The Great Curve

This narrative describes the virtual flow of oil through the system from discovery to production, providing a problem domain description to how an oil peak occurs. You can find the accompanying stochastic differential equations here.

Precondition: Oil sits in the ground, undiscovered.
Invariant: The amount of oil in each of its states cumulatively sums to a constant.

The states of the oil in the ground consist of fallow, construction, maturation, and extraction.

1. Someone makes a discovery and provides an estimate of the quantity extractable.
2. The oil sits in the fallow state until an oiler makes a decision or negotiates what to do with it. Claims must be made, etc. This time constant we call t1, and we assume this constitutes an average time with standard deviation equal to the average. This becomes a maximum entropy estimate, made with the minimal knowledge available.
3. A stochastic fraction of the discovery flows out of this fallow state with rate 1/t1. Once out of this state, it becomes ready for the next stage (and state).
4. The oil next sits in the build state as the necessary rigs and platforms get constructed. This time constant we call t2, and has once again an average time with standard deviation equal to the average.
5. A stochastic fraction of the fallow oil flows out of the build state with rate 1/t2. Once out of this state, it becomes ready for the next stage.
6. The oil sits in the maturation state once the rigs get completed. We cannot achieve the maximum flow instantaneously as the necessary transport, pipelines, and other logistics are likely not 100% ready. This time constant we call t3.
7. A stochastic fraction of the built rig's virtual oil flows out of the maturation state with rate 1/t3. Once out of this virtual state, it becomes ready for the next stage of sustained extraction.
8. The oil sits in the ready-to-extract state once the oil well becomes mature.
9. The oil starts getting pumped with stochastic extraction rate 1/t4. The amount extracted per unit time scales proportionally to the amount in the previous maturation state.
   

Post-condition: All oil eventually gets extracted at time=infinity. But because of the proportionality extraction rate assumed, this decline only asymptotically approaches zero at long time periods. Also, the cumulative amount extracted at time=infinity equals the total discovered. However, since we never achieve infinite time, cumulative extraction never matches cumulative discoveries.

We can consider each one of these states as a reservoir with a capacitive time lag associated with the time constant set for each stage. In stochastic terminology the flow approximates a Markovian Process.

The extraction from the final stage gives the average production level. Since Markov processes have well-behaved linear properties and remain conditionally independent of past states, we can apply an entire set of discoveries as forcing functions to this process flow and the result will reduce to a convolution of the individually forced solutions.

The final production profile over time approximates the classic Hubbert curve. This narrative explains in very basic terms how and why the peak gets shifted well away from the discovery peak. However we observe no symmetry in the derived curve, as the nature of time causality rules long negative tails out.

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Regarding the US crude oil production curves, Staniford was able to make a very good fit over a few orders of magnitude using a gaussian. As for temporal properties of this curve over time, Staniford noted it has the property that:

   dP/dt = K * (t0-t) * P(t)

where t0=PeakTime. This relationship reads that the production increase slows down over time linearly, but also scaled by the amount in production at that time -- kind of an linearly decreasing positive feedback turned into a linearly increasing negative feedback. At t=t0, the production increase turns into a production decrease. Unfortunately, I can't provide a problem-domain narrative for the gaussian formulation. Like the logistic curve formulation, I get brain-freeze trying to shape it into intuitive terms.