Hyperbolic Decline a Fat-Tail Effect

If the Gulf Oil spill shows results of a hyperbolic decline, the effects can go on for quite some time.

For a typical reservoir, oil depletion goes through either an exponential decline or a hyperbolic decline. Geologists by and large don't realize this, and definitely don't teach this, but hyperbolic decline constitutes a "fat-tail" effect that results from an aggregation of varying exponential declines summed together. As to the behavior of hyperbolic decline, one notices that the effects tend to drag out for a long time. The fast exponential decline finishes more quickly than the slower exponential components. That's where the fat-tail comes from and why the hyperbolic decline can proceed endlessly are at least as long as the longest exponential portion.

Derivation of hyperbolic decline as a one-liner:

The exponential has a rate of x, and x gets integrated over all possible values of r according to an exponential Maximum Entropy probability density function. You can see the fat-tail in the plot below:


This is just entropy at work because nature tends to want to disperse.

EDIT:

JB asked the question on the slope of the two functions. As plotted, these give the cumulatives. If we want to look at the probability density functions, then yes you will see that the hyperbolic gives a mix of these rates more in line with intuition, with a faster initial slope and then the fatter tail later. See the figure below: