An oil well when in production has a controlled flow of oil and natural gas from the producing reservoir in the subsurface to the surface, however this production during the day may have some fluctuation, which is why it will have a minimum and a maximum production.

Oil production in the oil industry is measured in barrels, the purpose of this post is to show you all how the oil production of an oil well can be expressed in the form of an absolute value inequality, which means that to find the minimum and maximum production during the day the solution is to solve the absolute value inequality, and logically to give a coherent interpretation of the meaning that our result will have.

In order for us to understand this application of absolute value inequalities in the production of oil from an oil well, the following example is proposed:

The estimated daily production P from an oil well is: |P-2250|<125, where P is measured in barrels of oil. Determine the highest and lowest production levels.

As a solution we must see that this inequation is of the absolute value type, by absolute value property we have that:

[1] P-2250<125
[2] P-2250>-125

We solve for each case and we are left with:

[1] P-2250<125

For this first case, the oil well would be producing a maximum of 2375 barrels per day.

[2] P-2250>-125

In this case the well would be producing a maximum of 2125 barrels.

Interpretation of results

Absolute value inequality presented for the production of the oil well is a linear inequality that can represent the optimization in the production of the well, i.e. when the linear inequality is solved we realize that it is desirable that the oil well produces between a minimum of 2125 barrels of oil per day and a maximum of 2375 barrels of oil per day.

This leads us to the following questions:

Why want to produce no less than 2125 barrels?

As the linear inequality is a representation of the optimization of the well in terms of productivity, it is convenient that the well produces more than 2125 barrels, since suddenly with a lower production it would not be possible to recover the investment costs in relation to its drilling and completion to put it into production.

Why would you want to produce no more than 2375 barrels?

The production of an oil well must be controlled, i.e. the higher its productive potential is logically better, however exploiting an oil well in the maximum of the ranges can cause the production equipment such as wellhead, valves, pumps and all surface and subsurface equipment to be compromised to the point that they can fail, which is why if the linear inequality represents the optimization of the production of the well, the most advisable is not to exceed a production of 2375 barrels per day.

If we express this production interval P on the real line, we would be left with:

As the intervals are open, it means that the production of the well can reach for example 2124 barrels but not 2125 barrels and that it can reach a maximum production of 2375 barrels but without being 2375 barrels, but for example it can reach a maximum of 2374 barrels, leaving perhaps a margin of 1 barrel of safety, both for the minimum value and for the maximum value.

Recommended and consulted bibliography

Calculus with Analytic Geometry book. Author: Larson and Hostetler. Volume I. 7th edition.

Note: The cover image was created using Microsoft PowerPoint's geometric shape insertion tools. The equations presented were created using Microsoft PowerPoint's equation insertion tools.