Numerical Investigation of the Applicability of Preferential Grade Deportment by Size

Abstract

The effective separation of ore is based on two fundamental processes: liberation and separability. Liberation involves the reduction of size, yielding smaller particles with enhanced compositional homogeneity. Understanding liberation requires an understanding of rock breakage, as it impacts mineral liberation and helps identify ores suitable for pre-concentration. Non-random breakage, influenced by textural and mineral properties, introduces heterogeneity in mineral distribution across size fractions. Physical attributes, including ore and gangue mineralogy and texture, influence fractionation tendencies during breakage. Notably, the presence of mineralization in veins substantially assists early-stage liberation in mineral processing. The aim of this study is to develop a methodology that allows the prediction of natural fractionation tendencies based on geological, mineralogical, and textural data using Discrete Element Method (DEM) modeling. DEM simulations provide insights into granular material behavior, capturing phenomena such as crack initiation and propagation. The use of DEM, particularly with models such as the Flat Joint Model (FJM), enhances our understanding of rock damage mechanisms. In this paper, DEM is used to predict preferential grade by size deportment, and a numerical model is developed to reflect grade distributions across size fractions. A fragmentation analysis is conducted after rock breakage simulations using DEM to analyze the fragment sizes and grades and calculate the Response Rankings of synthetic specimens.

1. Introduction

The mining industry has faced a productivity crisis since the end of the “Commodities Super Cycle” in 2011, following a decade-long period during which commodities traded above their long-term price trends [1]. Among other challenges, the mining industry must improve energy efficiency and reduce water requirements. As the trend of escalating production volumes became more pronounced, the issue of declining feed grades in mining operations became apparent [2]. This observation led to the development of the view that diminishing feed grades are an inevitable trajectory for the mining industry. Comminution, comprising crushing and grinding, refers to the process of reducing ore particles to the size at which the mineral particles are liberated from the host rock. To free the target minerals, the primary size reduction involves crushing, which is achieved by compressing the ore against rigid surfaces or impacting it within a constrained motion path [3]. Following crushing, the rock is reduced to a manageable size in millimeters, and subsequent grinding further decreases it to particle sizes of micrometers. This sized material is then processed using techniques such as flotation, gravity, magnetics, and dense medium separation. Comminution is the most energy-consuming process in mining operations and is a major contributor to overall global electricity consumption. In 2007, the U.S. Department of Energy stated that in the mining industry, crushing processes consumed approximately 4% of the total energy, whereas grinding processes accounted for about 40% [4]. Comminution consumes up to 4% of the world’s energy, and approximately 6700 kWh/kilo-tonne is consumed by comminution [3].

Early rejection of waste and uneconomic material is an alternative technology that has a positive impact on energy efficiency and productivity in the mining industry. By implementing coarse particle rejection techniques, targeting particle sizes of 10–100 mm, mines have the potential to counteract declining productivity by eliminating uneconomic material before it reaches costly and energy-consuming comminution. Pre-concentration using screening is a coarse separation method that enables the removal of uneconomic material prior to the energy and water-intensive grinding process. Exploiting the concept of preferential grade by size deportment offers one way to achieve this goal. This phenomenon refers to the propensity for some ore types to selectively distribute valuable metals into specific particle sizes during the breakage process. Preferential grade by size deportment is influenced by various factors such as rock mass properties, texture, ore paragenesis, and mineralogy at a range of scales [2].

The effective separation of ore involves two essential processes: liberation and separability. Liberation refers to the rock-based property that allows the extraction of valuable minerals from gangue material [5]. This process of liberation is a measurable property of the rock that can be linked to downstream separation techniques in a mineral processing circuit, the aim of which is to concentrate valuable material [6]. A coarser grain size of the target mineral phases correlates with a higher tendency to achieve more effective ore liberation [7]. Reduction of the rock size to the millimeter scale by crushing and subsequently reducing it to the micrometer scale by grinding allows the use of separation techniques such as flotation, gravity, magnetics and dense media. Separation techniques employ the physical properties of the ore such as density, shape and size. The particle size serves as a physical property of the rock for pre-concentration via screening. One of the most important aspects of understanding liberation is the process of rock breakage. The influence of breakage on mineral liberation is crucial in identifying ores that have favorable characteristics for pre-concentration. Understanding mineral liberation requires an understanding of crack propagation within the feed material. Properties of minerals and grain boundaries predominantly dictate crack propagation in rocks. Micro-scale crack data generally indicate two distinct modes of propagation: through mineral grains or along grain boundaries, as shown in Figure 1 [8]. Interfacial breakage occurs when cracks propagate along grain boundaries, while preferential breakage arises when cracks traverse mineral grains due to inherent physical properties such as brittleness, hardness, cleavage, and fracture.

Non-random breakage, which follows textural and mineral properties, has the potential to create heterogeneity in the distribution of minerals across various size fractions. The release of valuable material in finer size fractions is assumed to depend on the mineralization style and the type and intensity of breakage energy applied to the feed material [9]. The physical characteristics of the material, including the mineralogy and texture of the ore and gangue phases, may influence their tendency to fractionate during breakage [10]. Factors such as crystal size, paragenetic association, and spatial distribution of ore and gangue could play a role in the amenability of mineral phases to concentrate in different size fractions after breakage. Previous studies indicate that the presence of mineralization in veins contributes significantly to the liberation of sulfides in the early stages of mineral processing [11]. Carrasco, Keeney and Napier-Munn [6] introduced a Response Ranking parameter (RR), which serves as a measure of liberation at the course (mm) scale before the grinding process and quantifies the extent of metal deportment. Coarse separation normally includes two or more streams with different grades, and therefore, the first step in calculating the RR is to define the Response Factor (RF)—the ratio between the grade of the separated stream and the feed grade (Equation (1)).

$$Response Factor \left(RF\right)=\frac{New Stream Grade}{Feed Grade}$$

(1)

Response Ranking is the slope of the Response Factor as a function of cumulative mass in log–log space, as shown in the equation below.

$$Response Ranking \left(RR\right)=-200\frac{ln \left(RF\right)}{ln \left(CM\right)}$$

(2)

where: ln—natural logarithm, RF—Response Factor, CM—cumulative mass (%).

A high RR value indicates significant grade variability across different size fractions, with one of the fractions demonstrating the highest concentration of metallic content. RR can range from a negative value to a maximum response of 200. For samples with a Response Ranking between 20 and 80, a screening methodology can effectively be applied in combination with other methodologies to achieve effective pre-concentration of the material. Based on practical observations and testing, it is suggested that the physical characteristics of rocks, including mineralogy and texture in both ore and gangue phases, influence their tendency to fractionate during breakage, thereby affecting their distribution across size fractions [10]. Rocks with noticeable differences in mineral composition or texture are expected to have a higher Response Ranking than more uniform rocks [6]. There is potential to predict natural deportment tendencies by using available geological, mineralogical, and textural data for attribute development applicable to block models. The method of breaking rocks, whether in laboratory-scale tests or in the initial stages of comminution, may influence the coarse liberation of target mineral phases. Consequently, materials identified with a significant geological inclination for fractionation may not fractionate optimally if the comminution process hinders or dilutes natural deportment effects [12].

Preferential grade by size deportment is now an accepted and acknowledged concept for some types of orebodies. The gap in the literature, however, is a clear understanding of the pre-conditions required for successful application of the concept. The most successful reported applications are almost entirely for porphyry copper deposits, i.e., orebodies in which there is some lithological control on grades. [13] concludes that grade by size deportment does not apply to the Olympic Dam orebody, at least for values greater than 0.30 wt.% copper. Very low-grade mineralization below ~0.30 wt.% occurs along microfractures; above this value, the mineralization is disseminated. Olympic Dam is an iron oxide copper-gold ore deposit. This type of deposit is characterized by complex mineralogy with a complicated distribution of critical minerals. This appears to confirm that the applicability of grade-by-size deportment depends on some form of lithological control, such as the presence of veins, stockworks or angular fragments. Another important gap is in the prediction of preferential grade deportment by size based on a breakage model. There is obviously a relationship between mineral liberation mechanisms at the microscale and at the coarse scale (mm), and breakage will affect both scales. Breakage models to predict grade deportment can generally be more efficient in terms of time compared to laboratory tests. It is important to note that the calibration and validation of a breakage model are essential for rapid predictions and simulations of the breakage process. Thus, numerical modeling could be a complementary method for a more comprehensive understanding of the underlying mechanisms and behavior of breakage and preferential deportment of metal into specific size fractions, which would enable predictions and simulations of different scenarios.

Cundall and Strack [14] proposed the Discrete Element Method (DEM), which is a numerical modeling technique for simulating the behavior of granular materials. By discretizing the non-continuous medium into rigid discrete bodies that can have translational, rotational, interactive, and separative movements, this method enables a thorough examination of the defined contact mechanisms. DEM has a significant advantage over other techniques, such as the detection of new contacts during the simulation process. When the maximum stress surpasses predefined thresholds for tensile or shear strength, cracks initiate and propagate along the boundaries of blocks [14]. Later, the authors of [15] introduced a numerical model to represent rock as a compact arrangement of non-uniform circular or spherical particles where cementation serves as the bonding between particles. These particles are interconnected at their contact points, and their mechanical response is simulated using the discrete element method. This approach has found extensive application in the exploration of rock fracture and fragmentation processes [15].

Flat Joint Models (FJM) are widely used for simulating brittle rock, providing a close match to the mechanical behaviors observed in lab-scale rocks. The behavior of the flat-jointed Bonded Particle Model (BPM) contains aspects seen in the parallel-bonded BPM. However, it also surpasses the capabilities of the parallel-bonded BPM by allowing for partial damage within the interface. In a flat-jointed BPM, each rigid particle contact mimics the behavior of a finite-length interface between two particles with locally flat notional surfaces. This unique characteristic enables even a fully broken interface to resist relative rotation. This model effectively captures the intricate characteristics of rock damage, making it a valuable tool for studying micro- and macro-mechanisms of rock damage [16].

Lee et al. [17] used the DEM to develop a three-dimensional model comprising 10 aggregates, aiming to explore the breakage and liberation characteristics in response to varying numbers of free fall impacts. The findings revealed a consistent correlation between the simulated outcomes and experimental results in terms of breakage and liberation characteristics. Furthermore, Bahaaddini et al. [18] investigated the efficacy of the FJM in replicating the mechanical behavior of intact rocks, comparing it against laboratory experiments conducted on Hawkesbury sandstone. Similarly, Cheng and Wong [19] examined the application of the flat-jointed model to simulate the mechanics of Carrara marble. Collectively, the outcomes from these investigations indicate that the flat-jointed model demonstrates the capability to accurately replicate both the mechanical properties and the fracturing behavior, as well as the breakage characteristics, of the rock specimens.

Despite rapid advancements in computational power and the ability to model large-scale problems, many numerical models in mining and geotechnical engineering are still conducted in two dimensions. This is primarily due to time constraints and the complexity of interpreting 3D results. Bahrani et al. [20] performed sensitivity analyses on micro-properties and compared the outcomes of 2D and 3D BPMs to determine whether Particle Flow Code 2D (PFC2D) results could be used to interpret the three-dimensional nature of laboratory tests. The study found that the uniaxial compressive strength of 2D and 3D BPMs is comparable, although the tensile strength of the 3D BPM is lower than that of the 2D BPM. Nevertheless, the failure modes and crack propagation observed in unconfined compressive and direct tensile test simulations in 3D are very similar to those in 2D. Overall, while the BPM has fundamental limitations, the main characteristics of breakage behavior in PFC2D align well with those observed in laboratory tests [21].

The primary aim of the study presented here is to use the Discrete Element Method (DEM) as an effective tool for predicting rock breakage and to develop a numerical model capable of quantifying the distribution of grade across different size fractions based on rock texture, mineralization style, and hardness. The purpose of the proposed methodology is to construct a robust tool that can accurately capture the deportment of grades within the rock samples, thereby enhancing our understanding of rock breakage processes. The following sections of this paper cover the development of a methodology that applies a numerical model of preferential grade deportment by size, together with a comparison of the Response Rankings derived from the laboratory tests conducted by the University of Tasmania (UTAS), the Centre for Ore Deposit and Earth Sciences (CODES), and the Cooperative Research Centre Optimising Resource Extraction (CRC ORE) and the Response Rankings calculated from the numerical model described in this study.

2. Research Methodology

This study initially focused on predictive numerical modeling by replicating different rock textures and different mineralization styles and applying different sets of parameters of the bond strength. These models consisted of twelve two-dimensional BPMs, for which each model includes “matrix” and “inclusions” groups to represent the gangue material of embedded rock and the high-grade inclusions as veins, angular fragments, and disseminated crystals. The models are based on porphyry, volcanic-hosted massive sulfides (VHMS), and epithermal intermediate sulfidation. This study is not focused on determining the applicability of grade deportment by size for a specific deposit but rather on assessing its applicability to different mineralization styles. The aim of this study is to model and predict the breakage behavior and the applicability of preferential grade concentration into specific size fractions by replicating geological attributes of the rock textural features (e.g., presence of veins, alteration), mineralogical composition, and hardness using the DEM approach. The rock samples used in this study were obtained from the digital libraries of the UTAS, CODES, and CRC ORE projects [22]. The rock samples were described using detailed logging, rebound hardness mapping, geochemistry, hyperspectral mapping, magnetic susceptibility, breakage tests with a rock press, and 3D imaging, as well as a calculation of Response Ranking. Images of the rocks are available on the UTAS AusGeol website in the UTAS_RR collection [23].

In the numerical modeling of engineering problems, certain issues can be adequately represented by a finite number of well-defined components. The overall behavior of the system can be ascertained through the well-defined interrelations between these individual components (elements). A number of numerical models have been suggested for replicating a system with varying degrees of accuracy [24]. The discrete element method (DEM) is a numerical approach pioneered by Cundall [25], initially applied to address challenges in rock mechanics. Cundall and Strack [14] and Cundall [26] later used the BALL formulation of DEM to develop constitutive relationships for granular soils. DEM offers the advantage of explicitly analyzing each particle within the granular assembly and their interactions. To enhance computational efficiency, particle shapes are simplified as discs in 2D and spheres in 3D, enabling the analysis of many particles while capturing essential response characteristics of granular material behavior. A distinctive feature of DEM is its capability to apply loads and deformations virtually. This contrasts with physical laboratory tests, where accessing necessary information and data on the particle-scale mechanisms underlying the complex material response is challenging. The particle flow code (PFC) from the Itasca Consulting Group [27] is widely used in geomechanics as a DEM code and employs a granular assembly following Newton’s law of motion. These models operate based on the possibility of reproducing the microstructures of rocks, and their interactions allow for the emulation of the macro properties of intact rocks.

2.1. Bonded Particle Model

Rock behavior can be represented as densely packed cemented granular material with circular particles that are bonded together at their contact points and are deformable [15]. The bonded-particle model for rock (BPM) replicates this system and corresponds to the behavior of real rocks. The BPM provides both a scientific and a theoretical instrument for exploring the effect of micro-properties on macroscopic behavior as well as an engineering tool to predict these macroscopic behaviors. PFC2D is a commercially available particle flow code incorporating the DEM [27]. PFC 2D simulates the rock domain by modeling the interaction among an assembly of circular particles with specific statistical size distributions enclosed within four rigid walls. The particles are automatically generated, with radii distributed either uniformly or based on a Gaussian distribution. The bond between particles must be established to create an assembly. The general mechanical behavior of the assembly is controlled by the micro-properties of both the particles and the bond. The main steps in creating a bonded assembly are particle generation, packing stage, initialization of stress, and bond installation [27].

The mechanical behavior of both rock and BPM is dictated by the force chains that propagate from one grain to the next across the grain–grain contacts. Macroscopic force is applied to the grains and cement structure in the form of force chains developing from one grain to the next across the model. The motion of the particles follows Newton’s law of motion, and the interaction among particles is dictated by constitutive models applied at their contact points. Particle motion encompasses two distinct components: translational motion and rotational motion, as shown in Figure 2.

The contact force and moment that occur when two particles come into contact can be described as follows:

$$F_i={{\displaystyle \sum }}_{j=1}^n(F_{ij}^n+F_{ij}^s)+F_i^{app}$$

(3)

$$M_i={{\displaystyle \sum }}_{j=1}^n{F}_{ij}^s{r}_{ij}+M_i^{app}$$

(4)

where $r_{ij}$ is the distance from the center of particle i to the contact point between particle i and particle j; $F_i^{app}$ and $M_i^{app}$ are the additional force and moment applied to the particle i; $F_{ij}^n$ and $F_{ij}^s$ are the normal and shear contact forces in the local coordinate system between particle i and particle j (see Figure 1B).

The DEM defines the interaction of the particles as a dynamic process in which equilibrium can be reached whenever the internal forces are balanced. The contact forces and displacements of an assembly of particles are identified by recording the movements of the individual particles. Movements result from the propagation of stress induced by walls and externally applied forces. By modeling a synthetic rock sample as a collection of separate particles bonded at contact points, the simulated material can develop cracks as bonds between particles break under normal and shear loads. While particles are assumed to be rigid in PFC2D, the deformability of the assembly is derived from normal and shear bonds. Each bond has a strength representing intact bonding (cohesive strength), and a broken bond carries no tension when either a tensile or shear force limit is reached [27].

2.2. Properties of the Flat-Joined Model

The flat-jointed bonded-particle model is a particle model that can represent the real macroscopic behavior of rock and has a closer match of structural and microstructural rock. A flat-joint model incorporates both the assembly of loosely packed grains to represent the behavior of porous rock and an assembly of well-interlocked particles to represent intact hard rock. The rigid particles interact along deformable and breakable particle-particle interfaces [28]. The flat-joint contact model provides the macroscopic behavior of a finite-size, linear elastic, and either a bonded or a frictional interface that may sustain partial damage. A flat-joint contact simulates the behavior of an interface between two notional surfaces, each of which is connected rigidly to a piece of a body. A flat-jointed material consists of bodies (balls, clumps, or walls) joined by flat-joint contacts such that the effective surface of each body is defined by the notional surfaces of its pieces, which interact at each flat-joint contact with the notional surface of the contacting piece. The notional surfaces are called faces, which are lines in 2D and disks in 3D. A flat-joint contact has two main types of elements: bonded elements and unbonded elements. The bonded element’s strength envelope is shown in Figure 3.

The tensile strength is ${\sigma }_b$. When the normal stress ${\sigma }_{max}^e>{\sigma }_b$, then the element breaks with a tensile crack, and the element bond state changes to an unbonded state.

The shear strength ${\tau }_c$ follows the Coulomb criterion:

$${\tau }_c=c_b-\overline{\sigma }tan{\varphi }_b$$

(5)

where $c_b$ is the bond cohesion and ${\varphi }_b$ is the local friction angle.

When the bond is broken in shear, the element bond state is modified to the unbonded state, and the residual friction follows the equation below:

$${\tau }_r=-\overline{\sigma }tan{\varphi }_r$$

(6)

where ${\tau }_r$ is the residual friction strength and ${\varphi }_r$ is the residual friction angle.

Both the flat-joint model (FJM) and the bonded-particle model (BPM) replicate a finite-length grain–grain interface, operating as bond models. If the stress exceeds the contact strength, the contact breaks as a tensile or shear crack. In the BPM, the entire interface breaks, whereas in the FJM, individual elements break, causing partial damage due to the interface’s segmented nature. Post-breakage, the interface disappears in BPM, but in FJM, it persists, enabling continued rotation resistance even after breaking, a feature lacking in the standard BPM.

The rock samples tested by the CODES research group were processed in a laboratory jaw crusher until 99% of the material passed through a 3.5 mm sieve. The resulting crushed material was then divided into four size fractions using sieves of 2.36 mm, 1.7 mm, and 0.6 mm [28]. These sieve sizes were selected to ensure an approximately equal mass distribution across each size range. Similarly, in this study, four main size fractions are used in fragmentation analysis that is undertaken after crushing the synthetic samples in PFC2D. Similarly, this study uses four primary size fractions for fragmentation analysis following the crushing of synthetic samples in PFC2D.

The flow chart of the methodology of this study is shown in Figure 4.

2.3. Calibration and ReCalibration of MicroParameters

2.3.1. Mechanical Characteristics of Rock Samples

Several laboratory tests were carried out on rock samples during the collaborative CRC ORE and CODES projects. Rocks are represented by three different deposits with different mineralization styles and consist mainly of quartz, feldspar, muscovite, and silica cement. The elastic constants of Young’s modulus and Poisson’s ratio and the uniaxial compressive strength measured from unconfined compression tests are shown in

Table 1. Mechanical properties of tested rock samples.

Property	Sample A Volcanic Breccia	Sample B Quartz-Pyrite Vein Sandstone	Sample C Sphalerite-Galena Vein Rhyodacite
Young’s modulus, E (GPa)	3.2	15.2	11.1
Poisson ratio, ν	0.25	0.27	0.10
UCS, (MPa)	13.4	68	73.5

. The rock samples used as a base model for this study are volcanic breccia—synthetic sample A, sandstone—synthetic sample B and rhyodacite—synthetic sample C.

The process of calibrating microscopic parameters based on macro-properties, acquired either in a laboratory or in situ, is considered essential for conducting PFC simulations. FJM involves a calibration procedure to assign accurate values to local micro-properties, aiming to replicate the intended elastic properties. Typically, calibration can be achieved through a systematic trial and error approach [29] or a direct calibration process [30]. Given the substantial number of local input parameters in FJM and the intricate interdependencies influencing apparent behavior a proposed indirect calibration algorithm was implemented in this study. The initial calibration stage involves generating an assembled homogeneous model; however, subsequent alterations to the model’s geometry and the introduction of heterogeneity necessitate recalibration. The first calibration process was done on a 5 × 10 cm² rectangular sample consisting of around 20k discrete disks. The loading rate V was set to 0.005 m/s to achieve quasi-static loading conditions. The effect of the ratio of the maximum to minimum ball radius (r_max/r_min) has been extensively studied. [31] observed that UCS and E increase, while ν shows an opposite trend with an increasing r_max/r_min ratio. In contrast, [32] pointed out that the simulation results did not show any significant effects on E, ν, and UCS. Most of the published research shows ranges between 1.32 and 3.00 and the most typical selection is 1.66. Several studies [29,31,32] suggest a decrease in UCS, TS, and E with an increasing porosity n, while Poisson’s ratio—ν remains unaffected. In contrast, computational efficiency can significantly improve with an increase in n due to a reduction in the number of particles. In this investigation, for the sake of simplification, n was set to a constant value of 0.16.

The macro properties, such as Young’s modulus and the Poisson ratio, are determined by the local E* and ν*, whereas UCS is identified by both bond strength parameters and elastic micro parameters. [29] indicated that the effect of the stiffness ratio ν* on the macro-elastic and strength properties of the synthetic rock are significant. As ν* increases, the E of the BPM material decreases, but the ν increases. ν* is the main element for calibrating these two macro-elastic properties. Young’s modulus E is linearly dependent on E* for the FJM material, and UCS has a positive linear relationship with c—cohesion, one of the essential elements influencing the UCS.

Table 2. Summary of micro constitutive parameters from the first calibration for rhyodacite sample.

FJM Input Parameter		Unit
Effective modulus, E*	83.2	GPa
Stiffness ratio, ν*	2.35	-
Bond cohesion, c	23.6	MPa
Tensile strength, ${\sigma }_b$	85	MPa
Local friction angle, φb	35	[°]

shows the input micro-parameters used for calibration.

2.3.2. Sample Preparation and Grouping Method

In the PFC2D framework, the unconfined compression test can be simplified to a configuration involving two moving walls exerting pressure on the particle assembly, as shown in Figure 5. These walls were modeled as frictionless rigid surfaces. Three experimental specimens were generated, and their characteristics are shown in

Table 3. Characteristics of the synthetic specimens.

Sample ID	Width (mm)	Height (mm)	Number of Balls	rmin	rmax
Sample A	80	160	3893	0.75	1.2
Sample B	60	120	4010	0.75	1.2
Sample C	80	120	2912	0.75	1.2

.

A standard particle size distribution was employed, encompassing particle radii from 0.75 mm to 1.2 mm. The upscaling was used to minimize the particle size while preserving computational efficiency and reducing computational run time. The specimens are modeled at the meso-scale as small rock blocks or core specimens. The heterogeneity of the samples mirrors that of real rock formations. The study conducted by the CODES research group involved hardness measurements collected using an EQUOtip device, a portable electronically controlled velocity rebound hardness tester. The EQUOtip functions by using a projectile to impact the surface under examination. During each hardness assessment, the tungsten carbide tip strikes the test surface under spring force, generating an impact energy of approximately 11 Nm before rebounding. These measurements are recorded within a few seconds. Hardness readings were gathered from flat surfaces of the rocks, following a 1 cm $\times$ 1 cm grid pattern, and subsequently transformed into hardness maps of the rock surfaces.

Flat surfaces were prepared from bulk sample fragments by sawing, enabling hardness measurements to be conducted on one of the faces [28]. An example of rebound hardness data is shown in Figure 6.

The instrument used as a hand-held XRF is an Olympus Vanta M series that enables rapid, non-destructive elemental analysis of samples through X-ray fluorescence, with measurements recorded for 60 s per analysis. Analyses were performed on a standardized grid pattern on one face of all samples, with the spacing of the grid tailored to the shape and size of each specimen. For bulk sample specimens, the grid spacing was set at 2 cm $\times$ 2 cm, as shown in Figure 6. An example of pXRF data is shown in Figure 7.

Later in this study, the rebound hardness data will serve as the basis for representing the textures of real rock samples in the modeling of synthetic specimens. Additionally, portable X-ray fluorescence measurements will be used to assign grades to specific sections of the synthetic specimens.

Sample A (volcanic breccia) is characterized as polymictic and poorly sorted, featuring very angular clast shapes, with some fragments exhibiting cuspate shapes (Figure 8A). Fragment lithology comprises porphyritic and aphanitic rhyodacites and dacites, mudstones, and juvenile clasts, while the matrix consists of rock flour, white mica, and hematite. Hyperspectral mapping reveals higher quartz content in the breccia matrix compared with the fragments, with the latter exhibiting elevated levels of K-feldspar and intensified alteration to white mica. Rebound hardness measures range from 200–850 HLS, with lower values typically associated with highly altered clasts and harder values linked to the rock matrix. Mineralization is dispersed within the breccia matrix, occasionally concentrating within the angular fragments and clasts. The substantial variability in hardness between the breccia matrix and fragments arises from differences in their mineralogical compositions, with the matrix being notably harder due to its composition of fine quartz and K-feldspar, whereas the fragments display pronounced alteration to white mica, indicating softer mineral content. The mineralogical analysis of the sandstone samples reveals a predominant composition of quartz, accompanied by minor proportions of carbonates and clay (Figure 8B).

In terms of rebound hardness, Sample B (sandstone) exhibits a considerable range, spanning from 450 to 950 HLS. This variance can be attributed to the combination of high-density minerals constituting the matrix and the relatively lower hardness of pyrite within the samples. Notably, the sandstone specimens are characterized by an abundance of pyrite and quartz veins, with a correspondingly small spacing between these veins. Pyrite emerges as the primary sulfide mineral present. It is plausible that the iron and gold content within these samples is influenced by the presence of pyrite, as well as other minerals such as hematite and siderite. This mineralogical composition underscores the diverse geological processes and conditions that have influenced the formation of these sandstone samples.

Sample C is a biotite rhyodacite specimen consisting of K-feldspar, quartz, biotite, hornblende, and plagioclase, with the feldspars exhibiting a weak alteration to white mica (Figure 8C). In terms of hardness, the sample demonstrates a relatively uniform homogeneous consistency and exhibits a rebound hardness ranging from 700 to 860 HLS. Notably, mineralization within this sample is concentrated within veins containing sphalerite and galena. New rebound hardness and pXRF maps were generated by employing the nearest neighborhood method to construct two standard rectangular-shaped synthetic samples with dimensions of 160 mm height, 80 mm width, and 150 mm height and 60 mm width, respectively, to represent Sample A—volcanic breccia and Sample B—sandstone core sample. Also, one cubic-shaped synthetic sample with dimensions of 80 mm height and 80 mm width was generated to represent Sample C—biotite rhyodacite. These updated maps are illustrated in Figure 9.

The biotite rhyodacite specimen, sample C, demonstrates a relatively consistent hardness, which indicates its homogeneous nature. In the modeling process, this sample will be represented as homogeneous to reflect accurately its hardness characteristics. However, as pXRF data show high concentrations of the grades in the vein, the grouping method will be used to show the overall texture of the rock, while veins containing sphalerite and galena will be created to simulate the mineralization pattern observed in the sample. The grouping method, based on hardness and pXRF data, is used to introduce the heterogeneity to the synthetic sample. Minimum, maximum, and average hardnesses within the matrix zone are shown in

Table 4. Hardness data.

Sample ID	RH Min	RH Max	StdDev	Average Matrix Hardness
Sample A	211	679	106	550
Sample B	450	950	125	650
Sample C	700	857	36	800

.

Table 5. pXRF data.

Sample ID	Zn (ppm) Min	Zn (ppm) Max	Average Matrix Zn (ppm)
Sample A	500	10,000	2300
Sample C	500	35,000	1900

represents pXRF minimum, average and maximum values.

2.3.3. Scenarios and Recalibration

Incorporating the observed lithological composition and hardness distribution, three synthetic samples were generated in PFC 2D using a grouping method to depict different rock textures with the aim of simulating the behavior of the real rock samples under loading conditions. Synthetic samples display varied geometries that align with the actual geometries of real rock samples. The first scenario centers around the volcanic breccia sample A and is characterized by a complex matrix hosting multiple inclusions. Within this matrix, five soft inclusions, one strong inclusion, and one medium strength inclusion were strategically positioned according to the hardness. The resulting rebound hardness and pXRF maps are shown in Figure 9A. This configuration mirrors the heterogeneous nature often observed in brecciated formations. In the second scenario, sandstone specimen B, infused with pyrite-quartz veins, is a common occurrence in geological settings rich in sulfide minerals. The pyrite grains surrounding the quartz vein were integrated with a soft bond between the particles, contrasting with the quartz vein itself, which has a hardness equivalent to the surrounding matrix (Figure 10B). The replication of the vein width and location is taken from a picture of a real sample. The third scenario is a rhyodacite sample, distinguished by its homogeneous hardness profile (Figure 10C). Despite its uniformity, this sample featured a visible vein across the matrix, adding an element of complexity to its mechanical response. Despite the uniform hardness, the presence of the vein introduces heterogeneity in terms of grade distribution. These scenarios were used to simulate how distinct geological settings, mineralogical compositions, and texture types impact the mechanical properties and the response to external forces on the rocks. Figure 10 represents the textures of rock samples A, B, and C simulated by PFC 2D.

The PFC model employs contact bonds to support the previously established texture base. Different strengths of the contact bonds are used for this purpose. Specifically, the contact bonds within the inclusions are five times weaker for soft inclusions and veins and two times weaker for medium inclusions compared with the contact bonds of the matrix. The strength properties of the contact bond for each synthetic specimen are shown in

Table 6. Micro-properties of contacts for Samples A, B and C.

Sample ID	Inclusions Type	Flat Joint Stiffness Ratio, ν*	Flat Joint Effective Modulus, E* (GPa)	Flat Joint Tensile Strength, ${\mathit{\sigma }}_{\mathit{b}}\left(\mathbf{MPa}\right)$	Flat Joint Bond Cohesion, c (MPa)	Local Friction Angle, φb
Sample A volcanic breccia	Strong inclusion	2.3	32.5	141.5	163.5	32
	Soft inclusion			28.3	32.7
	Medium inclusion			56.6	65.4
	Matrix			141.5	163.5
Sample B quartz-pyrite vein sandstone	Strong vein	2.7	26	67.4	72.2	35
	Soft vein			13.4	14.4
	Disseminated inclusions			13.4	14.4
	Matrix			67.4	72.2
Sample C sphalerite-galena vein rhyodacite	Strong vein A	2.5	16	42.2	33.8	30
	Strong vein B
	Matrix

. The selection of the stiffness ratio, tensile strength, and cohesion as representative parameters stemmed from their significance in influencing the macroscopic behavior of the rocks. The stiffness ratio, represented by the ratio of local stiffness parameters (ν*), was found to significantly influence the macro-elastic properties of the simulated rocks. Similarly, tensile strength and cohesion played crucial roles in determining the uniaxial compressive strength (UCS) of the rocks. Due to the introduction of heterogeneity into the synthetic samples, it is necessary to readjust the micro-properties to maintain the consistency of the macro properties such as UCS, Young’s modulus, and Poisson ratio. Recalibration was undertaken for all three synthetic samples, and after multiple trial and error iterations, three to four distinct sets of micro-properties were obtained. According to various researchers, the uniaxial compressive strength of the model is predominantly controlled by micro-parameters such as tensile strength σ_b, and cohesion c, which vary across segments exhibiting differing strengths. Young’s modulus and Poisson’s ratio are affected by the effective modulus E* and stiffness ratio ν*; hence, these values are determined for the entire model. Table 6 shows microproperties of contacts applied in the synthetic samples.

For uniaxial compression tests, the top and bottom walls function as loading platens. The specimen undergoes loading as the top and bottom walls move toward each other at a predetermined velocity, ensuring a quasi-static loading condition. The macroscopic properties of the rocks measured in the laboratory and simulation results are shown in

Table 7. Experimental test data and simulation results.

	Sample A Volcanic Breccia		Sample B Quartz-Pyrite Vein Sandstone		Sample C Sphalerite-Galena Vein Rhyodacite
Property	Lab experiment	Simulation results	Lab experiment	Simulation results	Lab experiment	Simulation results
Young’s modulus, E (GPa)	3.2	3.4	15.2	16.3	11.1	11.2
Poisson ratio, ν	0.25	0.29	0.27	0.30	1.10	1.10
UCS, (MPa)	13.4	14.2	68	69.4	73.5	73.6

. A comparison of the laboratory measured stress–strain curve of the rhyodacite sample and the stress–strain results from the simulation is shown in Figure 11.

3. Validation of Numerical Results

While all textural characteristics are important, specific parameters have a greater influence during breakage and subsequent fractionation. Previous investigations into pre-concentration at a coarse scale underline the significance of the vein presence in the liberation of sulfides due to preferential breakage along these mineralized veins [28]. This phenomenon has also been observed at the micron scale, where vein-type mineralization demonstrated the highest degree of sulfide liberation, while disseminated grains exhibited the lowest degree of liberation across all particle sizes within the range of 50–150 µm [7]. In the context of PFC 2D, it is crucial to recognize that although the synthetic sample is depicted as an assembly of circular particles bonded together with contact bonds, these circular particles do not represent rock grains.

Pre-existing zones of weakness or distinct properties are significant factors in the initial localization of fractures and consequent mineral fractionation, as these structures have a propensity to reopen when subjected to stress. Sample A, representing volcanic breccia texture, showed crack propagation through soft inclusions and around them, which is described by the uneven texture of the white mica within the breccias that facilitated the fragmentation of white mica into finer fractions. From the description of Sample B and the modeling results, existing quartz-pyrite veins seem to be a critical factor influencing breakage, as pyrite has lower hardness values compared with the quartz grains around pyrite and the surrounding sandstone matrix. According to Morales Leiva [28] this type of rock sample showed that breakage occurred along the vein, resulting in broken particles visibly containing a substantial proportion of smaller grains identified as pyrite, which corresponds with the crack propagation in the model. In Sample C, which contains sphalerite-galena veins, cracks did not propagate along the vein and breakage propagated through the matrix. This was because this vein type has a similar hardness to that of the matrix and is less amenable to concentrating metal into specific-size fractions. Figure 12 shows the first breakage that occurred in models representing the textures of real rocks.

It was observed that the highest Response Ranking (RR) in the numerical simulations was exhibited by the breccia synthetic sample, which is consistent with the laboratory results as shown in

Table 8. The Response Factors and Response Rankings from PFC simulations were calculated according to Morales Leiva [28].

Sample ID and Texture	Size Fraction (mm)	Grade ppm, ppb	Cumulative Mass % from Lab Test	RF Calculated from Lab Test	Cumulative Mass % from PFC Simulation	RF Calculated from PFC Simulation	RR from Lab Test	RR from PFC Simulation
Sample A volcanic breccia	Head grade	3569					76	79
	>2.36	2310	100.0	1.00	100.0	1.00
	>1.70	3000	80.4	1.09	72.5	1.14
	>0.60	2900	63.7	1.17	58.9	1.22
	<0.60	5590	35.2	1.49	37.9	1.49
Sample B quartz-pyrite vein sandstone	Head grade	0.8					69	77
	>2.36	0.02	Not available	1.00	100.0	1.0
	>1.70	0.05		1.48	53.2	1.32
	>0.60	0.05		1.42	34.8	1.35
	<0.60	0.13		1.56	29.1	1.71
Sample C sphalerite-galena vein rhyodacite	Head grade	3188					49	51
	>2.36	2310	100.0	1.00	100.0	1.00
	>1.70	2600	94.7	1.01	89.3	1.05
	>0.60	2220	80.8	1.05	76.2	1.06
	<0.60	4040	47.1	1.29	49.2	1.11

. Samples with significant variations in hardness within texture groups demonstrated higher RR values.

A comparison of grade distributions across the size fractions of volcanic breccia and their synthetic replication in PFC is shown in Figure 13A. Response Factors of the volcanic breccia sample and Response Factors calculated from fragmentation analysis of PFC simulations are shown in Figure 13B. The graph of the grade distribution in a real rock sample and a synthetic specimen of volcanic breccia shows that PFC simulation and further fragmentation analysis can be an effective tool to predict grade concentration in particular size fractions. The effectiveness of applying PFC simulation of breccia texture can be explained by significant differences in grade and hardness between the matrix and inclusions. This difference results in quicker fragmentation of the softer inclusions and the subsequent distribution of high-grade particles into smaller size fractions.

Another example is a sandstone sample containing a quartz-pyrite vein, where the quartz has high hardness, and the pyrite is a soft mineral. After crushing, the pyrite content was enriched in the 0.6 mm size fraction in a real rock case and 0.36 mm in a synthetic replication relative to the rest of the size fractions. A potential mechanism that would explain the Au fraction behavior is that breakage occurs across the pyrite crystals in the vein and liberates the fine Au particles. The grade distribution across the size fractions and the RF calculations are shown in Figure 14A,B.

The presence of sphalerite-galena veins in a rhyodacite sample should favor the upgrade potential since breakage can propagate along the veins and liberate the mineralization. However, the rebound hardness map of Sample C in Figure 14A does not show a significant difference between measurements made along or around the vein or within the matrix. As previously mentioned, rebound hardness might not be effective in representing the hardness and cohesion of the rock texture. The upgrade potential of rhyodacites is highly dependent on hydrothermal alteration. This sample was simulated in PFC as homogeneous in terms of hardness and heterogeneous in terms of grade based on hardness and pXRF maps. The breakage shows that PFC can replicate the behavior of this sample. However, the absence of an understanding of the alteration type and difficulties with its integration into the model results in less effective prediction in this case. Grade distribution across the size fractions and RF calculations are shown in Figure 15A,B.

Breakage modes produced during the laboratory tests undertaken by UTAS, CODES, and CRC ORE are shown in Figure 16. The fragment shape, location, and size produced during the laboratory tests correspond with the fragment’s characteristics developed during the PFC simulation. Figure 17A shows that the fragmentation predicted by the modeling is similar to that generated in the test, with two large fragments separated by several smaller fragments. The simulation also predicted the production of fine particles within the vein and two large particles surrounding the vein, as shown in Figure 17B. Figure 17C shows how breakage occurred within the matrix and produced several pieces that do not break along the vein, as was the case for the rhyodacite sample. The comparison illustrates a similarity between the fragmentation predicted from the modeling and the actual results of the test revealing two significant fragments separated by several smaller fragments that represent soft inclusions in Sample A. Sample B shows breakage through the matrix, creating several big fragments, and Sample C indicates breakage along the vein, forming a number of small particles that occurred in both the laboratory test and the PFC simulation.

4. Discussions and Conclusions

The geological attributes of the host rock and the style of mineralization have been identified as crucial factors influencing the potential for the application of grade deportment by size [28]. Specific characteristics of rock texture can be used to classify materials in DEM modeling. Breakage tests conducted on samples revealed that the initial breakage occurred along pre-existing structures, such as fractures and veins or in areas with high concentrations of disseminated pyrite or hematite [28]. Notably, preferential breakage was evident in the PFC simulations, where host rocks composed of harder minerals exhibited larger fragment sizes, while pyrite within veins tended to produce smaller fragments. This behavior was further observed in volcanic breccia samples, in which smaller fragments containing higher grades were initially formed during breakage. In instances where host rocks and veins exhibited relatively similar properties, homogeneous breakage modes were observed; however, grades were still concentrated in fine-size fractions. Further investigation is needed to understand how the cement bonds in rocks affect their breakage and fragmentation, particularly for rocks with consistent hardness but varying levels of heterogeneity in composition. Laboratory experiments conducted by Morales Leiva [28] as part of a CRC ORE project and computer simulations performed in this study aimed to investigate the roles of rock textures in modeling preferential grade distribution by size using DEM. The study demonstrated the application of distinct element modeling in simulating breakage and predicting grade distribution by size at the meso-scale, provided adequate data are available.

Furthermore, the results of the simulations of breakage tests using additional rock texture settings and further RR calculations offer insights into techniques for predicting grade concentrations in specific size fractions using DEM. However, it is important to acknowledge the inherent variability and complexity of geological deposits. While the methods outlined in this study serve as a guide for the development of meso-scale models of preferential grade distribution by size, it is crucial to note that each deposit may present unique or more intricate rock texture characteristics. The methodology presented in this study may not be universally applicable or may only be partially applicable to other deposits. However, this methodology of numerical modeling of preferential grade deportment by size using the grouping method, along with its validation, is crucial for potential application on a production scale at mining sites. For example, if a mining unit demonstrates consistent geomechanical properties based on a few laboratory tests, this model can be used to accurately predict metal deportment for the whole unit.

The findings show the potential of DEM modeling to predict grade distribution by size and provide a means of assessing the mineralization concentration behavior and extraction strategies at the meso-scale. The results suggest that conducting simulations in 3D could provide more accurate insights, highlighting a crucial direction for future research. Overcoming current challenges in 3D modeling could significantly enhance our understanding of complex liberation processes in synthetic heterogenous samples. Additionally, Response Ranking might improve and more closely match, laboratory test results. Further research in this area is required to refine modeling techniques and adapt them to diverse geological settings.