LICENTIATE T H E S I S

Department of Civil, Environmental and Natural Resources EngineeringDivision of Minerals and Metallurgical Engineering

ISSN 1402-1757ISBN 978-91-7583-904-2 (print)ISBN 978-91-7583-905-9 (pdf)

Luleå University of Technology 2017

Mineral Processing

Particle Generation for Geometallurgical Process Modeling

Pierre-Henri KochPierre-H

enri Koch: Particle G

eneration for Geom

etallurgical Process Modeling

Particle generation for geometallurgical process modeling

Pierre-Henri Koch

piekoc@ltu.se

Licentiate thesis in Mineral Processing

Division of Minerals and Metallurgical Engineering (MiMeR)

Department of Civil, Environmental and Natural Resources Engineering

Luleå University of Technology

SE-971 87 Luleå

Sweden

Supervisors

Jan Rosenkranz

Cecilia Lund

Pertti Lamberg

Printed by Luleå University of Technology, Graphic Production 2017

ISSN 1402-1757 ISBN 978-91-7583-904-2 (print)ISBN 978-91-7583-905-9 (pdf)

Luleå 2017

www.ltu.se

Part I

i

ABSTRACT

A geometallurgical model is the combination of a spatial model representing an ore deposit and

a process model representing the comminution and concentration steps in beneficiation. The process model itself usually consists of several unit models. Each of these unit models operates at a given level of detail in material characterization - from bulk chemical elements, elements by size, bulk minerals and minerals by size to the liberation level that introduces particles as the basic

entity for simulation (Paper 1).

In current state-of-the-art process simulation, few unit models are defined at the particle level because these models are complex to design at a more fundamental level of detail, liberation data

is hard to measure accurately and large computational power is required to process the many

particles in a flow sheet. Computational cost is a consequence of the intrinsic complexity of the unit models. Mineral liberation data depends on the quality of the sampling and the polishing, the settings and stability of the instrument and the processing of the data.

This study introduces new tools to simulate a population of mineral particles based on intrinsic characteristics of the feed ore. Features are extracted at the meso-textural level (drill cores) (Paper

2), put in relation to their micro-textures before breakage and after breakage (Paper 3). The result is a population of mineral particles stored in a file format compatible to import into process simulation software. The results show that the approach is relevant for geometallurgy and can be generalized towards new characterization methods.

The theory of image representation, analysis and ore texture simulation is briefly introduced and

linked to 1-point, 2-point, and multiple-point methods from spatial statistics. A breakage mechanism is presented as a cellular automaton. Experimental data and examples are taken from a copper-gold deposit with a chalcopyrite flotation circuit, an iron ore deposit with a magnetic

separation process.

This study is covering a part of a larger research program, PREP (Primary resource efficiency by

enhanced prediction).

Keywords: Geometallurgy, ore texture, texture classification, breakage simulation, process modeling

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ACKNOWLEDGEMENTS

I wish to thank my supervisors for their support in both practical and technical aspects. Professor

Jan Rosenkranz for keeping this work on track by discussions and reviews. Professor Pertti Lamberg for his ideas and previous work on liberation and simulation. Cecilia Lund for her expertise in geology, daily discussions, and help for characterization.

This study is part of PREP (Primary resource efficiency by enhanced prediction), a project within the Swedish Mining Research program SIP-STRIM and partially funded by VINNOVA. I am

grateful to our partners LKAB, Boliden, Outotec, Lundin Mining and Chalmers University of Technology.

The help of Antti Remes and Antti Roine from Outotec, Therese Lindberg and Mattias Gustafsson from LKAB and Adam McElroy from Boliden was particularly appreciated.

My colleagues from the MiMeR department from Luleå University of Technology are also a powerful support. I would like to thank in particular Viktor Lishchuk for his ideas about PREP

and his work in spatial modeling as well as his help in characterizing the samples we collected and Mehdi Parian for providing interesting discussions about textures and providing data. Erdogan Kol for his understanding of geometallurgy and fruitful discussions, and Bertil Pålsson

for his experience in mineral processing and ideas about simulation are also kindly thanked. I wish to thank Ulf Nordström for his help in the laboratory.

I wish to thank my family and my fiancée Coralie for their patience and constant support.

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LIST OF PAPERS

This licenciate thesis is based on the following papers

- Paper 1

Koch, P.-H., Lamberg, P., Rosenkranz, J., 2015. How to build a process model in a geometallurgical program? in: Andre-Mayer, A., Cathelineau, M., Muchez, P., Pirard, E., Sindern, S. (Eds.), Proceedings of the 13th SGA Biennial Meeting. Nancy, p. 85.

- Paper 2

Koch, P.-H., Lund, C., Lishchuk, V., Lamberg, P., Rosenkranz, J., 2017. Automated classification of drill cores textures for geometallurgy. (Submitted to Minerals Engineering)

- Paper 3

Koch, P.-H., Rosenkranz, J., 2017. Texture-based liberation models for comminution, in:

Proceedings of the Conference in Minerals Engineering. Luleå, Sweden, pp. 83–96.

The following paper is not included in the thesis

Lishchuk, V., Koch, P.-H., Lund, C., Lamberg, P., 2015. The geometallurgical framework.

Malmberget and Mikheevskoye case studies., in: XVth Conference of PhD Students and Young Scientists. Szlarska Poreba (Poland).

Koch, P.-H., Lund, C., Lishchuk, V., Lamberg, P., Rosenkranz, J., 2017. Automated classification of drill cores textures for geometallurgy, in: Proceedings of Process Mineralogy 2017. Cape Town, South Africa.

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TOOLS DEVELOPED

- Matlab code for texture simulation by Markov Fields (stationary and non-stationary q-

Potts models with hyperparameter estimation by adaptive stochastic gradient descent) - Matlab code for feature extraction on RGB images - WEKA classification models for RGB images (random forests, k-NN, SVM and ANN) - Matlab code for texture breakage for generation of particles

- HSC Sim model for simulation of Aitik flotation circuit

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Contents 1. Introduction .................................................................................................................... 1

2. Process model in geometallurgy ....................................................................................... 4

2.1. Traditional approach ................................................................................................ 6

2.2. Approach based on proxies ...................................................................................... 6

2.3. Mineralogical approach ............................................................................................ 7

2.4. Mass balancing ......................................................................................................... 8

2.5. Process simulation ................................................................................................. 11

2.5.1. Elements ........................................................................................................ 12

2.5.2. Minerals ......................................................................................................... 12

2.5.3. Particles ......................................................................................................... 13

3. Ore textures ................................................................................................................... 14

3.1. Definitions ............................................................................................................ 15

3.1.1. Representation of a texture ............................................................................ 15

3.2. Acquisition of an image or a mineral map .............................................................. 16

3.3. Texture modeling .................................................................................................. 17

3.3.1. Random functions ......................................................................................... 17

3.3.2. One-point methods ....................................................................................... 18

3.3.3. Two-point methods ....................................................................................... 19

3.3.4. Harmonic methods ........................................................................................ 22

3.3.5. Multiple-point methods ................................................................................. 23

3.3.6. Wavelets ........................................................................................................ 26

3.3.7. Morphology .................................................................................................. 27

3.4. Texture classification ............................................................................................. 29

3.4.1. Feature extraction .......................................................................................... 29

3.4.2. Texture classification algorithms ..................................................................... 30

3.4.3. Pixel classification method ............................................................................. 32

3.5. Texture simulation ................................................................................................ 32

3.6. Markov fields methods ....................................................................................... 33

4. Breakage modeling ........................................................................................................ 38

5. Experimental results ....................................................................................................... 40

5.1. Meso-texture classification ..................................................................................... 40

5.2. Meso-texture pixel classification ............................................................................ 45

5.3. Micro-texture simulation ....................................................................................... 47

5.4. Micro-texture breakage ......................................................................................... 56

6. Conclusion .................................................................................................................... 58

7. References ..................................................................................................................... 60

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List of figures Figure 1 : Geometallurgical model based on particles (Lamberg, 2011a), modified ................... 2 Figure 2 : Overview of the process model ............................................................................... 4 Figure 3 : Example of flowsheet for the concentration of magnetite ........................................ 5 Figure 4 : Units and streams inside the process model .............................................................. 9 Figure 5 : Different level of details in process simulation ........................................................ 13 Figure 6 : Regular cubic tiling and equivalent graph ............................................................. 15 Figure 7 : 4-neighborhood (left) and 8-neighborhood (right) ................................................ 16 Figure 8 : LBP with P=8 and R=1 (Ojala et al., 2002), modified .......................................... 21 Figure 9 : Transformed image I (2) after convolution by kernel h. Yellow pixels are high values

and blue pixels are low values................................................................................................ 25 Figure 10 : Original image I (1) ............................................................................................. 25 Figure 11 : Transformed image I (2) after convolution by kernel h2. Deep blue pixels are low values and green pixels are high values. ................................................................................. 25 Figure 12 : Example of a scaling and wavelet functions (Daubechies wavelets db6) ................ 27 Figure 13 : Effects of binary dilation and erosion operators (ball with an 8 pixels radius) on a particle .................................................................................................................................. 28 Figure 14 : Texture classification process ............................................................................... 29 Figure 15 : Process of texture classification and simulation ..................................................... 37 Figure 16 : Chalcopyrite grade and recovery from simulation. The square marker indicates actual data from the plant survey. .................................................................................................... 44 Figure 17 : A pixel classifier after training on three different images. ...................................... 46 Figure 18 Validation of a pixel classifier trained on five different images of the same texture. . 46 Figure 19 : Comparison between the training image and the simulated texture, visual comparison (top) and grain size distribution by phase (bottom) ................................................................ 48 Figure 20 : Initial mineral map, courtesy of Mehdi Parian (LTU) .......................................... 50 Figure 21 : Training mineral map .......................................................................................... 51 Figure 22 : Simulated mineral map ........................................................................................ 51 Figure 23 : Scatterplot of the association matrices values for the training map and the simulated map ...................................................................................................................................... 52 Figure 24 : Scatterplot of the cumulative grain distribution values for the training map and the simulated map ....................................................................................................................... 53 Figure 25 : Piece of drill core from Malmberget iron ore deposit. The main phases are magnetite, amphibole and feldspar .......................................................................................................... 54 Figure 26 : Training image and one realization by MS-CCSIM ............................................. 55 Figure 27 : Example of preferential breakage simulation in magnetite ore .............................. 56 Figure 28 : Different simulated grain size distributions with the same modal mineralogy ........ 57 Figure 29 : Simulated degree of liberation per phase for the cases from Figure 27 .................. 57

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ABBREVIATIONS AND SYMBOLS

Abbreviations

1D one-dimensional

2D two-dimensional

3D three-dimensional

HSC HSC Chemistry software by Outotec

SEM Scanning Electron Microscope

EDS Energy-dispersive Spectroscopy

Auto-SEM Automated SEM

XRD X-Ray Diffraction

XRF X-Ray Fluorescence

AAS Atomic absorption spectroscopy

PSD Particle size distribution

FFT Fast-fourier transform

LBP Local binary patterns

GLCM Gray-level co-occurrence matrix

AI Association matrix

W Normalized association matrix

RF Random forest

SVM Support vector machine

kNN k-Nearest neighbors

ANN Artificial neural networks

MRF Markov random field

MS-CCSIM Multi-scale cross-correlation simulation

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Symbols

Ω Set of all possible realizations

A Set of events

Pr Probability measure

g grain size distribution

Z(x,ω) Random function

Z(x) Random variable

z(x) Realization of a random function

m(x) Mean

Variance

Covariance

C(h) Covariance of wide-sense stationary random function

V Volume

N(h) Number of pairs of points separated by a distance h

^ Fourier transform

* Estimator or convolution operator

E(X,B) Binary erosion of X by B

D(X,B) Binary dilation of X by B

M Margin

U Gibbs measure of energy

β Regularization factor

L Likelihood

φ Potential of the field

e Error

γ Mineral mapping

GIC Fracture toughness [Jm-2]

1

1. Introduction

This licenciate thesis presents introductory results and future directions about the design of a process model in the context of geometallurgy.

Geometallurgy is a multi-disciplinary approach that links geology, ore properties and behavior of the feed in metallurgical operations (Lamberg, 2011a; McQuiston and Bechaud, 1968; Vann

et al., 2011).

A geometallurgical program is the industrial result of geometallurgy in a mining operation context. It includes a spatial model that forecasts how the ore will behave in mineral processing operations (Aasly and Ellefmo, 2014; Alruiz et al., 2009; Lund et al., 2013; Schouwstra et al.,

2013). It consists of two main sub-units:

- Spatial model: 3D block model in which geometallurgical data is available. This model can be used for data management, visualization, for production planning and to test different production scenarios.

- Process model: a flow sheet or arithmetic model in which geometallurgical data is used to forecast the products. Depending on the approach, this model can predict the recovery rate, the element grade or mineral grade in the concentrate and tailings, particle size distribution,

water, and energy consumption.

Depending on the approach taken for geometallurgy and the level of geometallurgical information needed, the program describes the sample collection and characterization, the nature and number of tests needed for the process model, the definition of ore types and domains and the validation of the models based on case studies .

These approaches are usually one of the following:

- Traditional approach: the geological and process models are based on elemental grades from drill cores assays. Variability testing helps compute the recovery for a given element.

- Approach based on proxies: the process model is based on many small-scale tests that describe the metallurgical response of the ore in the process.

- Mineralogical approach: the geological and process models are based on quantitative

mineral characterization. This approach has three levels of information:

o One-dimensional (1D): Phases, chemical and modal composition o Two-dimensional (2D): Particle size classes, chemical and modal composition o Three-dimensional (3D): Chemical and modal composition, liberation

distribution by particles size classes

The value of geometallurgy resides in its capacity to cover the entire value chain of minerals

with error estimates for every step and its ability to give reliable predictions. This work aims at connecting the spatial and the process model.

A general approach based on particles and their properties (which derive from their mineralogy) has been described by Lamberg, 2011. The idea is to use the information from the geological

model (spatial model) to generate particles and simulate the comminution and concentration process by using the generated particles to forecast the products (i.e. the concentrate and tailings). The results can be converted into performance indicators relative to a specific deposit and plant. These indicators are stored in the spatial model as a tool for mine planning and process design.

The entire chain forms a geometallurgical model shown in Figure 1. Mineralogy, texture,

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particles and their behavior and performance indicator represent the data that can be

experimentally quantified (measured) at distinct stages of the ore value chain, whereas the spatial, texture breakage and process models as well as production forecast show the different models with their inputs and outputs. The difference between the models and the measured data can be

compared and an error is computed for every step.

Figure 1 : Geometallurgical model based on particles (Lamberg, 2011a), modified

It implies a slight modification of the very general definition of geometallurgy. I propose a new definition in which geometallurgy is a multi-disciplinary approach that links geology, process

mineralogy and metallurgy to establish and validate a predictive model of the ore value chain performance. The proposed definition now distinguishes characterization from modeling and accounts for current research activity in flexibility, sustainability, and variability of mineral

processing circuits. This presents geometallurgy as a continuation of past and current research at universities and in the mineral industry rather than a new discipline.

The choice of predictive modeling as opposed to explanatory modelling is where economical value lies. Another distinction is made by King (2012) between phenomenological models (based on the fundamental laws of physics and chemistry governing the process) and empirical models

(based only on experimental data). A predictive model can be either phenomenological or empirical whereas an explanatory model is phenomenological (but validated with experimental data). The domain of validity of a predictive model is usually larger when it is based on a good

understanding of the governing physics of both the ore and the equipment.

This work brings out techniques that cover the second step on Figure 1 by breaking a texture

to generate mineral particles. The main research questions studied are

• What classes of models are useful for geometallurgy (as a predictive discipline)?

• How can we model an ore texture?

• How can we derive quantified information about ore textures at different scales?

• Is it possible to take explicitly in account mineral texture in process simulation?

•Spatial model

Mineralogy

Textures

•Texture breakage model

Particles

•Process model

Particles behavior

•Production forecast

Performance indicators

3

The style chosen in this work is to favor practical results over theory. Mathematical tools from

different fields reflect the multidisciplinary of geometallurgy. Therefore, the introduction to some branches of mathematics will be very succinct, inviting the reader to consult more specialized literature.

A good predictive process model for geometallurgy has two desirable properties. The first one is coherence, the representation of information across all the steps must be compatible and the

error associated with it must be computable. This means that every variable used in the process model can be measured or estimated in the plant or in the lab and compared by computing an error. This error propagates through the model and measures the accuracy (or trueness) of the

geometallurgical model.

The second condition is a quantified power of prediction. This relies on proper sampling of the circuit at various times and the characterization of the streams. One should calculate the domain of validity of the model and its sensitivity relative to the nature of the feed but also operational parameters. This part will come in further works.

This licentiate thesis presents tools to ensure coherence through the particle model, its

representation and manipulation in models.

After this introduction, the second chapter will serve as a brief literature review of existing approaches in process modelling for geometallurgy. Several different methods have been either adapted from classical metallurgical tests or designed specifically with the process model in mind. A technique for mass balancing at the particle level developed by Lamberg and Vianna, 2007 is

also reviewed. It is a valuable tool that provides a ground truth to validate results of simulations. The chapter will cover different approaches for circuit simulation from chemical elements to particles. Particles will be introduced to structure information.

Chapter 3 will introduce textures by adopting definitions and tools from both geostatistics and

image processing. Definitions will help clarify the properties of textures but also particles. The simplest tools to the more complex and recent ones will fit into a texture classification scheme. A new method for texture simulation for process model in geometallurgy will be studied.

The study of some breakage models will follow in chapter 4 with a brief review of texture breakage models and the reasons why this area of research is so active since Gaudin (1939) and

cellular automata models will be introduced.

Experimental results will then illustrate the theory, covering the whole chain from characterization to the production of particles in a format that will allow particle level process simulation for geometallurgy.

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2. Process model in geometallurgy

Mineral processing as defined by Wills and Finch (2016) is the process that, after mining, liberates and concentrates valuable minerals. This definition, as opposed to ore dressing or ore

beneficiation, explicitly uses minerals as the basis of creation of value.

The recovery of valuable minerals from the gangue consists of four main operations:

• Handling: transporting the ore from the mine to the plant, conveying the ore, pumping

water, or pulp (containing water and ground ore, sometimes reagents) from one stage to

the other

• Comminution: reducing the size of the material and liberate the valuable minerals from the gangue. During this step, a parent particle will break into progeny particles of smaller

sizes

• Classification: modification of the particle size distribution of the material to select the widest range of sizes in which the valuable minerals are present and adequately liberated

• Concentration: particles are separated per physical properties (i.e. specific gravity, magnetic susceptibility, hydrophobicity, electrical conductivity, or shape)

Some stages may include several operations in one piece of equipment, as for instance internal classification in a mill.

Conceptually, as shown on Figure 2, the process takes a given ore as an input (Feed) and

produces concentrates containing valuable minerals and tailings containing mostly gangue minerals (and water depending on the operation). This will be the base of the process model in this work.

Figure 2 : Overview of the process model

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One can study the performance of such a process with the following variables (often computed

for the concentrate)

• Grade [wt. % of the element or mineral of interest]

• Recovery [wt.% of the element or mineral of interest]

• Throughput [t of ore processed/h]

• Energy consumption [kWh/t of ore]

• Water consumption [ton of water/ton of ore]

• Reagents consumption [kg or g/ton of ore]

• Concentrate and tailings specification [wt. % or ppm of penalty elements for quality requirements, wt. % passing for size requirements]

These primary variables provide a basis for higher level indicators such as

• Flexibility [score]

• Environmental impact [score]

The way the last two quantifiers are evaluated is currently an active area of research. Flexibility

covers the ability for the same process to be used for different ores or valuable minerals from different deposits. The environmental impact analysis studies a given process in terms of energy exchange, mass exchange with the boundaries of the system (including the nature, phase, and particle size distribution of the products). A systematic and semi-quantitative approach to

environmental impact are life-cycle assessment (LCA) studies (Durucan et al., 2006; Khoo et al., 2017; Norgate et al., 2007).

As an example, Figure 3 presents a flowsheet for the concentration of magnetite by wet low-intensity magnetic separation (WLIMS). ROM_1 is the feed consisting of magnetite and gangue, the SE3_CONC_14 is the concentrate and SEx_TAILy are the tailings where x and y depend

on the stage.

Figure 3 : Example of flowsheet for the concentration of magnetite

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Depending on the approach chosen for the geometallurgical program, several ways exist to

characterize and in some cases, estimate some of the process variables such as grade and recovery of a given element or mineral to the concentrate. The different approaches provide validation data for the simulation of the entire process as well. Each of these approaches corresponds to a

level of detail at which the data can be acquired and analyzed. The description of the following approaches comes from Koch et al. (2015) but the interested reader is invited to read a more comprehensive review in Lishchuk (2016).

2.1. Traditional approach

The main source of information are chemical assays coming from laboratory tests. In the traditional approach, models are built based on statistical analysis and turned into a function of several variables linking the grade of the element of interest in the ore with its grade-recovery

curve. Variability testing is used to compute the recovery for a given element or other metallurgical performance parameter and rely on repeated tests to ensure statistical significance of the results.

The most common assay values come from solution chemistry after dissolution of the minerals (e.g. fire assays, titration, absorption), spectroscopy (e.g. XRF, IR, AAS, ICP-OES, ICP-MS,

NAA). The challenges associated with the approach are the quality of the chemical assays but also the difference between the laboratory scale and the full-scale process.

2.2. Approach based on proxies

The process model is based on many small-scale tests that describe the metallurgical response of

the ore in the process directly or indirectly. The difference with conventional laboratory tests is that these tests are faster, use less material and can be applied to a high number of samples.

This approach has been applied for magnetic separation processes for example. The Davis tube tests are a proxy used to quantify the behavior of ore in magnetic separation (Murariu and

Svoboda, 2003; Niiranen and Böhm, 2012). It operates by creating a divergent magnetic field in a glass tube in which the sample is poured (dry or as a slurry). The Frantz isodynamic separator

test is similar and can separate more selectively weakly magnetic particles due to its more uniform magnetic field (Oberteuffer, 1974).

Other examples in comminution include breakage properties tests used as proxies for size

reduction and throughput of AG, SAG, and Ball mills. A drop-weight test (JKDWT) to describe accurately the breakability of the ore has been developed by the Julius Kruttschnitt Mineral Research Centre (JKMRC). The test provides an abrasion breakage parameter (ta) as well as impact breakage parameters (A and b) (JKTech, 1992; Napier-Munn et al., 1996). A second test

called the JK Rotary breakage test (JKRBT) can measure the specific breakage energy (Ecs) and the amount of material passing 10 % of the size of the initial material (t10)(JKTech, 2014). Another test called SMC can provide similar results to the JKDWT based on the same principles

(Morrell, 2004) . A model for AG and SAG mills has been proposed by Minnovex to measure a SAG power index (SPI) and a crushing index (CI) based on small-scale SAG mills (Amelunxen et al., 2014). A comprehensive review of tests available can be found in Mwanga et al. (2015).

For geometallurgy purposes Mwanga et al. 2014 have proposed the geometallurgical comminution test (GCT) that combines small-scale ball mill tests with product characterization

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to build geometallurgical domains based on comminution and mineral liberation properties. The

author also provides an in-depth literature review of comminution tests and suggests guidelines for comminution tests in a geometallurgical program (Mwanga, 2014).

Efforts have been done to build geometallurgical tests for flotation. One of them was based on a shaker test in which a test tube containing the pulp is shaken by a pneumatic device (Chudacek and Fichera, 1991). Vos et al. (2014) proposed a new small-scale test to determine flotation

performance based on the same principle: JK Mineral Separation Index (JKMSI). The results were consistent with the overall performance of batch flotation tests and plant processes, despite challenges to achieve a high recovery on the test case samples. No widely accepted proxy for

flotation exists currently to the author’s knowledge. Instead, common practice is to develop a standard laboratory test for each case individually.

The use of these proxies (alone or in conjunction with the traditional approach tests) has been described in the literature, mostly in the context of case studies (Bulled and McInnes, 2005; Montoya et al., 2011; Schouwstra et al., 2013).

2.3. Mineralogical approach

The process model in this case is based on quantitative mineral characterization. The identity and amount (wt. %) of each mineral phase present in the sample is called the modal mineralogy of the sample. Early works in geometallurgy demonstrated how particles consisting of mineral

grains can be used as objects for both the spatial and process model (Lamberg, 2011b; Lamberg and Lund, 2012)

This approach has three levels of information about the sample:

• Bulk parameters: Chemical assays and modal mineralogy

• Variables distributed over particle size: Chemical assays and modal mineralogy by particle size classes.

• Liberation: Chemical assays and modal mineralogy with liberation distribution by particles size classes

Characterization methods come from mineralogy, more specifically from process mineralogy.

Depending on the process but also the amount of minerals present, several techniques have been developed, the interested reader can consult (Adair et al., 2016). Some of the most common tools include:

• Optical microscopy (OM)

• X-Ray diffraction (XRD)

• Automated scanning electron microscope with energy dispersive spectroscopy (SEM, Auto-SEM-EDS)

• X-Ray computed microtomography (XCT)

• Synchrotron-based techniques

The link between the traditional approach and the mineralogical one is the ability to convert

chemical assays to modal mineralogy using a matrix transform and referred to as element-to-mineral conversion (EMC) (Herron et al., 1992; Parian et al., 2015). An essential ingredient for such a method is the chemical composition of minerals also called mineral matrix. To analyze

the chemical composition of one mineral, more precise techniques are required such as

8

• Electron probe microanalysis (EPMA)

• Energy-dispersive spectrometry (EDS)

• Wavelength-dispersive spectrometry (WDS)

• Laser ablation induced coupled plasma mass spectrometry (LA-ICP-MS)

The error associated with modal mineralogy measurements must be calculated against chemical

assays.

2.4. Mass balancing

With a conceptual model of a process, the equation for conservation of mass can be written. In a closed system without accumulation or losses, the variation of the total mass in the system with

time is zero.

which for a steady state system with one input and one output becomes

where M is the total mass flowrate [t/h]

This describes that we assume no mass is lost nor gained in the process.

With the fixed flow sheet model given in Figure 2, the process is further subdivided into unit

operations. Simulation of the entire model relies on separate unit models each performing one or several of the main operations in mineral processing (handling, comminution, classification, and concentration). Mathematically, a convenient representation of such a process is a graph,

which can in turn be represented by adjacency matrix.

9

Figure 4 : Units and streams inside the process model

Equation 1 : Adjacency matrix of figure 3 process model

In terms of mass balance equations, one obtains for the global process

(2)

If we form a system with the mass balance equation for each unit, we obtain the following system

(3)

But this only accounts for the total mass, if we have chemical assays the equation for the global process becomes

(4)

where x is the chemical assay of interest (expressed in % wt. or g/ton).

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If we have access to the variance of the measurements, the objective of mass balancing will be to minimize the error between the measured variables M and x (from plant survey) and the

estimated variables M* and x* (the results of mass balance). The function to minimize under the constraints of mass conservation equations is now

(5)

This technique can be generalized to modal mineralogy instead of chemical assays x, creating the

level 1 (bulk mineral mass balance). It can be used also by size fractions, with a special attention payed to the units that change the particle size distribution and is called level 2 (minerals by size mass balancing).

The highest level of mass balancing is at the liberation level (level 3). A technique developed by

Lamberg and Vianna (2007) allows the balance of particles flowrates while respecting the level 2 and level 1 mass balance. The first step is to adjust the mass proportion of each particle to respect the level 2 balance. In a second step, the particles are binned according to their liberation (fully liberated, binary, ternary, and more complex particles). This is done to standardize the different

classes of particles present in each stream. The third step is to extrapolate the liberation spectrum to the fine and coarse size fractions that can be challenging to measure. The last step is to balance the mass proportion of each particle class for the entire circuit. Similarly to the level 1 and 2 mass

balance, by minimizing the difference between the measured and the estimated mass proportion of each particle class in each stream(Adair et al., 2016).

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2.5. Process simulation

Since the value of the process model in geometallurgy is to be predictive, a simulation of the process flow sheet at the adequate level of detail is needed.

The simulation of ore dressing circuits historically derives from the simulation of chemical processes. Early works by Kehat and Shacham (1973); Ford (1979); Ford and King (1984) were

focused on the theoretical design of simulators, the classification of different simulation systems but also the design principles for a simulation environment to be used in a production context.

Table 1 : Classification of process simulation approaches

Specific simulator Section based simulator

Unit operation based simulator

Hypotheses Fixed flow sheet Sections of the flow sheet are fixed

Usually steady state process

Method Optimization of a global

equation sets under constraints

Optimization of several

equation sets for each section under constraints

Solve each unit operation

sequentially

In addition to the classification presented in Table 1 , each simulator can be operated in steady-state in which the variables are independent of time or in dynamic mode in which time is

discretized and each variable is updated according to variations either in the structure of the flowsheet, the characteristics of the equipment at a given time, in the characteristics of the ore or following simulated events in the case of operators training for example (Outotec, 2016a).

Nowadays, most of the commercial software seems to be based on unit operation working either in a steady state or dynamic mode.

Table 2 lists different levels of details in process modelling and calls for two remarks. First, the

level of information required to feed the unit models depends directly on the approach selected

for the geometallurgical model: there is no need for mineralogical information if the global model only relies on elemental grade. Secondly, each unit model operating at a given level depends on the knowledge of the modeler about the effect of the equipment on the streams. This links the

quality of the predictive model to the one of explanatory models.

12

Table 2 : Levels of detail for process simulation

Level Bulk Sized

Chemical elements fixed mineralogy

fixed PSD

Mineralogy fixed

PSD varies

Minerals Fixed liberation distribution

fixed PSD

Liberation fixed

PSD varies

Minerals with behavioral classes Fixed texture

fixed PSD

Fixed texture

fixed PSD

Particles Not applicable Association varies

Fixed shape

Microtextural information could be

used (Pérez-Barnuevo et al., 2013)

2.5.1. Elements

Using chemical assays for simulation is the most common approach motivated by the fact that in many cases, the weight percentage of each element measures the value of the concentrate. However, issues can arise when the feed mineralogy changes. For example, the chemical

composition of an iron ore can remain the same but the main iron mineral can change from magnetite to hematite, which has an impact on the process. In that case, a change in mineralogy can invalidate the simulation since the contribution of a mineral to the element grade of the feed

(deportment) is likely to change as well.

2.5.2. Minerals

Simulation based on minerals offers a higher level of detail by relating the physical properties of

the minerals to their behavior in the plant. It solves the problem of a change in the modal mineralogy of the feed. As an extension, minerals can be classified into behavioral classes for flotation (e.g. each mineral can be classified as non-floating, slow-floating or fast-floating for

flotation unit models) or comminution operations. One challenge in this approach is to take liberation into account. If liberation is not constant in the real process, the simulation can be invalidated. In comminution in particular, liberation will change the modal mineralogy of each size class.

13

2.5.3. Particles

Particles in the real process consist of grains of different mineral phases arranged in space. The

properties of these particles with the equipment performance govern the process. These particles inherit properties directly from the ore among which their modal mineralogy and texture. Particles as an object linking geology with the downstream processes has been introduced in Lamberg (2011b).

During each basic operation in the process, the entire population of particles will be transported,

split in different subpopulations, mixed with other particles, and undergo breakage at different

scales. The perspective of having individual particles instead of a limited number of variables to describe a stream may look difficult at first, given their very high number. Indeed, the main

challenge is a representative characterization of a population of particles. On the other hand, statistics is a science focused on inferring properties about a real distribution based on a limited number of observations which provides the right mathematical tools to make inference as reliable as possible.

The main tool currently used in characterizing particles by both the industry and academics is

the scanning electron microscope (Auto-SEM-EDS). A special attention must be given to the sampling and sample preparation operations (Kwitko-Ribeiro, 2012). The data obtained with this instrument is compatible with simulation software that is able to work at the particle level.

Figure 5 : Different level of details in process simulation

Recently, several unit models operating at the particle level have been designed for comminution (Schneider, 1995; Wei and Gay, 1999), flotation (Lamberg and Vianna, 2007; Savassi, 2006) and magnetic separation (Parian et al., 2016). Figure 5 shows the different level of details at which a

process can be simulated and the relations between them.

14

3. Ore textures

During comminution, the spatial distribution of mineral grains of the intact ore undergo a range of geometric transformations (rotation, translation, deformation, separation) by the action of different forces to yield a population of particles. The transformation is an operator that depends

on time, the nature, the mass flowrate of the ore, the presence of pre-existing damage in the ore, properties of the equipment and operational conditions.

The exact nature of the transformation is difficult to study due to the lack of appropriate tools. Several properties of the progeny particles can be studied like particle size distribution, modal mineralogy of particles, degree of liberation or their combination, the degree of liberation by

size. These measurements, however, do not provide enough information to reconstruct the entire transformation.

The link between particle size distribution and energy has been studied by several authors (Austin, 1973; Bond, 1952; Hukki, 1962; Kick, 1885; von Rittinger, 1867). Also, the relation between the size reduction and liberation has been examined (Mariano et al., 2016). Several

studies report the importance of texture in the understanding of the transformation and its importance for downstream processes. Different approaches have been used to describe textures, at a specific scale or for specific structures. This section presents definitions to clarify the meaning

of texture and methods to measure properties of a texture and its implications for mineral processing simulation.

The program is inspired by Barbery & Leroux (1988)

• Describe the texture: first qualitatively, then quantitatively

• Classify the obtained textures by using the descriptors

• Simulate the texture and validate the results

• Use the simulated texture to produce particles

• Use the particle population as an input for the separation unit models

One of the objective of comminution in mineral processing is to liberate the minerals of interest

at the appropriate particle size (Wills and Finch, 2016). Comminution models are used both for

process design and process simulation. Yet, few unit models operate at the liberation level.

In this work, texture is first studied as the spatial distribution of minerals and pores in a defined volume, then a notion of regularity is introduced. Texture has been carefully studied in different deposits, at meso-scale (Bonnici et al., 2008; Lund et al., 2015; Pérez-Barnuevo et al., 2016;

Wightman et al., 2014) but also at micro-scale (Pérez-Barnuevo et al., 2013, 2012; Pirard et al., 2007) or both (Bonnici, 2012). The behavior of an ore in processing circuit is influenced by the texture of the particles created during the comminution stages (Bushell, 2012; Donskoi et al., 2008; Lund et al., 2014; Pérez-Barnuevo et al., 2016; Tungpalan et al., 2015)

15

3.1. Definitions

The definition and description of a texture needs a proper definition of the space in which to work. The representation of the texture will determine the methods that can be used.

3.1.1. Representation of a texture

A particle is a solid and has a finite volume. The simplest way is to describe it as a subset of ℝ3. To partition space, we can use a regular tiling such as the cubic one. To each tiling, one can

associate a graph in which each vertex represents a tiling element and each edge represent adjacency relations between elements. Therefore, one has now two equivalent representations of the same set: either the cubic lattice with cubes filling the space or a graph of the vertices and

edges of the cubes. Formally, the following subgroup is a lattice:

ℤ𝑛 = (𝑥1, . . . , 𝑥𝑛) ∈ ℝ3: ∀𝑖, 𝑥𝑖 ∈ ℤ (6)

We can write the mapping γ between the tiling and the graph with vertices V and edges E by

∀𝑥 ∈ ℤ𝑛, 𝛾: 𝐱 → 𝑉𝑥, 𝐄𝐱 ∈ 𝐺 (7)

Figure 6 : Regular cubic tiling and equivalent graph

In image processing, an element of the lattice is a pixel in 2 dimensions (2D) or voxel in 3 dimensions (3D). A topology can be built from the edges of the graph, in 2D we have

𝐄𝑥 ⊂ ℤ2, 𝑉𝑥 ⊂ 𝐄𝑥2 (8)

And the neighborhood relation is

𝑥 ∼ 𝑦 ⇔ (𝑥, 𝑦) ∈ 𝐄𝑥 (9)

16

In 2D we can define two different neighborhoods:

• 4- neighborhood

• 8-neighborhood

Figure 7 : 4-neighborhood (left) and 8-neighborhood (right)

In 3D, the following neighborhoods can be defined

• 6-neighborhood

• 18-neighborhood

• 26-neighborhood

A mineral phase can be assigned to each vertex of the graph to form a mineral map

𝜙: ∀𝑉𝑥 ∈ 𝐺 → 𝑠 ∈ 𝑆 (10)

Where S is a set of symbols (either a natural used as a label or a string of characters “Py” for pyrite for example) or an integer. A distinction is made between images for which S can be real numbers and a mineral map in which S contains integers.

A grain can be seen as a subset where for each xi neighbors, S being constant.

3.2. Acquisition of an image or a mineral map

The acquisition of a mineral map can be done with a SEM or with distinct parts of the electro-magnetic spectrum on drill cores. Visible light has the advantage of minimizing the costs and maximizing the speed. The drawback of high-resolution visible light images is that there is no

guarantee that a single pixel will measure a single mineral phase. Nevertheless, it is a valuable tool to improve knowledge about texture at an early stage.

Like with every characterization method, the error associated with the measurements must be estimated to contribute to the global error of the model.

17

3.3. Texture modeling

A convenient framework to represent a texture is provided by the random function (or field) theory, for a geosciences approach the interested reader is referred to Chiles and Delfiner, 2009; Lantuéjoul, 2013; Matheron, 1970 ,and for an image processing perspective, to Serra and Cressie, 1984; Winkler, 2012. From a historical perspective, it is significant that Jean Serra (one of the

key persons in morphology) and Georges Matheron (one of the key researchers in geostatistics) worked both at the Ecole des Mines de Paris, in Fontainebleau. An account of the birth of mathematical morphology investigating the links between geostatistics and morphology is to be

found in Matheron and Serra, 2002.

The proposed texture model uses the simple case of a random function distributed on a

discretized finite volume and the associated sets of edges (Ex) and vertices (Vx).

3.3.1. Random functions

This short introduction about random function comes from reference books in geostatistics, in

particular Chiles and Delfiner (2009) and Lantuéjoul (2013).

A function

(11)

with domain

(12)

with a positive volume, and a probability space

(13)

Such that

(14)

is a random variable on the same probability space, is called a random function (or random field).

The notation Z(x) will be used for the entire random function Z(x, ω ). The reason it is well

suited for the analysis of texture is that a texture can be thought of as a realization z(x) = Z(x,ω0) of an underlying unknown random function Z. The next sections will focus on the inference of the random function from only one realization of it. Hypotheses need to be introduced to make

that inference possible.

Depending on Ω , the random function can associate an element of D to a real number (case of

images) or an integer (case of indexed mineral phases). Since Ω will be specified in each case, the notation Z(x) applies to Z(x,:).

The random function is a probabilistic representation of an image or a distribution of minerals

in a given space. In the next sections, some hypotheses will be introduced to precise what we mean by texture in the context of ore (intact or particles). A distinction is made between meso-texture (where one point of the domain is of the order of 1 mm) and micro-texture (where one

18

point of the domain is of the order of 1 μm) for clarity, but the same mathematical objects will

be used in both cases. The usual distinction will be done between an image and a mineral map. In an image, the values that a realization can take will be in R in general whereas in a mineral

map, each mineral is coded as an integer.

3.3.2. One-point methods

One-point methods refer to functions that depend only on one location of the field. To obtain global statistics, one can sum one-point functions over the field.

If one chooses a domain

(15)

And a realization space

(16)

One can define statistics of the random function. All these statistics are based on the histogram of the values the function can take, as a result they are sensitive to any change of contrast. The

empirical frequency of an element k of Ω is

(17)

where 1 is the indicator function that looks like

(18)

and can be generalized for conditions other than equality.

The first moment is the mean

(19)

by

(20)

and the second moment, the estimator of the variance

(21)

Higher order moments include the kurtosis and skewness, energy, and entropy.

Initial interest in textural descriptors seemed to come from biology and psychology. The aim of the research was to determine on which statistical properties is based our perception of a texture. The first attempts to describe textures focused on statistics on contrast, brightness, color and slope

of lines (Beck, 1973).

19

3.3.3. Two-point methods

The intuition about texture is that there is some form of relation between the values of a random function at distinct positions x and y. Two-point methods introduce functions and hypotheses

that clarify the nature of these relations. This is the basis of two-point geostatistics (Chiles and Delfiner, 2009).

The covariance of a random function is defined as

(22)

Which is related to the variance defined previously by

(23)

One important property of a random function is stationarity. A random function Z is strictly stationary if for a vector h in the spatial domain.

(24)

where Pr is a measure of probability.

A strictly stationary random function has moments that are invariant by translation

(25)

and the covariance becomes

(26)

A function that verifies only the two previous conditions is called wide-sense stationary (second-order stationary).

This property is very important when modeling a texture as a realization of a random function since a texture is expected to have some form of spatial homogeneity, which is expressed by

stationarity. Unless explicitly stated, the random functions we will consider from now on will

be wide-sense stationary.

If the mean of the field is equal to zero, the estimator of the covariance becomes

(27)

If the mean is constant over the field, one can define the variogram

(28)

The variogram is a measure of correlation between the values of points distant by a vector h. In

general, when h increases, the correlation tends to decrease.

A second important property is ergodicity, defined as

(29)

20

where V is a volume, subset of the spatial domain.

The main issue when we observe a texture and try to model it with a random function, is that

we need only have access to one realization of the function. If we try to estimate the mean, we would like to be in the case of an ergodic function. The problem is that even for a wide-sense stationary function we only have the following ergodic theorem

(30)

In that case, the mean is not a constant since it depends on the realization we observe from the

parent random function.

While the concept of random function helped clarify the different properties of the model for a

texture, two issues remain.

The first one is the difficulty of inferring the mean from only one realization, this is solved by choosing an ergodic random function as a model. The Slutsky ergodic theorem states that (Chiles and Delfiner, 2009)

(31)

is a sufficient and necessary condition for Z to be ergodic. In practice, it is verified for a covariance model that decreases to zero when h tends to infinity. This provides a class of covariance models appropriate in the context of an ergodic stationary random function. To

estimate the scale of the underlying phenomenon modelled by the random function, the integral range was introduced (Lantuéjoul, 1991). It is defined for a stationary ergodic random function as

(32)

The physical interpretation of A is the scale of the observed phenomenon while V is the scale at which it is observed (Lantuéjoul, 1991).

The second one is the fact that either with images or mineral maps we work in a limited portion of space which cannot be extend to infinity. If V is large enough and C(h) decreases fast enough,

then the Slutsky theorem applies and the mean can be estimated. If not, we can use the variogram

which is based on increments (Z(x+h)) instead (Chiles and Delfiner, 2009).

In terms of descriptors of a texture, the experimental variogram can be calculated without assumptions on the underlying model (random function)

(33)

Indeed, experimental semi-variograms have been used recently for image classification tasks

(Pham, 2016).

Another class of descriptors have been introduced by Haralick et al. (1973). They are based on

a gray-level co-occurrence matrix (GLCM) which, for two vectors h1 and h2 and a function z with domain [1,…,G] is defined as

(34)

21

Different statistics can be calculated from this matrix (e.g. contrast, energy, entropy, …). These

values provide textural descriptors that can be calculated either on the entire field or on subsets of the field. In the case of mineral maps, gray-level co-occurrence matrices (or normalized variants) are relevant and have been used in 2D and 3D (Becker et al., 2016; Jardine et al., 2014)

In the context of image processing, the value of the experimental variogram has been shown to be equivalent to the contrast of gray-level co-occurrence matrices (van der Sanden and

Hoekman, 2005).

Another technique relates to two-point statistics and gray-level co-occurrence matrices. Local binary pattern have been introduced by Ojala and Pietikainen (2002) and have found multiple applications in image processing such as face description (Ahonen and Hadid, 2006) and

specifically in texture analysis either as such (Li et al., 2015) or in a modified version (Hegenbart and Uhl, 2015; Liao et al., 2009).

The construction of the descriptors is based on gray-level images and a discrete circular neighborhood. The local binary pattern (LBP) operator is defined by two parameters

• P the number of pixels in the neighborhood (in our case 8)

• R the radius of the circular neighborhood (in our case 1)

Figure 8 : LBP with P=8 and R=1 (Ojala et al., 2002), modified

The texture T is defined by the joint distribution t of the gray levels g for P image pixels and gc the gray level of the center pixel:

(35)

which is estimated by

(36)

Replacing the gray level values differences by their sign yields

(37)

If a factor 2p is assigned to each term, a single LBP can be associated to a given neighborhood

(38)

22

These can be computed for cells in which the image is divided, yielding a single number for

each pixel in the neighborhood for each cell. The histograms over the whole image can be computed by concatenating the normalized histograms of the cells. The frequency of each bin is used as a single feature. The resulting feature vector has the size of the number of bin which

is given by 3+connectivity*(connectivity-1). Their different distributions characterize the texture of the image. Local binary patterns are invariant to a change of contrast and can be made invariant to rotation.

Two-point statistics introduced the idea of comparing values as pairs. Textures are not only defined by the global distribution of values on the field but also by the relations (interactions)

between pairs of points on the field. The contribution of the random function theory to the analysis of textures is that we can choose to model textures as a realization of a random function

with chosen properties depending on our observations. Stationarity and ergodicity are the properties of the model (and not of measurements) chosen depending on the objectives of our

model and indications about homogeneity of the texture. Two-point statistics introduced the possibility to study how correlation between values of the random function change when the distance or lag (h) between them changes. This adds another key factor to the understanding of

textures, linking the scale of observation to the scale of the phenomenon via the integral range.

From a texture descriptors point of view, since no model is inferred from the data in that case, experimental variogram, grey-level co-occurrence matrices (GLCM) or local binary patterns (LBP) provide valuable information about the relation between values in the mineral map or texture image. The difference between the descriptors is in their invariance to translation,

rotation, and scale transforms.

3.3.4. Harmonic methods

Harmonic methods are using the Fourier transform as the basis for analysis of a signal. Under appropriate hypotheses on Z, one can define the Fourier transform of a random function as

(39)

And the Fourier transform of the covariance as

(40)

The Wiener-Khintchine theorem (Khintchine, 1934; Wiener, 1930) links the covariance with the Fourier transform of the random variable by

In practice this means that we can estimate the covariance of a stationary random function by computing the module of its Fourier transform (spectrum). In the context of images, one can use the discrete Fourier transform of an image f[m,n] where m,n are pixels in the spatial domain [1,…,M]x[1,…,N] defined as

(41)

23

And the inverse transform

(42)

In practice, this means that the module of Fourier coefficients of an image can be used as textural

descriptors. It should be noted that these descriptors are invariant to translation. For an image of dimensions M by N, an efficient algorithm to compute a Fourier transform called (discrete) fast Fourier transform (FFT) with a computational cost of the order of NM log NM exists (Cooley and Tukey, 1965). This is of practical importance for cases where speed is needed (for example

drill core scanning).

Spectral methods contribute in two ways to texture analysis. First, they offer texture descriptors giving information about the frequency content of the texture. Secondly, they suggest an approach by analysis and synthesis: the Fourier coefficients (with module and phase) contain enough information to synthetize the original texture. By intuition it is a desirable property of

texture descriptors to contain enough information to simulate a new texture of similar statistical properties rather than just the ability to discriminate between different textures.

3.3.5. Multiple-point methods

The framework of random functions allows creating an ergodic wide-sense stationary model of the mineral map or image. Stationarity and ergodicity are not verifiable assertions but merely a choice. Moreover, the very concept of random function is an abstraction since what is observed is not the entire random function but only one field with one distribution of values. Another

approach, based on the data started to emerge (Mariethoz and Caers, 2014; Matheron, 1978). Its main objective is to reproduce the variability observed in a real field (facies or mineral map, image) in a simulation of the field. The model underlying the construction of the simulation is not always directly accessible and therefore these methods provide simulation techniques rather

than texture descriptors.

The general idea is to start with a training image (TI) deemed representative of the field under study and analyze patterns (set of values of the field in a neighborhood of a position x) to build

a database of patterns. Given a new node to simulate, the neighborhood of the node will be

extracted and used to derive the distribution of the value of the field Z(x) conditional to the values already in the neighborhood of x. A value is sampled from this conditional distribution and the value sampled is placed on the simulated field at position x (Mariethoz and Caers, 2014). Instead of one value Z(x) (pixel-based algorithms), the entire neighborhood of x can be simulated

as this gives generally faster methods (patch-based algorithms).

The contribution of these methods to texture analysis and synthesis are important. Instead of considering pairs of points as in two-points methods, the main object is now a (moving) neighborhood around the location x. This neighborhood can be extracted at different scales by up-scaling or downscaling the sampling grid (multi-grid techniques) which ensures that

structures at different scales can be reproduced.

One way to evaluate the frequency at which a pattern occur in the neighborhood of a value Z(x) in the training image is to compute the convolution between the pattern and the TI.

24

Convolution is widely used in image processing and an N by M image convoluted by a filter (or

kernel) gives, for each pixel of the image, the similarity between the neighborhood of that pixel and the convolution kernel. Mathematically it can be expressed as

(43)

This formulation is very close to the cross-correlation that is written

(44)

As an example, one image from the dataset containing amphibole, quartz, biotite, chalcopyrite

and pyrite is convoluted by a (5,5) kernel. This kernel can be used to identify high-intensity pixels in a noisy image. If a thresholding of the pixel intensity values follows this operation, it can be used to detect areas where bright mineral phases (such as some sulfides) are more likely

to occur. The second example uses a filter aimed at detecting horizontal edges by convoluting the image with a discrete version of a horizontal gradient. For our application, will be used on an image. The values of the pixels represent a color as a real number coded as a double.

Equation 45 : Example of a (5,5) convolution kernel

Equation 46: Example of a (3,3) convolution kernel

25

Figure 10 : Original image I (1)

An efficient way to compute convolutions is provided by the convolution theorem that states that, if F is the discrete Fourier transform,

(47)

So, the convolution can be computed in the frequency domain using a FFT instead of moving the convolution filter over the entire field.

I(2) = I(1) * h1

Figure 9 : Transformed image I (2) after

convolution by kernel h. Yellow pixels are high values and blue pixels are low values.

Figure 11 : Transformed image I (2) after

convolution by kernel h2. Deep blue pixels are low values and green pixels are high values.

I(2) = I(1) * h2

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3.3.6. Wavelets

Convolution is an example of a linear transform of an image. More generally one defines a linear

transform T of an image f as a function (Bloch et al., 2005)

(48)

With

(49)

This formula can be seen as a change of basis in a vector space. If one chooses

(50)

The resulting transform T is the Fourier transform, which was previously introduced in the discrete case for images. Compared to the convolution case, the filters are now a complex

combination of sine and cosine functions in 2D. The analysis of the image is done by comparing elements of the field with filters of different frequencies. The image is decomposed into the limit of a weighted sum of sinusoids in 2D.

An alternative can be to choose

(51)

With

(52)

Where Rθ is the 2D rotation matrix. Alternatively, one can limit the values of θ and set (Meyer, 1990)

(53)

Where φ is a 1D low-pass filter function localized in 0, referred to as scaling function (or father

wavelet) and ψ is a 1D wavelet function (or mother wavelet) that respect

(54)

and

(55)

27

The role of a in (51), is to stretch or contract the wavelet. By changing a, x0 and y0, one can

analyze the field by the result of its convolution by a wavelet of different frequencies but also scales. The main difference with the convolution introduced before is that the scale can change due to the scaling function. The drawback is the loss of invariance to translation.

Figure 12 : Example of a scaling and wavelet functions (Daubechies wavelets db6)

The contribution of the linear transform of an image (or mineral map) is to present convolution,

Fourier transform and wavelets as different filters with different properties. Convolution is generic and can be used to detect (match) different kind of patterns whereas the Fourier transform will detect different frequencies but globally and wavelet transforms will detect different frequencies at different scales but will lose the invariance by translation.

3.3.7. Morphology

Another set of methods based on multiple points was introduced in mathematical morphology (Serra, 1982). Tools introduced in morphology cover a wide area in image processing but in the

context of this work, granulometry is of interest and will be the only one presented.

The binary dilation of a set X (in this case, our texture or mineral map) by a structuring element B (one can think of it as an object similar to the kernel introduced for convolution on an image: a filter but in this case with binary values) is defined as

(56)

And the erosion is defined as

(57)

By combining dilation and erosion, a new operator can be created, the opening by a binary

structuring element B, defined as the successive application of an erosion and a dilation

(58)

The opening makes elements of X that are smaller than B disappear.

And the closing operator by a binary structuring element B

(59)

The closing fills holes in X that are smaller than the structuring element B.

28

Figure 13 : Effects of binary dilation and erosion operators (ball with an 8 pixels radius) on a particle

If we use a discrete binary ball in 2D as a structuring element and let its radius increase, elements smaller than B will disappear for each increment of the radius. Physically, it can be seen as sieving

elements of X with sieves of different openings. In the case of a mineral map, since to each phase corresponds a specific gravity, one can count the pixels that disappear for each increment and build a particle size distribution. Historically, one of the first application of the morphological

granulometry has been the study of porous media (Matheron, 1967).

29

3.4. Texture classification

Texture quantitative analysis has provided descriptors with different properties (invariance to translation, rotation or change of contrast) that can be used to discriminate textures. Since the texture of particles is inherited from a parent intact ore, one natural hypothesis is that the same

intact texture after comminution with the same equipment in the same operational parameters (same breakage) will result in similar particle populations.

The general procedure to classify a texture or a mineral map consists of three main steps. The first one is the acquisition (after calibration of the instrument and at the appropriate sampling

density) of an image or mineral map (at a large scale, drill core scanning and at a smaller scale, microscopy), the image obtained can be filtered to remove artefacts if necessary (in this case, particular care about the nature of the transformation is needed to avoid the creation of new artefacts) and cropped to standardized dimensions to ease feature extraction. The second step is

the extraction of texture descriptors itself, as demonstrated earlier, this can be done in many ways depending on the desired properties of the features. These features are organized as a n-dimensional vector and written to a data set. Once the feature vectors have been extracted for all the images, a classifier is a function that maps a feature vector to a unique class. These classifiers

can be cross-validated and their performance can be compared. The process is shown on Figure 14.

Figure 14 : Texture classification process

3.4.1. Feature extraction

In this study three texture descriptors are used: gray-level co-occurrence matrix (GLCM) statistics, local binary patterns (LBP) and convolution filters. GLCM descriptors have been introduced by Haralick et al. (1973) and recently applied to drill core characterization by Becker

et al. (2016). In my implementation, I combine a Haar wavelet multi-level decomposition (Antonini et al., 1992; Mallat, 1989) with statistics derived from the GLCM at each scale. LBP have been introduced by Ojala and Pietikainen (2002) and have found multiple applications in image analysis such as in face description (Ahonen and Hadid, 2006) and specifically in texture

analysis either as such (Li et al., 2015) or in a modified version (Hegenbart and Uhl, 2015; Liao et al., 2009). The use of convolution filters can be combined with texture classification itself in a method called convolutional neural networks (LeCun et al., 1995). In that case, the filters used

for feature extraction will not be pre-defined but learned directly from examples.

Image acquisition

•Calibration

•Filtering

•Cropping

•Fixed size patches

Feature extraction

•Compute descriptors

•Batch process

•Write datasets

Texture classification

•Classify the descriptors

•Cross-validate the classifier

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3.4.2. Texture classification algorithms

Two main approaches exist for classification: the unsupervised, which relies only on unlabeled

data as an input (images not associated with a class) and the supervised, which relies on two subsets of the initial data: the training set and the test set. The training set consists of labeled data (images associated with a class) from which the model will learn the classification function. The test set consists of unlabeled data to which the classifier will be applied.

3.4.2.1. Unsupervised classification

One method for unsupervised classification is called “k-means” where feature vectors are

represented as points in a n-dimensional space (where n is the number of features in the vector). At first, a pre-defined number of class centroids are randomly placed in the feature space. The second step computes Minkowski distances between points and the centroids

(60)

where q=1, 2, …, K

and assigns each point to the closest class centroid. The third step is to recalculate the position of the new centroids by averaging the position of points that belong to the class. Steps 2 and 3

are repeated until convergence (MacKay, 2003). The k-nearest neighbors (KNN) is a variation of k-means in which the point to classify is assigned to the class that contains the majority of points inside a neighborhood of specified dimensions (Cover and Hart, 1967). KNN can be

turned into a supervised method if the metric (distance) is learned from the data.

Unsupervised classifiers are useful during the definition of textural classes, when little or no

information is available a priori.

3.4.2.2. Supervised classification

Classification trees are an example of supervised classifier. A set of decision trees is built during the training phase by growing decision trees consisting of one binary test at each node. This test is either random (data independent) or chosen to maximize the information gain resulting of the classification at this node. Once the tree has reached its maximum depth or if the node does not

receive enough training examples, the growth is stopped (Bosch et al., 2007). Tree growth is based on random subsets of training data (Aggarwal, 2015). During the classification phase, each feature vector enters each tree until it reaches a terminal node. The posterior probabilities are

averaged over all the trees and the returned class is the one with the maximal probability.

The random forest method constructs a population of classification trees by building separate trees on part of the data selected by bootstrapping. Once the population of trees has been trained, the classification of a new sample is done by choosing the class that corresponds to the majority vote of the population of trees (Breiman, 2001; Liaw and Wiener, 2002).

31

Another example of supervised method is artificial neural networks (ANN) (Hebb, 1949;

Rosenblatt, 1961). It will be introduced for 2 classes so that the data consists of xk the feature vectors and yk the class indicator equal to 1 if xk belongs to class 1 and equal to -1 if xk belongs to class 2.

The decision to assign a point in the feature space to a class is modeled by a a linear function D(x) with D(x) > 0 if x belongs to the class, defined as

(61)

where wi are weighting factors, φi(x) a base of functions and b a parameter of the model.

A simple case is polynomial functions for φi(x) in which φi(x) are products of components of the feature vector x to classify. The learning phase in ANN is based on the computation of a loss function that measures the difference between the true class of a point of the training set with

the predicted value D(x). One technique to minimize the value of the loss is to apply gradient descent to the weights wi until the values of the loss functions are small (Hecht-Nielsen, 1989).

The scalar product wiφi(x) can be seen as a hyperplane separating the 2 classes. If such a hyperplane exists and M is the margin between the plane and points of the training set, all points of the training set satisfy

(62)

The objective of support vector machine (SVM) techniques (Boser et al., 1992) is to find the vectors w so that the margin M is maximized

(63)

Supervised classifiers are useful when knowledge about the classification exists already. This is the case when classifying mineral textures from drill core images: trained geologists are experts

in classification and are able to extract most of the relevant features in a limited time. Additional information can be included in the features including grinding tests results (BWi), modal

mineralogy or chemical assays. In that case, the classification is not only based on the visible

information, making it suitable for geometallurgy.

Classifiers have different performances and must be validated. A common way is to split the

initial data set between a training and a testing set. If the classifier is built on the training set then applied to the testing set, one can compare the predicted class to the actual one. A confusion matrix can be built which has the predicted classes as columns and the actual classes as rows, the

values are the number of instances predicted to be of each class.

True T1 True T2

Predicted T1 90 5

Predicted T2 10 95

Table 3 : Example of confusion matrix for 2 classes

32

3.4.3. Pixel classification method

Once the acquired image is linked to its textural class, the set of probable minerals can be

reduced. It is thus possible to classify pixels from the original image into minerals or phases consisting of a mixture of minerals.

A machine learning approach is chosen, based on a random forest classifier (Breiman, 2001) and features extracted from pixel values. The procedure is the same as texture classification algorithms (acquisition, feature extraction and classification), the only difference being the features that are

statistics computed in a neighborhood of different radius (similar to the one used in local binary patterns) of each pixel, or the result of filtering. These extracted statistics include the mean,

minimum, maximum, variance but also convolution by different filters (e.g. Gaussian blur,

derivatives of different orders). Each pixel of the image is described by one feature vector which can be classified.

Supervised learning is interesting here since a mineralogist or a geologist can provide labeled examples for training while the classifier learns from human experience. In image processing the activity of assigning a label to group of pixels of similar appearance is referred to as segmentation

for which several methods exist (e.g. graph-based methods (Nalina and Muthukannan, 2013; Shi and Malik, 2000) , Markov fields (Benboudjema, 2005) or convolutional neural networks (Long et al., 2015) and is an active field of research (Pal and Pal, 1993).

3.5. Texture simulation

To generate realistic particles that inherit textural properties from the intact ore, discrimination between textures does not help. Variability in the ore body causes differences in the modal mineralogy of the feed which in turn impacts the modal mineralogy of the particles generated after comminution. To take that variability into account, one can choose to distinguish the spatial

features (texture) from the modal mineralogy. By doing so, the same texture can apply to a changing modal mineralogy. The challenge in texture simulation in the process of mineral processing is that the modal mineralogy must be precisely controlled as the mineral or element

grades of the streams in the process derive from the modal mineralogy of particles population.

The objective of this section is to present an original application of a method to generate a mineral map and to compare different realizations with a ground truth.

The study of spectral methods for texture analysis suggested the use of textural descriptors that contain enough information to work in both analysis and synthesis. The option here is not to use the Fourier or wavelet transform but rely on a variation of GLCM to measure the association

of minerals and morphological granulometry to control the grain size in the simulated texture.

The multiple-point (geo)statistics (MPS) methods offer the concept of training image (TI) as a guide to extract textural properties. In this case, a training image is derived from a mineral map that is acquired by a SEM. The terminology of MPS will be used, this means that the initial data will be a TI and the resulting simulated mineral maps will be called realizations.

The simulation method in itself is derived from popular Markov Fields methods from statistical

physics. The interest of using this class of models is that they can be interpreted in terms of physical variables (temperature and energy), provide a simplification of the processes underlying the formation of crystal grains in intact textures and the grain size distribution is controlled.

33

3.6. Markov fields methods

A model based on Markov Random Fields for controlled texture simulation (Cross and Jain, 1983; Winkler, 2012) is proposed to generate realizations of similar textural properties while

changing the modal mineralogy.

Several methods have been proposed to simulate a realistic mineral texture (Gasnier et al., 2015; Klichowicz and Lieberwirth, 2016). In this study, the association matrix (AI) and the cumulative grain size distribution of each phase are used to control and validate the realization against the training image. Since the aim is to generate a stream with varying modal mineralogy, it is an

input parameter of the simulation.

Let us model a mineral map as the realization z(x) of a random function (or random field) Z(x,ω) on a spatial domain D. Z takes integer values from E, each number representing a mineral phase or a void. The introduction of Markov fields follows the presentation given in Sigelle and Tupin

(1999).

The field Z is a Markov field (MF) if and only if its local conditional probability at x is only a function of a neighborhood of x. If V is the neighborhood of x, it can be reformulated as

(64)

This means that the probability of the random function to have a value i given the knowledge of its value everywhere else is the same as the one calculated with the limited knowledge of a neighborhood.

One should note that there is no limitation regarding the size of the neighborhood which can

be arbitrarily large.

The Gibbs measure of energy U is defined as

(65)

With

(66)

Where C is a clique system defined on the neighborhood system V on D. A clique is either one

single point of the neighborhood or a set of points that are direct neighbors inside the neighborhood. The equation means that the global energy U(z(x)) can be decomposed into a sum of cliques’ energies.

The Gibbs field of potential associated to a neighborhood system V is the random field for which

the measure of probability is a Gibbs measure associated to the neighborhood system V.

For a random function with discrete values on a finite field, with a finite system of neighborhood

V, Z is a Markov field relative to V, and Pr(Z(x)=z(x))>0 if and only if Z is a Gibbs field of potential associated to V.

The generation of a realization of the field is based on the potential U which in turn is based on local cliques’ potentials. A simulated mineral map will be a sample drawn from a distribution

with the global Gibbs probability

34

(67)

With

(68)

Two main methods have been proposed to sample a Markov Field. The first one is called a Gibbs

sampler (Geman and Geman, 1984) and builds iteratively different realizations z(n) that converge to Z. For each step n, it picks a point at random, computes the probability of the point to have a certain value conditional to the value of its neighbors. The probability law used is the ratio

between the Gibbs energy of the point conditional to its neighborhood and the sum of the Gibbs

energies for every value the random function can take at the point conditional to its neighborhood. The last step updates the value of the realization at the point with the previous probability.

The second one is called Metropolis-Hastings algorithm (Hastings, 1970; Metropolis et al., 1953)

and is based on evaluating how the energy of the system is modified when a point changes value.

Here are the different steps of the Metropolis-Hastings algorithm (Bloch et al., 2005) for an image n.

1. Pick a random point x

2. Draw λ from E following a uniform law

3. Compute how the energy changes if z(x) is set to λ:

4. If ΔU < 0 then the energy is lowered by the substitution, set z(x)=λ . Else, accept

the change with probability exp(-ΔU)

This algorithm constructs successive images z(n) that converge to the law of the random function Z. In other words, if initial values of the field are generated randomly every time, and the

Metropolis algorithm is applied with a defined function U(x) until convergence, realizations of the random function Z are produced. An interesting modification of the Metropolis-Hastings algorithm is to replace step 2 where the new value is drawn from a uniform law by drawing it from another location on the field and propose to swap values between the current point x1 and

another random point x2. This will preserve the non-spatial distribution of the values of Z0 the initial field. For mineral maps, it offers a method of controlling the modal mineralogy and keep it constant for all the realizations.

So far, no explicit form for U(x) has been introduced. In statistical physics, the Ising’s model

(Ising, 1925) originally describes the behavior of spin at each site of a ferro-magnetic material as a function of the temperature (above or below Tc the Curie or critical temperature where magnetization is lost). The values the elements can take are spins in this case taking values in E = [-1;1]. To each clique, it assigns a potential Uc so that

(69)

Where beta is a coupling (or regularization) constant. If two neighbor spins have the same value,

the product xs. xt will be equal to 1 and the potential equal to -β. Its value describes the intensity of the influence between sites x: for a large value, the value of an element is heavily influenced by its neighbors. The total energy of the system can be written as a Boltzmann distribution

35

(70)

If β>0, different spins for neighbor sites will be favored (the energy E will be lower if xsxt=-1)

whereas for β>0, same spins for neighbors will dominate (the energy E will be lower if xsxt=1).

Ising's model can be generalized for multiple phases i.e. E = 0; N with 0 phase reserved for voids (in this case, pre-existing cracks and porosity) and is called a Potts model (Potts, 1952) . This approach has been used to study the grain growth (Zöllner and Streitenberger, 2006) with

a specific interest in the interface regions.

To generalize the Potts model one can start from the Gibbs field of potential

(71)

With

(72)

For which we choose

(73)

Where

(74)

Where 1 is the unit matrix, I the identity matrix and AI(i,j) is the association matrix, that contains

the empirical frequencies of a given phase i to be associated with a phase j in a given a weighted neighborhood. Weighting makes possible the calculation of an element of AI which is not a proper frequency but account also for points that are not direct neighbors of x in 4 or 8

connectivity. A typical choice for a weighted neighborhood is a matrix similar to a convolution filter with values equal to one around its center element and decrease as the inverse distance

towards the outwards elements of the filter.

The matrix βcontains information about the influence of the value of Z(x) on its neighbors (like the regularization term in the Ising’s model). It is a parameter of the model that can be tuned. It

has an influence on the smoothness of the Markov field. At high β values, the global energy U(z(x)) will be low. Physically, this represents the case of a cold system in which distinct phases form coarse grains with clear boundaries. In general, the values of the matrix β depend on the location x. If β is independent of x, then the model will be stationary since the interactions

between two different values will have the same intensity everywhere in the field.

(75)

A definition of a Markov field, a specified energy U(x) with a value for its parameters AI and β, and an algorithm to sample Z make it possible to produce realizations of Z for different values

of the parameters. The easiest parameter to control is β, but its impact is multiple: not only is it

related to the general regularity of the field, but it automatically changes the association index of

minerals in the realization as well. Another physical quantity modified by β, is the grain size of each mineral phase in the realization. As a result, there is a need for an optimization method that

36

finds the best value of β such that the differences in terms of association of minerals and grain

size distribution are minimal.

If we consider a function U(z(x)) = θ.φ(x), the expression of the likelihood (L) of θ is

(76)

One way to maximize it, is to use a stochastic gradient descent (SGD) on θ (Younes, 1998). In

this algorithm, θ is modified at each iteration according to

(77)

Where φ(z(x)(n)) is the value of the potential of a realization produced by a Metropolis algorithm

at step n , φ(z(x)) the value of the potential for the current realization and V a positive constant.

The identification of the parameter β can be done via the SGD, however, the convergence speed of the presented version is quite low. By using a slightly different version of the stochastic gradient descent, faster convergence can be achieved. In a broader perspective, aiming at real-

time simulation at the particle level, computational aspects matter.

A procedure to optimize β has been established but no metric for the error between the training image and the realizations has been introduced yet.

To measure the difference between the generated texture and the training image, a norm should be chosen, in this case, a weighted sum of a max-norm with a l2-norm on both AI and the cumulative grain size distribution g was chosen. At each iteration, the grain size distribution is

computed as a morphological granulometry and the association index of the realization is evaluated as described in , and an error e is evaluated

(78)

Where g are the grain size distributions for the current realization and for the training image.

This class of models are called auto-models in the image processing community (Besag, 1974; Winkler, 2012).

The method for mineral map simulation has desirable properties. The parameters needed to produce a realization can be derived from a measured mineral map (grain size distribution of

each phase, association index matrix). The proposed model simulates the physics of grain growth but also ensures the reproduction of the association index. Stationarity is not needed anymore

since in general, the regularization parameter β depends on the location x. The prerequisite to work with non-stationary models is the presence of a good training image (at the appropriate

scale, that contains enough variability). Using the swapping version of Metropolis-Hastings algorithm, changes in modal mineralogy can be integrated since the initial field can be chosen arbitrarily. An appropriate choice is randomly distributed values respecting the modal mineralogy

distribution.

One drawback is the absence of guaranteed convergence in general, for an arbitrary training image, an arbitrary neighborhood system. To control the error at each step, a complete granulometry is evaluated at each step along with an estimation of the association matrix. These

37

two steps slow down the entire algorithm, but the cost is not too high, as shown in the

experimental results.

The necessary tools to both analyze and synthetize an ore texture have been introduced. The missing piece is a breakage mechanism that will produce heterogeneous particles that inherit textural properties from the ore. Figure 15 presents a summary of the different methods from the acquisition of the image or mineral map to its classification or simulation.

Figure 15 : Process of texture classification and simulation

ObjectiveInformation

representationInformation extraction

RepresentationAcquisition

TextureImage or

mineral map

Model

Z(x,ω) Simulation

cλ

Simulation

Classification

DescriptorsFeature vectors

Classification

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4. Breakage modeling

In comminution, several breakage mechanisms have been identified (King, 2012)

• Shatter: when a particle is subject to a fast, compressive stress. Particles undergo a cascade of breakage events resulting in a population of smaller particles. Cracks propagate until

no energy is left.

• Cleavage: when cracks propagate along preferential paths. The particle is split between small fragments coming from the interface between the preferential paths and coarser fragments coming from the main domains delimited by preferential paths.

• Attrition and chipping: when the energy is not sufficient to propagate cracks. The particle

is eroded which results in very fine particles coming from the edges and the main

fragment of coarser size.

Among the open questions is the randomness of breakage: do cracks propagate randomly in

heterogeneous crystalline media or not? Observations by (Garcia et al., 2009) support a dependency of randomness in crack propagation to the strain rates (i.e. the speed of propagation of the cracks in the medium). This phenomenon could be analog to the situation in hydraulics

where information about an obstacle can propagate either downwards only (in the case of a supercritical flow) or downwards and upwards (in the case of a subcritical flow) (Ancey, 2017). If the speed of propagation is slow enough, the tip of the crack can receive information about the upcoming grains and adapt accordingly whereas for a high speed, the information does not

have time to propagate backwards, resulting in a crack propagation independent of the medium ahead of the tip. More generally, this underlines a general problem in modeling of breakage in comminution equipment: the boundary conditions, strain and stress fields change with time and

are generally unknown. This limits the tools that can be used in modeling to methods that do not require a perfect knowledge of the stresses acting on particles.

If random breakage was used in many previous models, recent studies have used FEM to infer breakage patterns (Wang, 2015). Another approach is to use cellular automata (Pan et al., 2009).

The model chosen relies on texture-controlled breakage. Since a graph can be associated to the mineral map, by assigning a weight w to each vertex, we can build a graph associated to a mineral

map.

(79)

(80)

For a fracture crossing the entire field, G(V;E) will be split into two particles P1 and P2 such that

(81)

(82)

To this partition one can associate a cost function defined as

(83)

Where w are the values on the edges of the lattice for a defined neighborhood system.

The choice of values for w will be guided by a minimum energy principle. A convenient way to propagate a crack is to use a minimum energy path. A fracture will propagate to the next cell

39

in its neighborhood that has the minimum weight. As soon as the total energy is equal to zero,

propagation of cracks is stopped.

The ideal case, where only phase-boundary fracture or detachment occurs is analogous to the min-cut problem. In that case, segmentation algorithms can be used to produce single particles. Otherwise, the value of w at a given site x is a function of mechanical parameters. As a first approximation, the fracture toughness GIC [Jm-2] is used (Tromans and Meech, 2004, 2002).

Dimension of fracture toughness gives a physical meaning to the cost function as the energy needed to propagate a crack through these edges.

To represent phase-boundary fracture, specific fracture toughness can be chosen (TG, trans-granular or GB, grain boundary) or w can be weighted with a gradient of the mineral map. To

represent boundary-region fracture, the metric defined on the mineral map can be used to compute a distance from the grain center to its edges. Multiplying w by a correcting factor from the center to boundary distance can lower the cost of the cut and favor a propagation close to the edge of the mineral grains.

In practice, two challenges appear. First, if the rule of the cellular automata is to propagate to

the weakest values, the representation of the texture should move from only the vertices (where the realization of the random function is seen as pixels) to a lattice with both vertices and edges (where the values w will be assigned to edges and the mineral phase assigned to the vertices).

One way to do this is to use the Kronecker product to scale up the mineral map, then assign the values of GIC to w.

(84)

(85)

The second challenge is to avoid losing pixels in the process. If cracks propagate a value outside the limits of mineral phases, it means that the values of the vertices need to be excluded from

the path of the cracks. This is done by assigning a maximum value to vertices after the Kronecker multiplication. Since the propagation rule is to go for the minimal energy it means that maximal values will never be chosen. Maximum values can also be assigned to existing cracks to ensure

that cracks will not propagate only where cracks are already present. The resulting map is referred to as the cost map.

To account for grain boundary breakage, the gradient of the image is calculated and its normalized absolute value is subtracted from the cost map. This results in lower values at mineral

40

grain boundaries. Another phenomenon that can be modeled is the long-range impact of

structures. This is achieved by using convolution filters on the cost map.

Once cracks have propagated and no energy is available, particles need to be separated from each other and stored. This is done by blob analysis (Soille, 2013) which is based on morphological opening of the particles using cracks as background.

To build a realistic stream for simulation purposes, the particle size distribution must be like the one measured on the process. To calibrate the liberation model, the particle size distribution can

be tracked for each propagation step of cellular automaton. To compute the particle size distribution, the fracture path is masked, then a morphological sieve is applied. Therefore, particle size distribution can be plotted as a function of the total energy used for fracture

propagation.

As a result, a stream is created (as a XML file structured like a QEMSCAN result file) with a defined modal mineralogy, controlled particle size distribution, and a liberation spectrum that derives from intact textures.

5. Experimental results

The techniques introduced in the previous sections were used in case studies. The focus is to demonstrate what can be done at different scales and bring the textural information from the intact ore to the particles. One important assumption made here is that meso-scale texture is related to micro-scale texture.

5.1. Meso-texture classification

The material selected for meso-texture classification comes from the Aitik Cu-Au-Ag mine, operated by New Boliden, located south of Gällivare, northern Sweden which is the largest copper mine in Europe. Aitik operates with a head grade of 0.23% Cu, 1.8 g/t Ag and 0.13g/t

Au with chalcopyrite as the main copper mineral (Wanhainen et al., 2012). All the rock units of the deposit were studied in literature (Wanhainen et al., 2012, 2006) with a focus on its lithology

and genetic evolution. Each lithology is described in terms of mineralogy and qualitative textural

description such as grain size, shape, and mineral association.

The dataset consists of high-resolution images taken with a Canon Pro 815 camera and a pixel size of 0.12 mm2 (the scale was kept constant during acquisition) with a stable light source stored as RAW files then converted to TIFF format. For each ore type between 5 and 10 m of drill cores from different depths were scanned to obtain a representative sample of each type.

The data set was then randomly cropped and rotated to form the complete data set of 1509

images in 13 classes.

For image processing purposes, Fiji (Schindelin et al., 2012) was used. WEKA (Hall et al., 2009), Python and MATLAB (Mathworks, 2016) were used for the machine learning part of the work. HSC Sim 9 (Outotec, 2016b) was used for process simulation.

Spatially distributed drill core samples of all rock units in the mine were sampled and mapped based on their geological features and their predicted metallurgical process performance.

Thirteen units representing the hanging wall, ore zone and footwall were, on a macroscopic

41

scale, classified into different textural classes giving broad ranges of textures for each rock unit

(Table 4). The goal of the classifier is to extract relevant descriptors and train a classifier to automate the classification of drill cores by type.

Table 4 : Rock units from Wanhainen et al. (2012, 2006) divided into geometallurgical textural classes

Rock units Textural classes

Footwall Feldspar-biotite-amphibole gneiss 3,5

Intrusive rocks Quartz monzodiorite 10,11,12,13,14,15

Micro-quartz monzodiorite 10, 12

Ore zone Garnet-bearing biotite schist 7,8,9

Quartz-muscovite-sericite schist 4,6

Pegmatite veins 1

Hanging wall Feldspar-biotite-amphibole gneiss 2

Pegmatite veins 1

Mafic intrusions Meta-gabbro -

An example of qualitative description of textures is shown on Table 5. While these descriptions are important to understand the geology, and classify drill cores, the lack of objective data limits

the possibility of building a model.

Table 5 : Two texture classes used for pixel classification.

Textural class 11 Quartz monzodiorite – Footwall

A medium gray-white, with grain size about 0.3–0.6 mm, and a

plagioclase-porphyritic matrix of plagioclase, biotite, K-feldspar,

and quartz. The granoblastic texture is partly sericite altered and microcline grains are usually intergrown within the matrix.

Chalcopyrite, pyrite, or magnetite, occur as veinlets and as

disseminated grains

Textural class 9 Garnet-bearing biotite schist/gneiss – Ore zone

The microcrystalline matrix of feldspar biotite, amphibole and

quartz with a grain size of average < 0.2 mm contains approximately 40% of plagioclase.

Finely disseminated chalcopyrite and/or larger patches of chalcopyrite and pyrite are the most common mineralization styles in the biotite schist/gneiss

42

The capacity of texture indicators to be automated depends on the resources needed to extract

features. In Table 6 the total time for feature extraction is presented for the whole image data set but also for each image.

In an industrial context, one strategy to classify images can be to acquire smaller portions of the drill core, send the data to the classifier and get a texture class in return. Depending on the size of the patches, the texture class can be presented as an overlay on the initial drill core. Time

performance is critical if a large amount of drill core is being processed on a daily basis. The entire process of feature extraction should be as fast as possible but also provide reliable information. This results in a trade-off between the predictive capacity and resources.

Table 6 : Time performances for the selected feature sets

Method Number of features Total CPU time [s] CPU time [ms/image]

LBP 32 65.22 43.22

HaarLevel2 128 217.98 144.45

HaarLevel3 192 299.23 198.29

LBP+HaarLevel2 160 268.94 178.22

The GLCM features are extracted from wavelet decomposition at different levels (2 or 3), which

increases their CPU time.

Each data set was tested with each classifier using 15-fold cross validation. This is the list of selected data sets

• LBP: Local binary patterns

• HaarLevel2: GLCM statistics from Haar wavelet decomposition at level 2

• HaarLevel3: GLCM statistics from Haar wavelet decomposition at level 3

• LBP+HaarLevel2: Combination of LBP and HaarLevel2 descriptors

and the following supervised classifiers

• RF: Random Forest

• SVM: Support vector machine

• kNN-p2: k-Nearest Neighbors with a Euclidean distance (q=2)

• kNN-p1: k-Nearest Neighbors with a Manhattan distance

• kNN-p1.5: k-Nearest Neighbors with a Minkowski distance (q=1.5)

• ANN: Artificial neural network with 3 hidden layers

The experiment was aimed at determining which combination of features would work best with

a given classifier. The choice of descriptors is still being studied, in particular the choice of the wavelet family is subject to caution.

Different metrics can be used to compare classifiers, in this case Matthews correlation coefficient is presented in Table 7. It is a description of the confusion matrix that indicates the correlation between the predicted class and the observed class (Matthews, 1975).

43

Table 7 : Matthews correlation coefficient. Values in bold are significantly different at p=0.05, values between

parentheses are the standard deviation values for 250 repeats.

RF SVM kNN-p2 kNN-p1 kNN-p1.5 ANN

LBP 0.97(0.06) 0.80(0.14) 0.95(0.07) 0.95(0.07) 0.96(0.07) 0.91(0.11)

HaarLevel2 0.97(0.06) 0.97(0.05) 0.90(0.10) 0.90(0.10) 0.92(0.09) 0.95(0.08)

HaarLevel3 0.97(0.06) 0.97(0.06) 0.91(0.09) 0.91(0.09) 0.92(0.09) 0.96(0.07)

LBP+HaarLevel2 0.98(0.05) 0.98(0.05) 0.91(0.09) 0.91(0.09) 0.94(0.08) 0.98(0.05)

By adding the time needed to extract features from one image to the time needed to classify that image, the total time to process one image can be estimated (feature extraction and classification). If these values are transformed into a time score (ranked) with value 0 for the longest time and 1 for the shortest processing time, a global score can be built as the product of Matthews

correlation coefficient and the time score. The meaning of the global score is the trade-off between classification performance and time.

Table 8 : Global score for classifiers and data sets. Values in bold indicate the best scores.

RF SVM kNN-p2 kNN-p1 kNN-p1.5 ANN

LBP 0.87 0.72 0.85 0.85 0.82 0.82

HaarLevel2 0.64 0.64 0.54 0.56 0.28 0.63

HaarLevel3 0.52 0.52 0.41 0.43 0.00 0.51

LBP+HaarLevel2 0.57 0.57 0.48 0.48 0.25 0.57

The results shown on Table 8 indicate that the Random Forest classifier and local binary patterns

is a suitable combination for classifying Aitik textures with a reasonable time and performance balance.

A common risk associated with machine learning is over-fitting. If the initial data set has a small variability and the model has many parameters, if the model is trained until the classification

performance is very close to 100 %, it means that the model might be memorizing the training set and be unable to generalize the results. One solution is to use an ensemble approach in which several classifiers are built then combined as in the Random Forest classifier (Cawley and Talbot, 2010).

Classification of drill cores can be used to predict the process performance. As an example, a

process model has been built for Aitik in HSC Sim 9, based on the plant configuration observed during a plant survey. After mass balancing and data reconciliation at the bulk mineral level, the chalcopyrite grade and recovery were calculated and used as a reference to build a steady-state simulation based on kinetic models for flotation and fixed particle size distribution for the mills.

The current version assumes fully liberated particles. In the future, this will be changed to include liberation data and use geometallurgical tests results for a dynamic simulation.

After classification, each image belongs to a textural class for which the approximate modal mineralogy is known. The modal mineralogy of the class is converted into a stream file that is

used as the feed to the model. In this case, the modal mineralogy was drawn from the modal

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mineralogy population for all classes. At the beginning of the simulation, all the streams are first

emptied (to remove any results from previous simulation rounds) then are calculated until convergence. The grade and recovery of chalcopyrite is computed and presented on Figure 16. Different cases were tested, the first attempt was to validate the model against the survey data.

The points labelled as sim_high, sim_low and sim_avg represent the interval of confidence for the simulation using survey data. The results show a difference which can be due to characterization or element-to-mineral conversion.

The labels A4, A8, A11, A14, A16 and A18 are different textural classes that have been characterized by chemical assays and scanning electron microscopy. The simulation results are

generally in good agreement with the survey data. The simulation model itself can be improved by more tests to fit the flotation kinetic values but also by validating the simulation with another

survey or on-line measurements from the plant. Since the simulation was steady-state, the impact of operational conditions is not present in the simulation results. The reason for the deviations

not only in grade but also recovery is the impact of classification in hydrocyclones and kinetic coefficients defined by size classes.

Figure 16 : Chalcopyrite grade and recovery from simulation. The square marker indicates actual data from the

plant survey.

Automated drill core classification can save time but also allow the use of textural information for the downstream process. If a link can be established about meso- and micro-texture, then each texture class can be linked to mineral maps. These mineral maps can be used as training

images for mineral map simulation that can produce particles for simulation.

A4Blend1

A8

A11 A14

A16

A18

Sim_low

Sim_high

Sim_avg

Survey

89

89.5

90

90.5

91

91.5

92

92.5

93

0 10 20 30 40 50 60 70 80

Ccp Rec. [%]

Ccp [wt.%] in the concentrate

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5.2. Meso-texture pixel classification

If high resolution images of drill cores have been classified, the set of likely minerals present in each class might be reduced compared to the entire deposit. In that case, another classifier can assign a mineral to each pixel. The idea presented here is to use the expertise of a geologist or

mineralogist to train a model to detect minerals at the surface of drill cores.

A machine learning approach is chosen, based on a random forest classifier and local features (in this case, the minimum, maximum, mean and variance of pixel value and different filters). This procedure allows the classifier to learn from the geologist example and improve over time.

Using Fiji/ImageJ image processing software and its trainable Weka segmentation plugin for pixel classification the following steps were taken (Table 9).

Table 9 : Steps to train and validate WEKA trainable pixel classifier on drill core images.

Pixel classifier training and validation procedure

1 Open Fiji/ImageJ

2 Open a digital image in tiff format

3 Start the trainable Weka segmentation plug-in

4 Select the mineral phases used for training by using point, polygon, or freehand selection tool. Use the original drill core to ensure valid training data set

5 Train the model with the defined training areas from step 4

6 Improve the classification by adding new training areas in the image

7 Repeat stage 5 until satisfactory results are achieved

8 Save the model and the training data set

9 Validate the model with a new image of the same texture and load the existing model/classifier

10 Apply the model. If necessary, add or remove a mineral class and re-train the model

As an example, a pixel classifier was trained (using the procedure described in Table 9) to classify

pixels into mineral phases for coarse-grained plagioclase-porphyritic quartz monzodiorite rock type. The mineral list contained plagioclase, quartz and biotite and chalcopyrite as small patches. To train the classifier, three different images of the same texture were used (Figure 17) until the

classification error was low. The results can be presented either as a color overlay or as a probability map for each mineral. In general, the overlay is easier to understand.

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Figure 17 : A pixel classifier after training on three different images.

The procedure described above was applied to the garnet-bearing biotite schist/gneiss rock type.

The mineralogy is similar to the previous case except for magnetite that was present as veins in the second sample. Five different images of the same texture were used to train the classifier. In this case, a new image presented in Figure 18 was presented to the classifier. A matrix of fine grained feldspar and biotite was identified with larger veins and patches of chalcopyrite, pyrite,

as well as irregularly distributed magnetite (Figure 18). The pixel classifier produces results similar to a naturally formed geological texture for both smaller features, such as disseminated grains of chalcopyrite but also larger veinlets of magnetite.

Figure 18 Validation of a pixel classifier trained on five different images of the same texture.

Estimation of modal mineralogy from drill core surfaces is challenging. Therefore, it should not

be used as the only characterization method. However, if drill cores are automatically classified based on high resolution images, additional information is available through pixel classification. The presence of a new mineral can be problematic since the models have been trained based on

the mineralogy of one class.

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5.3. Micro-texture simulation

The model based on MRF introduced previously can be used to simulate a wide range of mineral maps. Simulation of these maps is motivated by the variability in the ore body. If the modal mineralogy of the feed changes in the process, how can particles that still inherit the texture be

generated for simulation purposes?

The experiments consist of two parts: the learning phase in which the modal mineralogy, association matrix (AI) and grain size (g) are extracted from the training image and the simulation itself where the distribution will be sampled using Metropolis-Hastings algorithm with a gradient

descent to minimize the error.

The first step is to start with a random realization that respects the modal mineralogy. The

association index is used to generate the matrix W which is used to compute the energy U(x).

The user provides the first estimate of the regularization map β(x) in the case of a non-stationary

model, or a single value of β in the stationary case.

Pairs of points are randomly proposed for a swap. The change is accepted only if the global

energy decreases or with a probability of exp(-dU). This procedure is repeated for a specified number of pixels, the error is evaluated afterwards. The value of beta is modified according to gradient descent procedure, then a new set of pairs of points is selected and the procedure is applied iteratively.

The following examples present an easy (stationary) and more difficult (non-stationary) case for

texture simulation from actual measurements.

The first case was a macro-texture acquired from a high-resolution photograph of an Aitik sample with a disseminated texture. The image was cropped, then segmented using a k-means algorithm to obtain a mineral map.

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Figure 19 : Comparison between the training image and the simulated texture, visual comparison (top) and grain

size distribution by phase (bottom)

Size of the structuring element Size of the structuring element

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Table 10 : Comparison between the training image AI and the simulated result AI

Training image AI Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

Phase 1 0.0919 0.0147 0.0137 0.0153 0.0165

Phase 2 0.0147 0.1433 0.0162 0.0178 0.0194

Phase 3 0.0137 0.0162 0.1118 0.0161 0.0172

Phase 4 0.0153 0.0178 0.0161 0.1427 0.0197

Phase 5 0.0165 0.0194 0.0172 0.0197 0.1773

Simulated texture AI Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

Phase 1 0.0947 0.0143 0.0133 0.014 0.0157

Phase 2 0.0143 0.147 0.015 0.0169 0.0181

Phase 3 0.0133 0.015 0.1149 0.0152 0.0166

Phase 4 0.014 0.0169 0.0152 0.1472 0.0182

Phase 5 0.0157 0.0181 0.0166 0.0182 0.1815

Visual inspection is not the most reliable way to work with textures, but in this case, it seems to indicate that the realization is close to the measurement. The association matrices are presented in Table 10. The diagonal represents the tendency to form grains of the same phase while the

other elements tell about the proportion of one phase that is sharing a neighborhood with another phase. By construction, the modal mineralogy is preserved which can be seen by summing the association matrix by row or column.

The first case was modelled with a stationary field. However, the assumption of stationarity is not always valid. In the second example,the scale is very different (micro-texture) and the mineral

map was acquired with a SEM. In that case, β was different at each point. The estimate for β provided by the user is distributed in the field (beta map) and the gradient descent procedure

now updates β locally, for each pair of points selected for a swap.

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Figure 20 : Initial mineral map, courtesy of Mehdi Parian (LTU)

The result presented in Figure 22 reveals the strength and weaknesses of the method. The general look of the

texture seems similar but the shape of the grains is not well respected. This can be an issue if the realization is

used for breakage, if cracks tend to propagate along grain boundaries. On the other hand, the mismatch between

the association matrix and the grain size distribution per phase presented on

Figure 23 and Figure 24 is small. Another feature of the realization is the impact of noise, since the method is based on pixels, even the smallest details have the same probability of being sampled. As a result, very small structures are also reproduced (being noise or actual measurements).

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Figure 21 : Training mineral map

Figure 22 : Simulated mineral map

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Figure 23 : Scatterplot of the association matrices values for the training map and the simulated map

Values in AI for the training map

Val

ues

in

A

I fo

r th

e sim

ula

ted m

ap

imag

e

Values in AI for the training map

53

Figure 24 : Scatterplot of the cumulative grain distribution values for the training map and the simulated map

It appears from the examples that the auto-model is suitable for the simulation of a wide range of mineral maps. In the current version of the software, the user needs to specify the size of the

neighborhood which is equivalent to asking the scale of the phenomenon (as opposed to the scale of observation). This makes the results sensitive to the user choices and an automatic detection of the spatial range of interactions should be developed in the future.

Values in AI for the training map

Val

ues

in

A

I fo

r th

e sim

ula

ted m

ap

imag

e

Values in AI for the training map

54

Simulation of textures can also be done by other methods, among which MPS ones. Multi-scale

cross-correlation simulation (MS-CCSIM) is an pattern-based (or patch-based) algorithm developed for geostatistics (Tahmasebi et al., 2014). It is multi-scale because it uses a multi-grid approach and is suitable for images because it has been accelerated by using the FFT. The calculation of the cross-correlation is a convolution (filtering), and the convolution theorem

makes it possible to work in the Fourier space, which replaces a convolution by a multiplication, the change to and from the Fourier space can be done efficiently with the FFT. The application of this algorithm for mineral maps is currently being investigated.

Figure 25 : Piece of drill core from Malmberget iron ore deposit. The main phases are magnetite, amphibole,

and feldspar

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Figure 26 : Training image and one realization by MS-CCSIM

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5.4. Micro-texture breakage

Propagation of cracks with a cellular automaton rule results in a mineral map with cracks coded as a new phase (the maximum phase +1 to distinguish it from actual mineral phases). After a blob analysis, individual particles are stored. Once the particles are obtained, they can be used in

HSC Sim for simulation purposes.

The experiments presented here demonstrate the use of the breakage algorithm. Figure 27 shows

the propagation of cracks around a grain of magnetite in an ore from Malmberget. The type of breakage is non-random and a couple of phenomena can be seen. The contrast between distinct phases makes the gradient of the cost map to be lower around grain boundaries, which prevents

the crack from randomly entering mineral grains. The second feature of the model is the ability to account for previous damage in the ore. The previous generations of cracks are kept during breakage.

Figure 27 : Example of preferential breakage simulation in magnetite ore

The second result is based on three different synthetic mineral maps (Figure 28) generated by an auto-model to control the grain size. For the same modal mineralogy, the texture changes from a disseminated one to a coarse-grained one. If the energy available for breakage is kept constant, the difference in liberation will only be due to the texture. This is observed on Figure 29 where

the degree of liberation of each of the 4 phases is displayed as a function of the texture.

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Figure 28 : Different simulated grain size distributions with the same modal mineralogy

Figure 29 : Simulated degree of liberation per phase for the cases from Figure 28

The proposed breakage model seems to produce realistic results. Even without using preferential breakage, the particles produced will be at least as realistic as the ones produced by a random

breakage model. However, the physics of fracture propagation are not fully simulated. The

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resulting model is physics-inspired rather than fully built on fracture mechanics. The method of

calibration is still being studied but the idea is to first reproduce the same particle size distribution as the experimental data, then adjust the length of cracks and the cost map until the liberation spectrum is the same as the measured one.

6. Conclusion

If the main value of geometallurgy lies in its predictive capacity, then the process model is an essential part of it. Particle-based simulation models in mineral processing can provide more detailed information about the behavior of ore in the plant (in terms of the quality of the

products, throughput but also energy and reagent consumption). On the other hand, process

mineralogy has developed tools that can be used to validate the simulations. Texture has been shown to play an important role in both comminution and concentration processes. Therefore, understanding, quantifying the variability in textures and link the texture of the intact ore to the

particles is essential.

This work contributes to a better integration of different techniques into geometallurgy and provides a complete path from the drill core to the particles. Each step can be validated and an error estimate is provided. The final result is a population of particles with the correct modal

mineralogy, size distribution and texture.

The main research questions have been answered

• What classes of models are useful for geometallurgy as a predictive discipline?

Predictive models operating at the appropriate level of detail for each particular situation. For research, focus on designing and validating particles-based unit models seems to be relevant. The difference between the level of detail used in process mineralogy and the level of detail used in process simulation should be as small as possible. If the information is available, it should be used

also for simulation purposes in the context of geometallurgy.

• How can we model an ore texture?

Random functions appear to be a powerful tool for texture modeling. The choices about the properties of the random functions must be done after careful study of the phenomenon itself.

A good model should work both in analysis and synthesis, since this indicates that the full range

of information has been captured from the measurements.

• How can we derive quantified information about ore textures at different scales?

Different descriptors exist depending on the purpose. For drill core classification, the tests showed satisfactory results with local binary patterns on images. Wavelet-based descriptors can be used to study different scales. For mineral maps, the modal mineralogy, association index and

grain size distribution offer a good quantification but also a basis for simulation.

• Is it possible to take explicitly in account mineral texture in process simulation?

Yes, by defining textural classes, assign each drill core interval to a class by extracting features from images and geometallurgical tests, then establish correlations between meso-scale textures and micro-scale textures. Once the micro-scale textures have been measured (as mineral maps)

then variation of modal mineralogy can be simulated, the simulated mineral maps can be broken to particles. These particles can be stored in a stream file for use in particle-based process simulation.

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Orientations for further research are suggested by literature and the current limitation of the

tools proposed.

The first task after this licenciate will be to validate the models both for texture simulation and breakage to ensure that they are reliable. This will be done by SEM measurements and high resolution visible light drill core scanning.

In mineral processing simulation, the continuation of this work will be to design full circuits operating at the particle level and include textural information from the block model to the

process. This is part of PREP and done in collaboration with Viktor Lishchuk and Cecilia Lund at LTU.

The integration of characterization tools and models is another relevant topic since in the current

situation, techniques from image analysis are not well integrated in mineral processing simulation

software to the author’s best knowledge. The question of the propagation of the error is important for the applicability of geometallurgy as a whole.

For texture descriptors, the study of the transforms to which descriptors are invariant could offer innovative ways to study and analyze textures.

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68

Part II

Paper 1

How to Build a Process Model in a Geometallurgical Program? Koch PH, Lamberg P, Rosenkranz, J MiMeR - Minerals and Metallurgical Research, Department of Civil, Environmental and Natural Resources Engineering, Lulea University of Technology,Sweden Abstract. This work presents a literature review on ways to acquire relevant experimental data for the process model of a geometallurgical program. It identifies the needs in several unit models and proposes ideas for future developments. Keywords. Geometallurgy, process model, magnetic separation, flotation, gravity separation.

1 Introduction: Approaches in geometallurgy Geometallurgy is defined as a multi-disciplinary science that links geology, mineral properties and behavior of the material in metallurgical operations, more specifically in minerals processing (McQuiston and Bechaud 1968; Lamberg 2011; Vann et al. 2011).

A geometallurgical program is the industrial application of geometallurgy in a mining operation context. It includes a spatially-based model that forecasts how the ore will behave in mineral processing operations (Alruiz et al. 2009; Lund et al. 2013; Schouwstra et al. 2013; Aasly and Ellefmo 2014). It usually consists of two main sub-units: • Geological model: 3D block model in which

geological and geometallurgical data is available. • Process model: A flow sheet model in which data

from geological model is used to forecast the metallurgical performance. Depending on the approach, this model can predict the recovery rate, the element grade or mineral grade in the concentrate and tailings, particle size distribution, water and energy consumption. Combining the two models the information can be

used for data management, visualization, for production planning and to test different production scenarios.

Depending on the approach taken for geometallurgy and the level of geometallurgical information needed, the program describes the sample collection and characterization, the nature and number of tests needed for the process model, the definition of ore-types and domains and the validation of the models based on case studies.

Usually one of the following approaches is used: o Traditional: the geological and process models are

based on chemical grades from drill cores assays. Variability testing is used to develop functions to predict metallurgical performance parameters.

o Proxies: the process model is based on a large number of small-scale tests that describe the metallurgical response of the ore in the process more or less indirectly.

o Mineralogical: the geological and process models are based on quantitative mineral characterization. This approach has three levels of information : Unsized (1D) : Phases, chemical and modal

composition Sized (2D): Chemical and modal composition

by particle size (class). Liberation (3D): Chemical and modal

composition, liberation distribution by particles size classes

A review of the use of geometallurgy in current operations can be found in Lishchuk et al. (in prep.). 2 Process models used in geometallur-

gical programs The simulation of ore dressing circuits historically derives from the simulation of chemical processes. Early works Kehat and Shacham 1973; Ford 1979; Ford and King 1984 were focused on the theoretical design of simulators, the classification of different simulation systems but also the design principles for a simulation environment to be used in a production context.

Table 1 presents the most common computer simulation environment types. Nowadays, most of the commercial software seems to be based on unit operations working either in a static (steady state) or dynamic mode.

The level of information required to feed the unit models depends directly on the approach selected for the geometallurgical model: there is no need for mineralogical information if the global model relies on elemental grade only. Table 1. Classification of simulating environments.

Specific simulator

Section based simulator

Unit operation

based simulator

Hypotheses The flow sheet is fixed

Sections of the flow sheet are

fixed

Usually steady state

process

Method

Optimization of a global

equation sets under

constraints

Optimization of several equation

sets for each section under

constraints

Solve each unit

operation sequentially

3 Experimental requirements To model and simulate a process at any above listed levels, experimental data is needed where the streams are described on corresponding level. For developing and calibrating models the experimental data must be mass balanced. This gives challenging when aiming to

liberation level since the mass balancing must be done also on mineral liberation level. Table 2. Assumptions in unit models using different level of information.

Unsized Sized Chemical elements

-Mineralogy is fixed -Particle size distribution (PSD) is fixed

-Mineralogy is fixed -PSD varies

Minerals

-Liberation distribution is fixed -PSD is fixed

-Liberation fixed -PSD varies

Behavioral classes : minerals by floatability components

-Association is fixed -PSD is fixed

-Association is fixed -PSD varies

Particles

(not applicable) -Association varies -Shape is fixed -Micro-textural information could be used (Pérez-Barnuevo et al. 2013)

Experimental data is acquired by performing 3 types

of tests: o Geometallurgical tests: small-scale tests using a

limited number of samples designed to predict the particles behavior.

o Variability tests: small-scale tests using a simplified flow sheet, an extensive number of samples and duplicates designed to describe how variations in the process impact the products.

o Laboratory tests: small to medium scale tests using a limited number of samples, designed to build a flow-sheet that will be up-scaled first to the pilot level, then used in a full-scale process.

The following section details the tests available so far for each level, provides a short literature review of each of them. Some tests might not be included in this work due to the abundance of various proprietary tests. 3.1 Comminution

The basic test for comminution is the Bond test (Bond and Wang 1950; Bond 1961) that gives a measure of the Bond Work index (BWi). However, despite its wide acceptance and use, several limitations have made it less attractive (Powell and Morrison 2007).

For AG and SAG mills variability and laboratory testing, the Julius Kruttschnitt Mineral Research Centre (JKMRC) has been developing a drop-weight test (JKDWT) to describe accurately the breakability of the ore. The test provides an abrasion breakage parameter (ta) as well as impact breakage parameters (A and b) (JKTech 1992; Napier-Munn et al. 1996). A second test called the JK Rotary breakage test (JKRBT) can measure the specific breakage energy (Ecs) and the amount of material passing 10 % of the size of the initial material (t10; JKTech 2014). Another test called SMC can provide similar results to the JKDWT based on the same principles (Morrell 2004) . A model for AG and SAG mills has been proposed by Minnovex to measure a SAG power index (SPI) and a crushing index (CI) based on small-scale SAG mills (Amelunxen et al. 2014).

For geometallurgy purposes Mwanga et al. 2014 has proposed the geometallurgical comminution test (GCT) that combines small-scale ball mill tests with product characterization. The author also provides an in-depth literature review of comminution tests and suggested guidelines for comminution tests in a geometallurgical program (Mwanga 2014). 3.2 Magnetic separation The use of dry low intensity magnetic separators (dry LIMS) like the Mörtsell (1955) and wet low intensity magnetic separators (wet LIMS) for variability and laboratory testing is well established but magnetic deportment has also been integrated in geometallurgical models as a bulk property (Philander and Rozendaal 2013; Philander and Rozendaal 2014).

The Davis tube is another test that can be used as a proxy to describe the behavior of ore in magnetic separation (Murariu and Svoboda 2003; Niiranen and Böhm 2012). It operates by creating a divergent magnetic field in a glass tube through which the sample is poured (dry or as a slurry). 3.3 Gravity separation Several methods can be used at variability testing or laboratory testing level to reproduce gravity separation units: jigs, shaking tables, small-scale spirals, dense/heavy media separation and centrifugal methods (Knelson). Dense media separation (DMS) tests have been applied to choose a gravity separation method according to the density of the ore (Mills 1980). However, the use of these tests in a geometallurgical context seems to be limited. Possible development in this area could include a gravity separation step coupled with a real-time characterization method like flow-cell multispectral measurements (Leroy et al. 2014). This could allow a fast acquisition of both mineralogical and liberation data at a particle level, based on gravity separation. 3.4 Flotation For variability and laboratory testing, flotation tests are repeated with a large number of samples under various experimental conditions (changes in collector, frother concentration, particle size distribution, pulp density, pH, Eh, air flow and agitation speed) using a simplified sequence similar to the full scale process. These tests can be done either as one batch or in a cycle mode, closer to the plant operating conditions but time consuming and prone to errors (Wills 2011).

Several testing procedures have been developed to standardize these tests and ease their up scaling as summarized in following chapters. A detailed review of variability and laboratory test procedures can be found in Yianatos et al. (2010).

JK Floatability Index test (JKFIT, developed at the JKMRC) uses a sum of the floatability component of each mineral (fast, slow or non-floating) weighted by their mass proportion in the sample to compute a floatability index that can be scaled up. The basic test is

at the unsized mineral level but can be improved by integrating particle size distribution and liberation data from MLA measurements. As a result, this test is suitable for laboratory and variability testing but can also be linked with a geometallurgical model using JKSimFloat and JKSimMet.

MinnovEX Flotation Test (MFT, developed by SGS) uses a continuous flotation rate distribution (k distribu-tion) to distribute pulp particles of different sizes and degree of liberation into several classes. Once the k distribution is known by doing a small-scale test, it can be scaled up. Based on a mass conservation relation, one k distribution propagates from the feed to the other streams in the process. Since the test is at the sized mineral level, changes in the particle size distribution can be reproduced in simulation (Dobby and Savassi 2005). A similar test has been developed by (Amelunxen and Amelunxen 2009). Since liberation data is not explicitly taken into account, this test is suitable for laboratory and variability testing.

Efforts have been done to build geometallurgical tests for flotation. One of them was based on a shaker test in which a test tube containing the pulp is shaken by a pneumatic device (Chudacek and Fichera 1991). Vos et al. 2014 proposed a new small-scale test to determine flotation performance based on that principle: JK Mineral Separation Index (JKMSI). The results were consistent with the overall performance of batch flotation tests and plant processes, despite challenges to achieve a high recovery on the test case samples.

In flotation, collectors are fixed to the minerals to make their surface hydrophobic. Hydrophobicity is related to the liquid-mineral contact angle θ (measured between the liquid-air interface and the solid-liquid interface) by Young’s equation.

During flotation the contact angle between water and the mineral should decrease and the air-mineral contact angle should increase as the surface of a floatable mineral is hydrophobic. Therefore, proper contact angle measurements could be related to the floatability of a given mineral and provide a proxy for flotation in a geometallurgical context. However, this kind of measurement is often done by optical instruments and in non-ideal conditions (non-smooth and non-flat surface, reactive minerals) resulting in an apparent contact angle rather than the actual (intrinsic) one (Chau 2009).

Another technique called Time-of-flight Secondary ion mass spectrometry (TOF-SIMS) is based on the bombardment by a primary ion beam resulting in the detachment of secondary ions at the surface of the particle. The main issue with this method is the collection and storage of samples which are required to be degassed with nitrogen then frozen in liquid nitrogen (Brito e Abreu and Skinner 2011; Chehreh Chelgani and Hart 2014). For geometallurgical purposes, a possible flotation proxy could be developed through reliable optical contact angle measurement technique which could then be validated with TOF-SIMS. 3.5 Sensor-based tests for particles Developments in sensor technology and improved computational power increase the use of real-time signal

processing to classify particles. A wide range of the electromagnetic spectrum can now be measured either as a transmitted, diffracted, reflected or polarized signal (Wotruba and Harbeck 2012). Further extensions to sensors might be expected in the Terahertz range as well in a near future (Yu et al. 2010).

Limitations for a laboratory use include the high costs and dimensions of the equipment. So far, the main purpose of sensor-based equipment has been to track changes in the streams and adapt process parameters adequately. As such, they cannot be regarded as laboratory nor variability testing methods.

These techniques are in between characterization of particles and separation tests and are de facto suitable for geometallurgical testing if they provide reliable online measurements.

Table 3. Classification of small-scale tests Geometallurgical

tests Variability

tests Laboratory

testing Scale-up

factor

Comminution GCT SPI, JKRBT, JKDWT

Bond test, simplified Bond test

Test-specific

Gravity separation - DMS

DMS, jigs, spirals, shaking table variants

Test-specific

Flotation JKMSI, Contact angle measurement

MFT, JKFIT

Traditional flotation test work

Between 1.5 and 3 for classical tests (Yianatos et al. 2003)

Magnetic separator

Davis tube and Frantz isodynamic test

Mörtsell, Small-scale LIMS

Small-scale LIMS Not needed

4 Conclusion Despite a trend towards standardization of variability tests and various laboratory scale devices, the actual number of tests satisfying the requirements of low costs and ease of use for geometallurgy is very limited. This shows an urgent need for developing such tests. Alternative approach is to use particle approach where both ore and process models are taken to particle, i.e. mineral liberation level. In testing this means that laboratory tests must be analyzed in mineral liberation level. Thereafter in modeling and simulation a validated assumption that similar particles behave in process in similar manner regardless of their origin, can be used. Acknowledgements The authors wish to thank Mehdi Parian and Abdul Mwanga who have assisted and have shown great support in this study. This survey is part of PREP project funded by VINNOVA. References Aasly K, Ellefmo S (2014) Geometallurgy applied to industrial

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Paper 2

Automated Classification of Drill Core Textures for Geometallurgy

Pierre-Henri Koch*, Cecilia Lund, Viktor Lishchuk, Jan Rosenkranz, Pertti Lamberg

* Corresponding author: pierre-henri.koch@ltu.se

Luleå University of Technology

Division of Minerals and Metallurgical Engineering (MiMeR)

Luleå, Sweden

Abstract

In geometallurgy, a process model operating at the mineral liberation level needs quantitative textural information about the ore. The utilization of this information within process modeling and simulation will increase the quality of the predictions.

In this study, we use descriptors derived from RGB images and machine learning algorithms to group drill core intervals into textural classes and estimate mineral maps by automatic pixel classification. Different descriptors and classifiers are compared, based on their accuracy and capacity to be automated. Integration of the classifier approach with mineral processing simulation is also demonstrated. The quantification of textural information for mineral processing simulation introduces new tools towards an integrated information flow from the drill cores to a geometallurgical model.

The approach has been validated by comparing traditional geological texture classification against the one obtained from automatic methods. The tested drill cores were sampled from a porphyry copper deposit located in Northern Sweden.

1. Introduction Drill core logging by geologists commonly includes a qualitative description of the lithology, mineralogy and ore textures. It is not only a cost and time consuming method occupying several geologists, it also suffers from human interpretation. With recent technology in drill core scanning and industrial devices (e.g. HyLogger and SisuROCK) the fast identification of some of the mineral phases to generate mineral maps at macro scale has been facilitated. This information gives the opportunity to quantify this textural information for comparison and classification purposes (Schodlok et al., 2016; Specim, 2013).

Quantified mineralogical information (modal mineralogy and mineral texture) enables simulation to forecast the process performance. On-line measurement methods exist for elemental analyses e.g. XRF (Estrada Calderon and Zagerholm, 2012) and methods to quantify the mineralogy by using the routine chemical assay to convert the elemental components into mineral grade has been developed and tested (Lund et. al 2013). Mineral (micro-) textures are difficult to measure and quantify. Moreover, the high-resolution hyperspectral scans still do not provide adequate additional mineralogical information for the technique to be implemented at a mine production scale due to its reliance on human interpretation. Therefore it is associated with an interpretation bias (Arqué Armengol, 2015). A new logging methodology is therefore necessary to facilitate the daily tasks of geologists which also allow cost-effective, non-destructive and fast logging of large intervals of drill cores.

1.1 Establish and evaluate analytical tools The development of technology in image acquisition and image processing combined with the increasing computing power enables the integration of a wide range of sensors in drill core scanning devices. However, the acquired signals should be relevant to the problem at hand. A geometallurgical model consists of two main parts: a spatial model and a process model. Drill core samples are adequate for geometallurgy since they are collected in any case for exploration and resource estimation purposes. Different small-scale tests can be conducted on such samples, for example in comminution (GCT, Mwanga 2014) but also in concentration (JKMSI, Jos et al. 2014).

Texture is among the parameters that influence the processing behavior of an ore and has been carefully studied in different deposits (Bonnici, 2012; Bonnici et al., 2008; Lund et al., 2015; Pérez-Barnuevo et al., 2013; Tungpalan et al., 2015).

A general approach based on particles and their mineralogy has been described by Lamberg (2011). The idea is to use the information from the geological model (spatial model) to generate particles and simulate the comminution and concentration process to forecast the products. The results can be converted into performance indicators relative to a specific deposit and plant. These indicators can then be stored in the spatial model as a tool for mine planning and process design. The entire chain forms a geometallurgical model shown in Figure 1.

Figure 1 : Geometallurgical model based on particles (Lamberg, 2011), modified

In this context, the present work aims at using drill cores to acquire information about texture. Recently, interest in automated drill core scanning has increased, with extensions towards 3D textural analysis (Becker et al., 2016), and correlations between the textural class and their behavior in the process (Nguyen et al. 2016, Pérez-Barnuevo et al., 2016).

A systematic approach to texture classification is developed and used to study the potential of several textural descriptors (or features) and classifiers in an automated logging process. The first step is to acquire images of the drill cores with sufficiently high resolution. Features are then extracted from the image and stored in a data set. Depending on the values of the descriptors, a classification algorithm assigns each image into a texture class. Once the class is known, the adequate pixel classifier is chosen and the mineral phases of the drill core are identified. If the textural class is known and the modal

• Spatial model

MineralogyTextures

• Texture breakage model

Particles• Process

model

Particles behavior

• Production forecast

Performance indicators

mineralogy is estimated, the knowledge is sufficient to build a stream file for process simulation. The simulation software (in this case HSC Sim, version 9 Outotec 2016) uses the constructed feed to calculate every stream including the concentrate and tailings stream (in terms of recovery per mineral, flow rate, particle size distribution). Figure 2 illustrates the procedure.

Figure 2 : Texture classification process

2. Material and methods 2.1 Data set and software

Images of the original dataset were taken with a Canon Pro 815 camera and a pixel size of 0.12 mm2, with a stable light source and images were stored as RAW files then converted to TIF files. For each ore type between 5 and 10m of drill cores from different parts of the ore body were scanned to obtain representative RGB images of each type. The scale was kept constant.

The data set was then randomly cropped and rotated to generate images of different sizes and 3 color channels for feature extraction and classification purposes. The complete dataset includes 1509 images in 13 textural classes. In general, the definition of classes depends on the deposit and its variability.

For image processing purposes, Fiji (Schindelin et al., 2012) was used. WEKA (Hall et al., 2009), Python and MATLAB (Mathworks, 2016) were used for the machine learning part of the work. HSC Sim 9 (Outotec, 2016) was used for process simulation.

2.2 Description of the textural classes The selected case study is the Aitik Cu-Au-Ag mine, operated by New Boliden, located just to the south of Gällivare, northern Sweden which is the largest copper mine in Europe. Aitik operates with a head grade of 0.23% Cu, 1.8 g/t Ag and 0.13g/t Au with chalcopyrite as the main copper mineral (Wanhainen et al., 2012). Previous studies have described and classified in detail all the rock units of the Cu-Au-Ag deposit (Wanhainen et al., 2012, 2006) to unravel the genetic evolution. Each lithology is described in terms of mineralogy and qualitative textural description such as grain size, shape and mineral association.

Spatially distributed drill core samples of all rock units in the mine were sampled and mapped based on their geological features and their estimated metallurgical performance. Thirteen textural classes representing the hanging wall, ore zone and footwall were, on a macroscopic scale, classified into different textural classes giving broad ranges of textures for each rock unit (Table 1).

Image acquisition•Color

calibration•Filtering•Cropping

Feature extraction•Batch process•Write datasets

Texture classification•Fixed size

patches•Deposit

specific

Pixel classification•Segmentation

algorithm•Mineral

identification

Feed construction•Modal

mineralogy•Comminution

properties

Table 1 : Rock units from Wanhainen et al. (2006, 2012) divided into geometallurgical textural classes

Mine units Rock units/lithology Geometallurgical textural classes

Footwall Feldspar-biotite-amphibole gneiss 3,5

Felsic intrusive rocks Quartz monzodiorite 10,11,12,13,14,15

Micro-quartz monzodiorite 10, 12

Pegmatite veins 1

Ore zone Garnet-bearing biotite schist 7,8,9

Quartz-muscovite-sericite schist 4,6

Hanging wall Feldspar-biotite-amphibole gneiss 2

Mafic intrusion rock Meta-gabbro -

In this case study, two textures, one from the mineralized footwall and one from the main ore zone are described in detail and used to demonstrate pixel classification (Table 2).

Table 2 : Two texture classes used for pixel classification.

Textural class 11 Quartz monzodiorite – Footwall

A medium gray-white, with grain size about 0.3–0.6 mm, and a plagioclase-porphyritic matrix of plagioclase, biotite, K-feldspar, and quartz. The granoblastic texture presents partial sericitic alteration and microcline grains are usually intergrown within the matrix.

Chalcopyrite, pyrite or magnetite, occur as veinlets and as disseminated grains.

Textural class 9 Garnet-bearing biotite schist/gneiss – Ore zone

The microcrystalline matrix of feldspar biotite, amphibole and quartz with a grain size of average < 0.2 mm contains approximately 40% of plagioclase.

Finely disseminated chalcopyrite and/or larger patches of chalcopyrite and pyrite are the most common mineralization styles in the biotite schist/gneiss.

2.3 Feature extraction Among the most popular textural features two main approaches were selected: grey-level co-occurrence matrix (GLCM) statistics and local binary patterns (LBP). GLCM descriptors have been introduced by Haralick et al. (1973) and recently applied to drill core characterization by Becker et al. (2016). In our implementation, we combine a Haar wavelet multi-scale decomposition (Antonini et al., 1992; Mallat, 1989) with statistics derived from the GLCM at each scale. LBP have been introduced by Ojala and Pietikainen (2002) and have found multiple applications in image analysis such as in face description (Ahonen and Hadid, 2006) and specifically in texture analysis either as such (Li et al., 2015) or in a modified version (Hegenbart and Uhl, 2015; Liao et al., 2009).

The result of feature extraction is a vector (feature vector) that contains the texture descriptors associated with a given input image.

2.4 Texture classification algorithms Classifiers are functions that assigns to each feature vector a texture class. K-Nearest Neighbors methods (Aggarwal, 2015; Bandyopadhyay and Pal, 1991), Random Forests (Breiman, 2001), Support vector machine (Pérez-Ortiz et al., 2016) and artificial neural networks (Hebb, 1949; Rosenblatt, 1961) were selected as classifiers for this study.

2.5 Pixel classification method Once the acquired image is classified, the set of probable minerals can be reduced. It is thus possible to classify pixels from the original image into minerals. A machine learning approach is chosen, based on a random forest classifier and image features. This procedure allows the pixel classifier to learn from the geologist and improve every time new training data is provided.

Using Fiji/ImageJ image processing software and its trainable Weka segmentation plugin (Arganda-Carreras et al., 2017) for pixel classification the following steps were taken (Table 3).

Table 3 : Steps to train and validate WEKA trainable pixel classifier on drill core images.

Pixel classifier training and validation procedure

1 Open Fiji/ImageJ

2 Open a digital image in tiff format

3 Start the trainable Weka segmentation plug-in

4 Select the mineral phases used for training by using point, polygon or freehand selection tool Use the original drill core to ensure valid training data set

5 Train the model with the defined training areas from step 4

6 Improve the classification by adding new training areas in the image

7 Repeat stage 5 until satisfactory results are achieved

8 Save the model and the training data set

9 Validate the model with a new image of the same texture and load the existing model/classifier

10 Apply the model. If necessary, add or remove a mineral class and re-train the model

3 Experimental results In this section, we present the results of the numeric experiments for each of the steps presented in Figure 2. The focus besides the accuracy of the classification is to evaluate the potential of each method in terms of automation. The main factors considered are the speed of feature extraction and the speed of classification once the features have been extracted.

3.1 Feature extraction performances The capacity of texture indicators to be automated depends on the resources needed to extract features. In Table 4 we present the total time for the whole image data set but also the time taken to extract a feature vector for each image.

Table 4 : Time performances for the selected feature sets

Method Number of features Total CPU time [s] CPU time [ms/image]

LBP 32 65.22 43.22

HaarLevel2 128 217.98 144.45

HaarLevel3 192 299.23 198.29

LBP+HaarLevel2 160 268.94 178.22

The GLCM features include the wavelet decomposition which increases their CPU time.

3.2 Texture classification Each data set was tested with each classifier using 15-fold cross validation. This is the list of selected data sets

• LBP: Local binary patterns • HaarLevel2: GLCM statistics from Haar wavelet decomposition at level 2 • HaarLevel3: GLCM statistics from Haar wavelet decomposition at level 3 • LBP+HaarLevel2: Combination of LBP and HaarLevel2 descriptors

and classifiers:

• RF: Random Forest • SVM: Support vector machine with sequential minimal optimization (SMO) • kNN-p2: k-Nearest Neighbors classifier with a Euclidean distance • kNN-p1: k-Nearest Neighbors classifier with a Manhattan distance • kNN-p1.5: k-Nearest Neighbors classifier with a Minkowski distance (p=1.5) • ANN: Artificial neural network with 3 hidden layers

Different metrics can be used to compare classifiers, in this case Matthews correlation coefficient is presented in Table 5. It is a description of the confusion matrix that indicates the correlation between the predicted class and the observed class (Matthews, 1975).

Table 5 : Matthews correlation coefficient. Values in bold are significantly different at p=0.05, values between parentheses are the standard deviation values for 250 repeats.

RF SVM kNN-p2 kNN-p1 kNN-p1.5 ANN

LBP 0.97(0.06) 0.80(0.14) 0.95(0.07) 0.95(0.07) 0.96(0.07) 0.91(0.11)

HaarLevel2 0.97(0.06) 0.97(0.05) 0.90(0.10) 0.90(0.10) 0.92(0.09) 0.95(0.08)

HaarLevel3 0.97(0.06) 0.97(0.06) 0.91(0.09) 0.91(0.09) 0.92(0.09) 0.96(0.07)

LBP+HaarLevel2 0.98(0.05) 0.98(0.05) 0.91(0.09) 0.91(0.09) 0.94(0.08) 0.98(0.05)

By adding the time needed to extract features from one image to the time needed to classify that image, the total time to process one image can be estimated. If these values are transformed into a time score with value 0 for the longest time and 1 for 0 second of processing, a global score can be built as the product of Matthews correlation coefficient and the time score. This helps rank the performance of a given combination of features and classifiers.

Table 6 : Global score for classifiers and data sets. Values in bold indicate the best scores.

RF SVM kNN-p2 kNN-p1 kNN-p1.5 ANN

LBP 0.87 0.72 0.85 0.85 0.82 0.82

HaarLevel2 0.64 0.64 0.54 0.56 0.28 0.63

HaarLevel3 0.52 0.52 0.41 0.43 0.00 0.51

LBP+HaarLevel2 0.57 0.57 0.48 0.48 0.25 0.57

The results shown on Table 6 indicate that the Random Forest classifier and local binary patterns is a suitable combination for classifying Aitik textures with a reasonable time and performance balance.

3.3 Pixel classification As an example of pixel classification, we present the two classes introduced in section 2.2.

3.3.1 Training of pixel classifier The first pixel classifier was trained using the coarser grained plagioclase-porphyritic quartz monzodiorite rock type. In total four different minerals were identified. The major minerals are plagioclase, quartz and biotite occurring as granular matrix cross-cutting by smaller veins of quartz. Minor amount of chalcopyrite exists as smaller patches. To train the classifier, three different images of the same texture class were used (Figure 3) until the out-of-bag error was small.

Figure 3 : A trained pixel classifier using three different images of quartz monzodiorite textural type.

3.3.2 Validation of pixel classifier The stepwise procedure described above was applied to the garnet-bearing biotite schist/gneiss rock type. Five different images of the same texture were used to train the classifier and the image presented in Figure 4 was used only to validate the trained pixel classifier. A matrix of fine grained feldspar and biotite was identified with larger veins and patches of chalcopyrite, pyrite, as well as irregularly distributed magnetite (Figure 4). The pixel classifier produces validation results in accordance to a naturally formed geological texture for both smaller features, such as disseminated grains of chalcopyrite but also larger veinlets of magnetite.

Figure 4 Validation of a pixel classifier trained on five different images of the same texture.

The results suggest that visible light image (RGB) can be used for phase identification after textural classification constrained the mineralogy. Training the segmentation model requires mineralogists’ and geologists’ expertise but once trained, it can be automated.

4 Process simulation and performance Textural classes can be used for process simulation. If chemical assays of the current drill core are available, the appropriate element-to-mineral conversion (EMC) recipe can be selected (with correct minerals and the exact mineral compositions is available) (Lund et al. 2013). In that case, the quality of

the modal mineralogy estimate is improved. In absence of chemical assays for every drill core, its textural class can be used to choose the appropriate pixel classifier and estimate the mineralogy at the surface of the drill core. Grinding tests results and modal mineralogy provide enough information to simulate the concentration processes at the 2D level (i.e. minerals by size, Lamberg and Lund, 2012).

A process model has been built for Aitik in HSC Sim 9, based on the plant configuration observed during a plant survey. After mass balancing and data reconciliation, the chalcopyrite grade and recovery was calculated and used as a reference to build a steady-state simulation. The current version is basic and assumes fully liberated particles. In the future, this will be changed to include liberation data and use geometallurgical tests results for a dynamic simulation.

Once the drill core image has been classified (assigned a geometallurgical texture), the modal mineralogy of the texture class is used to build a stream file. This file is the stream representing the feed for the model in HSC Sim. When the simulation is started, all the streams are calculated until convergence (in this case 45 iterations) and concentrate (and tailings) streams are saved. The grade and recovery of chalcopyrite (as an example, in the case of copper concentration) is computed and presented on Figure 5. The labels indicating Sim_low and Sim_high represent the upper and lower bounds for modal mineralogy of the plant survey (low and high chalcopyrite grade), whereas the Sim_avg is an average of the two extreme cases. The simulation results are in good agreement with the plant data. The labels A4, A8, A11, A14, A16 and A18 are different textural classes that have been characterized by chemical assays and scanning electron microscopy. This illustrates the different behaviors of different classes in the process model as well as the potential of the method.

The main limitations of the current model are the assumption of liberated particles (this eliminates the need for an explicit texture-based liberation model for now) and the lack of validation for single texture performance (due to blending during mining and stockpiling operations). As an example, blending was tested and reported under the label Blend1 which corresponds to a mix of 60 % of A8 and 40 % of A16.

Figure 5 : Chalcopyrite grade and recovery from simulation. The square marker indicates actual data from the plant

survey.

5 Integration of process simulation to the block model Knowing spatial distribution of the metallurgical properties inside the ore body is key to a reliable production forecast. Production forecast is important for the mines with deposits of complex mineralogy and high fluctuations of the beneficiation performance. Since operating costs of the mining operations are significantly higher than capital investments, these fluctuations can contribute to a loss of profitability.

Integration of the mineral processing parameters into a spatial model has multiple approaches due to their variety: recovery, throughput, energy and reagent consumption, degree of liberation, texture etc. Having different natures, mineral processing parameters or geometallurgical parameters will have different properties when propagated in the spatial model. These variables can be classified as continuous (e.g., elemental and mineral grades, Davis tube recovery, A*b, solubility ratio, liberation) or categorical (e.g., lithology, hydrothermal alteration, ore type, textural class). For sampling and modelling purposes it can be important to distinguish intrinsic properties called primary parameters (e.g., mass, color, specific gravity) from process-dependent one, called response (e.g., throughput, recovery, grindability) parameters. This classification was introduced by Coward et al. (2009) and is known as the “Primary-response framework”.

Limitations of spatial modelling of the geometallurgical parameters caused by nonlinearity or non-additivity have been highlighted by multiple scholars (Coward et al., 2009; Deutsch et al., 2016; Keeney and Walters, 2011; Newton and Graham, 2011; Richmond and Shaw, 2009; Walters, 2011). Therefore, many (such as Richmond and Shaw (2009), Keeney and Walters (2011), Dunham and Vann (2007), van den

A4 Blend1A8

A11 A14

A16

A18

Sim_low

Sim_high

Sim_avg

Survey

89

89.5

90

90.5

91

91.5

92

92.5

93

0 10 20 30 40 50 60 70 80

Ccp Rec. [%]

Ccp [wt.%] in the concentrate

Boogaart et al., 2013) agree that traditional geostatistical methods which are based on weighted averages have limited application for the geometallurgical block model. Both Newton and Graham (2011) and Walters (2009) distinguish two possible origins of the additivity problem: mathematical and machine influence. Boogaart et al. (2013) also mention how non-linearity originates in non-linear scales at which those parameters were measured.

One possible solution for overcoming non-additivity issue is simulation (Carrasco et al., 2008; Deutsch, 2013; van den Boogaart et al., 2013). The advantage of the simulation approach over the conventional one is that value calculated from the multiple simulations is an average and therefore is a more reliable estimate of the nonlinear parameter. In addition, van den Boogaart et al. (2013) suggest to use compositional and geometric geostatistics.

Table 7 Geometallurgical indices deployment into the 3D spatial model – current state

Parameters Measured/claimed* impact

Method Comments Case study

Reference

Throughput (via low A*b and high BWi)

Open pit optimization, 300 days of the operation forecast (probabilistic). Monte Carlo simulation was used.

Multivariate geo- statistical simulation using an intrinsic super secondary approach, Gaussian sequential simulation, global kriging,

Metallurgical properties are frequently unequally sampled and sampled at scales much larger than typical assay measurements. Handling multivariate data was a challenge.

South American copper– molybdenum porphyry deposit.

Deutsch et al., 2015

Grade of unspecified element/mineral, liberation grain sizes

Gain per ton was optimized

Conditional simulations Correct modelling of the complete random field of geometallurgical properties is required.

Non-spatial simulated data

Boogaart et al., 2013

Material composition (deleterious, hematite, quartz, and shale)

Gain of selling at selected choice for the minable blocks (sell as lump, sell as high quality, sell as low quality)

Co-kriging and geostatistical simulation

- A high-grade iron ore deposit

Tolosana-Delgado et al., 2015

Multivariate mineralogical groups (MMG) defined via PCA, A*b, BMWi

Mine planning, scheduling activities

Nearest neighbor, indicator kriging, stochastic, ordinary kriging trend analysis (with and without zone control for A*b and BMWi)

- Cadia East, Red Dog

Newton and Graham, 2011

Cu recovery, U3O8 recovery, acid consumption, net recovery, drop weight index, and bond mill work index

Mine planning, mine optimization, and plant performance optimization*

Sequential Gaussian simulation with a subsequent regression fit to predict plant performance. Linear regression model was based on four super variables derived from 112 original variables.

Reliance on the multivariate Gaussian distribution after univariate transformation was a limitation factor.

Olympic Dam

Boisvert et al., 2013

Throughput, recovery

Net present values, internal rate of return

Process simulation applied to every block. Nearest neighbor, linear prediction for data propagating.

- Synthetic deposit model

Lishchuk et al., 2016a, 2016b

Most common techniques for propagating geometallurgical parameters into a block model include: geostatistical interpolation (kriging, inverse distance), geostatistical simulation, regression, machine learning, domaining, classification or their combinations. Some of those are listed in the Table 7 and mention relevant case studies. Until recently, not much work has been conducted on deployment of the textural data in a spatial model. Texture as it is understood in this study, is a multivariate descriptor of the ore and multivariate predictor for the process simulation. Texture carries information about both primary and response parameters and that is why it cannot be regarded as another dimension reducing method (such as principle component analysis method for example). This poses three important requirements to textural classes. The first one is that texture should include a distinctive description of the intrinsic ore properties thus being easily separated from other textures either by a trained geologist or by the automation system. Next, it should represent a significantly difference in processing properties. The third one is to have a representative volume of the defined texture in the ore body. Without fulfilling those requirements spatial modelling will not yield into a reliable forecast. The easiest way to approach solution is testing hypotheses on a synthetic deposit models such as proposed by Lishchuk (2016). An advantage of using synthetic data is the complete knowledge of the simulated ground truth.

6 Discussion The case study has revealed several critical steps. The first one is the definition of appropriate geometallurgical classes based on rock types but also process performance (either estimated from laboratory tests or plant survey). The underlying assumption is that a correlation exists between the meso texture observed on the drill core surface, its micro texture as seen under a microscope and its behavior in comminution and concentration. There is a need for a measure of similarity between different scales of textures.

The second challenge is to define the boundaries of the approach: in this case study, a good knowledge of the geology was needed as well as variability in textures and enough drill core images to use machine learning tools. This challenge is directly linked to the process model: each geometallurgical class must be tested at a level sufficient to build a stream file and run the simulation.

Benchmarking different combinations of textural features and classifier is a tool for decision. The first requirement is a suitable metric that includes both texture discrimination accuracy and speed requirements. We proposed a global score in this study, but other options should be investigated.

7 Conclusion An approach with a limited number of features and a simple but efficient classifier allows reaching performances close to a real-time drill core texture classification system. The value for geometallurgy is to maximize the information extracted from samples available at an early stage. Building a stream file for an integrated simulation is possible, however the link between meso- and micro-texture should be studied in more detail to simulate the process at the liberation level. The combination of automated drill core scanning with a process and a spatial model inside a geometallurgical model should be validated at full-scale with all textural types.

8 Acknowledgments The authors wish to thank Boliden for their support in this study and Prof. C. Wanhainen for interesting discussions about the Aitik deposit. This work is part of PREP project funded by VINNOVA.

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Paper 3

TEXTURE-BASED LIBERATION MODELS FOR COMMINUTION

Pierre-Henri Koch *, Jan Rosenkranz

Luleå University of Technology Division of Minerals and Metallurgical Engineering (MiMeR)

Luleå, Sweden

* Corresponding author: piekoc@ltu.se

ABSTRACT

The relation between breakage mechanisms and liberation is critical in mineral processing. Recent studies underline the importance of texture in liberation. This study reviews relevant liberation models and proposes a new method for generating particles using image processing algorithms. One new texture simulation method and its relevance for liberation simulation is also introduced.

1 Introduction

One of the objective of comminution in mineral processing is to liberate the minerals of interest at the coarsest particle size (Wills and Napier-Munn, 2011). Comminution models are used both for process design and process simulation. Yet, few unit models operate at the liberation level.

In this work, texture is defined as the distribution of minerals and pores in a defined volume. Texture has been carefully studied in different deposits, at meso-scale (Bonnici et al., 2008; Lund et al., 2015; Pérez-Barnuevo et al., 2016; Wightman et al., 2014) but also at micro-scale (Pérez-Barnuevo et al., 2013, 2012; Pirard et al., 2007) or both (Bonnici, 2012). The behavior of an ore in processing circuit is influenced by the texture of the particles created during the comminution stages (Bushell, 2012; Donskoi et al., 2008; Lund et al., 2014; Pérez-Barnuevo et al., 2016; Tungpalan et al., 2015)

A general approach based on particles and their mineralogy has been described by Lamberg, 2011. The idea is to use the information from the geological model (spatial model) to generate particles and simulate the comminution and concentration process to forecast the products (i.e. the concentrate and tailings). The results can be converted into performance indicators relative to a specific deposit and plant. These indicators can then be stored in the spatial model as a tool for mine planning and process design. The entire chain forms a geometallurgical model shown in Figure 1.

Figure 1 : Geometallurgical model based on particles (Lamberg, 2011), modified

This work offers a technique that covers the second step on Figure 1 by breaking a texture to generate particles. These particles can then be turned into a stream file and used for simulation purposes.

2 Texture representation

A convenient framework to represent a texture is provided by the random function (or field) theory, for a geosciences approach the interested reader is referred to Chiles and Delfiner, 2009; Lantuéjoul, 2013; Matheron, 1970 ,and for an image processing perspective, to Serra and Cressie, 1984; Winkler, 2012. From a historical perspective, it is significant that Jean Serra (one of the key figures in morphology) and Georges Matheron (one of the key figures in geostatistics) worked both at the Ecole des Mines de Paris, in Fontainebleau. An account of the birth of mathematical morphology clarifying the links between geostatistics and morphology is to be found in Matheron and Serra, 2002.

The proposed texture model uses the simplest case of a discretized finite volume and the associated sets of edges (Ex) and vertices (Vx).

(1)

(2)

If the lattice is equipped with a distance, it becomes a metric space.

A particle is defined as a subset

(3)

Each element of Pl can be associated to scalar physical properties, for example a mass by unit volume element

(4)

At the particle level, a symbol set S can be associated to each particle, in this case we number mineral phases.

(5)

(6)

The function φ equipped with a probability measure and an algebra is a random function.

The n-neighbors of an element is the set formed by the union of all the elements that are n-connected to this element. For each x, s is an element in its neighborhood is denoted s ~ x.

•Spatial model

MineralogyTextures

•Texture breakage model

Particles•Process

model

Particles behavior

•Production forecast

Performance indicators

Figure 2 : 6-neighborhood

Let x be an element of phase i of a mineral map, then we can count the number of elements of phase j in s the neighborhood of x

(7)

With

In practice, we can use neighborhoods to compute a global association index for a given finite mineral map. As an example, on a mineral map with 3 phases, one can randomly pick up an element x and count the number of each different phases in its n-neighborhood, repeat this step many times, then divide it by a normalization constant N2. The law of large numbers ensures convergence to a frequency and the association matrix (AI) can be built (equation (8)).

(8)

Along with other measurements such as the grain size distribution, the association matrix provides a convenient way to quantify association of mineral phases both on particles and intact textures. The association matrix is similar to what has been used in the past as phase-coordination matrix used by Steiner, 1975. One major difference is that AI is a multi-resolution operator since the size of the neighborhood can be modified.

3 Predictive models of liberation

Several models of liberation have been proposed over the years by many scholars. The most important ones that have reached a usable form and their main properties are presented in Table 1.

The mode of breakage is classified between random and non-random according to the recommendations of Mariano et al., 2016. A model is classified as non-random if it implements at least one of the non-random breakage mechanisms described in King and Schneider, 1998.

The first observation is that most of the early models focused on binary ore-gangue system, which reduces their practical use for simulation of more complex mineralogy. Recent focus on mineral processing simulation consider multi-minerals systems.

The mode of breakage was considered random for a long time and Barbery and Ledoux note in 1988 that “all researchers agree on the assumption that particle and texture are not related” (Barbery and Leroux, 1988). However, two recent studies have revealed the importance of phase boundary fracture for liberation (Leißner et al., 2016; Little et al., 2016; Xu et al., 2013).

Not all the models presented have been used for mineral processing simulation purposes. In this regard, one model has been implemented in commercial simulation software (King and Schneider, 1998).

Some of the models do not use explicitly particles, as such the shape of the particles produced is not applicable. On most of the other models, shape is identified as a parameter often overlooked and restricted by many assumptions (at best convexity).

Table 1 : Literature review of liberation models

Reference Grains morphology Grain size Phases Mode of breakage Dimensions

Shape of particles

Gaudin, 1939 Cubes Constant 2 Random 3 Cubes

Wiegel and Li, 1967 Cubes Variable 2 Random 3 Cubes

Andrews and Mika, 1975 Any Variable 2 Random 1 NA

Steiner, 1975 Any Variable 3 Random 1 NA

King, 1979 Any Variable 2 Random 1 NA

Klimpel and Austin, 1983 Any Variable 2 Random 1 NA

Meloy and Gotoh, 1985 Spheres Variable 2 Random 2 Any

van Egmond and Meloy, 1989

Rods, plates, diamonds Constant 2

Non-random 2 Any

Barbery and Leroux, 1988 Polyhedra Variable 2 Random 2 Any

Ferrara et al., 1989 Bands, rods and spheres Constant 2 Random 3 Any

Hsih and Wen, 1994 Squares Variable 2 Non-random 1 NA

Bonifazi and Massacci, 1995 Pixels Variable 2+1 Random 2 Cubes

King and Schneider, 1998 Any Variable 2 Non-random 2 NA

Wei and Gay, 1999 Any Variable 2 Random 3 Cubes

Gay, 2004, 1999 Disks Variable 2 Random 2 Disks

Stamboliadis, 2008 Spheres Constant 2 Random 3 NA

Djordjevic, 2013 Pixels Variable Any Non-random 2 Any

Zhang and Subasinghe, 2013 Polyhedra Variable 2 Random 3 Convex

Evans et al., 2013 Cubes Variable Any Random 3 Cubes

Ghazvinian et al., 2014 Voronoi polyhedra Constant 1 Non-random 3 NA

Wang, 2015 Pixels Variable Any Non-random 3 Any

van der Wielen and Rollinson, 2016 Voronoi polygons Variable Any Random 2

Voronoi polygons

This study Pixels Variable Any Non-random 2 Any

4 Proposed model

A model combining Markov Random Fields for controlled texture simulation (Cross and Jain, 1983; Winkler, 2012) and cellular automata for breakage is proposed as a predictive model of liberation.

4.1 Texture generation

The first step is to generate a texture that reproduces the observed one for a given ore. Several methods have been proposed to simulate a realistic mineral texture (Gasnier et al., 2015; Klichowicz and Lieberwirth, 2016). In this study, the association matrix (AI) and the cumulative grain size distribution of each phase are used to build a stream file for process simulation. Since the aim is to generate a stream with varying modal mineralogy, it is an input parameter of the simulation.

From statistical physics, the Ising’s model (Ising, 1925) originally describes the behavior of spin at each site of a ferro-magnetic material as a function of the temperature (above or below Tc the Curie or critical temperature where magnetization is lost). The values the elements can take are spins in this case taking values in E = [-1;1]. To each clique, it assigns a potential Uc so that

(9)

Where beta is a coupling (or regularization) constant of dimension of [T-1]. Its value describes the intensity of the influence between sites x: for a large value, the value of an element will be heavily influenced by its neighbors. If two neighbor spins have the same value, the product xs. xt will be equal to 1 and the potential equal to -β. The total energy of the system can be written as

(10)

Ising's model can be generalized for multiple phases i.e. E = 0; N with 0 phase reserved for voids and is called a Potts model (Potts, 1952) . This approach has been used by to study the grain growth (Zöllner and Streitenberger, 2006) with a specific interest in the interface regions. This model can be used to synthetize mineral maps of specified mineral grades. The potential can be changed to

(11)

The expression of the potential can be generalized to a given neighborhood w

(12)

Where

(13)

With I the identity matrix, 1 the matrix of ones and AI the association matrix defined in (8).

Using the fact that negative values of beta indicate a lowering of the energy, we can use a normalized version of AI as a beta matrix. To obtain a simulation, only a training image is needed from which the parameters for the Potts simulation will be extracted (AI matrix and cumulative grain size distribution for each mineral phase). Alternatively, the user can input a cumulative grain size distribution and an association matrix manually.

To measure the difference between the generated texture and the training image, a norm should be chosen, in this case, a weighted sum of a max-norm with a l2-norm on both AI and the cumulative grain size distribution was chosen.

The identification of the parameter beta can be done via the stochastic gradient descent (Younes, 1998), however, the convergence speed of the vanilla version is quite low. By using a modified version of the gradient descent, faster convergence can be achieved. In a broader perspective, aiming at real-time simulation at the particle level, computational aspects matter. Two examples of texture simulation are presented, one is a macroscopic disseminated texture from a porphyry copper deposit (Table 2), the other one is a microscopic coarse grained iron ore. The first figure is the simulated texture compared to the training one, then the values of their association matrices AI are compared along with the difference in their grain size distribution.

Visually the training image and the simulated texture are very similar. The grain size distribution per phase differs slightly phases of similar distribution in the original image are identical in the simulated result. The association matrices are presented in Table 3. Again, the difference is very small. The sum of each line of each AI gives the modal mineralogy, which is identical in both cases.

5 cm

Figure 6 : Example 1 training map Figure 3 : Example 1 simulated map

Figure 5 : Cumulative grain size distribution for example 1 training map

Figure 4 : Cumulative grain size distribution for example 1 simulated map

The error metric used in the simulation algorithm is a weighted sum of squares of the residuals between the cumulative grain size distribution matrices and the residuals between the association matrices.

Table 2 : Comparison between the training image AI and the simulated result AI

Training image AI Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

Phase 1 0.0919 0.0147 0.0137 0.0153 0.0165

Phase 2 0.0147 0.1433 0.0162 0.0178 0.0194

Phase 3 0.0137 0.0162 0.1118 0.0161 0.0172

Phase 4 0.0153 0.0178 0.0161 0.1427 0.0197

Phase 5 0.0165 0.0194 0.0172 0.0197 0.1773

Simulated texture AI Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

Phase 1 0.0947 0.0143 0.0133 0.014 0.0157

Phase 2 0.0143 0.147 0.015 0.0169 0.0181

Phase 3 0.0133 0.015 0.1149 0.0152 0.0166

Phase 4 0.014 0.0169 0.0152 0.1472 0.0182

Phase 5 0.0157 0.0181 0.0166 0.0182 0.1815

The first case was stationary, with a very close grain size distribution, however it is not always the case. In the second example, a mineral map from an iron ore is acquired from a scanning electron microscope. In that case, as can be seen on Figure 9, the geometry of the phase is well respected, but the shape of the boundaries is slightly different. However, association matrices and cumulative grain distributions are still very close.

Figure 7 : Initial mineral map, courtesy of Mehdi Parian (LTU)

Figure 8 : Training mineral map

Figure 9 : Simulated mineral map

Figure 10 : Scatterplot of the association matrices values for the training map and the simulated map

Figure 11 : Scatterplot of the cumulative grain distribution values for the training map and the simulated map

4.2 Breakage model

If random breakage was used in many previous models, recent studies have used FEM to infer breakage patterns (Wang, 2015). Another approach is to use cellular automata (Pan et al., 2009).

Since a graph can be associated to the mineral map, by assigning a weight w to each vertex, we can build a graph associated to a mineral map.

Each fracture will partition G (V; E) into two particles P1 and P2 so that

The cost for the partition can be calculated as

A convenient way to propagate a crack is to use a minimum energy path. A fracture will propagate to the next cell in its neighborhood that has the minimum weight. As soon as the total energy is equal to zero, propagation of cracks is stopped

The ideal case, where only phase-boundary fracture or detachment occurs is analogous to the min-cut problem. In that case, segmentation algorithms can be used to produce single particles. Otherwise, the value of w at a given site x is a function of mechanical parameters. As a first approximation, the fracture toughness GIC [Jm-2] is used (Tromans and Meech, 2004, 2002).

To represent phase-boundary fracture, specific fracture toughness can be chosen (TG, trans-granular or GB, grain boundary) or w can be weighted with a gradient of the mineral map. To represent boundary-region fracture, the metric defined on the mineral map can be used to compute a distance from the grain center to its edges. Multiplying w by a correcting factor from the center to boundary distance can lower the cost of the cut and favor a propagation close to the edge of the mineral grains.

Figure 15 : Synthetic texture 1 Figure 12 : Synthetic texture 2

Figure 13 : Synthetic texture 3 Figure 14 : Synthetic texture 4

Figure 16 : Degree of liberation per phase for the synthetic cases from Figures 12 to 15

A synthetic case is presented in Figures 12 to 15. While preserving the same modal mineralogy, the value of the regularization factor has been changed to produce textures varying from fine to coarse grained. The different textures are then broken with the same total energy and fracture parameters with assigning a different fracture toughness to each phase. Thus, the liberation spectrum is modified by the initial grain size distribution as shown on Figure 16. In Figure 17, the ore from the second example is broken using preferential breakage. The propagation of the tip of the crack goes preferably around the magnetite grain in this case.

Figure 17 : Example of preferential breakage simulation in magnetite ore

4.3 Building HSC Streams

To build a realistic stream for simulation purposes, the particle size distribution must be like the one measured on the process. To calibrate the liberation model, the particle size distribution can be tracked for each propagation step of cellular automaton. To compute the particle size distribution, the fracture path is masked, then a morphological sieve is applied. Thus, particle size distribution can be plotted as a function of the total energy used for fracture propagation.

As a result, a stream can be created (as a XML file) with a defined modal mineralogy, controlled particle size distribution, and a liberation spectrum that derives from intact textures.

5 Discussion

The flexibility of the model allows a detailed control of both the texture used (either measured or simulated) and the breakage mechanism (continuum from random to non-random). Moreover, no hypothesis is done on the shape of the mineral grains in the intact texture nor the particles generated after comminution (in particular, none on convexity).

The tools developed to characterize, simulate and break the initial texture are compatible with future 3D models. The computational cost might prevent a practical use of these methods for now, but combined with X-Ray tomography measurements, simulation of realistic 3D particles will be possible.

While the use of the proposed model as a unit model in a simulation flow sheet is not obvious, the data generated is very similar to the one obtained from automated mineralogy measurements. As such, it can be validated and liberation information has the potential to be integrated in existing mills models. For example, using a modified version Andrews-Mika diagrams derived from breakage simulation data.

Both the texture simulation and breakage mechanisms are based on statistical physics models. This choice may increase the number of input parameters but at the same time ensures realistic results within the validity boundaries of the models.

A disadvantage of the texture model is that there is no guarantee that it will be able to simulate an arbitrary texture with an arbitrary precision. So far, many textures have been successfully tested but difficult cases may exist, the texture simulation part may be then replaced by a more adapted method. Regarding the breakage model, validation work is needed regarding the way a crack propagates in heterogeneous materials.

6 Conclusion

A brief literature review has been presented and a model to simulate liberation for simulation purposes has been introduced. The next steps will focus on the validation and application of the model to test cases.

7 Acknowledgements

The authors wish to thank all PREP partners for their support in this study and Mehdi Amiri Parian for the SEM section used as an example. This work is part of PREP project funded by VINNOVA.

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