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Sunday, August 23, 2009

Deviations from Hubbert Linearization

If dispersive discovery proceeds with an exponentially accelerated search over a spatially dispersed volume then we see the classic Hubbert curve -- a Logistic sigmoid for cumulative growth. Applying the technique of Hubbert Linearization to that formulation, we plot a straight line, as shown to the right (plotting cumulative production U against fractional production P/U).

What happens if at some point in the accelerating search we apply an even more aggressive search policy? Say that we super-accelerate by applying a Gompertz-like growth term exp(kt2) instead of the linear exponential in the classic Logistic. The previously straight line develops a bulge that initially looks like a shallower HL slope but which eventually slopes downward to the URR cumulative intersection U0. Note that the URR gets normalized to 1 in the figure and amounts to a geological limit.

Conversely, what happens if the accelerating search stabilizes and transforms into a steady, linear growth? In this case, the HL linearization plummets before asymptotically approaches the same URR.

This deviation from the linear HL behavior may have happened already, but the noise in the discovery history likely has obscured the effect. In terms of the deviation direction, this could go either way. An aggressive search acceleration would come about if oil prospectors had confidence that they could apply a huge, albeit transient, investment into their infrastructure. On the other hand, the deceleration would obviously come about if the oil industry collectively started to give up and thus either reduce their search effort or resort to maintaining their previous rate.

As a key to understanding classical economics one really only has to understand the concept of continual growth. An implicit Ponzi scheme exists within our system, but remains largely unobservable since the acceleration occurs over decades. In the absence of alternative resources to fuel the engine of growth, we can observe directly the hoisting of our own petard as economic growth starts to mutate. Two possible outcomes will prevail: either a feverish last attempt to maintain pace in the face of a wall looming ahead, or a resignation to the idea that continuous growth can not sustain itself.


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