Techniques of prediction of future events range from the completely irrational to the semi-rational to the highly rational. Rational techniques of predicting the behavior of a system require first an understanding of its mechanism and of the constraints under which it operates and evolves. This permits the development of appropriate theoretical relations, which, when applied to the data of the system, permit solutions of its future evolution with varying degrees of exactitude. Such a theoretical analysis provides an essential criterion for what data are significant and necessary for the solution.from M.K. Hubbert, TECHNIQUES OF PREDICTION AS APPLIED TO THE PRODUCTION OF OIL AND GAS, in Oil and Gas Supply Modeling, Proceedings of a Symposium held at the Department of Commerce, Washington, DC, June I5-20, 1980, Edited by: Saul l. Gass, Nat. Bur. Stand. (U.S.). Spec. Publ. 631.778 pages (May 1982) (excerpt from Abstract)

After his 1956 paper, Hubbert admitted that he ran into another problem that forced him to develop another predictive methodology, as he described in his 1980 paper:

The weakness of this analysis arose from the lack of an objective method of estimating the magnitude of Q∞ from primary petroleum-industry data. The estimates extant in 1956 were largely intuitive judgments of people with wide knowledge and experience, and they were reasonably unbiased because of the comfortable prospects for the future they were thought to imply. When it was shown, however, that if Q∞ for crude oil should fall within the range of 150~200 billion barrels the date of peak-production rate would have to occur within about the next 10 to 15 years, this complacency was shattered. It soon became evident that the only way this unpleasant conclusion could be voided would be to increase the estimates of Q∞, not by fractions but by multiples. Consequently, with insignificant new information, within a year published estimates began to be rapidly increased, and during the next 5 years,successively larger estimates of 250, 300, 400, and eventually 590 billion barrels were published.

This lack of an objective means of estimating Q∞ directly, and the 4-fold range of such estimates, made it imperative that better methods of analysis, based directly upon the primary objective and publicly available data of the petroleum industry, should be derived..... "Techniques" p. 43-44

In "Techniques" (p. 45-50) Hubbert describes these better methods, which are mainly based on the idea that it was better to examine the production rate (dP/dt) as a function of Q, the cumulative oil produced, rather than as a function of time:

...It is convenient, therefore, to consider the production rate P, or dQ/dt, as a function of Q, rather than of time. In this system of coordinates, dQ/dt is zero when Q = 0, and when Q = Q∞. Between these limits dQ/dt> 0, and outside these limits, equal to zero. While it is possible that during the production cycle dQ/dt could become zero during some interval of time, for any large region this never happens. Hence we shall assume that for

0< Q < Q∞, dQ/dt > 0. (19)

The curve of dQ/dt versus Q between the limits 0 and Q∞ can be represented by the Maclaurin series,

dQ/dt = c_{0}+ c_{1}Q + c_{2}Q^{2}+ c_{3}Q^{3}+ ..... (20)

Since, when Q = 0, dQ/dt = 0, it follows that c_{0}= 0.

ThendQ/dt = c_{1}Q + c_{2}Q^{2}+ c_{3}Q^{3}+ ....., (21)

and, since the curve must return to zero when Q = Q∞ , the minimum number of terms that will permit this, and the simplest form of the equation, becomes the second-degree equation,

dQ/dt = c_{1}Q + c_{2}Q^{2}. (22)

By letting a = c_{1 }and -b = c_{2 }this can be rewritten as

dQ/dt = aQ - bQ^{2}. (23)

Then, since when Q = Q∞, dQ/dt = 0,

aQ∞ - bQ∞^{2}= 0,orb = a/Q∞,

anddQ/dt – a(Q – Q^{2}/Q∞). (24)

After noting that equation (24) defines a parabola, Hubbert further noted:

It is to be emphasized that the curve of dQ/dt versus Q does not have to be a parabola, but that a parabola is the simplest mathematical form that this curve can assume. We may accordingly regard the parabolic form as a sort of idealization for all such actual data curves, just as the Gaussian error curve is an idealization of actual probability distributions.

Hubbert also recognized that equation (24) can be converted into a linear equation with respect to dQ/dt and Q:

One further important property of equation (24) becomes apparent when we divide it byQ.We then obtain

(dQ/dt)/Q = a - (a/Q∞)Q. (27)

This is the equation of a straight line with a slope of-a/Q∞which intersects the vertical axis at(dQ/dt)/Q=aand the horizontal axis atQ=Q∞.If the data,dQ/dtversusQ,satisfy this equation, then the plotting of this straight line gives the values for its constantsQ∞anda.

The linear equation (27) was of critical importance to Hubbert because it allowed him to estimate

*Q*∞ and hence*Q*∞/2, without the need to rely upon guesstimates from experts, as he did in 1956, and which Hubbert suspected were bogus after his 1956 paper. Hubbert comments:The virtue of the first of these two equations lies in the fact that it depends only upon the plotting of primary data, (dQ/dt)/Q. versus Q. with no a priori assumptions whatever. Using actual data for Q and dQ/dt, it is to be expected that there will be a considerable scatter of the plotted points as Q Þ 0, because in that case both Q and dQ/dt are small quantities and even small irregularities of either quantity can produce a large variation in their ratio. For larger values of both quantities, as the production cycle evolves, these perturbations become progressively smaller and a comparatively smooth curve is produced. If the data satisfy the linear equation, then a determinate straight line results whose extrapolation to the vertical axis as Q Þ 0 gives the constant a, and whose extrapolated intercept with the Q-axis gives Q∞. However, even if the data do not satisfy a linear equation, they will nevertheless produce a definite curve whose intercept with the Q-axis will still be at Q = Q∞. "Techniques" (p. 52)

I will not show Hubbert’s derivation here, but Hubbert also showed that a form of the logistic equation could be derived from equation (24) above:

Q = Q∞ /(1+N

Q = Q∞ /(1+N

_{o }e^{-at}), where N_{o}= (Q∞-Qo)/Qo (38)Hubbert notes:

In equation (38) the choice of the date for t = 0 is arbitrary so long as it is within the range of the production cycle so that N_{o}will have a determinate finite value.

This is another important point for Hubbert, because it shows that Hubbert’s basic assumption that the simplest way to model the dQ/dt versus time, using an equation that defines a parabola, is equivalent to assuming a logistic growth curve with respect to the change in Q (cumulative oil production) with time.

Hubbert goes on to derive several other linear equations and other relationships which apparently he used in 1962, (National Academy of Sciences-National Research Council Publication 1000-D, 1962) to reanalyze the oil production, proven reserves and discovery data for the USA :

from "Techniques" p. 66

After a detailed review of the analysis of this, and additional previous data, using these models, Hubbert concludes:

The present cumulative.statistical evidence with regard to crude oil leads to a figure of approximately 163 ± 2 billion barrels for the ultimate cumulative production in the Lower-48 states.... However there still remain geological uncertainties regarding the occurrences of undiscovered oil and gas fields, yet those are being severely restricted by the extent of exploratory activity. In the case of crude oil, there is also the uncertainty regarding the magnitude of future improvements in extraction technology.

With due regard for these uncertainties, estimates for crude oil that do not exceed that given hereby more than 10 percent may still be within the range of geological uncertainties; estimates that do not exceed this by more than 20 percent may be within the combined range of geological and technological uncertainties. Estimates for natural gas that do not exceed the upper limit of the range given above by more than 10 percent may likewise be regarded as possible although improbable. But estimates for either oil or gas, such as those that have been published repeatedly during the last 25 years, which exceed the present estimates by multiples of 2, 3, or more, are so completely irreconcilable with the cumulative data of the petroleum industry as no longer to warrant being accorded the status of scientific respectability.