stanford 2009

Eight International Geostatistics Congress. Santiago, Chile 1

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


Goal: Improve the mineral resources evaluation of a deposit in the initial

stages of exploration



How: Through experimental design techniques applied to variogram

based estimation methods.



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



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.

Figure 2 a

Figure 2 c

Figure 2 b


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

Minimum(-) Central(0) Maximum(+) Range(NS) 180 200 240 Range(EW) 280 300 340 Range(vertical) 7,5 9 10,5 Sill 0,119 0,139 0,149 Nugget effect 0,000 0,01 0,030

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



a) A three dimensional model of the selected region of the deposit, shown in Figure 1, was built using the Gauss Simulation algorithm as implemented in the package GSTAT of the R environment. It consists of 5400 blocks with support 10m x 20m x 5m. The semivariogram used in the simulation was the one with the central values, as in Table 1.



a) The simulated variable is the grade in percentage. Density is considered constant and equal to 1Kg per cubic meter. After simulating the grades one can easily calculate the amount of nickel through the basic equation R= VDT (where V stands for volume, D for density and T for grade). This formula is applied to each block and then summing it up. The obtained value was 31,365 Kg.



a) In the Figure 3 it is depicted the vertical section of the simulated deposit. In figures 4 to 6 it is shown horizontal sections at different depths.

Figure 3



Horizontal Section

Figure 4



Horizontal Section

Figure 5



Horizontal Section

Figure 6



b) Experimental design principles can be used in uncertainty assessment for more realistic sensitivity analysis. There are hundreds of different schemes that can be adopted. We opted for the Box-Behnken design with four factors and three levels (the nugget effect will be kept constant and equal to 0.01) which focuses in the central values, for instance: N-S range, E-W range, Vertical range and Sill. As response variable we took the total resources of the deposit, coded as V.



b)For each interpolator the sequence presented in

Table 2 was applied, obtaining diferent 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.



N-S range(-)



Table 3



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.


c) The first experiment was done with ordinary kriging

to generate a three dimensional model of the deposit

with the same dimensions, that is 5400 blocks with

support 10m x 20m x 5m. In the sequence we can see

horizontal sections for some levels obtained from run

1 for variable Ni. We have obtained all levels for every

run, but for sake of conciseness, just a few are shown.




Figure 7 Horizontal Section at level -5m , Kriged according to run 1

Horizontal Section



bbbbbbbbbbbbbbb


Horizontal Section




Horizontal Section



c) Next, we show the same level for different runs.

Although, the difference between the same levels,

from the different runs, are less visible, there may

be large discrepancies for the total volumes, with

some up to 2,500 Kg. ! In the Table 4 we show the

total amount of Ni as calculated for each run.



Figure 10 (a) Run 2

Figure 10 (b) Run 9

Figure 10 (c) Run 19



Volumes 9, 18 and 27 are equal because they correspond to the same choice of parameters, central values.

Table 4



c) As mentioned before we calculate the average of the volumes 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 volumes indicated by its run number. That is, for N-S RANGE(-) we take the average between vol[1], vol[2], vol[10], vol[11], vol[19] and vol[20], that happens to be 30,539Kg. The same procedure is repeated for each row, generating a designed average.



true value = 31365



The results stress the relationship between the quality of the semivariogram and that of the resource estimation. The N-S and E-W semivariograms are not good and their impact on estimation is straightforward. Conservative choices of the range imply in better estimations.



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 sill also is based on the best semivariogram and the central value is the more effective for estimating the overall resources.



It is important to remind that the estimation is done by averaging a selection of runs, instead of just one kriging interpolation. The experimental design give us a new range of interpolators based on the quality of the semivariograms. In this case, the average of fifteen different kriging interpolations, see row “V range (0)” in Table 4, which gives an amount of 30,625 Kg of Ni.



The combined estimation was superior to the ordinary kriging alone. For example, a natural choice would be the central values for every semivariogram parameter, that is, run 9 with 30,558 Kg which is poorer than the former average (30,625 Kg), as the true value is 31,365 Kg.



Besides improving resources estimation, the designed averaged kriging (DAK) does better than ordinary kriging to estimate blocks, which enable us to produce more realistic maps of the deposit. Consider the histogram of the errors of blocks, obtained by comparing the true amount of Ni in each block with the one estimated by DAK (V range(0)).



Figure 15 Histogram of Errors (DAK)) Figure 16 Histogram of errors - run 9



The summary of the errors is given in Table 5, where density is equal to 1kg per m3, therefore the maximum amount of Ni in a block being 1000kg.

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.



The methodology may be applied to improve the performance of any semivariogram based interpolator. The aleatorization of the runs is quite natural in the case of simulation as interpolator, but it may also be done with “kriging like” interpolators by using any technique of “resampling”. In the next section we present an analogous experiment with simulation as an interpolator.



d)Following the same guidelines used in the

previous section, 27 simulations of the deposit

were generated according to Tables 1 and 2. As

the seed of the random generator was initialized

only once, we did not obtain the same values for

runs 9, 18 and 27. For each run just one

simulation was obtained.



d) In the sequence one can observe horizontal

sections for some levels obtained from the run 1

for variable Ni. We have obtained all levels for

every run, but for the sake of conciseness, just a

few are shown.



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



The equivalent runs 9, 18 e 27 now have different valuesbecause of the aleatorization of the simulation process.



Figure 18 (a) Figure 18 (b)

Figure 18 (d)Figure 18 (c)

31365 31365

31365



The extension of the method to simulation will be called designed averaged simulations (DAS). The results did not follow exactly the same patterns observed before, but preserved the relation between the quality of the semivariogram and that of the resource estimation.



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 than exclusive simulation.



The DAS for the vertical range shows a total of 30,726 Kg, while the average of the equivalent simulations runs 9, 18 and 27 shows 30,463 Kg, with the true value equal to 31,365 Kg.



Both present good results, but DAS (V(0)) is lessbiased and has lower variance than V(0).

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.



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.



THANK YOU !

stanford 2009

Education

mathematical geosciences

chileinternational association

experimental designluis

experimental design

total resources

lateritic ni deposit

adjusted semivariogram

mineral resources evaluation