# Hubbert linearization

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]

## Principle

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:

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

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

${\displaystyle {\frac {P}{Q}}=K\left(1-{\frac {Q}{URR}}\right)\qquad {\mbox{(2)}}\!}$
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.

## Examples

### US oil production

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

### Second Hubbert linearization

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

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

the left term is often called the decline rate.

### Hubbert parabola

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

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

## References

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.