### Part I: A Micro Peak Oil Model

I recently got involved in a discussion about Hubbert Curve denier Michael Lynch and his latest article at the PeakOil.com message board. In the past, Lynch's disputes with depletion experts such as Colin Campbell had to do with a seemingly trivial feature of the oil production profile -- in particular that the curves showed too much symmetry. I would argue that much of the rationale for the argument arises (in the first place) from assorted media people who refer to the curves as describing a Bell shape or having a symmetric Normal or Gaussian distribution. From 2003, John Attarian pointed out Lynch's attack angle:

Michael Lynch's July 14 article is a peevish exercise in intellectual dishonesty. To begin with, he makes an utterly misleading fuss because "oil production rarely follows a bell curve." Much ado about nothing! Hubbert's main point was that a fossil fuel's endowment is fixed; therefore its production curve "will rise, pass through one or several maxima, and then decline asymptotically to zero." "Energy from Fossil Fuels" also stated explicitly that such a curve may have "an infinity of different shapes." How can Lynch not know this? The bell-shaped curve is simply a stylized, idealized representation of the phenomenon of rise, peak, and decline of output, amenable to mathematical expression and analysis, useful as a pedagogical and forecasting device--in fact, the sort of thing economists do all the time. As Lynch should know, real-world data don't necessarily conform to idealized shapes generated by mathematics--and aren't expected to. The shape does handily illustrate the general phenomenon. So real-world data aren't a smooth bell curve. Big deal. What matters is the general pattern of rise, peak, and decline.I agree that Lynch latches on to this description and takes the symmetry shorthand too literally, and in the grand tradition of an anal retentive techy nerd, parades around with his arguably correct interpretation like a propeller stuck on top of his beanie hat. For certain, real curves can't show this degree of symmetry -- for the simple fact that time-based processes have a non-negative starting point while symmetric curves contain tails that eventually go negative. Good statisticians will never misuse the Normal distribution in this way; instead they use it in situations where the law of large numbers applies. In that case, the long tails quickly dissipate to zero away from the mean; something that does not occur for the Hubbert curves. And of course, Lynch does not talk about that fact, demonstrating intellectually dishonesty to his right wing core.

Further, I do not mean to imply that depletion experts (those not named "Lynch") actually use the symmetric Bell curves for their analysis. In fact, the Hubbert curve gets expressed as the derivative of a logistics curve

*dP/dt = rP(K-P)*, an unquestionably non-symmetric profile:

Wikipedia has good reviews on the various distributions: Like a few analysts, I have problems with the derivation of the logistics curve as it applies to depletion. In a hand-wavy fashion, I can understand how the differential equation can empirically match a physical process; unfortunately it contains the non-linear factors that typically do not follow from any theory. I also don't like it because it corresponds more to a population growth scenario than a depletion scenario.

- the rate of reproduction is proportional to the existing population, all else being equal
- the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the population growth

## The Model

Given that (1) I don't much like that Lynch uses depletion analysts as romper room punching bags, and (2) that we can do better on an understandability level than the logistics curve, I propose my own model which uses a minimal set of assumptions.I use as an implicit assumption that any rate of extraction or flow is proportional to the amount available and nothing more; past and future history do not apply. This describes a first-order linear Markov approximation that allows one to either calculate analytically (in the simple cases) and computationally for more elaborate scenarios, the stochastic trend of resource depletion over time.

The simple case reduces to the exponential model. Here, we assume two states: an

**undepleted**state #1 that transforms into a

**depleted**state #2 according to a Markovian rate term.

For the right-brained people out there, we can visually depict this as a state-transition diagram -->

And given a value for the rate parameter assuming a particular time-scale, we can easily automatically solve these differential equations through straightforwardly-derived numerical integration routines:

We provide detail to the model by adding rate terms that describe the other state transitions that occur during the oil production life-cycle.

Each transition follows a Markov rate, with the strength of the transition proportional to how quick we can "turnover" the amount in the previous state. In general, approximating the strength of extraction on the proportion left allows us to intuitively model such effects as the small amount taken from stripper wells and the infrequency of shipping small volumes of oil.

- Extracting
- 0.2/year
- Shipping
- 2/year
- Refining
- 1/year
- Transporting
- 1/year
- Storing
- 5/year
- Trucking
- 5/year
- Pumping
- 5/year

The initial conditions place all states at 0.0 except the

**InGround**state which we normalize to 1.0 representing the full capacity of the reservoir.

For the rate parameters chosen above, we can calculate the profile after 4 years of extraction (each state gets scaled by the rate going out of that state to capture the "in-the-pipeline" effect, something the consumer can most closely identify with):

The snap-shot for the state diagram at 4 years shows the maximum available at the pump. Note that a maximum in the extracted state had already occurred.

After 20 years, the depletion at the pump becomes clearly visible:

I will further interpret this analysis in Part II, but a few things to note from what we have modeled and simulated so far:

- Asymmetric peak from single reservoir depletion.
- Depending on how we define the peak, it may depend on where we look in the state transition "pipeline"
- Imagine sets of these curves laying on top of each other, representing independent reservoir depletion profiles.
- Any reduction in the rates at any stage will push the peak to the right along the timeline
- High relative rates in any of the transitions affect the peak location very little as these act as efficiently pass fluid flow quickly to the next state

*macro-economics*) simply by distributing oil discoveries along an empirical Bayesian timeline, but the shorter time frame (i.e.

*micro-economics*) of the Markov process better describes affects due to local perturbations, including:

- pipeline sabotage
- refinery explosions
- transportation bottlenecks

*To be continued in Part II...*

## 9 Comments:

Nice! Looks like my old friend black body radiation!

Great work.

I actually had an extended Lynchian argument with a friend last year who just couldn't get past the "It looks like a Bell curve ! Why does it have to be a Bell curve !" stage. So I can see how you got the motivation to embark on this exercise...

For blackbody radiation we would also have to invite our mutual friend Mr. Physics to participate.

The curves look a bit like the Planck radiation law but that's a quantum/statiscal mechanics route to a derivation.

Of course I don't mean to imply a similar physical mechanism but rather the interesting way that these wonderful forms seem to repeat themselves in totally unrelated fields.

SW,

I guess I took the literal instead of the figurative interpretation on your comment.

Interesting stuff.

Have you read Car Busters' Peak Oil Issue?:

http://carbusters.org/

This exact topic was covered in a paper presented in Lisbon a few weeks ago by Schoppers and Murphy...

It was linked on energybulletin.net here:

http://www.cge.uevora.pt/aspo2005/abstracts.php

and

http://www.cge.uevora.pt/aspo2005/abscom/Abstract_Lisbon_Schoppers.pdf

Someone more enterprising than I can post some of the more interesting slides and graphs...

Worth the time to read...

You might consider a log-normal distribution, where the logs of the values are distributed normally, which is typically used to model stock prices. There is a lower limit of zero at the left tail, and an infinite upper limit on the right tail. For oil, the zero lower limit is 1859 when Edwin Drake drilled the first commercial oil will in Titusville, PA. And we will never reach the other limit by extracting every barrel.

And as you correctly point out, distributions are idealizations. To get real life to look a bit like the model, try a centered moving average.

numen, I like the centered moving average. That is something that signal processing algorithms use quite a bit. The window could be a model of discoveries made over time. The mathematics becomes a simple convolution of the windowing function with the state-diagram Markov model results.

I also think that extraction rates increase with time as technology improves. This could also be added to the averaging process.

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