Compartmental Models

I have mentioned the concept of compartmental models in past comments and during a discussion at peakoil.com with the commenter named EnergySpin, primarily with respect to the data flow construction of the Oil Shock model. I had not pursued the method too deeply because it essentially reinforces what the shock model tries to achieve in the first place, and the on-line literature has proven a bit spotty and discipline-specific, hidden behind publishing house firewalls. For example, this link on Stochastic Compartmental Models likely relates to drug delivery models in the field of pharmacokinetics. As I recall, EnergySpin happened to have an MD and perhaps and engineering or math degree and had expertise in such matters.

Let me take a moment and look back at what EnergySpin said in the PeakOil.com thread from 2 years ago:

I was pleasantly surprised to notice that you are using a form of compartmental analysis without knowing it in your models (Smile)
In the following post at your blog:
http://mobjectivist.blogspot.com/2005/06/part-i-micro-peak-oil-model.html
you actually make some really good points about the shape of the depletion curve (we seem to agree that the mathematics constrain this curve to have at least one point where its derivative goes to zero i.e. a maximum and nothing else!), but I'm more interested in the following statement:

Quote: --This is me
I use as an implicit assumption that any rate of extraction or flow is proportional to the amount available and nothing more; past and future history do not apply.

This is basically the equation for the output compartment in a multi-compartment system communicating with the real world. In actuality all the steps that you mention in your post try to model reality as fluxes of material between distinct states, with rates proportional to the amount of material in different stages of the material flow graph.
For example if one has N compartments linked in a unidirectional chain graph, the output can be shown to be be given as the weighted sum of N exponential decay terms or go directly to ODE models that relate the constants to the physical properties of the systems (e.g. volumes of quantities, flux constants etc because under certain conditions of connectivity the two formulations are equivalent)

I bring this up again because we might have another connection that may prove more worthwhile to pursue, if for nothing else that it further substantiates the assumptions that I have made in building up the foundation of the Oil Shock model. Based on some discussions of the concept of "fractional flow" at TOD, with a petroleum engineer poster named Fractional_Flow, I googled this keyword phrase: "fractional flow" "compartmental model". Lo and behold, we come up with a bunch of links to the same pharmacology disciplines that came up in the earlier EnergySpin discussion. So evidently the general mathematical concept known as fractional flow applies to compartmental models. This probably does not come as a great revelation to those petroleum engineers studying fractional flow theory during their university coursework, but it does help build some confidence in the intuition behind the oil shock model.

Simply put, if fractional flow places diffusion-like properties on top of the oil extraction process, the original premise behind the Oil Shock model still stands:
"I use as an implicit assumption that any rate of extraction or flow is proportional to the amount available and nothing more"
To first-order, diffusion properties derive from concentration differentials and the flow stays proportional to the magnitude on the high side. I would suspect the second-order nature of oil extraction may kick in as a reservoir starts depleting and we start hitting the reserve growth components. At that point a more complicated diffusion law would probably prove more effective. I suspect fractional flow and my musings concerning self-limiting parabolic growth have something in common. I just don't see the need (nor do I have the time dedication) to get into the arcanities of 3-D flow to achieve a proper fractional flow analysis when a statistical macro view of the behavior would suffice.

BTW, Khebab has done yours truly a great service and has started to review the Oil Shock model in great detail, both at his blog and at TOD. He has some very intriguing interpretations and posters have left some good comments, including Memmel (who triggered me to review the compartment model). A commenter named Davet left this appraisal of the assumptions behind the Oil Shock model:

There's a principle which I think you call convenience and I call doability: certain approximations are routinely made in order to keep the problem tractable. Gaussian distributions using central limit considerations even when the statistical basis set is small, Markovian conditional probabilities, stationarity, etc, simply because the idea of non-Markovian or worse yet nonstationary dynamics is just too exhausting to entertain; and besides, often our precision isn't good enough to warrant the extra effort -- that is, the simpler assumptions are "good enough for government work" and at least do not lull us into thinking we have the correct mechanism. (Disclosure: I do it all the time). And yet ...

The underlying principle determining Markovian behavior is that the individual events contributing to the behavior of the statistical ensemble have very short timescales relative to the ensemble timescale and have small amplitude. So the decay of a U235 atom is essentially instantaneous but the decay of a kg of U235 takes millennia; the timescale of the Chicxulub event was minutes but the amplitude so large that the future even now is affected by that event 65 MYA. Radioactive decay follows an exponential beautifully; the history of life does not. I would very much like to know what the h functions are and how they satisfy this.

The reason for picking on this (and why I keep asking about the h functions) is that it is important to get the mechanism right or at least realistic. A model with the wrong mechanism can have good descriptive ability but have poor predictive ability, as you know. As we should all know from arguing about HL. Correlation is not causation.

That comment brilliantly displays the scientific counterpart to the political "concern trolls" who routinely show up on popular blogs. In this variation, we have the concerned scientist who probably has our best interests in mind, but tries to repudiate the direction of the analysis because it doesn't prove "deep" enough. Well, screw that. I contend that the lack of simplified, yet realistic, models remains the big barrier to making progress in the oil depletion analysis field. You just don't dismiss and then abandon a promising avenue of research because of some perceived impediment that gets in the path. That gets us into a state of statis, hoping that somebody will come along to do what we affectionately refer to as an "end-to-end" model. Well, we know that will never happen. Instead, you basically have to make sound engineering judgments at every turn, knowing when to dismiss something if the math proves to unwieldy, or when to make some grand simplifying assumption if your intuition tells you. I replied to Davet's concern with this comment:

The thing that you may find confusing is that Khebab and I use the Exponential function for 2 things: estimating latencies and also for the final extraction. The Uranium decay you speak of refers more to the latter type of exponential decay. I have the amplitude taken care of. Bigger wells proportionately generate greater output, but the decay is the same. Oil extraction is like pigs feeding at the trough -- the bigger the trough, the more pigs can feed (or the more holes you punch in a reservoir/trough), but the pigs can only extract the slop at a relatively fixed rate. Heck, we're not talking about orders of magnitude differences like U decay!

I can sum up the history of life, greed=pigs=humans, so if you take the perspective that humankind is greedy, then we will extract any reservoir, big or small, with the same relative voracious appetite, plus or minus some unimportant differences in the greater scheme of things.

That about summarizes my most elementary level of intuition on the matter; I equate oil extraction to the diffusivity of pig slop.