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.

Peak Fossil Fuels

Just over a year ago, Bob Hirsch’s presentation at ASPO was something like “there isn’t an energy problem; there is a liquids problem”.  He went on to recall the societal disruption of the 1973 oil embargo in the USA, and suggested that peak oil disruptions would, by analogy, be far more than simply disruptive.

While there have been significant strains, so far there hasn’t been a collapse.  Indeed, oil production continues to edge upwards year after year.  I don’t think there will be a collapse.  As I've said privately, those arguing for peak oil should declare a victory and move on.  The consistently higher price for oil over the past decade is evidence that the peak oil predictions were essentially correct.  But the fact is that the predictions were only valid for the boundary conditions, and these have now changed.

Whether the economic setting for oil is measured by price, EROI, rig count, or in some other way, there can be no doubt that the oil exploitation environment has changed significantly over the past decade.  This tells me that the entire peak oil analysis has a significant economic component.  It is not that there is suddenly an increased physical supply of crude oil as the result of higher prices; rather the higher prices are but one manifestation of the fact that the entire economic environment has changed in ways that are very complex.  By analogy, the bulk chemistry of a slate and a gneiss are similar, but the mineral assemblages are very different as a result of the differing pressure and temperature conditions.  In the energy economy the relative prices of oil, coal, natural gas and non-fossil fuels are changing; and the energy assemblage of our economy is reorganizing itself. 

The past 60+ years have been a relatively steady state with respect to energy economics.  Oil has been cheap, with additional supplies being available without much relative price increase.  That is no longer true.  As the economy adjusts, the the relative costs of different forms of energy are temporarily out of balance.

Of course, what we are all really interested in is the future.  It seems to be part of the human condition that we look forward, project, and otherwise plan for tomorrow.  And we would all like to know how the changes that the world energy economy is currently experencing will play out. 

I think one of the ways to look at this issue is to consider not peak oil, but rather to look at fossil fuels as a single energy source.  What can we say about “peak fossil fuel”?

This is the combined data for fossil fuels over the past 150+ years.  When this data is analysed using the mathematics that Deffeyes, following Hubbert, used to predict the oil peak the result is that the peak is already significantly behind us.  But the data clearly shows that this is not the case.

Note that Hubbert’s original paper foresaw a smooth transition from oil to nuclear power, which is probably the reason that we tend to use a roughly symmetrical curve to describe peak oil.  If the economy becomes truly constrained with respect to energy due to shortage of the supply of oil, or I would argue of total fossil fuels, then there will be no smooth transition to something else.  The result will be more like Bob Hirsch’s predictions for oil, and the downside will be quite abrupt as the economy shuts down.  Of couse, this means that almost half of the ultimate recoverable reserves would then be left in the ground, unexploited.

My own thinking is that the total fossil fuel peak is still some ways away.  And whether the peak will be symmetric or a rather sudden collapse is still unknown.  In brief, the peak can be symmetric only if fossil fuels are replaced by new non-fossil fuel energy sources that cost less.  If these are not available, the result will be collapse with a more sudden decrease in production levels.

The importance of relative costs can be seen within the total fossil fuels mixture.  

The data is from the long time series that Rembrandt Koppelaar published in The Oil Drum (http://www.theoildrum.com/node/8936) – thanks Rembrandt.  The gas numbers have probably been significantly under-reported, as even today they don’t include much of the gas that is flared in association with oil production.

As the above figure shows, and as we all know, the initial fossil fuel was coal, followed by exploitation of oil, and finally of natural gas.  This reflects the relative difficulty, i.e. cost, of transporting solids, liquids and gasses. 

One of the interesting things about fossil fuels is that the three types are relatively interchangeable.    Note, I do say relatively.  This both supports aggregating them for total energy analysis, and provides some insight into energy analyses by looking at the details of how one substitutes for another.  For example, natural gas is replacing coal for electricity generation in North America and the reverse is happening in Europe.  This is due to differences in transportation costs for supplying the respective fuels in these two markets, and the speed of the change has been quite rapid.  The technology is similar, making it both familiar and substitutable.  Banks, governments and utility companies like this, as it means the technologic risk is low.

Transportation – ocean shipping, land transport, air – can also substitute, probably more than is generally anticipated.  For example, North America, with more than a 4:1 energy cost benefit driving the process, is moving to use natural gas for fleets and long-distance trucking.  These are the markets which have the lowest infrastructure support investment, and these markets likely will provide the foundation for more widespread use.  Furthermore, as is the case for electricity generation, the technologies for transporting, storing and using natural gas instead of gasoline or diesel are in use, so there is minimal technologic risk to investments.

Which brings us back to the role of economics in determining the energy mix for the economy.  The present change from an oil-dominated energy economy to something different seems, at present, to be towards a combined fossil fuel energy economy.  This raises concerns both about when will we reach the inevitable fossil fuel peak, and for the amount of global warming that will be engendered by the time such a peak is reached.  But the continuing steep rise of total fossil fuel production makes me sceptical that the economic system has yet found an alternative for society’s energy demand.