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Saturday, October 02, 2010

Lake Size Distributions

Our environment shows great diversity in the size and abundance in natural structures. Since we extract oil from our environment, it stands to reason that many of the same mechanisms leading to oil formation could also reveal themselves in more familiar natural phenomena. Take the size distribution of lakes as an example.

Freshwater lakes accumulate their volume in a manner analogous to the way that an underground reservoir accumulates oil. Over geologic time, water drifts into a basin at various rates and over a range in collecting regions. In the context of oil reservoirs, I have talked about this behavior before and the Maximum Entropy prediction of the size distribution leads to the following expression:

P(Size) = 1/(1+Median/Size)

Surveys of lake size show the same reciprocal power law dependence, with the exponent usually appearing arbitrarily close to one. In Figure 1 below, the data plotted on a ranked plot clearly shows this dependence over several orders of magnitude.

Figure 1: Northern Quebec lakes [1]

More revealing, in Figure 2 we can observe the bend in the curve that limits the number of small lakes in exact accordance to the equation. The agreement with such a simple model suggests that a universal behavior links the statistics between environmental phenomena as seemingly distinct as those of lakes and oil reservoirs.

Figure 2: Amazon lakes [2]

This provides other intuitive clues to how to think about reservoir sizing. Consider the fact that very few freshwater lakes reach gigantic portions, the Great Lakes serving as a prime example. Similarly, the rare occurrence of “super-giant” reservoirs follow from the same principles. We clearly won’t find any new huge freshwater lakes, while the future occurrence of super-giant oil reservoirs remains very doubtful just from the statistics of oil reservoirs found so far. Finding substantial numbers of super-giant reservoirs would result in deviations from the size distribution plot, making it very unlikely.

[1] K&C Science Report – Phase 1 Global Lake Census

[2] Estimation of the fractal dimension of terrain from Lake Size Distributions


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