Stanford 2009

Modeling of mineral deposits using geostatistics and experimental design Luis P. Braga(UFRJ) Francisco J. da Silva(UFRRJ) Claudio G. Porto(UFRJ) Cassio Freitas(IBGE) International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009 [email_address] http://www.slideshare.net/bragaprof/stanford-2009-1872654

Goal: Improve the mineral resources evaluation of a deposit in the initial stages of exploration. International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009 [email_address] http://www.slideshare.net/bragaprof/stanford-2009-1872654

How: Through experimental design techniques applied to variogram based estimation methods. International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009 [email_address] http://www.slideshare.net/bragaprof/stanford-2009-1872654

Outline of the presentation a)Creating a synthetic study case: Simulate the grade on a regular 3D mesh based on data of a lateritic Ni deposit and calculate the amount of resources. b)Applying designed experiments: Varying the values of the four main parameters of the semivariogram, according to an experimental design. c)Testing the method with kriging: Using a sample, estimate the total resources with kriging by changing the semivariogram parameters values, according to an experimental design. Calculate the different resource totals and compare with a). d)Testing the method with simulation: Repeat c) with simulation as an interpolator. e)Discussion. International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009 a) Simulate the grade on a regular 3D mesh based on a sample of a lateritic Ni deposit, and calculate the amount of resources. Figure 1

a)The data consists of 76 drillholes, located in a grid of 100mx100m having in total 2021 drillholes samples which were collected downhole at 1m interval. The experimental and the adjusted semivariogram in the principal directions were obtained by a geologist . International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009 Figure 2 a Figure 2 c Figure 2 b

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 a) The central values(0) of a spherical semivariogram model parameters, as well as, the minimum(-) and maximum(+) acceptable values to the geologist are presented in Table 1. Table 1

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 3

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Horizontal Section Figure 4

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 b)For each interpolator the sequence presented in Table 2 was applied, obtaining different resources values. For each range and level an average of the resources obtained are calculated, as shown in the next slide, leading to the average effect table.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 N-S range(-)

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Table 3 Average Effect Table

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 b)From Table 3 we build the average effect plots that indicate the best combination regarding the estimation of the resources. We are not proceeding the full designed experiments phases, that is, the regression between the variable with its factors, but only keeping the factors levels assignment strategy.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Horizontal Section

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 bbbbbbbbbbbbbbb Horizontal Section

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 10 (a) Run 2 Figure 10 (b) Run 9 Figure 10 (c) Run 19

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Resources 9, 18 and 27 are equal because they correspond to the same choice of parameters - central values. Table 4 30558.84 R[27] 30558.84 R[18] 30558.84 R[9] 30608.18 R[26] 30379.65 R[17] 30328.81 R[8] 30610.49 R[25] 30712.45 R[16] 28159.78 R[7] 30522.38 R[24] 30277.72 R[15] 30660.73 R[6] 30525.41 R[23] 30612.68 R[14] 30669.34 R[5] 30579.27 R[22] 30809.97 R[13] 30846.89 R[4] 30912.85 R[21] 30811.65 R[12] 30594.60 R[3] 30346.11 R[20] 30547.45 R[11] 30591.69 R[2] 30686.57 R[19] 30549.52 R[10] 30500.94 R[1]

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 c) As mentioned before we calculate the average of the Resources for each set of runs, as arranged by type of parameter and its value. It will allow us to build the so called “average effect plot”. For each row we calculate the average of the corresponding Resources indicated by its run number. That is, for N-S RANGE(-) we take the average between R[1], R[2], R[10], R[11], R[19] and R[20], that happens to be 30,539Kg. The same procedure is repeated for each row, generating a designed average.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 true value = 31365

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 0 0 0 0 9 Sill Vert. range E-W range N-S range run

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 15 Histogram of Errors (DAK) ) Figure 16 Histogram of errors - run 9

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 The histogram of the errors of the geologist choice of parameters (central values) , which corresponds either to run 9, 18 or 27, represents a good match, but DAK performs better. Table 5 Errors statistics for DAK(Vrange(0)): True - Estimated Table 6 Errors statistics for single Kriging (Run 9): True - Estimated

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 17 (a) Horizontal Section level -10m Run 1 Figure 17 (c) Horizontal Section level -10m Run 3 Figure 17 (b) Horizontal Section level -10m Run 2

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Table 7 29763.12 R[27] 30040.92 R[18] 31586.04 R[9] 30548.88 R[26] 30124.75 R[17] 30318.43 R[8] 31650.80 R[25] 31533.80 R[16] 28775.79 R[7] 30719.53 R[24] 30478.49 R[15] 29796.91 R[6] 30981.64 R[23] 30496.09 R[14] 30125.19 R[5] 30582.42 R[22] 31114.96 R[13] 30499.56 R[4] 31184.91 R[21] 30626.19 R[12] 30588.43 R[3] 30400.86 R[20] 30521.16 R[11] 30836.71 R[2] 30541.56 R[19] 30754.89 R[10] 30649.90 R[1]

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 18 (a) Figure 18 (b) Figure 18 (d) Figure 18 (c) 31365 31365 31365

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 The extension of the method to simulation will be called designed averaged simulations (DAS). The results preserved the relation between the quality of the semivariogram and that of the resource estimation.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 The vertical semivariogram is the best one and there is no relevant difference between conservative and central choices, but for optimist choices the impact is negative. The resources estimation is better achieved with DAS(Vrange(0)) than simulation.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 0 0 0 0 9, 18, 27 Sill Vert. range E-W range N-S range run

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 We also evaluated the simulated values for each block, as we did before with kriging, comparing each designed averaged simulated block (DAS) with the true one. Table 8 Errors Statistics for DAS(Vrange(0)): True - Estimated Table 9 Error Statistics for simulation (Vrange(0)): True - Estimated

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 d) conclusions: 1)The gain in the evaluation process was almost 10% 2)The method orientates which simulations or interpolations must be kept. 3)The method allows a better selection of directional semivariograms and its parameters levels. 4) Future work includes tests with other samples and simulation methods.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Send comments to: [email_address] http://www.slideshare.net/bragaprof/stanford-2009-1872654 THANK YOU !

Stanford 2009

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Stanford 2009