42 comments on The limit of the statistic R/P in models of oil discovery and production
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This post was discussed between TOD contributors, here is a copy of my comments:
It is said that "In order to have a mathematical description of R/P, we must have models of both discovery and production", however discovery does not lead directly to proven reserves. Proven reserves are supposed to represent discoveries that have been developed or are planned to be developed in the near future (let's say within 5 years). The initial discovery strike has to go through some standard oil development phases (decision to develop, planning phase, build phase, etc.) which is introducing a dispersive/smoothing and lagging effect. This relationship between the discovery curve and reserve additions is at the heart of the Shock model. In the proposed approach, reserve is approximated by:
R(t)= CumulativeDiscovery(t) - Scaled_and_shifted_CumulativeDiscovery(t)
It's not clear how good is this approximation.
The fact that the R/P is misleading is well known despite being widely used. The best example is maybe the US, where the R/P has been around 10-15 years since 1920:
I think that reserve growth must be important in fact, but what I notice from the paper is the shape of the curves having the effect of interest. Could it not be that what we are seeing there is that each time you calculate reserves, your production rate has also fallen so that the ratio won't converge to zero?
Chris
Good point. I have always said that Production PDF has to be the convolution of a Discovery PDF with some other function. Unfortunately, there is no function that when convolved with another one will give either a Logistic or Lorentzian or Gaussian for that matter. This means that the use of these curves is a contrivance that does not prove anything.
So your point on production rate falling forcing a finite convergence makes a lot of sense because it makes a lot of intuitive sense. It's like pigs sucking out of constantly shrinking trough. The pigs are furiously slurping but as the trough gets smaller, the number of pigs decreases as well.
The convolution of a Gaussian with a Gaussian gives another Gaussian. The sigmas add in quadrature. This is because the product of two Gaussians is a Gaussian and the Fourrier transform of a Gaussian is a Gaussian. It is a special function. I've derived analytically what minimal Gaussian kernels are needed to bring two arbitrarily oriented 2-D Gaussians ellipsiods of differing dimensions into coincidence with the least required distortion to each in an attempt to speed up a program I wrote which derives such kernels for galaxy images obtained with different seeing by means of minimization. The problem there is very degenerate so I wanted to give it a head start. From working on the problem, I can say that you can always take a delta function into another function but after that it gets tricky. Things that look quite narrow sometimes don't have a kernel that takes them into a broader function. But, there is always a pair of kernels that take both into a common function (each other in the trival case) and it is possible to say something about what the best pair looks like.
I think the beginning of the logistics function has a sound basis. If you are making money and you want to make more, you reinvest based on the amount of money you are making. This produces an exponential. The center of the logistics function also has a sound basis: diminishing returns. At some point, putting more effort in isn't getting you anywhere so you put your money somewhere else. I don't know why depletion would look like the logistics function. Perhaps it has to do with wanting to keep some of the people employed who made the whole thing go in the first place. If the coal seam, for example, is running out though, you can only put so many miners at the face. And if an oil well breaks down that is not producing much, you might decide not to fix it. Perhaps depletion is sentimental so it does not get a clear cut off that you might expect.
If you like, I can see if there is a kernel that takes discovery into production via convolution, but convolution is not necessarily the proper method because the time-scale of production probably depends on what has been learned from prior experience. A time varying kernel might be a better representation. It may be possible to take production curves as a function of time and derive a projection that is helpful for estimating the behavior of future or recent discoveries. One catch there though is that the recent curves which are most constraining of future curves may not be complete enough to learn from.
Chris
You are technically correct but this brings up two points:
1. The Gaussian convoluted with a Gaussian is a Gaussian only if it stretches to + and - infinity on the tails. Same thing with a Lorentzian. I see with your background in Fourier optics that this is pretty obvious, but then again spatial diffraction is different than temporal time series in that the former stretches + or - in the limit, whereas temporal has a clear one-sided forcing function due to time causality. (i.e. effects at time at -infinity cannot happen)
So this only means that it is a non-analytic Gaussian and probably approximates it only depending on how bad the temporal tail truncation is. This is even more obvious for Lorentzians convolved with each other, which have even bigger tails.
2. Yet if we may say that it approximates a Gaussian after a convolution with itself, this means that the Discovery profile is a Gaussian! Then we have to explain the rationale for this behavior. And we have another problem to deal with, as we are only switching it from the Production profile to the Discovery profile.
The temporally causal nature of the problem is why I came up with the Dispersive discovery model and the components of the Shock model which use temporally one-sided exponential functions to model the production delays and extraction rates. I am definitely not seeking kernel or windowing functions of the sort you are describing.
I used a simple model, not differentiating between proven and possible reserves, so the analysis would also be simpler. Maybe my analysis could also be applied to your reserve function, but the Shock model described in your link looks a lot more complicated than just using R(t)= CumulativeDiscovery(t) - Scaled_and_shifted_CumulativeDiscovery(t).
Are there models where possible reserves either change to proven reserves or disappear, possibly randomly?
Hi Dudley,
For U.S. oil, the simplest model for the EIA reserves is just 10 times the production in that year. That number is within 20% of the actual reserves for the last 50 years.
For coal, the pattern is different. Countries typically carry unrealistically large reserves, until someone in a resources agency decides that there are no good prospects for new mines. Then the reserves to collapse to the coal that is accessible at working mines. For example, German bituminous coal reserves were reported as 23Gt in the 2001 World Energy Council Survey, but in the next Survey in 2004, they dropped to 183Mt.
Dave
That's interesting. So for oil R/P is about 10 but for coal R/P jumps around. Does it jump around some median value in a white noise or Gaussian kind of way. In Deffeyes' book he uses anthracite coal (I think) as an example which does follow a Hubbert-type curve.
Hi Dudley,
Here is PA anthracite, with the cumulative and the gaussian fits. The reserves are indicated by circles. The reserves are too high, and they do not come down until too late.
Dave
DAve
Interesting that applying the shock model to a Kroneker delta function for a single discovery would add a fixed width production so the scaling on width does not really jive with that model.
In some sense R/P=constant is a pure Markovian rate formulation. The amount of extraction is always proportional to the amount of perceived reserve available. And the reason it starts out high and then damps is because of the transient lags for the shock model phases (decision, plan, build) before the production hits that long-term steady state.
I hazard a guess that this will play out a bit differently using the HSM (Hybrid Shock Model) because the extraction rate builds up with cumulative discovery in HSM. It would be interesting to make the comparison.
Otherwise good use of pragmathematics. Using L'Hopital's rule is indeed clever.
Yes, the scaling on width wouldn't work for a delta function but it could work on a pulse function approximating a delta. The scaling is more about the time delay between discovery and production. It sounds like the Shock model models that time delay in a more elaborate way and is probably more accurate.
The delay and width scaling comes about from convolutions in the shock model. Even taking the Lorentzian example and the Gaussian examples, you get summed (not multiplied) scalings if convolution is at the root. Two infinite Gaussians convoluted together give a new width that is sqrt(w12+w22) and two infinite Lorentzians convoluted together goes as w1+w2. That's at least what I think is at the root of the scalings.