### 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:

## 6 Comments:

WHT,

Shouldn't you normalize those plots? That is, the hyperbolic decline would initially be more rapid but would then have the fat tail.

JB

Almost on to something. These are the cumulatives. If you look at the probability density functions, then yes you will see this effect. I added an edit to show this.

Have a question about entropy in this fat tail situation.

Let's say I have a PDF composed of two exponentials, fast and slow. Can I say anything about individual entropy contributions for each exponential? I.e. does it make sense that the fast process might be "diabatic" (hence more entropy?) vs slow "adiabatic".

Entropy is only maximized subject to some constraints. Actually, if you calculate the entropy for that combination it will be less than if you had a single one with the same mean.

The hyperbolic only has a higher entropy because the mean diverges.

This is informational entropy but I think you are referring to thermodynamic entropy.

You have to instead think about the entropy in a dispersion of rates and not time constants. When you have this uncertainty in rates and and uncertainty in volume that these rates work at, then you can get twice the entropy with a hyperbola than you can get with an exponential because you have twice the degrees of freedom (rate + volume) vs only volume.

That is why people get confused about the topis.

You've been 'complaining' that nobody thinks of entropy and this kind of stuff.

For this one, it seems that finally you are not alone:

http://pre.aps.org/abstract/PRE/v66/i6/e067103

It;s behind paywall, but abstract is self-explanatory

Thanks, I have seen Reed's work before. He has a different spin that doesn't quite match the way I think about these matters.

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