Editing Hubbert peak theory (section)

==Hubbert's theory==

===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>

===Use of multiple curves===
{{Expand section|date=June 2008}}
The sum of multiple Hubbert curves, a technique not developed by Hubbert himself, may be used in order to model more complicated real life scenarios. When new production methods, namely [[hydraulic fracturing]], were pioneered on the previously unproductive oil-bearing Shale formations, the sudden, dramatic increase in production necessitated a distinct curve. Advances in technologies such as these are limited, but when a paradigm shifting idea impacts production and causes a need for a new curve to be added to the old curve, or the entire curve to be reworked. It should be noted, & it is well documented, that production from shale wells are unlike that of traditional well.  A traditional oil well's rate of decline is shallow, & exhibits a slow, predictable rate of decline as the reservoir is drawn down (Drinking of Milkshake). Whereas production from shale wells, assuming successful fracturing, will see its peak production at the moment the well is brought in, with a drastic rate of decline shortly thereafter.  However, one revolutionary aspect of these types of production methods are the ability to refracture the well.  Production may be brought back up, to near peak levels with a reapplication of the fracturing technology to the subject formation. Once again releasing the hydrocarbons trapped tightly within the shale & allowing them to be drawn to the surface. This process allows the for an outward manipulation of the curve, simply by purposefully neglecting to rework the well until the operator's desired market conditions are present.  <ref>{{cite web|url=http://dieoff.org/page191.htm|title=The Hubbert curve : its strengths and weaknesses|last=Laherrère|first=J.H.|date=Feb 18, 2000|website=dieoff.org|access-date=September 16, 2011|archive-url=https://web.archive.org/web/20181009083419/http://www.dieoff.org/page191.htm|archive-date=October 9, 2018|url-status=dead}}</ref>

===Reliability===

====Crude oil====
[[File:Hubbert Upper-Bound Peak 1956.png|thumb|Hubbert's upper-bound prediction for US crude oil production (1956), and actual lower-48 states production through 2016]]
Hubbert, in his 1956 paper,<ref name="Hubbert1956"/> presented two scenarios for US crude oil production:
* most likely estimate: a logistic curve with a logistic growth rate equal to 6%, an ultimate resource equal to 150 Giga-barrels (Gb) and a peak in 1965. The size of the ultimate resource was taken from a synthesis of estimates by well-known oil geologists and the US Geological Survey, which Hubbert judged to be the most likely case.
* upper-bound estimate: a logistic curve with a logistic growth rate equal to 6% and ultimate resource equal to 200 Giga-barrels and a peak in 1970.
Hubbert's upper-bound estimate, which he regarded as optimistic, accurately predicted that US oil production would peak in 1970, although the actual peak was 17% higher than Hubbert's curve. Production declined, as Hubbert had predicted, and stayed within 10 percent of Hubbert's predicted value from 1974 through 1994; since then, actual production has been significantly greater than the Hubbert curve. The development of new technologies has provided access to large quantities of unconventional resources, and the boost of production has largely discounted Hubbert's prediction.{{Citation needed|date=October 2018}}

Hubbert's 1956 production curves depended on geological estimates of ultimate recoverable oil resources, but he was dissatisfied by the uncertainty this introduced, given the various estimates ranging from 110 billion to 590 billion barrels for the US. Starting in his 1962 publication, he made his calculations, including that of ultimate recovery, based only on mathematical analysis of production rates, proved reserves, and new discoveries, independent of any geological estimates of future discoveries. He concluded that the ultimate recoverable oil resource of the contiguous 48 states was 170 billion barrels, with a production peak in 1966 or 1967. He considered that because his model incorporated past technical advances, that any future advances would occur at the same rate, and were also incorporated.<ref>M. King Hubbert, 1962, "Energy Resources," National Academy of Sciences, Publication 1000-D, p. 60.</ref> Hubbert continued to defend his calculation of 170 billion barrels in his publications of 1965 and 1967, although by 1967 he had moved the peak forward slightly, to 1968 or 1969.<ref>{{cite journal |last1=Hubbert |first1=M. King |title=National Academy of Sciences Report on Energy Resources: REPLY |journal=AAPG Bulletin |date=1 October 1965 |volume=49 |issue=10 |pages=1720–1727 |doi=10.1306/A66337C0-16C0-11D7-8645000102C1865D }}</ref><ref>{{cite journal |last1=Hubbert |first1=M. King |title=Degree of Advancement of Petroleum Exploration in United States |journal=AAPG Bulletin |date=1 November 1967 |volume=51 |issue=11 |pages=2207–2227 |doi=10.1306/5D25C269-16C1-11D7-8645000102C1865D }}</ref>

A post-hoc analysis of peaked oil wells, fields, regions and nations found that Hubbert's model was the "most widely useful" (providing the best fit to the data), though many areas studied had a sharper "peak" than predicted.<ref name=Brandt2007>{{cite journal |last1=Brandt |first1=Adam R. |title=Testing Hubbert |journal=Energy Policy |date=May 2007 |volume=35 |issue=5 |pages=3074–3088 |doi=10.1016/j.enpol.2006.11.004 |bibcode=2007EnPol..35.3074B }}</ref>

A 2007 study of oil depletion by the [[UK Energy Research Centre]] pointed out that there is no theoretical and no robust practical reason to assume that oil production will follow a logistic curve. Neither is there any reason to assume that the peak will occur when half the ultimate recoverable resource has been produced; and in fact, empirical evidence appears to contradict this idea. An analysis of a 55 post-peak countries found that the average peak was at 25 percent of the ultimate recovery.<ref>Steve Sorrell and others, ''Global Oil Depletion'', UK Energy Research Centre, {{ISBN|1-903144-03-5}}.{{page needed|date=January 2021}}</ref>

====Natural gas====
[[File:Hubbert US Lower 48 Gas Prediction - 1962.png|thumb|Hubbert's 1962 prediction of US lower 48-state gas production, versus actual production through 2012]]
Hubbert also predicted that natural gas production would follow a logistic curve similar to that of oil. The graph shows actual gas production in blue compared to his predicted gas production for the United States in red, published in 1962.<ref>M. King Hubbert, 1962, "Energy Resources," National Academy of Sciences, Publication 1000-D, pp. 81–83.</ref>