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Monday, July 10, 2006

Immaculate Conception Peak Oil Models

Staniford revisited his Hubbert Linearization (HL) formulation in some recent TOD posts. In the latter comment, he makes a mention of the good fit over a range of data of US oil production using a Gaussian curve.

I don't recall pointing this out in anything more than a passing comment, but one can represent a Gaussian curve in terms of a rate equation. The familiar bell-shaped curve of a Gaussian follows from this partial differential equation:
dP(t)/P(t) = K(T1-t)
where P = production, t = time, T1 = the peak date, and K parameterizes the width of the Gaussian. I suppose one can read that as exponentially growing production with the throttle linearly pulled back through to where it switches sign at the peak. However, the T1 number looks suspiciously preordained as opposed to coming out of some intuitive process.

Although Staniford has some good empirical fits using Hubbert Linearization, he still doesn't know why they work out so well:
However, we don't, at the moment, have a very good theoretical understanding of exactly why, so there remains room for doubt about whether it will continue to work as well in the future.
Where HL works well (eg the US), it seems to be because the production curve is close to Gaussian. Presumably there is some kind of central limit theorem "adding lots of random variables together" or "random walk through oil exploration space" kind of reason for this. If so, the asymmetry of individual field profiles may not necessarily give rise to an asymmetric overall shape. However, since we lack a clear and persuasive account of why the Gaussian shape arises, it's hard to say.
I will start to label this style of analysis under the category of Immaculate Conception Peak Oil Models. The empiricism leads straight to the fact that no one has ever suggested a forcing function to cause the temporal behavior in any of these models, including the logistic curve. They remain at best a set of heuristics that tend to at least shadow the production data. Without a clearly identified forcing function, I half-jokingly suggest we cook up analogies to Spontaneous Human Combustion to get better insight.

For a more practical model, go here.

On the other hand, I could side with Staniford and look at the Gaussian from a law-of-large-numbers/ central-limit-theorem-flavored approach, which would mean for me to give up and punt away my understanding, while waving my hands wildly. A hand-waving heretic? Not me. An Immaculate Conception believer? Not me. An agnostic function forcer? That's the ticket.


Professor Blogger Robert McLeod said...

I have trouble not laughing at all the fitting fun because none of the modeling systems utilize the initial values and boundary conditions of the system. The curves we keep seeing are simply a least squares fit of some preconceived function on empirical data. Certainly the normal distribution doesn't reflect reality because if you integrate it from negative infinity to infinity it would return an infinite amount of fuel reserves.

It's not clear why a beta function (the logistical curve) is better than a Gaussian or Lorentzian function. It may give a better fit but it's still a function invented by an economist that They are still just curve fitting, not physics.

Realistically, I think that one might be able to depict a differential equation to represent flow rates from a single oil field based on the number of wells drilled, oil viscosity, porosity of the rock, etc.. A region's or country's production would be a linear superposition of all these curves for each oil field. The quality of data is clearly not high enough to do it.

8:20 AM  
Professor Blogger WHT said...

I agree with this 100%.

The quality of data is not high enough for individual fields but mean values applied to aggregate sets of these fields is likely a valid stochastic approach.

4:47 PM  
Professor Blogger Robert McLeod said...

Engineers deduce empirical laws from empirical data all the time. However, these functions are never considered valid outside the outlying data.

Consider a thermocouple. The relationship between current and temperature is relatively linear for some temperature range. As such, I can construct an equation to relate the measured current to the temperature at the thermocouple. However, what happens if I try to extrapolate outside the boundaries of the original data that I fit my curve to? I will get gibberish; the properties of the semiconductor will change and the response will not longer be linear.

Scientifically, this sort of empirical fitting is valid only for interpolation. However, we consistently see it used to extrapolate a result. This is dismal science. How come we only ever see the fit curve as it progresses into the future? Wouldn't it also be interesting to see how well the curve correlates to oil production before the first datum? I.e. I want to know how much oil so-and-so's model claims we produced in 1200 AD.

9:27 AM  
Professor Blogger WHT said...

You are again absolutely right. The key thing missing is discovery data, which provides the causal forsing function which prevents models from producing preposterous results for 1200 AD.

In predicting the future, we have much better hope, as we can use infinite impulse response techniques to at least guess at the temporal response.

11:52 PM  

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