Editing Hubbert peak theory (section)

===Hubbert curve===
[[File:Hubbert curve.svg|thumb|The standard [[Hubbert curve]]. For applications, the ''x'' and ''y'' scales are replaced by time and production scales.]]
[[File:US Crude Oil Production and Imports.svg|thumb|U.S. Oil Production and Imports 1910 to 2012]]
In 1956, Hubbert proposed that fossil fuel production in a given region over time would follow a roughly bell-shaped curve without giving a precise formula; he later used the [[Hubbert curve]], the derivative of the [[logistic curve]],<ref>Bartlett A.A 1999 ,[http://www.hubbertpeak.com/bartlett/hubbert.htm "An Analysis of U.S. and World Oil Production Patterns Using Hubbert-Style Curves."] Mathematical Geology.</ref><ref>M. King Hubbert, 1962, "Energy Resources," National Academy of Sciences, Publication 1000-D, p. 57.</ref> for estimating future production using past observed discoveries.

Hubbert assumed that after fossil fuel reserves ([[oil reserves]], coal reserves, and natural gas reserves) are discovered, production at first increases approximately exponentially, as more extraction commences and more efficient facilities are installed. At some point, a peak output is reached, and production begins declining until it approximates an exponential decline.

The Hubbert curve satisfies these constraints. Furthermore, it is symmetrical, with the peak of production reached when half of the fossil fuel that will ultimately be produced has been produced. It also has a single peak.

Given past oil discovery and production data, a Hubbert curve that attempts to approximate past discovery data may be constructed and used to provide estimates for future production. In particular, the date of peak oil production or the total amount of oil ultimately produced can be estimated that way. Cavallo<ref name="cavallo">{{cite journal |last1=Cavallo |first1=Alfred J. |title=Hubbert?s petroleum production model: an evaluation and implications for World Oil Production Forecasts |journal=Natural Resources Research |date=December 2004 |volume=13 |issue=4 |pages=211–221 |doi=10.1007/s11053-004-0129-2 |bibcode=2004NRR....13..211C |s2cid=18847791 }}</ref> defines the Hubbert curve used to predict the U.S. peak as the derivative of:

:<math>
Q(t) = {Q_{{\rm max}}\over {1 + ae^{-bt}}}
</math>

where <math>Q</math><sub>max</sub> is the total resource available (ultimate recovery of crude oil), <math>Q(t)</math> the cumulative production, and <math>a</math> and <math>b</math> are constants. The year of maximum annual production (peak) is:

:<math>
t_{{\rm max}} = {1\over b}\ln \left({a} \right).
</math>

so now the cumulative production <math>Q(t)</math> reaches the half of the total available resource:

:<math>
Q(t) = Q_\text{max}/2
</math>

The Hubbert equation assumes that oil production is symmetrical about the peak. Others have used similar but non-symmetrical equations which may provide better a fit to empirical production data.<ref>{{cite journal |last1=Malanichev |first1=Alexander |title=Limits of Technological Efficiency of Shale Oil Production in the USA |journal=Foresight and STI Governance |date=30 December 2018 |volume=12 |issue=4 |pages=78–89 |id={{ProQuest|2239256388}} |doi=10.17323/2500-2597.2018.4.78.89 |doi-access=free }}</ref>