Arunn's Notebook

A Walk Down Hubbert’s Peak

Arunn Narasimhan

Ninety nine percent of our energy comes from the Sun. The commercial energy that we pay for is about one percent of the energy we use. Commercial energy mostly (82 %) comes from non-renewable sources like oil (32%), coal (21%) and natural gas (23 %). About 24 percent of consumed energy globally is for transportation, 40 percent for industry, 30 percent for domestic and commercial purposes and the remaining 6 percent for other things including agriculture [1].

Since 1859, the year Edwin Drake struck the first oil well in the USA, for about 150 years we have savored and devoured our oil resources dry. While generations of us were busy slipping into an energy dependent society, unaware of the finiteness of oil, Marion King Hubbert (1903 - 1989) in 1969 wrote a paper in which he predicted the peak and the subsequent decline in production of oil in the USA.

Using methods used for analyzing population growth, Hubbert correctly predicted the bell shape of the US oil production curve with the inevitable peak in 1970 and the subsequent decrease. In 1982, in a subsequent paper [2] using the same method he predicted 2005 as the year the World oil prediction to peak. Now in 2006, we know Hubbert was right about the 1970 peak and subsequent decline of the US oil production. For the World, the scaling of the peak is slightly delayed from the predicted 2005.

We discuss here the math behind the Hubbert’s method to equip ourselves of the right knowledge to ward off baseless skepticism about the method itself. Prof. Kenneth Deffeyes of the Princeton University, one time associate of Hubbert at Shell Oil, has written two excellent books on the Hubbert’s peak - Hubbert’s Peak and Beyond Oil. I follow here, his explanation from Beyond Oil [3]. 

To explain Hubbert’s method, we use the USA crude oil production data between 1859 and 2006 available from the EIA [4] . Oil production is expressed in barrels (1 barrel = 169 liters) and the following figure graphs the USA crude oil production P in billion barrels (or $latex 10^9 $ barrels or Gigabarrels) per year since 1859.

hub_pvsyear1_small.png

 

Figure 1 [click on above thumbnail for a bigger picture - opens in new tab]

Evidently, the US production has peaked by 1970 and since then is steadily falling down. To analyze further the implications, from the data in Fig. 1, for every year, one can find the cumulative oil production Q until that year (i.e. Q for an year is obtained by adding all the P values in Fig. 1, until that year). Doing so, we can rearrange the data of Fig. 1 as P/Q versus Q as shown in the graph of Fig. 2 below.

Figure 2 [click on above image for bigger picture - opens in new tab]

As Q is plotted in the X axis, proceeding from left to right is analogous to marching in time (year). If we choose 1859 as the first year in which the first barrel of oil was unearthed in the USA (a result that cannot be verified but doesn’t alter any of our conclusions), we would obviously have P/Q = 1 for the first data point. I have reduced the P/Q axis to show only until 0.1 only for clarity. Further, in the initial years, the production P would be comparable to that of the cumulative Q for each year and hence we can expect the data set to crowd at higher P/Q values (see top left corner of the graph). After some initial oscillations, from year 1958, the P/Q data settles down to a nice straight line fit.

The slanted red line in Figure 2 is such a straight line curve fit of the data set for P/Q between 1958 and 2006. And that straight line is Hubbert’s theory.

We have extended the line either way (extrapolated) to touch the X and Y axis to obtain the maximum Q value (X value, when Y = 0) and maximum P/Q value (Y value when X = 0). Mathematically, these values, Q_T = 228.4 and c = 0.0536 respectively, are useful to find the slope of the straight line. Contextually, the Q_T = 228.4 is the expected maximum oil that could be produced in the USA. Because, that is the point at which P/Q = 0, meaning no more oil is available to be produced beyond that point. Simple analysis of raw data yielding a result of enormous implication.

As a sidelight, compare this 228.4 Gigabarrels with the 362 billion barrels estimate given by the US Geological Survey in 2000. The slanted red line has to do the remarkable thing of following the dashed red line in the future to make this estimate a reality. Either that or I quote Prof. Kenneth Deffeyes from his book

Either that straight line is going to make a sudden turn, or the USGS was counting on bringing in Iraq as the fifty-first state.

Incidentally according to available reports [5], three countries are yet to peak in their oil production: Kuwait (expected to peak in 2013), Saudi Arabia (2014) and Iraq (2018). India, for instance, peaked by 1997.

As shown in Fig. 2 above, the algebraic representation of the straight line can be written by the familiar equation Y = mX + c . For Fig. 2, this translates to (P/Q) = c - (c/Q_T)Q where the negative sign is because the P/Q value begins at a maximum ( = c = 0.0536) and decreases for increase in Q. This can be rearranged further to yield P = c(1 - Q/Q_T)Q. This equation, when plotted, would result in a bell shaped curve for P called a “logistic” [6] curve. The term inside the parenthesis (1 - Q / Q_T) or (Q_T - Q) / Q is the remaining fraction of oil to be produced in the USA.

Observe that by flipping the equation for P (which is in Gigabarrels per Year) and writing as 1/P, we could get Years per Gigabarrels produced. We know from EIA records [4] that the cumulative US oil production (Q) equaled 100 Gigabarrels in about 1972.1 (or 169 Gigabarrels in 2002 according to the ASPO’s newsletters [8]). To obtain the bell curve, we follow this procedure: Compute P from the equation in Fig. 2 above. Compute 1/P from that. Set the clock as 1972.1 for Q = 100 Gigabarrels and extrapolate into the past and future from that instant by successively subtracting or adding the 1/P for each Gigabarrel of oil produced. By plotting the calculated P against the time period (in Years) obtained by this procedure would result in the bell shaped logistic curve for US Oil production. This is shown as the curve formed by open red circles in Fig. 3 below. The blue square symbols are the original recorded data of P (from Fig. 1), included here for comparison.

hub_pvsyear3_small.png

Figure 3 [click on above image for bigger picture - opens in new tab]

We are now in 2006 and the US oil production reaches zero by 2075.

A similar analysis can be done for the oil production for the World. Similar to Fig. 1, the oil production for the World between the years 1965 and 2006 (obtained from [7]), when plotted between P/Q and Q would resemble the graph below. Unlike the US oil data, the World data includes not only crude oil production but also shale oil, oil sands and natural gas in liquid form.

Figure 4: [click on above image for bigger picture - opens in new tab]

As seen from Fig. 4, the World data after 1983 (green symbols) settles into a nice straight line fit (red slanted line). Applying a similar extrapolation and analysis used for Fig. 2, we can estimate from the straight line fit of Fig. 4, the cumulative total World oil as about 2170 Gigabarrels.

The inset in Fig. 4 shows P versus Year for the World oil production. The peak is estimated by Prof. Deffeyes (in his book) to happen by 2006. Most computer studies, similar in style to Prof. Deffeyes (and Hubbert’s) analysis suggest the World oil production peak will occur between 2010 and 2020. Geologists agree that there are no major oil fields left to discover (it would take another essay to explain the theory behind this prediction). This means, we have already extracted and used at least half of the 2170 Gigabarrels of World oil.

We don’t have to wait until the bell curve in Fig. 3 or in the inset of Fig. 4, meets the X axis (zero production). The imminent crisis happens much earlier when the energy expended in the extraction of the remaining oil exceeds the energy yield from the extracted oil. The oil crisis will affect almost all walks of our energy dependent daily life as about 500,000 goods currently derived from petroleum would experience shortage [1]. For instance, global food security will be destabilized due to lack of fertilizers and pesticides, both byproducts of petroleum. Compared to that, being denied of a BMW drive on the asphalt (byproduct of crude oil) road is less worrisome.

Let me save the doomsday implications and energy alternatives for a future essay. At this stage, as an informed citizen, I should know Mother Earth is finite sized. If I don’t believe in the finiteness of her natural resources, I am deluded or worse - an armchair analyst who believe in the persistence of exponential growth. Over the last 150 years, thanks to the industrial revolution, as a society keen on converting the natural energy resources irreversibly into action and entropy, we have successfully managed to scale the Hubbert’s peak. What remains is the economically disruptive, societally uncomfortable, descent.

By Walk.

References

  1. see page 121 - 122 in Environmental Studies: from crisis to cure by R. Rajagopalan, Oxford Uty. Press, 2005.
  2. M. King Hubbert, “Techniques of Prediction as Applied to the Production of Oil and Gas,” in S. I. Gass, ed., Oil and Gas Supply Modeling, Special Publication 631, Washington, D. C., NBS 1982.
  3. Kenneth S. Deffeyes, “Beyond Oil,” Hill and Wang Pub., 2005.
  4. EIA - Petroleum Data, Reports, Analysis, Surveys: U.S. Crude Oil Field Production (Thousand Barrels)
  5. see at the end of the peak oil wikipedia page
  6. a logistic curve looks like an elongated S and initially raises exponentially and slows down as it peaks. Logistic Function was discovered by Pierre Francois Verhulst
  7. from BP Global - Reports and publications - Statistical Review of World Energy 2007
  8. ASPO’s Newsletters [pdf file]
© Arunn Narasimhan | Original version written ~ 2009 | Last revision on Apr 01, 2012

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