Part 2: A Macro Peak Oil Model
Based on feedback I received on Part 1, I created a few more plots which demonstrate trends if we change parameters of the model.
The assumption of first-order rate depletion (i.e. rate proportional to how much remains) has problems when considering extremely large reservoirs, where a constant depletion rate (or even increasing rate) can occur for a long period of time until we start hitting hard limits. Same thing applies for natural gas reservoirs (i.e. the "whipping cream container" phenomenon described in the Micro Peak Oil model).
Otherwise I believe a first-order rate remains a valid assumption; consumers historically have shown greediness in plundering any resource discovery. Market forces will tend to maximize the extraction in proportion to the amount available. Conservation of petroleum use during the late 70's caused a huge decrease in demand; before this time, people treated oil like an endless supply of water. In other words, in the old days, the spigot effectively had an opening proportional to the size of the reservoir.
So even though I think the first order exponential removal remains a valid assumption, the following graph shows the zero-order approximation -- extraction gets fixed to a constant rate for new discoveries. However, the total extracted remains the same as the first-order rate.
You can see the disappearance of the long tails as we remove the difficulty of oil extraction for depleted reservoirs. No, none, nada stripper wells in this zero-order model.
We can also change the discovery profile a bit to aid our intuition. Initially I set it as a symmetric profile (RED=forcing, YELLOW=response) which means that the peak discoveries occur at the midway point of the discovery lifecycle. However, we should equally consider the cases where we discover many of the reservoirs relatively early on (the "low hanging fruit" and "hunting elephants" phenomena). Here GREEN=forcing and BLUE=response. For completeness, I also added the late discovery profile to the chart, where VIOLET=forcing and TURQUOISE=response.
Even though I don't have belief in a forthcoming "just-in-time" late discovery model, the shape freaks me out a bit. Intuitively it means that once we discover the last reservoir, production starts its inexorable decline almost immediately. In general, the scales somewhat mirror the following (1) Early discovery - USA, (2) Symmetric discovery - The World, and (3) Late discovery - Parts unknown of Saudi Arabia and Iraq.
Another thing to note: these models all have the stationary process property. Simply put, the rates don't change with time. One can argue this point as we expect that technology and population growth to exert an ever upward growth to extraction rates. I don't include it as it defeats the purpose of providing a simple underpinning to the understanding of the curves. Higher-order effects like this will simply change the tails at a scale of 10's of years. Clearly important for the actual peak oil date but not for understanding the general shape.