Hubbert Linearization:
As I described in my previous entry, Hubbert linearization is important because it simplifies the process for building oil depletion models. I will explain the mathematics behind linearization and then apply the technique to
If the mathematics behind the calculations are not of interest to you, feel free to skip to the next section which details the implications of the model.
Hubbert’s curve is the first derivative of a logistic function. The area under the curve represents all of the oil that will be produced from the country or field in question. The peak of the curve represents the time period at which half of the available oil has been produced. The Hubbert curve makes it possible to estimate what percentage of oil has been recovered at any point in time. It also predicts annual production volumes.
Since the curve is the first derivative of a logistic function, the most important piece of information is the total amount of oil in place. If this bit of information is known, it is possible to construct the curve based on annual production and the area under the curve (the total oil in place.)
Because of the complexity in Hubbert’s analysis, it is easier to use linearization to estimate the total oil in place. Only one variable is needed: production data. The first step in the process is to plot the production data (P) as a fraction of cumulative production (Q) on the vertical (y) axis. Cumulative production (Q) is placed on the horizontal (x) axis.
As the chart to the left shows, the relationship between P/Q and Q becomes linear as production matures. This linear relationship follows the simple y = mx + b format for graphing a line in a plane. Hubbert’s logistic differential equation has a linear property – we can exploit this relationship with the linear representation of P/Q versus Q.
After the denominator Q becomes sufficiently large, P/Q begins to decline linearly. In the chart of
[One interesting side note on the regression: In this example the analysis is conducted from 1958-2007 with an adjusted R2 of 98.5%. A logical question to ask is whether or not this kind of model is possible earlier in the life of a country’s oil production. The answer (as Hubbert demonstrated) is yes. For example, I ran a linear regression of variables from 1958-1978. The results are almost identical. In the first example, the intercept equals 0.0569 and the slope equals -0.000247. In the second case, the intercept equals 0.0566 and the slope equals -0.000244. This results in an almost identical amount of oil in place: 230.3 billion barrles and 232.0 billion barrels respectively. The main difference is a reduction in R2. It falls to 85.7%. There is a logical reduction in certainty when fewer data points are used. However, the end results are almost identical.]
Plotting the line represented by the regression equation gives x and y intercepts. The x intercept is crucial. The x axis represents the total quantity of oil produced (Q). The intercept is an estimate of the total oil that will be produced (Qf). Remember, this is the most important piece of information needed to build Hubbert’s logistic curve! The
The slope in the regression equation is also important. It represents the annual production as a fraction of cumulative production. This completes the set of variables needed to form a production function. The chart to the left shows the translation of the y = mx + b equation to the oil production variables.
Algebraic manipulation (demonstrated in the video) yields an equation for annual oil production: P = Qa(1 – Q / Qf )
The inverse of this equation gives a measure of time: years per billion barrels. To complete the linearization process and build a Hubbert curve, plot the projected production and this inverse value in 1 billion barrel increments of cumulative oil production. To add actual years to the plot, look at the actual cumulative production (Q) and match it to the corresponding value in the model. Then, subtract or add 1/P for values above and below the target year respectively. An example of this process is shown to the right.
In my next entry I will apply this methodology directly to the United States' historical oil production levels. This will demonstrate how the model works and help illustrate the decline in domestic oil production. It will also provide the foundation from which we will explore global production and its potential to unbalance global supply and demand.
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