Number-average size model for geological systems and its application in economic geology

Abstract. Various natural objects follow a number-size relationship in the fractal domain. In such relationship, the accumulative number of the objects beyond a given size shows a power-law relationship with the size. Yet in most cases, we also need to know the relationship between the accumulative number of the objects and their average size. A generalized number-size model and a number-average size model are constructed in this paper. In the number-average size model, the accumulative number shows a power-law relationship with the average size when the given size is much less than the maximum size of the objects. When the fractal dimension D_s of the number-size model is smaller than 1, the fractal dimension D_s of the number-average size model is almost equal to 1; and when D_s > 1, the D_m is approximately equal to D_s. In mineral deposits, according to the number-average size model, the ore tonnage may show a fractal relationship with the grade, as the cutoff changes for a single ore deposit. This is demonstrated by a study of the relationship between tonnage and grade in the Reshuitang epithermal hot-spring gold deposit, China.