Hubbert curve Hubbert peak theory

the standard hubbert curve. applications, x , y scales replaced time , production scales.



u.s. oil production , imports 1910 2012

in 1956, hubbert proposed fossil fuel production in given region on time follow bell-shaped curve without giving precise formula; later used hubbert curve, derivative of logistic curve, estimating future production using past observed discoveries.

hubbert assumed after fossil fuel reserves (oil reserves, coal reserves, , natural gas reserves) discovered, production @ first increases approximately exponentially, more extraction commences , more efficient facilities installed. @ point, peak output reached, , production begins declining until approximates exponential decline.

the hubbert curve satisfies these constraints. furthermore, symmetrical, peak of production reached when half of fossil fuel produced has been produced. has single peak.

given past oil discovery , production data, hubbert curve attempts approximate past discovery data may constructed , used provide estimates future production. in particular, date of peak oil production or total amount of oil produced can estimated way. cavallo defines hubbert curve used predict u.s. peak derivative of:

q
(
t
)
=



q


m
a
x




1
+
a

e

−
b
t







{\displaystyle q(t)={q_{\rm {max}} \over {1+ae^{-bt}}}}

where



q


{\displaystyle q}

max total resource available (ultimate recovery of crude oil),



q
(
t
)


{\displaystyle q(t)}

cumulative production, ,



a


{\displaystyle a}

,



b


{\displaystyle b}

constants. year of maximum annual production (peak) is:

t


m
a
x



=


1
b


ln
⁡

(

a

)

.


{\displaystyle t_{\rm {max}}={1 \over b}\ln \left({a}\right).}

so cumulative production



q
(
t
)


{\displaystyle q(t)}

reaches half of total available resource:

q
(
t
)
=

q

max



/

2


{\displaystyle q(t)=q_{\text{max}}/2}






^ bartlett a.a 1999 , analysis of u.s. , world oil production patterns using hubbert-style curves. mathematical geology.
^ m. king hubbert, 1962, energy resources, national academy of sciences, publication 1000-d, p. 57.
^ hubbert’s petroleum production model: evaluation , implications world oil production forecasts, alfred j. cavallo, natural resources research, vol. 13, no. 4, december 2004 [1]