Finding Needles in a Haystack
Posted by Khebab on June 27, 2007 - 11:38am
Topic: Supply/Production
Tags: oil discoveries, oil production modeling, Shock model [list all tags]
This is a guest post by WebHubbleTelescope.
In school, we used to do horrendously difficult mathematical "word" problems routinely. I remember occasionally getting one right, but more often ended up punting on the problem, and then waiting for the teacher to explain the solution in all its elegant simplicity. Of course, just about every real-world problem contains inherent ambiguities and incomplete information. So we rarely get to see the elegant solution in our day-to-day work life. Sometimes we get lucky and nail a problem, but in the majority of cases, we eventually resort to creating a limited model of the problem domain and deal with that.The problem that I have recently wrestled with has to do with predicting future oil discoveries based on historical dynamics. Ideally, I want to reduce it to a solution that has the elegance of a word problem, and not have to deal with messy economic and geologic factors that would quickly turn it into a rat's nest of complexity. Call me an optimist in this regard, but my intuition tells me that the solution remains as simple as ... finding needles in a haystack.
So I present a starter word problem:
Given a large number of needles dispersed in a random spatial manner throughout a good-sized haystack, at what point in time would we find the maximum number of needles? As a nod to technology we get to monotonically increase our search efficiency as we dig through the stack, and we can add human helpers as we progress.
Answer:
Obvious,
and we don't have to even lift a pen. On average, the maximum discovery of
needles occurs as we sift through the last of the volume, and once finished,
the discovery rate drops to nil. So the instantaneous "discovery" rate looks
similar to the curve at the right. The acceleration upward in the curve occurs
as we get more proficient over time and can attract some help. Note that if we
mixed larger nails and smaller pins with the needles and instead measured
total weight or volume instead of quantity, we would have the same curve (this
has implications for the oil discovery problem).
Next, let's make the word problem a bit more sophisticated. Say that instead of dispersing the needles randomly through the entire haystack, we only do it to a certain depth, and to top it off, we do not reveal to the needle and pin searchers this depth. They basically have to oversample the haystack to find all the needles. If you look at the following figure, we separate out the "easy" part of the search from the "difficult" part (i.e. difficult as in not finding much even though we expend the effort). The boxes represent monotonically increasing sampling volumes, which we use to sweep out the volume of the haystack.
Hand-Wavy Answer:
Suffice to say, if we search top to bottom, we will similiarly reach a peak,
but the peak will also contain a gradual backside. Intuitively, we can sense
that the sharpness of the peak reduces as the sampling volume overlaps the
region that contains the needles with the region absent of needles. And then
as the sampling volume drifts even deeper, the amount discovered drops closer
and closer to zero.
For us to draw the peak as a smooth curve, we need to add stochastic behavior to the search process. This can occur, for example, if the individual searchers have varying skills.
a stochastic variable is neither completely determined nor completely random; in other words, it contains an element of probability. A system containing one or more stochastic variables is probabilistically determined.
What really makes the haystack problem different than the global oil discovery
doesn't lie in the basic word problem but rather in the application of
randomness or dispersion to the problem. We have much greater uncertainties in
the stochastic variables in the oil discovery problem, ranging from the
uncertainty in the spread of search volumes to the spread in the amount of
people/corporations involved in the search itself. We don't just deal with a
single haystack, but multiple haystacks all over the world. So the sharply
defined geometric discovery profile shown to the right gets washed out as a
result of the statistical mechanics of the oil industry ant-people hard at
work.
Final Exam Answer: Let's jump from haystacks to oil discovery. We solve the problem by making the generally useful assumption that the current swept volume search has an estimated mean, and a variance equal to the square of the mean. In other words, in the absence of having any knowledge in the distribution of instantaneous swept volumes, we assume a maximum entropy estimator and set the standard deviation to the mean. A damped exponential probability density function follows this constraint with the least amount of bias, maximum uncertainty, and a finite bound (the latter factor would rule out something like a log-normal distribution). The following curve demonstrates how the spread in values gets expressed in terms of error bars.
In a nutshell, we want to solve the discovery success rate of a swept volume
realizing that part of the volume straddles empty space. In other words, to
account for the effects of the dispersion of oversampled volume, we have to
integrate the exponential probability density function (PDF) of volume over
all of space, and determine the expected value of the cross-section. To solve
the problem by baby-steps, we first take a look at the one-dimensional version
of the problem, then extend it to three-dimensions, and finally add the time
variation.
I originally used the following single-dimension equation derivation to solve the reserve growth "enigma" of a single reservoir.
In the three-dimensional case, the stochastic variable lambda
represents current mean swept volume, the term x integrates over all
volumes, and L0 represents the finite container volume
Vd. The outcome L-bar represents a kind of pro-rated
proportion of discoveries made for the dispersed swept volume at a particular
point in time.
By itself, the function corresponding to L-bar doesn't look like
anything special, and indeed looks a lot like the cumulative of the
exponential PDF. However, the fact that lambda monotonically increases
with time, together with L-bar appearing in the denominator, gives it
interesting temporal dynamics, of which I contend follows the empirical
observations of cumulative oil discovery and that of reserve growth as well.
From first principles, we would expect that swept volume growth approaches a power-law, and likely a higher-order law. For example, considering the "gold-rush" attraction of prospecting resources alone, we would expect that linear growths in (a) oil exploration companies, (b) employees per company, and (c) technological improvements would likely contribute at least a quadratic law.[1] In terms of the bottom-line, multiplying two linear growth rates generates a quadratic growth[2], and multiplying more linear rates leads to higher order growth laws. As an example, you can see this power-law increase play out as evidenced by the historical increase in average oil well depth over the years (see [3] for data point references).
But of course, this only accounts for one dimension in the sampling volume. So if we make the assumption that the effective horizontal radius of the probe also increases with a quadratic law, we end up with a power-law order of n=2*3=6, where the 3 refers to number of dimensions in a volume. Because we actually use cumulative volume in the stochastic derivation, the order becomes 6 in the result shown below. When we make an assumption that the parameter k denotes a fraction of the swept volume that results in a cumulative discovery D(t), we can replace Vd with Dd, where Dd is essentially equivalent to a URR for discoveries.
D(t) = kt6*(1-exp(-Dd/kt6))and the derivative of this for instantaneous discoveries (e.g. yearly discoveries) results in:
dD(t)/dt = 6kt5*(1-exp(-Dd/kt6)*(1+Dd/kt6))For a family of power-law growth functions, the trend looks like the following set of curves. The salient point to note relates to how we trend toward an asymptotic limit at the volume Vd as the power-law index gets larger.
To briefly summarize how dispersion of prospecting effort affects the discovery process, consider the curve below. Initially, as the sampling probe stays well within the Vd limit, the dispersed mean comes out as expected since we do not oversample the volume. However, as the standard deviation excursions of the cumulative volume starts to bleed past Vd, the two curves start to diverge and a rounded discovery peak results.
Scores of depletion analysts, including Laherrere, have pointed out the similarity of yearly discovery curves to the classic Hubbert curve itself. For the following discovery curve from Shell Oil (courtesy of a TOD post from Rembrandt) one can see the same general trend, albeit buried in the noisy fluctuations of yearly discoveries.
To remove the noise, we can generate a cumulative discovery curve. Apart from missing out on the cumulative data from the years post-1858 to the initial year of collected data, we can generate a good fit to the curve with an n=6 power-law dispersive growth function. (Note that the curve has a constraint to start in 1858, i.e. t=0, the "official" date which signalled the beginning of serious oil exploration)
Applying this modelled discovery curve to the Oil Shock production model (see the m o b j blog and a review by Khebab here at TOD), we come up with the following production extrapolation
The oil shock parameters include a fallow latency of 6 years, a construction latency of 8 years, and a maturation latency of 10 years. It also includes the following extraction rate shock profile
Interesting that this gives a production peak around the year 2010, even
though the effective URR from the Shell discovery data amounts to 3.5 trillion
barrels -- much higher than the lowball 2+ trillion estimate commonly bandied
about by pessimistic peak oil analysts (note that the shell estimates uses the
somewhat ambiguous "barrels of oil equivalent").
We can further substantiate the discovery fit by applying it to the USA data subset. For instance, let's consider what would happen if we used the same parameters from the global data to estimate U.S. discoveries. Note that the same constants (i.e. k and n=6) are used, but we change the Dd to reflect a fractional area of the US in comparison to the world.
World Land Area = 150,000,000.0 km2So to first-order, the Dd for USA is 1/15th that of the world's Dd (Roland Watson posted a similar sanity check recently on TOD with reference to USA and world URR). The following figure lays the cubic-quadratic discovery curve on top of Laherrere's data.
USA Land Area = 10,000,000.0 km2
Within an order-of-magnitude, the fit doesn't look out-of-place. In the context of swept volume, it means that the USA reached its limit of easily discovered oil quicker than the rest of the world, which makes sense as serious oil exploration started in the USA.
After the equations have been solved, the result can be translated back into the ordinary language.
As for as other criticisms, I suppose one could question the actual relevance of a power-law growth as a driving function. In fact the formulation described here supports other growth laws, including monotonically increasing exponential growth. Furthermore, one could question whether we can sustain a power-law growth in the future, which together with extraction rate extrapolations, will have a significant impact on how future production will conceivably pan out. And to account for any further reserve growth, the fact that much of the fit curve occurs before the peak happens means that past discovery estimates have had a chance to mature and we have more confidence in the discovery decline profile. In my opinion, this makes it a fairly conservative estimator -- to substantiate this take a look at the huge effective URR for the Shell discovery data, which in all likelihood includes reserve growth, and note how it only impacts the peak date a few years from my previous shock model prediction of 2004 (which had no extrapolated future discovery data and used solely Laherrere's discovery data which had a much lower effective URR of around 2000 GBls).
Or, one could question the impact of super-giant discoveries on the smoothened discovery plot. Statistically, super-giants get treated like anything else in this model and they populate the volume with the same randomness. Predictably, one could also question the absence of deep geologic or economic considerations in the model. The canned response to that line of questioning is second nature to a seasoned statistical mechanic: physicists and other scientists apply such stochastic approximations all the time without a lot of fundamental problems. Why should this stochastic model become an exception to the rule?
I also have not opened up the future possibility of a levelling out or even general decline in discovery search effort. I gave this some serious effort in past blog postings, but realized that this would give too pessimistic a prediction and perhaps too much of an artificial constraint.
Finally, one could question why no one else in the oil industry thinks in terms of this kind of discovery model, in other words, why hasn't someone else found this proverbial needle in a haystack? Don't ask me; for all I know, an analyst in some energy corporation's back room has come up with the same idea and it has transformed into filing-cabinet intellectual property with no hope of seeing the light of day (i.e. what good would it do them financially?). Or perhaps, a similar idea remains buried in some academic journal, for which I lack the resources to discover on my own. But if my approach indeed has some originality and correctness to it, I can rationalize this with a more mundane explanation that comes from, in part, my experiences in solving problems in the research and software world. Occam says to rely on the simplest explanation to a problem; but what happens when two sufficiently separate but equally fundamental explanations contribute to a greater understanding? In these cases, we have to overcome the inertia of conventional wisdom.
To explain this rather philosophical point, I consider an oil depletion model as a two-stage word problem. The first part of the word problem relates to production (illustrated by the Oil Shock model) and the second part provides a model of the discovery input used to feed production (i.e. the basis of the Cubic-Quadratic discovery model desribed in this post). The relationship of two interacting models has some similarity to an aspect of software debugging instanced by the occasional defect that takes enormous resources to resolve. Or resembles in some ways to the laboratory anomaly that no one can pin down precisely by experiment. Invariably, the most difficult bugs to resolve result from two or more interacting defects. In my opinion, these remain the most elusive problems to solve simply because you don't normally think that more than one fundamental issue contributes to the cause of a root problem. And there you have an example of a real-world word problem. While everyone and their cousin wants to figure out oil depletion with a single freakin' logistic curve (excepting R2), as though that contains THE key to the kingdom, we realize that oil depletion may have two underlying forces at work -- namely, the discovery process followed by the extraction process. And so we rely on the wisdom of a divide-and-conquer strategy -- figure out the extraction/production problem all the while knowing that the discovery problem lays in waiting, or vice-versa. Now think back to the original "needle in the haystack" problem; notice that in that case, discovery and extraction occur at the same time. Once you find the needle you can extract it. But not so with oil, as discovery only starts the process that culminates in extraction and production. In my opinion, when we can understand the two problems individually, we can then solve the penultimate word problem of our times.
[1] Note that parabolic growth is not the same as quadratic growth. Due to some historic conventions inherited from Silicon Valley, parabolic growth actually follows a fractional power-law growth, more precisely a square-root of time dependence.
[2] See growth in wiki words for another real-world example of quadratic growth that occurs as we speak.
[3] I gathered the max depth well chart from these sources: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Khebab,
As usual, A+ on your article.
I think I could find needles in a haystack more easily than I could find much new oil--because with a powerful magnet I could get through that stack in a day or three.
Your assumptions are clear, your logic is valid, and I think your conclusions are compelling.
This masterpiece is actually from WebHubbleTelescope, not me. I'm just a mere messenger.
Whoops! I see the A+ should have gone to webhubbletelescope. But Khebab still retains his grade, because all his articles (in my estimation) are in the top tenth of the top one percent of quality of postings found on TOD. Indeed the quality of postings this past month has been extraordinarily high. Could it be that the most excellent Oildrum is getting even better?
Or maybe its Peak Oildrum excellence....lol. Seriously though, it will take several passes and some college algebra review for me to fully understand. So far, excellent work Webhubbletelescope!
I have been interested in the following method, since I read about it in "Ship of Gold in the Deep Blue Sea"
http://en.wikipedia.org/wiki/Bayesian_statistics
I am working to flatten the Lower 48 HL plot, by looking for small overlooked oil fields. While small, the fields can be quite valuable. For example, with the right reservoir, a 500 acre area could produce 5 mb of oil. If Simmons is right about oil prices, this 500 acre area could generate a gross cash flow to the working interest owners and royalty owners of a billion dollars, in constant 2005 dollars. Which of course is fine, until the rioters appear at the gates of the mansions of the energy producers.
Private security is one of the most rapidly growing industries in the U.S. Globally, I wonder how many private security men are employed in the oil industry? My suspicion is that the number is quite large and growing in double digits each year. This factor is getting to be a significant cost of doing business.
Bayes rocks.
I know a decent amount of it, but not enough to express competence. I've done a couple of conference papers using Bayesian models, but that's because a buddy is much harder core Bayes than myself.
The frequentist world is a lot easier (central limit theorem, yadda, yadda), but Bayes makes a lot more intuitive sense. The math, however, is a lot harder.
I recommend Jeff Gill's Bayes book, if you're interested:
http://www.amazon.com/Bayesian-Methods-Behavioral-Sciences-Approach/dp/1...
(the estimation of priors, on the other hand, by this crowd, would be pretty good. *laugh*)
I'll probably get the book.
If you haven't read about the SS Central America, here is the link: http://www.amazon.com/Ship-Gold-Deep-Blue-Sea/dp/0349110999/ref=pd_bbs_s...
It's a remarkable story, from the sinking of the ship, and the rescue of some of the passengers, to the search for the wreck.
Prof. Goose,
IMO, the best introduction to Bayesian statistics is still the classic by Leonard J. Savage, "Foundations of Statistics." After sixty years, that text has stood the test of time. I remember struggling with the book back when I was fifteen years old and smarter than I am now, but after a few rereadings and working of problems I finally got it. (I may not be exceptionally bright, but I am exceptionally persistant.)
I find the combination of Bayes, diffuse priors, and the Kalman Filter to be very appealing & a useful linear model.
Neat stuff.
The basic idea behind the Kalman filter, IMO at least, is to have a good model for the uncertainty in the data. In other application areas, this uncertainty can be related to noise or other fluctuations which is then used to for example extract a signal from noise. In the model presented here the uncertainty about the mean is really meant to represent fluctuations in the volume sampled, or also in terms of what we think the volume that we sampled. So in this regard we can try to extract the growth in discoveries from the underlying dispersion.
The latter uncertainty is also very critical as input to extraction models, because our estimation of, e.g., how much reserve we have, is crucial input to the amount of effort we expend on getting the stuff out.
Can you use a substitute model for comparison? Such as the number of dry holes per successful hole, with a slope adjusted for increasing data confidence due to better imaging and drill guidance?
"Dig your heels firm unto dirt; and where is the dirt going..?" -Frank Herbert, "The Jesus Incident"
Web -
To add to your KF description, I learned the KF (long ago) via recasting the standard OLS problem in the KF framework. I liked being able to 'see' the impact on the parameter estimates as data points were added to the time series: I found the explicit signal-noise decomposition the KF provides to be 'illuminating'.
Wow, that was brilliant! It is exciting to read something that makes total sense and is also surprisingly novel.
I suspected that the key to understanding oil production lay in the discovery part. It now makes sense that oil production and fishing show similar curves, as they both involve searching for a resource. I wonder where this leaves the logistic. Models of infective agents also involve searching for an uninfected host, for example.
It makes some weird sense since King Hubbert was known to be a fanatic fisherman, and he made several friendships with other sports fisherman who also wrote about peak oil.
Craig Bond Hatfield : http://www.theoildrum.com/node/2364/0
George Pazik : http://mobjectivist.blogspot.com/2005/05/our-petroleum-predicament.html
I think that these guys spent a lot of time thinking on the water what the future held for a finite resource. I can't think of anything that a fisherman hates worse than a "fished-out" body of water.
Of course, we still have to deal with an issue of some complexity (as if we needed more!). that being: Suppose you are lookin for needles in a haystack that has certain sections of the haystack off limits?
The issue with oil is exactly that, in that some areas are pretty much forbidden to be drilled in for environmental reasons or cannot be correctly explored due to geo-political reasons. Who is going to spend money looking for oil in areas that they will not be allowed to produce in anyway?
I am asking this question for a reason. Recent remarks from the oil industry both worldwide and domestic U.S. seem to indicate that this is the argument the industry intends to make, i.e., the issue is not lack of oil out there to be drilled, it's an issue of "access" to that oil.
Christophe de Margerie has been adament on this, with his "120 million barrels a day, never", remarks. Some people took this as an endorsement of Peak Oil, but it was NOT an endorsement of conventional (Hubbert) peak, but instead an argument that the politics and geopolitics are the limiting factor.
In his nationwide road show, the President of Shell Hofmeister has been making the same argument.
The access argument is the only one that can rationalize the problem that while the oil industry continues to assert there is enough oil so that we "never run out, never" (Mr. Hofmeister of Shell), or that there is no danger of peak until sometime after 2050 if ever (API chief), or that the oil is out there, we just cannot get it out of the ground fast enough due to logistical/political constraints.(Christophe de Margerie of Total) Who looks for oil in areas that they will not be allowed to produce in anyway? How long would the shareholders put up with that?
Christophe de Margerie of Total, Mr. Hofmeister of Shell, and the American API all seem to be lining up on this. One assumes that the other oil companies will soon follow, or that I simply have not heard thier speeches, and they are already lined up on this position.
Either way, for the consuming nations, the whole debate may not matter. A "logistical peak" can be just as destructive as a "geological peak" (i.e. Hubbert peak) for the buyers, driving prices higher and finally resulting in real shortages. My guess has always been that we will be forced to alternatives long before we ever have to worry about a Hubbert geological peak, and that at least half or more of the worlds oil will be recovered at a very slow rate compared to history, mainly as an industrial/chemical raw material and not as fuel to be burned (always a waste of crude oil and natural gas anyway), but will be thusly very expensive compared to historical standards.
On that assumption, we can conclude that more oil has already been discovered than we (the consuming nations) will be allowed to drill, at least for some time. That would make future discovery a moot point. Why would an oil company search for oil it cannot drill when it already knows the location of more oil than it is allowed to drill, or can logistically support the drilling of due to lack of labor, capital or machinery?
Roger Conner Jr.
Remember, we are only one cubic mile from freedom
The amount of "off limits" area is statistically tiny compared to the overall surface. Further, much of this so-called off limits area has been explored but not developed, either because of restrictions or because the area was devoid of oil producing rock. It would seem that you are more worried about the speck in your brother's eye than the log in your own.
Ghawar Is Dying
The greatest shortcoming of the human race is our inability to understand the exponential function. - Dr. Albert Bartlett
GreyZone:
The entire eastern half of the Gulf of Mexico, The entire Atlantic Coast of the US, and Offshore Pacific Coast areas have been off limits since the Santa Barbara Blowout of Chevron's in 1978, as I remember. Since then the frontiers in the drillable part of the Gulf has expanded from about 500 ft. water depth to the current limit of 7,000 ft.subsea. In addition the drilling methods have changed radicially with huge advances in directional drilling, and exploration has changed because of 3-d seismic to an economic success ratio of about 2/3rds on wildcats. There's a map in the pocket of the second edition of Halbouty's Saltdomes Gulf Region of United States and Mexico that shows dozens of these prospects.
Don't misinterpret this to mean that we can drill our way out of 21 million barrels of daily consumption in the by opening these areas up, but there is probably huge amounts of oil and natural gas in these areas. In addition there are the very difficult areas of the Artic Ocean, Greenland and the Antartic which have never been explored. Russia has claim to about 2/3rds of the Artic Ocean, the US and Canada and Denmark much of the rest.
These aren't "statistically tiny" areas
From what I've read, the estimates that have been done so far on oil in Antarctica suggest that optimistically it wouldn't be economical at anything less than $100/barrel, even though there may well be super-giant fields there.
No drilling has actually been done though.
The problem with such figures is that they are based implicitly on the ceteris paribus assumption. If oil was $100 and all the other prices stayed the same then maybe this activity would be economical. The problem is that if oil is $100 then the rest of the price structure in the economy adjusts and the a higher price is needed to justify the activity. It's not quite Zeno's paradox but the same rosy predictions for the tar sands were made that they would be very profitable with $50 oil. This has been brought up on this board several times before.
+ the fact that we have a continuum between the ideal locations for exploration and the most inhospitable, and we have not filled that in with any major discoveries suggests that we know the trend. In other words, the model does not have a clean break between the Texas's of the world and the Anatarctica's. The world is continuous and the model does not show discontinuities, and likely neither will the data (apart from fluctuations).
WebHubbleTelescope
That would be true only if we had a sampling of the inhospitable areas equal to that of the easy to drill areas, and the factors for oil and gas formation were known to be equivalent.
The current thinking is that much of the oil source rock is organic rich shales deposited in cold 500 meter deep range waters in anaerobic conditions, which strikes me as a good description of the Artic-Antartic regions. But this is an hypothesis, a little more proven than dinosaur farts, but still no where near an established consensus.
The point made above about economics is valid. No way drilling in 7,000 ft of water on the outer continental shelf or in the high latitudes is going to be the substitute for oil produced at $5/bbl lifting costs, and the transportation is going to be exhorbitant too. I don't think we can ever overcome the cornucopians arguements by answering every pie in the sky scheme individually. But we can answer them that the Peak is here in economicly produced crude.
The current thinking is that much of the oil source rock is organic rich shales deposited in cold 500 meter deep range waters in anaerobic conditions, which strikes me as a good description of the Artic-Antartic regions.
In general the climate and location of the oil we are extracting today was quite different from that when the source rock was formed and the later heating event is very important.
I suspect most of the dry holes people talk about are more no useful quantities of oil or gas but almost any sedimentary basin probably has at least trace quantities of oil and gas.
Its commercially exploitable reserves that are rare.
I'm from Arkansas but we hit methane drilling water wells through the buried shell layer often. These are shallow wells but they often produce problematic amounts of methane
coming from natural fractures.
As far as wild theories about oil production my theory is the oil shale is laid down and buried during global warming events. The scenario is as follows.
The glob warms drying the land leading to huge dust storms that fertilize the ocean this leads to blooms and anoxic conditions leading to a high rate of shale formation.
Next these same condition cause peirodic and devastating super storms hurricanes are very strong monsoon events that strike the land masses and since they are mainly desert you get extensive erosion from these events. This sediment is spread over a large area by the outflow of water after the storm passes burying the mud under a deep layer and also filling in nearby areas with future sandstone. This continues until the CO2 levels drop but we now have conditions primed to start oil formation.
A bit further out but reasonable is that as the climate goes from cold to hot conditions the melting ice and warming seas
cause increased forces in the basin areas and rising land where the ice is melting this speeds up plate tectonics by creating and relieving stresses over a pretty short period of time. Not sure the end effect of this but you would expect a increase in volcanic activity with in a reasonable amount of time of going from cold to warm environment and vice versa.
Crackpot theories over :)
Roger,
If were to read the following speech by Exxon's Tillerson You will see that he is saying more or less the same thing. He did mention OPEC directly but it can be inferred from the context.
The speech was made last week, in London, at the Royal Institute for International Affairs.
off limits
There are three significant advantages to the off limits argument:
money - It opens up the US market to places currently off-limits (national parks, wilderness areas, etc). The US is one of the most generous nations in the world regarding oil and gas leases. Even if the total volume is modest - the ROI is quite attractive and the risks are low.
imperialism - it supports an aggressive investment in policies and strategies designed to open up markets via extreme measures.
CYA - we can always shift the blame to Saudis, Chavez, Putin, et al rather than examine how our own policies, investments and decisions may have contributed to the current challenge.
Thank you, WHT! As for your discovery graph, I believe it includes natural gas as well (just as Shell now includes all the Omani natural gas to offset their "reserves" loss due to the collapse of Yibal's crude production). This is relevant because the natural gas is being burned at the same time as the crude oil so if you are going to model production (and consumption) along similar lines you would have to merge the natural gas and petroleum production and consumption data to get a full picture.
Ghawar Is Dying
The greatest shortcoming of the human race is our inability to understand the exponential function. - Dr. Albert Bartlett
Yes, the boe - "barrels of oil equivalent" - metric. I know it can't include all the NG discoveries or it would really bloat the curve. See the following USGS estimates of boe:
http://www.usgs.gov/newsroom/article.asp?ID=636
So they in fact tend to cherry pick the sources. Makes a lot of sense, particularly when we realize that Shell Oil came up with the graph.
As far as the logistic model goes or HL its greatest weakness is it effectively assumes discovery is finished well before significant extraction starts and is focused on exploiting a known resource. This is why the shock model is a much better model for real world oil extraction. However since peak oil happens well after the discovery curve had dampened down the effect or echo of discovery times should decrease as the system approaches peak and passes peak oil. So increases in production because of new discoveries becomes less and less of a factor.
As far as mapping this to a disease model I'd guess the closest would be the spread of a epidemic. Once the rate of infection drops below a certain level the epidemic dies out.
Its interesting to note that the worst epidemics generally only kill less than 50% of the hosts before the rate of infection drops to zero.
http://en.wikipedia.org/wiki/Black_Death
In the case of the black death the rate was 70%.
So simply assuming infection is similar to finding oil.
We will extract anywhere from 30-70% of the oil before stopping.
So at least with this model questioning how much of the oil we will extract post peak makes sense.
Yes, many of these predator-prey and birth-death models show an approximately 90 degree phase shift between the effects. In some ways these mirror a form of non-linear trigonometry and comes about for essentially the same reason why a cosine wave is shifted by 90 degrees from a sine wave.
I still can't explain how it has a valid relationship to oil though.
So hopefully we can eliminate the need for it. I do not like the logistic for oil production. But its not something that can be trivially dismissed.
Your on the right track to either removing it or finding a physical basis that causes logistic behavior. But any reasonable physical model is far better than empirical logistic fitting that happens to work and the models will obviously be close in the cases where the logistic works.
Web, I rejoice in finally having a post from you on TOD. You’ve been around longer than I, and you surely had an important role in making TOD what it is today.
I don’t think that everyone’s relying on a single Logistic like you imply, any modeler should at least look at discovery trends before endorsing blindly the results of HL. For me it’s the fact that several essentially different techniques point to a close result, that make me think that this problem is quite well understood. And if HL is not the key, at least looks like a very simple candidate to that (thinking of Occam).
I’ll not pretend that I’m savvy enough in mathematics to criticize your conclusions, but I can’t help to ask you what the ultimate word problem of our times is.
Last question first: the ultimate word problem is "What is the meaning of life?"
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I basically have said what you recommend in comments to a recent post by WestTexas -- that HL and the Logistic model should be used as a supplement to other techniques. That it should not be used isolation, which is why adding in discovery and bottom's up techniques is so important.
The mathematical problem with the Logistic/sigmoid equation is that it is non-decomposable. If you accept the premise that total production is some convolution of the driving stimulus of discoveries along with the dynamics of extraction, then you need separately derived functions that work together. I assert that you cannot find me two functions when convolved will give a sigmoid function or its derivatives. In fact, the derivation of the Logistic comes about from a very simple nonlinear differential equation which by itself does not decompose very nicely (because it is in fact so simple).
Now, it may be that discoveries themselves could be represented by a Logistic model, but for me its a head-scratcher to set up as a word problem that does not somehow involve population dynamics. I believe that I may be too tainted by looking at birth-death and predator-prey models to think outside the box or to crowbar the Logistic model into a different application area.