The basic parts of the derivation that we can substantiate involve the L-bar calculation in the figure below (originally from):
The key terms include lambda, which indicates cumulative footage, and the L-bar, which denotes an average cross-section for discovery for that particular cumulative footage. This represents Stage-1 of the calculation -- which I never verified with data before -- while the last lines labeled "Linear Growth" and "Parabolic Growth" provide examples of modeling the Stage-2 temporal evolution.
Since the results come out naturally in terms of cumulative discovery, it helps to integrate Hubbert's yearly discovery curves. So the figure below shows the cumulative fit,
while the original numbers came from this data set:
I did a least-squares fit to the curve that I eyeballed from the previous post and the discovery asymptote increased from my estimated 175 to 177. I've found that generally accepted values for this USA discovery URR ranges up to 195 billion barrels in the 30 years since Hubbert published this data. Which in my opinion indicates that the model has potential for good predictive power.
So at a subjective level, you can see that the cumulative really shows the model's strengths, both from the perspective of the generally good fit for a 2-parameter model (asymptotic value + cross section efficiency of discovery), but also in terms of the creeping reserve growth which does not flatten out as quickly as the exponential does. This slow apparent reserve growth matches empirical-reality remarkably well. In contrast, the quality of Hubbert's exponential fit appears way off when plotted in the cumulative discovery profile, only crossing at a few points and reaching an asymptote well before the dispersive model does.
Just like we were taught in school, provide a hypothesis and then try to verify with data. Unlike the wing-nuts who believe that school only serves to Indoctrinate U. (seriously, click on the link if you want to read my review of one of the worst documentaries in recent memory)