The geostatistical approach for reserves (Article published in Mining in Mining Magazine - May 1996) J. Deraisme (Geovariances) C. De Fouquet (ENSMP)
Geostatistics provides estimation methods which are based on the assumption which a mineralized phenomenon can be considered as the realization of a random process. This approach must not lead us to forget the deterministic parameters which are responsible for the distribution of the reserves. On the contrary the first step which comes before any geostatistical study is a sound "exploratory data analysis", where geological, statistical and geostatistical approaches must be combined together.
Once this essential step is achieved, the relevant variables and populations should have been identified, as well as their t heir relations with the geological structures involved. Let's consider the two following examples:
Cross-sections and geological maps established from exploratory boreholes enable
the geologist to understand the structure of the mineralization. However, trying to quantify reserves from this data will introduce an important bias in the estimation, on one hand because lateral extensions of the mineralized intercepts between the boreholes are very much subjective, on the other hand because the distributions of grades in the cores are completely different of those t hose of the mining units. However, the geological interpretation helps in analyzing the structures of spatial correlation, by taking into account the main directions of mineralization, tectonics and lithologies. It also provides information on possible heterogeneities and explains their origin.
Fig. 1: Geological interpretation of a stockwork uranium deposit, from borehole data
The second example relates to an iron deposit containing two types of ore: hematite and itabirite. These two ore materials are deeply embedded at a metric scale, moreover their relative proportions change with the depth. Since the exploitation method expects to extract the hematite selectively from the itabirite it is essential to estimate a model of grades for both of them. A detailed statistical analysis by zones and layers, combined with the geological interpretation given on maps and cross-sections is necessary to identify the statistically homogeneous populations, as well as relevant variables (Figs. 2 and 3).
The aim: from samples to reserves
A good knowledge of tonnages and grades within an orebody is essential in order to assess the economical feasibility of putting the mine into production, or when choosing equipments or plants. A orebody is always made of several types of ore and waste minerals. Separating the ore from the waste is a difficult task which is almost impossible to achieve, for many reasons: The geological boundary between ore and waste is seldom clear in nature. Boundaries which are defined on economical criteria (such as a cutoff grade) do not even match any geological reality. Depending on the dimensions of the shovels, more or less waste material is mixed with the ore when loading trucks. This results in grades lower than expected, yet from an economical point of view this loss is counterbalanced by an increase in productivity. The exploitation technique applies constraints which compel extraction of waste in order to give access to the ore, for example the stripping ratio in open-pit mining. These constraints in the mining process, as well as others, make it more difficult to integrate the tonnage/grade relationships observed on the samples when one tries to forecast recoverable reserves.
Fig. 2: Bimodal histogram of Iron and Silica grades within a given granulometric class. (graphs made with Isatis software)
Fig. 3: Cross-plot between Iron (Fe) and Silica (SiO2) within two given granulometric classes (after gaussian anamorphosis). The figure clearly shows two populations of samples.
Among other benefits of using geostatistics in such a study is the fact which these problems can be taken into account through 3 basic concepts: support effect, information effect and constraints effect. Support Effect
Any mining engineer knows that recovered grades are lower when selectivity is poor, in other words the bigger the mining units, the lower the grades. The average grade of a huge block of several thousands of cubic meters can be considered as the average of the grades of smaller blocks of a few cubic meters contained in the big block. The distribution of grades of huge blocks is obviously less scattered than that of the small samples.
Fig. 4: Support Effect
The only grades which are known experimentally are those of the samples; in order to forecast the distribution of grades for blocks of different dimensions geostatistics provides models of change of support, which are based on the exper imental histogram of sample grades as we ll as their spatial correlations through the variogram. Let's consider the two grade profiles shown in Fig. 5, which both share the same experimental histogram. The spatial correlations of the grades, however, are different with less correlation for the left-hand profile, hence a certain amount of nugget effect on its variogram. The distributions of grades estimated for blocks and the selectivity will t herefore be different.
Information Effect
During the exploitation of the mine, the true grade of the mining block is still unknown. Therefore the decision to send the material to the ore plant or to the waste dump is still taken from the estimated grade and not from the true grade. As a consequence it is not possible to avoid sending blocks to the wrong destination: rich blocks will end up on the waste dump, because they are estimated as "poor", while poor material will feed t he ore plant, as illustrated in Fig. 6.
Fig. 6: This cross-plot of estimated grades vs true grades of the mining units illustrates the Information Effect.
This information effect results in a degradation of the tonnage/grade relationship, therefore a loss of selectivity, in the same way as the support effect. This consequence is more drastic when the density of samples is poor. Non-linear geostatistical methods such as Disjunctive Kriging can quantify the a mount of loss of selectivity due to the information effect. These techniques allow one to dimension an adequate sampling mesh for pre-mining boreholes. Constraints Effect
Technical constraints imposed by the exploitation method add even more penalties on the recovery of the ore material. It may prove necessary to leave in place some rich parts of the orebody if the technical infrastructures involved would cost too much (e.g. in underground mining) or if the stripping ratio is much too high in open-pit mining. Moreover the minimum opening width of stopes as well as blasts are responsible for dilution which again lowers the grades and the select ivity. When technical constraints are simple, such as the stability of slopes in open-pit or footwall and hanging wall, so that the whole seam is extracted without further vertical selectivity, a technical parameterization of reserves can be applied to optimize ore recovery. When constraints are more complex, it is necessary to use a realistic numerical model of the orebody which reproduces a spatial distribution of grades similar to the true distribution. This
model built through geostatistical conditional simulations can then be used to estimate recoverable reserves.
The aims of a study
The geostatistical methods which are used during a study obviously depend on the type of problem to be solved:
Kriging is the geostatistical estimation method which gives an estimate of the in-situ resources from which one can identify low and high grade areas. Moreover the kriging variance quantifies the accuracy of the estimation and can therefore be used to choose an optimal drilling grid. Disjunctive Kriging with a model of change of support enables one to estimate recoverable reserves given a particular selectivity support, corresponding to the mining equipment and the amount of information. Grade-tonnage curves for different supports are then used to select the economically optimal selectivity. This estimation method does not take into account the exploitation constraints, it is based on the hypothesis of "free selection". Finding the optimal pit and sequence of exploitation in the mine requires a technical parameterization of the reserves under the constraints given by slope stability. Using the technique of convex analysis a whole family of optimal pits is produced which enable one to recover the maximum amount of metal for different total volumes of material. Conditional Simulations are used when other technical constraints must be taken into account. The numerical models of orebody that are built by these simulations can be input to a simulator of exploitation processes, in order to optimize the exploitation method: for example, select the optimal dimension of an homogeneization stockpile, select the geometries of workings in order to optimize the grade quality, find the optimum working faces and bench heights.
Whatever its final aim, the success of a geostatistical study relies essentially on one common step shared by all approaches: the analysis of spatial correlation of the relevant variables from the exploration samples, in other words the Variographic Analysis. The introductory recommendations concerning the use of the geological interpretation must not be forgotten, as well as the following principles: Variables must be additive, otherwise calculations which involve linear combinations will be biased. For instance the average grade on a core is not additive unless all cores have the same length. If cores have different lengths it is necessary to regularize the grades along boreholes in order to work with composite samples on a regular support. When studying grades computed from a layer of variable thickness, one must work with the two additive variables: thickness and accumulation of metal, (i.e. the product of grade time thickness). Moreover correlations between several variables should be taken into account while kriging or simulating these variables, so that the different estimations are consistent. This requires a multivariate geostatistical analysis.
A sound analysis of histograms and cross-plots must be performed to detect possible distinct populations which should be studied separately. When populations are mixed in space, a detailed characterization of this mixing is necessary.
The stationarity of the variables must be assessed in order to use appropriate models of correlation. If the variables are non-stationary one will try to identify areas where a "local stationarity" can be assumed. Choosing the lags and directions for computing the experimental variograms depends mainly on the spatial location of the samples as well as the geological interpretation. Several directions of calculation must be used in order to check for possible anisotropies in the mineralization. When fitting a model to the experimental variograms, those variogram points which have been computed from very few pairs of samples should be ignored. In the same way, variogram points that have been computed for distances above half the field's extension are not reliable and should be ignored when fitting.
Conclusion
The use of Geostatistics is increasing very quickly in various fields thanks to the spectacular evolution of computer power which makes geostatistical software available to a large number of people. Although these packages are more user-friendly and powerful, one must not forget to perform a rigorous data analysis backed up by a good geological interpretation, build hypotheses and check them at the different steps of the study, then use the appropriate method for solving the problems brought by the exploitation of the mine. The quality of results depends much more on the initial structural analysis of the data (in particular fitting properly t he model of variogram of the phenomenon) than on the next steps of Kriging or Simulations. Therefore to perform a sound geostatistical study which will rely on a good model describing the spatial characteristics of the phenomenon, one will prefer a software which provides powerful and user-friendly tools for exploratory data analysis and structural modeling.
Fig. 5: Grade profiles along galleries.
