When confronted with such data, one can hardly avoid wondering how long such growth and production rates can be sustained. A powerful method of analysis is that illustrated in Figure 8 (Hubbert, 1956, Fig. 11). For any exhaustible resource, such as coal or petroleum, the curve of the annual rate of production must begin initially at zero. The production rate tends to increase exponentially for a limited period. Next, as the re­source becomes progressively depleted, the production curve must reach one or more maxima and finally decline gradually to zero as the resource becomes exhausted. A mathematical property of this graph is that when the production rate, dQ/dt, is plotted against time on an arithmetic scale, the area under the curve is proportional to the cumulative production. Hence, this area must always be equal to or less than the quantity of the resource initially present. Therefore, if by geological and other means the amount of coal or oil initially present in a given area can be estimated, the production curve analogous to those shown heretofore can be approxi­mated for the future, subject to the stringent condition that the area under the curve must not exceed the estimate of the initial magnitude of the re­source.

What is required, therefore, is an independent estimate of the amount of the resource initially present. For coal, both for the United States and for the major geographical areas of the world, such estimates have recent­ly been published by Paul Averitt (1969) of the United States Geological Survey. With an allowance of a 50 per cent loss of coal in place during mining, these estimates are shown graphically in Figure 9.

Of the world’s initial supply of 7.6 X 1012 metric tons of producible coal, about 65 per cent occurs in Asia (including European Russia), about 27 per cent in North America, about 5 per cent in Europe, and the remaining 2 per cent on the three entire continents of Africa, South America, and Australia. Of the North American initial supply of 2.0 X 1012 metric tons, 1.5 X 1012, or three-quarters, occurred in the United States.

Using these data in conjunction with the previous production histo­ries, and the technique shown in Figure 8, estimates can be made of the fu­ture possible coal production for both the world and the United States. In each instance, two curves are drawn, the first based on Averitt’s total esti­mate, and a second of about half that amount. The lower figures result from eliminating the deeper and thinner coal seams included in Averitt’s estimates.

Western

Europe

South and Central ЩЁ *14*X lO^rnetric tons

^ Ш

For the world production estimate (Fig. 10), the curve based on the higher figure is shown reaching its peak at about the year 2140; the peak of the curve based on the lower figure is shown at a few years earlier — at about 21

Academy of Sciences.) versa. Also shown in Figure 10 is a curve of what the world production rate would be if the growth rate of 3.61 per cent per year which has pre­vailed since the end of World War II should continue for another half cen­tury. In the view of the limitations imposed by the area under the curve, such a continuation is manifestly impossible.

Corresponding curves for the future coal production of the United States (Fig. 11) show the curve based on the higher figure of about 1.5 x 1012 metric tons of initial producible coal peaking at about the year 2220; the lower curve peaks at about 2170. Here again, higher rates of produc­tion than those shown would cause the curves to reach their peaks earlier, whereas lower production rates would delay the peak dates somewhat. However, a conclusion that is consistent with all of the curves shown in Figures 10 and 11, or any likely variation of these curves, is that the maxi­mum length of time that the coal reserves will be sufficient to supply a ma­jor part of the world’s, or of the United States’, energy needs can hardly exceed about three centuries.