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# Hubbert's Peak Mathematics

## by Luís de Sousa

M. King Hubbert's early work was founded on somewhat complex differential equations. That earned him some criticism; the method was as impenetrable as a monolith, which only those with profound mathematical knowledge could understand.

Based on population work from the 1960s decade, Hubbert presented in his 1982 article "Techniques of Prediction as Applied to the Production of Oil and Gas" an alternative method, much more accessible. Such method is clearly described in Professor Kenneth Deffeyes' book "Beyond Oil". What follows is an adaptation from this book using data available on-line.

## United States

In order to determine the Hubbert's Peak one needs two sets of data, annual production, called P, and the cumulative production, known as Q. Let's start by applying this method to the lower 48 states of USA which, as we know, passed their peak in 1971. Annual production figures for these states are available in the Energy Information Administration (EIA - Energy Information Administration) web site. We can't find cumulative production here, but we can use the value published in ASPO's newsletter no. 23 that, for 2001, indicates a cumulative production of 169 gigabarrels.

With the help of a calc sheet we can determine for each year the value of P/Q. Next we can draw the graph of P/Q versus Q, getting something like this:

For the first years there is some disorder, but from 1958 onwards the dots take a negative trend towards the xx axis. Let us fit a straight line to this set of dots from 1958 on, using this formula:

Y = mX + a

In this case Y is P/Q and X is Q, a is the value P/Q gets when Q is zero and m is the line's inclination. The line fitted to this dots has a value of 0.061 for a and -3x10E-4 for m. With this straight line we can find the value of Q for which P/Q is zero, in this case 198.395, usually called Qt. This value is the maximum cumulative production that will ever be achieved, knowing that the peak occurs exactly amidst this total; we can easily put it in 1973, with 99.198 gigabarrels produced.

Hubbert's theory is simply the assumption that the relation between P/Q and Q follows a straight line, all the rest is pure mathematics. Let's resolve the line's equation in order to P and see what appends:

P/Q = mQ + a

P/Q = -aQ/Qt + a

P/Q = a(1 - Q/Qt)

P = a(1 - Q/Qt)Q

The bit inside the parenthesis  (1 - Q/Qt) is the fraction of the total oil left to produce, meaning that the capacity we have to produce oil at a given moment in time is linearly dependent on the amount of oil still available to produce. The lowermost equation is a logistic curve that stands a bell shape.

In order to get this curve we must use again our calc sheet. First of all let's get the expression for 1/P:

1/P = 1/[a(1 - Q/Qt)Q]

With 1/P we now have years per gigabarrel instead of gigabarrel per year. We go back to the calc sheet, and create a new column for Q which we fill with 1 gigabarrels increment, starting in 1 and ending in Qt. In another column we compute the value of 1/P and in another P, using the expressions achieved before. Finally we have to adjust Time in here, easy to do it knowing that by the end of 2001 169 gigabarrels had been produced. We add another column and in the line that Q stands 169 we insert 2002, for the other cells of this new column we just add or subtract successively the values in 1/P. The result will be similar to this table:

All we need to do now is to draw the graph with Time in the xx axis and P in the yy axis; we can add the original data too:

## The World

For the whole world we can use the data available in the BP Statistical Review, that has production figures from 1965 to present. Once more for the cumulative values we must recur to the ASPO's newsletters, that for 2004 show a cumulative production of 1040 gigabarrels. The data we used for the US only yielded conventional oil, the BP data also include oil sands, heavy oils and Liquefied Natural Gas (LNG). Here's the P/Q versus Q graph:

Again we find the chaotic start, but after 1983 the dots align is a smooth downward trend. Fitting a straight line as before we get this:

For this line the equation is P/Q = -2.36x10E-5 Q + 0.051, resulting in a value of 2164.86 for Qt. And the magic is done; let's compute 1/P to get our beloved P versus Time graph:

All of this to get the conclusion that applying Hubbert's method to the data available until 2004 we have a production peak in 2006 Summer Solstice.

## Conclusions

Knowing how the Hubbert method works we can now understand the reasons behind the different dates pointed to the Oil Peak . The differences are mainly related with the data used:

• Considering only conventional oil we'll get the peak in 2004, as indicated in the ASPO newsletters.
• Using all the production, but excluding tar sands and LNG, we get the peak by the end of 2005. This is the data used by professor Deffeyes in his book.
• As we saw before, using all the liquids we get the peak in the mid of 2006. In older models this can turn out to be 2007, also small differences in a or m can shift the peak some months.
• Studies that put the peak beyond 2010 do not use this method; usually they consider the declining rates of existing fields and the projected production of developing fields. These studies are often called bottom-up analysis.
• Last but not least, it is worth mentioning the model published by ASPO, for all liquids, driven by Dr. Colin Campbell and the Uppsala Hydrocarbon Depletion Study Group, lead by Professor Kjell Aleklett. Presently the peak year is indicated to be 2010. This apparently results from the combination of Hubbert's method with a bottom-up analysis, considering a major development in deep-water exploration.

Of all these models, Professor Deffeyes' might be the most meaningful. There's a lot of a difference between conventional oil and the rest, the easiness with which is extracted from the soil. Global production can continue to increase for some more years, but recurring to resources much more difficult to explore, and much more expensive. Cheap oil might be already a thing of the past.

In his book, Professor Deffeyes took the chance to create a symbolic date to the Oil Peak , Thanksgiving 2005. It's the right time to thank for a century and a half of cheap oil that Nature gave us. And it's also the right time to once and for all face this challenge, undoubtedly the greatest faced by Mankind.

For examples of the use of these techniques in a school or college situation, visit this US college page.

Luís de Sousa has a Portuguese website on peak oil at PicoDoPetroleo.net

 Contents The USA The World Conclusions
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