08/06/2021
This picture shows the flow of oil leaking from the Deepwater Horizon platform in the Gulf of Mexico in 2010.
A very important question is the following:
How much oil in total was leaked during the 87 days of the BP oil spill?
Experts estimated the flow rate of the oil spill from pictures like the one shown here.
On April 20, 2010, the oil drilling rig Deepwater Horizon in the Gulf of Mexico, exploded and sank resulting in the death of 11 workers on the Deepwater Horizon and the largest spill of oil in the history of marine oil drilling operations.
It was estimated that around 4.9 million barrels of oil leaked from the damaged well over an 87-day period, before it was finally capped on July 15, 2010.
The government report cited below provides data for the flow rate of the oil spill at 8 time points, given in the table on the next slide.
| day | flow rate |
|---|---|
| 1 | 0 |
| 2 | 0 |
| 3 | 62000 |
| 45 | 57500 |
| 46 | 59800 |
| 84 | 55200 |
| 86 | 53000 |
| 87 | 0 |
The idea is to use these data and create a continuous mathematical function of the flow rate \(\text{flow}(t)\) as a function of time \(t\), which interpolates the data linearly, meaning all data points are connected by line segments.
Then, to find the total volume of the oil leaked during this 87-day period, we just need to integrate this function over this time period.
\[\text{Volume} = \int_1^{87} \text{flow}(t) dt\] We can create a continuous, piece-wise linear function \(\text{flow}(t)\) for the flow rate by linearly interpolating the 8 data points, using the connector() function from the mosaic R package, authored by Daniel Kaplan, Randall Pruim and Nicholas Horton. We also use the developmental version of the mosaicCalc R package.
First, we need to convert the given data points into a data frame.
# given data points key_days <- c(1,2,3,45,46,84,86,87) flow_rates <- c(0,0,62000,57500,59800,55200,53000,0) flow_data <- data_frame(key_days,flow_rates)
We can use connector() to create a piece-wise linear function from the data:
flow<-connector(flow_rates ~ key_days, data=flow_data)
The connector() returns a function of time in days, named flow. This is the R function representing the piece-wise linear, continuous mathematical function \(\text{flow}(t)\) that connects linearly the data points. We can visualize the given data points along with the interpolating function \(\text{flow}(t)\).
gf_point(flow_rates ~ key_days, data=flow_data, size=2, col="blue",
xlab="days", ylab="flow rate") %>%
slice_plot(flow(day) ~ day, domain(day=c(1,87)), size=1, col="blue", alpha=0.6, n=1000)
To integrate the flow rate, we compute numerically the anti-derivative function volume of the function \(\text{flow}(t)\), using antiD() from the mosaic R package.
volume = antiD(flow(t) ~ t)
Mathematically, this is equivalent to:
\[\text{volume}(t) = \int \text{flow}(t) dt\]
The antiD() has a default value for the constant of integration \(C=0\). By the fundamental theorem of calculus, the total volume of the oil spill is:
\[\text{Vol} = \int_1^{87} \text{flow}(t) dt = \text{volume}(87) - \text{volume}(1) = 4.91885\times 10^{6}\]
We can visualize the volume of the oil spill as a function of time:
slice_plot(volume(t) ~ t, domain(t=c(1,84)), col="blue", size=1, n=500) %>% gf_labs(x= "time", y = "volume of oil spill")
The main idea of this project was to interpolate linearly the flow rate data and create a continuous mathematical function that can be integrated numerically to compute the total volume of the oil spill over the 87-day period.
The students can also compute the volume of the oil spill by hand, given the piece-wise linear nature of the flow rate function, which would be a good exercise to supplement the computational side of the project.
The total volume of the oil spill that we obtained is 4,918,850 barrels of oil.
The report from the US Department of the Interior, cited above, gives an estimate of 4.9 million barrels of oil, with estimated uncertainty of \(\pm 10\%\).
The Deepwater Horizon oil spill is the largest oil spill in the history of marine oil drilling operations.
