One objective of the curve modeling is to try to infer the Ultimate Recoverable Resource (URR) from the observed oil production profile. Assuming a logistic model, the Hubbert linearization is a convenient way to estimate the URR from a simple Prod. / Cum. Prod. vs Cum. Prod. (see Stuart Staniford's post for more details). Another approach, less employed, is to estimate the URR from the observed distribution of oil field sizes. Jean Laherrère explored that avenue and came up with the Parabolic Fractal Law (PFL). This law is quite general and can be applied to various natural phenomenon where there is a certain level of self-similarity and is a generalization of the Pareto distribution. The PFL is well verified for the distribution of galaxy sizes, town populations, Revenues, Internet traffic, etc.. These distributions are very sparse (i.e. lots of small size objects, a few large ones) and are better visualized in a log-log plot as shown on Fig. 1.

Fig 1. The distribution of AOL users' visits to various sites on a December day in 1997. One can observe that a few sites get upward of 2000 visitors, whereas most sites got only a few visits (70,000 sites received only a single visit). On a log-log scale the distribution shows itself to be linear. This is the characteristic signature of a power-law. Log-log scale plot of the distribution of users among web sites

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.

Fig 2. Individual oil field production profiles color coded according to the first year of production.

Fig 3. Hubbert Linearization technique applied on the total UK crude oil production (excluding condensate and refinery gains). The green points are those involved in the fit. The URR estimated is 28.4 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.

Fig 4. Cumulative production of each oil fields against their rank at various points in time. Observe how the distribution converge toward a parabolic law.

Fig 5. Cumulative production of each oil fields against their rank. Each point is color coded according to the first year of production.

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.

Fig 6. Estimation of the URR from the Parabolic Fractal Law as a function of the maximum field rank used for the regression.

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!

Fig 7. Influence of the minimum oil field size (cutoff size) on the URR value assuming a Parabolic Fractal law. The observed number of fields is 228. Doubling the number of fields (456) by exploiting oil field with a size > 6 Mb will only increase the URR by less than 10%.

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.

Fig 9. Influence of the minimum oil field size (cutoff size) on the world URR value assuming the Parabolic Fractal model proposed by Jean Laherrere. The initial number of fields is 2,092. Multiplying by 16 the number of fields (33,472) by exploiting oil fields with a size > 1 Mb will increase the URR by 22%.

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.