I was going to just leave the oil spill in the gulf topic alone. Not because it isn’t important, obviously it is. Rather, I wasn’t going to do anything because I didn’t really have anything to add to the topic. After a couple of readers requested it, I think I do have something to add. How exactly do you estimate the amount of oil flowing into the gulf?

What do I have to start with? A video. Here is a video of the undersea oil leak.

Now, I am not the first to estimate the oil flow rate (NPR on Purdue Prof’s estimation and here a commenter make a quick calculation). Like I said, I am really focusing on how to do this estimation.

Looking at the video, there are some issues.

- First, it may be difficult to use the video to determine the speed of the flow. You don’t really see any objects in the flow that can be tracked.
- Even if you could get a good measure of the speed from the outside. The inside of the stream is likely going even faster.
- What about the density? Looking at the video, it doesn’t look like pure oil (but what do I know?)
- The video looks like it has duplicate frames – maybe the frame rate isn’t quite right.

I will proceed anyway. Now that I have a video, I will use my favorite tool Tracker Video Analysis. Very briefly, the video analysis steps are as follows: (here are some other posts that use video analysis)

- Scale the video. For now, I am going to call the width of the stream “1 unit”. This way I can get an estimate of the speed in terms of the width.
- Set the coordinate so that the flow is in 1 dimension (just to make my life easier).
- I don’t need to adjust the coordinates throughout the video since the background is stationary with respect to the camera (win).
- Now, I just just find some points in the stream that I can pick out and mark in several frames.
- Once I get position time data, I can find the speed.

Here is a shot of a fit from Tracker Video:

I fit a total of 6 “features”. From this I get the following speeds (where the U unit is the diameter of the stream):

- 6.74 U/s
- 5.65 U/s
- 5.77 U/s
- 3.27 U/s
- 3.54 U/s
- 3.94 U/s

This gives an average of 4.82 U/s with a standard deviation of 1.42 U/s. Really, to look at uncertainty stuff I should do this more than 6 times. With this in mind, I will just estimate the velocity as:

I increased the flow rate on the assumption that it is faster in the center (a calculation I will do later).

Others have estimated the diameter at about 19 inches. I will use:

One other estimate – the density of oil (or what percent of that flow is oil). Off the wall estimate of 80% +/1 10% of the flow is oil (I am going to call this variable “p”). Now, what I want is a volume flow rate. How many cubic meters of oil flow out every second. So, suppose I look at some time interval, delta t. How much oil would this be? Here is a diagram:

The volume of this cylinder will be how much oil came out in that time interval. Here the height of this cylinder increases at a rate of v_{flow}. This gives a volume of:

And a volume flow rate (I will just call this “f”) of:

Oh! I forgot about the density. If only 80% of this is oil, then it will change the flow rate to:

Now, I could put in my estimates. However, I think it would be useful to have a value for the uncertainty in the volume flow rate. I am not going to talk about uncertainty here (but this is a good example if you are interested). Instead, I am just going to calculate the uncertainty in the flow rate due to the uncertainty in p, d, and v. I get:

If I put in my values, I get:

How does this compare to the According to Steve Wereley from Purdue, he estimates 70,000 barrels a day. BP estimates 5,000 barrels a day. So, I need to convert units to compare. Wikipedia lists an oil barrel as 55 US gallons. So, here is the conversion:

Now to convert this to barrels a day:

Wow. That is way higher than the other estimates. Something must be wrong. Well, let me convert the uncertainty also. This gives:

That is a little better – I guess. Well, there is something else I can do to check this sucker. How fast would the oil slick grow? First, how thick is an oil slick? I don’t really know, for now I will just call this “s”. Let me make a couple of assumptions. First that the oil slick is circular. Second that the oil slick is continuous (no holes). Diagram time:

The key: The time rate of change of the volume of the oil slick should be the same as the flow rate (where does the slick get it’s volume of oil from?). The volume of the oil slick would be (where *r* is the radius of the slick and changes with time)

Now, taking the derivative of this with respect to time (like I said – *s* is a constant):

The rate the radius changes I can call the radial velocity of the slick, *v _{r}* (don’t confuse this with V for volume). Solving for

*v*, I get:

_{r}Check – this has the right units and would get smaller as the oil slick gets bigger. That is good. Ok, let me put in some numbers. How big would the slick be after 3 weeks? Wikipedia lists the thickness of oil with color as 0.0003 millimeters

There is no need to integrate the radial velocity. I can just calculate the volume of oil and see how big of a pancake it would make.

This would give a slick radius of:

This would be about 800 miles. I don’t think it is that big. Of course, if the oil is thicker, it would be a smaller circle. I think I went too far with this.