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Using Differential Equations to Model the Spread of an Oil Slick

Applying mathematical techniques to an environmental problem

Data for Change

Image by Roberto on Unsplash
Image by Roberto on Unsplash

We will be investigating a hypothetical scenario of an oil slick spreading at sea. From time to time, but irregularly, a helicopter is dispatched to photograph the oil slick. On each trip, the helicopter arrives over the slick. The pilot takes a picture, waits 10 minutes, takes another picture, and heads home. On each of seven trips, the size (area in square miles) of the slick is measured from both photographs.

We will use differential equations to model the spread of the oil slick, predict the square mileage of it at various times, plot the prediction function with respect to time, and determine the times at which the observations in the table below were recorded. To note, we will walk through completing a series of specific tasks, which are derived from a source referenced at the end of this post.

Part (a): Build a model for the size of the oil slick

First we can add a column to the table above that describes the rate of change, also equal to ΔA(t)/Δt, of the area of the slick over a 10-minute time period (where A(t) represents the area of the slick at time t). This is calculated by:

Thus, we achieve the following table, appended with the column ΔA(t)/Δt.

Now, we can plot the initial observations with the rate of change of area of the oil slick.

As calculated from Microsoft Excel (and displayed above), the line of best fit is y = –0.001x +0.0102, meaning ΔA(t)/Δt = –0.001x +0.0102. This can also be represented as the first-order linear differential equation model:

Dividing both sides of this differential equation by –0.001x +0.0102 and then integrating it with respect to t to isolate A(t) yields:

Assuming that the first observation occurred at time t = 0, we can substitute the initial condition (t, A(t)) = (0, 1.047) and solve for c₆. When simplified, we get c₆ = 0. Thus:

Part (b): Predict the future size of the oil slick at t = 10 minutes, t = 20 minutes, and t = 120 minutes.

We must substitute our given t-values of 10, 20, and 120 into the function A(t) found previously.

Part (c): Plot your model of the size of the oil slick as a function of time.

Below is the model based off of the function A(t).

Part (d): Find the time at which the oil slick is 8 square miles.

We must set A(t) equal to 8 and solve for t.

Part (e): Determine the time of each of the observations for the first, third, fifth, and seventh initial observations.

We can use the same process as demonstrated in part (d) to find the times for the first, third, fifth, and seventh initial observations.

First initial observation

Third initial observation

Fifth initial observation

Seventh initial observation

Conclusion

Through this project, we explored how modeling the spread of oil slicks can be achieved by first plotting initial observations with the rate of change of area of the slick, computing an accurate trend line (one with a very high R² value of 0.9967!), and use Mathematics to generate a prediction equation for oil slick area. When performing a "reality check" with our function A(t) and our initial data, we see that this function is logical, because it makes sense for an oil slick to begin to grow rapidly and then quickly slow down to a much smaller growth rate as time increases. We were also able to calculate the times at which certain observations were taken as well as estimate future sizes of the oil slick.

Another method we could have used to model the spread of the oil slick is to plot initial observation values against observations taken 10 minutes later, rather than against rate of change of size. This, when carried out, is another route to a differential equation. Although we could expect these results to be slightly different than those offered by our current method, they should be quite close.

One limitation of this approach is that it assumes that the change in area of the oil slick follows just one differential equation. Secondly, it does not take into account any environmental factors that could affect how much the oil slick spreads. This makes it difficult to expand to other scenarios.

Overall, we practiced analyzing and modeling oil slick areas by applying skills of first-order differential equations to a real life scenario. In the process of deriving an accurate Model based on the given data, we used integration, mathematical modeling, and the solving of differential equations given initial values.

References

[1] Judson, T. W. (2020, August 1). The Ordinary Differential Equations Project. Stephen F. Austin State University. http://faculty.sfasu.edu/judsontw/ode/html-20200801/odeproject.html

[2] Winkel, B. (n.d.). Spread of Oil Slick. Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations. https://www.simiode.org/resources/2038/download/1-5-S-OilSlick-StudentVersion.pdf


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