Tuesday, October 5, 2010

Refining the Peak Oil Rosy Scenario Part 5: Preliminary nonlinear logistic modeling of the USA oil production curve

Historic Source Data
My primary source data for yearly oil production and consumption will be BP's "Statistical Review of World Energy 2010 ("Statistical Review") which provides historical yearly production and consumption data for most countries from 1965 and on in an EXCEL work file.

One problem here for the USA is that 1965 is pretty "late" in the production curve; recall that Hubbert predicted a peak in production in 1965 and the actual peak occurred in 1970.

Fortunately, the EIA's Table 5.1 entilted, "Petroleum Overview" provides yearly production and consumption data back to 1949 for the USA.  Moreover, the columns of data under the headings "Field Production Total" and "Petroleum Products Supplied" are in close agreement with the production and consumption data for the USA, as reported in BP's Statistical Review.  I am not aware of any earlier historic data that is online.

I already have shown the USA's production data from Table 5.1 in part 2 of this series—but here it is again in Figure 13, in a slightly different format.

Since I am interested in determining yearly changes in the exponential rate constant "a," the production data, originally presented as millions of barrels per day has been converted to billions of barrels per year. 

A preliminary NLLS analysis of USA production data  
As an initial exploration of using the NLLS process, described in Part 4, I selected the entire data base of dQ/dt versus t(years) from 1949 to 2009.  The red solid line in Figure 1 shows the best fit curve obtained to the source data points from 1949 to 2009 (I used seed values of 250, 35 and 0.07, respectively for Q∞, Qo and "a"). 

The best-fit values for the parameters in the nonlinear logistic equation are as follows:
            Q∞ =  298 bbls;
             Qo =    53 bbls; and
            “a” = 0.0511 yr-1

Notice that Qo is large because it is an estimate of the oil accumulation through 1948.

Next, I repeated the NLLS fitting to progressively 10-year smaller data sets, that is, 1949 to 1999, 1949 to 1989, 1949 to 1979 and 1949 to 1969 (all using the best fit from the full data set as the seed value).  These results are shown in Figure 14, with different colored curves corresponding to the different data ranges as defined in the legend. Also included for reference are the source data and the best-fit curve using the full data set.

The best-fits to the progressively smaller data sets are pretty similar to the best-fit using the full data set until we get to the 1949-1969—I extended the predicted curve in the figure beyond the data range so that the trend was clear to see.  The curve fit to the 1949-1969 data actually looks pretty good—I can’t expect the NLLS procedure to do anything else than estimate the best fit to the data presented to it. 

The estimated parameter values for the 1949-1969 range depart wildly from the estimates made from the larger time ranges:



The implications of this are clear: I can not expect the nonlinear logistic equation to give reasonable predictions of Q∞ unless either:

1) The data on the growth side of the production curve is sufficiently scatter-free that roll-over towards the plateau in production (dQ/dt) can be discerned by the NLLS procedure.  That is clearly not the case for the USA production data in the 1949-1969 range.  Other than taking moving averages there is nothing I can do to smooth the data.  However, just looking at the data in the 1949-1969 range suggests that a moving average would not likely have helped too much here.
2) Enough data is included from the decline side of the production curve that the plateau in production can be discerned NLLS procedure.  For instance, for the 1949-1979 data set, including about ten-years worth of data that is past the actual year of maximum production (1970) gives predictions of Q∞, Qo and "a" that are in reasonably good agreement with the predictions made using the data set over a larger time span. 

Performing (2) would still allow the remainder of the decline curve (i.e., 1980-2010) to be analyzed using NLLS with Q∞ fixed, as described in procedure developed at the end of Part 4.  There is a broader problem with this type of approach, however.  I expect that there may be production curves for at least some countries for which the plateau in production has not yet occurred.  For such data sets, the method outlined in (2) would not be available.  So (2) is not a general solution that can be relied on.

Perhaps, there is a third option:

3) Estimate Q∞ and "a" on the growth side of the production curve using Hubbert’s logistic linear equation and then apply NLLS to analyze the decline side.

Ahaaa...back to modeling....

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