Norway and the Parabolic Fractal Law
URR= Ultimate Recoverable Ressource
PFL= Parabolic Fractal Law
HL= Hubbert Linearization
Mb= Millions of barrels
Gb= Billions of barrels
Norway's oil production is one of the most well documented production with complete data freely available on the Internet . Production has been booming in the late 80s with a 50% growth in production between 1987 and 1992. However, production has peaked in 2001 and has declined dramatically afterward with a decline rate that mirrors the previous strong growth.
Fig. 1- Norway oilfields (crude oil only). GraphOilogy.
As we can see on Fig. 2, the Hubbert approach gives rather good results. The estimated URR is 26.54 Gb (crude oil only, no condensates) and the observed cumulative production is 19.2 Gb (72.5% of the URR).
Fig. 2- Hubbert Linearization technique applied on
(green points are the data used for the fit).
We can appreciate how well the curve fitting is working for Norway by comparing with some reserve estimates given in Table I.
|Reserve (Gb)||URR (Gb)|
|ASPO ||End 2001||15.2
|Oil & Gas Journal ||end
|World Oil ||end
Reserve estimates may include NGL.
The logistic growth rate (or decline rate) K for Norway is very large at 17% . One reason for that steep production growth and decline is maybe due to the very large decline rates observed on the top 5 fields as shown on Table II. Four of the top fields have values for K ranging from 25% to 37%! Fig. 3 gives the result of the logistic curve modeling on Statfjord which is at 98% of its URR. Only Ekofisk (Fig. 4), one of oldest field, can be seen as the remaining backbone of Norway's production with only a logistic decline rate at 8% and only 47% of its URR produced.
|Field||Q(2005) (Gb)||Starting Year||URR (Gb)||K (%)|
URR and logistic growth (K) estimates using the HL technique.
Fig. 3- HL technique applied on the Statfjord oilfield. The logistic curve peak position is computed
in order to have the same cumulative production in 2005.
Fig. 4- HL technique applied on the Ekofisk oilfield. The logistic curve peak position is computed
in order to have the same cumulative production in 2005
The data for Norway contains production profiles for 47 oilfields. In order to get a more precise estimate of the PFL parameters, we perform the HL technique on each of the top 16 fields represented as empty circles on Fig. 5. Note that this refinement step does not change much the top of the PFL curve which is already very mature.
Fig. 5- Field cumulative production values displayed
in a log(size)-log(rank) plane at different points in time.
The empty circles are the URR estimates for the 16 top fields using the HL technique.
The next step is to try to
fit a quadratic polynome using the
data representation shown on Fig. 6. Only the top 16 fields are used for the fit. The resulting
curvature is surprinsingly low at -0.31
which is more than four times the value obtain for the UK and the world
(-0.07) . I
don't have a definitive answer on why this value is so low, one
possibility is the dataset contains too few oil fields (only 47). One
clue: In 
page 26, Jean Laherrère reports the following:
In UK only 52% of the discoveries are developed representing 89% of the reserves. In Norway, only 33% of the discoveries are developed representing 83% of the reserves.
Table III is from the same document and gives some details on the distribution of reserves for the UK and Norway (NW).
Fig. 6- Estimation of
various PF Laws with different
fixed curvature values.
Each data point is color coded according to the oil field age..
Table III. from Jean Laherrère , page 26: O= crude oil, C= Condensate, nb= number of fields.
Now, let's assume that we have a total number of fields around 200. This number will intersect the different Parabolic Fractal curves and is represented as an orange dotted line on Fig. 7. The intersection of this line with the different curves gives different URR and minimum size values as shown on Fig. 8. We can see that the world curvature (the orange disc) leads to an URR close to the ASPO and the official reserve numbers (the minimum field size is then around 7 Mb).
Fig. 7- Derived URR from
the PFL shown on Fig 6
for different curvature values and number of fields.
The orange dotted line is the isoline corresponding to 200 fields.
Fig. 8- Derived URR from Fig. 7 by fixing the number of fields
to 200 (dotted orange line on Fig. 8).
The disc size and color is function of the minimum field size.
HL is the URR estimate from the Hubbert Linearization shown on Fig. 2.
- the Hubbert linearization gives an URR at 26.5 Gb
- the number of fields discovered is around 200
- the PFL + world curvature gives an URR between 26 and 32 Gb
- the PFL + world curvature + 200 fields gives an URR at 31 Gb
References TOD: An Attempt to Apply The Parabolic Fractal Law to Saudi Arabia
 GraphOilogy: What Can We Learn From The Oil Field Size Distribution?
 GraphOilogy: Some Detailed Views on Norway's Oil Production
 Jean Laherrère: “Parabolic fractal” distributions in Nature. (in French)
 Jean Laherrère: Estimates of Oil Reserves
 ASPO: Newsletter 62 (2006/02)
 ASPO: Newsletter 25 (2003/01)
 World Proved Reserves of Oil and Natural Gas, Most Recent Estimates
 BP: Statistical Review of World Energy 2006
 Norwegian Petroleum Directorate