Abscissa-Transforming Second-Order Polynomial Functions to Approximate the Unknown Historic Production of Non-renewable
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Abscissa-Transforming Second-Order Polynomial Functions to Approximate the Unknown Historic Production of Non-renewable Resources J. Müller · H.E. Frimmel
Received: 17 May 2010 / Accepted: 23 April 2011 / Published online: 1 July 2011 © International Association for Mathematical Geosciences 2011
Abstract For many non-renewable resources, reliable production data are only available from a certain point in time but not from the beginning of production periods. In order to constrain the unknown historic production of such resources for those ancient times for which no reliable annual production data are available we present a novel mathematical technique, based on Verhulst’s logistic function. The method is validated by the United States’ crude oil production for which the complete production cycle, starting in 1859, is well documented. Assuming that the oil production in the USA between 1859 and 1929 is unknown, our method yields values for this period of 16.0 gigabarrels (Gb) based on a second-order polynomial fit and 13.5 Gb based on a third-order polynomial fit of post-1929 production data, respectively. Especially the latter amount compares well with the actual value of 12.1 Gb, thus illustrating the strength of the method. For global gold (Au) production, our method yields an ancient production up to the year 1850, when official and reliable production statistics began, of approximately 10,000 metric tons (t) based on a second-order polynomial fit. For mercury (Hg) a production of 64,000 t was determined for the time up to the year 1900, when annual production figures started to become available, again using a second-order polynomial fit. While the results obtained by the application of secondorder polynomial functions could be confirmed by higher-order polynomial functions in the cases of both USA oil as well as global Au production, this was not possible in the case of global Hg production because of a highly irregular production curve. Keywords Hubbert-linearization · Ultimate recoverable amount · Historic production figure · Finite resources · Mercury, gold J. Müller () · H.E. Frimmel Geodynamics & Geomaterials Research Division, Institute of Geography & Geology, University of Würzburg, Am Hubland, 97074 Würzburg, Germany e-mail: [email protected] H.E. Frimmel Department of Geological Sciences, University of Cape Town, Rondebosch 7701, South Africa
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Math Geosci (2011) 43:625–634
1 Introduction For any non-renewable resource, such as most mineral resources, there exists a pattern of increasing production, one or more production peaks, and eventually a decrease in production. As production of such a resource progresses, the energy required for both the production and a sustainable environmental remediation increases. The dependence on growth, the cumulated production to date and the future production can all be described by the differential equation dQ = k · Q(t) · Q∞ − Q(t) , dt where dQ/dt is the annual production of a resource, k is a growth rate, Q(t) the cumulated production and [
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