Fig. 1 Example of a simple self-similar object size distribution represented in a log(rank)-log(size) plane. The red line has a slope equals to -1.
The Parabolic Fractal Law (PFL) is an unperfect self-similar law where a quadratic term is added:
log(Size(i))= a + b.log(i) + c.log(i)^2
URR= Sum_i(a + b.log(i) + c.log(i)^2)
with: a + b.log(i) + c.log(i)^2 > Size_Min
Application to Saudi Arabia
Unfortunately, there is no public database on Saudi Arabia oilfields. We don't need to get an exhaustive dataset but only a few estimates about the size of the largest fields. I found some data about the top 9 fields from various sources on the web and from Simmons's book (Twilight in the Desert).
|Field||URR (Gb)||Discovery Date|
Because we have so little data, it will be difficult to reliably estimate a valid parabolic curvature. So we proceed asfollowing:
- The Field URR are ranked according to their size and represented in a log(rank)-log(URR) plane as shown on Fig. 2. When only an interval is available, we take the center value.
- a robust linear fit (no curvature) is applied on the data points (red line on Fig. 2).
- A parabolic model is then fitted using the slope established in 2) as first guess for the linear term b and a fixed curvature value c (blue lines on Fig. 2). The algorithm used for this step is the Levenberg-Marquardt algorithm.
- The URR values are estimated from the areas under the PFL curves conditionally to a particular minimum oil field size as shown on Fig. 3.
Fig. 2. Estimation of various PF Laws with different fixed curvature values. Each data point is color coded according to the oil field age.
Fig. 3. Derived URR from the PFL shown on Fig 2. The URR value is function of the minimum oil field size considered.
Fig. 5. Hubbert Linearizations on Saudi Arabia Production profile (data from BP, Crude + NGL). The blue points are the ASPO forecast which see a constant production level for the next 20 years at 9.5 mbpd (newsletter 66).
Fig. 6. Contributions from oil fields using the PFL law for Saudi Arabia with the world curvature (green line on Fig. 2 and Fig. 3). The top 10% of oil fields (size > 67 Mb) contributes to 97% of the total URR.
- combined with the Hubbert Linearization technique, the PFL could be useful for tortuous production profiles from immature countries such as Saudi Arabia, Iraq and Iran.
- only the top fields are necessary for the fit which is interesting because they are usually the most mature and the most documented. However, we implicitly assume that the discovery of large fields has peaked early in the production history and that no giant or super-giants will be discovered.
- the PFL integrates naturally contributions from small fields and the derived URR is dependent on the minimum field size. Therefore, some reserve growth can be simulated by changing the small field cutoff value.
- Jean Laherrère: “Parabolic fractal” distributions in Nature. (in French but has many interesting graphs),
- Ada A. Adamic. Zipf, Power-laws, and Pareto - a ranking tutorial
- William J. Reed. The Pareto, Zipf and other power laws
- Narushige SHIODE. Power Law Distributions in Real and Virtual Worlds.