Multigaussian and plurigaussian simulation to quantify uncertainty in estimating ore grades and rock types in copper deposits

The main objective of this researching was to quantify the uncertainty of a copper deposit using multigaussian simulation of grades of rock types. The developed algorithms for simulation of categorical variables, usgsim for continuous variables and algorithms programmed in Python 3.0. The dimensions of a block are 20x20x15m indexed 81x109x58 in xyz; making a total of 512 082 blocks. The results of the categorical simulations show that there is a probability (p.v. ≥ 0.8) of 19 987, 227 387 and 49 036 blocks for rock type 1, 2 and 3, respectively. Consequently, the quantified uncertainty for this threshold (0.8) is 20 %; representing 58 % of the blocks. On the other hand, the quantification of the uncertainty of continuous variables (associated with grades of Cu and Mo), not less than 90, 80 and 70 %, based on cut-off thresholds of variance per block, resulted in lithology One 353, 1900 and 7553 blocks with Cu grades, which represents 0.07, 0.37 and 1.49 % of the total blocks, respectively; a total of 24, 139 and 582 blocks with grades of Mo, Please cite this article as: Cotrina M.A., Vega J.A., Multigaussian and plurigaussian simulation to quantify uncertainty in estimating ore grades and rock types in copper deposits, Journal of Sciences and Engineering, vol. 6, no 1, 2022, pp. 11-19. https://doi.org/10.32829/sej.v6i1.157 SEj Journal of Sciences and Engineering A publication of CINCADER Vol. 6, N° 1, 2022 Centre of Research and Training for Regional Development Online at www.journals.cincader.org Copyright © 2022, CINCADER. ISSN 2523-9503 DOI: https://doi.org/10.32829/sej.v6i1.157 12 representing 0.006, 0.027 and 0.114 % of the total blocks; for lithology Two there were 106, 648 and 2739 blocks with Cu grades, which represents 0.021, 0.127 and 0.535 % of the total blocks; a total of 32, 129 and 357 blocks with grades of Mo, which represents 0.006, 0.025 and 0.070 % of the total blocks; finally for lithology 3 there were 478, 2324 and 8804 blocks with Cu grades, which represents 0.093, 0.454 and 1.719 % of the total blocks; a total of 93, 355 and 982 blocks with grades of Mo, which represents 0.018, 0.069 and 0.192 % of the total blocks. Finally, it is concluded that it was possible to quantify the uncertainty in the estimation of mineral grades and rock types in a copper deposit.


Introduction
Companies that exploit metal and non-metal deposits in the world need geological resources and mineral reserves in order to continue their operations. Geological resources are estimated in quantity (tonnage) and quality (grades or metallic content) through different estimation techniques (Chilès & Delfiner, 2012). There are two techniques, the traditional ones such as: distance inverse, polygons, profiles or others; and the probabilistic ones such as: simple or ordinary kriging. The latter are good estimators of a value at one site using known values at other sites. However, although the estimation techniques are validated in different ways, they have a lot of error. The foregoing will be verified by means of the following example, a mine that we will call "alpha" plans its exploitation thus determining that at the end of the period, generally annual, it will extract "beta" tons of ore with a "theta" grade. Said "theta" grade translates into metal content sold by the mine, which is seen in economic terms as a NPV. This NPV is never fulfilled in the planned amount, the ideal is that there is a variation of ±10%. However, the variation reaches thresholds of ±35%. Consequently, it makes the mining business critical.
There are certain factors that produce this error in the estimate. One of them and the most important is the little information or the reduced number of samples resulting from the diamond drilling that are available to estimate the ore's grade.
All of the above is based on uncertainty, that is, how much error our estimate of geological resources will have. Estimating uncertainty is no small problem. Today, due to powerful software, simulations can be done that result in millions of simulated values, only one simulation process can deliver values that weigh up to a terabyte.
For this reason, it is necessary to use simulations to be able to quantify the uncertainty. In most geological deposits, the metal of interest is never found alone, but is associated with another metal (with directly or inversely proportional values), which are considered secondary due to their economic value. There are several examples, Blenda (ZnS) with Galena (PbS). For the present study, Copper (Cu) is associated with Molybdenum (Mo). Mery (2016) carried out a geostatistical modeling of the uncertainty of grades and rock types in a copper deposit to obtain a master's degree in mining, where the quantification of resources was obtained through a multigaussian model for ore grades, loss of fire and granulometry, and to determine rock types I use a plurigaussian model.

Materials and Methods
The study material consisted of the composite grades of copper, molybdenum and types of rocks from the copper deposit.

Statistical analysis of composite samples
In Table 1, the descriptive statistics of the grades of copper and molybdenum composites at 15 meters length are presented. In total there are 5625 data where the maximum grade for copper is 2.949 and for molybdenum it is 0.237. In Table 2, the number of blocks with lithologies (or rock types) 1, 2 and 3 is observed, it is observed that rock 2 has 55.74%. Histograms of copper and molybdenum grade In Figure 2, the distributions of copper and molybdenum grades are observed, with a positive asymmetry coefficient, and in Figure 3, the distribution of normalized Cu and Mo grades is shown; The foregoing is done with the purpose of transforming the regionalized variable into Gaussian data in order to continue with the study.
The application of anamorphosis of Cu and Mo grades, allowed normalizing the regionalized variables transforming them into Gaussian variables since the multigaussian model requires that the spatial distribution of the random function be Gaussian (Figure 3), as indicated by Journel & Huijbregts (1978) and Dunn (2011). In Figure 4, (a) you can see the spatial correlation cloud of the transformed grades of copper for the geological domain of rock type 1. This is corroborated by a positive slope. As is known, correlation clouds are one of the Bigausianity tests, described by Sucapuca (2018). Figure 4. (b) shows the experimental directional variogram for copper, a geostatistical tool that allows us to know the spatial continuity of the variable of interest. Figure 4. (c) shows the modeled variogram for the copper grade in rock 1. Figure 4. (d) shows the variogram of indicators for rock 1.  In Figure 6 (a) the proportion according to the domain in each realization (scenario) is observed, clearly rock type 2 has a greater dominance over the other rock types, followed by rock type 3 and finally in lesser proportion. type 1 rock (Portilla Lazo, 2018;Díaz & Casar, 2009;Gálvez & Emery, 2012;Mery, 2016). Figure 6 (b) shows the probability of occurrence for rock 1, red color maximum probability and blue color very little probability (rock 2 and 3). Figure 6 (c) shows the probability of occurrence for rock 2, red color maximum probability and blue color very little probability (rock 1 and 3). Figure 6 (d) shows the probability of occurrence for rock 3, red color maximum probability and blue color very little probability (rock 1 and 2).

Uncertainty in fine copper and molybdenum content
The quantification of the uncertainty was established based on the variance of the Cu and Mo grades for each block, establishing 10 cut-off thresholds, where the greatest variability is represented by the highest variance thresholds, for an uncertainty not less than 90%. , 80 and 70%, there are 353, 1900 and 7553 blocks with grades of Cu, which represents 0.06893, 0.37123 and 1.49440% of the total blocks; a total of 24, 139 and 582 blocks with Mo grades, representing 0.00569, 0.02714 and 0.11365% of the total blocks, in rock 1.  1, 2 and 3), facilitating its analysis and simulation. Twenty scenarios were considered, showing a clear relationship between them, whose results also show a relationship in each scenario by lithology, with higher grades in domains 2 and 3 (rock type 2 and 3).
The research managed to quantify the uncertainty in each domain, for a p.v. ≥ 0.8 with an uncertainty of 20%. Above this threshold, the number of blocks obtained is 19,987, 227,387, and 49,036 for each domain (rock type 1, 2, and 3), respectively. The quantification of the uncertainty associated with grades of Cu and Mo based on cut-off thresholds of their variance per block resulted in lithology One of 353, 1900 and 7553 blocks with grades of Cu, which represents 0.06893, 0.37123 and 1.49440% of the total blocks; a total of 24, 139 and 582 blocks with Mo grade, representing 0.00569, 0.02714 and 0.11365% of the total blocks; for lithology two there were 106, 648 and 2739 blocks with grades of Cu, which represents 0.02070, 0.12657 and 0.53487% of the total blocks; a total of 32, 129 and 357 blocks with Mo grades, which represents 0.00644, 0.02539 and 0.06972% of the total blocks; finally, for lithology 3, there were 478, 2,324, and 8,804 blocks with Cu grades, which represents 0.09334, 0.45383, and 1.71926% of the total blocks; a total of 93, 355 and 982 blocks with Mo grades, which represents 0.01816, 0.06932 and 0.19177% of the total blocks, this represents an uncertainty of not less than 90, 80 and 70%, respectively.

Acknowledgments
Special thanks to Dr. Mauricio Garrido, academic from the University of Chile, for his extraordinary support in the development of this research.