G E B W E B

UPDATE: This widget no longer works. Please consider the Commodities Widget instead.

Apple Mac OS X (version 10.4 and above) has this neat feature called dashboard. It lets tiny programs, called widgets, run as an overlay to your screen when you press ‘F12′. There are widgets that show the current time in any timezone, hurricane advisories, stock quotes, your computer’s vital stats, and much more.

I’ve got a certain desire to stay updated on the price oil, since it’s often related to world events. For instance, a spike in the oil price might mean a hurricane is headed towards the Gulf of Mexico or that there is more unrest in the Middle East. So to satisfy this urge for oil price updates, I’ve created the Oil Price Widget. It works on Mac OS X Tiger and gathers information from 321energy.com, which is displayed in a small window on the dashboard.

Download: Oil Price Widget.

Instructions: Mac OS X 10.4 Tiger is required. If you’re using Safari, click the download link. When the widget download is complete, Show Dashboard, click the Plus sign to display the Widget Bar and click the widget’s icon in the Widget Bar to open it. If you’re using a browser other than Safari, click the download link. When the widget download is complete, unarchive it and place it in /Library/Widgets/ in your home folder. Show Dashboard, click the Plus sign to display the Widget Bar and click the widget’s icon in the Widget Bar to open it.

Assume and are the amount of oil in the ground and the production rate respectively. We model the behaviour of the oil field with a system of linear ordinary differential equations (ODEs).

In clear text, this means that the amount of oil in the ground decreases by the amount produced, and the production capacity increases if it is small compared to the amount of oil left in the field, and decreases if it is large compared to this amount.

Now, the system is easily transformed into a single ODE by differentiating the second equation and inserting for from the first equation. This gives

which is has a simple analytical solution. The general solution is

Now we impose the initial conditions

which translates to some finite amount of oil in the reservoir, and the production capacity 0 at the time we start. These two conditions are used to determine the coefficients and . The solution is then

This solution reveals that the model has some obvious flaws. The first thing that becomes apparent when plotting the solution (or simply noticing the sine factor), is that the production becomes negative at times. Another flaw, is that if we integrate the production from the start to the time it becomes negative, the amount of oil extracted is larger than the amount that was originally in the field. Clearly, some modifications to the model are needed. A fix is to replace the first equation by

This new equation makes the system a lot harder to solve by hand. Using a computer program and Euler’s method for explicit time-integration, it was easy to plot the result, however.

This time, the integral is 1 (at least with numeric integration), and the production is never negative.

Compared to real-life oil fields, the model production is ramped up too fast. The model does not incorporate the effects of limited manpower, investment and equipment. A sharp cliff is present where the technical production capacity exceeds the geological capacity of the field. This cliff should not be observed in any well-planned oil project, since no-one would invest in bringing production capacity beyond what the field can handle.