GEOMETALLURGICAL SIMULATION OF THE
WORK INDEX IN A PORPHYRY COPPER DEPOSIT
USING GEOSTATISTICAL TECHNIQUES
SIMULACIÓN GEOMETALÚRGICA DEL ÍNDICE DE TRABAJO
EN UN DEPÓSITO PÓRFIDO CUPRÍFERO UTILIZANDO
TÉCNICAS GEOESTADÍSTICAS
Ramos Armijos Nelson Jesus
Universidad Nacional Mayor de San Marcos, Perú
Calderón Celis Marilú
Universidad Nacional Mayor de San Marcos, Perú
pág. 807
DOI: https://doi.org/10.37811/cl_rcm.v8i3.11288
Geometallurgical Simulation of the Work Index in a Porphyry Copper
Deposit Using Geostatistical Techniques
Nelson Jesus Ramos Armijos1
nramos_5215@hotmail.com
nelson.ramos1@unmsm.edu.pe
https://orcid.org/0000-0001-9188-6422
Unidad de Posgrado
Facultad de Ingeniería Geológica
Minera, Metalúrgica y Geográfica
Universidad Nacional Mayor de San Marcos
Lima – Perú
Calderón-Celis Marilú
jcalderond2@unmsm.edu.pe
https://orcid.org/0000-0002-1374-9307
Unidad de Posgrado
Facultad de Ingeniería Geológica
Minera, Metalúrgica y Geográfica
Universidad Nacional Mayor de San Marcos
Lima – Perú
ABSTRACT
The spatial variability in the geometallurgical attributes of the deposits is a crucial parameter from the
exploration stage, which conditions and influences the mineral processing. Consequently, the objective
of this research is to elaborate the geometallurgical simulation of the Bond Work Index for a porphyry
copper deposit. For this purpose, information of primary and response attributes corresponding to ore
zones, lithologies and BWi contained in 1,449 samples of exploratory drill holes were used. An
exploratory data analysis of this information was carried out, and geometallurgical units were defined
based on the geological and processing knowledge that validates the behavior of each one of them
within the deposit; then Sequential Gaussian Simulation was applied, running 100 realizations in each
GMU, those that best reproduce the statistics of the original samples were chosen. The results show that
the lithology of the deposit controls the BWi variability and according to the rock competence the ore
zones are classified from the softest to the hardest in oxides, mixed and sulfides.
Keywords: geometallurgy, mineral deposit, geostatistics, simulation, processing
1
Autor principal
Correspondencia: nramos_5215@hotmail.com
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Simulación Geometalúrgica del Índice de Trabajo en un Depósito Pórfido
Cuprífero Utilizando Técnicas Geoestadísticas
RESUMEN
La variabilidad espacial en los atributos geometalúrgicos de los depósitos es un parámetro crucial desde
la etapa de exploración, lo cual condiciona e influye en el procesamiento mineral. En consecuencia, el
objetivo de esta investigación es elaborar la simulación geometalúrgica del Índice de Trabajo de Bond
para un depósito pórfido cuprífero. Para esto se utilizó información de atributos primarios y de respuesta
correspondientes a zonas minerales, litologías y de BWi contenidos en 1,449 muestras de sondajes
exploratorios. Se realizó el análisis exploratorio de datos de dicha información y se definieron unidades
geometalúrgicas fundamentándose en el conocimiento geológico y de procesamiento que valida el
comportamiento de cada una de ellas dentro del depósito; luego se aplicó Simulación Secuencial
Gaussiana, ejecutando 100 realizaciones en cada UGM y se eligieron aquellas que reproducen mejor
las estadísticas de las muestras originales. Los resultados muestran que la litología del depósito controla
la variabilidad del BWi y de acuerdo a la competencia de la roca las zonas mineralizadas se clasifican
desde la más blanda a la más dura en óxidos, mixtos y sulfuros.
Palabras clave: geometalurgia, depósito mineral, geoestadística, simulación, procesamiento
Artículo recibido 10 abril 2024
Aceptado para publicación: 20 mayo 2024
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INTRODUCTION
Currently, the exploratory stage of deposits faces challenges associated with geometallurgical
uncertainty (Lishchuk et al., 2020; Mwanga et al., 2015) which impacts mineral processing (Dominy et
al., 2018). In this context, response attributes influence the resource value because up to 70 % of the
total energy consumption is used in comminution (Mohammadi et al. 2021); therefore, this impacts
porphyry copper deposits characterized by high lithology hardness (Bilal, 2017). In this regard, the
Bond Work Index “BWi” is frequently used for the calculation of the energy requirement (Aras et al.,
2019); consequently, it is essential to evaluate its spatial variability within the deposit, however, its
limited information in early stages of mining (Garrido et al., 2020) and high cost of metallurgical tests
(Ranjbar et al., 2021) hinder its characterization.
In this sense Harbort et al. (2011) performed geometallurgical modeling for a porphyry copper-gold
deposit located in Peru. Nghipulile et al. (2023) studied the effect of mineralogy on the milling of copper
oxides and sulfides in a deposit located in Namibia. Harbort et al. (2013) applied geometallurgy to
estimate comminution attributes in porphyry copper deposits.
Therefore, the objective of this work is to elaborate the geometallurgical simulation of the Bond Work
Index, by incorporating primary and response attributes of a porphyry copper deposit, using
geostatistical techniques.
Geometallurgy
Geometallurgy incorporates geological, metallurgical, mine planning information to improve decision
making in mining projects (Mu & Salas, 2023), for which it makes use of primary and response variables
(Castro et al., 2022), and predictive spatial models (Castillo et al., 2022).
A geometallurgical variable is defined as any attribute of the rock that positively or negatively affects
the value of a mineral deposit and is classified into primary and response variables (Coward et al.,
2009). Primary variables are intrinsic to the rock, directly measured and are used to predict metallurgical
response and are additive, e.g., ore grade, lithologies, mineralized zones (Morales et al., 2019).
Whereas, response variables are rock attributes that describe the response to a processes or energy
application (Morales et al., 2019), they are non-additive and the characterization of their spatial
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variability is elaborated by geostatistical simulation (Hosseini & Asghari, 2015), e.g., Bond Work
Index, throughput, reagent consumption, processing capacity, recovery.
The Bond Work Index is a variable which represents a measure of an ore´s resistance to grinding and it
represents the energy (kWh/t) required to reduce the material of one short ton from a theoretically
infinite feed size to size at which 80 percent of material passes through sieve with square aperture of
100 micrometers in size (Todorovic et al., 2017). Bilal (2017) classifies BWi values according to rock
competence into soft (7-9), medium (9-14), hard (14-20), and very hard (>20).
On the other hand, the geometallurgical model is a 3D space that is typically synthesized from early-
stage small-scale samples to predict the process response based on the location of samples in a deposit
(Lishchuk et al., 2020).
Geostatistical simulation
It is a technique that allows obtaining realizations that reproduce the statistics and spatial variability of
the original data (Narciso et al., 2019) achieving an unsmoothed representation of reality (Abzalov,
2016). In the study of geometallurgical variables Sequential Gaussian Simulation "SGS" is used
(Hosseini & Asghari, 2015), which requires normal distribution in the samples. The simulated value
is determined by Equation 1 (Abzalov, 2016).
( 1 )
Where
is SGS simulated value,
is Simple Kriging “SK” estimate, is standard deviation of
the Kriging estimate and is a random normal function. It should be noted that Sequential Gaussian
Simulation is applied to normal random functions, however, geometallurgical variables are generally
not symmetrically distributed (Adeli, 2018), therefore, their gaussian anamorphosis must be performed
beforehand. The Simple Kriging is used to calculate the conditional cumulative distribution function
for SGS, which requires knowing the mean value of the variable under study, expressed through
Equation 2 (Abzalov, 2016).
( 2 )
Where are the SK weights assigned to each sample . SGS involves modeling variograms to
establish directions of anisotropy and model performance is evaluated by cross-validation (Ekolle et al.,
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2022), ensuring that the correlation coefficient in its scatter plot between predicted and actual values is
close to 1 and that its error histogram tends to symmetric behavior (Rossi & Deutsch, 2014).
METHODOLOGY
Type and design of research
This is an applied and non-experimental research in which the variables have not been deliberately
manipulated, that is to say, the phenomenon has been observed in its natural context (Hernández-
Sampieri & Mendoza, 2018). The design is cross-sectional correlational. In addition, considering the
type of data, the approach of this study is quantitative and qualitative.
Unit of analysis and study population
The unit of analysis is a porphyry copper deposit and the study population consist sixty-one exploratory
drill holes.
Size and selection of the sample
The sample size is 1,449 data points for Bond Work Index, ore zones and lithologies.
Data collection techniques
The data were collected in formats established worldwide for the elaboration of the geometallurgical
simulation. According to Rossi & Deutsch (2014), the information should be systematized in: Header,
Survey, Assays, Lithology, Minzone.
Analysis and interpretation of information
The geological and block model for the mineral deposit was developed in RecMin software. Exploratory
data analysis, definition of geometallurgical units “GMUs”, sample visualization and results were
performed in Jupyter Notebook. Whereas, the simulation was carried out in SGeMS software.
RESULTS
Geology of the study area
The mineral deposit is a porphyry copper located in Peru, for confidentiality, the coordinates have been
modified. Its mineralization (Figure 1a) is formed by three zones: the highest zone of oxides which is
underlain by a mixed zone and below this the primary sulfides zone; while the lithological model
(Figure 1b) is composed of intrusive and extrusive igneous rocks.
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Exploratory data analysis
Figure 2 shows the spatial distribution of the BWi samples. Their overall statistics (Figure 3) indicate
that there are 1,449 data points with mean 17.07, standard deviation (Std) 2.35, minimum value 12.13.
The first, second, and third quartile (Q) is 15.66, 16.40, 18.80 and its maximum value is equal to 21.99.
The skewness factor (Skew) is 0.27 and kurtosis (Kurt) -0.65; therefore, the distribution of its histogram
has a platykurtic behavior (Figure 3).
Figure 1. Visualization of BWi samples. a) Three-dimensional. b) Plan view
Figure 2. Histogram and samples statistics of BWi
Sample statistics (Figure 4) by ore zones show that oxides and mixed contain the most data, while
sulfides contain the least. In addition, the mean BWi in each zone is well differentiated (Table 1 and
Figure 5).
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Figure 3. Visualization of BWi samples by ore zones. a) Three-dimensional. b) Plan view
Table 1. BWi statistics by ore zones
Ore Zones
# Samples
Mean
Std
Min
Q1
Q2
Q3
Max
Oxides
741
15.42
1.25
12.13
15.38
15.76
16.28
17.35
Mixed
473
17.77
1.51
14.76
16.21
18.35
18.88
20.20
Sulfides
235
20.90
0.57
19.58
20.45
20.78
21.47
21.99
Figure 4. a) Histogram of BWi by ore zones. b) BWi sample mean plot
Statistics for BWi samples in lithologies (Figure 6) indicate that breccia and granite rock types are the
least competent, while andesite, biotite granodiorite and granodiorite are the hardest. (Table 2 and
Figure 7).
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Figure 5. Visualization of BWi samples by lithologies. a) Three-dimensional. b) Plan view
Table 2. BWi statistics by lithologies
Lithologies
# Samples
Mean
Std
Min
Q1
Q2
Q3
Max
Andesite (L1)
407
17.31
1.85
15.00
15.67
16.20
18.66
20.95
Biotite Granodiorite (L2)
347
18.05
2.13
15.20
15.85
18.41
19.80
21.99
Breccia (L3)
282
14.59
1.50
12.13
13.00
15.12
15.80
17.83
Granite (L4)
45
14.83
1.44
13.00
13.60
13.90
16.47
16.75
Granodiorite (L5)
368
18.07
2.11
15.45
16.40
16.73
19.95
21.93
Figure 6. a) Histogram of BWi by lithologies. b) BWi sample mean plot
Definition of geometallurgical units
To develop the BWi simulation, geometallurgical units "GMUs" of the deposit were considered, so that
each GMu is a 3D spatial section of a mine body with similar geological and metallurgical
characteristics.
Since each mineralized zone has a different distribution in the response variable (Figure 8a), samples
from different ore zone will not be considered to define GMUs. Furthermore, as specified
in Figures 8b, 8c and 8d, lithology controls the distribution of BWi in the ore deposit; therefore,
considering the number of mineralized zones and lithologies there could be thirteen GMUs. However,
pág. 815
when verifying the rock types present per mineral zone there can be up to thirteen GMUs and among
these five GMUs (Table 3) have been defined taking into account the similarity in the behavior of the
BWi.
Figure 7. Violin plots. a) Ore zones. b) Oxides – Lithologies. c) Mixed – Lithologies.
d) Sulfides – Lithologies
Table 3. BWi statistics by GMUs
GMUs
Ore
Zones
Lithologies
# Samples
Mean
Std
Min
Q1
Q2
Q3
Max
GMU 1
Oxides
L1, L2, L5
584
16.01
0.45
15.00
15.67
15.90
16.40
17.35
GMU 2
L3, L4
157
13.19
0.54
12.13
12.86
13.05
13.57
15.26
GMU 3
Mixed
L1, L2, L5
303
18.79
0.64
17.40
18.40
18.70
19.31
20.20
GMU 4
L3, L4
170
15.95
0.62
14.76
15.50
15.81
16.34
17.83
GMU 5
Sulfides
L1, L2, L5
235
20.90
0.57
19.58
20.45
20.78
21.47
21.99
Geometallurgical simulation of BWi
Preliminary gaussian anamorphosis was performed on the BWi data shown in Table 3, to obtain a
symmetrical distribution in the samples (Table 4 and Figure 9).
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Table 4. Statistics of BWi transformed into gaussian variable
GMUs
# Samples
Mean
Std
Min
Q1
Q2
Q3
Max
GMU 1
584
0
1
-2.927
-0.665
0.011
0.676
2.927
GMU 2
157
0
1
-2.493
-0.665
0
0.665
2.493
GMU 3
303
0
1
-2.717
-0.664
0.016
0.669
2.717
GMU 4
170
0
1
-2.521
-0.634
0
0.665
2.521
GMU 5
235
0
1
-2.633
-0.648
0
0.674
2.633
Figure 8. Histograms of original and transformed samples to gaussian variable by GMUs
Note: BWi_ T (BWi transformed)
Exponential-type variograms were modeled in each GMU (Table 5 and Figure 10) to identify the spatial
continuity and directions of anisotropy (major axis, semimajor axis and minor axis) that will form the
search ellipsoid. In addition, a leave-one-out type cross-validation (Figure 11) of the modeled
variograms was performed, that is to say, each known sample is successively removed from the dataset
and a new value is predicted by Simple Kriging at that location using the other samples, indicating the
difference between the actual and predicted value to what extent the data value fits the neighborhood
of the nearby samples. Subsequently, the BWi simulation was performed on the block model developed
for the GMUs (Figure 12).
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Table 5. Parameters for modeled variograms
GMUs
Major Axis
Semimajor Axis
Minor Axis
Azimuth
(°)
Dip
(°)
Range
(m)
Sill
Azimuth
(°)
Dip
(°)
Range
(m)
Sill
Azimuth
(°)
Dip
(°)
Range
(m)
Sill
GMU 1
67.5
0
250
0.9
157.5
0
200
0.9
0
-90
150
0.9
GMU 2
135
0
230
1.1
45
0
200
1.1
0
-90
150
1.1
GMU 3
22.5
0
290
1
112.5
0
180
1
0
-90
160
1
GMU 4
0
0
230
1
90
0
170
1
0
-90
120
1
GMU 5
67.5
0
260
1
157.5
0
225
1
0
-90
135
1
Figure 9. Modeled variograms (MV) by GMUs
Note: S (Sill), R (Range)
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Figure 10. Histogram of errors and scatter plot between true values and predicted values by GMUs
Figure 11. Geometallurgical block model
Note: The size of each block is 10 meters
Sequential Gaussian Simulation was applied to elaborate BWi realizations using its data transformed
to a normal distribution with a mean of 0 and variance of 1. The number of realizations for BWi in each
GMU was 100 as recommended by Bai & Tahmasebi (2022); then the values were brought to their
initial units through inverse anamorphosis and finally, the optimal simulations were defined,
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considering those that best reproduce the mean and histogram of the original samples. The results
obtained are shown in Table 6 and Figures 13 and 14.
Table 6. Statistics for original samples and optimal realizations by GMUs
Description
# Samples
Mean
Std
Min
Q1
Q2
Q3
Max
GMU 1
584
16.01
0.45
15.00
15.67
15.90
16.40
17.35
Realization #72
21,224
16.01
0.45
15.00
15.66
15.87
16.39
17.35
GMU 2
157
13.19
0.54
12.13
12.86
13.05
13.57
15.26
Realization #25
11,247
13.20
0.57
12.13
12.87
13.06
13.58
15.26
GMU 3
303
18.79
0.64
17.40
18.40
18.70
19.31
20.20
Realization #18
16,276
18.80
0.63
17.40
18.41
18.70
19.33
20.20
GMU 4
170
15.95
0.62
14.76
15.50
15.81
16.34
17.83
Realization #63
13,024
15.95
0.57
14.76
15.53
15.87
16.30
17.83
GMU 5
235
20.90
0.57
19.58
20.45
20.78
21.47
21.99
Realization #99
6,710
20.89
0.62
19.58
20.38
20.75
21.50
21.99
Figure 12. Mean plots for BWi realizations and histograms of optimal realizations
Note: Sky blue color indicates mean of original samples in each GMU
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Figure 13. Spatial (left) and 2D (right) visualization of optimal realizations for GMUs
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DISCUSSION
The application of Sequential Gaussian Simulation shows results in accordance with the reality of the
phenomenon, since the statistics and spatial variability of the original BWi samples are better
reproduced; in this sense, the degree of uncertainty is significantly reduced.
Likewise, through the findings found in this research it is established that the mineralized zones in the
porphyry copper deposit studied in terms of hardness, the primary sulfide zone is the most competent,
followed by mixed and finally oxides; which is related to the works developed by Harbort et al. (2013),
Harbort et al. (2011) and Nghipulile et al. (2023) ; however, in each study, there is a variation in the
values for BWi due to the specific characteristics of deposits (Figure 15).
Figure 14. BWi values determined for different porphyry copper deposits by ore zones
CONCLUSIONS
Through the research carried out, it was determined that the lithology of the porphyry copper deposit
studied is a geometallurgical attribute that influences and controls the variability related to comminution
in each ore zone.
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According to the results obtained and considering the competence of the mineralized zones by way of
BWi, these are classified from the softest to the hardest in oxides, mixed and sulfides.
By means of the simulation elaborated in GMUs, it has been possible to obtain multiple realizations of
the BWi and evaluate its variability, adequately representing the proportion of high and low values, the
spatial complexity of the deposit and the continuity of the geometallurgical variable three-
dimensionally.
ACKNOWLEDGMENTS
To the Vice Rectorate of Investigation and Postgraduate Studies of National University of San Marcos,
for the support provided to the Research Project with Code C231609735e, for the academic publication
of this scientific article.
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