Hubbert linearization

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The Hubbert Linearization is a way to plot production data to estimate two important parameters of a Hubbert curve; the logistic growth rate and the quantity of the resource that will be ultimately recovered. The Hubbert curve is the first derivative of a Logistic function, which has been used in modeling depletion of crude oil, predicting the Hubbert peak, population growth predictions[1] and the depletion of finite mineral resources.[2] The technique was introduced by Marion King Hubbert in his 1982 review paper.[3] The geologist Kenneth S. Deffeyes applied this technique in 2005 to make a prediction about the peak production of conventional oil.[4]


The first step of the Hubbert linearization consists of plotting the production data (P) as a fraction of the cumulative production (Q) on the vertical axis and the cumulative production on the horizontal axis. This representation exploits the linear property of the logistic differential equation:

\frac{dQ}{dt}=P=KQ\left(1 - \frac{Q}{URR}\right) \qquad \mbox{(1)} \!

where K and URR are the logistic growth rate and the Ultimate Recoverable Resource respectively. We can rewrite (1) as the following:

\frac{P}{Q}=K\left(1 - \frac{Q}{URR}\right) \qquad \mbox{(2)} \!
Example of a Hubbert Linearization on the US Lower-48 crude oil production.

The above relation is a line equation in the P/Q versus Q plane. Consequently, a linear regression on the data points gives us an estimate of the slope and intercept from which we can derive the Hubbert curve parameters:

  • the K parameter is the intercept with the vertical axis.
  • the line slope is equal to -K/URR from which we derive the URR value.


US oil production[edit]

The chart on the right gives an example of the application of the Hubbert Linearization technique in the case of the US Lower-48 oil production. The fit of a line using the data points from 1956 to 2005 (in green) gives a URR of 199 Gb and a logistic growth rate of 6%.

Alternative techniques[edit]

Second Hubbert linearization[edit]

The Hubbert linearization principle can be extended to the second derivatives[5] by computing the derivative of (2):

\frac{dP}{dt}\frac{1}{P}=K\left(1 - 2\frac{Q}{URR}\right) \qquad \mbox{(3)} \!

the left term is often called the decline rate.

Hubbert parabola[edit]

This representation was proposed by Roberto Canogar[6] and applied to the oil depletion problem:

P=KQ-\frac{K}{URR}Q^2 \qquad \mbox{(4)} \!


  1. ^ Roper, David. "Projection of World Population". 
  2. ^ Roper, David. "Where Have All the Metals Gone?" (PDF). 
  3. ^ M. King Hubbert: Techniques of Prediction as Applied to the Production of Oil and Gas, in: Saul I. Gass (ed.): Oil and Gas Supply Modeling, National Bureau of Standards Special Publication 631, Washington: National Bureau of Standards, 1982, pp. 16-141.
  4. ^ Deffeyes, Kenneth (February 24, 2005). Beyond Oil - The view from Hubbert's peak. Hill and Wang. ISBN 978-0-8090-2956-3. 
  5. ^ Khebab (2006-08-18). "A Different Way to Perform the Hubbert Linearization". The Oil Drum. 
  6. ^ Canogar, Roberto (2006-09-06). "The Hubbert Parabola". GraphOilogy. 

External links[edit]