An Oil Production Model from Roger Bentley

Posted by Chris Vernon on August 8, 2008 - 10:46am in The Oil Drum: Europe
Topic: Supply/Production
Tags: oil production, oil production modeling [list all tags]

This is a guest article by Dudley Stark, Reader in Mathematics and Probability in the School of Mathematical Sciences, Queen Mary, University of London.

Bentley introduced the following model of oil production on page 204 of Global oil & gas depletion:an overview, and it is dicussed in the book The Last Oil Shock by David Strahan. This posting is meant to explain his model and some results I obtained for it. Consider the following oil production curve:

It rises quickly to it's peak at time t=1 and decreases slowly until no oil is produced at time t=6. The idea is that the natural pressure of the oil field causes rapid production initially, after which decline is more gradual. Before and after the peak the curve is linear, so it looks like a triangle.

Suppose the next oil field looks the same as the first one, but oil production begins one unit of time later and the total amount of oil produced is only 75% of the oil in the first field. It looks like this:

Adding the production of the two oil fields together gives this production curve:

If you do this eight times, each time shifting the start of production by one time unit from the previous oil field and also making the amount of oil produced 75% of the previous oil field, you get a curve like this: It is starting to look like a plausible oil production curve. Note, however, that it is not too realistic because, for one thing, the curve is linear in between integers.

In my paper Peak Production in an Oil Depletion Model with Triangle Field Profiles, accepted to appear in Journal of Interdisciplinary Mathematics, I analyze Bentley's model. The production curve of the ith oil field is assumed to look like this:

Note that the amount of time from the start of production until peak is delta for each curve and the amount of time from peak to the end of production is lambda. We assume that delta is smaller than lambda. The ith field begins production at time (i-1)delta. The amount of production at peak is determined by a parameter m_i which also determines the total amount of oil produced by the ith field.

I show that in this model, assuming that the m_i are decreasing, the resulting total oil production curve is concave on the interval [0,lambda] and is decreasing on the interval [lambda,infinity). The oil production curve therefore takes it's maximum on the interval [0,lambda]. If the m_i decrease geometrically, then in addition the oil production curve is convex on the interval [lambda,infinity).

I also show that if delta=alpha/n for a constant alpha and a parameter n and if the m_i=f(i/n) for a decreasing function f, then as n goes to infinity the oil production curve converges to a smooth (meaning differentiable) curve which is again concave on [0,lambda] and decreasing on [lambda,infinity). Here is an example of a smooth curve:

These curves are concave on the interval [0,lambda] and attain their unique peaks there. If f is a negative exponential function (which is the continuous analogue of a decreasing geoometric), then the oil production curves are convex on [lambda,infinity). I show that for these limiting curves it is not possible to have 1/2 of the total oil produced at the time of peak production and in fact they have at most about 35% of the total oil produced at the time of peak production.

A related paper is "Oil Production: A probabilistic model of the Hubbert curve" by Bertrand Michel, who spoke at the ASPO-V conference, to appear in Mathematical Geosciences. In his model the field sizes are chosen uniformly at random from a truncated Pareto distribution. Conditional on the size of the field, the starting time for oil production is Gamma distributed with parameters depending on the size. The field profile shapes are constant; in his application they are given by splines. So Michel's model is similar to Bentley's but harder to analyze and probably more realistic. He uses it to model North Sea oil production (see Figure 11 on page 18). The fit is pretty good except for a decrease in production caused by the "Piper disaster", which couldn't be anticipated by the model.

I perform a similar analysis for Bentley's model of gas production, in which the ith field production profile has a trapezoidal shape:

The trapezoidal shape is suppposed to be more realistic for gas profiles because gas production is constrained by the pipes which transport it.

Conclusion

Bentley's model generates plausible looking asymmetric production curves in which less than 50% of the total oil has been produced at the time of peak production. The fact that they are asymmetric is not surprising: see the paper "Testing Hubbert " in which it is found that the asymmetric exponential curve is most often the best fit when six different types of curves are fitted to real data.

I'd also like to point out that, though I think papers like this one are important because oil production is important, no one else seems to be doing this kind research in the mathematics of oil production and therefore many math journals wouldn't consider publishing a paper like this. The reason is that they only want to publish papers that reference other published papers in respected journals and this paper only references oil industry papers and peak oil books, as well as another paper written by myself. I was pleased to learn about Michel's paper, which is published in Mathematical Geology, but references a lot of oil industry papers and is like an applied statistics paper.

Scoring the 'Green' in Green Energy

By Nick Hodge
Friday, August 8th, 2008

Has green lost its luster?

Of course not. Yet that seems to be the attitude of more than a few.

Sure, solar stocks have been battered over the past few months. But didn't the Dow (Index: DJI) go from over 13,000 to below 11,000 in the same time? Indeed, it did.

And hasn't Exxon Mobile (NYSE: XOM) gone from nearly $95 to to about $75 in the same time? Indeed, it has.

Even the incessantly-talked-about Transocean (NYSE: RIG) is down 15%. . . in just the past two months.

So for renewables to be dismissed as bad investments by bulls of other energy sectors is not only wrong, it's quite hypocritical.

Indeed, with the world's largest oil fields being depleted—some by as much as 15% per year—and natural gas facing a similar long-term plight, we're going to need all the energy we can get. And there's plenty of money in all of it.

But you must realize, new oil discoveries, and even arctic and offshore drilling, are certainly no catholic cure. In fact, the amount of oil they're providing—and could potentially provide—is absolutely not enough to offset rising demand and oil field depletion, not to mention that oil is increasingly more expensive to extract. This is an often overlooked aspect of new oil finds.

Yes, there's money to be made from the remaining oil and the companies that refine the ever more heavy and sour crude. And I'm not against making that money.

But to think that there will be no serious energy transition to include a large share of renewables coupled with efficiency is naïve at best, and moronic at worst.

That said, let's look at the billions of dollars that continue to pour into the green sector even during this progressively bearish market.

Renewable Energy Markets: Investment Magnet

According to New Energy Finance, one of the world's most highly respected renewables analytics firms, private equity and venture capital deals in cleantech surged to a record $5.8 billion in the second quarter.

If the technology had no future, would it be bringing in record investment dollars at the most foundational levels? Hardly. Surely you can remember the internet naysayers of the 1990s. And look how that turned out. . . you're reading this on the internet, and it's marvelous.

These green technologies that are being invested in today on a very nascent level are going to be the energy juggernauts of the future. From my perch, it's inevitable.

But back to that second quarter $5.8 billion investment, which, by the way, is more than double the $2.6 billion invested in the first quarter.

You know where that money went: wind, solar, and second generation biofuels.

And speaking of oil companies and second generation biofuels, check out where BP (NYSE: BP) thinks transportation fuels are headed. On Wednesday, they announced a $90 million partnership with Verenium (NASDAQ: VRNM), to speed the development of cellulosic ethanol.

Verenium shot up over 75% on the news. . . in two days.

More affirmation of the ethanol market came this week when the EPA decided to uphold the increased Renewable Fuel Standard (RFS) signed into law last December, which increases the amount of renewable transportation fuels used annually to 36 billion gallons by 2020—16 of which must be cellulosic ethanol.

Texas Governor Rick Perry had requested that the EPA cut the RFS mandate by 50%. He lost.

Back to green energy investment. . .

Beyond second generation biofuels, wind and utility-scale solar continue to be hotbeds of investment. According to a recent Wall Street Journal Environmental Capital posting:

Wind energy in the U.S. is still going strong, despite the lingering uncertainty over congressional extension of tax credits for clean energy. The American Wind Energy Association today said second-quarter wind-power installations fell slightly from the first quarter, to 1,194 megawatts. But the wind lobby said 2008 should be another record year overall, with more than 7,500 megawatts installed, provided Congress finally renews the tax credits.

Power companies in the U.S. and Europe are increasingly looking to new types of solar power for big clean-energy installations, rather than do-it-yourself rooftop arrays. Biofuels have gotten a bad rap, but the next generation-made from stuff you can't eat like waste wood and algae-is drawing multi-million dollar investments.

We now know that pre-public cleantech companies are still attracting gobs of investment.

So what's the problem with publicly traded companies?

Publicly Traded Renewable Energy Companies

Part of the problem stems from the fact that renewable energy companies are finding increased resistance to listing publicly. This is do to instability in the overall equity markets fueled by a credit crunch, a mortgage mess, and a weak dollar.

That precarious situation has shelved several cleantech IPOs, making it harder for early-stage investors to recoup their investments.

Even though companies raised $5.2 billion in public markets during the second quarter, over half of that came from just one listing: Portugal's EDP Renovaveis.

Nonetheless, that $5.2 billion was monstrous jump over the dejected $1 billion during the first quarter—a sign that conditions may be starting to improve.

Until conditions completely improve for new entrants, one must embrace the current culture of the market.

There are plenty of bargains to be had in these troubled waters. Great companies, across all renewable sectors, have been beaten down as a function of broader market conditions.

Chinese solar companies, for example, are painfully oversold. And their rebound to levels of old will certainly provide gains congruous with the amount of of their recent losses—about 46%, on average.

I'm talking about companies here like JA Solar (NASDAQ: JASO), Solarfun (NASDAQ: SOLF) and Yingli (NYSE: YGE).

The same holds true for other renewable sectors.

Indeed, the bull market in energy is long from over. In fact, we're just in the first inning. The downturn in energy stocks due to negative external conditions is only a bump along the way.

As long as demand continues to rise exponentially—and we all know it will—then the energy, and particularly the renewable energy, bull market is destined to press on.

Call it like you see it,

Nick