nero wrote:{I think this thread has gone way off the original topic, and it might be worthwhile if a moderator came along and separated out the discussion of the hubbert curve et al into a separate thread with an apropriate name.}
decent discussion thou. i think if we are to critic lynch or anyone else its worthwhile getting to the nitty gritty
Ok the logistics curve sounds like a starting point for a theoretical basis for the hubbert bell curve. That makes alot more physical sense than the gaussian curve. First because physically there is no justification for the gaussian curve (IMO). We are not modelling the frequency of something occuring that has some inherent uncertainty around a constant value. maybe we could make a gamma distribution for rate of discovery of oil under ideal conditions, I haven't thought about that carefully but anyways what the hubbert curve is modelling is the production not the discovery.
The logistics curve has the advantage of having some physical reason for its use. Physically it is most easily understood by the differential equation
P' = rP(K-P)
This describes the classic S curve where something starts growing exponentially until it nears a constraint at which point the constraint starts to limit the rate of growth and the growth slows down to nothing. The problem of using the logistics curve is that you can't make a physical justification for it's use in this case. For oil production P' = rate of production, P is the cumulative production, K is the ultimatly recoverable reserves and r is a constant determining the growth rate.
But the rate of increase of oil production has not been limited physically by cumulative production. More realistically it is limited by the current rate of production. (ie the percentage increase of production is limited due to physical constraints on the ability of the industry and the economy to grow). I would suggest the differential equation such as
P'' = rP'(K-2P)
makes more sense. It has been a while (and differential calculus was never my strong suit) but that also will physically look like an S curve and the first derivative also is a bell curve like shape. It also is symetrical looking although it has boundary conditions at t=0.
So why isn't this model used instead ofthe logistics curve? Can anyone explain why the logistics curve is a good model for historical oil production?
I can't.. i am on the rivet of my understanding here. nut i am going to have a layman's stab at it conceptually.. i struggle just understanding why it should be a peak.
i see some intuitive sense to why there should be some peak but the actual idea of a bell curve or any other model of rates of production is at first a mystery
Unless its just a fundamental nature of growth when a imperative of efficiency or demand is placed on it.. am i naive here?
if the ultimate production or end product has some physical limit (in place oil) what reasons are there for a peak rather than a ramp with some abrupt end point? because surely recovery has sweet spot associated with the geographic location of any incidence
the physicality issue in logistic curve for growth constants must be something to do with areas volumes and incidence?
you would expect some inverse cube or square over time rule for pressure to well so as production increased due to incidence of wells a point is reached where incidence(number of wells) is counterproductive.
rates of growth in incidence is tied to a work force dynamic that is very iffy to model
the ability to push up production on a field may be tied to a constant rig count .. or at least to a workforce that doesn't change in size by orders of magnitude for a given area of production?
therefore the lifetime of previous wells becomes the dynamic in the curve as additional wells come online quicker than older ones die out. this must be why you get a peak of some sort... you still have dying wells when your last new one goes in and as the recovery per well is a pressure dynamic we are looking at the addition of thousands of little peaks.
it must ultimately come down to pressure and physics heavily modified by a quadzillion other factors
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