Peak oil analyzed with a logistic function and idealized Hubbert curve

Energy Policy 39 (2011) 790–802

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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

a b s t r a c t

Article history: Received 19 July 2010 Accepted 29 October 2010 Available online 20 November 2010

A logistic function is used to characterize peak and ultimate production of global crude oil and petroleumderived 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.

Keywords: Peak oil Logistic function Hubbert curve

1. Introduction In 2007, the US Government Accountability Office released a report titled: Crude Oil: Uncertainty about Future Oil Supply Makes It Important to Develop a Strategy for Addressing a Peak and Decline in Oil Production, (GAO, 2007). The report included references to twenty-two well-known organizations and researchers that have expressed their opinions on this issue and included their forecasts of when peak oil might occur. Some forecasts were very broad or open-ended in their peaking estimates. Using a criterion of conciseness in peak year predictions, twelve forecasts were selected by the author for a simple analysis. The median year for the estimated peak oil occurrence was 2011, while the mean year was 2014. Three years later, the US still does not have a national plan or priority about the peak oil. The only thing that has changed is that the uncertainty has been removed. Today most informed researchers and many government and private organizations agree that peak oil is close at hand, although some ardent optimists profoundly disagree. Many of the leading researchers have predicted that oil production will peak soon (Campbell and Laherre re, 1998; Duncan, 2000; Campbell, 2003; Laherre re, 2003). Some researchers claim that oil production has already peaked (Bakhtiari, 2004; Deffeyes, 2005; Simmons, 2005) or may peak in a plateau-like manner (Heinberg, 2005; Shah, 2005). Peak oil estimates frequently include a bell-shaped curve analysis created by Hubbert (1956). The famous Hubbert forecast on the peaking of the US lower 48-State production of oil was

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proven correct and is the primary basis and motivation for the ensuing work, including this analysis. Hubbert provided an extraordinary insight into the peak oil situation, using a simple graphical analysis. Although very useful, the Hubbert analytical approach has limitations (Laherre re, 1997, 2000; Bardi, 2005; Guseo et al., 2007; Maggio and Cacciola, 2009). Laherre re (1997) discussed variants of the Hubbert curve and a modified approach for handling multiplepeaking curves encountered when discovery and/or production occur/s over several different cycles. He uses the phrase ‘‘Multi-Hubbert Modelling’’ to explain this approach in his 1997 paper of the same title. In his paper on the strengths and weaknesses of the Hubbert Curve, Laherre re (2000) further elaborated on constraints to using symmetrical curves and provides numerous examples of how a multi-Hubbert approach can provide models that better fit the raw data. In both papers, Laherre re (1997; 2000) develops and explains the cumulative (logistic) function, but prefers to use its derivative Hubbert function as the primary modelling tool. He emphasizes the importance of using multiHubbert models, whenever necessary, to produce more reliable results. When applying this approach, Laherre re (2000) forecasts that the world production of liquid fuels will peak before 2010. Maggio and Cacciola (2009) collected extensive data on estimated ultimate resources (EUR) and relied heavily on the multi-Hubbert technique to develop global oil peaking forecasts based on an EUR estimates of 2250–3000 billion barrels. The authors show that probable scenarios of peak production start in about 2009 (for an EUR of 2.25 trillion barrels or less); and with higher EURs will occur most likely prior to 2015, with a lesser chance of occurring up to 2021. Bardi’s stochastic models (Bardi, 2005) strongly imply that the world oil production will be asymmetrical and decline at a steeper rate than results modelled by the symmetrical Hubbert

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

function. For example, Bardi’s modelling suggests that advances in the technology for discovering and extracting oil may have accelerated its total production rate and perhaps the impending steep decline. In their paper on world oil depletion models, Guseo et al. (2007) use advanced statistical methods to forecast peaking and declining characteristics of global oil. The authors relied on a Generalized Bass Model (GBM) to treat global oil growth as a natural diffusion process linked with key exogenous factors of price, technology and strategic interventions. The GBM represents a class of equations that include advanced logistic functions that do not depend on knowledge of an EUR, but instead use autoregressive correlation techniques to dissect historical production time-series data into elements of diffusion, long-memory interventions, and stochastic white-noise components. Using data through 2005, the authors forecast a 2007 peak followed by a precipitous resource depletion of 90% by 2019. In effect, this paper supports and quantifies Bardi’s sharply skewed decline of global production, due to the technologically accelerated growth. The Guseo paper will be later used in the results section to portray global oil depletion events that could occur in the near future. The researchers identified above, and others, have advanced the Hubbert function, variant models and sophisticated statistical simulations, which have provided useful oil production peaking and decline profiles. Many of these methods use annual oil production data and advanced mathematics to model future oil production profiles. Some data are less than reliable, especially reserves information from several major oil exporting countries that prefer to keep their data proprietary and perhaps overly optimistic for economic reasons. This paper suggests a complementary approach based on cumulative data and logistic curves for gaining further insight into the peak oil. The logistic curve is commonly used to describe bounded exponential growth and is the foundation for the Hubbert bell-shaped model, which is the derivative of the logistic function. In this paper, a simple methodology is developed to provide additional illumination into the peak oil phenomenon, using a visual approach similar to Hubbert’s. The major differences from Hubbert’s approach include focusing on the logistic curve and the use of modern computer technology and electronic spreadsheets that were not readily available in the 1950s. The approach herein involves very little complex mathematics and relies heavily on a relatively simple computerized graphical technique. The approach to the subject is twofold. First an extended logistic curve is constructed from the annual production data, and then deconstructed into a new type of curve called an IHC, which can be considered a potential production curve. The logistic function, widely used in simulating exponential growth of natural populations, is generally constrained by a resource limitation. This limitation can be a composite of several limiting factors, such as food, habitat, environmental conditions, etc. Quantification of the limiting factors is usually based on external data; but when the data are 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 the case for global oil production. A logistic curve can be easily modelled from cumulative production data formed by a simple summation 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 using three controlling parameters including a composite limiting factor, K. For the analogy with oil, a composite limiting factor could consist of multiple production factors, such as resource availability, technology, market forces, unstable conditions, etc. It is not possible to assess all of the limiting factors that have influenced the global oil production system for the past 150 years. However, it is possible to extract information about the combined effect of these

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influences, to gain insight into a composite ultimate limitation. When reliable data are available on actual ultimate recoverable resources, these can be used to more precisely define the modelled logistic curve. Since 1859, when the first commercial oil production well was successfully completed in Pennsylvania (Paleontological Research Institution, 2010), global oil production has expanded worldwide. The aggregate global oil production has been shaped by various resource limitations and market conditions, whose historical effects are embedded in the cumulative production curve assembled from all worldwide production data. The cumulative oil production data and simple logistic curve can be examined to reveal the past, present and future story of oil. The model logistic curve tells a story of an ideal production potential and the IHC visualizes the results.

2. Premise The premise for this analysis is that a logistic curve produced from one or more cumulative data patterns can be used to approximate future production of applicable resources. A beginning logistic curve reflects an exponential growth that can be modelled, including the probable peak-production point and ultimate limits to production. The end-term prediction of the ultimate limit assumes a mirror-image logistic curve, although this may not occur in reality. Once a sufficient data pattern is available to reliably model a complete logistic curve, it can then be deconstructed (differentiated) to generate a theoretical production curve or an IHC. The IHC does not predict actual production rates, since they are subject to the vagaries of the present; but it does provide a pattern of idealized production data based on an embedded information in the logistic curve. The subject method is not a rigorous mathematical procedure and is not intended to replace the precise analytical methods described in the brief literature review. However, the IHC method offers a simple complementary procedure for developing resource depletion patterns that can provide further insight into future peaking and decline of energy and mineral resources. Moreover the subject procedure can be carried out by anyone with a standard computer and spreadsheet program, who has access to annual production data. If historical cumulative data can be used to predict future events, then partial cumulative data ending before present should be able to predict similar events, if conditions have not changed appreciably. Upon testing this premise, cumulative data obtained from production data ending significantly before the present (2010) appear to reasonably predict production patterns observed today. The estimates get better as more cumulative data become available to tell their story as we approach the actual peaking point. Intuitively, this effect would be expected. The IHC predicts what the theoretical production pattern should be if the past conditions that formed the logistic data stay constant, at least on an average. Significant changes in the production conditions such as encountering separate regions, where oil production is already peaking, will result in a readjustment of the IHC pattern. However, normal variations in annual production rates are usually dampened out in the cumulative formation process, unless they represent a very large change or paradigm shift. For smaller geographical areas, an example would be a shift to deeper reservoirs, thanks to advances in technology. But when viewed in the context of a large diversified global resource, even this major local change is dampened out in the total cumulative data composed from statistically varied worldwide sources. Use of an IHC allows insightful comparison with actual production data and patterns to understand better large consistent deviations in production that do occur.

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3. Methodology A logistic function was used to characterize cumulative global oil production. The function selected was a standard population growth function originally described by Verhulst (1838). This function is the solution to the differential equation that describes the population growth. 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 such that dP/dt¼rP. The well-known solution to this differential equation is P(t)¼P0 ert, where P0 is the initial size or population, and r is a growth-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 dominates growth, 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/m provides the basic differential equation for limited population growth:



dP ¼ rPð1P=KÞ: dt

3.1. Spreadsheet ð1Þ

Eq. (1) is used to describe the rate of cumulative growth. The constant r is sometimes referred to as the biotic potential and K as the environmental resistance. The r term is considered a natural positive force that fundamentally controls growth (or reproduction), while K is a negative feedback factor that provides a limit to growth. In ecology, K is usually called the carrying capacity and represents food, habitat or other necessary conditions required to sustain growth. We can apply this same equation to the cumulative growth of global oil production, where m is defined as the end of production (all economically recoverable oil is depleted) and K will represent the estimated ultimate resource (EUR). The solution to Eq. (1) can be found by separating the variables and then integrating both sides. First, the P and t terms are separated dP ¼ rPð1P=KÞdt

ð2Þ

Integrating both sides: Z  Z dP ¼ r Pð1P=KÞ dt

ð3Þ

The solution is     K PðtÞ ¼ K= 1 þ 1 ert P0

represent growth rather than r due to Excel cell name constraints on 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 by experimental curve-fitting methods when compared with a beginning cumulative curve formed from actual annual production data. Initial curve-fitting values selected for the equation parameters were 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 separate coefficients for handling more complex situations (to be discussed later). If the cumulative data are sufficient, a unique curve can be obtained by adjusting the control parameters until a best-fit curve and minimum root-mean-square deviation (RMSD) error are obtained 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 complete logistic curve as explained next.

ð4Þ

where P(t) ¼cumulative oil production in barrels as a function of time and: K is the composite resource limit (carrying capacity) in barrels of oil; P0 is the initial population (or oil production in barrels) at t ¼0; r is an exponential factor that controls growth (%); and t is time (years in this case). Note in Eq. (4) that at t ¼0, P(t)¼ P0, and at t ¼N, P(t)¼K. Eq. (4) is used to describe cumulative global oil production as a function of time. This equation will be used to develop a complete logistic curve based on incomplete cumulative oil production data. Converting into computer terminology provides an Excel spreadsheet version of Eq. (4) as follows: Cumulative production ¼ K=ð1 þ ðK=P0 1Þ EXPðrateðAi1860ÞÞÞ

Table 1 contains an abbreviated set of the data used in this analysis as an example for discussion. A1 in Eq. (5) represents the first cell in Column A, starting in 1864 and extending to 2152 in six-year increments. The actual spreadsheet model includes 300 rows and extends from 1860 to 2160. Column B is annual oil production data and Column C calculates cumulative oil production data by summing up the annual data. Let Bi represent cells of annual data and Ci represent cells of cumulative data. Then, equation Ci ¼cell Bi + cell Ci  1 will produce the cumulative data for Column C. This process integrates the production data and produces a beginning logistic curve. Conversely, if the cumulative data are modelled with Eq. (5) in Column E, the model production data can be derived by Di ¼Ei + 1–Ei. This is equivalent to differentiating the logistic curve into an IHC. Column E of Table 1 contains Eq. (5) used to develop simulated logistic curve data. The cumulative logistic curve always rises (unless the production falls to zero) similar to a space ship that climbs upward on a predictable path and begins to level off (but never comes back down!). The logistic curve approaches its asymptotic limit referred to as the EUR. This value is equal to the area under the IHC and represents the integral of the IHC function. Column F is the squared deviation error between the simulated cumulative production data (Column E) and the actual cumulative production data (Column C). Column F will be summed and averaged and then its square root divided by the cumulative production amount at the end of the series. This produces a least RMSD error in percent. The RMSD is minimized by adjusting the parameters of Eq. (5) to maintain the slope of the simulated logistic curve equal to the slope of the beginning cumulative production curve.

4. Annual and cumulative production data Annual oil production data, excluding non-petroleum-derived liquid fuels, from 1860 to 2008 were assembled from several sources for the following years:

 1860–1949: Personal communication, R. J. Andres (CDIAC,



ð5Þ This equation is used to model a complete logistic curve based on the three controlling parameters of K, P0 and r. ‘‘Rate’’ is used to



2010). Original data in metric tons were converted by one tonne¼7.333 barrels. 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).

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Table 1 Abbreviated data subset showing every 6th data value from 1864 to 2152. (A) Data reference year

(B) Annual oil production data Mb/y

(C) Cumulative oil production data Mb

(D) Annual oil production model Mb/y

(E) Cumulative oil production model Mb

(F) Cumulative oil curve deviation (col E  col C)^2

1864 1870 1876 1882 1888 1894 1900 1906 1912 1918 1924 1930 1936 1942 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002 2008 2014 2020 2026 2032 2038 2044 2050 2056 2062 2068 2074 2080 2086 2092 2098 2104 2110 2116 2122 2128 2134 2140 2146 2152

2 6 11 37 51 88 148 212 348 499 1038 1432 1804 2091 3423 5020 7680 12,627 19,602 23,132 21,070 23,909 25,530 27,186 29,885

11 35 91 251 495 982 1724 2902 4757 7417 12,463 20,218 29,355 41,977 58,585 84,409 124,041 185,226 285,221 416,481 548,013 683,658 830,552 992,041 1,168,491

28 39 55 78 111 156 221 311 439 619 872 1226 1721 2416 3403 4878 7339 11,976 19,012 22,334 21,926 23,531 26,439 28,954 30,128 29,578 27,396 24,037 20,109 16,163 12,580 9549 7111 5221 3793 2735 1961 1401 998 709 503 357 253 179 127 90 64 45 32

495 699 988 1397 1974 2790 3942 5568 7863 11,098 15,656 22,068 31,075 43,717 61,493 86,759 123,947 182,777 278,978 408,050 540,405 676,451 827,665 995,578 1,174,200 1,353,896 1,524,433 1,677,481 1,808,086 1,914,827 1,999,028 2,063,611 2,112,088 2,147,890 2,174,015 2,192,911 2,206,492 2,216,209 2,223,138 2,228,067 2,231,568 2,234,052 2,235,812 2,237,059 2,237,942 2,238,567 2,239,010 2,239,323 2,239,544

2.34E+ 05 4.41E+ 05 8.05E+ 05 1.31E+ 06 2.19E+ 06 3.27E+ 06 4.92E+ 06 7.11E+ 06 9.64E+ 06 1.36E+ 07 1.02E+ 07 3.42E+ 06 2.96E+ 06 3.03E+ 06 8.45E+ 06 5.52E+ 06 8.71E+ 03 6.00E + 06 3.90E+ 07 7.11E+ 07 5.79E+ 07 5.19E+ 07 8.33E+ 06 1.25E+ 07 3.26E+ 07 RMSD¼ 0.3233%

Fig. 1 illustrates the historical global oil production data from 1930 to 2008. These data show an annual oil production value of 81.8 Mb/d (29.9 Gb/y) in 2008 with a total cumulative production of 1.168 trillion barrels. The figure also displays EIA liquid fuels data (EIA, 2010a) and EIA crude oil data (EIA, 2010b) for comparison. The Oil Production Data used in this figure are intended to characterize marketable liquid fuels refined from crude oil and natural gas liquids that originate from crude oil production activities. Fig. 1 displays a slight ‘‘bend’’ upward in the cumulative production curve beginning in 1970, and then downward in 1980. This bend causes difficulty in forming a logistic model that will reliably forecast future production. The bend is due to the data ‘‘hump’’ between 1970 and 1980, the result of several large oil regions peaking during this period (including North America and the Former Soviet Union). This effect would be considered a major change that cannot be dampened out. This condition represents a two-cycle production process as explained by Laherre re (2000),

and requires a multi-Hubbert modelling approach (or multilogistic model for the subject methodology). Therefore, the data were separated into two groups: (1) a plot of production that peaked 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 to generate the assumed first peak data. As suggested by Laherre re (2000) and applied by Maggio and Cacciola (2009), the coefficients were selected to make the first peak cumulative data total 150 billion barrels (Gb). These coefficients are summarized at the end of Table 2. Fig. 2 depicts the modified production and cumulative data curves. By separating out the early peaking data, the modified cumulative production curve E is significantly improved without a conspicuous bend. The first peak cumulative curve F is already a complete and perfect logistic curve, since it was formed by a theoretical Hubbert function. These changes facilitate a more

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

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

Table 2 Coefficients used in modelling equations. Data set and analysis Type

Figure nos.

Model Eq. (s)

Initial values (P0)

Growth (rates) (%)

Separate EURs (K)

Combined EUR (K)

Separate RMSDs (%)

Combined RMSD (%)

Global oil 1860–2008 2-cycle

3 and 4

Global oil 1860–1980 2-cycle

7 and 8

Global oil 1860–1960 1-cycle USA oil 1860–2008 1-cycle USA oil 1860–1955 1-cycle

9 and 10 12 and 13 15 and 16

23.020 5.770 23.020 5.780 23.020 5.791 5.800 5.534 6.900

1.508E + 11 2.090E+ 12 1.508E + 11 2.080E+ 12 1.508E + 11 2.160E + 12 2.480E + 12 N/A N/A

0.6633 0.3869 0.6633 0.3869 0.6633 0.3869 0.3869 N/A N/A

0.3233

5 and 6

5.420E  1 3.927E + 8 5.420E  1 3.927E + 8 5.420E  1 3.927E + 8 3.700E +8 3.496E + 6 1.101E+ 8

2.2401E + 12

Global oil 1860–2000 2-cycle

(6)a First peak (6)b Main peak (6)a First peak (6)b Main peak (6)a First peak (6)b Main peak (6)b Main peak (6)b Main peak (6)b Main peak

2.2301E + 12 2.3101E + 12 2.6301E + 12 2.3700E +9 2.2100E +9

0.2999 0.5228 0.8189 1.0185 1.1625

First peak data FP(t) ¼Pm/(2 + 2  cosh(c  (year  pkyr)))  365.25(Mb/py); period¼ 1942–2008. Pm (peak multiplier)¼ 100; c ¼ time modifier¼ 0.24336; pkyr ¼peak year set at 1975. Peak production of 25 Mb/pd occurs in 1975; total cumulative production ¼ 150 GBO in 2008.

reliable curve-fitting process that generates two separate simulated logistic curves. The modified cumulative data (E) and the first peak cumulative data (F) are used to generate model logistic curves using Eq. (5). Once minimum RMSD’s are obtained for both logistic curves, their separate curve equations and respective control parameters 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. This IHC is then compared with the original production data. The cumulative oil production data were used to create a beginning logistic curve shown in Fig. 3. Also shown is the final model logistic curve, generated by curve matching techniques to obtain a ‘‘best fit’’ with a least RMSD. Fig. 4 compares actual oil production data with the modelled IHC. The IHC peak occurred in 2009 at 82.6 Mb/d (30.2 Gb/y). An actual oil production reached a maximum in 2008 of 81.8 Mb/d (29.9 Gb/y) and then declined in 2009. The fact that peaking did not occur in 2009, as predicted, could be due to a significantly reduced demand resulting from the economic downturn.

5. Testing the IHC concept Qualitative tests were conducted to evaluate the concept of an IHC. Truncated data sets of global oil production data were created by simply removing the recent data. These data sets include

1860–2000, 1860–1980 and 1860–1960 data. The first two data sets were analyzed by creating a double-cycle logistic curve and then differentiating it by de-constructing into an IHC. The 1860–1960 data were analyzed by a single-cycle logistic curve and subsequent IHC model. The results of the truncated data set analyses are included in Table 3 in the Summary of Analyses Section (paragraph 7). Figs. 5–10 show the logistic curves and IHC’s produced from all abbreviated data sets for comparison with each other and with the current data set of 1860–2008 (Figs. 3 and 4). Simulated logistic curves and IHC’s are displayed as smooth lines with actual production data shown as individual data points. The data points sometimes appear as continuous lines due to the resolution limits of the small graphs. The 1860–2000 data set (Figs. 5 and 6) shows a production profile very similar to the current data set of 1860–2008. Both indicate a 2009 peak year, but the shorter set (to 2000) indicates a slightly lower production value at the peak year of 82.3 versus 82.6 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 for the same peak year of 2009, along with a slightly higher EUR of 2.31 trillion bbls (Tb). Deviations do occur from the IHC, but the IHC curves predict plausible production patterns including peak periods 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

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

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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.

logistic curve of about 25% completeness as shown in Fig. 9. Yet this partial curve was sufficient to model a completed logistic curve that produced 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 set peak production in 2005 is 22% higher than the current data set result (100.8 versus 82.6 Mb/d). This may be due to the 1860–1960 data reflecting the potential of a more unfettered production era prior to the first peaking data of the 1970s and 1980s.

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Table 3 Summary of graphical analytical results. Data set description

Data set length years

Predicted peak year

Peak value (Mb/d)

Peak value (Gb/y)

EUR K

RMSD error (%)

Global oil 1860–2008, 2-cycle analysis Global oil 1860–2000 2-cycle analysis Global oil 1860–1980, 2-cycle analysis Global oil 1860–1960, 1-cycle analysis USA oil 1860–2008, 1-cycle analysis USA oil 1860–1955, 1-cycle analysis

148 140 120 100 148 95

2009 2009 2009 2005 1978 1972

82.57 82.32 85.64 100.81 8.982 10.443

30.159 30.067 31.282 36.821 3.282 3.814

2.24E +12 2.23E +12 2.31E +12 2.28E +12 237E +09 221E +09

0.3233 0.2999 0.5228 0.8189 1.0185 1.1625

Fig. 5. Logistic model 1860–2000 data. Fig. 7. Logistic model 1860–1980 data.

Fig. 6. IHC model 1860–2000 data.

Fig. 8. IHC model 1860–1980 data.

The 1860–1960 data obviously would not have the early peaking information of the 1970s and 1980s embedded in the cumulative production data. In 1961, these data would have been analyzed with a single-cycle logistic curve, as shown in Figs. 9 and 10. This analysis shows an early peaking in 2005 at a sharper and higher peak value of 100.8 Mb/d, for the reason described above. As the data sets approach the actual time of peaking, a pattern of decreasing peak production estimates seems to occur when using this methodology. This was especially evident from other single-cycle analyses of all data analyzed by the double-cycle logistic curves (not shown here). The conclusion of these tests was that the IHC results appear reasonable if the cumulative curve is generated by the appropriate logistic model (single-cycle, doublecycle, etc.).

6. Hubbert’s peak versus an idealized Hubbert peak The IHC theory was also applied to US oil production, using a single-cycle approach. Fig. 11 shows US annual and cumulative oil production data from 1860 to 2008 (EIA, 2009). The cumulative production and model logistic curves are shown in Fig. 12. The 148 years of US oil data generated a very good logistic curve that is about 75% complete. The differentiated logistic model curve data produced an IHC that peaked in 1978 at 9.0 Mb/d (Fig. 13), while the actual peak occurred in 1970 at 9.6 Mb/d. The actual peak was asymmetrical due to a sharp rise (1966–1970) followed by a sharp decline and overshoot (1971–1976). If this aberrant sharp increase

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

and decline had not occurred, the peak pattern would likely have been smoother and perhaps closer to the IHC value. The IHC method produces smoothed data that portray potential production values 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.

797

for US production based on a best fit of 1860–2008 data, which includes Alaska Prudhoe Bay oil production from 1977. This is an important change in production factors that must be considered in evaluating 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 production data between 1860 and 1955 (EIA, 2009) produced an IHC that peaked in 1972 at 10.4 Mb/d (Figs. 14–16), considerably closer to the actual Hubbert peak of 1970 (Fig. 13). The reason for this is the best fit 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 peaked in 1987 (Alyeska Pipeline Service Company, 2010) at approximately 744 Mb/y (2.04 Mb/d). Based on an average production of about 533 Mb/y over a 30-year period (1977–2007), the Alaska pipeline oil produced to-date totals approximately 16 billion barrels. In conclusion, the 1972 peak prediction using the IHC method was based on a smaller EUR value, since the embedded logistic curve data did not include the Prudhoe Bay Alaska oil resource. In this case, recent data changed the prediction based on the additional Alaska resources data embedded in the more complete logistic curve. A closer look at Fig. 13 shows that US oil production actually reached three high points over a 15-y peaking period of 1970, 1978 and 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 a broad peaking pattern between 1970 and 1985 and has been steadily declining ever since, as illustrated by the IHC and annual production data. A broad production peaking pattern of an essential national resource makes more sense than a single sharp peak, since extraordinary efforts would normally be made to keep production levels as high as possible for as long as possible. Finally, the cumulative oil produced in the US by 2008 was approximately 198 billion barrels. Allowing for another 3 billion barrels produced since then suggest that approximately 36 billion barrels of US oil remain to be recovered at this time, including the remaining Alaska pipeline oil, unless major new resources are developed or an oil recovery technology is significantly improved.

7. Limitations

Fig. 10. IHC model 1860–1960 data.

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

Fig. 11. Annual and cumulative US oil production 1860–2008 data.

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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.

The Illinois oil production data represent three distinct phases and peaks as illustrated in Fig. 18. The first phase from 1904 to 1936 reflects shallow oil reservoir development, which peaked very soon and then declined. The second phase began around 1937 when the deeper geological oil formations were developed using the seismographic technology. This phase peaked almost immediately and then started to decline rapidly. Phase 3 started in about 1945 and reflects secondary oil production based on water flooding. Phase 3 has a broad production peak between 1957 and 1967. These phases can also be detected in the bending of the cumulative curve around 1938 and 1955 (Figs. 17 and 19), due to the changes in production methods. Curve fitting shown in Fig. 19 is not very good (an RMSD

was 2.94%), since the model relies on a single exponential growthrate and the Illinois cumulative oil production growth rates varied considerably. The IHC produced by the subject methodology is shown in Fig. 20 along with the annual production data. The IHC cannot indicate a reliable peak since there are three distinct peaks, including a sharp peak, while the IHC peaks broadly around 1955. However, the area under the IHC curve of 3700 million barrels is still a valid EUR based on average production data. The EUR amount is equal to the K factor entered into Eq. (5). This value was verified by separating the production data into three phases shown in Fig. 18, and then analyzing each phase individually. The Illinois historical oil data represent an example that will

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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.

not form smooth logistic curves, and therefore the IHC method is not applicable. However, when this type of production is part of a much larger and diversified data set (such as US or global oil production), there is no problem. The discontinuous production data are a statistically small contribution to the total data and do not significantly affect the cumulative data and curve shape. These data can be analyzed by the multi-Hubbert method developed by Laherre re (1997, 2000) and recently applied by Maggio and Cacciola (2009).

8. Summary of analyses Table 2 summarizes the values of coefficients used in Eq. (5) to model the cumulative oil production based on obtaining a minimum RMSD with respect to the actual cumulative oil data. Also shown are values of coefficients used to create a Hubbert curve for the ‘‘first peak’’ production data between 1942 and 2008.

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Fig. 19. Cumulative data and logistic model.

Fig. 21. Recent production and IHC peaking profile.

Fig. 20. Illinois oil production data and IHC.

9. Results and discussion This relatively simple methodology applied to existing global oil production data produced a logistic curve (and derived IHC) that produced 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 actual production for 2009 (30.16 versus 29.22 Gb/y). The subject method provides an estimate of a ‘‘potential production’’ curve, which appears close to being exceeded. The reported annual oil production values have been moving mostly sideways since 2005 just beneath the boundary of the IHC (see Fig. 21). Considering data and system error allowances, it is possible that world oil production has been in a broad peaking mode for the past five years. Actual peaking

conditions are dependent on the elusive ultimate recoverable resource value (URR). The IHC method suggests that an alternative to a reliable URR for forecasting remaining resources is the size, shape and timing of the cumulative production curve, which is readily available. The IHC represents an ideal smoothed production data similar to a moving average filter or trendline, but it is not the same. The IHC can be considered as a potential production curve which cannot provide a high-resolution peaking estimate. Actual oil production appeared to peak in 2008 and then decline in 2009, but no definite peak pattern has yet been discerned. The production decline in 2009 is likely associated with a decrease in demand, due to the economic downturn; or perhaps the maximum production recorded in 2008 may turn out to be a true peak. As explained earlier, a broad peaking of global oil production appears more likely than a sharply defined peak. The consistency of test results in obtaining similar peaking profiles between 2005 and 2009 with diverse data inputs over a 50-year period suggests that this potential peak production period should be seriously considered, even though a definitive peak has not yet occurred, as far as known at this time. A broad peak zone of approximately 10-years width between 2004 and 2014 (similar to the US oil peaking pattern) makes more sense and is within the range of many respected peak oil forecasts (GAO, 2007). Analyses using truncated data produced similar results with differences that can be rationalized. It is essential to use a multicycle logistic curve model, when the data show multiple cycles of production. This is exactly the same requirement as found by Laherre re for Hubbert curve analyses (Laherre re, 1997, 2000). Single-cycle logistic curve analyses of the data sets used in this paper result in earlier peaking estimates than the required 2-cycle analysis. This might explain why some prior forecasts of peak oil have shown slightly premature peaks. Analysis of 1970–1980 data that did not account for the first peaking effect would be based on the above-average production patterns and result in earlier peaks and higher resource limitations. Annual oil production actually decreased after 1980 due to political and economic reasons, which

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caused an additional change in the production factors. Also, the apparent 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. The IHC requires a compromise in the modelling process to split the difference among varying exponential growth rates. The IHC method will frequently result in a much higher initial condition (P0) than shown by the formula solution at t ¼0. This is the result of P0, compensating for a widely varying growth-rate during the early years. Errors caused by a compromised exponential growth-rate are reduced by adjusting the initial production coefficient, P0. The IHC procedure is not exact, but will provide reasonable estimates of future production provided production conditions remain relatively unchanged. In contrast, US oil production patterns were altered after developing the Alaska pipeline oil fields, which affected the IHC composite limitation for total US oil production and produced a revised EUR (K factor). The advantages of this methodology originate from the simplicity and smoothness of the cumulative data curve that forms a beginning logistic curve. The procedure of incrementally adding a relatively small amount to a larger, constantly increasing cumulative value provides an intrinsic smoothing and damping effect to data variances. Once a spreadsheet is developed to organize all of the data and models as explained in Section 3.1, curve-fitting procedure is quite easy after an initial learning curve. In curvefitting procedure, the RMSD must be minimized as best as possible to maintain the proper slope in the model curve or significant errors will occur. This usually requires careful and iterative input adjustments out to 4-place decimal accuracies. If the IHC method is truly meaningful, global oil production should begin to decline within a few years at about one to two percent annually, and then begin to accelerate in its decline. This estimate is based on the initial post-peak slope of the current IHC shown in Fig. 4, which assumes a symmetrical curve. As Bardi (2005), Guseo et al. (2007) and others have indicated, the decline curve could become much steeper. If demand strongly recovers, global oil production could continue to move sideways temporarily, due to aggressive oil exploration and recovery efforts along with a willingness to pay higher prices. This will only prolong and exacerbate the inevitable collapse when extraordinary efforts to maintain high oil production reach their limits. We can then expect to experience sharply declining oil production rates and rapidly increasing prices as predicted by many leading researchers in this field (GAO, 2007). More work is needed to explore further the methodology discussed herein. A mathematical approach to assuring that the IHC represents a unique logistic curve is necessary and a computer algorithm to minimize the RMSD error automatically in curve matching would be a big help. This type of procedure could be applied to other important resources, including other fossil fuels, to estimate their peaking patterns and remaining life. It is most important that more research be applied to the potential production decline characteristics and economic effects once peak production does occur. For example, taking results from Fig. 2 of the Guseo, 2007 paper, and from Fig. 4 of the subject paper, Fig. 22 portrays two production and decline profiles of global oil forecast data normalized as percent of the peak production. The data normalization was required, since the subject and Guseo papers use production data of slightly different oil categories. The two modelling methods produce good fits with actual production data and almost identical patterns up to their respective peaking periods (2007 for the Guseo paper and 2009 for the subject paper). The Guseo GBM model much better simulates the erratic production swings in the 1970s. By 2015, the forecast of the GBM begins to sharply deviate below the optimistic mirror image assumption of the IHC forecast. The GBM steep decline forecast represents a startling insight into a near-term global oil production

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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.

collapse. If the two declining profiles, shown in Fig. 22, are perceived as potential upper and lower bounds of the post-peak production, then either one (or anywhere in between) represents a scenario of the ´lypsis end of oil age apocalypse. Apocalypse from the Greek apoka 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 by false perceptions of an unlimited resource. Apocalypse can also refer to an end of the world, at least as we have come to know it. These harsh revelations are defined as apokalupsis eschaton, literally ‘‘revelation at the end of the æon, or age’’.

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