The measurement and inclusion of a stochastic ore-grade uncertainty in mine valuations using PDEs

Mining companies world-wide are faced with the problem of how to accurately value and plan extraction projects subject to uncertainty in both future price and ore grade. Whilst the methodology of modelling price uncertainty is reasonably well understood, modelling ore-grade uncertainty is a much harder problem to formulate, and when attempts have been made the solutions have taken unfeasibly long times to compute. This paper provides a new partial differential equations approach to the problem, and achieves this by treating the grade uncertainty as a stochastic variable in the amount extracted from the resource. We show that this method is well-posed, since it can realistically reflect the geology of the situation, and in addition it enables solutions to be derived in the order of a few seconds. The paper also provides for ore-grade parameter estimation by using Maximum Likelihood Estimates on the estimated ore-grade data set, thus generalising the approach. A comparison is made between a real mine valuation where the prior estimate of ore grade variation is taken as fact, and our approach, where we treat it as a stochastic uncertain estimate.