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Monday, January 09, 2006

Would You Believe?

In my previous post, I made an offhand reference to how the initial increase of crude oil discoveries might follow a certain well-understood pattern:
I then put a "gold rush" mentality on the frequency of discovery strikes; this essentially started with 8 strikes per year and rising to 280 strikes per year at the peak.
You can either think of the discovery growth as a steady year-to-year increase or as an accelerating increase. The latter refers to a quadratic growth law commonly found in many situations where increasing numbers of resources get applied to a problem over time. Much like gold spawns a fevered rush of interest which seems to accelerate through a parabolic boom before finally busting, I offer that oil strikes might follow the same swarming pattern.

Consider a very recent example of quadratic growth which comes from the wonderful world of the Wiki. At least one team of academics has noticed that the rate of increase of Wikipedia words follows a quadratic growth law. I extracted the following relationship from the Wikipedia statistics table:

Note that when we take the square root of the growth, it tracks a straight line. Also remember that quadratic growth does not equate to exponential growth. Exponential growth occurs when the rate of increase of a quantity proportionally scales to the amount of quantity at that specific time. This amounts to a much different swarming activity.
Quadratic Growth : d2Q(t)/dt2 = k
Exponential Growth : dQ(t)/dt = aQ(t)

In the context of crude oil discovery, I curiously haven't seen much written about the number of discoveries over time. From my last post you can clearly see how noise obscures much of the trend (which the oil shock model effectively filters out -- more on that in a future post).

However, over the weekend, Staniford at the TOD blog brought up the topic of USA production curves dating back to the first discovery made in 1859. He created some quite amazing fits to the entire USA oil production profile using a gaussian function, which looks like an inverted parabola on a semi-log plot:

That particular parabola, a quadratic in fact, I have little interest in. But, I do get excited about encountering some new data to pound on.

I used the USA lower-48 discovery table as a reference point and then eyeballed a quadratic growth factor to generate an accelerating rate of oil discoveries.

The discovery peak hits about 1930 and then decreases after that point -- basically a boom and bust cycle in full view.

With the basic assumption that the size of strikes remains independent over time, I took a shot at applying the oil shock model to the artificially constructed quadratic discovery data. I used about the same extraction rate that I previously used for the lower-48 model increasing it slighty to 0.08. I modified the fallow, construction, and maturation constants a bit from 8 years to 12 years. I did not apply any oil shocks, because they would not show up on this kind of scale in any case. I just wanted to see if I could match that gaussian characteristic that Staniford observed.

I will let you pick the winner. Note the EIA data points from 1859 and 1860 that Staniford omitted because he considered them "outliers".

As an assumption I had to give the model an initial discrete stimulus to promote the discovery value above zero in 1859 (alternatively, I could have backed up the starting point a bit). This transient has little effect other than to keep the numbers on the graph. However, you can see that even the real data seems to plunge toward zero at the discovery -- something that the gaussian curve cannot handle. (Some people treat outliers as garbage, I prefer to treat them with a modicum of respect :-) Moreover, as I pointed out on the TOD comments, the extrapolation of the gaussian would show 100 of barrels of oil in production many years before somebody officially discovered the oil! In other words, the gaussian curve does not consider causality correctly.

Interestingly, the initial point that the EIA gives, 5000 barrels, has a lengthy historical narrative.
On a brilliant Saturday afternoon, August 27, at 69 feet down, the drill suddenly dropped six inches into a crevice. Uncle Billy fished it out, wiped it off carefully, and knocked off for the Sabbath. But Monday seemed a long way off, and on Sunday Smith was back at the well, peering down the pipe, wondering if he really saw something glistening on the surface below. He grabbed a leftover end of pipe, plugged it up like a dipper, and thrust it down on a stick. It came back up filled to the brim with oil. A wild shout brought several mill hands running. Young Sammy raced off to town to notify Colonel Drake.

The whole village was buzzing; even townsmen who still couldn't imagine what might come of the find were eager to see it. A man from the nearby town of Franklin, on the Allegheny River, who visited Drake's well the following day, joining the eager crowds streaming in on every road in wagons, on horseback, and on foot, reported, "It comes out a flowing dark grease with a heavy white froth."

By then, the few pine barrels Drake had provided were already full. Drake took Margaret Smith's washtub from the engine-house shanty (she complained later she never could get it clean after that), then commandeered old whiskey barrels and sperm oil containers. And still Uncle Billy kept pumping and the oil kept coming; so did the crowds.
The story also gives some older historical background. I did not know this, but arguably, we shouldn't even attach the discovery of oil in Titusville to any individual person. Many settlers had seen the residue in the oil over the years. So yes, in fact, it likely showed a fallow period, followed by Drake's construction period, and finally a maturation period.

Funny how things scale.


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