No cigar

This falls under the "nice try" department. Khebab resurrected a derivation of a single reservoir oil depletion calculation and posted it in a TOD comment section here. I distinctly remember seeing this months ago, kind of dismissed it at the time, but now I have the benefit of a gestation period, so I took Khebab's bait and gave it a serious reconsideration.

As I understand it, the original analyst, Abrams, clearly wanted to see if he could get a Logistic style curve from the solution of the quasi-static fluid dynamics from a depleting reservoir. He tried his darndest, enlisting the aid of the Maple symbolic equation solver, but I think he mainly accomplished a further muddying of the waters. First off, I do buy the mathematical premises he set up, but he made a serious mistake in trying to create a good fit by varying the diameter of the bore hole, A(t), as a function of time. Eventually, it ended up looking like the middle green curve below:

What do you know, but the green curve itself looks like a Logistic, which basically subverts the whole analysis. For instance, I could have also created a Logistic peak if I assumed a constant flow of oil out of the reservoir and modulated the aperture with a diameter that follows a Logistic curve.

In retrospect Abrams should have taken this fundamental differential equation,

and looked for an inflection point due to a change in sign of the second derivative. If he did try, he wouldn't have found one because none exists, and in fact, the rate of change of volume monotonically changes, showing no signs of a Hubbert-like peak. Unless of course, he generated a contrived A(t) curve that does show a Hubbert peak!

Everyone makes mistakes, but this kind of stuff really irks me because of the sleight-of-hand that goes on. Granted, I like the original foundational premise, but somehow the agenda got hijacked to the temple of Hubbert.

As an interesting side-note, the fluid dynamics and hydrostatics described by Abrams' equations provides an interesting alternative to the diffusion model I set up previously (which arises from a statistical mechanics view of things). That derivation shows that reserve growth follows a square root law (contrarily called parabolic growth), and I have a suspicion that the actual Abrams curve lies closer to this one than a Logistic curve. In certain regimes, it looks like it goes like approximately t^2/3 instead of t^1/2. Worth another with the aid of a numerical solver.


Update: Luis de Sousa at TOD/Europe talked about the Gompertz curve as a possible Logistic replacement and one that shows asymmetry in peak shape. But once again, no cigar, as the long tail at minus infinity violate all rules of causality. Like the Logistic curve, it doesn't reflect any physical model and only serves as a potential empirical fit.