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
caDipartimentodiChimica–UniversitàdiFirenze,ViadellaLastruccia3,SestoFiorentino[Fi],Italy bICCOM-CNR,PoloscientificodiSestoFiorentino,50019FirenzeItaly
cSocietyforPetroleumEngineers,10777WestheimerRd.,Suite1075,Houston,UnitedStates
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.
© 2011 Elsevier B.V. All rights reserved.
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
finan-0304-3800/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2011.05.021
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
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
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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