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## Friday, January 27, 2006

### Grove-like growth

I looked at the dynamics of fossil fuel "reserve growth" some more and I do not think it demonstrates compound growth by any stretch of the imagination. If it did in fact show such behavior, the growth would fly off the chart in a Ponzi scheme fashion.

Compound growth in the traditional sense has a fixed proportional rate. This would give an accelerating slope. However, reserve growth has an apparently monotonically decreasing proportional rate over time which leads to a decelerating slope. Think of it this way -- if the growth rate follows 1/x, then any increase in x gets counterbalanced by a smaller proportional amount or ~(1+1/x)x. I plotted a 0.5/x curve (in green) on top of the moving-average fit below.

Having been away in Silicon Valley on business the past few days, I haven't had a chance to follow-up more quickly, but oddly enough, I sense an analogy to silicon in the way that reserve growth actually works. Take for example, the work of Andy Grove, one of the co-founders of Intel, who did his PhD thesis in diffusion-limited oxide growth, a physical process critical to building integrated circuits. In a nutshell, silicon dioxide needs a source of silicon to form, but as the SiO2 layer gets thicker, it becomes harder and takes longer for the Si atoms to diffuse to the surface and react with oxygen. This leads to a law of the following form, where F(t) is thickness as a function of time:
dF/dt = k/F(t)
F = sqrt (2kt)
Note that the fractional rate reduces to:
dF/dt / F = 0.5/t
Note that this follows the "reserve growth" curve fit fairly well, where the fractional growth rate slows down inversely proportional to time. Microelectronics engineers refer to this as the parabolic growth law (a parabola sitting on its side, see the overlay on the green curve below).

Now if we suddenly became stupid semiconductor neophytes living in the 1950's and thought that the oxide growth could only be "guessed" at, then we would never have advanced through the microelectronics revolution and process unpredictability would have killed us. We would still be working with crystal radio sets. None of the multi-million gate circuits would have ever gotten made!

But the fact that material scientists and engineers like Andy Grove quickly characterized the phenomena within a few years time (mid 1960's) and got their process down to a gnat's eyelash speaks volumes about the difference between real engineers and the geologists who believe in magical, enigmatic reserve growth. (I don't know a fab engineer in a bunny suit who believes in "enigmatic" oxide growth)

I have a suggestion for the geologists and petroleum engineers. Figure out what the heck your measurements and estimates mean, and then perfect the formula to eliminate the magical guess work. The more I look at it, the more I seriously think that no one has figured out how to do estimates of oil reservoir volume correctly. Might they all measure the volume as an approximation to how much they have extracted, with the increase over time caused by diminishing returns? Much like a thick SiO2 layer prevents fast oxidation, that drilling "deeper" into a field slows further depletion and you need to work harder and wait on average longer times to get at it? It almost sounds as if no one wants to admit that a parabolic growth law has any kind of importance.

If done correctly, reserve growth would transform from enigmatic magic to a measure of extractability over time.