The fundamental equation of mineral production
allows to model and design the dynamics of mineral production, however complex
they are or could be. It considers not only the case of a constant production
to reserves ratio for given intervals of time, but with a piecewise approach, it
is also enabled to account the variation on time of this ratio. With a constant
production to reserves ratio, the limit expression of the fundamental equation
takes the form of an Erlang distribution with a fixed shape parameter. The rate
parameter equals the scale factor. The discrete piecewise version, instead of
considering the reserves and the production to reserves ratio being constant
through certain intervals of time, updates both variables by units of time. This
version, using either lineal or non lineal functions for the variables involved,
let to model known production profiles or to forecast them by experimental
design. The Hubbert’s linearization updated with recent data and the p-box
method applied to determine ultimate recovery of U.S. crude oil reserves
indicate official accounts underestimate them. The analysis of the ideal model
of production based on Hubbert’s linearization and curve, can be made by decomposing
it in the distribution with time of the reserves and of the production to
reserves ratio. The distribution of reserves with time is synchronized for both
the ideal Hubbert’s curve and real profiles, disregarding whether they match or
not. The departure of real profiles from the ideal Hubbert’s curve lies on the
differences or correspondences of the distribution with time of the production
to reserves ratio. The MonteCarlo simulation applied to forecast US crude oil
production for the next five years points to a slow decline, with average annual
yields presenting a difference lower than 10% between the start and the end of
S. Perez Rodriguez, “Methodology and Equations of Mineral Production Forecast. Part I. Crude Oil in the UK and Gold in Nevada, USA. Prediction of Late Stages of Production,” Open Journal of Geology, September 2013.
S. H. Mohr and G. M. Evans, “Combined Generalised Hubbert-Bass Model Approach to Include Disruptions When Predicting Future Oil Production” Natural Resources, Vol. 1, No. 1, 2010, pp. 28-33.
D. Berleant and C. Goodman-Strauss, “Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals,” Reliable Computting, Vol. 4, No. 2, 1998, pp. 147-165.
D. Karanki, H. Kushwaha and A. Verma, “Uncertainty Analysis Based on Probability Bounds (p-box) Approach in Probabilistic Safety Assessment,” Risk Analysis, Vol. 29, No. 5, 2009, pp. 662-675.