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:
Canada, Mexico, US.
SOUTH AND CENTRAL AMERICA:
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:

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:

EgyptGabonRep_of_CongoBrunei
IndiaIndonesiaVientnamDenmark
FSUItalyNorwayRumania
other_EuropeOmanSyriaMexico
USArgentinaPeruTrinidad

TOO EARLY TO SAY COUNTRIES:


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

AngolaChadEquatorial_GuineaSudan
Taylandiaother_AsiaQatarBrazil
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.

AlgeriaCameroonLibyaNigeria
Tunisiaother_AfricaAustraliaChina
MalaysiaUKIranIraq
KuwaitOther_MESaudi_ArabiaUAE
YemenCanadaEcuadorOther_S_and_Cent_America
Venezuela

19 comments:

  1. 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

    ReplyDelete
  2. great post. looking forward on this post.

    ReplyDelete
  3. Wood flooring is one of the best choices to consider when constructing and designing your floors. This is a huge help for those who want to consider different sources.Wholesale Garcinia cambogia

    ReplyDelete
  4. I am currently teaching a class and part of the curriculum involves this subject matter. Do you have any other posts I can look at regarding this local marketing

    ReplyDelete
  5. Thanks first time I've read such a good article, really very good,I hope there still can continue to read such a good article.Cheap Towels Wholesale

    ReplyDelete
  6. I think this one's a great place to spend my holidays with my family. I should probably consider this one. Thanks! Waterssound real estate

    ReplyDelete
  7. I merely have a look here and emerge big to hunt out out this journal. created content conceiving hand and very cooperative location. I crave most of we've got a bent to UN agency ar to hunt out these varieties of things, here we are going to understand everything auto title loans for you. i am with the content esteem associated do esteem him as an honest provider. Thanks for your labor and you too.

    ReplyDelete
  8. Thanks a lot very much for the high quality and results-oriented help. I won’t think twice to endorse your blog post to anybody who wants and needs support about this area.game iwin

    ReplyDelete
  9. Our philosophy is based in the knowledge that we are generally the first health professionals who are asked to make decisions that can enhance or diminish our patients’ facial appearance.Escort Svizzera

    ReplyDelete
  10. I read the complete blog and it was really nice for me. I have also published one article on that topic.
    schifffonds

    ReplyDelete
  11. This website has very good content. Thank you for the great article I did enjoyed reading it, I will be sure to bookmark your blog and definitely will come back from again

    ReplyDelete
  12. good morning I hope to spread the knowledge you gain more knowledge for the benefit of all Pest Control

    ReplyDelete
  13. “Thank you, I have been searching for facts about this topic for ages and yours is the best I’ve discovered so far.At the same first, I would support a state of grace to this article illuminating. Secondly, I prefer to ask me where I can learn a lot more information about testing.
    Energiefonds

    ReplyDelete
  14. I need this article to complete my assignment in the college, and it has same topic with your article. Thanks, great share.

    ReplyDelete
  15. I read the complete blog and it was really nice for me. I have also published one article on that topic.
    schifffonds

    ReplyDelete
  16. A mind blowing article is provided here. And it is written with great skill and the words directly explain the thought of author.

    ReplyDelete
  17. I can not stop reading this. And 'so fresh, so full of information, I do not know. I'm glad that people actually write the smart way to show the different sides of him.

    ReplyDelete
  18. I think this one's a great place to spend my holidays with my family. I should probably consider this one. Thanks puresilk salt cell

    ReplyDelete
  19. Hi there, You've done a great job. I will certainly digg it and in my view recommend to my friends. I'm confident they will be benefited from this website.key performance indicator

    ReplyDelete