42 comments on The limit of the statistic R/P in models of oil discovery and production
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42 comments on The limit of the statistic R/P in models of oil discovery and production
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Hi Dudley,
Thank you for posting your work. Could you please share with us your thinking on why you used the language of probability theory for this? It is true that fossil-fuel production history often follows a normal curve well. Ken Deffeyes showed us in his books that a cumulative normal curve fits U.S. oil production "like a glove." Clearly the Central Limit Theorem is at work, but how? Production is a time series, which is not the same thing as a probability function. Can you help us out on this?
The other issue, which we have often discussed at the Oil Drum, is that it is not always clear what information we get from reserves. As Jean Laherrere's graph shows, U.S. oil reserves have consistently been about 10 years worth of current production. We get no new information from a number that can be accurately estimated by multiplying current production by ten. He quipped that when the last barrel of oil is produced in the U.S., there will still be 9 barrels of reserves that would then be converted to resources. On the other hand, coal reserves have historically been higher then future production, which means something is wrong with them. Reserves should be a lower bound on future production, and there should be additons over time due to discovery and improved technology.
Dave
Hi Dave,
I used probability theory language because it's what I'm used to, because it seemed natural to express a shifted rescaled function using random variables, and because having amount of discovered oil at least as much as amount of produced oil for any time t corresponds to a property of random variables called stochastic domination. It was not necessary for me to use probability theory language and I did not use any limit theorems from probability theory.
It's fascinating that the normal curve fits oil production, but it's not at all clear how the central limit theorem applies. It's not even clear why the logistic distribution should fit the data. Deffeyes waves his hands a little about applying the logistic equation to finding oil, but, as he says, it's not convincing.
Bentley has a model described in the Strahan book which results in asymmetric depletion curves. I've written a paper about it which has been submitted to a journal. At a Parliamentary meeting on peak oil last week a BP representative said that Hubbert uses a bell shaped curve because of the central limit theorem. Bentley objected to that as well as many other things the BP rep said.
I think looking at simple models can clarify the reasons for a peak and why something like the R/P statistic is nonsense. At the same time, it would be interesting to apply ideas developed for simpler models to more complicated models.
Sorry, I can't help you out with the coal reserves!
Hi Dudley,
Thanks for your comments. Does anyone recall Hubbert mentioning the Central Limit Theorem in his work? My sense is that Hubbert was more comfortable with the logistic function than the normal curve. In his earlier work, he thought in terms of exponential growth, and this made it natural to progress to a logistic function, because the beginning of a logistic function is exponential. Also, before the development of personal computers, the logistic function was much more tractable mathematically than a normal curve.
Dave
We tend to find and--far more importantly--we tend to develop the largest oil fields first. When the large oil fields start declining in a given region, we can't offset the declines with new smaller oil fields.
Caveat is that many things can "fit like a glove" over a certain range.
Thinking in terms of a time-series, the Gaussian or Normal curve has the property of dP(t)/dt = k*(tpeak-t)*P(t) which means that if Production is equal to P(t), then the rate of production is in proportion to the time from initial production. The sign of this rate changes sign at the peak! You have to roll that one around in your mind whether to believe if this is significant to our understanding.
The statistical view of the normal is more difficult since the gaussian tails extend to negative times, which doesn't make pragmatic sense.
The Dispersive Discovery model combines the deterministic time series aspects with a spread of statistical rate values which essentially replace the gaussian formulation with something that makes a lot more intuitive sense.
Hi WHT,
This figure for US oil production is probably one you have seen. The normal curve is constrained to match the actual production in the latest year for which there is data. This leaves only two parameters, a production scale (the ultimate production), and a time scale (the standard deviation), to be determined by a least-square fit. As Deffeyes says, for US oil, it "fits like a glove." And this takes us back to the question about how the Central Limit Theorem is at work.
Dave
Unfortunately it doesn't fit well at all in the first few years after oil was first discovered. That is, the fitted curve also requires a starting point with initial conditions. So a Gaussian actually requires 3 parameters. But it breaks time causality that a time series curve is largely set by the initial values of its data set without a forcing function to drive it. The fit may be impressive but it is more than likely purely coincidental.
As a counter-example, this curve fits much better and it uses really only two parameters, with a starting point fixed in the vicinity of original oil discovery. Notice the strong bending around 1850, something that a parabolic mapping cannot accomplish.
(from the Dispersive Discovery post referenced earlier)
I know that this may all seem a rather intricate argument, but I am after an understanding of what is going on, not some heuristic fit to the data that comes from scavenging through a junk drawer of mathematical curves.
--
I saw a little bit of the PBS Nova show on the hunt for Bose-Einstein Condensation the other day. If you know anything about this topic you find that both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations. But the distinctions in the assumptions leading to their derivation make all the difference in the world. It basically explains what is going on at low temperatures and how the properties of BE particles are so different than that of FD particles. And no one would ever make any progress in understanding if you didn't junk FD when looking for the key to BE condensation.
So I argue that we really need to junk the heuristics of oil depletion because no really substantial theory stands behind it.
Hi WHT,
Thanks for showing your very interesting graph again. Your points are well taken. I think that they apply to the future as well as the past, because gaussians roll off faster than in your dispersive discovery model. Would it be possible for you to do a run of historical fits to the ultimate using your model to see how stable predictions of the ultimate have been with time? For comparison here are the historical fits to the cumulative gaussian for US oil. They start to give useful results, within a factor of two of where we are now, in the 40's. The current fit is 225Gb for the ultimate, and it has been relatively stable, but looking at the curves, one could easily imagine it turning up and ending up at 250Gb, which whould be near you. Would your model give a flatter result than this?
I get more stable fits for the ultimate when I fit cumulative curves rather than straight production curves, so it might be worth trying both ways.
The curve labeled reserves is reserves plus cumulative. It is close to the gaussian fit now, but you get the sense that it is correct only in the sense that a clock that has stopped is right twice a day.
Thanks,
Dave
I haven't read up on the Shock or Dispersive Discovery models, but I will just say that my guess is that if there was some nice interpretation of the differential equation for the normal distribution, which is so simple, then it would have been found already. I suspect that differential equations are not the right way to think about the problem.
I've also thought about the fact that the normal approximation doesn't make sense for t before some initial point. One way to avoid that problem might be to note that the normal curve is approximated by Gamma distributions with increasing parameters, and the densities of Gamma distributions are zero on the negative axis. Moreover, the curves from the Bentley model I wrote about above look somewhat like Gamma distributions (but they aren't).
I'll have to read about the Shock and Dispersive Discovery models.
I am putting all my stock in differential equations, and more precisely pragmatic use of stochastic differential equations which are useful for these kinds of problems.
Interesting that the Gamma distribution comes out of the Oil Shock model fairly cleanly, with an order determined by the number of convolution stages you use:
http://mobjectivist.blogspot.com/2005/11/gamma-distribution.html
Like you said it also has the property that "densities of Gamma distributions are zero on the negative axis". See red curve below compared against the Logistic in yellow.
In reality, we don't see the Gamma because the spread in the Discovery profile will wipe out the intrinsic shape. Like the Bentley model you mention, it may kind of look like a Gamma but it isn't.
If you read up on the Dispersive Discovery model and the Shock model, remember that they describe largely orthogonal features of the oil life-cycle. The Shock model requires a forcing function that is provided by the Dispersive discovery model. In other words, the Shock is applied post-discovery.
Also look at Khebab's Hybrid Shock Model, which has the benefit of a somewhat independent take on the original premise. This also tries to unify the Shock model with the conventional wisdom of the Logistic, which is a great service to a more complete understanding of how we have evolved in curve fitting.
Sure I could do this. This is actually a good way to estimate how good a predictor someone has.
Cumulative curves have a built-in filter so I like fitting these better as well. They only suffer from the lack of a good number for the initial cumulative value (since production data is very sketchy early on).
Instead of the stopped clock analogy, it's that everything kind of looks like a Gaussian if it has been convolved enough. It's the understanding of how this happens which piques my interest.
Hi Dave,
I am wondering if the central limit theorem really has relevance. You suggested to me a while back that it might be operating in the many business decisions taken by an ensemble of people, but I wonder if that would be more appropriate to the noise on the curve rather than the curve itself. I can see a large potential for straight determinism rather than a random variable explaining the shape of the curve. In the beginning, money starts to flow so more wells can be drilled. After a while, the edges of the field are found so no more wells are drilled and production levels off until the field starts to be exhausted. Money shifts to discovering new fields once all the wells are built and you have a self-similar process growing until no new fields are found, or, rather, they are found at such a low rate that the discovery effort is considered too expensive.
On coal, discovery is not as difficult so it may not build in the same way. Perhaps production is set more by saturated demand rather than the desire to produce more so that even more can be produced?
Chris
Hi Chris,
Thank you for an interesting comment. To get a gaussian as a deterministic function of time, wouldn't we need to able to interpret the production in terms of the differential equation WHT just gave us in a physical way?
dP(t)/dt = k*(tpeak-t)*P(t)
Dave
Hi Dave,
WHT points to a departure from this form at the beginning, though it would be hard to pick up in fitting without log weighting I think. But, we might try to separate the rate of increase looking like the function iself (exponential) multiplied by a linear term by considering that an oil company reinvesting profits (yielding the exponential) looks like a going concern so that others might get into the game as an increasing function of time. Most folks in the solar industry right now are saying that now is the time to establish market share so there is a rush to get in. Some who are already in are posturing saying that it is already too late and no one can compete with them.
I remember at the time of the peak, oil companies were giving away all kinds of things the way cereal companies do to try to retain market share. Our steak knives at home all came from Shell. The companies would advertize how freindly and helpful their service people were. Now such things seem to only be nostalgic rather than a real marketing effort.
In the function that WHT shows, the decline looks more exponential than Gaussian. The oil companies don't seem to be growing in number now except perhaps very slowly around the Bakken play, so we don't get the extra linear term. So, perhaps we are seeing something physical about how an oil resource that has been exploited in the prior manner drips out its dregs.
This is just a narrative, rather than a demonstration of deterministic trajectory but it points towards the possibility that a random distribution does not underlie the shape of the curve. Rockefeller might be randomly exchanged with du Pont in their businesses but the shape would be the same perhaps.
Chris
"This is just a narrative, rather than a demonstration of deterministic trajectory but it points towards the possibility that a random distribution does not underlie the shape of the curve. Rockefeller might be randomly exchanged with du Pont in their businesses but the shape would be the same perhaps."
I agree "that a random distribution does not underlie the shape of the curve", but I do think that stochastic elements contribute to the profile of the shape. The stochastic elements are basically described by the spread in rates as various reservoirs become discovered and then extracted. The forcing function has some determinism due to the monotonically increasing march of technology and human adoption, which I think completely wipes away any hope that a single random PDF underlies the shape of the curve.
The only caveat is that a convolution of a many gaussians approaches a gaussian due to effects as mdsolar pointed out earlier in the thread. But then again the shock model does multiple convolutions and in accordance to the Central Limit Theorem everything starts to look normal.