Hubbert’s Legacy: A Review of Curve-Fitting Methods to Estimate Ultimately Recoverable Resources

Hubbert�s Legacy: A Review of Curve-Fitting Methodsto Estimate Ultimately Recoverable Resources

Steve Sorrell1,3 and Jamie Speirs2

Received 11 November 2009; accepted 2 July 2010Published online: 22 July 2010

A growing number of commentators are forecasting a near-term peak and subsequent ter-minal decline in the global production of conventional oil as a result of the physicaldepletion of the resource. These forecasts frequently rely on the estimates of the ultimatelyrecoverable resources (URR) of different regions, obtained through the use of curve-fittingto historical trends in discovery or production. Curve-fitting was originally pioneered byM. King Hubbert in the context of an earlier debate about the future of the US oil production.However, despite their widespread use, curve-fitting techniques remain the subject of con-siderable controversy. This article classifies and explains these techniques and identifies boththeir relative suitability in different circumstances and the level of confidence that may beplaced in their results. This article discusses the interpretation and importance of the URRestimates, indicates the relationship between curve fitting and other methods of estimatingthe URR and classifies the techniques into three groups. It then investigates each group inturn, indicating their historical origins, contemporary application and major strengths andweaknesses. The article then uses illustrative data from a number of oil-producing regions toassess whether these techniques produce consistent results as well as highlight some of thestatistical issues raised and suggesting how they may be addressed. The article concludes thatthe applicability of curve-fitting techniques is more limited than adherents claim and that theconfidence bounds on the results are wider than usually assumed.

KEY WORDS: Petroleum resource estimation, peak oil.

INTRODUCTION

A growing number of commentators are fore-casting a near-term peak and subsequent terminaldecline in the global production of conventional oilas a result of the physical depletion of the resource(Campbell and Heapes, 2008; Aleklett and others,2009). These forecasts frequently rely on the esti-mates of the ultimately recoverable resources

(URR) of different regions, obtained through theuse of curve-fitting techniques.4 Curve fitting wasoriginally pioneered by M. King Hubbert in thecontext of an earlier debate about the future of USoil production (Bowden, 1985). However, despitethe widespread and increasing use of these tech-niques, they remain the subject of considerablecontroversy.

In a seminal article, Hubbert (1956) forecast thefuture of US oil supply by fitting a curve5 to his-torical production and projecting this forward under

1Sussex Energy Group, SPRU (Science & Technology Policy

Research), Freeman Centre, University of Sussex, Falmer,

Brighton BN1 9QE, UK.2Imperial College Centre for Environmental Policy and Tech-

nology (ICCEPT), Imperial College London, South Kensington

Campus, Exhibition Road, London SW7 2AZ, UK.3To whom correspondence should be addressed; e-mail:

[email protected]

4These involve selecting and parameterising a mathematical

function that has the �best� statistical fit to a set of data points,

possibly subject to constraints.

5Hubbert�s original curve was hand drawn, but his later articles

used more formal techniques.

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1520-7439/10/0900-0209/0 � 2010 International Association for Mathematical Geology

Natural Resources Research, Vol. 19, No. 3, September 2010 (� 2010)

DOI: 10.1007/s11053-010-9123-z

the assumption that, first, production must eventu-ally decline exponentially; and second, the area un-der the curve must equal the URR of the UnitedStates. Using the industry consensus estimates forthe US URR (150–200 billion barrels), Hubbertforecast that the US oil production would peaksometime between 1965 and 1971. Many commen-tators considered Hubbert�s approach to have beenvindicated when the US production peaked in 1970and began its long decline (Strahan, 2007). However,subsequent analysis has shown that the accuracy ofhis forecast was partly fortuitous (Kaufmann andCleveland, 2001; Cavallo, 2005a, b).

Shortly after the publication of Hubbert�s article,the consensus on the US URR estimates evaporated.Numerous commentators disputed Hubbert�s ap-proach, and a series of more optimistic estimates ofthe US URR began to appear. Most famously, the USGeological Survey (USGS) estimated a value of 580billion barrels (Gb) for the lower 48 states, based onforecasts of future drilling activity (Zapp, 1962). Suchdiscrepancies motivated Hubbert to develop moreformal methods to estimate the URR based on curve-fitting to production and discovery data. Hubbertdeveloped and applied several such techniques be-tween 1962 and 1982, all of which produced estimatesin the range 150–200 Gb. These techniques havesince been adopted and developed by numerousauthors (Laherrere, 2000a, b, 2002a, b, 2004a, b;Campbell and Heapes, 2008; Mohr and Evans 2008),and the results have been used to underpin forecastsof near-term peaks in regional and global oil pro-duction. Although dismissed by critics as �trendology�(Charpentier, 2003; Lynch, 2003), curve-fitting tech-niques have much in common with more sophisti-cated methods of estimating resource size such asdiscovery process modelling. They may also have animportant role to play when, as is generally the case,only aggregate data on production or discovery in aregion is available—as opposed to detailed geologicalinformation or data on individual fields.

This article classifies and explains these curve-fitting methods and identifies both their relativesuitability in different circumstances and the level ofconfidence that may be placed in their results. Thearticle is based on a comprehensive review bySorrell and Speirs (2009). The article begins by dis-cussing the interpretation and importance of theURR estimates, indicating the relationship betweencurve-fitting and other methods for estimating URRand classifying the curve-fitting techniques into threegroups. The subsequent sections investigate each

group in turn, indicating their historical origins,contemporary application, and major strengths andweaknesses. The article then uses illustrative datafrom a number of oil-producing regions to assesswhether these techniques produce consistent results,highlights some of the statistical issues raised andsuggests how they may be addressed. The articleconcludes that the applicability of curve-fittingtechniques is more limited than many adherentsclaim and that the confidence bounds on the resultsare wider than those that are usually assumed.

METHODS FOR ESTIMATINGULTIMATELY RECOVERABLERESOURCES

The ultimately recoverable resources (URR) ofa region represent the amount of resources that areanticipated to be recovered from when productionbegins to when it finally ends—although for practicalpurposes, estimates are usually made for a shorterperiod of time (e.g. up to 2050). Estimates of theURR require specification of the boundaries ofthe relevant region, the hydrocarbons covered, thetimeframe for which the estimate is made, the rele-vant technical and economic assumptions and theassociated range of uncertainty. Lack of clarity oversuch issues, together with the tendency to producesingle value estimates, explains much of the con-troversy over this topic (Rogner, 1997). While notalways treated as such, the URR is an endogenousvariable. For example, increasing prices shouldmake marginal resources more profitable as well asinducing technical improvements that reduce pro-duction costs, improve recovery factors and allowaccess to previously inaccessible resources. At issueis whether such developments can be expected tosignificantly increase the recoverable resources froma region, or whether only marginal increases may beexpected. This in turn will depend on the charac-teristics of the region, the extent to which it has beenexplored, the technology currently used and thetimeframe under consideration.

Estimates of the URR may be developed forlevels of aggregation ranging from individual fieldsto the whole world, with different techniques beingmore or less suitable for different levels. Aggregateestimates may be derived from the estimates devel-oped at lower levels (e.g. for individual fields), butthe latter can only be summed arithmetically if theyrepresent mean estimates (Pike, 2006). Estimates

210 Sorrell and Speirs

are frequently made at the country or regional level,but these typically encompass several geologicallydistinct areas that may also extend into neighbouringcountries or regions. The lack of geological homo-geneity within such boundaries can greatly compli-cate resource assessments when only aggregate datais used (Charpentier, 2003).

There are a variety of methods for estimatingthe URR and many variations on the basic tech-niques. The appropriate choice depends on the nat-ure of the region under study, and the data andhuman resources available. The most reliable esti-mates are likely to be derived from a combination ofmethods, and evidence suggests that the results canvary widely (Divi, 2004; Ahlbrandt and Klett, 2005).Most of the methods associated with Hubbert may becharacterised as producing single-value estimatesfrom the extrapolation of curves fitted to historicdata on production or discoveries for aggregateregions such as an oil-producing country. There isrelatively little use of geological or other informa-tion, and the methods are simple to apply using datathat is either available in the public domain oravailable at reasonable cost from commercial sourcessuch as IHS Energy. In contrast, the methods used bythe USGS and others produce probabilistic estimatesfrom geological assessments of disaggregate regions,with extensive use of geological information andstatistical techniques (USGS, 2000). These methodsare complex and resource intensive and rely onextensive data sources that are usually inaccessible tothird parties. This characterisation is an oversimpli-fication, however, as there are considerable overlapsbetween the two, especially for regions that are at arelatively mature stage of exploration and produc-tion (Drew and Schuenemeyer, 1993).

For the less-explored regions, estimates mustrely on the geological analysis of seismic and otherdata. A traditional approach, usually applied at thebasin level, is to estimate hydrocarbon volumes bymultiplying the estimated sedimentary volume by anestimated yield in barrels per cubic kilometre(Weeks, 1952; White and Gehman, 1979; Gautier,2004). For unexplored areas, the values for suchcalculations are based on measurements from geo-logically similar regions where more information isavailable. Other approaches are applied at lowerlevels of aggregation and typically use Monte Carlomethods to multiply estimates or measurements ofvariables such as pore volume, porosity and oil sat-uration. Data may only be available for a subset ofthese variables and, for unexplored areas (e.g. the

Arctic), such estimates must necessarily have largeconfidence bounds. In Hubbert�s view:

‘‘…it is easy to show that no geological information

exists, other than that provided by drilling, that will

permit an estimate to be made of the recoverable oil

obtainable from a primary area that has a range of

uncertainty of less than several orders of magni-

tude.’’ (Hubbert, 1982)

Estimates may also be made by combining theexpert judgment of several geologists (Baxter andothers, 1978). Typically, each geologist reviews therelevant information and then estimates either asingle value or a probability distribution for each ofthe relevant factors which are then combined into aprobability distribution that reflects the full range ofopinions (White, 1981; Gautier, 2004). This methodis appropriate for all the levels of aggregation anddata availability, and can accommodate explorationconstraints and other factors that may be poorlyhandled by other methods (Charpentier and others,1995). However, it lacks transparency and reliesheavily on the knowledge and objectivity of theindividual assessors.

For well-explored regions, more reliable esti-mates can be obtained by using data on discoveredfields. There are three approaches:

� Field-size distributions: Estimates of theURR may be derived by combining data onthe sample of discovered fields with assump-tions about the size distribution of theunderlying population of fields. For example,if the field population is assumed to take a�power-law� distribution, a plot of the numberof fields exceeding a particular size on loga-rithmic scales should approximate a straightline. In this case, undiscovered resources maybe estimated by plotting a cumulative fre-quency distribution on a log scale, fitting alinear regression, extrapolating this to smallerfield sizes and calculating the area under thecurve (Cramer Barton and La Pointe, 1995).This estimate is sensitive to the point at whichthe observed size distribution is curtailedwhen fitting the curve, as well as to assump-tions about the minimum viable field size andthe appropriate functional form for the pop-ulation size distribution (which is an enduringfocus of controversy) (Charpentier andothers, 1995; Laherrere, 2000a, b; Kaufman,2005). An alternative approach plots cumu-lative discoveries as a function of the rank of

211Hubbert�s Legacy

the field (where the largest field is rank 1) andestimates the URR from the asymptote towhich the curve is trending. This approachhas much in common with the �discoveryprojection� technique described later.6

� Discovery process modelling: The discoveryprojection techniques involve statistical anal-yses of the number and size of discoveredfields as a function of either time, the discoverysequence or some measure of exploratory ef-fort, such as the number of exploratory wellsdrilled (Arps and Roberts, 1958; Kaufman,1975; Meisner and Demirmen, 1981; Formanand Hinde, 1985). This technique is sometimescombined with assumptions about the fieldsize distribution and/or information aboutfield location (Schuenemeyer and Drew,1994). Several of these models simulate aprobabilistic law governing the process of newfield discovery and can be used to provideforecasts of the number, size and sequence offuture discoveries together with the antici-pated success rate of exploratory drilling(Power, 1992; Power and Fuller, 1992a, b).

� Curve-fitting: The curve-fitting methods useregression techniques to fit curves to the his-toric trends in discovery or production in aregion and extrapolate the curves to estimatethe URR. The explained variables may becumulative production, the rate of production,cumulative discoveries or the rate of discov-eries, while the explanatory variable may ei-ther be time or some measure of exploratoryeffort, such as the cumulative number ofexploratory wells. A major advantage ofcurve-fitting techniques is that they do notrequire data on individual fields—which frommany regions of the world is unavailable,unreliable or very expensive to obtain. Curve-fitting was pioneered by Hubbert (1956, 1959,1962, 1982) and has subsequently been adop-ted and developed by numerous analysts,including in particular those concerned about�peak oil� (Cleveland and Kaufmann, 1991;Campbell, 2002; Laherrere, 2003; Imam andothers, 2004; Mohr and Evans 2008).

The above three �extrapolation� techniques allassume that: the field size distribution is highlyskewed, with the majority of oil being located in asmall number of large fields; and that the large fieldstend to be discovered relatively early, with sub-sequent discoveries being progressively smaller andoften the product of increasingly greater effort.These features should be reflected in both the sizedistribution of the discovered fields and the �shape�of the discovery or production cycle. For example,Figure 1 plots cumulative discoveries in a region as afunction of time. As the average size of the newlydiscovered fields falls, the curve trends towards anasymptote which can be taken as an estimate of theregional URR. All of the curve-fitting techniquesrely on patterns such as this.

The extrapolation techniques are best applied togeologically homogeneous areas that have had a rel-atively unrestricted exploration history (e.g. withoutareas being closed to exploration for legal or politicalreasons). If this is not done, then the mixing of dif-ferent populations of fields or the opening up of newareas for exploration (e.g. new plays within a basin)can lead to inconsistencies in the time-series andundermine the basis for estimating size distributionsand extrapolating historical trends (Wendebourg andLamiraux, 2002). However, even if the region isgeologically homogeneous, various technical,economic and political factors (e.g. the advent ofhorizontal drilling) can lead to structural breaks in atime-series (Harris, 1977).7 Since all the historical

Figure 1. Cumulative discovery in a region as a function of time.

Source: IHS Energy. Note: Name of region withheld on grounds

of data confidentiality.

6If the discovery rate was constant and if fields were discovered

precisely in the descending order of size, the field-rank and

discovery projection techniques would be identical. But the

advantage of discovery projection is that it does not require data

on individual fields.

7Exploration is very rarely unrestricted in aggregate regions. For

example, the US imposes far fewer restrictions than most

countries, but the Arctic National Wildlife Refuge, the eastern

Gulf of Mexico, much of the western offshore and many onshore

areas in the Rockies are off-limits for environmental reasons

(Mills, 2008).


trends reflect the net effect of physical depletion,technical change and numerous economic, politicaland institutional influences, the extrapolation ofthose trends can only be expected to provide reliableURR estimates if physical depletion provides thedominant influence and continues to do so in the future.This condition will not apply in all oil-producingregions although depletion should become increas-ingly important as exploration proceeds.

Curve-fitting techniques are very popular owingto their simplicity and the relative availability of therelevant data. These techniques estimate the URRfor a region by extrapolating historical trends inaggregate data, but they vary in their choice anddefinition of the explained and explanatory variables(Table 1). Other explanatory variables can andshould be included in the specification, but generallyare not. The following sections discuss each group oftechniques in turn.

PRODUCTION OVER TIME TECHNIQUES

The simplest, although not the most reliablemethod of estimating the URR uses non-linearregression8 to fit a curve to time-series data on

cumulative production. This curve may take a vari-ety of forms with its shape being defined by three ormore parameters, one of which corresponds to theURR. Hubbert (1982) assumed a logistic model(Box 1) which implies that cumulative productionwill initially grow exponentially, but the rate ofgrowth will fall and eventually decline to zero as theURR is approached.9 The curve is defined by threeparameters, representing the URR, the �steepness�of the curve and the midpoint of the growth trajec-tory, with the production cycle being obtained fromthe first differential of this curve. The URR may alsobe estimated by fitting a curve to the productiondata, although the result may well be different(Carlson, 2007). While a �bell-shaped� productioncycle is usually referred to as a �Hubbert curve�,Hubbert repeatedly states that it need not take thisform (Hubbert, 1982).

Curve-fitting to production trends should bemore reliable if production has passed its peak and isonly viable if the rate of increase of production haspassed its peak (i.e. the point of inflection on therising production trend). The US data fits the logisticmodel relatively well (Fig. 2) despite covering aperiod that includes two world wars, several reces-sions, two oil shocks, revolutionary changes in

Table 1. Classification of curve-fitting techniques by their choice of explained and explanatory variables

Group Technique Explained variable Explanatory variable

Production over time Cumulative production

projection

Cumulative production Time

Production projection Rate of production Time

Production decline curve Rate of production Cumulative production

Discovery over time Cumulative discovery

projection

Cumulative discovery Time

Discovery projection Rate of discovery Time

Discovery decline curve

(time)

Rate of discovery Cumulative discovery

Discovery over

exploratory effort

Creaming curve Cumulative discovery Exploratory effort

Yield per effort curve Rate of discovery wrt

exploratory effort

Exploratory effort

Discovery decline curve

(effort)

Rate of discovery wrt

exploratory effort

Cumulative discovery wrt

exploratory effort

Notes: (1) The terms used to label these techniques are not standardised. (2) Rate of production is the first derivative of cumulative

production with respect to time. Alternative terms are the rate of change of cumulative production, or more simply production. Similar

comments apply to the rate of discovery, although here the derivative may be with respect to either time or exploratory effort. See Sorrell

and Speirs (2009) for mathematical definitions of the explained an explanatory variables

8Non-linear regression is straightforward with modern computer

technology, but the earlier literature uses simpler methods such as

the linear transformation of the functional form followed by a

linear regression (Hubbert, 1982). If the production or discovery

cycle is well advanced, then it is possible to estimate the URR

through visual identification of the asymptote to which the curve

is trending (Bentley, 2009).

9Hubbert (1982) begins with an assumed parabolic relationship

between production and cumulative production and uses this to

derive a logistic equation for cumulative production over time.

However, this formal derivation came more than 20 years after he

first referred to the logistic model (Hubbert, 1959; Sorrell and

Speirs, 2009).


technology and the opening up of new oil-producingregions. In contrast, the logistic model provides arelatively poor approximation to the productioncycle for most other oil-producing regions, especiallywhen this occurs over a shorter period of time(Brandt, 2007).

The logistic is one of a family of symmetric andasymmetric curves that are widely used to modelgrowth processes (Meade, 1984; Tsoularis andWallace, 2002). Alternatives include the generalisedlogistic (Nelder, 1971), Bass (1969), Gompertz(Moore, 1966) and bi-logistic (Meyer, 1994) as wellas the cumulative lognormal, Cauchy and Weibull

distributions (Wiorkowski, 1981; Meade, 1984).Oneof the few that has been applied to oil depletion isthe cumulative normal, which Bartlett (2000) fittedto the US production data. While there is no robusttheoretical basis for choosing between these models,there are a variety of reasons for expecting theproduction cycle to be asymmetric. For example,production could decline rapidly after the peak asthe large fields are depleted, or could decline moreslowly if enhanced oil recovery (EOR) techniquesare used. Brandt (2007) analysed 74 oil-producingregions and found that the rate of productionincrease exceeded the rate of decline in over 90% of

Box 1. The Hubbert logistic model of oil depletion

The key features of Hubbert�s logistic model are that:

- Cumulative production is modelled with a logistic function;

- Production is modelled with the first derivative of the logistic function;

- The production profile is symmetric (i.e. maximum production occurs when the resource is half depleted and its functional form is

equivalent on both sides of the curve);

- Production increases and decreases in a single cycle without multiple peaks.

� Hubbert (1959, 1982) noted frequently that these were only simplifying assumptions to allow tractable analysis.

� Hubbert�s model for production is defined mathematically as follows:

QðtÞ ¼ Q11þe�aðt�tm Þ Q0ðtÞ ¼ dQðtÞ

dt ¼aQ1e�aðt�tm Þ

1þe�aðt�tm Þð Þ2

� Where Q(t) is cumulative production; Q¢(t) is production; Q1 is the URR; a defines the �steepness� of the cumulative production curve;

and tm specifies the time when cumulative production reaches one half of the URR

� The diagram shows the resulting cumulative production cycle (top) and production cycle (bottom)


cases, suggesting that an asymmetric model may bemore appropriate. However, when Moore (1962)fitted an asymmetric Gompertz function to the USdata he obtained an URR estimate that was almosttwice as large as that from the logistic model for acomparable goodness of fit. Very similar resultswere later obtained by Wiorkowski (1981)10 andCleveland and Kaufmann (1991).11 This highlights ageneric weakness of curve-fitting techniques, name-ly: different functional forms often fit the data com-parably well but give very different estimates of theURR (Ryan, 1966).

Production cycles often have more than onepeak as a result of economic, technical or politicalchanges or the opening up of a new region (Laherrere,2000a, b). For example, Illinois experienced two

production cycles as a consequence of early devel-opments in exploration technology (Hubbert, 1956).Laherrere (2004a, b) argues that most countrieshave several cycles of discovery and production andare best modelled by two or more curves. If eachcurve represents the production of resources from ageologically homogeneous region, then the aggre-gate URR may be derived from the sum of estimatesfrom the individual curves. Several authors havefollowed this approach (Imam and others, 2004;Mohr and Evans, 2007; Patzek, 2008; Reynolds andKolodziej, 2008) and any cumulative productiontrend could in principle be decomposed into the sumof an arbitrary number of logistic curves (Meyer andothers, 1999). However, the better fit of a morecomplex model may not be justified statistically,owing to the risk of �over-fitting�. A useful rule ofthumb is to ensure that there are at least 10 datapoints per parameter, but many published examplesof curve-fitting do not meet this criterion (Laherrere,2002a, b; Campbell and Heapes, 2008). Also, theresults will be unreliable if further cycles areexpected in the future. This highlights a secondgeneric weakness of curve-fitting techniques: theirinability to anticipate future cycles of discovery andproduction in aggregate regions.

If cumulative production grows logistically,then a plot of the ratio of production to cumulativeproduction as a function of production should beapproximately linear (Hubbert, 1982). If a linearregression is fit to this data, then the URR may beestimated by extrapolating and identifying theintersection with the cumulative production axis(Fig. 3). This straightforward technique was popu-larised by Deffeyes (2005) and is sometimes termed�Hubbert Linearisation� (HL). While methodologically

Figure 2. A fit of the logistic model to US oil production. Source:

IHS Energy. Note: Data for all US states, including crude oil,

natural gas liquids, condensate and heavy oils.

Figure 3. �Hubbert Linearisation� of US oil production. Note:

Data for all US states, including crude oil, natural gas liquids,

condensate and heavy oils.

10Wiorkowski (1981) compared a �Generalized Richards� model

(which can take an exponential, logistic, or Gompertz form

depending on the parameters chosen) with a cumulative Weibull

and found that they fit the US cumulative production data equally

well but led to significantly different URR estimates (445 and

235 Gb, respectively).

11Cleveland and Kaufmann (1991) fitted a logistic curve to the US

production data through to 1988 and found that the adjusted R2

changed only from 0.9880 to 0.9909 as the value of the URR

varied from 160 to 250 billion barrels.


straightforward, it is equivalent to fitting a logisticcurve to cumulative production and hence will beunreliable if (as is usually the case) cumulativeproduction departs from the logistic model.

The linearisation technique is closely related tothe decline curve analysis used by reservoir engi-neers to project the future production of individualwells or fields (Arps, 1945; Arps, 1956; Gowdy andRoxana, 2007). If production declines exponentially,then the URR for the field may be estimated byplotting production against cumulative production,fitting a linear regression and extrapolating this untilit crosses the cumulative production axis (Fig. 4).This technique is widely used, but alternative func-tional forms for decline curves should also beinvestigated since the exponential model canunderestimate the URR (Kemp and Kasim, 2005; Liand Horne, 2007). Nevertheless, a �productiondecline curve� for an individual field may be expectedto be lead to more reliable estimates of URR than anequivalent technique for an oil-producing region.

In sum, curve-fitting to production trends isstraightforward and relies on data that are readilyavailable, relatively accurate and free from thecomplications of reserve growth (see below). How-ever, while these techniques may sometimes providereliable estimates in regions that are well past theirpeak of production, they have important drawbacks

including the lack of a robust basis for the choice offunctional form; the sensitivity of the estimates tothat choice; the risk of �over-fitting� multi-cyclemodels; the inability to anticipate future productioncycles; and the neglect of economic, political andother variables that have shaped and will continue toshape the production cycle. These drawbacksare also shared by the discovery-based techniquesdescribed shortly.

DISCOVERY OVER TIME TECHNIQUES

Curve-fitting to discovery trends was firstintroduced by Hubbert (1962, 1966, 1982) and hassince been employed by other authors, includingLaherrere (1999, 2003, 2004a, b, 2005). These tech-niques have much in common with those describedabove and raise a comparable set of issues andconcerns. In principle, the extrapolation of discoverytrends should provide more reliable estimates of theURR because the discovery cycle is more advanced(Fig. 5). However, discovery data is less accessibleand reliable than production data and, unlike thelatter, is estimated to different levels of confidence.Of particular importance is (a) whether the discov-ery estimates are based on proved reserves or provedand probable reserves; and (b) how revisions to

Figure 4. Linearisation of exponential production decline for the UK Forties field. Source: Gowdy

and Roxana (2007). Note: The introduction of enhanced oil recovery (EOR) techniques in 1986

appears to have only temporarily increased production in this field without having a significant

impact on the URR. Gowdy and Roxana (2007) observe similar patterns in the Yates field in Texas

and at Prudhoe Bay in Alaska, where EOR appears to have increased production at the expense of

steeper decline rates in later years. Whether this conclusion applies more generally is a topic of

considerable dispute.


reserve estimates are reflected in the discovery data.These points are reviewed in Box 2 and Box 3.

Hubbert�s discovery projections were based onthe idealised life-cycle model illustrated in Figure 5(Hubbert, 1959). Hubbert assumed that both cumu-lative discovery and cumulative production grewlogistically, with the former preceding the latter bysome time interval. As the peak rate of discoveryprecedes the peak in production, identification of theformer could form a basis for predicting the latter.

Hubbert (1962) fitted a logistic curve to the USdata on cumulative proved discoveries and used thisto estimate an URR of 170 Gb. Subsequent studiesconfirmed this estimate (Hubbert, 1966, 1968, 1974,1979, 1982) but several authors questioned Hubbert�sresults. For example, Ryan (1965, 1966) fittedlogistic curves to the US production and discoverydata and found they led to widely different estimatesfor the URR. He also showed that much largerestimates could be cited with equal justification andthat the estimates increased rapidly with the addi-tion of only a few more years of data. Cavallo (2004)recreated Hubbert�s original dataset and found that

the R2 for the best-fitting models changed only from0.9946 to 0.9991 as the value of URR varied from150 to 600 Gb. Similarly, Cleveland and Kaufmann(1991) found Hubbert�s results to be highly sensitiveto the length of data series chosen.

The assumption that the discovery cycle takesthe same form as the production cycle appears nei-ther necessary nor plausible—although it worksfairly well for the US when proven reserve data areused.12 The factors influencing discovery at differentpoints in time are likely to be different from thoseinfluencing production at a later point in time andthe skewed field-size distribution would be expectedto (and frequently does) lead to a sharply risingcumulative discovery cycle and an asymmetric dis-covery cycle (Nehring, 2006a, b, c). For similarreasons, there is unlikely to be a predictable time lagbetween the peaks in discovery and production.

While Hubbert used proven (1P) reserves toform his cumulative discovery estimates, subsequentauthors use proven and probable (2P) reserves(Campbell, 1997; Laherrere, 2004a, b; Bentley andothers, 2007). Since these are generally larger than1P estimates, they should lead to a higher estimateof the URR. Also, cumulative discovery estimatestend to increase over time even if no new fields arefound as a result of reserve growth at existing fields(Box 2). In order to take account of this, Laherrere(1996, 2002a, b) and others use backdated estimateswhich are typically larger than those made at thetime of field discovery as a result of reserve growthin the intervening period (Box 2). Hubbert (1967)was one of the first persons to develop backdateddiscovery estimates, but did not use these for dis-covery projection. A discovery cycle based onbackdated estimates will be a different shape fromone based on current estimates and will have a dif-ferent date for the peak in discoveries.

Backdated estimates provide a more accuratepicture of what was found at a particular time andare also more suitable for estimating the URR sincethe cumulative discovery curve is more likely totrend to an asymptote. However, they can bemisleading since the sizes of fields discovered at

Figure 5. Hubbert�s idealised lifecycle model of an oil-produc-

ing region. Source: Hubbert (1982). Note: Cumulative discov-

eries are calculated from the sum of cumulative production and

declared reserves. When reserves reach their maximum value,

the rate of production (which is still increasing) is equal to the

rate of discovery (which is decreasing).

12�…Because petroleum exploration in the US began very early,

because the initial exploration and discoveries occurred in what

has proved to be relatively minor basins, because early drilling

technology was very limited in its drilling depth capabilities, and

because discoveries in the major basins only hit their stride

between 1910 in 1950, the US comes closest to a symmetric

discovery curve of any major oil producing country or region�(Nehring, 2006a, b, c).


different times will not have been estimated on aconsistent basis (i.e. they will reflect differingamounts of reserve growth). For the same reason,both the height and shape of the curve will changeover time, and they will not represent the ultimateresources that were found since more reserve growthcan be anticipated in the future (Fig. 6). Hence, toprovide reliable estimates of the URR, backdateddiscovery data should be adjusted to allow for futurereserve growth (Drew and Schuenemeyer, 1992;Root and Mast, 1993). While there will be uncer-tainty regarding the appropriate growth function to

use (Box 2), the failure to do so is likely to lead tounderestimates of the URR. Lynch (2002), forexample, likens the neglect of future reserve growthto: ‘‘…comparing old orchards with newly plantedsaplings and extrapolating to demonstrate decliningtree size.’’ Campbell and Laherrere claim thatadjustments are unnecessary, since 2P estimatesshould not change much following field discovery.However, this is inconsistent with the available evi-dence (Sorrell and others, 2009) and also contra-dictory, since, if 2P estimates are relatively stable,then there should be no advantage in backdating.

Box 2. Understanding reserve estimates

Oil reserves are those quantities of oil in known fields which are considered to be technically possible and economically feasible to extract,

under defined conditions. Reserve estimates are inherently uncertain and are quoted to three levels of confidence, namely proved reserves

(1P), proved and probable reserves (2P) and proved, probable and possible reserves (3P). However, these terms are defined and

interpreted in different ways and regulatory bodies have made only limited progress towards standardisation (Thompson and others,

2009a, b).

Publicly available data sources are poorly suited to studying oil depletion and their limitations are insufficiently appreciated (Bentley and

others, 2007). The databases available from commercial sources are better in this regard, but are also expensive, confidential and not

necessarily reliable for all regions. Only a subset of global reserves is subject to formal reporting requirements and this is largely confined

to the reporting of highly conservative 1P data for aggregate regions. Country-level 2P reserve estimates are available from commercial

sources but only 1P data is available for the US. In the absence of audited estimates for individual fields, analysts must rely on

assumptions whose level of confidence is inversely proportional to their importance—being lowest for those countries that hold the

majority of the world�s reserves.

The common practice of adding 1P estimates to form a regional or global total is statistically incorrect and likely to significantly under-

estimate actual 1P reserves. Aggregation of 2P estimates should introduce less error but this may be either positive, negative or zero

depending on the interpretation of the estimates and the shape of the underlying probability distributions—which is rarely available.

Reserve estimates of known fields commonly undergo repeated revisions as a result of better geological understanding, improved extraction

technology, variations in economic conditions and changes in reporting practices. Changes in reserve estimates over time are usually

referred to as reserve additions, although the changes could be either positive or negative.

Box 3. Understanding cumulative discovery estimates

The cumulative discoveries in a region in a particular point in time may be estimated from the sum of cumulative production and declared

reserves. Depending upon the data available, it may be possible to estimate cumulative 1P, 2P or 3P discoveries. The rate of discovery, or

more simply discovery, is the first derivative of cumulative discovery with respect to time.

Cumulative discoveries are not changed by production since this merely transfers resources from one category (reserves) to another

(cumulative production). However, cumulative discoveries may be either increased or reduced by revisions to the reserve estimates for

known fields. The latter is usually referred to as reserve growth since estimates are normally revised upwards rather than downwards

(Thompson and others, 2009a, b). However, a more accurate term is cumulative discovery growth, since reserves are continually being

depleted by production. An alternative terms is �ultimate recovery growth� since what is growing are the estimates of what will ultimately

be recovered from the field or region. A growth function is a plot of the growth of cumulative discovery estimates for a particular category

of field over time (Verma, 2003; Klett and Gautier, 2005; Verma, 2005). In some cases, the growth can be substantial, especially if 1P data

is used. For example, Lore and others (1996) found that the estimated size of offshore fields in the Gulf of Mexico doubled within six

years of discovery and quadrupled within 40 years. In principle, the reserve growth observed using 2P estimates should be less than this -

although analysis of country-level data shows that this is not necessarily the case (Sorrell and others, 2009).

Some data sources record reserve revisions in the year in which they are made and make no adjustment to the discovery data for earlier

years. Others backdate the revisions to the year in which the relevant fields were discovered. The logic of the first approach is that the

reserves did not become �available� for production until the estimate was revised and therefore should only appear at the time of the

revision. The logic of the second approach is that the reserves are contained in a field that was discovered many years earlier, so

backdating provides a more accurate indication of what was �actually� found at that time as well as what will ultimately be recovered from

that field. Both of these approaches have their merits, but the difference between them is not always appreciated. When cumulative

discovery estimates are backdated, the �shape� and �height� of the cumulative discovery curve will change as revisions are made (Fig. 6).


The complications introduced by reservegrowth are illustrated by Nehring�s study of thePermian Basin and San Joaquin valley in theUS—which both have been producing oil for morethan 80 years (Nehring, 2006a, b, c). Nehring em-ploys backdated cumulative proved discovery esti-mates and corrects these with Hubbert�s (1967)growth function to estimate the ultimate resourcesdiscovered in each time interval. When using datathrough to 1964, the corrected cumulative discoverycurve for the Permian Basin suggests an URR of27.5 Gb, compared to only 19 Gb with the uncor-rected data. However, when using data through to2000, the URR estimate is 37% larger, and theestimated date of peak discovery has moved back intime. While Hubbert�s growth function predictssubstantial reserve growth, it nevertheless underes-timates the growth that actually occurred—especially for the older fields. Nehring comments:

‘‘…the continuous upward movement in the [cor-

rected] cumulative discovery curve makes this curve

useless as a tool for predicting the ultimate recov-

ery. Estimates of ultimate recovery derived from

cumulative discovery curves are only valid if one

can guarantee that there will be no further increases

in the ultimate recovery of discovered fields … no

such guarantee can be made.’’ (Nehring, 2006a, b, c)

This example could be unrepresentative, how-ever. Proven reserve estimates would be expected togrow by more than 2P estimates since they representa more conservative estimate of recoverable re-serves. As a result, the estimates derived from 2Pdata should be more reliable. Also, Nehring relieson a growth function that is nearly 40 years oldand is only applied to the most recent 30 years of

data—despite more recent growth functions beingavailable (Verma, 2003, 2005). In addition, the ob-served reserve growth derives primarily from CO2

injection in large fields of low permeability and suchtechniques may neither be suitable nor available inother fields and regions. However, Nehring high-lights an important problem that is common to alltechniques that rely on discovery data.

DISCOVERY OVER EFFORT TECHNIQUES

If data are available, then exploratory effortshould provide a better explanatory variable thantime, since the corresponding rate of discoveryshould be less affected by time-varying factors suchas changes in tax regimes. Hubbert (1967) was oneof the first persons to fit curves to discovery data as afunction of exploratory effort and variants of thisapproach have subsequently been employed byother authors (Cleveland, 1992; Campbell, 1996;Ivanhoe, 1996; Laherrere, 2002a, b). This approachis also the basis of discovery process modelling,which was first introduced by Arps and Roberts(1958) and has subsequently been widely used(Power and Fuller, 1992a, b; Drew and Schuenemeyer,1993; Drew, 1997). Both methods rely on backdateddiscovery estimates and require some method forestimating future reserve growth. However, only thelatter needs data on individual fields.

There are several ways of measuring explor-atory effort,13 although the choice will be largelydictated by data availability. The most commonmetric (the cumulative number of �new field wildcat�wells (NFWs)) may not be the best, however, sincemuch reserve growth derives from developmentrather than exploratory drilling (Cleveland, 1992).There are also difficulties when accounting for thedelays between drilling and reserve additions, indistinguishing between the search for oil and thesearch for gas resources and in allowing for spatialand temporal variations in drilling patterns (Byrdand others, 1985).

Time

FutureReservegrowth

Future discoveries

Cumulative discoveries

Cumulative discoveriesYear X

Cumulative discoveriesYear X+ N

Estimated ultimate discoveries

Figure 6. The impact of reserve growth on discovery

projections.

13Including the cumulative length of exploratory drilling

(Hubbert, 1967), the total number of exploratory wells (Ryan,

1973), the number of successful exploratory wells (Moore, 1962),

the cumulative length of successful exploratory wells (Stitt, 1982)

or the cumulative length of all wells (i.e. both exploratory and

development) (Cleveland, 1992). A distinction may also be made

between the first exploratory well to be drilled (�new field

wildcats� or NFWs) and subsequent wells.


A �creaming curve� is a plot of backdatedcumulative discoveries against exploratory effort(Fig. 7),14 while a �yield per effort� (YPE) curve is aplot of the rate of discovery against exploratory ef-fort (i.e. the first derivative of the creaming curve)(Fig. 8). Provided the �yield� from drilling declines asexploration proceeds, an estimate of the URR maybe derived from the asymptote of the former, theintegral of the latter or the corresponding parame-ters in the fitted curves. Changes in yield representthe net effect of changes in the success rate (thefraction of exploratory wells drilled that yield com-mercially viable quantities of oil) and changes in theaverage size of discovered fields. Evidence suggeststhat the success rate in most regions has declinedonly relatively gently, if at all, indicating thatimprovements in exploration technology have par-tially or wholly offset the anticipated decline in thesuccess rate as a result of the declining number ofundiscovered fields (Meisner and Demirmen, 1981;Forbes and Zampelli, 2000).15 However, since large

fields generally occupy a larger surface area, theytend to be found relatively early even if drilling israndom (Arps and Roberts, 1958). In contrast, theaverage size of discovered fields in many regionshas fallen by an order of magnitude since theearly days of exploration. Hence, declining YPEis most likely to be the result of falling averagefield-sizes.

Hubbert�s investigation of YPE curves wasdeveloped in response to Zapp (1962), who usedexploratory effort as an explanatory variable butassumed that the yield would remain unchanged,leading to an unrealistically large estimate for theUS URR (590 Gb). Zapp�s approach was subse-quently adopted by Hendricks (1965), who simplyassumed that the yield would decline linearly. Incontrast, Hubbert (1967) based his forecast of futureyield on a detailed analysis of past trends, whichshowed a negative exponential decline.

Hubbert (1967) fitted a negative exponentialcurve to his estimates of YPE in the lower 48 USstates and estimated a URR of �170 Gb, consistentwith his estimates from production and discoveryprojection However, Harris (1977) showed thatHubbert�s method violated standard statistical pro-cedures, placed excessive weight on the last (andmost uncertain) data point and led to systematicallybiased estimates. The only reason Hubbert�s esti-mate was consistent with his earlier study was thatthe discovery rate had increased—something whichHubbert considered to be both anomalous andtemporary. Harris also showed that a YPE curve foran aggregate region such as the US will not neces-sarily be exponential, even if the trends for indi-vidual regions are exponential. As a result, the URRestimated from a curve fit to aggregate data will bedifferent from the sum of the estimates from curvesfit to regional data.

A few recent studies have used creamingcurves rather than YPE curves. Laherrere (2001,2002a, b, 2004a, b) has estimated creaming curvesfor all the regions of the world and found that theytend to rise steeply in the early stages of explora-tion, reflecting the discovery of small number oflarge fields. He fits �hyperbolas� to these data, butrarely provides either the functional form or thegoodness of fit. Smooth curves (e.g. Fig. 1) may bethe exception rather than the rule, however. Forexample, Sneddon and others (2003) show how theYPE in an exploration play typically exhibits two orthree plateaus and provide some technical reasonsas to why this may be the case. While diminishing

0

2

4

6

8

10

12

14

16

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Cumulative number of new field wildcats

Cu

mu

lati

ve D

isco

very

(G

b)

Figure 7. Example of a �creaming� curve. Source: IHS Energy.

Note: Name of region withheld on grounds of data confidentiality.

14Laherrere (2004a, b) states that the creaming curve was

invented by Shell in the 1980s, but variants of this approach have

been used in the oil industry for much longer period (Arps and

Roberts, 1958; Arps and others, 1971; Odell and Rosing, 1980;

Harbaugh and others, 1995). Two employees of Shell published a

paper on �the creaming method� in 1981, but this describes a highly

sophisticated (and not widely used) discovery process model that

relies on Monte Carlo simulation of trends in both success rates

and average field sizes and assumes a lognormal field size

distribution (Meisner and Demirmen, 1981).

15The IEA (2008) reports that, over the last 50 years, the global

average success rate has increased from one in six exploratory

wells to one in three. Similarly, Lynch (2002) reports that the

average success rate in the US increased by 50% between 1992

and 2002 and Forbes and Zampelli (2000) report that the US

offshore success rate doubled between 1978 and 1995. In an

econometric analysis, Forbes and Zampelli (2000) estimate that,

over the period of 1986–1995, technological progress increased the

US offshore success rate by 8.3%/year.


returns to exploratory effort are widely observed atthe play level, the same may not always be observedat larger geographical scales because regions arefrequently developed in the order of ease ofexploration and development rather than size(Charpentier, 2003). For example, a combination ofgeological accessibility and improvements inexploration technology led to the largest play inthe Michigan basin being developed relativelylate (Charpentier, 2003). Similar phenomena arereported by Wendebourg and Lamiraux (2002), whofind two exploration cycles in the Paris basin. Whilea creaming curve estimated using data through to1986 leads to an URR estimate of 15Mt, a similarcurve estimated using data through to 1996 leads toa much larger estimate of 46Mt. The larger thegeographical region, the more significant thisproblem could become. Laherrere (2003, 2004a, b)addresses this through the use of multi-cycle mod-els, but provides little statistical support for hischoice of curves and, in some cases, the appropriatenumber is unclear. For example, Laherrere (2003)models the oil and gas resources of the Middle Eastwith two creaming curves but in a subsequent arti-cle, this has increased to four (Laherrere, 2004a, b).The choice appears to be largely determined by the�shape� of the data, with interpretation in terms ofexploration history being made ex post.

The potential for further exploration cyclescannot be established from the statistical analysis ofhistorical data, but only from a detailed evaluation

of geological potential and exploration history. Forsmall, geologically defined regions where explora-tion is well advanced, the probability of new cyclesmay be relatively low, while for large, politicallydefined regions, which are partly unexplored (e.g.owing to the depth of drilling required, or geo-graphical remoteness, or political restrictions) theprobability may be much higher. The reliabilityof curve-fitting, therefore, depends heavily onthe assumption that any new exploration cycleswill have only a small impact on aggregateresources—either because there will be no or fewsuch cycles, or because the discovered resources willbe relatively small. While this judgement may bereasonable for many regions around the world, itremains problematic for key regions such as Iraq.Unfortunately, these are precisely the regions thataccount for a significant proportion of the globalURR.

In sum, while exploratory effort should pro-vide a better explanatory variable than time, curve-fitting techniques must still be used with care.Common difficulties include the inaccuracy of thedata used for discovery estimates, the uncertaintyabout future reserve growth, the apparent sensi-tivity of the results to the choice of functionalform, the existence of multiple exploration cyclesand the inability to anticipate new explorationcycles in the future. Overall, these difficultiesappear more likely to lead to underestimates of theregional URR.

Figure 8. Example of a yield per effort (YPE) curve. Source: IHS Energy. Note: Name of region

withheld on grounds of data confidentiality.


CONSISTENCY OF CURVE-FITTINGTECHNIQUES

We investigated the reliability of curve-fittingtechniques with the help of illustrative data from 10oil-producing regions.16 The data were taken from adatabase supplied by IHS Energy and includes an-nual estimates of production, backdated 2P discov-eries and the number of new field wildcat wells forall oil-producing countries. The selected regions in-clude both individual countries and groups ofcountries, with all but one (Region B) apparentlypast their peak of discovery, and five past their peakof production. Since the objective was to test thereliability of curve-fitting techniques as currentlyused (e.g. Laherrere, 2004a, b), we did not correctthe discovery estimates to allow for future reservegrowth.

We estimated the URR for each region usingproduction decline curves, cumulative discoveryprojections and creaming curves. In each case, weinvestigated the consistency of the estimates ob-tained with different lengths of data series, differentchoices of functional form and different numbers ofcurves, and also compared the estimates producedby each technique. For illustrative purposes, wejudged two sets of results to be consistent if themean URR estimates differed by less than 20% ofthe cumulative production (Q2007) or cumulativediscoveries (D2007) in the region through to 2007. Amore or less stringent definition of consistencywould not significantly change the results, since mostestimates were found to be either broadly consistentor substantially different. The full results of thesetests are given in Sorrell and Speirs (2009) and aresummarised below.

First, the results raise concerns about the reli-ability of the URR estimates from curve-fittingtechniques, at least when (as is usually the case) theyare applied at the country or regional level with datathat has not been corrected for future reservegrowth. In particular, we observed that: (a) in onlyone of the regions examined were the mean URRestimates consistent between all three curve-fittingtechniques; (b) variations in the length of time ser-ies, functional forms and number of curves led toinconsistent results more often than consistent re-sults; and c) the degree of inconsistency in the URRestimates was frequently very large. While estimates

were more likely to be consistent for regions at alater stage of their discovery and/or production cy-cle, inconsistent results were frequently obtained formature regions as well.

Different functional forms were often found tofit the data equally well,17 but to provide substan-tially different estimates of URR. This is illustratedin Figure 9 which shows cumulative discovery pro-jections for each region using both logistic andGompertz functional forms. Taking Region E as anillustration, the difference between the R2 for eachmodel is only 0.003 but the URR estimates differ bya third. The mean difference in the URR estimatesfrom the two models was 59% of the cumulativediscoveries through to 2007 (ranging from 1% to362%), but the mean difference in R2 estimates wasonly 0.001. The estimates only converge when re-gions are at a relatively late stage in their discoverycycle when the asymptote of the curve is clearlyapparent (e.g. Region I). As a result, estimates madeat earlier stages in the discovery cycle can lead tosignificantly different results (see Fig. 10). Contraryto expectations, a logistic or Gompertz functionalform was found to be more appropriate than anexponential in all the regions examined, but thechoice can bias the results (e.g. the Gompertz modelprovides higher estimates in all cases).

The production decline (�Hubbert Linearisa-tion�) technique was found to be particularly unre-liable and the results suggest a systematic tendencyto underestimate the URR. As an illustration,Figure 11 shows the results for Region J. While theonshore data �settles� into an approximate linearrelationship that can be modelled by a single linearregression, the corresponding data for both offshoreand the region as a whole exhibits �trend breaks� thatmay result from additional cycles of exploration andproduction. If this technique had been used at anearlier stage of the production cycle (i.e. prior to thetrend break), then it would have led to a significantunderestimate of the regional URR. Trend breaks ofthis form were observed for six of the ten regionsusing aggregate data and for all of the regions usingeither onshore or offshore data (or both). The fre-quency of such breaks gives little confidence that thedecline curves will remain stable in the future.

16The names of the regions are withheld due to data confiden-

tiality.

17Goodness of fit was measured using simple R2. However, we

recognise that this is not the best measure to use when comparing

�non-nested� models, such as a logistic versus a Gompertz

(Kennedy, 2003).


Figure 9. Cumulative discovery projection results.


Our results also do not support the claim thatcreaming curves are generally more reliable thandiscovery projections. Notably, the creaming curvesfor four of the regions do not exhibit asymptoticbehaviour, although, in two of these cases, the corre-sponding discovery projection was asymptotic. Onceagain, the results were sensitive to the particularfunctional form assumed (see Fig. 12) and while threeof the regions could be fit with either one or twocreaming curves, the corresponding URR estimateswere significantly different (see Fig. 13). Without adetailed knowledge of the exploration history of aregion, it is difficult to justify one choice over theother.

The primary reason for these inconsistent resultsis that the techniques are being applied to large andgeologically diverse regions that lack a consistentexploration history. In addition, we did not alwaysdistinguish between onshore and offshore regions,the data source did not classify exploratory drilling assearching for either oil or gas, and we did not correctthe discovery data to allow for future reserve growth.Future applications of curve-fitting should, therefore,address each of these problems as far as the availabledata permits. Nevertheless, the results are sufficientto demonstrate the limitations of curve-fitting tech-nique as currently used and suggest that the associ-ated URR estimates should be treated with caution.

RECONCILING CURVE-FITTINGWITH ECONOMETRICS

Curve-fitting techniques assume that the �shape�of the production or discovery cycle can be

estimated from the historical data and that thisshape will not be significantly affected by any futurechanges in prices, technology and other relevantvariables. As a result, there has been a tendency toneglect these variables, despite the potential errorsthat may result. For example, low oil prices andpolitical constraints may restrict production whenresources are abundant, while high prices maystabilise or increase production when depletion isadvanced. As a result, many applications of curve-fitting techniques are likely to suffer from missingvariable bias and/or serial correlation of the errorterms. This could lead to biased estimates of model

Figure 10. Cumulative discovery projections for

Region E using logistic model—sensitivity to

length of data series.

Figure 11. Production decline curves for Region J.


parameters (including the URR), underestimates ofthe associated standard errors and overestimates ofthe model goodness of fit. Several examples of thisare provided in Sorrell and Speirs (2009).

These problems may potentially be addressedby including one or more �lags� of the dependentvariable within the model specification (e.g. makingproduction in the current year a function ofproduction in one or more previous years), but the

re-specified model may not necessarily lend itself tothe estimation of URR. A more promising approachis to include some of the economic and politicaldeterminants of discovery and/or production withinthe model specification. This effectively gives a �hy-brid� of a curve-fitting approach to estimating URRand an econometric approach to estimating futureoil discoveries (Walls, 1992, 1994). The latter origi-nated with Fisher (1964), who estimated equationsfor exploratory activity, success rate and the averagesize of discovered fields as a function of oil prices,past average discovery size and lagged dependentvariables. Many variants have followed,18 but thesehave mostly focused on forecasting discoveries ra-ther than estimating URR and are also likely to bebiased since they ignore the geological determinantsof discoveries and production (Power and Fuller,1992a, b).

A good example of a hybrid model is Kaufmann(1991), who fits a logistic model to the US cumula-tive production and then uses an econometric modelto account for the deviations between predicted andactual production. This formulation assumes thatgeologic and physical factors cause oil production torise and fall over the long term, while economic andpolitical variables (e.g. legal restrictions on produc-tion)19 lead to short-term variations around thisunderlying trend. This model provides a much betterfit to US production over the period 1947 to 1985,but estimates of the URR require assumptions aboutthe future values of the relevant economic andpolitical variables. Pesaran and Samiei (1995) arguethat Kaufmann�s model is biased because the esti-mation of URR in the first stage does not take intoaccount the effect of economic factors which onlyenter the analysis in the second stage. They avoidthis by modifying the logistic model to allow for thedependence of URR on a number of economic andother variables and re-formulating it to eliminateproblems of serial correlation. Their model explainsover 98% of the variation in the US production overthe period of 1948–1990 and leads to a higher esti-mate of the URR. However, the oil price elasticityof the URR is both symmetric and independent oftime (e.g. the same before and after the peak) whichseems implausible.

Figure 12. Creaming curves for Region A using exponential and

hyperbolic functional forms.

Figure 13. Creaming curve for Region E fitted with one (top)

and two (bottom) hyperbola.

18For example, Epple and Hansen (1981), MacAvoy and Pindyck

(1973), Walls (1994) and Mohn and Osmundsen (2008).

19For example, between 1957 and 1968, the �prorationing�decisions of the Texas Railroad Commission shut in more than

50% of Texan oil-producing capacity.


In a significant innovation, Kaufmann andCleveland (2001) include the average US productioncosts as an explanatory variable and hence removethe need to assume a particular functional form forthe production cycle. Their empirically estimatedU-shaped average cost curve mirrors the bell-shapedproduction curve and represents the net effect ofresource depletion and technical change—withsteeply rising costs after 1970 indicating acceleratingdepletion (Cleveland, 1991). This model accountsfor most of the variation in the US oil productionbetween 1938 in 1991 and highlights the importanceof economic and political variables in determiningthe �shape� of the discovery or production cycle:

‘‘…Hubbert was able to predict a peak in US pro-

duction accurately because real oil prices, average

real cost of production, and decisions by the Texas

Railroad Commission coevolved in a way that

traced what appears to be a symmetric bell-shaped

curve for production over time. A different evolu-

tionary path for any of these variables could have

produced a pattern of production that was signifi-

cantly different from a bell shaped curve…. In ef-

fect, Hubbert got lucky.’’ (Kaufmann and

Cleveland, 2001)

However, while this model overcomes a keyweakness of curve-fitting techniques (i.e. the arbi-trary choice of functional form), it may not beapplicable for other regions owing to the lack of dataon production costs. Moreover, if it is to be used toestimate the regional URR, then some assumption isrequired about future trends in production costs. Inaddition, while Hubbert may have �got lucky� inforecasting the date of peak production, it is muchless clear as to whether accounting for economicvariables will make a significant difference to theestimated URR.

If the required data are available, then verysimilar approaches can be used to modify and im-prove any of the curve-fitting techniques. Forexample, Cleveland and Kaufmann (1991, 1997)modified Hubbert�s exponential model of YPE toaccount for short-term changes in oil prices and therate of drilling. Their equation provides a muchbetter fit to historical trends and shows how periodsof relative stability or increases in YPE were asso-ciated with changes in the rates of drilling and/or oilprices. However, again, depletion dominates in thelong-term, implying that the revised model may notsignificantly change the estimates of URR (Kauf-mann and Cleveland, 1991).

Some authors attempt to separate the effect oftechnical change from that of resource depletion

(Power and Jewkes, 1992; Iledare and Pulsipher,1999). For example, in their study of YPE in the Gulfof Mexico, Managi and others (2005) model deple-tion by cumulative discoveries and technical changeby an index of the annual number of innovationsadopted by the offshore industry, weighted by theirrelative importance (NPC, 1995; Managi and others,2004). The results show that the pace of technicalchange has increased since 1975, greatly expandingthe area of exploration and leading to a YPE in 2000that is comparable to that achieved 50 years previ-ously. While the geological diversity of the Gulf ofMexico contributes to this result, a more likely rea-son is that exploration has been geographically re-stricted in the past, owing to a combination of thetechnical difficulties of deep-water drilling andchanging licensing regimes (Priest, 2007). As a result,fields have not been found in the approximatedeclining order of size that is normally assumed.

In sum, while hybrid models improve on stan-dard curve-fitting, they may be more suitable forshort-term supply forecasting than for estimatingURR. The latter requires assumptions about thefuture values of the relevant explanatory variablesand despite their better fit to historical data, it is notobvious that hybrid models lead to substantiallydifferent estimates of URR. However, they do allowthe dependence of the URR on energy prices andother factors to be directly explored.

Hybrid models may still lead to misleadingconclusions if applied to regions that lack eithergeological homogeneity or a consistent explorationhistory and may still be vulnerable to missing vari-able bias since it is impractical to include more thana subset of the variables that could affect productionand/or discovery trends (e.g. tax rules, leasing deci-sions, geographical restrictions on exploration, pro-duction/import/export quotas etc.) (Lynch, 2002).Also the required data are frequently either lackingor unreliable and are rarely available at the level ofdisaggregation required. Problems such as these maypartly explain why there are so few �hybrid� studies,and why the available studies are largely confined tothe United States.20

SUMMARY

M. King Hubbert has left an important andinfluential legacy in many areas, including in

20A notable exception is Dees and others (2007).


particular, the use of curve-fitting methods to esti-mate URR. These techniques were central to the1960s debate on the US oil production and remainequally relevant to the contemporary debate on�peak oil�. This article has reviewed these techniquesand compared them with other approaches to esti-mating URR. The main conclusions are as follows.

First, curve-fitting forms part of a broadergroup of techniques based on the extrapolation ofhistorical trends in production and discovery. Thesetechniques differ in degree rather than kind andshare many of the same strengths and weaknesses.However, a key practical difference is that field-sizedistribution and discovery process techniques re-quire data on individual fields, while simple curve-fitting only requires aggregate data at the regionallevel. All these techniques assume a skewed fieldsize distribution and diminishing returns to explo-ration, with the large fields being found relativelyearly. However, these assumptions will only hold ifdepletion outweighs the effect of technical changeand if the region is geologically homogeneous andhas had a relatively uninterrupted exploration his-tory. Unfortunately, this is rarely the case.

Second, many applications of curve-fitting takeinsufficient account of the weaknesses of this tech-nique, including: the inadequate theoretical basis;the sensitivity of the estimates to the choice offunctional form; the risk of overfitting multi-cyclemodels; the inability to anticipate future cycles ofproduction or discovery; and the neglect of eco-nomic, political and other variables. In general,these weaknesses appear more likely to lead tounderestimates of the URR and have probably con-tributed to excessively pessimistic forecasts of oilsupply. Curve-fitting to discovery data introducesadditional complications such as the uncertainty inreserve estimates and the need to adjust estimates toallow for future reserve growth. The common failureto make such adjustments is likely to have furthercontributed to underestimates of resource size.

Third, tests of curve-fitting techniques usingillustrative data from a number of regions haveshown how different techniques, functional forms,length of time series and numbers of curves can leadto inconsistent results. However, while this raisesconcerns about the reliability of such estimates,the degree of uncertainty declines as explorationmatures. Hence, as more and more regions becomeextensively explored, the accuracy of regional andglobal URR estimates should improve. In addition,some of the limitations of curve-fitting may be

overcome with the use of hybrid models that incor-porate relevant economic and political variables.However, despite their better fit to historical data,such models may not lead to substantially differentestimates of the URR.

Finally, while these limitations do not mean thatcurve-fitting should be abandoned, they do implythat the applicability of these techniques is morelimited than some adherents claim and that theconfidence bounds on the results are wider than isusually assumed. Where possible, resource assess-ments should employ multiple techniques andsources of data and be informed by knowledge of thegeological characteristics and exploration history ofthe region. They must also acknowledge the con-siderable uncertainty in the results obtained.

ACKNOWLEDGMENTS

This article is based on a comprehensive reviewof the curve-fitting literature by Sorrell and Speirs(2009) which in turn formed part of wide-rangingstudy of global oil depletion by the UK EnergyResearch Centre (Sorrell and others, 2009). Theauthors would like to thank the UK ResearchCouncils for their financial support, IHS Energy forallowing the publication of data from their PEPSdatabase, Fabiana Gordon for her help on statisticalissues, and Roger Bentley, Richard Miller, JeanLaherrere and Robert Kaufmann for their helpfulcomments on earlier drafts. The usual disclaimersapply.

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Hubbert’s Legacy: A Review of Curve-Fitting Methods to Estimate Ultimately Recoverable Resources

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