Hubbert linearization
The Hubbert Linearization is a way to plot production data to estimate two important parameters of a Hubbert curve, the approximated production rate of a nonrenewable resource following a logistic distribution:
The Hubbert curve is the first derivative of a logistic function, which has been used for modeling the depletion of crude oil in particular, the depletion of finite mineral resources in general[1] and also population growth patterns[2]. The linearization technique was introduced by Marion King Hubbert in his 1982 review paper.[3]
Principle

The first step of the Hubbert linearization consists of plotting the yearly production data (P in bbl/y) as a fraction of the cumulative production (Q in bbl) on the vertical axis and the cumulative production on the horizontal axis. This representation exploits the linear property of the logistic differential equation:
with
- k as logistic growth rate and
- URR as the ultimately recoverable resource.
We can rewrite (1) as the following:
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 line slope calculated by -k/URR and intercept from which we can derive the Hubbert curve parameters:
- The k parameter is the intercept of the vertical axis.
- The URR value is the intercept of the horizontal axis.
Examples
The geologist Kenneth S. Deffeyes applied this technique in 2005 to make a prediction about the peak of overall oil production, which has since been found to be premature. He did not make a distinction between "conventional" and "non-conventional" oil produced by fracturing, aka tight oil, which has continued further growth in oil production.[4]
US oil production
The charts below 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%.
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Hubbert Linearization on US's oil production
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Hubbert curve on US's oil production
Norway oil production
The Norwegian Hubbert linearization estimates an URR = 30 Gb and a logistic growth rate of k = 17%.
Alternative techniques
Second Hubbert linearization
The Hubbert linearization principle can be extended to the second derivatives[5] by computing the derivative of (2):
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:
Logit transform
David Rutledge applied the logit transform for the analysis of coal production data,[7] which often has a worse signal-to-noise ratio than the production data for hydrocarbons. The integrative nature of cumulation serves as a low pass, filtering noise effects. The production data is fitted to the logistic curve after transformation using e(t) as normalized exhaustion parameter going from 0 to 1.
The value of URR is varied so that the linearized logit gives a best fit with a maximal coefficient of determination .
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British coal production of two centuries: rise, plateau, decline
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Cumulative production in green as sigmoid curve (logistic growth)
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logit transform; when the line crosses the zero, that's half time (~1920)
External links
- Does the Hubbert Linearization Ever Work? - The Oil Drum
- Hubbert's Peak, The Coal Question, and Climate Change - Peak Oil, Peak Coal, Peak fossil-fuels]
- Excel Workbook - Hubbert's Peak, The Coal Question, and Climate Change[permanent dead link]
References
- ^ Roper, David. "Where Have All the Metals Gone?" (PDF). Archived from the original (PDF) on 2007-09-28.
- ^ Roper, David. "Projection of World Population". Archived from the original on 2007-02-18.
- ^ 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.
- ^ Deffeyes, Kenneth (February 24, 2005). Beyond Oil - The view from Hubbert's peak. Hill and Wang. ISBN 978-0-8090-2956-3.
- ^ Khebab (2006-08-18). "A Different Way to Perform the Hubbert Linearization". The Oil Drum.
- ^ Canogar, Roberto (2006-09-06). "The Hubbert Parabola". GraphOilogy.
- ^ Rutledge, David (2011-01-01). "Estimating long-term world coal production with logit and probit transforms". International Journal of Coal Geology. 85. Elsevier: 23–33. doi:10.1016/j.coal.2010.10.012.