Hubbert linearization

The geologist Kenneth S. Deffeyes applied this technique in 2005 to make a prediction about the peak of overall oil production at the end of the same year, which has since been found to be premature.[5] He did not make a distinction between "conventional" and "non-conventional" oil produced by fracturing, aka tight oil, which has continued further growth in oil production. However, since 2005 conventional oil production has not grown anymore.[6]

The charts below gives an example of the application of the Hubbert Linearization technique in the case of the US 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%.

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  Hubbert Linearization on US's oil production
• 
  Hubbert curve on US's oil production

The Norwegian Hubbert linearization estimates an URR = 30 Gb and a logistic growth rate of k = 17%.

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  Hubbert Linearization on Norway's oil production
• 
  Hubbert curve on Norway's oil production

The Hubbert linearization principle can be extended to the first derivatives of the production rate[7] by computing the derivative of (2):

${\displaystyle {\frac {dP(t)/dt}{P(t)}}=k\cdot \left(1-2{\frac {Q(t)}{URR}}\right)}$

(3)

The left term, the rate of change of production per current production, is often called the decline rate. The decline curve is a line that starts at +k, crosses zero at URR/2 and ends at −k. Thus, we can derive the Hubbert curve parameters:

• The growth parameter k is the intercept of the vertical axis.
• The URR value is twice the intercept of the horizontal axis.

This representation was proposed by Roberto Canogar[8] and applied to the oil depletion problem. It is equation (2) multiplied by Q.

${\displaystyle P(t)=kQ(t)-{\frac {k}{URR}}Q(t)^{2}}$

(4)

The parabola starts from the origin (0,0) and passes through (URR,0). Data points until t are used by the least squares fitting method to find an estimate for URR.

David Rutledge applied the logit transform for the analysis of coal production data,[9] which often has a worse signal-to-noise ratio than the production data for hydrocarbons. The integrative nature of cumulation serves as a low pass, filtering noise effects. The production data is fitted to the logistic curve after transformation using e(t) as normalized exhaustion parameter going from 0 to 1.

${\displaystyle e(t)=Q(t)/URR}$

(5)

${\displaystyle \operatorname {logit} (e(t))=-\ln \left({\frac {1}{e(t)}}-1\right)=-\ln \left({\frac {URR}{Q(t)}}-1\right)}$

(6)

The value of URR is varied so that the linearized logit gives a best fit with a maximal coefficient of determination ${\displaystyle R^{2}}$.