### Linearization

The technique of Hubbert linearization for estimating oil URR works perfectly for only one class of models: those which obey the logistic curve. Although a linearization technique may apply for other models, the fact that no one uses other models likely means that one does not exist -- as of yet.

The fact that a logistic curve forms a straight line with negative slope if plotted as Rate/Q vs Q (where Q = cumulative), means that it gets a lot of head-nodding agreement when data seems to fit the linearization. Of course the oil shock model will not linearize the same way as a logistic curve will (as the curves themselves have distinct difference regarding symmetry, etc). In any case, I find it instructive to plot a typical delta-input (single discovery, all rates equal) oil shock depletion model with the same data transformation as the logistic curve.

If plotted on a semi-log chart, you can see the salient feature of the depletion model in a linearization context. At some point, further incremental additions of production will not affect the cumulative amount. This forms the oil extraction analog of "the law of diminishing returns".

Michael Lynch talks about (I think) linearization in his paper "The New Pessimism about Petroleum Resources: Debunking the Hubbert Model (and Hubbert Modelers)".

Concerning the above chart (Figure 4 in his paper), Lynch says:

Finally, Campbell and Laherrere use production data to estimate field size, "improving" on the IHS Energy data. By graphing production against cumulative production, as in Figure 3, they claim that a clear asymptote can be seen, allowing for a more accurate estimate of ultimate recovery from the field. The first problem with this is that there is no explanation for how often the method is employed.I see the problem in that Lynch does not plot the curve in the classical Hubbert sense, i.e. he forgets to divide production by cumulative along the vertical-axis. So basically, once again, either (1) Lynch gets something horribly wrong, or (2) the traditional analysts have become lazy in not rigorously using the Hubbert linearization, forcing Lynch to call them on it. Somebody may yet sort this out.

[...]

Examining this data does confirm that some fields display a clear-cut asymptote. However, out of 21 fields whose peak production was above 2 mt/yr (or 40 tb/d), only 7 show such behavior. The rest do not show a clear asymptote (as in Figure 4), or worse, show a false one, as Figure 5 indicates. Clearly, this method is not reliable for estimating field size.

So what happens if we plot Lynch's Figure 4, the UK North Sea Cormorant field, the correct way? It looks like this:

Conclusion: It may not obey a true Hubbert linearization but it sure looks similar to the oil shock model along much of its range. If we plot the same delta-function driven shock model in the same way as Lynch does, it looks like:

Now, take a look at some of the other curves in Lynch's paper:

Kind of spooky? Look at the superposition of the curves.

**Update:**For yucks, here is another shock model with the first two time constants (Fallow period and Construction period) removed. This tends to make the curve more asymmetric and brings the peak in closer. (I gave all these curves the intuitive eyeball fit. And I normalized the curves by eye as well, since I had no discovery data.) Overall, I think this kind of "integration" linearization has some validity but it does transform data by compression, which tends to make the fit look better than if we kept the time domain in there and fit the original data as an "uncompressed" set of points. In essence, using the cumulative as an axis does a good job of filtering via integration, as discussed in a PeakOil.com thread. And the same holds true for Hubbert Linearization.

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