Mathematics and Peak Oil

Transcription

1 Mathematics and Peak Oil Richard W. Beveridge Clatsop Community College

2 Historical Background The conception of exponential growth has been around for a long time. Problem #79 from the Rhind Papyrus (17 th century BC) There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all enumerated things?

3 Growth by Doubling In 1256, the author Ibn Kallikan published the story of the Grand Vizier Sissa ben Dahir, the inventor of the chessboard. He brought the board to his king, King Shirham of India.

4 Growth by Doubling To reward Sissa, the king asked what he wanted in return. Sissa replied that he simply wanted one grain of wheat on the first square of the board, two grains on the second, four grains on the third, eight grains on the fourth and so on for all 64 squares on the board.

5 Fibonacci The famous Fibonacci sequence originated in a question of population. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,

6 Fibonacci A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive?

7 Mathematics and Social Science Thomas Malthus ( ) Adolphe Quetelet ( ) Pierre Verhulst ( )

8 Thomas Malthus Wrote An Essay on the Principal of Population (1798) Based his work on observations related to doubling time for populations.

9 Thomas Malthus This fundamental relationship is that the RATE of growth of a population is proportional to the size of the current population. More people=more babies=even more people and so on

10 Calculus in Brief What is commonly known as calculus is generally comprised of two pieces: Differential Calculus Integral Calculus

11 Differential Calculus Differential Calculus is generally concerned with finding the rate of change (or slope) of a given curve at a given point. A straight line has a constant rate of change, whereas a curve has a different rate of change at different points on the curve.

12 Year Population Change from last census ,929, ,308,483 1,379, ,239,881 1,931, ,638,453 2,398, ,866,020 3,227, ,069,453 4,203, ,191,876 6,122, ,443,321 8,251, ,558,371 7,115, ,189,209 11,630, ,979,766 12,790, ,212,168 13,232, ,228,496 16,016, ,021,537 13,793, ,202,624 17,181, ,164,569 8,961, ,325,798 19,161, ,323,175 27,997, ,302,031 23,978, ,542,199 23,240, ,709,873 22,167, ,421,906 32,712,033

13 Year Population Change from last census % change ,929, ,308,483 1,379, % ,239,881 1,931, % ,638,453 2,398, % ,866,020 3,227, % ,069,453 4,203, % ,191,876 6,122, % ,443,321 8,251, % ,558,371 7,115, % ,189,209 11,630, % ,979,766 12,790, % ,212,168 13,232, % ,228,496 16,016, % ,021,537 13,793, % ,202,624 17,181, % ,164,569 8,961, % ,325,798 19,161, % ,323,175 27,997, % ,302,031 23,978, % ,542,199 23,240, % ,709,873 22,167, % ,421,906 32,712, %

14 Exponential Growth Given a population of size P, then the rate of change of the population with respect to time ( t) is called dp. dt So, if the rate of change of the population is proportional to the size of the population, then

15 Exponential Growth dp = dt kp dp = P kdt dp P = k dt ln P = kt+ c kt ae = P

16 Exponential Growth y x y x

17 300,000, ,000, ,000, ,000, ,000,000 50,000, U.S. Census Data

18 Exponential Growth The Malthusian exponential growth model is only useful for limited time periods as it seems clear that nothing in the natural world can grow without limits.

19 Logistic Models From data gathered in 1919 by researchers growing sunflowers at University of California, Riverside. Days Height (cm)

20 Logistic Models y x

21 England Census Data ,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000,

22 1,200,000 1,000, , , , , Rhode Island Census Data

23 9,000,000 8,000,000 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000, New York City Population

24 Pierre Verhulst Pierre Verhulst conceived of a method to improve on the Malthusian or unlimited exponential growth model. Verhulst s idea was to introduce a counterbalancing factor into the equations that govern the population growth model.

25 Pierre Verhulst Verhulst theorized that population growth would be limited by the relationship between the size of the population and a theoretical carrying capacity.

26 Pierre Verhulst This was the foundation for the logistic model. dp = kp dt 1 P C dp = kp k dt P C 2

27 Exponential Growth dp = dt kp Logistic Growth dp = kp dt 1 P C

28 Pierre Verhulst In this case the population is decreased by its relationship to the carrying capacity

29 Pierre Verhulst The solution of the previous differential equation dp = kp dt 1 P C is given as P( t)= C 1+ Ae kt

30 Pierre Verhulst P( t)= C 1+ Ae kt where C is the carrying capacity and A and k are constants derived from observable data. This is known as the logistic function.

31 Logistic Models This model was found to be more realistic and useful for modeling the growth of living organisms and populations.

32 Logistic Models Taking a very basic form of the logistic equation P( t)= e t we can examine the graph

33 Logistic Models y x P( t)= e t

34 Logistic Models y x

35 Logistic Models If we go back and look at the sunflower data again, we can examine the rate of growth rather than the height of the plant. Days Growth (cm)

36 Logistic Models y x Days Growth (cm)

37 Logistic Models We can also examine the graph of the derivative or rate of change. dp dt = e t e t

38 Logistic Models dp dt = e t e t y x

39 Annual US Oil Production (in thousands) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,000 0

40 Logistic Models y x P( t)= e t

41 Cumulative U.S. Oil Production (in thousands) 250,000, ,000, ,000, ,000,000 50,000,000 0

42 M. King Hubbert Marion King Hubbert ( ) Hubbert was a geophysicist who worked for Shell Oil from In 1956, Hubbert presented a paper to the Spring Meeting of the American Petroleum Institute in San Antonio.

43 M. King Hubbert In Hubbert s 1956 presentation, he was mainly concerned with the exponential growth in oil production. At the time of his talk, world oil production was increasing by 7% per year and, consequently, doubling every 10 years.

44 M. King Hubbert These facts alone force one to ask how long such rates of growth can be kept up Thus in ten doubling periods, the production rate would increase by a thousandfold; in twenty by a millionfold No finite resource can sustain for longer than a brief period such a rate of growth of production.

45 Calculus in Brief What is commonly known as calculus is generally comprised of two pieces: Differential Calculus Integral Calculus

46 Integral Calculus Integral calculus is generally concerned with finding the area under a curve. This is often useful when the curve itself represents a rate of change then the area under the curve will be the total amount.

47 Logistic Models dp dt = e t e t y x

48 Logistic Models y x P( t)= e t

49 Annual US Oil Production (in thousands) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,000 0

50 Cumulative U.S. Oil Production (in thousands) 250,000, ,000, ,000, ,000,000 50,000,000 0

51 M. King Hubbert In his 1956 talk, Hubbert used a method estimating total oil reserves in the lower 48 states as between 150 and 200 billion barrels. Since the area under the production curve represents the total amount produced, this area can never exceed the total reserves.

52 M. King Hubbert This restriction on the area under the curve creates an initial estimate for peak production. By fitting the remaining oil available under the remaining piece of the curve, Hubbert arrived at an estimate for peak oil production in the lower 48 states.

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54 M. King Hubbert He estimated that U.S. oil production would peak some time between 1965 and He was correct.

55 U.S. Oil Production In fact U.S. oil production peaked in 1970 at just over 3.5 billion barrels that year. In 1985, there was a second, lower peak of about 3.3 billion barrels that year as a result of Alaskan oil production.

56 Annual US Oil Production (in thousands) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,000 0

57 U.S. Oil Production Since 1985, U.S. oil production has declined by about 45%. In 2008, U.S. oil production was about 1.8 billion barrels.

58 M. King Hubbert In 1962, Hubbert refined his methods to improve his analysis of oil production. His new method used data sets for U.S. oil production and U.S. oil discoveries.

59 M. King Hubbert The cumulative production when plotted as a function of time, will increase slowly during the early stages of petroleum exploitation, increase more and more rapidly with time to about the halfway point, and then continue its ascent by rising more and more slowly, finally leveling off to the ultimate figure [of total oil available] as production ceases.

60 Logistic Models y x P( t)= e t

61 M. King Hubbert As oil must be found before it can be produced, the curve of cumulative proved discoveries must closely resemble that of cumulative production, except that it must plot ahead of the production curve by some time interval

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66 5 4 US Lower 48: annual oil discovery & production with Hubbert discovery model discovery smooth 5 yr model Hubbert disc. production Hubbert disc. shift 35 yr 3 2 deepwater Jean Laherrère Jan year

67 US lower 48 cumulative oil production & mean discovery with logistic models Jean Laherrere year L1 U=150 Gb L1+ L2 U=200 Gb mean discovery logistic U=200 Gb production

68 2,500,000 2,000,000 1,500,000 1,000, , Iran Annual Oil Production (in thousands)

69 France oil discovery & production modeled with 2 Hubbert curves discovery CD=930 Mb prod. Petroconsultants production USDOE H1 U=350 Mb H1+ H2 U=800 Mb Jean Laherrere year

70 2 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 World cumulative conventional oil discovery & oil production with a logistic model discovery oil+condensate L1 discovery U=2 Tb production liquides production crude oil L2 production U=2 Tb USGS 3012 Gb Jean Laherrere year

71 Saudi Arabia Annual Oil Production (in thousands) 4,500,000 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,

72 1,000, , , , , , , , , , Nigeria Annual Oil Production (in thousands)

73 1,600,000 1,400,000 1,200,000 1,000, , , , , Venezuela Annual Oil Production (in thousands)

74 1,600,000 1,400,000 1,200,000 1,000, , , , , Mexico Annual Oil Production (in thousands)

75 35,000,000 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000, World Annual Oil Production (in thousands)

76 Annual US Oil Production (in thousands) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000, ,000 0

77 U.S. Oil Production Data From the U.S. Energy Information Administration. D_CRPDN_ADC_MBBLPD_A.htm

78 World Oil Production Data BP Statistical Review of World Energy June icle.do?categoryid= &contenti d=

79 References Hubbert, M.K., Energy from Fossil Fuels, Science, February 1949, v. 109, n. 2823, p , Nuclear Energy and the Fossil Fuels, Drilling and Production Practice, 1956, Presented before the Spring Meeting of the Southern District of the American Petroleum Institute, San Antonio, Texas, March 7-9, 1956.

80 References Hubbert, M.K., Energy Resources A Report to the Committee on Natural Resources of the National Academy of Sciences National Research Council, Publication 1000-D, National Academy of Sciences National Research Council, Washington, D.C., 1962.

81 /rbeveridge/research/papers_and _Presentations.htm