**Summary**

*There are several ways to estimate the amount of oil that will ultimately be pumped from the ground — that is, the Ultimate Recoverable Resource (URR). The usual way is to apply a curve-fitting technique following W. King Hubbert’s approach. *

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Another way is to study the distribution of oil field sizes, something that is well described by a Parabolic Fractal Law.

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*The latter approach is illustrated on the UK field-by-field data set. We find that bringing smaller oilfields online will not make a significant difference. Small oilfields will not save us.*

What is interesting in that model, is that the law parameters can be estimated from only the top fields (the King, the Queens, etc.) which are usually more mature because exploited first (see for instance Fig. 2 for the UK). We can use a quick analogy for this approach: the total size of an iceberg (i.e. the URR) can be easily estimated from the emerged visible part (i.e. the top fields). Below, we will explore that approach for the case of the United Kingdom using the production data for 228 oil fields (Fig. 2). The Hubbert Linearization technique applied for the UK is shown on Fig. 3 and leads to an URR equals to 28.3 Gb.

The parameters of the Parabolic Fractal Law will be derived from a fit of a parabol in the log(size)-log(rank) domain. Of course, oil fields have not reached the same level of maturity an the observed curve will evolve in time as shown on Fig. 4. However, because top fields are also the oldest, we can observe that they have reached some kind of asymptote simply because their cumulative production is approaching their ultimate size. Consequently, we can reasonably fit the parabola from the top 32 oil fields as shown on Fig. 5.

From the PF Law we can estimate the URR (i.e. the area under the parabolic curve) which gives 24 Gb. One issue, is the sensitivity of the URR estimate to the data points used for the fit. Fig. 6 gives the the various URR estimates when the maximum rank considered varies between 4 and 32. We can see that the estimate is reasonably stable (24-32 Gb) beyond the rank 15 and is sligthly decreasing probably because some immature fields are included.

Will small fields make a difference?

Well, the URR value will depend on the minimum oil field size that will be developped. Assuming that only small fields have yet to be discovered, the size cutoff value will not impact significantly the URR value as shown on Fig. 7. Even if we add 228 new small fields, the Parabolic Fractal Law predicts only an increase of 2 Gb in reserves!

What about the world?

The application of Parabolic Fractal Law is interesting in case the top oild fields have been exploited first and are mature enough to provide a reliable estimate of the model parameters. Giant and super-giant oil fields are important to watch because they are the tip of the iceberg that can help us assess what is liying beneath! Jean Laherrère has actually applied the PF law to the world as shown on Fig. 8. Using his parameters, we can compute a world URR (excluding the US and Canada, conventional oil) equals to 1.250 Trillions of Barrels (Tb) without considering oil fields with sizes below 50 Mb. The URR value will be dependent on the minimum oil field size as shown on Fig. 9.

References:

[1] Lada A. Adamic. Zipf, Power-laws, and Pareto – a ranking tutorial

[2] Jean Laherrère. “Parabolic fractal” distributions in Nature.

[3] William J. Reed. The Pareto, Zipf and other power laws.