Sunday, December 03, 2006

The Loglet Analysis Applied to the United Kingdom's Oil Production

Last week, Euan published on TOD a great compilation of UK forecasts. The UK production profile is a notoriously difficult case to model because it has two pronounced Hubbert cycles. The loglet analysis with two Logistic curves seems appropriate and gives the following result:

Fig. 1 Result of the loglet analysis (bi-logistic) in the standard Hubbert Linearization representation. The red line is the standard mono-logistic approach. Data is for crude oil plus condensate (source: DTI). mbpd= millions of barrels per day.

Fig. 2 Result of the loglet analysis (bi-logistic) and the mono-logistic curve derived from the Hubbert Linearization. Data is for crude oil plus condensate (source: DTI). mbpd= millions of barrels per day.

We can appreciate how accurate is the loglet analysis (Fig. 2) compared to the standard mono-logistic approach. The Ultimate Recoverable Resource (URR) is around 25 Gb for the loglets whereas the Hubbert Linearization gives 28.77 Gb. The production is predicted to reach 0.6 mbpd in 2010 (almost 50% of current levels). The parameters of the two logistic curves (or loglets) are the following:

Loglet 1Loglet 2
URR (Gb) 7.921 16.85
K (%)
42.85 22.72
Peak Date 1984.4 1998.7

The decline in production is dominated by the second logistic curve which has a fairly large logistic growth rate (K) at 22% especially compared with the result of the HL technique (K= 12.9%). When modeled by a logistic curve, the decline rate in production accelerates linearly with cumulative production and ultimately reaches the value K which can be considered as the ultimate decline rate value:

(dP/dt)/P=K(1-2Q/URR)=K(Reserve Fraction - Cum. Prod. Fraction)

For instance, when 75% of the URR has been extracted (Q=0.75xURR), the reserve fraction is equal to 25% and the cumulative production fraction is equal to 75% which implies that the decline rate reaches K/2.

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:


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:

Algeria, Angola, Cameroon, Chad, Egypt, Equatorial_Guinea, Gabon, Libya, Nigeria, Rep_of_Congo, Sudan, Tunisia, other_Africa.
Australia, Brunei, China, India, Indonesia, Malaysia, Taylandia, Vientnam, other_Asia.
Denmark, FSU, Italy, Norway, Rumania, UK, other_Europe.
Iran, Iraq, Kuwait, Oman, Qatar, Saudi_Arabia, Syria, UAE, Yemen, Other_ME.
Canada, Mexico, US.
Argentina, Brazil, Colombia, Ecuador, Peru, Trinidad, Venezuela, Other_S_and_Cent_America.

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


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:



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



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.