Issue 034 - Oil spill - Area and thickness scaling
How much oil does it take to create a 2,000 km2 slick?
A grounded tanker off Oman was carrying about 800,000 barrels of crude oil. Reuters reported that independent satellite analysis estimated the resulting slick at more than 2,000 km2, much larger than Oman's earlier official estimate.
The problem
Estimate the average thickness of the oil layer if the tanker's entire 800,000-barrel cargo were spread uniformly across a 2,000 km2 slick.
Then work backward using your own estimate of how thick a real ocean oil slick might be: what fraction of the tanker cargo would need to have leaked, and how much oil would that represent by mass or volume?
Does the enormous geographic area of an oil spill imply an equally enormous volume of spilled oil, or can a relatively small fraction of a tanker's cargo spread across a surprisingly large area?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I know that oil can form extremely thin films on the order of about 10^-6 m in thickness, but that would appear transparent or be barely recognizable. I expected that in order for the slick to be observable by satellite analysis, it would need to be much thicker, so I set a lower boundary around 10^-4 m, about the thickness of a hair, and thought it might very likely be thicker than that.
Starting with that lower bound:
slick area ~= 2,000 km2
~= 2 x 10^9 m2
assumed thickness ~= 1 x 10^-4 m
oil volume ~= 2 x 10^9 x 1 x 10^-4
~= 2 x 10^5 m3
~= 2 x 10^8 LI believed a barrel of oil contains about 80 gallons, and at 3.8 L/gallon, that is about 300 L/barrel. Dividing into the volume estimate above:
barrels spilled ~= 2 x 10^8 L / 300 L/barrel
~= 7 x 10^5 barrelsThat would mean most of the tanker's volume spilled.
However, if it is possible to spot the slick by satellite with a minimum thickness of 10^-6 m, then:
oil volume ~= 2 x 10^9 m2 x 1 x 10^-6 m
~= 2 x 10^3 m3
~= 2 x 10^6 L
barrels spilled ~= 2 x 10^6 L / 300 L/barrel
~= 7 x 10^3 barrelsThat would be only about 0.9% of what was being transported. Ultimately, I thought the answer would depend on the thickness necessary to make the slick visible, and my intuition was that it would need to be much thicker than the theoretical minimum.
Calibration Score
Matt's Calibration Score: 50 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 0/30. Observable slick thickness and barrel-volume pegs were the major misses.
Model: 30/30. Area times thickness is the right model.
Math: 0/10. The barrel-size error substantially changed the spill-volume interpretation.
Result: 20/30. The final range still contained the right order of magnitude because the thin-film case was considered.
Grounding facts
The full-cargo thickness is useful because it gives an upper-bound intuition:
800,000 barrels spread over 2,000 km2 ~= 64 micrometers thickA 1-micrometer average film over the same area is only about 1/64 of the cargo. That is the whole lesson: a thin film turns volume into area very efficiently.
After checking sources
Check and recalibrate
The first correction is the oil-barrel peg. A petroleum barrel is 42 U.S. gallons, about 159 liters, or 0.159 m3.
cargo volume ~= 8 x 10^5 barrels x 0.159 m3/barrel
~= 1.27 x 10^5 m3The slick area is:
area ~= 2,000 km2
~= 2,000 x 10^6 m2
~= 2 x 10^9 m2If the entire cargo were spread uniformly over that area:
average thickness ~= volume / area
~= 1.27 x 10^5 m3 / 2 x 10^9 m2
~= 6.4 x 10^-5 mThat is about 64 micrometers, or 0.064 millimeters. Even the whole cargo, spread over 2,000 km2, would be thinner than a sheet of paper on average.
Now reverse the calculation. NOAA notes that oil changes appearance as it spreads from dark oil to iridescent or silver sheen. NOAA also explains that synthetic aperture radar can detect slicks because oil dampens tiny wind-driven waves, reducing radar backscatter. That means satellite-detected slick area does not require optically thick oil everywhere.
If the broad slick averages only 1 micrometer thick:
volume ~= 2 x 10^9 m2 x 1 x 10^-6 m
~= 2 x 10^3 m3
barrels ~= 2 x 10^3 m3 / 0.159 m3/barrel
~= 1.3 x 10^4 barrels
fraction of cargo ~= 1.3 x 10^4 / 8 x 10^5
~= 1.6%If the mapped slick averages 5 micrometers thick:
barrels ~= 5 x 1.3 x 10^4
~= 6.3 x 10^4 barrels
fraction ~= 6.3 x 10^4 / 8 x 10^5
~= 8%Using the IMO-reported cargo mass of roughly 130,000 metric tons, the 1-micrometer case corresponds to about:
mass leaked ~= 0.016 x 1.3 x 10^5 metric tons
~= 2 x 10^3 metric tonsSo a 2,000 km2 slick can plausibly be produced by a few thousand to perhaps tens of thousands of barrels, depending on average thickness. Most of the cargo does not necessarily need to escape in order to create a huge geographic footprint and serious local environmental impact.
Post-check reflection
Matt's reflection
Looks like my intuitions were not quite right on this one. The main way they failed me was the assumed slick thickness required to be observable by satellite. I was not considering that satellite detection can occur by radar observation of wave-dampening, not only optical detection of the oil itself. By this inference, even thicknesses as small as 10^-6 m are detectable.
The next major intuition fail was the size of a barrel: 42 gallons is about half what I thought I remembered. That is a good peg to keep track of in the future.
Ultimately my calculations did land me in the right order of magnitude, and if I had a more appropriate intuition for slick thickness I would have correctly answered that less than a few percent of the crude oil cargo might need to have spilled to produce a slick of the described size.
I think this is a really important example. Before doing the math, I might have considered this to be a much more severe incident in terms of tanker cargo loss than it may actually be. The ecological stakes can still be serious, especially near sensitive coastlines and marine reserves, but the area alone does not prove that most of the ship's cargo was lost. That gives me a different framing of the news item.
Recommended memory peg
For oil-slick Fermi estimates, remember 1 oil barrel = 42 gallons ~= 159 L ~= 0.159 m3, 1 km2 = 10^6 m2, and thickness = volume / area. A 1-micrometer film over 1 km2 is about 1 m3 of oil.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.