The Sigmoid Fraud
Unlike Sigmund Freud, I don't do psychology as a career but I do see something seriously disturbing about the fact that a majority of depletion analysts view the Logistic Function as something that contains some deep and significant meaning.
On the contrary, the Sigmoid curve --as the simplest manifestation of the Logistic-- remains a cheap empirical relationship that describes a value that increases and then saturates below some constrained limit. It indeed does follow from the solution of a non-linear differential equation, but this equation describes the temporal dynamics of a simplistic birth-death model used to describe interacting entities. One can choose populations of biological creatures or concentrations of chemical reagents to plug in to the equation. But you don't insert oil molecules into the equation and expect it to make any sense.
Take a look at this post at TOD on whale oil harvesting in the 1800's. Although the original poster does not bring up the Logistic to describe the saturation, plenty of commenters do. Fair enough, whales do fall into a biological classification, and they do give birth and die. But whale oil harvesting never tracked a population rise in whales themselves. It actually tracked the reverse. So, instead of calling it a "birth-death" model we should refer to it as a "death-birth" model. The parameter "death" represents the culling of the whale population for oil and any residual "birth" comes about because the whales can reproduce themselves based on the size of their population. Then as an exercise for the reader, one can plug some values into the birth-death equations as described here: Derivation of Logistic Growth.
But then we get to the real twist. Since whales do reproduce, if we play our cards right, then the amount of whale oil that we can harvest has no limit! The URR of whale oil essentially becomes infinite since the cumulative never abates. And unless we harvest the whales to extinction, the Logistic Function will fail miserably in describing whale oil production. (In actuality, cumulative whale oil production likely saturated because crude oil replaced whale oil as a harvestable resource.) See passenger pigeons if you want to get closer to a saturated harvest driven to extinction.
This whole analysis when incorrectly applied to oil exploration and production can induce early psychosis. On the one hand, oil does not reproduce like a biological entity nor does it act like a chemical reagent. So the equations themselves make no sense. But since oil only gets consumed and obeys the rules of a finite resource (abiotic-oil-mental-midgets notwithstanding), it will eventually saturate. So the Sigmoid falls into our lap in spite of itself. The fraud survives in effect only because it looks like an S-curve !
To avoid this mental anguish, I prefer to use the Dispersive Discovery formulation for discoveries and the Oil Shock model for extraction/production dynamics. This approach makes intuitive sense, the math falls out naturally, and you don't have to continue to psychoanalyze insane ramblings of people that live in some freakish world where square pegs fit into round holes and empiricism has the dynamic range of a stupid heuristic. The rise and fall of the oil culture deserves a better understanding than the Logistic can ever offer.
End promised rant;