The Frustration of Flow Rate

Since the Deepwater Horizon explosion on April 20th, there has been a lot of speculation about the amount of oil that is actually flowing into the Gulf. The estimates have varied considerably. Yesterday, the US government announced that the amount of oil is likely to be double what was originally estimated, approximately 25,000 to 30,000 barrels per day. But we still do not know for certain, and this is frustrating to many who feel that with the technology we have and the amount of time this has been going on, we should be able to come up with definitive numbers.

Leaving aside speculation about whether or not BP is being upfront about their data, let's take a moment to look at how flow rate is generally calculated and why some of these estimates have been drastically different.

Whether you're a plumber or an oil driller, there is a general formula used to understand the flow rate of a liquid leaving a pipe:

Velocity  X  Pipe Area = Flow Rate

Velocity is a vector quantity and is measured by the speed of an object over time. When we talk about the speed/time, it refers to constant velocity -- both the size of the object and the direction remain the same. Because a pipe is a fixed object, the direction of the oil within it is obviously not going to change. But when it leaves the pipe, it encounters the pressure of water one mile beneath the surface (approximately 500 times greater than at sea level) and other particulates, which can deflect the direction of the flow. How scientists are calculating velocity thus far has been a bit murky. Measuring it within the pipe will generate one number; outside the pipe, another.

Now, the pipe area. This is one number that should actually be pretty straight-forward. The area of a cylinder is the perimeter of the opening multiplied by the height. In this case, it's the perimeter of the pipe opening multiplied times the length of the pipe. One of the reasons we include pipe area in calculating flow rate is because the friction of the inside of the pipe applies force on the liquid.The longer the pipe or shorter the diameter, the greater the force applied. Cutting the pipe increased the perimeter because it removed the valve that was slowing the flow down -- so the flow rate has actually increased since this was done.

In the case of the Gulf, we want to know more than the flow rate -- we want to know the mass flow rate. We want to know how much oil is coming into the area, not just how quickly.

In order to calculate mass flow rate, we need to incorporate density into the equation:

Mass Flow Rate = Density X  Velocity X Pipe Area

Density is measured by dividing the mass of an object by its volume (for more on density, click here). When you have a fixed object, calculating density is rather straight-forward: find its mass, measure its volume, do the math, et viola. Unfortunately, the crude oil coming out of the sea bed in the Gulf is not uniform. It is composed of solid particulate, gas and liquid. And the concentrations of these will vary over time. Add to the mix that BP is injecting dispersants into the pipe, and density becomes even more difficult to calculate. 

Clearly, the fact that this leak is occurring in three places 5,000 feet below the surface of the ocean means that there are more factors at play than simply multiplying a few numbers together. It becomes a matter of getting numbers that accurately reflect the averages for density and velocity, which has been quite difficult considering the locations of the three leaks.

With luck, the leaks will be stopped soon enough so that we no longer have to worry about whose numbers are correct.

Image courtesy of BP/AP