# Modelling deep-sea oil spills

(by Rachel Philip.)

In a deep-sea oil spill, a broken pipe near the seabed (containing a mixture of oil and water) can result in the release of a high-speed jet into the surrounding ocean. The jet’s shearing motion will typically cause the oil to break up into smaller droplets, which are then more easily dispersed and decomposed by sea microbes.

Our aim is to model the distribution and size of the oil droplets. This involves coupling the motion of the large-scale jet (hundreds of metres in height) with the breakup of small droplets (millimetres in diameter). A key result, which enables us to connect these disparate scales, is to estimate the energy controlling droplet

breakup as

E=E_0/(d_0 + 4 C z)^4

where E_0 is a known constant, d_0 is the pipe diameter, C is an experimentally determined parameter, z is the jet height and (d_0 + 4 C z) is the jet width.

As the oil stream emerges from the pipe, it mixes with surrounding seawater, absorbs it, and expands in width. We make the modelling assumption that, at any height, the speed of seawater flowing into the jet, U, is proportional to its upward velocity, W, with constant of proportionality C. Thus U = C W. How can C be determined in real-life situations? Modelling the jet, we find its width is linear, with a slope proportional to C. Therefore, simply measuring the jet’s angle determines its width and the parameter C. How does this relate to droplet breakup? Using well-known models, we deduce the energy controlling droplet breakup is inversely proportional to the fourth power of the jet width, which yields the formula above for E.

What does this mean in an oil spill? Higher up the jet, when the width is greater, the energy is smaller and droplets don’t have enough energy to break up. The natural clean-up process becomes ineffective. Therefore, we must add other mechanisms to the jet to stimulate droplet breakup; our next step is to explore how these response options can be best applied.

Rachel Philip is a member of Cohort One of the InfoMM CDT at the Mathematical Institute, University of Oxford.