Mathematics and Peak Oil - PDF Free Download

Mathematics and Peak Oil Richard W. Beveridge Clatsop Community College

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?

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.

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.

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

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?

Mathematics and Social Science Thomas Malthus (1766-1834) Adolphe Quetelet (1796-1874) Pierre Verhulst (1804-1849)

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

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

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

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.

Year Population Change from last census 1790 3,929,214 1800 5,308,483 1,379,269 1810 7,239,881 1,931,398 1820 9,638,453 2,398,572 1830 12,866,020 3,227,567 1840 17,069,453 4,203,433 1850 23,191,876 6,122,423 1860 31,443,321 8,251,445 1870 38,558,371 7,115,050 1880 50,189,209 11,630,838 1890 62,979,766 12,790,557 1900 76,212,168 13,232,402 1910 92,228,496 16,016,328 1920 106,021,537 13,793,041 1930 123,202,624 17,181,087 1940 132,164,569 8,961,945 1950 151,325,798 19,161,229 1960 179,323,175 27,997,377 1970 203,302,031 23,978,856 1980 226,542,199 23,240,168 1990 248,709,873 22,167,674 2000 281,421,906 32,712,033

Year Population Change from last census % change 1790 3,929,214 1800 5,308,483 1,379,269 26.0% 1810 7,239,881 1,931,398 26.7% 1820 9,638,453 2,398,572 24.9% 1830 12,866,020 3,227,567 25.1% 1840 17,069,453 4,203,433 24.6% 1850 23,191,876 6,122,423 26.4% 1860 31,443,321 8,251,445 26.2% 1870 38,558,371 7,115,050 18.5% 1880 50,189,209 11,630,838 23.2% 1890 62,979,766 12,790,557 20.3% 1900 76,212,168 13,232,402 17.4% 1910 92,228,496 16,016,328 17.4% 1920 106,021,537 13,793,041 13.0% 1930 123,202,624 17,181,087 13.9% 1940 132,164,569 8,961,945 6.8% 1950 151,325,798 19,161,229 12.7% 1960 179,323,175 27,997,377 15.6% 1970 203,302,031 23,978,856 11.8% 1980 226,542,199 23,240,168 10.3% 1990 248,709,873 22,167,674 8.9% 2000 281,421,906 32,712,033 11.6%

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

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

Exponential Growth y 8 6 4 2-4 -3-2 -1 1 2 x y 400 300 200 100-8 -6-4 -2 2 4 6 8 x

300,000,000 250,000,000 200,000,000 150,000,000 100,000,000 50,000,000 0 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 U.S. Census Data 1790-2000

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.

Logistic Models From data gathered in 1919 by researchers growing sunflowers at University of California, Riverside. Days Height (cm) 0 0 7 17.93 14 36.36 21 67.76 28 98.1 35 131 42 169.5 49 205.5 56 228.3 63 247.1 70 250.5 77 253.8 84 254.5

Logistic Models y 250 200 150 100 50 10 20 30 40 50 60 70 80 90 x

England Census Data 1811-2001 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000,000 0 1801 1821 1841 1861 1881 1901 1921 1941 1961 1981 2001

1,200,000 1,000,000 800,000 600,000 400,000 200,000 0 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Rhode Island Census Data 1790-2000

9,000,000 8,000,000 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 0 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 New York City Population 1790-2000

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.

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

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

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

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

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

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.

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

Logistic Models Taking a very basic form of the logistic equation P( t)= 1 1 + e t we can examine the graph

Logistic Models y 1-3 -2-1 1 2 3 x P( t)= 1 1 + e t

Logistic Models y 250 200 150 100 50 10 20 30 40 50 60 70 80 90 x

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) 7 17.93 14 18.43 21 31.4 28 30.34 35 32.9 42 38.5 49 36 56 22.8 63 18.8 70 3.4 77 3.3 84 0.7

Logistic Models y 30 20 10 20 40 60 80 x Days Growth (cm) 7 17.93 14 18.43 21 31.4 28 30.34 35 32.9 42 38.5 49 36 56 22.8 63 18.8 70 3.4 77 3.3 84 0.7

Logistic Models We can also examine the graph of the derivative or rate of change. dp dt = e t 1 2 + e t

Logistic Models dp dt = e t 1 2 + e t y -3-2 -1 1 2 3 x

1860 1867 1874 1881 1888 1895 1902 1909 1916 1923 1930 1937 1944 1951 1958 1965 1972 1979 1986 1993 2000 2007 Annual US Oil Production (in thousands) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000 0

Logistic Models y 1-3 -2-1 1 2 3 x P( t)= 1 1 + e t

1860 1867 1874 1881 1888 1895 1902 1909 1916 1923 1930 1937 1944 1951 1958 1965 1972 1979 1986 1993 2000 2007 Cumulative U.S. Oil Production 1860-2008 (in thousands) 250,000,000 200,000,000 150,000,000 100,000,000 50,000,000 0

M. King Hubbert Marion King Hubbert (1903-1989) Hubbert was a geophysicist who worked for Shell Oil from 1943-1964. In 1956, Hubbert presented a paper to the Spring Meeting of the American Petroleum Institute in San Antonio.

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.

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.

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

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.

Logistic Models dp dt = e t 1 2 + e t y -3-2 -1 1 2 3 x

Logistic Models y 1-3 -2-1 1 2 3 x P( t)= 1 1 + e t

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.

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.

M. King Hubbert He estimated that U.S. oil production would peak some time between 1965 and 1975. He was correct.

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.

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.

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.

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.

Logistic Models y 1-3 -2-1 1 2 3 x P( t)= 1 1 + e t

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

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 1 0 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 Jean Laherrère Jan. 2003 year

200 180 US lower 48 cumulative oil production & mean discovery with logistic models 160 140 120 100 80 60 40 20 0 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 Jean Laherrere year L1 U=150 Gb L1+ L2 U=200 Gb mean discovery logistic U=200 Gb production

2,500,000 2,000,000 1,500,000 1,000,000 500,000 0 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 Iran Annual Oil Production (in thousands) 2001 2004 2007

40 35 30 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 25 20 15 10 5 0 1940 1950 1960 1970 1980 1990 2000 2010 2020 Jean Laherrere year

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 0 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 Jean Laherrere year

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 1,000,000 500,000 0 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007

1,000,000 900,000 800,000 700,000 600,000 500,000 400,000 300,000 200,000 100,000 0 1965 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 Nigeria Annual Oil Production (in thousands)

1,600,000 1,400,000 1,200,000 1,000,000 800,000 600,000 400,000 200,000 0 1965 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 Venezuela Annual Oil Production (in thousands)

35,000,000 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000,000 0 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 World Annual Oil Production (in thousands)

U.S. Oil Production Data From the U.S. Energy Information Administration. http://www.eia.doe.gov/ http://tonto.eia.doe.gov/dnav/pet/pet_cr D_CRPDN_ADC_MBBLPD_A.htm

World Oil Production Data BP Statistical Review of World Energy June 2009 http://www.bp.com/sectiongenericart icle.do?categoryid=9023770&contenti d=7044467

References Hubbert, M.K., Energy from Fossil Fuels, Science, February 1949, v. 109, n. 2823, p. 103-109., 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.

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.

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