by WebHubbleTelescope » Thu 07 Jun 2007, 23:03:19
$this->bbcode_second_pass_quote('pup55', '
')As to the HL method, I always thought it was a really interesting calculation, and gives elegant graphs and results, but it is a big world, the thing we are trying to measure is underground, and so difficult to apply this globally. I think it ought to work great for smaller fields with finite boundaries.
Hubbert Linearization relies on the mathematical characteristics of the logistic curve. The logistic is a nonlinear variant of a stochastic differential equation which requires a rather large aggregate of entities to hold true. As fields get smaller and finiteness looms, the laws of stochastic processes give way to determinism. Why again would HL work better on smaller fields?
Let me put this another way: If we have a finite cube of oil containing crust, say 1000 meters on a side, how would you from first principles derive anything that derived a logistic equation?
This can't be that hard to do! Yet know one has done this exercise. You know why? Because you can't. The fundamental equation describing the logistic curve does not describe anything on this scale or these units. The only thing close it describes is population dynamics, and population dynamics does not play out on a smaller field.
Birth and death rates, P = population:
B = B0 - B1*P
D = D0 + D1*P
Population dynamics:
dP/dt = (B-D)*P
expanding
dP/dt = (B0-B1*P-D0-D1*P)*P = (B0-D0)*P - (B1+D1)*P[sup]2[/sup]
correlating to the logistic:
dP/dt = rP*(1-P/P[sub]infinity[/sub])
So where again does oil fit into this derivation?