A fractal approach to the relationship between ore grade and tonnage

It is now recognized that a variety of natural processes exhibit scale invariance over a wide range of scales. Scale invariant processes often exhibit a fractal behavior. There are various ways to define fractal behavior; one is to relate the frequency of occurrence to size. Tonnage and grade relations for economic ore deposits exhibit a fractal behavior if the tonnage of ore with a mean grade is proportional to the mean grade raised to a power. We show that such a relation is a good approximation for mercury, copper, and uranium deposits in the United States; the respective fractal dimensions are 2.01, 1.16, and 1.48. The renormalization group approach is a quantitative method for studying scale-invariant processes. If it is assumed that the concentration of elements in ores is statistically scale invariant, the renormalization group approach can be used to derive a fractal relationship between tonnage and mean grade.