dQ/dt = P = kQ(1 - Q/Qt) ......................(1).
Here, P is the production (number of barrels of oil) per year, Q is the cumulative production (i.e. the total amount of oil recovered from the source to date), Qt is the total amount of oil that will ever be recovered from it and k is the logistic growth rate (described by Kenneth Deffeyes [3] as a "sort of compound interest"). Equation (1) is a quadratic (and describes a parabola or bell-shaped curve, Figure 1) but may be re-written in linear form, by dividing through by Q to give equation (2):
P/Q = k - kQ/Qt ......................................(2).
Thus a plot of P/Q (i.e. the number of barrels of oil produced each year divided by the total amount of oil extracted to date) versus Q (the total amount of oil extracted to date) directly, gives a straight line (Figure 2) with a y-axis intercept equal to k and a slope -k/Qt. From k and Qt, values for P can be estimated using equation (1), for each unit value of Q, from which it is apparent that the ability to produce oil depends entirely on the "unproduced fraction", (1-Q/Qt), i.e. how much oil there is remaining in the well... and on nothing else. Qt is also given by the intercept on the x-axis, since it corresponds to the point at which the resource is exhausted and P/Q = 0.
To make a plot of P against time - i.e. a classic production curve - it is necessary to replace Q as the x-axis unit by time (e.g. by year). This can be done by noting that P = dQ/dt, as in equation (1).
Hence, 1/P = dt/dQ. By using equation (1), values of P can be predicted for each barrel of oil (billion barrels of oil are a more convenient unit), by increasing or decreasing Q by increments of one (billion barrel) unit from cumulative production at a specified year (to act as a "clock", e.g. Q = 169 billion barrels by 2002 for the US). By then dividing the P values into 1, we get the reciprocals (1/P) which are in units of years/billion barrels rather than of billion barrels per year (P). Then for each value of P, we calculate a year-fraction* (i.e. how long it took to produce each billion barrel unit) and make a production plot of P versus year-fraction, giving the curve in Figure 1, the area under which is equal to the total volume of the resource, Qt.
[*i.e. the division does not come out conveniently in round year units, but is usually fractional. For example, for the US production, for which we obtain k=0.061 and Qt=198.395 Gb. When Q=169 Gb, P=1.532, 1/P = 0.653, and we set the year at 2002, by when production data shows that 169 Gb had been produced. This is our "clock". We then calculate for Q=168, P=1.574, 1/P=0.635, and so the "year fraction" is 2002-0.653=2001.347. For Q=167, P=1.902, 1/P=0.526, and the year fraction is 2001.347-0.526=2000.821. The points can be extended above the "clock" year too, e.g. for Q=170, P=1.488, 1/P=0.672, and the "year"= 2002+0.672=2002.672. The procedure is continued for all values of Q to obtain a good data set for the plot of P/"year"].
For world oil reserves, the analysis predicts a value for Qt of around 2 trillion barrels, which would suggest we are close to (or past) the half-way point, i.e. we have used around half our original bestowal of oil.
The method has been extended to using second derivatives [4], e.g. in the form of equation (3):
(1/P)dP/dt = k(1 - 2 Q/Qt) ....................(3).
In equation 3, the term before the equals sign is often called the decline rate (of a resource). Use of this formula has been called the "Second Hubbert Linearization". A plot of delta-P/P versus 2Q gives a value of 2634 billion barrels for Qt and k = 4.6%. There are two potential matters of import here, if this analysis is correct: (1) we may have another 600 billion barrels of oil available to us, (2) the date of peak oil is shifted from around 2006 [as is obtained from equations (1) and (2)] to around 2013.
According to the summary of a recent oil conference, consensus on peak oil is that it will be with us by 2012 [5]. This is in accord with the prognosis made by the CEO of Shell who, earlier this year, stated that he expected to see a gap between demand and supply for oil at some time between 2010 - 2015.
That additional 600 billion barrels if real may not help us much though, because it is the rate of recovery that matters in closing the demand-supply gap for oil. If more oil cannot be pumped-out per day and refined fast enough to match demand, high prices will remain and there will be shortfalls in supply... somewhere or another, both in terms of fuel and chemical feedstocks for industry.
Related Reading.
[1 M.K.Hubbert, “Nuclear Energy and the Fossil Fuels.” Presented before the Spring meeting of the Southern District, American Petroleum Institute, Plaza Hotel, San Antonio, Texas, March 7-9, 1956.
[2] M.K.Hubbert, “Techniques of Production as Applied to Oil and Gas,” in S.I.Glass, ed., Oil and Gas Supply Modelling, Special Publication 631 (Washington D.C.: National Bureau of Standards, 1982), pp. 16-141.
[3] K.S.Deffeyes, “Beyond Oil”, Hill and Wang, New York, 2005.
[4] http://www.theoildrum.com/story/2006/8/16/102942/337
[5] D.Low, “ASPO conference confirms a peak in global oil production by 2012.” http://www.energybulletin.net/35127.html
Figures are available at the following URL's:
Figure 1. Logistic (Hubbert) curve fitted to crude oil production from the lower-48 US states. From S.Foucher: http://en.wikipedia.org/wiki/Image:Hubbert_US_Lower48.svg
Figure 2. Hubbert linearization of lower-48 states data, according to equation 2, yielding fit-parameters k=6.1% and URR = Qt = 199.07 Gb (billion barrels), used to derive the logistic curve in Figure 1. From S.Foucher: http://en.wikipedia.org/wiki/Image:HubbertLin_US_Lower48.svg
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