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## Saturday, March 31, 2007

### Central Limit Theorem and The Oil Shock Model

Frequently I will see a reference to the peak oil Hubbert curve describing it as a Gaussian or normal distribution. The reference to a Bell-shaped curve usually signifies some connection to a law of large numbers or central limit theorem argument -- or less frequently to some type of rate law. I have never bought into the classical hand-waving justification for a Hubbert curve following a normal distribution, as it violates causality in my opinion. However, I can throw in a bone as to how one can follow a path to the central limit theorem just by using the oil shock model.

First consider this statement describing the central limit theorem:
The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound, under the conditions stated above.
Let us parse the preceding paragraph from the Wiki definition. The first sentence basically reiterates the premise of the oil shock model. We made the assumption that the temporal dynamics of oil production rely solely on a set of random variables representing delays in extraction occurring after the discovery point. The densities (i.e. the probability density functions) of these delays follow a declining exponential in which the standard deviation equals the mean for each random variable. The second sentence indicates that the density of the sum (i.e. the sum of the variable delays) leading to a peak comes from the repeated convolution of the individual variable densities. The convenient1 Wiki definition shows how even an arbitrarily shaped density function when convolved against itself several times leads to a curve that has a "normal" shape:

As a refresher, let me give examples of the random variables that the central limit theorem refers to, in specific terms relative to the oil shock model.
`TimeDelay = (T1 + ... + Tn)` random variables
So for n=4, we may find for a specific well that
1. The discovery lays fallow for 3 years (T1=3) while negotiations take place for ownership, rights, permits, etc.
2. Next, construction of the oil rigs and infrastructure takes 8 years (T2=8)
3. After completion, it takes 5 years (T3=5) for the reservoir to reach maturation (toss in reserve growth considerations)
4. Once pumping at full rate, the reservoir drains with a time constant of 10 years (T4=10).
According to a deterministic setting, the sum of these values equals 3+8+5+10=26 years; or 26 years until the reservoir drops to its 1/e original value.

But in a stochastic world, the individual delays turn into density functions that we characterize completely by treating the delays as averages with a maximum entropy standard deviation, i.e. a decaying exponential.

So if we pair up the four stages as two sets of convolutions, we can generate intermediate density profiles.
`exp(-at) * exp(-bt) = (exp(-bt) - exp(-at))/(a-b)` where * is the convolution operator
The first convolution pairs the fallow stage with the construction stage. The peak of this curve occurs away from Time=zero even though the individual exponentials have peaks at zero. This demonstrates the initial impact of the central limit theorem.

The second convolution pair calculates the maturation+extraction shift. Note that the shift away from Time=0 illustrates the impact of a continuous maturation probability density function -- extraction will not hit a peak until the area matures to an expected value.

Next, the two pairs get convolved together to show the total shift from the initial discovery delta, with a peak at around 18 years. Note that the profile continues to sharpen and become more symmetric. (We can also generate the density profile via Laplace transforms, i.e. characteristic functions in central limit theorem parlance).

But remember that I generated this final production profile for a single discovery delta. Throw in a range of discoveries, perhaps following the quadratic or cubic discovery model I presented recently, and you will see the classical Bell-shaped curve emerge without requiring too many convolutions.

The outcome of this derivation suggests that we can use central limit theorem arguments to "prove" the existence of a roughly Bell-shaped curve without having to precisely match a Gaussian/normal profile. And again, the causal nature of the discovery/production process prevents us from achieving an exact match in the first place.

1 The Wiki definitions for math concepts look surprisingly useful, likely due to the fact that right-winged home-schoolers have yet to find a political agenda embedded in the symbol-heavy text.