁndthemselvesshortofcapitalandmust

reduceinvestments.Thisprocessleads,eventually,to“peaking”of

productionandtoitssuccessivedecline.

In the present paper we examine how a simple model can

describehowEROEIvariesasafunctionofproduction.Themodel

hasbeendescribedinapreviouspaper(BardiandLavacchi,2009)

∗ Correspondingauthor.

E-mailaddress:ugo.bardi@uniﬁ.it(U.Bardi).

andisrelatedtothewellknownLotka–Volterra(LV)model(Lotka,

1925; Volterra, 1926), also known as the “predator–prey” and

“foxesandrabbits”model.Thismodelcanprovideaquantitative

ﬁttingofthehistoricalproductiondataforanumberofcasesof

exploitationofnon-renewableresources,suchascrudeoil,orof

slowlyrenewableones,suchaswhaleoil.Wewillshowherehow

themodeldescribesthedeclineofEROEIduringtheexploitation

processandconﬁrmsthattheHubbertbehaviorisrelatedto

declin-ingenergyyield.

2. EROEIandnetenergy

Exploitinganenergyresourcecanneverbeen100%efﬁcient.For

instance,inordertoexploitthechemicalenergystoredinanoilwell

wemustexpendsomeenergyinoperationssuchasprospecting,

drilling,extracting, processing, and transporting.EROEI (energy

returnforenergyinvested)is deﬁnedastheratiooftheenergy

obtainedfromtheresourcetotheenergyexpendedinproduction

(Halletal.,1986,2008,2009).Arelatedconceptisthatofnetenergy,

deﬁnedastheenergyproducedminustheenergyexpended.When

theEROEIisequalto1orlower,thenetenergyiszeroorlower.

EROEIisausefulconceptforunderstandingtherealvalueofa

resourceineconomicandenergyterms.Obviously,largerEROEIs

arepreferableandanEROEIsmallerthanonecorrespondstoanet

lossofenergy.However,someprocessesmaybecarriedouteven

atlowEROEIs,evensmallerthanone,asaresultofspeciﬁcchoices

oftheeconomicsystem.Asanexample,biofuels,andin

particu-larethanol,havealowEROEI(PimentelandPatzek,2005)butfor

politicalreasonsgovernments(especiallyintheUS)provide

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cialsubsidiesthatresultinanettransferofenergyfromfossilfuels

toethanolproduction.

Despiteitsusefulness,thedeﬁnitionofEROEIsuffersofsome

uncertainty,mainlyintermsoftheboundariesofthesystembeing

considered.Theseboundariesmaybedeﬁnedaccordingtothe“Life

CycleAnalysis”(LCA)concept.Therelatednormsaredeﬁnedin

protocolssuchas,forinstance,theASTME1991-05.If,however,one

wantstotakeintoaccounteverythingthatisdonewithanenergy

source;fromproducingfuel,tothecarsandroadstouseit,allthe

waytocathedralsandpoetrythenwemayspeakof“fullEROEI”or

“societalEROEI”(Halletal.,2008,2009).ThevalueofsocietalEROEI

determinesthesurplusthatcanbeutilizedforallthoseactivities

thatareconsideredpartofwhatwecall“civilization”.Theproblem

doesnotexistaslongasconsistentEROEIdeﬁnitionsareusedwhen

comparingdifferentenergysources,butitmustbekeptinmind

whenmodellingeconomicsystems.

3. Asimplemodelforresourceexploitation

Lotka(1925)andVolterra(1926)developedawellknownmodel

of“predator–prey”relationshipinsimplebiologicalsystems.This

modelisintuitivelyandmathematicallyattractive,andisoftenused

inecologytoconceptualizetherelationbetweenpredatorandprey

species.TheLotka–Volterra(LV)modelworkswellenoughwith

simplelaboratorysystemsandalsoappearedtoworkinnature;

howeverHall(1988)showedthat mostexamplesgivenin

text-books,includingthewellknownoneofharesandlynx,infactdo

notsupporttheuseofsucha simplemodelin realecosystems.

Nevertheless,modelsbasedontheLVapproachhavefound

appli-cationsineconomics.Thesemodelsaresometimesknownas“Free

Access”model(see,e.g.Smith,1968).

BardiandLavacchi(2009)developedamodiﬁedversionofthe

LVmodelinordertodescribetheeconomicexploitationofnon

renewable(orslowlyrenewable)resourceswhereitisassumed

thattheprey(rabbits)donotreproduce.Themodelinvolvestwo

mainstockvariables:resourcesandcapital.Theamountofavailable

resourceisdeﬁnedasthe“resourcestock”,R.Theothermain

vari-ableofthemodelistheaggregateamountofeconomicresources

beingutilizedintheexploitation;thatisequipment,land,

knowl-edge,humanwork,andsimilar.Wecalledthisaggregateamount

“capitalstock”,C.RandCaredeﬁnedastheﬂow(thevariationas

afunctionoftime)of,respectively,resourcesandcapital.Further

parametersofthemodelaretheinitialstocksofresource(Ro)and

ofcapital(Co).

AsintheoriginalLVmodel,it isassumedthatresource (the

“prey”) can be extracted in proportion to the available

capi-tal(the“predator”)and, atthesametime, inproportiontothe

amountoftheresourcestock.Implicitly,thisassumptioninvolves

thatresourcesare“graded”and thatthe“easy”(lessexpensive)

resourcesareextracted[orproduced]ﬁrst.

Theotherfundamentalassumptionofthemodel–again

corre-spondingtotheoriginalLVmodel–isthatcapitalisgeneratedin

anamountproportionaltotheamountofextractedresources.In

otherwords,theresourcestockispartlytransformedintocapital

stock;letussaythattheextractedoilisusedtoprovidetheenergy

necessarytobuildmoreoilrigsandotherfacilities.Inmoregeneral

terms,thistransformationismediatedbythemarketsystem.That

is,theenergyre-investedisgeneratedviathesaleoftheresource

onthemarketandtheproﬁtsareusedtocreatetheequipmentand

facilitiestoproducemoreresource.

Aﬁnal assumptionof themodel isthat capital is dissipated

overtimeinproportiontotheamountofcapitalitself,thesame

assumptionmadeintheoriginalLVmodel.Thesigniﬁcanceofthis

assumptionisoftenmisunderstood.“Dissipation”mightbeseenas

correspondingto“depreciation,”thatisthedeclineinvalueofthe

Fig.1. QualitativesolutionsofthemodelobtainedusingtheVensimsoftware.The parametersshownareresource,capitalandproduction.

capitalstock,mainlyintermsofobsolescence.Inthebiological

ver-sionofthemodel,itwouldmeanthatitdescribeshowfoxesgetold

anddieofaging.Butthatisonlyapartialview:thecapitalstock

(orthefoxesstock)isanenergystockandthistermdescribeshow

muchofthisenergyislostasafunctionoftimewithoutspecifying

bywhatmechanism.Inpractice,theassumptionisaboutthe

funda-mentalfeedbackofthesystemthathascapitalallocatesomeenergy

inordertocreatemorecapital(thatis,howfoxesexpendenergy

bychasingrabbits).Thispointisfundamentalinthemodellingof

thesystem’sEROEI,asitwillbeshownlateron.

Themodelcanbedescribedinmathematicalformastwo

cou-pleddifferentialequations.Thesetwoequationsarebasicallythe

sameasthoseoftheoriginalLotka–Volterramodel,exceptthatone

termismissing,thatofthereproductionoftheprey.The“ks”are

constantswhichdescribethequantitativebehaviorofthemodel

R₌₋_k

1CR (1)

C₌_k

2CR−k3C (2)

Qualitativeresultsofthemodel– obtainedusingtheVensim

software–areshowninFig.1.Inorderforthemodeltobeable

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Fig.2. FittingofthedataforoildiscoveryintheUS48lowerstatesandofthenumberofwildcats.Inthiscase,thenumberofwildcatsisproportionaltothecapitalusedby theoilindustryintheeffortofdiscoveringtheresource(oilwells).FromBardiandLavacchi(2009).

someofthemainparametersintheequations.Inapreviouspaper

(BardiandLavacchi,2009)weusedhistoricaldataonproduction

andonaggregatecapitalaccumulation.Thelatterweremeasured

intermsof“proxies”;e.g.usingthetonnageofthewhalingﬂeetas

proportionaltothetotalcapitalavailabletothewhalingindustry.In

thisway,wefoundthatthissimpliﬁedmodelcandescribeanumber

ofhistoricalcasesofresourceexploitation,whentheresourceisnon

renewable(e.g.crudeoil)orslowlyrenewable(e.g.whaleoil).

AnexampleoftheresultsobtainedfortheUS-48lowerstates

historicaldataoncrudeoilproductionisshowninFig.1.Here,the

“discoveries”parameterisusedasaproxyforresourceproduction

andthenumberofwildcatsasaproxyfortheaggregatecapitalused

bytheoilindustryforprospectingintheareaconsidered(Fig.2).

Othercasesforwhichthemodelwasfoundtogiveagoodor

accept-abledataweregoldproductioninSouthAfricaandinCalifornia,

whaleoilproductioninthe19thcenturyandcrudeoilproduction

inNorway(BardiandLavacchi,2009).

4. EROEIandnetenergyintheexploitationofmineral resources

Anymodelattemptingtodescribeaphysicalsystemmust

sat-isfythelawsofphysicsand,inparticular,thoseofthermodynamics.

TheLotka–Volterramodelisnoexception.Initssimplest

imple-mentation,thatoftwobiologicalspecies(foxesandrabbits),the

twostocksinvolvedcanberegardedasenergystocks.Thesame

interpretationispossibleforenergyresourcessuchascrudeoil.

Intheabsenceofanexternalenergyﬂux,themodeldescribes

theﬂowofenergyfromonestock(resources)toanother

(capi-tal).Thetransformationiscompletelyirreversibleanditoccursas

theresultoftheincreaseinentropyofthesystem(Karnaniand

Annila,2009).Eventually,thedissipationtermofthemodel(−k3C)

willbringtozerotheamountoffreeenergystoredinthecapital

stock;thusmaximizingtheentropyinaccordancewiththesecond

principleofthermodynamics.

Parameters suchas EROEI and netenergy are not explicitly

expressedin theequationsof themodel,but canbecalculated

fromtheavailableparameterswhich,asmentionedbefore,include

energyandenergyﬂow.Inapreviousstudy(BardiandLavacchi,

2009),wedeﬁnedthe“yield”ofextractionas(R/C),thatistheratio

ofproductiontocapital.However,theconceptofEROEIrequiresa

moredetailedexamination.

Ifweapplythemodeltoanactualeconomicprocess,weare

interestedinmaximizingproductionandminimizingcosts.

Pro-ductionisexplicitlydeﬁnedinthemodelas“R”.The“costs”in

themodelaredeterminedbytheonlytermthatproducesalossof

capital(−k3C).Asdescribedearlieron,thesecostsaretheresultof

thesumoftheenergyspentinordertoexploittheresourceand

theenergylostbecauseofobsolescenceandotherfactors

(depre-ciation).Inotherwords,thek3termisthesumoftwoterms,one

relatedtoexploitation,theothertodepreciation.Boththese

fac-torsshouldbeincludedinanEROEIcalculationsbasedonlife-cycle

analysis,soitisjustiﬁedtousetheaggregatedk3 factoras

pro-portionaltotheoverallexploitationcosts.Hence,wecandeﬁne

theeconomicyieldoftheprocessastheratioofenergyproduction

(k1RC)toenergyloss(k3C).Analternativedeﬁnitioncouldbeto

usecapitalaccumulation(k2RC)asthenumeratoroftheratio,but

usingproductionappearstobemorecloselyrelatedtotheconcept

ofEROEIasitisusuallydeﬁnedinpractice.

Thereforewecanwrite:

EROEI=Rk1

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Fig.3.Qualitativesolutionsofthemodelequationssystemobtainedusingthe Ven-simsoftware.Theparametersreportedareproduction,netenergyandEROEI.

ThisratiodescribeshowtheEROEIofanonrenewableenergy

sourcevarieswithtime.Forinstance,itdescribeshowthe

aver-ageEROEIofsingleoilwellsvariesastheoilresourcesofaregion

areexploited.Notethatthe“capital”stockdoesnotappearinthis

expression.Thisappearstobecorrect,sincetheEROEIderivesfrom

thecharacteristicsoftheresourcebeingexploited.However,note

thatthetwoparametersk1andk3aredeﬁnedinthemodelwith

respecttobothresourcesandcapital,sothatthebehaviorofthe

capitalstockdoesaffectEROEI.

Alimitationofthisconceptisobviouswhenweapplythe

con-cepttoarealeconomicprocess.ThedeﬁnitionofEROEI=Rk1/k3

isrigorouslyvalidonlyforasystemwherealltheenergygained

fromtheexploitationofaresourceisusedforfurtherexploitation.

Thatmightbetrueinasimplebiologicalsystem,sayfoxesand

rab-bits,butsurelynotfortheworld’seconomy.However,theconcept

thattheEROEIisproportionaltoR(theresourcestock)remains

usefulinthereasonableassumptionthatthefractionofproﬁts

re-investedbytheindustryintheexploitationofaresourceremains

approximatelyconstantovertheresourcelifetime.This

assump-tionisultimatelyjustiﬁedbythefactthatthemodelcanreproduce

thehistoricaldataonproduction(BardiandLavacchi,2009)fora

numberofcases.

Notethatthereisnoelementintheformulathatwouldstop

processingwhentheEROEIbecomessmallerthanone;whenthat

occurs,exploitationwillcontinueutilizingpreviouslyaccumulated

energyresources.Notealsothat,sinceproductionisgivenbyR,a

maximumintheproductioncurvewillcorrespondtoaninﬂection

pointintheEROEIcurve.

Fromthisresult,wecanproceedwiththedeterminationofthe

formfornetenergy(NE)whichwemaydeﬁneasproductionminus

dissipation.Thatis

NE=C(k1R−k3)

WeseethatforRsufﬁcientlysmall,thatisintheﬁnalstagesof

exploitation,thenetenergyofthesystembecomesnegativeandit

remainsso.Wecanalsowritethisrelationas:

NE=CR(k1−k3/R)

AssumingthatRisrelativelylarge,thatiswearenotattheﬁnal

stagesoftheexploitationprocess,wecanapproximateNEasequal

tok1CR,thatisequaltoproduction.Therefore,weexpectnetenergy

tohaveamaximumthattakesplace,approximately,nearthe

pro-ductionpeak.QualitativesimulationsperformedusingtheVensim

softwareconﬁrmboththisstatementandtheonethatEROEIshould

haveaninﬂectionpointincorrespondencetotheproductionpeak;

asshowninFig.3.

Fig.4.HistoricalEROEIofcrudeoilextractioninthelower48USstates.Figurefrom Murphy(2009).

These calculations qualitatively correlate with the historical

dataontheEROEI ofoilextractionfromtheUS-48lowerstates

(Murphy,2009;Cleveland,2005)asshowninFig.4.Despitethe

smallnumberofpointsavailable,itispossiblethattheinﬂection

pointintheEROEIcurveoccursaround1970andthereforeit

cor-respondstotheproductionpeakintheregion.

Theseconsiderationsraisethequestionofwhetheritispossible

tousethemodeldescribedhereinordertodeterminetheEROEI

insteadofusingthestandardLCA.Therearetwoproblemsinthis

sense:theﬁrstistheuseofproxydatafortheﬁttingofthe

histor-icaltrends,thesecondisthatEROEI,asdeﬁnedwithinthemodel,

impliesthatalltheenergyproducedbythesystemisallreusedto

producemoreenergy,anobviouslynonrealisticassumption.

Regardingtheuseofproxydata,itispossibletonormalizethe

equationsofthemodelandtoobtainavalueoftheEROEIasa

func-tionofthemeasuredparameters.Forthesecondproblem,however,

themodelcannotsayanythingaboutthesocietaldecisionofwhich

fractionoftheproﬁtsfromtheexploitationofaresourcehavetobe

allocatedtofurtherexploitation.Indeed,testsmadewiththeUS-48

crudeoilsystemstudiedinapreviouspaper(BardiandLavacchi,

2009)showthattheEROEIcalculatedfromthemodelissmaller

thantheLCAcalculatedEROEI(Cleveland,2005),asexpected.Inany

case,furtherdataandanalysisarenecessaryinordertodetermine

thepotentialofthismethodasatoolforEROEIcalculations.

5. Conclusions

Themodelpresentedhereoffersusefulinsightsonthe

mecha-nismofresourceexploitationanditmayofferarouteformodelling

andunderstandingeconomicprocesseswhicharevital for

soci-ety.TheroleandtheimportanceofEROEI areclariﬁedbythese

resultswhichconﬁrmthat,indeed,EROEIisthedrivingforceinthe

exploitationprocess.Furthermore,themodelshowsthat,utilizing

accumulatedresources,theuseanonrenewableenergyresource

maycontinueevenforEROEIssmallerthanoneand–hence–for

negativenetenergyproduction.Whetherthiswillbethe

behav-ioroftherealworld’seconomy remainstobeseen,butit isat

leastapossibility;indicatingthattheeconomicmechanismswhich

arejudgedappropriatetodayfordeterminingexploitation

priori-tiesmaynotbesuchwhenappliedtononrenewableresources.

Althoughonlyqualitative,themodeloffersasimplementaltool,a

“mindsizedmodel”(Papert,1980),thatcanbeusedtounderstand

themechanismoftheexploitationofnaturalresources(including

theabilityoftheatmospheretocontainCO2withoutoverheating

theecosystem.

Understanding(andactingon)thesemechanismsisthemain

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