Intercept theorem

2014-01-29T07:50:00Z

83.179.61.22: Swedish link

The '''Hubbert Linearization''' is a way to plot production data to estimate two important parameters of a [[Hubbert curve]]; the [[Logistic function|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 [[Oil depletion|depletion of crude oil]], predicting the [[Hubbert peak]], [[population growth]] predictions<ref name="Roper1">{{cite web | url = http://arts.bev.net/roperldavid/WorldPop.htm | title = Projection of World Population | first = David | last = Roper}}</ref> and the depletion of finite mineral resources.<ref name="Roper2">{{cite web | url = http://www.roperld.com/minerals/metalgon.pdf | title = Where Have All the Metals Gone? | first = David | last = Roper|format=PDF}}</ref> The technique was introduced by [[Marion King Hubbert]] in his 1982 review paper.<ref name="Hubbert82">"Techniques of Prediction as Applied to the Production of Oil and Gas", in the collection Oil and Gas Supply Modeling, edited by Saul I. Gass (published as NBS Special Publication 631)</ref> The geologist [[Kenneth S. Deffeyes]] applied this technique in 2005 to make a prediction about the peak production of conventional oil.<ref name="Deffeyes1">{{cite book |last= Deffeyes |first= Kenneth |authorlink=http://www.princeton.edu/hubbert/index.html |coauthors= |title=Beyond Oil - The view from Hubbert's peak |date= February 24, 2005|publisher= Hill and Wang|location= |isbn=978-0-8090-2956-3 }}</ref>

== 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:

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

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

: <math>\frac{P}{Q}=K\left(1 - \frac{Q}{URR}\right) \qquad \mbox{(2)} \!</math>

[[File:HubbertLin US Lower48.svg|right|thumb|400px|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 [[Continental United States|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%.

<gallery caption="Other Examples">
Image:HubbertLin_Norway.svg|Hubbert Linearization on [[w:Norway|Norway]]'s oil production
Image:Hubbert_Norway.svg|Hubbert curve on [[w:Norway|Norway]]'s oil production
Image:HubbertLin_US_Lower48.svg|Hubbert Linearization on [[w:US|US]]'s oil production
Image:Hubbert_US_Lower48.svg|Hubbert curve on [[w:US|US]]'s oil production
</gallery>


==Alternative techniques==

===Second Hubbert linearization===
The Hubbert linearization principle can be extended to the second derivatives<ref name="Khebab1">{{cite web | author=Khebab | title=A Different Way to Perform the Hubbert Linearization | publisher=[[The Oil Drum]] | date=2006-08-18 | work= | url=http://www.theoildrum.com/story/2006/8/16/102942/337 | accessdate= }}</ref> by computing the derivative of (2):

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

the left term is often called the decline rate.

===Hubbert parabola===

This representation was proposed by Roberto Canogar<ref name="Canogar">{{cite web | author=Canogar, Roberto | title= The Hubbert Parabola | publisher=GraphOilogy | date=2006-09-06 | work= | url=http://graphoilogy.blogspot.com/2006/09/hubbert-parabola.html | accessdate= }}</ref> and applied to the oil depletion problem:

: <math>P=KQ-\frac{K}{URR}Q^2 \qquad \mbox{(4)} \!</math>


== References ==

{{Reflist}}

==External links==
* [http://www.theoildrum.com/node/2389 Does the Hubbert Linearization Ever Work?] - [[The Oil Drum]]
* [http://rutledge.caltech.edu/ Hubbert's Peak, The Coal Question, and Climate Change] - Peak Oil, Peak Coal, Peak fossil-fuels]
* [http://www.its.caltech.edu/~rutledge/Hubbert%27s%20Peak,%20The%20Coal%20Question,%20and%20Climate%20Change.xls Excel Workbook - Hubbert's Peak, The Coal Question, and Climate Change]

[[Category:Peak oil]]

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Intercept theorem