# Hubbert Curve: One Peak?

Hubbert, in a famous 1956 paper, contended that oil has to be found first before it is produced and that the production pattern has to be similar to the discovery pattern. He stated that production starts from zero, rises to a peak, and declines back to zero, but he did not give any equation except the bell­shaped curve. He said that the curve will likely be symmetrical. His Hubbert curve was obviously drawn by hand with an abacus, and he measured the volume by the surface below his curve, using as a unit a square in billions of barrels in the corner of the graph. He did not show any graph with several peaks, although he did not dismiss such a possibility.

Hubbert’s 1956 forecast was based on the assess­ment that the U. S. oil ultimate could be 200 billion barrels, giving a peak in 1970 (but also 150 billion barrels). It was not until the 1980s that he wrote that his curve was the derivative of the logistic function. The logistic function was introduced in 1864 by the Belgian mathematician Verhulst as a law for popula­tion growth. The equation for the cumulative
production (CP) for an ultimate U is

CP = U/{1 + exp[— b(t — tm)]},

where tm is the inflexion point (corresponding to the peak time for the annual production). A population with a constant growth rate (e. g., bacteria) displays an exponential growth until a point where the population reaches the limit of resources and starts a constant decline to stabilize at a constant level. A constant growth is impossible in a limited universe. Bacteria doubling each half-hour, without being constrained by the food resource, will reach the limit of the solar system in a week and the limit of the universe in 11 days.

The derivative of the logistic curve is a bell-shaped curve very close to a normal (Gauss) curve. Its equation for the annual production P, peaking at a value Pm for the time tm, is as follows:

P = 2Pm/{1 + cosh[— b(t — tm)]}.

When plotting the annual/cumulative production in percentage versus the cumulative production, if the production follows a derivative of logistics, the plot is linear. Using the mean values for the continental United States (48 states) as indicated previously, the plot is nearly linear from 1937 to 2001. The linear extrapolation toward zero indicates an ultimate of approximately 200 billion barrels. Hubbert was right on the peak in 1970 when he used an ultimate of 200 billion barrels (a rounded-up value because he knew that the accuracy was low). In Fig. 2, the linear trend

from 1938 to 1955 (at the time of the Hubbert forecast) already indicates this 200-billion barrel ultimate. Hubbert was lucky that the real value was close to a rounded number.

Plotting the annual mean discovery, it is easy to draw a Hubbert curve that fits the discovery data (smoothed over a 5-year period) and has an ultimate of 200 billion barrels (Fig. 3). U. S. discovery in the 48 mainland states peaked around 1935 (the largest oilfield, in East Texas, was discovered in 1930) at a level of 3.2 billion barrels/year. This Hubbert discovery curve, shifted by 30 years, fits the produc­tion curve perfectly. Why is oil production in the continental United States as symmetrical in the rise as in the decline?

Plotting cumulative production versus time shows a perfect fit using one logistic curve with an ultimate of 200 billion barrels, whereas cumulative discovery needs two logistic curves: the first with an ultimate of 150 billion barrels and the second with an ultimate of 50 billion barrels.

Oil production in the mainland United States comes from more than 22 000 producers (a very large number), and randomness has to be taken into account, as in the air where there is a very large number of molecules with a random Brownian move that gives a perfect law among pressure, volume, and temperature. According to the central limit theorem (CLT), in probability, the addition of a large number of independent asymmetrical distributions gives a normal (symmetrical) distribution. The large number of U. S. independent producers leads to random behavior, and the aggregation of the very large
number of fields is normal. In terms of the previous modeling of production by a bell-shaped curve, it occurs that the model seems wrong around 1930 (Great Depression), 1950 (pro-rationing); and 1980 (oil prices) as well as when political or economic events obliged all operators to act in the same direction so that randomness cannot apply.

It is easier to work on cumulative discovery (CD) and production (CP) as small details are smoothed. In the continental United States, the discovery can be well modeled with two logistic curves for an ultimate of 200 billion barrels when only one logistic curve fits the production well (Fig. 4).