Fig. 6 in Parker's recent article1 in Oil and Gas Journal places the peak of world oil production in the year 2035.
However, a modification of Parker's equation (see equation 3 below) leads to a simple graphical method that makes a minimum number of assumptions. That graph strongly implies a world production peak around 2005 (Fig. 2).
Several of M. King Hubbert's followers conclude that world oil production will start to decline in this decade,2 in contrast to other analyses that place the peak in the distant future.3 This is an enormously important issue. Four or 5 years is not enough time to get additional oil flowing from unconventional sources or to implement many of the proposed alternative energy schemes.
Equations 1 and 2, from Parker's article, describe exponential growth, compound interest, or radioactive decay:
dQ/dt = rQ (1)
Q=Qstart ert (2)
The initial size Q
dQ/dt = r Qstart (1 – Q/Q∞) (3)
As written, r times Qstart would act as a single constant, and Parker explains that equation 3 would describe an exponential decay in production.
In 1982, Hubbert's last long paper4 contained an equation similar to equation 3 but with Parker's constant Qstart replaced by the variable Q. In that form,
dQ/dt = r Q (1 – Q/Q∞) (4)
differential equation 4 leads to the bell-shaped logistic curve and its associated "S" shaped cumulative curve. Hubbert's 1982 paper derives the logistic equation by a long process of integration, but a simpler, one-page treatment by differentiation is available.5
In equation 4, "Q" is the cumulative oil production, in contrast to equation 1 where "Q" is the momentary balance in a compound-interest bank account. Equation 4 was proposed by Verhulst6 for populations whose growth rate depended on the unoccupied fraction of the environmental carrying capacity. In population biology, "Q" is births minus deaths. In Hubbert's sense, "Q" is the cumulative oil production. No barrel of oil ever dies.
Population biologists7 and Hubbert4 rearrange equation 4 to generate coordinates for a linear graph:
(dQ/dt)/Q = r –(r/Q∞) Q (5)
Y=a - b X (6)
Using (dQ/dt)/Q as the vertical axis and Q as the horizontal axis causes a bell-shaped logistic curve to plot as a straight line. The graph has several handy properties:
- The intercept of the straight line with the vertical axis gives "r."
- The intercept with the horizontal axis gives Q∞, the total recoverable oil.
- The midpoint, at Q∞/2, is the peak year of maximum oil production.
Hubbert defined cumulative "discoveries" as the cumulative oil produced up to a given year plus the known reserves in that same year.8 Cumulative discoveries lead cumulative production by about 11 years in the US and by roughly 34 years for the world. Hubbert's hypothesis states that the cumulative production and cumulative discovery curves are identical in shape. On a (dQ/dt)/Q vs. Q graph, discoveries and production should fall on a single straight line.9
As an example, the US production (solid dots) and discoveries (open circles) are shown on Fig. 1. From the year 1958 onward, the production dots form a reasonably straight line. The straight line drawn on Fig. 1 gives "r" in equation 4 to be 0.055 (5.5%), Q∞ is 220 billion bbl, and the best-fitting peak year is 1975. The actual year of greatest US production was in 1970 during a small blip above the best-fitting line.
The open circles on Fig. 1 correspond to Hubbert's definition of discoveries. As expected, for any year there are 10 or 11 open circles to the right of the solid dots that show US production. Although the discovery circles are more scattered than the production numbers, having the straight line go through the cluster of recent discovery points indicates that no huge recent discoveries are expected to enhance US production during the next few years.
Have I influenced the answer by choosing the axes? In using an unfamiliar graph, am I cooking the books? Of course, the graph is unfamiliar only because none of us is a population biologist. Trying the appropriate graph is simply a test for a particular mathematical expression.
For equations 1 and 2, I would choose a semi-log plot. If I want to test whether the Arrhenius equation describes the change of a lubricant's viscosity with temperature, I plot the log of the viscosity against the reciprocal of the absolute temperature. In both cases, if the data plot as a reasonably straight line then I use the underlying mathematical expression as a working hypothesis to explain the data. In Figs. 1 and 2, a straight line suggests that the logistic equation 4 is consistent with the production and discovery history.
Most economists distrust Hubbert's approach because the price of oil never enters the analysis. In a way, graphs based on equation 4 do allow price to have an influence. At times of high prices, we can take larger steps along the straight line, but the constraint remains that exploration success depends on the undiscovered fraction of Q∞.
World production and discovery histories are plotted on Fig. 2. The production numbers are from the last issue each year of Oil & Gas Journal.
There is a problem with the reported reserves, which are a component of Hubbert's discoveries. As Campbell pointed out,10 many of the OPEC countries announced abrupt increases in their reserves during the 1980s.
In plotting the open circles for discoveries in Fig. 2, I have subtracted out the sudden increases. In addition, a few OPEC countries are listed with constant reserves, despite continuing production and no major new discoveries. In those instances, I have estimated that the normal reworking, recompletions, new horizons, and field extensions will replace 40% of the production.
In the absence of new field discoveries, my guess is that 60% of production, on average, is accompanied by the drawdown of reserves.
It is important to notice that Figs. 1 and 2 do not require estimating beforehand the total amount of discoverable oil. Also, nothing in the way of production is excluded. Current production from tar sands and wells in deep water are included in the production data. Natural gas liquids are simply a different topic; you don't "find" oil by abruptly enlarging the definition of oil. It would be possible to start with the Drake well and work forward with the entire history of "oil plus natural gas liquids."
The startling conclusion from Fig. 2 is the short distance from the most recent production year, 2001, to the midpoint. The years are about equally spaced; annual production has not changed drastically. The time from 2001 to the center year is 3 or 4 years, placing the world peak in 2004 or 2005.
Remember that the mathematical peak for the US was 1975, while the single year of maximum production was 1970. A small amount of year-to-year jitter near the maximum can cause any year near the maximum to claim the record for largest production. Fig. 2 is not predicting which single year will be the largest, but it does predict that world production will turn down in the middle of this decade.
The discoveries, plotted in open circles on Fig. 2, give some aid and comfort to the placement of the straight line. However, my adjustments to the reserves influence the discoveries after 1984.
The major news about discoveries is that after the mid-1970s, discoveries have been modest compared to the earlier years. In the 1986 compilation of giant fields by Carmalt and St. John,11 the most recent finds larger than 5 billion bbl are the Cantarell Complex off Mexico (20 billion bbl, 1976) and Majnoon field in Iraq (7 billion bbl, 1976).
There are rumors of a 60 billion bbl new field in the Caspian area, but that oil is not going to be in the pipeline to alleviate a world production peak in 2004 or 2005.
You can borrow money at the bank in anticipation of future earnings, but the bank doesn't lend crude oil.
References
- Parker, H.W., "Demand, supply will determine when world oil output peaks," OGJ, Feb. 25, 2002, pp. 40-48.
- Hatfield, C.B., "Oil Back on the Global Agenda," Nature, Vol. 387, 1997, p. 121; Kerr, R.A., "The Next Oil Crisis Looms Large–And Perhaps Close," Science, Vol. 281, 1998, pp. 1,128-31; Campbell, C.A., and Laherrere, J.H., "The End of Cheap Oil," Scientific American, March 1998, pp. 78-83.
- US Geological Survey World Energy Assessment Team, "US Geological Survey World Petroleum Assessment 2000," USGS Digital Data Series DDS-60 (4 compact disks), 2000.
- Hubbert, M.K., "Techniques of Prediction as Applied to the Production of Oil and Gas," in Oil and Gas Supply Modeling, National Bureau of Standards Special Publication 631, 1982, pp. 16-141.
- Deffeyes, K.S., "Hubbert's Peak: The Impending World Oil Shortage," Princeton University Press, 2001, p. 201.
- Verhulst, P.F., "Notice sur la loi que la population suit dans son acroissement," Corr. Math. et Phys., Vol. 10, 1838, p. 113.
- Smith, F.E., "Population Dynamics in Daphnia magna and a New Model for Population Growth," Ecology, Vol. 44, 1963, pp. 651-663.
- Hubbert, M.K., "Degree of Advancement of Petroleum Exploration in the United States," AAPG Bull., Vol. 51, 1967, pp. 2,207-27.
- Deffeyes, K.S., "Hubbert's Peak: The Impending World Oil Shortage," Princeton University Press, 2001, p. 152.
- Campbell, C.J., "The Coming Oil Crisis," Multi-Science Publishing Co. and Petroconsultants, 1997, p. 73.
- Carmalt, S., and St. John, B., "Giant Oil and Gas Fields," AAPG Memoir 40, 1986, pp. 11-52.
The author
Kenneth S. Deffeyes ([email protected]) was a colleague of M. King Hubbert at the Shell Development laboratories in Houston. He holds degrees from the Colorado School of Mines and Princeton and is currently emeritus professor of geology at Princeton University. His book "Hubbert's Peak: The Impending World Oil Shortage," was published in 2001.
