Energy Policy 39 (2011) 790–802

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Energy Policy

0301-42

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Peak oil analyzed with a logistic function and idealized Hubbert curve

Brian Gallagher n

Ecotonics Incorporated, 1801 Century Park East Suite 2400, Los Angeles, CA 90067, USA

a r t i c l e i n f o

Article history:

Received 19 July 2010

Accepted 29 October 2010Available online 20 November 2010

Keywords:

Peak oil

Logistic function

Hubbert curve

15/$ - see front matter & 2010 Elsevier Ltd. A

016/j.enpol.2010.10.053

+1 310 556 9698; fax: +1 323 874 8109.

ail address: bgallagher@ecotonics.net

a b s t r a c t

A logistic function is used to characterize peak and ultimate production of global crude oil and petroleum-

derived liquid fuels. Annual oil production data were incrementally summed to construct a logistic curve

in its initial phase. Using a curve-fitting approach, a population-growth logistic function was applied to

complete the cumulative production curve. The simulated curve was then deconstructed into a set of

annual oil production data producing an ‘‘idealized’’ Hubbert curve. An idealized Hubbert curve (IHC) is

defined as having properties of production data resulting from a constant growth-rate under fixed

resource limits. An IHC represents a potential production curve constructed from cumulative production

data and provides a new perspective for estimating peak production periods and remaining resources. The

IHC model data show that idealized peak oil production occurred in 2009 at 83.2 Mb/d (30.4 Gb/y). IHC

simulations of truncated historical oil production data produced similar results and indicate that this

methodology can be useful as a prediction tool.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In 2007, the US Government Accountability Office released areport titled: Crude Oil: Uncertainty about Future Oil Supply MakesIt Important to Develop a Strategy for Addressing a Peak andDecline in Oil Production, (GAO, 2007). The report includedreferences to twenty-two well-known organizations and research-ers that have expressed their opinions on this issue and includedtheir forecasts of when peak oil might occur. Some forecasts werevery broad or open-ended in their peaking estimates. Using acriterion of conciseness in peak year predictions, twelve forecastswere selected by the author for a simple analysis. The median yearfor the estimated peak oil occurrence was 2011, while the meanyear was 2014. Three years later, the US still does not have anational plan or priority about the peak oil. The only thing that haschanged is that the uncertainty has been removed. Today mostinformed researchers and many government and private organiza-tions agree that peak oil is close at hand, although some ardentoptimists profoundly disagree. Many of the leading researchershave predicted that oil production will peak soon (Campbell andLaherr�ere, 1998; Duncan, 2000; Campbell, 2003; Laherr�ere, 2003).Some researchers claim that oil production has already peaked(Bakhtiari, 2004; Deffeyes, 2005; Simmons, 2005) or may peak in aplateau-like manner (Heinberg, 2005; Shah, 2005).

Peak oil estimates frequently include a bell-shaped curveanalysis created by Hubbert (1956). The famous Hubbert forecaston the peaking of the US lower 48-State production of oil was

ll rights reserved.

proven correct and is the primary basis and motivation for theensuing work, including this analysis. Hubbert provided an extra-ordinary insight into the peak oil situation, using a simple graphicalanalysis. Although very useful, the Hubbert analytical approach haslimitations (Laherr�ere, 1997, 2000; Bardi, 2005; Guseo et al., 2007;Maggio and Cacciola, 2009). Laherr�ere (1997) discussed variants ofthe Hubbert curve and a modified approach for handling multiple-peaking curves encountered when discovery and/or productionoccur/s over several different cycles. He uses the phrase‘‘Multi-Hubbert Modelling’’ to explain this approach in his 1997paper of the same title. In his paper on the strengths andweaknesses of the Hubbert Curve, Laherr�ere (2000) further elabo-rated on constraints to using symmetrical curves and providesnumerous examples of how a multi-Hubbert approach can providemodels that better fit the raw data. In both papers, Laherr�ere (1997;2000) develops and explains the cumulative (logistic) function, butprefers to use its derivative Hubbert function as the primarymodelling tool. He emphasizes the importance of using multi-Hubbert models, whenever necessary, to produce more reliableresults. When applying this approach, Laherr�ere (2000) forecaststhat the world production of liquid fuels will peak before 2010.Maggio and Cacciola (2009) collected extensive data on estimatedultimate resources (EUR) and relied heavily on the multi-Hubberttechnique to develop global oil peaking forecasts based on an EURestimates of 2250–3000 billion barrels. The authors show thatprobable scenarios of peak production start in about 2009 (for anEUR of 2.25 trillion barrels or less); and with higher EURs will occurmost likely prior to 2015, with a lesser chance of occurring up to2021. Bardi’s stochastic models (Bardi, 2005) strongly imply thatthe world oil production will be asymmetrical and decline at asteeper rate than results modelled by the symmetrical Hubbert

B. Gallagher / Energy Policy 39 (2011) 790–802 791

function. For example, Bardi’s modelling suggests that advances inthe technology for discovering and extracting oil may haveaccelerated its total production rate and perhaps the impendingsteep decline. In their paper on world oil depletion models, Guseoet al. (2007) use advanced statistical methods to forecast peakingand declining characteristics of global oil. The authors relied on aGeneralized Bass Model (GBM) to treat global oil growth as anatural diffusion process linked with key exogenous factors ofprice, technology and strategic interventions. The GBM representsa class of equations that include advanced logistic functions that donot depend on knowledge of an EUR, but instead use autoregressivecorrelation techniques to dissect historical production time-seriesdata into elements of diffusion, long-memory interventions, andstochastic white-noise components. Using data through 2005, theauthors forecast a 2007 peak followed by a precipitous resourcedepletion of 90% by 2019. In effect, this paper supports andquantifies Bardi’s sharply skewed decline of global production,due to the technologically accelerated growth. The Guseo paperwill be later used in the results section to portray global oildepletion events that could occur in the near future. The researchersidentified above, and others, have advanced the Hubbert function,variant models and sophisticated statistical simulations, which haveprovided useful oil production peaking and decline profiles. Many ofthese methods use annual oil production data and advanced mathe-matics to model future oil production profiles. Some data are less thanreliable, especially reserves information from several major oilexporting countries that prefer to keep their data proprietary andperhaps overly optimistic for economic reasons.

This paper suggests a complementary approach based oncumulative data and logistic curves for gaining further insight intothe peak oil. The logistic curve is commonly used to describebounded exponential growth and is the foundation for the Hubbertbell-shaped model, which is the derivative of the logistic function.In this paper, a simple methodology is developed to provideadditional illumination into the peak oil phenomenon, using avisual approach similar to Hubbert’s. The major differences fromHubbert’s approach include focusing on the logistic curve and theuse of modern computer technology and electronic spreadsheetsthat were not readily available in the 1950s. The approach hereininvolves very little complex mathematics and relies heavily on arelatively simple computerized graphical technique. The approachto the subject is twofold. First an extended logistic curve isconstructed from the annual production data, and then decon-structed into a new type of curve called an IHC, which can beconsidered a potential production curve. The logistic function,widely used in simulating exponential growth of natural popula-tions, is generally constrained by a resource limitation. Thislimitation can be a composite of several limiting factors, such asfood, habitat, environmental conditions, etc. Quantification of thelimiting factors is usually based on external data; but when the dataare unknown or unreliable, the methodology discussed herein can

provide an estimate of the effective limitation based on precise curve

fitting. It may be necessary to develop several independent limits,which are then combined into a composite limit. This is exactly thecase for global oil production. A logistic curve can be easilymodelled from cumulative production data formed by a simplesummation process. The summed data create a smooth curve,which is similar to a beginning logistic curve in many cases,although exceptions exist. A logistic curve is simulated by usingthree controlling parameters including a composite limiting factor,K. For the analogy with oil, a composite limiting factor could consistof multiple production factors, such as resource availability,technology, market forces, unstable conditions, etc. It is notpossible to assess all of the limiting factors that have influencedthe global oil production system for the past 150 years. However, itis possible to extract information about the combined effect of these

influences, to gain insight into a composite ultimate limitation.When reliable data are available on actual ultimate recoverableresources, these can be used to more precisely define the modelledlogistic curve.

Since 1859, when the first commercial oil production well wassuccessfully completed in Pennsylvania (Paleontological ResearchInstitution, 2010), global oil production has expanded worldwide.The aggregate global oil production has been shaped by variousresource limitations and market conditions, whose historicaleffects are embedded in the cumulative production curveassembled from all worldwide production data. The cumulativeoil production data and simple logistic curve can be examined toreveal the past, present and future story of oil. The model logisticcurve tells a story of an ideal production potential and the IHCvisualizes the results.

2. Premise

The premise for this analysis is that a logistic curve producedfrom one or more cumulative data patterns can be used toapproximate future production of applicable resources. A begin-ning logistic curve reflects an exponential growth that can bemodelled, including the probable peak-production point andultimate limits to production. The end-term prediction of theultimate limit assumes a mirror-image logistic curve, although thismay not occur in reality. Once a sufficient data pattern is availableto reliably model a complete logistic curve, it can then bedeconstructed (differentiated) to generate a theoretical productioncurve or an IHC. The IHC does not predict actual production rates,since they are subject to the vagaries of the present; but it doesprovide a pattern of idealized production data based on anembedded information in the logistic curve. The subject methodis not a rigorous mathematical procedure and is not intended toreplace the precise analytical methods described in the briefliterature review. However, the IHC method offers a simplecomplementary procedure for developing resource depletion pat-terns that can provide further insight into future peaking anddecline of energy and mineral resources. Moreover the subjectprocedure can be carried out by anyone with a standard computerand spreadsheet program, who has access to annual productiondata.

If historical cumulative data can be used to predict futureevents, then partial cumulative data ending before present shouldbe able to predict similar events, if conditions have not changedappreciably. Upon testing this premise, cumulative data obtainedfrom production data ending significantly before the present(2010) appear to reasonably predict production patterns observedtoday. The estimates get better as more cumulative data becomeavailable to tell their story as we approach the actual peaking point.Intuitively, this effect would be expected. The IHC predicts what thetheoretical production pattern should be if the past conditions thatformed the logistic data stay constant, at least on an average.Significant changes in the production conditions such as encoun-tering separate regions, where oil production is already peaking,will result in a readjustment of the IHC pattern. However, normalvariations in annual production rates are usually dampened out inthe cumulative formation process, unless they represent a verylarge change or paradigm shift. For smaller geographical areas, anexample would be a shift to deeper reservoirs, thanks to advancesin technology. But when viewed in the context of a large diversifiedglobal resource, even this major local change is dampened out inthe total cumulative data composed from statistically variedworldwide sources. Use of an IHC allows insightful comparisonwith actual production data and patterns to understand betterlarge consistent deviations in production that do occur.

B. Gallagher / Energy Policy 39 (2011) 790–802792

3. Methodology

A logistic function was used to characterize cumulative global oilproduction. The function selected was a standard population growthfunction originally described by Verhulst (1838). This function is thesolution to the differential equation that describes the populationgrowth. The term rP is defined as an exponential growth-rate, where r

is a rate constant and P is something increasing in size or numbers(like algae or a population). Therefore, rP is the rate of change of P suchthat dP/dt¼rP. The well-known solution to this differential equation isP(t)¼P0 � ert, where P0 is the initial size or population, and r is agrowth-rate constant. Since unlimited growth becomes a problem,Verhulst developed a modification by adding a mortality term, m,such that dP/dt¼rP–mP2. Initially, when P is small, r dominatesgrowth, since mP2 is very small. However, as P becomes large, mP2

becomes the dominant factor. Growth ceases when rP is equal to mP2.Taking out the rP term to get dP/dt¼rP(1–mP/r) and setting K¼r/mprovides the basic differential equation for limited population growth:

dP

dt¼ rPð1�P=KÞ: ð1Þ

Eq. (1) is used to describe the rate of cumulative growth. Theconstant r is sometimes referred to as the biotic potential and K asthe environmental resistance. The r term is considered a naturalpositive force that fundamentally controls growth (or reproduc-tion), while K is a negative feedback factor that provides a limit togrowth. In ecology, K is usually called the carrying capacity andrepresents food, habitat or other necessary conditions required tosustain growth. We can apply this same equation to the cumulativegrowth of global oil production, where m is defined as the end ofproduction (all economically recoverable oil is depleted) and K willrepresent the estimated ultimate resource (EUR).

The solution to Eq. (1) can be found by separating the variablesand then integrating both sides. First, the P and t terms areseparated

dP¼ rPð1�P=KÞdt ð2Þ

Integrating both sides:ZdP¼ r

ZPð1�P=KÞ

� �dt ð3Þ

The solution is

PðtÞ ¼ K= 1þK

P0�1

� �e�rt

� �ð4Þ

where P(t)¼cumulative oil production in barrels as a functionof time

and:

K is the composite resource limit (carrying capacity) in barrelsof oil;P0 is the initial population (or oil production in barrels) at t¼0;r is an exponential factor that controls growth (%); andt is time (years in this case).

Note in Eq. (4) that at t¼0, P(t)¼P0, and at t¼N, P(t)¼K. Eq. (4)is used to describe cumulative global oil production as a function oftime. This equation will be used to develop a complete logisticcurve based on incomplete cumulative oil production data. Con-verting into computer terminology provides an Excel spreadsheetversion of Eq. (4) as follows:

Cumulative production¼ K=ð1þðK=P0�1ÞEXPð�rateðAi�1860ÞÞÞ

ð5Þ

This equation is used to model a complete logistic curve basedon the three controlling parameters of K, P0 and r. ‘‘Rate’’ is used to

represent growth rather than r due to Excel cell name constraintson using r by itself. Ai represents years starting with 1860 in cell A1,which produces a zero-time starting value (from the equation‘‘Year—1860’’). Values of controlling parameters are determined byexperimental curve-fitting methods when compared with a begin-ning cumulative curve formed from actual annual production data.Initial curve-fitting values selected for the equation parameterswere 6% for rate, 5E+08 for P0 and 2.5E+12 for K. Sometimes Eq. (5)must be replicated into two or more equations with separatecoefficients for handling more complex situations (to be discussedlater). If the cumulative data are sufficient, a unique curve can beobtained by adjusting the control parameters until a best-fit curveand minimum root-mean-square deviation (RMSD) error areobtained simultaneously. A relatively simple but long spreadsheet(300 rows) was created to handle all of the data, production models,control parameters and RMSD errors for simulating a completelogistic curve as explained next.

3.1. Spreadsheet

Table 1 contains an abbreviated set of the data used in thisanalysis as an example for discussion. A1 in Eq. (5) represents thefirst cell in Column A, starting in 1864 and extending to 2152 insix-year increments. The actual spreadsheet model includes 300rows and extends from 1860 to 2160.

Column B is annual oil production data and Column C calculatescumulative oil production data by summing up the annual data. LetBi represent cells of annual data and Ci represent cells of cumulativedata. Then, equation Ci¼cell Bi+cell Ci�1 will produce the cumu-lative data for Column C. This process integrates the production dataand produces a beginning logistic curve. Conversely, if the cumu-lative data are modelled with Eq. (5) in Column E, the modelproduction data can be derived by Di¼Ei + 1–Ei. This is equivalent todifferentiating the logistic curve into an IHC. Column E of Table 1contains Eq. (5) used to develop simulated logistic curve data. Thecumulative logistic curve always rises (unless the production fallsto zero) similar to a space ship that climbs upward on a predictablepath and begins to level off (but never comes back down!).

The logistic curve approaches its asymptotic limit referred to as theEUR. This value is equal to the area under the IHC and represents theintegral of the IHC function. Column F is the squared deviation errorbetween the simulated cumulative production data (Column E) andthe actual cumulative production data (Column C). Column F will besummed and averaged and then its square root divided by thecumulative production amount at the end of the series. This producesa least RMSD error in percent. The RMSD is minimized by adjustingthe parameters of Eq. (5) to maintain the slope of the simulatedlogistic curve equal to the slope of the beginning cumulativeproduction curve.

4. Annual and cumulative production data

Annual oil production data, excluding non-petroleum-derivedliquid fuels, from 1860 to 2008 were assembled from severalsources for the following years:

�

1860–1949: Personal communication, R. J. Andres (CDIAC,2010). Original data in metric tons were converted by one tonne¼7.333barrels. � 1950–1964: Compiled by Worldwatch Institute from US

Department of Defense and US Department of Energy data(Worldwatch Institute, 2010).

� 1965–2008: BP Statistical Review of World Energy 2009

(BP, 2010).

Table 1Abbreviated data subset showing every 6th data value from 1864 to 2152.

(A) Datareference year

(B) Annual oilproduction data Mb/y

(C) Cumulative oilproduction data Mb

(D) Annual oilproduction model Mb/y

(E) Cumulative oilproduction model Mb

(F) Cumulative oilcurve deviation(col E�col C)^2

1864 2 11 28 495 2.34E+05

1870 6 35 39 699 4.41E+05

1876 11 91 55 988 8.05E+05

1882 37 251 78 1397 1.31E+06

1888 51 495 111 1974 2.19E+06

1894 88 982 156 2790 3.27E+06

1900 148 1724 221 3942 4.92E+06

1906 212 2902 311 5568 7.11E+06

1912 348 4757 439 7863 9.64E+06

1918 499 7417 619 11,098 1.36E+07

1924 1038 12,463 872 15,656 1.02E+07

1930 1432 20,218 1226 22,068 3.42E+06

1936 1804 29,355 1721 31,075 2.96E+06

1942 2091 41,977 2416 43,717 3.03E+06

1948 3423 58,585 3403 61,493 8.45E+06

1954 5020 84,409 4878 86,759 5.52E+06

1960 7680 124,041 7339 123,947 8.71E+03

1966 12,627 185,226 11,976 182,777 6.00E+06

1972 19,602 285,221 19,012 278,978 3.90E+07

1978 23,132 416,481 22,334 408,050 7.11E+07

1984 21,070 548,013 21,926 540,405 5.79E+07

1990 23,909 683,658 23,531 676,451 5.19E+07

1996 25,530 830,552 26,439 827,665 8.33E+06

2002 27,186 992,041 28,954 995,578 1.25E+07

2008 29,885 1,168,491 30,128 1,174,200 3.26E+07

2014 29,578 1,353,896 RMSD¼0.3233%

2020 27,396 1,524,433

2026 24,037 1,677,481

2032 20,109 1,808,086

2038 16,163 1,914,827

2044 12,580 1,999,028

2050 9549 2,063,611

2056 7111 2,112,088

2062 5221 2,147,890

2068 3793 2,174,015

2074 2735 2,192,911

2080 1961 2,206,492

2086 1401 2,216,209

2092 998 2,223,138

2098 709 2,228,067

2104 503 2,231,568

2110 357 2,234,052

2116 253 2,235,812

2122 179 2,237,059

2128 127 2,237,942

2134 90 2,238,567

2140 64 2,239,010

2146 45 2,239,323

2152 32 2,239,544

B. Gallagher / Energy Policy 39 (2011) 790–802 793

Fig. 1 illustrates the historical global oil production data from1930 to 2008. These data show an annual oil production value of81.8 Mb/d (29.9 Gb/y) in 2008 with a total cumulative productionof 1.168 trillion barrels. The figure also displays EIA liquid fuels data(EIA, 2010a) and EIA crude oil data (EIA, 2010b) for comparison. TheOil Production Data used in this figure are intended to characterizemarketable liquid fuels refined from crude oil and natural gasliquids that originate from crude oil production activities.

Fig. 1 displays a slight ‘‘bend’’ upward in the cumulativeproduction curve beginning in 1970, and then downward in1980. This bend causes difficulty in forming a logistic model thatwill reliably forecast future production. The bend is due to the data‘‘hump’’ between 1970 and 1980, the result of several large oilregions peaking during this period (including North America andthe Former Soviet Union). This effect would be considered a majorchange that cannot be dampened out. This condition represents atwo-cycle production process as explained by Laherr�ere (2000),

and requires a multi-Hubbert modelling approach (or multi-logistic model for the subject methodology). Therefore, the datawere separated into two groups: (1) a plot of production thatpeaked in the 1970s and early 1980s (‘‘first peak’’ data) and(2) original data modified by subtracting out the first peak data.A Hubbert function of the form P(t)¼Pm/(2+2 cosh(ct)) was used togenerate the assumed first peak data. As suggested by Laherr�ere(2000) and applied by Maggio and Cacciola (2009), the coefficientswere selected to make the first peak cumulative data total 150billion barrels (Gb). These coefficients are summarized at the end ofTable 2.

Fig. 2 depicts the modified production and cumulative datacurves. By separating out the early peaking data, the modifiedcumulative production curve E is significantly improved without aconspicuous bend. The first peak cumulative curve F is already acomplete and perfect logistic curve, since it was formed by atheoretical Hubbert function. These changes facilitate a more

Fig. 1. Global annual and cumulative oil production data through 2008.

Table 2Coefficients used in modelling equations.

Data set and analysis Type Figurenos.

Model Eq. (s) Initial values(P0)

Growth(rates) (%)

SeparateEURs (K)

Combined EUR (K) SeparateRMSDs (%)

CombinedRMSD (%)

Global oil 1860–2008 2-cycle 3 and 4 (6)a First peak 5.420E�1 23.020 1.508E+11 2.2401E+12 0.6633 0.3233

(6)b Main peak 3.927E+8 5.770 2.090E+12 0.3869

Global oil 1860–2000 2-cycle 5 and 6 (6)a First peak 5.420E�1 23.020 1.508E+11 2.2301E+12 0.6633 0.2999

(6)b Main peak 3.927E+8 5.780 2.080E+12 0.3869

Global oil 1860–1980 2-cycle 7 and 8 (6)a First peak 5.420E�1 23.020 1.508E+11 2.3101E+12 0.6633 0.5228

(6)b Main peak 3.927E+8 5.791 2.160E+12 0.3869

Global oil 1860–1960 1-cycle 9 and 10 (6)b Main peak 3.700E+8 5.800 2.480E+12 2.6301E+12 0.3869 0.8189

USA oil 1860–2008 1-cycle 12 and 13 (6)b Main peak 3.496E+6 5.534 N/A 2.3700E+9 N/A 1.0185

USA oil 1860–1955 1-cycle 15 and 16 (6)b Main peak 1.101E+8 6.900 N/A 2.2100E+9 N/A 1.1625

First peak data FP(t)¼Pm/(2+2� cosh(c� (year�pkyr)))�365.25(Mb/py); period¼1942–2008.

Pm (peak multiplier)¼100; c¼time modifier¼0.24336; pkyr¼peak year set at 1975.

Peak production of 25 Mb/pd occurs in 1975; total cumulative production¼150 GBO in 2008.

B. Gallagher / Energy Policy 39 (2011) 790–802794

reliable curve-fitting process that generates two separate simu-lated logistic curves. The modified cumulative data (E) and the firstpeak cumulative data (F) are used to generate model logistic curvesusing Eq. (5). Once minimum RMSD’s are obtained for both logisticcurves, their separate curve equations and respective controlparameters are combined to form a final model logistic curve.The IHC data represent the differentiating of the final logistic curve,by the incremental differencing process of the spreadsheet. ThisIHC is then compared with the original production data.

The cumulative oil production data were used to create abeginning logistic curve shown in Fig. 3. Also shown is the finalmodel logistic curve, generated by curve matching techniques toobtain a ‘‘best fit’’ with a least RMSD. Fig. 4 compares actual oilproduction data with the modelled IHC. The IHC peak occurred in2009 at 82.6 Mb/d (30.2 Gb/y). An actual oil production reached amaximum in 2008 of 81.8 Mb/d (29.9 Gb/y) and then declined in2009. The fact that peaking did not occur in 2009, as predicted,could be due to a significantly reduced demand resulting from theeconomic downturn.

5. Testing the IHC concept

Qualitative tests were conducted to evaluate the concept of anIHC. Truncated data sets of global oil production data were createdby simply removing the recent data. These data sets include

1860–2000, 1860–1980 and 1860–1960 data. The first two datasets were analyzed by creating a double-cycle logistic curve andthen differentiating it by de-constructing into an IHC. The1860–1960 data were analyzed by a single-cycle logistic curveand subsequent IHC model. The results of the truncated data setanalyses are included in Table 3 in the Summary of Analyses Section(paragraph 7). Figs. 5–10 show the logistic curves and IHC’sproduced from all abbreviated data sets for comparison with eachother and with the current data set of 1860–2008 (Figs. 3 and 4).Simulated logistic curves and IHC’s are displayed as smooth lineswith actual production data shown as individual data points. Thedata points sometimes appear as continuous lines due to theresolution limits of the small graphs.

The 1860–2000 data set (Figs. 5 and 6) shows a productionprofile very similar to the current data set of 1860–2008. Bothindicate a 2009 peak year, but the shorter set (to 2000) indicates aslightly lower production value at the peak year of 82.3 versus82.6 Mb/d for the complete 2008 data set. The 1860–1980 data(Figs. 7 and 8) show an increased peak production of 85.6 Mb/d forthe same peak year of 2009, along with a slightly higher EUR of 2.31trillion bbls (Tb). Deviations do occur from the IHC, but the IHCcurves predict plausible production patterns including peak per-iods and values based entirely on the embedded historical information

in the cumulative production data.

Figs. 9 and 10 illustrate the predictive potential of this method.The relatively short data set of 1860–1960 formed an immature

Fig. 2. Modified annual and cumulative oil production data through 2008.

Fig. 3. Global cumulative oil production and model logistic curves.

Fig. 4. Global oil annual production and IHC model.

B. Gallagher / Energy Policy 39 (2011) 790–802 795

logistic curve of about 25% completeness as shown in Fig. 9. Yet thispartial curve was sufficient to model a completed logistic curve thatproduced a reasonable forecast of global oil peaking in 2005(Fig. 10) at 100.8 Mb/d with an EUR of 2.28 Tb. For the 1980,2000 and 2008 data sets, the peaking years are the same and the

peak production values are less than 4% apart. The 1960 data setpeak production in 2005 is 22% higher than the current data setresult (100.8 versus 82.6 Mb/d). This may be due to the 1860–1960data reflecting the potential of a more unfettered productionera prior to the first peaking data of the 1970s and 1980s.

Table 3Summary of graphical analytical results.

Data set description Data setlength years

Predictedpeak year

Peak value(Mb/d)

Peak value(Gb/y)

EUR K RMSD error(%)

Global oil 1860–2008, 2-cycle analysis 148 2009 82.57 30.159 2.24E+12 0.3233

Global oil 1860–2000 2-cycle analysis 140 2009 82.32 30.067 2.23E+12 0.2999

Global oil 1860–1980, 2-cycle analysis 120 2009 85.64 31.282 2.31E+12 0.5228

Global oil 1860–1960, 1-cycle analysis 100 2005 100.81 36.821 2.28E+12 0.8189

USA oil 1860–2008, 1-cycle analysis 148 1978 8.982 3.282 237E+09 1.0185

USA oil 1860–1955, 1-cycle analysis 95 1972 10.443 3.814 221E+09 1.1625

Fig. 5. Logistic model 1860–2000 data.

Fig. 6. IHC model 1860–2000 data.

Fig. 7. Logistic model 1860–1980 data.

Fig. 8. IHC model 1860–1980 data.

B. Gallagher / Energy Policy 39 (2011) 790–802796

The 1860–1960 data obviously would not have the early peakinginformation of the 1970s and 1980s embedded in the cumulativeproduction data. In 1961, these data would have been analyzedwith a single-cycle logistic curve, as shown in Figs. 9 and 10. Thisanalysis shows an early peaking in 2005 at a sharper and higherpeak value of 100.8 Mb/d, for the reason described above. As thedata sets approach the actual time of peaking, a pattern ofdecreasing peak production estimates seems to occur when usingthis methodology. This was especially evident from othersingle-cycle analyses of all data analyzed by the double-cyclelogistic curves (not shown here). The conclusion of these tests wasthat the IHC results appear reasonable if the cumulative curve isgenerated by the appropriate logistic model (single-cycle, double-cycle, etc.).

6. Hubbert’s peak versus an idealized Hubbert peak

The IHC theory was also applied to US oil production, using asingle-cycle approach. Fig. 11 shows US annual and cumulative oilproduction data from 1860 to 2008 (EIA, 2009). The cumulativeproduction and model logistic curves are shown in Fig. 12. The 148years of US oil data generated a very good logistic curve that isabout 75% complete. The differentiated logistic model curve dataproduced an IHC that peaked in 1978 at 9.0 Mb/d (Fig. 13), whilethe actual peak occurred in 1970 at 9.6 Mb/d. The actual peak wasasymmetrical due to a sharp rise (1966–1970) followed by a sharpdecline and overshoot (1971–1976). If this aberrant sharp increase

B. Gallagher / Energy Policy 39 (2011) 790–802 797

and decline had not occurred, the peak pattern would likely havebeen smoother and perhaps closer to the IHC value. The IHCmethod produces smoothed data that portray potential productionvalues that cannot reflect individual year deviations. Furthermore,the limiting EUR value of K was determined to be 237 billion barrels

Fig. 9. Model logistic curve 1860–1960 data.

Fig. 10. IHC model 1860–1960 data.

Fig. 11. Annual and cumulative US o

for US production based on a best fit of 1860–2008 data, whichincludes Alaska Prudhoe Bay oil production from 1977. This is animportant change in production factors that must be considered inevaluating the IHC methodology, as further explained.

Hubbert’s famous paper was presented in 1956 (Hubbert, 1956)and most likely relied on data up to 1955 prior to the Trans Alaska

pipeline development. The 95 years of truncated US oil productiondata between 1860 and 1955 (EIA, 2009) produced an IHC thatpeaked in 1972 at 10.4 Mb/d (Figs. 14–16), considerably closer to theactual Hubbert peak of 1970 (Fig. 13). The reason for this is the bestfit of 1860–1955 data show an EUR of 221 billion bbls or 16 billion

bbls less than the most recent analysis. Trans Alaska pipeline oil peakedin 1987 (Alyeska Pipeline Service Company, 2010) at approximately744 Mb/y (2.04 Mb/d). Based on an average production of about533 Mb/y over a 30-year period (1977–2007), the Alaska pipeline oilproduced to-date totals approximately 16 billion barrels. In conclu-sion, the 1972 peak prediction using the IHC method was based on asmaller EUR value, since the embedded logistic curve data did notinclude the Prudhoe Bay Alaska oil resource. In this case, recent datachanged the prediction based on the additional Alaska resourcesdata embedded in the more complete logistic curve.

A closer look at Fig. 13 shows that US oil production actuallyreached three high points over a 15-y peaking period of 1970, 1978and 1985 at 9.6, 8.7 and 9.0 Mb/d, which together average 9.1 Mb/d.The most important conclusion is that US oil production was in abroad peaking pattern between 1970 and 1985 and has beensteadily declining ever since, as illustrated by the IHC and annualproduction data. A broad production peaking pattern of an essentialnational resource makes more sense than a single sharp peak, sinceextraordinary efforts would normally be made to keep productionlevels as high as possible for as long as possible. Finally, thecumulative oil produced in the US by 2008 was approximately 198billion barrels. Allowing for another 3 billion barrels produced sincethen suggest that approximately 36 billion barrels of US oil remainto be recovered at this time, including the remaining Alaskapipeline oil, unless major new resources are developed or an oilrecovery technology is significantly improved.

7. Limitations

The IHC methodology has limitations on the data to be analyzed.Oil production that is intermittent or has sharply changing growthrates will generally not produce smooth logistic curves, and theresulting IHC will be unusable as an indicator of peak production.An example case is illustrated for the State of Illinois oil productiondata, modified from Mast (1969), as shown in Fig. 17 below, alongwith a modified production graph in Fig. 18.

il production 1860–2008 data.

Fig. 12. Cumulative US oil production and logistic model 1860–2008 data.

Fig. 13. Annual US oil production and IHC model 1860–2008 data.

Fig. 14. Annual and cumulative US oil production 1860–1955 data.

B. Gallagher / Energy Policy 39 (2011) 790–802798

The Illinois oil production data represent three distinct phasesand peaks as illustrated in Fig. 18. The first phase from 1904 to 1936reflects shallow oil reservoir development, which peaked very soonand then declined. The second phase began around 1937 when thedeeper geological oil formations were developed using the seismo-graphic technology. This phase peaked almost immediately andthen started to decline rapidly. Phase 3 started in about 1945 andreflects secondary oil production based on water flooding. Phase 3has a broad production peak between 1957 and 1967. These phasescan also be detected in the bending of the cumulative curve around1938 and 1955 (Figs. 17 and 19), due to the changes in productionmethods. Curve fitting shown in Fig. 19 is not very good (an RMSD

was 2.94%), since the model relies on a single exponential growth-rate and the Illinois cumulative oil production growth rates variedconsiderably. The IHC produced by the subject methodology isshown in Fig. 20 along with the annual production data.

The IHC cannot indicate a reliable peak since there are threedistinct peaks, including a sharp peak, while the IHC peaks broadlyaround 1955. However, the area under the IHC curve of 3700million barrels is still a valid EUR based on average production data.The EUR amount is equal to the K factor entered into Eq. (5). Thisvalue was verified by separating the production data into threephases shown in Fig. 18, and then analyzing each phase individu-ally. The Illinois historical oil data represent an example that will

Fig. 15. US cumulative oil production and logistic model 1860–1955 data.

Fig. 16. Annual US oil production and IHC model 1860–1955 data.

Fig. 17. Illinois annual and cumulative oil production.

Fig. 18. Three phases of Illinois oil production.

B. Gallagher / Energy Policy 39 (2011) 790–802 799

not form smooth logistic curves, and therefore the IHC method isnot applicable. However, when this type of production is part of amuch larger and diversified data set (such as US or global oilproduction), there is no problem. The discontinuous productiondata are a statistically small contribution to the total data and donot significantly affect the cumulative data and curve shape. Thesedata can be analyzed by the multi-Hubbert method developed byLaherr�ere (1997, 2000) and recently applied by Maggio andCacciola (2009).

8. Summary of analyses

Table 2 summarizes the values of coefficients used in Eq. (5) tomodel the cumulative oil production based on obtaining a mini-mum RMSD with respect to the actual cumulative oil data. Alsoshown are values of coefficients used to create a Hubbert curve forthe ‘‘first peak’’ production data between 1942 and 2008.

Fig. 19. Cumulative data and logistic model.

Fig. 20. Illinois oil production data and IHC.

Fig. 21. Recent production and IHC peaking profile.

B. Gallagher / Energy Policy 39 (2011) 790–802800

9. Results and discussion

This relatively simple methodology applied to existing global oilproduction data produced a logistic curve (and derived IHC) thatproduced the following results:

�

Year of the inflection point occurrence:2009 � Peak production value in year 2009:30.16 billion barrels oil � Cumulative production value at an inflection point:1.20 trillion

barrels of oil

� Ultimate limit (carrying capacity) of curve:2.24 trillion barrels

of oil

The predicted peak of 2009 is 3% higher than the actualproduction for 2009 (30.16 versus 29.22 Gb/y). The subject methodprovides an estimate of a ‘‘potential production’’ curve, whichappears close to being exceeded. The reported annual oil produc-tion values have been moving mostly sideways since 2005 justbeneath the boundary of the IHC (see Fig. 21). Considering data andsystem error allowances, it is possible that world oil production hasbeen in a broad peaking mode for the past five years. Actual peaking

conditions are dependent on the elusive ultimate recoverableresource value (URR).

The IHC method suggests that an alternative to a reliable URR forforecasting remaining resources is the size, shape and timing of thecumulative production curve, which is readily available. The IHCrepresents an ideal smoothed production data similar to a movingaverage filter or trendline, but it is not the same. The IHC can beconsidered as a potential production curve which cannot provide ahigh-resolution peaking estimate. Actual oil production appearedto peak in 2008 and then decline in 2009, but no definite peakpattern has yet been discerned. The production decline in 2009 islikely associated with a decrease in demand, due to the economicdownturn; or perhaps the maximum production recorded in 2008may turn out to be a true peak.

As explained earlier, a broad peaking of global oil productionappears more likely than a sharply defined peak. The consistency oftest results in obtaining similar peaking profiles between 2005 and2009 with diverse data inputs over a 50-year period suggests thatthis potential peak production period should be seriously consid-ered, even though a definitive peak has not yet occurred, as far asknown at this time. A broad peak zone of approximately 10-yearswidth between 2004 and 2014 (similar to the US oil peakingpattern) makes more sense and is within the range of manyrespected peak oil forecasts (GAO, 2007).

Analyses using truncated data produced similar results withdifferences that can be rationalized. It is essential to use a multi-cycle logistic curve model, when the data show multiple cycles ofproduction. This is exactly the same requirement as found byLaherr�ere for Hubbert curve analyses (Laherr�ere, 1997, 2000).Single-cycle logistic curve analyses of the data sets used in thispaper result in earlier peaking estimates than the required 2-cycleanalysis. This might explain why some prior forecasts of peak oilhave shown slightly premature peaks. Analysis of 1970–1980 datathat did not account for the first peaking effect would be based onthe above-average production patterns and result in earlier peaksand higher resource limitations. Annual oil production actuallydecreased after 1980 due to political and economic reasons, which

Fig. 22. Normalized global oil forecasts. Data for the normalized GBM model

forecast modified from Guseo et al. (2007) and used with permission from the

authors.

B. Gallagher / Energy Policy 39 (2011) 790–802 801

caused an additional change in the production factors. Also, theapparent offset in the modelled data that occurs in several figures(and cannot be removed) is due to the use of a single exponential

growth-rate that does not occur in the actual production data. TheIHC requires a compromise in the modelling process to split thedifference among varying exponential growth rates. The IHCmethod will frequently result in a much higher initial condition

(P0) than shown by the formula solution at t¼0. This is the result ofP0, compensating for a widely varying growth-rate during the earlyyears. Errors caused by a compromised exponential growth-rateare reduced by adjusting the initial production coefficient, P0. TheIHC procedure is not exact, but will provide reasonable estimates offuture production provided production conditions remain rela-tively unchanged. In contrast, US oil production patterns werealtered after developing the Alaska pipeline oil fields, whichaffected the IHC composite limitation for total US oil productionand produced a revised EUR (K factor).

The advantages of this methodology originate from the simpli-city and smoothness of the cumulative data curve that forms abeginning logistic curve. The procedure of incrementally adding arelatively small amount to a larger, constantly increasing cumu-lative value provides an intrinsic smoothing and damping effect todata variances. Once a spreadsheet is developed to organize all ofthe data and models as explained in Section 3.1, curve-fittingprocedure is quite easy after an initial learning curve. In curve-fitting procedure, the RMSD must be minimized as best as possibleto maintain the proper slope in the model curve or significant errorswill occur. This usually requires careful and iterative input adjust-ments out to 4-place decimal accuracies.

If the IHC method is truly meaningful, global oil productionshould begin to decline within a few years at about one to twopercent annually, and then begin to accelerate in its decline. Thisestimate is based on the initial post-peak slope of the current IHCshown in Fig. 4, which assumes a symmetrical curve. As Bardi(2005), Guseo et al. (2007) and others have indicated, the declinecurve could become much steeper. If demand strongly recovers,global oil production could continue to move sideways tempora-rily, due to aggressive oil exploration and recovery efforts alongwith a willingness to pay higher prices. This will only prolong andexacerbate the inevitable collapse when extraordinary efforts tomaintain high oil production reach their limits. We can then expectto experience sharply declining oil production rates and rapidlyincreasing prices as predicted by many leading researchers in thisfield (GAO, 2007). More work is needed to explore further themethodology discussed herein. A mathematical approach to assur-ing that the IHC represents a unique logistic curve is necessary anda computer algorithm to minimize the RMSD error automatically incurve matching would be a big help. This type of procedure could beapplied to other important resources, including other fossil fuels, toestimate their peaking patterns and remaining life.

It is most important that more research be applied to thepotential production decline characteristics and economic effectsonce peak production does occur. For example, taking results fromFig. 2 of the Guseo, 2007 paper, and from Fig. 4 of the subject paper,Fig. 22 portrays two production and decline profiles of global oilforecast data normalized as percent of the peak production. Thedata normalization was required, since the subject and Guseopapers use production data of slightly different oil categories.

The two modelling methods produce good fits with actualproduction data and almost identical patterns up to their respectivepeaking periods (2007 for the Guseo paper and 2009 for the subjectpaper). The Guseo GBM model much better simulates the erraticproduction swings in the 1970s. By 2015, the forecast of the GBMbegins to sharply deviate below the optimistic mirror imageassumption of the IHC forecast. The GBM steep decline forecastrepresents a startling insight into a near-term global oil production

collapse. If the two declining profiles, shown in Fig. 22, are perceivedas potential upper and lower bounds of the post-peak production,then either one (or anywhere in between) represents a scenario of theend of oil age apocalypse. Apocalypse from the Greek apokalypsis

means ’’lifting of the veil’’ or revelation of something hidden from the

majority of mankind in an era dominated by falsehood and misconception

(Wikipedia, 2010). Peak oil reality has been hidden from most of us byfalse perceptions of an unlimited resource. Apocalypse can also referto an end of the world, at least as we have come to know it. Theseharsh revelations are defined as apokalupsis eschaton, literally ‘‘reve-lation at the end of the æon, or age’’.

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