Geology Reference
In-Depth Information
R
Q
=
Pdt
0
Time(t)
Fig. 13.1 Hubbert's bell shape curve of the production cycle of any exhaustible resource (Hubbert,
1956).
Hubbert's bell-shaped curve can be mathematically described through the
generic gaussian curve shown in Eq. (13.1).
f(t) = y
0
e
2
t
t
0
2
(13.1)
b
0
The integral of the gaussian curve is equal to the reserves (R) of the commodity:
Z
+1
f(t)dt = R
(13.2)
0
And the integral of Eq. (13.1) is given by Eq. (13.3).
Z
1
e
2
t
t
0
2
p
dt = b
0
2
(13.3)
b
0
0
Combining Eq. (13.2) and Eq. (13.3) and taking into account that the curve is
symmetric, the reserves can be expressed as:
p
y
0
b
0
2 = R
(13.4)
Hence, the model of the curve to be adjusted is given by Eq. (13.5):
2
e
2
t
t
0
2
R
p
f(t) =
(13.5)
b
0
b
0
whereby the parameters b
0
and t
0
are the unknowns. The function's maximum is
given by parameter t
0
, and it verifies that f(t
0
) =
R
p
2
.
The relationships expressed in this model (curve) were successful in predicting
peak oil extraction between the late 1960s and early 1970s in the lower 48 states of
the U.S. and a subsequent decline in production.
The Association for the Study of Peak Oil & Gas (ASPO) is actively engaged
in disseminating studies on this matter. ASPO's founder Colin Campbell defines
peak oil as
1
: “the maximum rate of the production of oil in any area under consid-
eration, recognising that it is a fin
ite natural resource, subject to depletion”. Some
1
See: http://www.peakoil.net/. Accessed Jan. 2013.
b
0
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