Mathematical problems with Peak Oil

M. King Hubbert’s classic 1956 paper has influenced the way we think about oil. Hubbert’s forecast, and most others following, have presumed a bell-shaped curve, with production increasing up to some point in time and then decreasing. The shape of the curve has led to the term “peak oil”, which is now commonly used in both popular and specialist discussions and articles.

The shape moulds our thinking about the availability of oil, although there is much discussion in the technical literature about details of the shape, how to use the associated mathematics, data definitions, and a host of additional details.  But the mathematical properties of the shape itself impose some constraints that can be tested against actual oil production data.  These tests show that the bell shape does not fit well.

Which raises the uncomfortable issue:  are we asking the right questions?

Hubbert himself used the bell-shaped curve defined by the logistics function.  The more familiar bell-shaped curve is the Gaussian normal distribution of elementary statistics.  But all bell-shaped curves have in common the fact that they are continuous functions which start at zero, rise to a peak, and then descend again to zero. This means that both the first and second derivatives of the bell-shaped curve must have topologically similar shapes.  Examples of these shapes for a symmetrical Gaussian normal curve look like this:

Fig. 1.  Gaussian distributions (from Glynn at http://research.stowers-institute.org/mcm/efg/R/Statistics/MixturesOfDistributions/index.htm)

So when we think about oil production as having a peak, we are also thinking that, for example, the second derivative of production will reach a minimum when the peak production is reached.

Looking at the data for oil production, we see the following:

Fig. 2a.  Oil production (data from Koppelhaar, 2012 and BP Statistical Data, 2013). Scale is 10^6 barrels per year.

The rate of change in oil production is

Fig. 2b. Rate of change in oil production.

And the rate of change in the rate of change (2nd derivative) is

Fig. 2c. Rate of change in the rate of change of oil production.

The derivative curves don't plot as expected for a bell-shaped curve.  Departures from the theoretical models have been noted before; and anyone who studies the oil industry will expect the data to be somewhat “noisey”. Furthermore, the fact that the earth is a finite body means that oil production will have to decrease at some point; when can be debated, but decline it must. Thus, some sort of bell-shape function for oil production seems reasonable. But, at best, it seems that factors other than simply the physical occurrence parameters for oil are of at least equal importance when it comes to how much is produced.

My point is NOT that there is abundant oil just waiting to be found because ‘peak oil’ is wrong. No. The point is that the models we have been using, and which are embedded in much thinking (including my own) may not be leading us in the most productive direction. In particular, it seems probable to me that political and economic factors influencing oil production have played a much more important role than we may have thought.  Indeed at first glance it appears that non-geologic factors have predominated since at least the 1970s.

The challenge is to find a model that will more fully explain the data than the physical supply, bell-shaped curves that have dominated the discussions. We need this new model so that our thinking will be more productive as we address both energy and environmental problems going forward.

Credit where credit is due: The above thoughts have been inspired considerably by Hall and Klitgaard’s 2012 book “Energy and the Wealth of Nations” and by discussion of a first draft of this posting with Roger Bentley. Thanks. Errors in the above discussion are all mine, not theirs.