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When Oil Peaked
By Kenneth S. Deffeyes Hill and Wang
Copyright © 2010 Kenneth S. Deffeyes
All rights reserved.
ISBN: 978-1-4299-8132-3
CHAPTER 1
Bell-Shaped Curves
Hubbert's original analysis, in 1956, was about U.S. oil production. That's not a matter of patriotism; the United States is the most intensely drilled area in the world. He began with two different estimates for the total amount of recoverable oil beneath the United States. The two estimates came from two senior petroleum geologists, Wallace Pratt and Lewis Weeks. Hubbert said that a graph of annual U.S. oil production versus time would have a bell shape, with a fast growth at the beginning, a curved peak in the middle, and a tailing off at the end: in round numbers, about one hundred years from the start to the peak and one hundred years of decline from the peak until the last well runs dry. There were two important constraints:
• The area under the bell-shaped curve had to match the estimate for the total U.S. oil.
• The early part of the curve, 1859 to 1956, had to fit to the historical U.S. oil production.
Hubbert then had two curves, one for each of the estimates, one from Pratt and one from Weeks. The lower estimate, from Pratt, of 150 billion barrels had its peak around 1965, and Weeks's more optimistic estimate of 200 billion barrels peaked around 1970. U.S. oil production did peak in 1970, and Hubbert moved up from being a crank to a folk hero. However, there are two substantial lessons from this episode:
• Hubbert had no way of knowing which estimate was correct. Most analysts have given him credit for a direct hit when using the higher estimate. (Long after the fact, I reworked his data; he had no grounds for rejecting the lower estimate using the production data up to 1956.)
• Critics of Hubbert's method have repeatedly said that all Hubbertian analyses require a prior guesstimate of the total oil. By 1960, the U.S. oil-production history was far enough along for Hubbert to dispense with the guesswork and sweat the total amount of oil out of the data. None of the modern analyses involve guessing the total recoverable oil.
Even with sufficient data, there are choices to be made. There are several different kinds of bell-shaped curves. The best-known bell curves are symmetrical. The history before the peak is a mirror image of that happens after the peak. (Later, I'll explain the arguments for accepting symmetry.) However, some cornucopians have challenged the symmetry arguments, hoping for a long and very gradual decline after the peak. I have a warning: Around 1980, I worked alongside the Princeton statistics department analyzing oil problems. The statisticians found an equation for a bell-shaped curve that was not symmetrical; the later downside could be either more gradual or more abrupt than the upside. They dumped the equation and the oil-production numbers into the computer and out came a disaster. The statistically best-fitting result was an abrupt crash down to zero production soon after the peak.
The Gaussian Versus the Logistic
By far the most famous bell-shaped curve is the Gaussian. It is used so frequently that it is also known as the "normal" curve. Unfortunately, that implies that all other bell-shaped curves are "abnormal." Over my workshop desk is a framed (pre-euro) German ten-mark bill, with a portrait of Carl Friedrich Gauss (1777–1855) and a graph showing the Gaussian bell-shaped curve. Gauss made important discoveries all across the field of mathematics; his Gaussian curve for statistics was a small part of his total contribution.
In his original 1956 paper, M. King Hubbert used a different bell-shaped curve called a "logistic" curve. Typical of Hubbert, he did not explain his reasons until 1982, when he was seventy-eight years old. Hubbert was a difficult character. Here's an example: When Hubbert and I were both working at the Shell research lab in Houston, a graduate student from Cal Tech gave a talk because he was being considered for a job at the lab. The student's doctoral thesis was a microscopic examination of a sample of rock about 6 inches square. He put a photograph on the screen showing the rock face, about 6 feet high, with the location of the sample square marked near the top of the outcrop. Hubbert asked a question about something lower on the photograph and the student said that it was outside his thesis area. Hubbert was furious. The student did not get a job offer from Shell. Further, Hubbert contacted the chairman of the Cal Tech geology department with the suggestion that the student not be allowed to graduate. The student eventually did graduate, but all of us learned a lesson about dealing with Hubbert.
The logistic curve that Hubbert chose is, at first glance, similar to the Gaussian. The rounded top of the Gaussian is wider than the logistic; in compensation, the Gaussian has narrower tails on the far left and right. For me, the choice between the two came up early. Around 1962, I considered leaving the Shell research lab and taking a university faculty appointment. Should I bet my career on changing my way to earn a living? A major consideration was Hubbert's 1956 prediction about U.S. oil production. If Hubbert was right, by the year 2000, when I was scheduled to retire, the U.S. oil industry would shrink to about half of its peak size. I didn't want to live in Algeria or Nigeria; I'm an American.
As is typical of scientists, I keep two mental lists. One list is scientific conclusions that I have read about and seemed reasonable; they are filed under "probably okay." My other list is shorter; it is problems that I have worked with on my own terms. Before deciding to switch from a research lab to teaching, I needed to work directly from the raw data. At the Shell lab, all I had to do was walk down one floor and ask Hubbert's assistant for a photocopy of the raw U.S. oil-production data. Hubbert had used the logistic curve; I decided to try the Gaussian. I got essentially the same results as Hubbert: a U.S. peak around 1970 and a reduction to half of the peak size when I retired. A year later, I was a professor at the University of Minnesota.
In 2000, happily retired, I returned once again to the logistic-versus-Gaussian outlook on oil production. By a narrow margin, the Gaussian made a better fit to the historical data. However, most of the difference between the Gaussian and the logistic was the narrower tails of the Gaussian. Because the U.S. oil industry grew very rapidly from 1859 to 1910, the Gaussian made a slightly better fit. But are we going to bet the ranch based on the happenings during the early days of oil? There is a deeper issue here. Is there some reason why Hubbert preferred the logistic curve? Does the Gaussian have a hidden flaw? Or, as some critics believe, are both of the curves somewhat off?
The Logistic Versus the Gaussian
The logistic curve has a logical story attached to it. At least for oil production, the Gaussian has lacked a logical justification. At the end of this section, I'll remedy that problem by introducing my homemade story.
We can begin by plotting the U.S. oil production on two similar, but not identical, graphs. The upper graph is contrived to plot a straight line if the production history follows a logistic curve. The lower graph gives a straight line for a Gaussian history.
On these graphs, P is the annual crude-oil production and Q is the cumulative production, beginning with the Drake well of 1859. (In this racket, Q stands for cumulative.) P' is the increase (or decrease) in production from one year to the next. (In nerdish, P' is the time derivative dP/dt.) The points a and b on both graphs indicate where the best-fitting straight line hits the vertical and the horizontal axes.
If we take the equation of a straight line (Y = a + bx) and substitute the symbols from each graph, we get these equations:
P = a (1 - Q/b) Q (Hubbert, logistic)
P' = a (1 - t/b) P (Gaussian)
What jumps off the page is inside the parentheses in the Hubbert logistic equation: 1 - Q/b, which is the fraction of the total oil that remains to be produced. If, instead of producing, you are exploring for oil, 1 Q/b is the fraction of the oil that remains undiscovered. Early on, when most of the oil hasn't been found, exploration is really easy. Toward the end, when most of the oil has been discovered, exploration is not very rewarding. A fellow who sees a friend fishing at the edge of a pond shouts, "How's the fishing?" The reply comes back: "The fishing is great. The catching stinks." That's the present-day story about oil exploration, the catching stinks. A few years ago, Chevron ran a series of ads saying that we are burning two barrels of oil for every new barrel we find. Today, we may be burning five barrels for each newly discovered barrel.
The logistic equation was developed in 1838 by Pierre-François Verhulst to describe population growth. Imagine a freshly bulldozed building lot that is left untouched for a while. A few weeds grow first. Eventually plants cover the entire lot and the population stops increasing. The logistic equation says the rate of plant growth depends on the fraction of the space that is still open for colonization. It was this behavior that Hubbert utilized to say that the ease of finding oil depends on the fraction of the total oil that remains undiscovered.
In 1967, the meaning of the logistic equation was refined by Robert MacArthur and E. O. Wilson. They pointed out that the two constants in the equation, a and b, were related to two different strategies. Weeds (and mosquitoes during the brief Arctic summer) are dominated by a, although they used the letter k in their equation. Natural selection for large a leads to reproductive strategies involving rapid maturity, a large number of offspring, and dispersal strategies for scattering the offspring. (After a hike, you find weed seeds stuck in your socks.) At the other extreme, selection for b, which MacArthur and Wilson called R, happens in mature and stable environments. Elephants are the extreme example: large body size, few offspring, the rarity of twins, long life, long gestation period (22 months), and a major investment in parental care. In the case of elephants, "parental care" involves the mother and a cluster of sisters and aunts.
In the oil business, there is a rough analogy. When a new area starts being explored, there are boomtowns and drilling rigs all over. In a mature area, there may be a few ultradeep wells being drilled.
Hubbert's logistic model has a nonbiological peculiarity: No barrel of oil ever dies. Although we may burn up the oil while driving the kids to Yellowstone, the oil is listed permanently in the cumulative roster of produced oil. To Verhulst, as well as to MacArthur and Wilson, population growth was an excess of births over deaths.
On the logistic curve, the initial growth stage is the mirror image of the later decay period. There are two ways to reach that conclusion; you are welcome to use either or both:
• If the logistic graph shows a straight line, then the bell-shaped logistic curve will be mirror-image symmetrical.
• If oil-finding success depends exclusively on the fraction of currently undiscovered oil, then the bell-shaped logistic curve will be mirror-image symmetrical.
It is important to notice that either of these conditions lock in mirror symmetry. Some of Hubbert's critics have claimed that mirror symmetry has to be proved by additional observations.
When we turn to the Gaussian, the first condition will work. If the Gaussian linear production graph shows a straight line, the Gaussian bell-shaped curve will be symmetrical. However, for the second logistic argument (discovery depends on the undiscovered fraction) there is no equivalent Gaussian story. Why is the story important? Philosophers may list it somewhere under "epistemology," but any used-car dealer knows it as a "sales pitch." It makes us feel better if there is an equivalent of Kipling's Just So Stories.
Here's an interesting story, but one that doesn't apply to petroleum: The vertical axis of the Gaussian graph is P'/P. Since P' is the annual change in production, then P'/P is an interest rate. If you have a savings account yielding 4 percent, the balance in your bank account gets multiplied by 1.04 each year. Now let's look at the situation where the interest rate goes down linearly with time. At first the interest rate is high and your account grows rapidly. Gradually and steadily, the interest rate goes down and your account grows more slowly. Then when the interest rate goes to zero, your bank balance ceases to grow. I know, I know, by then you would close your account and move the money elsewhere, but we're doing a thought experiment. What if you left the account alone as the interest rate moved into negative territory? Your account would shrink slowly at first, and then more rapidly. Eventually, the balance in your account would become very small. Here's the surprise: The balance in your account over time would draw a perfect bell-shaped Gaussian curve.
Can we use the savings account analogy to describe the oil business? If God were upstairs, keeping the books and doling out oil discoveries at a linearly decreasing annual rate, then we would have a story. It doesn't seem to me that God would want to act like an enlarged version of Bank of America. We'll have to look elsewhere.
Quite separate from the bank account analogy, there is another way to generate a Gaussian curve. It's called the central limit theorem, which sounds impressive; you could try to work it into your party conversation. When some guy is talking loudly about housing prices, ask him whether the central limit theorem explains it. The central limit theorem is easiest to understand through an analogy. Let's say that we have a rifleman shooting at a target. The bullet can hit the target to the left or right of the center for a number of reasons:
• There is a gusty wind blowing from left to right.
• The rifleman can't hold the rifle absolutely steady.
• Some of the bullets are a bit heavier, or lighter, than the others.
• The sights on the rifle are a bit out of adjustment.
• Some of the bullets are backed by a little more, or less, gun-powder.
Each of these variations needs to be independent of the others. There is no Murphy's Law saying that the rifleman always selects a lighter bullet on windy days. The central limit theorem says that adding up a bunch of these independent variations gives rise to a bell-shaped Gaussian distribution of bullet holes to the right and left of the center.
I'll put on my Rudyard Kipling hat and try to develop a Gaussian story for oil exploration. Be warned: Kipling and I made up these stories as we went along. There are obvious clues and subtle clues for oil discovery. If there are a number of clues, and if they are independent of one another, the central limit theorem would then get us a Gaussian history of oil discovery. We need at least five or six independent attributes, but the more the merrier. There are eight in the list below. The idea being tested says that the typical, average, plain-vanilla oil field would have been discovered in 1963. However, each of the eight independent conditions could advance or delay an oil field's discovery before or after 1963. These are not on-off toggle switches; there is every gradation from obvious properties that lead to early discovery, to subtle hints that don't drastically change the discovery date, to situations that substantially delay finding the oil field. In the extreme, the time leads or time lags can be as much as one hundred years.
• Surface Indications of Oil or Gas Early discovery: oil seeps, tar pits, or natural-gas leaks at the surface, as in the Bible (the bush that was burning and not consumed, a pillar of cloud by day and of fire by night, and the fiery furnace in the Book of Daniel). Late discovery: oil fields beneath nonleaky subsurface salt layers, as in offshore Brazil.
(Continues...)
Excerpted from When Oil Peaked by Kenneth S. Deffeyes. Copyright © 2010 Kenneth S. Deffeyes. Excerpted by permission of Hill and Wang.
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