How Much Oil's Spilling? It's Not Rocket Science

May 23, 2010 — -- How much oil has leaked into the Gulf of Mexico? Many, including BP, have made estimates and remarked on the difficulty of determining the answer.

As Steve Wereley, an engineering professor at Purdue University, has shown, however, and as many others would have shown had pictures of the leak been released earlier, an approximate estimate is quite easy to come by and indicates a vastly greater oil spill than BP has admitted.

Wereley and a few other scientists, who have come to the same general conclusions, have performed a public service.

Basically, the method for determining oil spillage boils down to common sense and high school (or even middle school) geometry, specifically the formula for the volume of a cylinder.

Let's first assume that the oil is spilling out through the pipe and maintaining its cylindrical shape. It's not of, of course. The oil is an ever-moving, shapeless blob, but the volume of this blob of oil is the same as the volume of a long cylinder of oil extending out of the pipe, and this latter volume is much easier to calculate.

So what is the ever-growing volume of this cylinder? Geometry tells us the volume of a cylinder is the circular cross-sectional area of the cylinder (BP certainly knows this because it knows the radius of the pipe is 10.5 inches) multiplied by the height (or because it's horizontal here, the length), h.

The Formula

And what is its height (or length), h? Again elementary math comes to the rescue.

The length of the imagined lengthening cylinder of oil equals the rate of flow of the oil multiplied by the time it's been flowing. That is, h = v*t.

We certainly know the time. The break occurred on April 20.

The calculation is then simply a matter of plugging the right values into the formula for the volume of a cylinder: V=pi*r^2*v*t, where, once again, V is the volume of a cylinder, t is the time the oil has been leaking, and v is the flow rate of the oil through the cylinder.

The flow rate is not known exactly, so what Wereley did was to turn the poor video snippet (Wereley suggested that BP released what they thought was the least helpful clip) into a series of still pictures. Then he calculated the velocity of particles in the flow by employing software that analyzed the still pictures frame by frame and pixel by pixel to come up with an estimated flow rate for the oil of 2 feet per second.

The Bottom Line

Once one converts all the units and multiplies these numbers, the calculations show that a bit more than 70,000 barrels of oil have been leaking out of the pipe daily. He's since upped his estimate by another 20,000 barrels a day because of other smaller leaks.

He admitted his is a rough estimate for three reasons. One, as he wrote me, is that the flow along the visible edge of the flow might differ a bit from the flow in the middle of the pipe. Another is the uncertain amount of (very compressed) natural gas mixed in with the oil, and the third is the poor quality of the video that was only recently and reluctantly released on May 12.

Wereley said that the leak may be 20 percent more or less than his best estimate, but it's still far, far above the 5,000 barrels per day that BP kept proclaiming until very recently -- about 20 times as great.

Troubling Questions

Questions remain why it's taken this long for evidence of a much bigger spill to come to light. The calculations above are elementary and could have been undertaken as soon as BP obtained pictures of the leak.

Perhaps the executives of BP, Halliburton and Transocean are innocent of all technical details and maybe even of elementary geometry. (It would have been interesting if at the Senate hearing where each of them blamed the other two they were asked the formula for a cylinder and its relevance to the size of the spill.)

But even if we excuse the executives' lack of technical knowledge, it's very hard to believe that the smart engineers who work for them didn't have a good feel for the extent of the leak early on. If they did, why didn't they speak up?

Had the video and pictures been obtained when they first became available, they would have alerted us much earlier to the truly catastrophic nature of the spill, perhaps induced an even greater urgency to stopping it, and allowed us to plan more realistically for the environmental consequences. Sticking to their estimate of 5,000 barrels per day and talking about surveillance of the ocean surface from the sky seemed to be counterproductive at the very least. Did BP and others think they could rely forever on an innumerate public to take their word for the size of the spill?

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as, most recently, "Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up."He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.