Geostatistics is a powerful technique in ore reserve valuation and addresses the estimation of ore reserves. It also determines confidence intervals on reserve estimates and optimal drillhole spacing, as well as assisting in contour mapping and detailed mine-planning. The information required for reserve estimation is the grade of samples taken at specific locations in the deposit. The geologist or engineer needs a precise grade estimate at locations between the sampled locations. To do this, the estimation method of kriging (named after South African mining engineer Dr D. G. Krige) was developed in the 1960s by the French mathematician George Matheron. Kriging is precise because it takes into account the number of samples available, the position of the samples, the distance between the sample location and the zone being estimated, and the spatial continuity of the grades.
Gold deposits exhibit several important characteristics:
* Geological controls, which can influence the distribution of the grade values within the deposit. Zones of high porosity and permeability are examples of important geological controls. In these zones, fracture systems, shears and faults develop in several stages, each of which concentrates (in the form of veins), disperses, limits or displaces the gold mineralization. Other principal geological controls are folding, the nature of the host rocks and alteration (including oxidation).
* Heterogeneity in a gold deposit signifies that several distinct types of mineralization are present. Each type may be confined to an area of the deposit or several types of mineralization may overlap.
* Complex geometry is characteristic of small deposits, especially those affected by several fracture systems and tectonic deformation. To draw the outline of an orebody from a few drillholes is difficult at best, and underground exposure is required to reveal the detailed geometry of the orebody.
* Long tail grade distribution. Because of the heterogeneity and very low content of gold in a typical gold deposit, the grade distribution always has a long tail. If grade values are averaged to determine the mean grade of an entire deposit, high-grade samples have a pronounced effect. Combining Geostats with Geology
Many small-lode gold orebodies are characterized by pronounced geological controls, heterogeneity, complex geometry and long tail grade distribution. Because of these features, it is dangerous to apply geostatistics in isolation for ore reserve estimation. The following geological steps should be combined with the geostatistical approach: * Identify the geological controls.
* Separate the types of mineralization.
* Define the outline of the orebody.
* Study the effect of high grades.
With a maximum use of the geological information and a close collaboration between the geostatistician, geologist and mining engineer, geostatistics will gain more popularity and respect within the gold mining community. Otherwise, the appellations of geomagician (for the geostatistician) and computer-driven orebodies (for relying completely on computer programs for the ore reserve estimation) may continue to spread.
We will now examine cases where geostatistics has been used, successfully in two instances and one case in which failure was the result.
Several geostatisticians worked on South African gold deposits to test if geostatistical estimations could be verified in practice. At the Hartbeesfontein mine (one of the top five gold producers in South Africa with an annual output of about 1 million oz, or 31.1 million g), about 5,000 sample grades were used to test the geostatistical estimations. The data were available on a 25-ft (7.5-m) grid (see Fig. 1). A block size of 125×125 ft (37.5×37.5 m) was considered, or 25 samples per block. The sample at the centre of the block was separated from the other 24 samples surrounding it in the block.
* The true grade of the block was assumed to be the average of the 24 samples (all samples taken, less the central one). To compare independent estimates, the central sample grade was excluded from the calculation of the true block grade.
* Estimated grades of the block were calculated using its central sample (and central samples of adjacent blocks).
The following methods were used to estimate the block grades (Fig. 2):
* Polygons of influence, which assigns the grade of the single central samples to the entire block;
* Inverse distance weighing, which uses the central sample and central samples of adjacent blocks, and assigns a weight to each sample according to its distance to the centre of the block to estimate; and
* Kriging methods, which use the central sample and central samples of adjacent blocks. A weight is assigned to each sample according to number available, position of the samples, distance between the sample location and the zone being estimated, and spatial continuity of the grades.
By moving the centre of the “sample” grid, a total of 4,808 blocks was estimated. For comparison purposes, the “true” block grades were plotted against the estimated ones for all estimation methods considered (see Fig. 3). If the estimated grades are identical to the “true” grades, all the points will fall on the 45 degrees line (perfect correlation). The kriging methods came closer to this than the methods of polygons of influence and inverse distance weighing.
At the Golden Sunlight gold mine in Montana, the mill head grade was compared with the grade estimated by kriging. Samples from blastholes drilled 15 ft apart were used to obtain kriged estimates for 25x25x25-ft (7.5×7.5×7.5-m) mining selection units. Kriging proved to be the most accurate and practical method of routinely estimating ore grade during production and selecting material by grade range.
Where it failed to work was on a gold deposit in the western U.S. The feasibility study was based on samples from diamond drillholes and reverse circulation holes drilled at a 50-ft (15-m) average spacing. To define the ore-waste boundaries as closely as possible, the mine planning engineers kriged blocks (20-ft, or 6-m, cubes), but these were small, compared with the spacing of the holes (50 ft, or 15 m). Because the grades show a geostatistical range of influence of fewer than 20 ft (fewer than 6 m), the estimated block grades are over-smoothed and give a false impression of a very homogenous distribution of the block grades within the orebody. When the mining operation started, the mill head grades were significantly more variable than anticipated. This, combined with low metallurgical recoveries, has prevented the mine from making money.
This example shows how geostatistics may be used incorrectly. When the range of influence of the grades is small relative to the size of the block being estimated, geostatistics provides poor estimates and should not be used for mine-planning. To achieve better estimates, additional sample grades are needed or, if certain grade distribution ass
umptions are verified, more sophisticated goestatistical techniques may be utilized. Normand Champigny is a consultant with The Coopers & Lybrand Consulting Group in Toronto.
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