Economy
& Energy 
Small State & Great Enterprises The technological prospective: prediction with a simple mathematical modeling (2) Evolution of Investments in Brazil

Text for Discussion: Omar Campos Ferreira. Advisor of the Minas Gerais Science and Technology Secretariat The petroleum issue is a recurrent theme in the media and it has already been examined before by the e&e periodical staff (“Depletion of the World Oil Reserves”, n0 1, Feb/Mar 1997 and “Oil depletion”, n0 4, Sept/Oct 1997). The recent price increase of petroleum products in the world market has stimulated a revision of the previous articles and the proposal of a more refined interpretation of the data that have been used. The methodology used has already been described in previous e&e articles (“Futurology or playing with the logistic function” n0 0 Dec/1996 and “Technological Forecasting”, n0 30, Jan/Feb 2002). The refinement to be introduced was suggested by a study concerning the Brazilian economy, where the correlation between the accumulation pattern and the important economic events are more easily perceived. In general terms it means to differentiate between long term transient fluctuations and pulses. The data are the same used in the previous articles regarding the world petroleum reserves ^{ [1]}^{.} Graphic 1 shows the accumulated discoveries of conventional petroleum[2] from 1930 to 1992 and the annual rate of discoveries. One can perceive leaps before (1938/1939) and after (1947/1948) the Second World War and less significant variations (6364 and 7879). The curve shows no more surprises relative to the expected behavior of a closed system such as the petroleum original reserve[3] since the substitution rate, if it exists, cannot be measured. Examining the curve of discoveries by year (Graphic 2) which is more sensitive regarding short term variations, one can notice a set of variations that can be interpreted as pulses of discoveries related with political or economical facts: start of the Marshall Plan (1948), the Suez Canal attack (1956), the Six Day War (1968), increase of petroleum prices by OPEP (1973), etc. However, there are fluctuations that apparently do not fit to the pulses and that could be explained by the discovery of giant fields, by exceptional prospecting efforts, etc.
Graphic 1. In order to filter isolated oscillations, we have grouped the data in constant time intervals, sufficiently long to smooth the data distribution and keeping the largest possible number of points in order to assure the fitting reliability. The choice of the amplitude is tentative and depends on the experience one has regarding the logistic curve; the tentative choices are compared through correlation coefficients between the observed and smoothed series. In the first article (“Depletion of the World Oil Reserves”) 10year intervals were used but now they seem too long to identify the possible pulses and this is the reason why the subject has been reexamined In the first tentative filtering, the annual discoveries were grouped in fiveyear intervals, represented by “centroids” shown in Graphic 2. The curve seems a logistic one whose differential equation is: dN/dt = a N (N* N), where N is the accumulated reserve until time t, N* is the final values and a is a kinetic parameter. The method used consist of approximating the differential equation by a finite difference equation: ΔN/Δt = a N (N*  N) and adjusting this equation in order to obtain an image of the differential equation from which the value of N* can be deduced by examining the maximum value of the function: y(N) = dN/dt = a N (N*  N). Actually, dy/dN = a (N*  2N) and, for dy/dN = 0, Nmax = N*/2^{.}^{[4]} Graphic 2 shows the accumulated discoveries represented by the fiveyear centroids and Graphic 3 is the representation of the y(N) function. From the equation of adjusted curve y(N) one calculates N* = 1690 Gb.
Graphic 2
Graphic 3. The finite (integrated) form of the logistic law is: N/N* = 1/(1 + e^{at + b}), where b is an integration constant determined by the initial condition of the problem. The linear form of the logistic law (ln F/1F = at + b , with F = N/N*), that determines both constants a and b is in Graphic 4; from the adjusted expression (ln F/1F = x), the discovery curve is rebuilt, solving the linear equation to determine N: N = N* ex / (1 + ex)
Graphic 4
Graphic 5 The adjusted distribution is shown in Graphic 5, together with the original distribution of the annual discoveries. The result obtained by grouping the data in five year centroids (1690 Gb) is not much different from the tenyear grouping (1630 Gb). The second filtering was carried out using a threeyear centroid, with the sequence of calculations and showed two threeyear pulses of discoveries variation rate (Graphic 6).
Graphic 6. One has applied to each pulse the treatment previously described, by calculating the corresponding maximum discovery (Graphic 7) and reproducing the part corresponding to the discovery distribution.
Graphic 7. One can observe an overlapping region of two pulses that should alter the shape of the second one. After the extrapolation of the first pulse, it was verified that the alteration is smaller than the incertitude concerning the second pulse which eliminates the correction. The adjusted curve is shown in graphic 8. The maximum value of the second pulse corresponds to the calculated original reserve (1870 Gb), a modest difference relative to the previous results.
Graphic 8.
Comparison of the results with other studies The data used in the present report comprise the 19301992 interval. In spite of the fact that the projected final value of reserves to not support expectations of a substantial change of the scenario regarding additional reserves discoveries, we have carried out a data search in publications from other countries and from independent authors, but the data series do not comprise the same time interval and when they are compared they do not show proportionality that would justify to consider the discrepancies as being systematic, in which case it would be possible to combine the sources. Among the independent authors publications, the article by Jean Laherrère (“Estimates of Oil Reserves”, EMF/IEA/IEW Meeting, Laxenburg, Austria, 2001) critically analyzes the more known projections (USGS, DOE, World Petroleum Council, British Petroleum…), pointing out methodological inconsistencies. The author is a retired geologist (who considers himself free from injunctions) and an usual collaborator of Campbell who uses a method similar to ours but uses the probability of discovery of giant field fitted to a “fractal parabola”. The article does not details the method so that one could analyze in detail the similarities and differences. A summary carried out by Laherrère of the maximum reserves estimates since 1960 shows an average value of 2000 Gb and 1600 and 3500 as extreme values. Quoting a result carried out by Perrodon A., in 1998 and updated in 2001, Laherrère and Campbell mention the average value of 1800 Gb, a minimum value of 1700 and a maximum one of 2200. Our estimate of 1870 Gb is close to the above mentioned average value. Accumulated consumption Consumption data from the same source were treated in the same way. Three consumption pulses were identified but not shown because they do not present any methodological novelty. Graphic 9 shows the discovery adjustment results, the accumulated consumption and the reserve/consumption ratio. The extrapolations in both cases were obtained from the last pulse of the respective distributions.
Graphic 9. As can be seen in graphic above, the ratio between the accumulated discovery and the accumulated consumption has continually decreased from 1930 on and came close to 2.2 in 1992. If we take into account the extrapolated parts, the value of this ratio will be 1.5 in 2020. The price increases of the product since the Gulf War were knowingly circumvented by the industrialized countries but it is now agreed in the petroleum circles that the present price, US$ 53 per barrel, will not return to the US$ 22 – US$ 28 plateau; however it is admitted that there is a speculative component and it is estimated that the price in the next years will be close to the US$ 40 – US$ 45 plateau. The attitude of the Brazilian authorities is one of expectation since incorporating the international price in the petroleum products price calculation will have important reflexes in the inflation rate. In a next issue of this periodical we will present a study regarding the Brazilian reserve, the consumption trends and the possibility opened to the country, during the crisis, to use more intensively biomass fuels. [1] Campbell, C. J. – “An oil depletion model”, Petroconsultants, 1994. [2] Conventional petroleum is the current designation for petroleum having characteristics similar to those of the petroleum that is presently extracted using the existing or predictable technology and with costs compatible with the competition from other energy resources. [3] The original reserve formed million of years ago remain practically unchanged since the then existing physicalchemical conditions (biomass buried under a thick clay layer, absence of oxygen, high pressure, etc.) possibly will not occur again. The international technical literature denominates in its projections the original reserve as the ultimate one (final value). [4] This procedure prevents handling the more complex function y(t) that is similar to a normal distribution curve but which is asymmetric.

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