## Wednesday, September 06, 2006

### The Hubbert Parabola

This is my first post in Graphoilogy, and first I would like to thank Khebab for including me as a contributor.

The logistic curve, and its derivative the hubbert's curve, has been widely used to model population growth. And it has been applied to model oil production by M. King Hubbert. The model comes from the following differential equation:

dQ/dt=kQ(1-Q/URR)

where Q(t) is a function of time (measured in years) and it is defined as the cumulative production of a region until the end of year t. The parameter URR is the "Ultimately Recoverable Resources" or the maximum cumulative production that can be reached. K is the Malthusian parameter or the maximum cumulative production growth.

The value dQ/dt for a specific year s, can be approximated by

(Q(s)-Q(s-1))/(s-(s-1))= Q(s)-Q(s-1)

which is the production of year s, so let us define P(t)=Q(t)-Q(t-1). Let us assume that we have a region where the oil production follows strictly the logistic model, and that we have the information of Q(t) for all years. If we had P(t) for all years then we could obtain Q(t), since Q(t)=P(t)+P(t-1)+P(t-2)+.... What happens if we place a point at position (Q(t),P(t)) in the plane for all years t? Well, since we have seen that P(t) is approximately KQ(t)(1-Q(t)/URR), then the points would follow very closely the parabola:

KQ(1-Q/URR) = KQ - (K/URR)*Q^2.

This parabola passes through the origin (0,0) and through (URR,0). Now we are going to experiment with 50 different regions of the world. For each one we are going todo two plots. In the first one we will place all the data points (Q(t),P(t)) until year 2005 (both measured in Giga-barrels) and then find the parabola that passes through the origin that better approximates the data points (by the least squares fitting method). The intersection of this parabola with the x-axis will give us the estimated URR. In the second plot we want to show how this estimated URR has changed through time. For this plot, we define the function URR(t) as the estimated URR by the prior method if we had used the data points up until year t, and discarding later years. In the second plot we place points at (Q(t),URR(t)). Clearly Q(t) < URR(t) should hold by definition (just note that with very strange data set this could be false). So all points in the second plot should be above the URR=Q line. The dashed line URR=2Q has an interesting property. If point (Q(t),URR(t)) lies above this line, then according to the logistic model, t is before the peak year (as calculated at year t), i.e. Q(t) < URR(t)/2. If (Q(t),URR(t)) lies below the dashed line then we are after peak year.

The data from 1857 till 1958 is from "API Facts and Figures Centennial edition 1959" (thanks to Laherrere and Stuart Staniford). For 1959-1964 it is from "Twentieth Century Petroleum Statistics2004" of DeGolyer & MacNaughton. And for 1965-2005 it is from BP Statistical Review of World Energy.

First I will put links to all the images of regions ordered by continents:

AFRICA:
Algeria, Angola, Cameroon, Chad, Egypt, Equatorial_Guinea, Gabon, Libya, Nigeria, Rep_of_Congo, Sudan, Tunisia, other_Africa.
ASIA:
Australia, Brunei, China, India, Indonesia, Malaysia, Taylandia, Vientnam, other_Asia.
EUROPE:
Denmark, FSU, Italy, Norway, Rumania, UK, other_Europe.
MIDDLE EAST:
Iran, Iraq, Kuwait, Oman, Qatar, Saudi_Arabia, Syria, UAE, Yemen, Other_ME.
NORTH AMERICA:
SOUTH AND CENTRAL AMERICA:

After a quick inspection of the plots we can divide countries in three groups:

### COUNTRIES THAT BEHAVE WELL:

By this I mean countries where the early points of the second plot start increasing above the dashed line, and after they cross the dashed line the points tend to stabilize into a fixed estimated URR. These include the following 20 (out of 50) countries:

 Egypt Gabon Rep_of_Congo Brunei
 India Indonesia Vientnam Denmark
 FSU Italy Norway Rumania
 other_Europe Oman Syria Mexico

### TOO EARLY TO SAY COUNTRIES:

Probably they haven't reach peak production yet. These group includes 9 countries.

 Taylandia other_Asia Qatar Brazil
 Colombia

COUNTRIES THAT DO NOT BEHAVE WELL:

Countries for which the points of the second plot just keep increasing between the two lines. Normally they increase in a very strait line. These group has 21 countries.

 Algeria Cameroon Libya Nigeria
 Tunisia other_Africa Australia China
 Malaysia UK Iran Iraq
 Kuwait Other_ME Saudi_Arabia UAE
 Venezuela

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