Evaluating uncertainty in kimberlite pipe volume by simulating geometry in cylindrical coordinates
A significant source of uncertainty in diamond resources is the volume of kimberlite pipes. Often, all material within the pipe is mined as ore, leaving the actual geometry of the pipe as the largest source of uncertainty. It is desirable to use stochastic techniques, rather than deterministic methods, to assess the pipe geometry because stochastic techniques generate multiple realizations that are reproducible and can be used to quantify uncertainty in pipe geometry. The proposed methodology uses sequential Gaussian simulation (SGS) of pierce points interpreted from drill holes through a kimberlite pipe. Kimberlite pipes are approximately cylindrical in shape; therefore, the data are transformed to cylindrical coordinates to facilitate the simulation of pipe geometry. Rather than using xyz coordinates to define the pierce points, z, q and the pipe radius are used. In cylindrical coordinates the radius variable can be simulated in z-q space to generate multiple realizations of the pipe volume. Uncertainty in the volume of ore can then be evaluated. This methodology is demonstrated with a data set from a kimberlite pipe at BHP Billiton’s EKATI diamond mine.
After transforming the available pierce point data (drill hole intersections through the kimberlite pipe) to cylindrical coordinates, traditional SGS can be employed to generate realizations of the kimberlite pipe geometry. The major contribution of this methodology is in the transformation to cylindrical coordinates and in wrapping of the q direction. Once in cylindrical coordinates, it must be considered that 0º is equivalent to 360º. Using a modified search for conditioning data that wraps in the q direction, realizations are generated.
Specifically, the proposed methodology is to:
Remove trends in the data, such as a decreasing pipe radius with depth;
Establish statistical parameters for detailed modelling, including a representative histogram, uncertainty in that histogram and a variogram quantifying the irregularity of the pipe;
Simulate multiple realizations of the pipe geometry in cylindrical coordinates using SGS;
Calculate the volume of the pipe and associated uncertainty from the realizations.
By means of a case study, it was found that the major source of uncertainty in the deposit is in the input statistical parameters. SGS can effectively capture uncertainty in prediction at unsampled locations, but it does not consider that the actual input histogram (the data) also contains uncertainty. The spatial bootstrap is presented to consider the uncertainty in the input histogram. Multiple histograms are generated and used in SGS to capture this parameter uncertainty in the input histogram.
Using the proposed methodology, the uncertainty in the geometry of a kimberlite pipe can be quantified. A number of areas of future work arise: (1) inherent secondary data could be used to better model the pipe geometry; (2) wrapping of the coordinates in the x and y directions has also been implemented with the intention of using a similar methodology to model a spherical-shaped deposit in spherical coordinates; and (3) this methodology could be used in 3-D (z-q -r space) to generate a locally varying mean model for diamond grade, which could generate a trend in either the z, q, or r directions.