Review of Numerical Simulations for Parameter Control in Heap Bioleaching of Copper Sulfide Ore

Abstract

Heap bioleaching is widely used to extract copper from low-grade sulfide ores thanks to its operational simplicity, low cost, and environmental sustainability. However, current control strategies rely primarily on single-factor optimization and often overlook the synergistic interactions of multiple key parameters, such as ore particle size, pore structure, pH, temperature, microbial activity, and oxygen transfer efficiency. As a result, issues such as low recovery rates, extended leaching periods, and high operational costs persist. Moreover, the “gray-box” nature of heap systems impedes real-time monitoring of internal physical, chemical, and biological processes. In addition, empirical multi-parameter optimization is time-consuming and inadequate for capturing complex interdependencies. This review was conducted to systematically examine the key factors influencing heap bioleaching efficiency and critically evaluate recent advances in numerical simulation and intelligent control strategies. As a result, we identified a major research gap: the existing models—including microscale shrinking core models (SCMs), mesoscale pore-network models based on CT reconstruction, and macroscale continuum models—have inherent limitations. SCMs assume idealized spherical particles with uniform mineral distribution while neglecting pore structure evolution and biofilm dynamics. Mesoscale models offer detailed pore characterization but lack robust multi-physics coupling (thermal–hydro–mechanical–chemical–biological, or THMCB). Macroscale models rely on homogenization assumptions that oversimplify spatial heterogeneity and temporal variations in permeability. This analysis covers the relevant literature from 1985 to 2025, with a focus on three methodological scales (micro, meso, and macro) and their integration with machine learning approaches. A notable finding is that hybrid neural network models (e.g., BP and RBF architectures) outperform purely physics-based models in predicting leaching kinetics under varying operational conditions. However, their accuracy depends heavily on high-quality field data—a limitation rarely addressed in prior reviews. By clearly delineating these model-specific limitations and scale-dependent trade-offs, this review makes two unique contributions: a structured framework for selecting and coupling numerical methods according to process requirements and a roadmap for integrating artificial neural networks with multi-physics simulations to achieve real-time intelligent control of heap bioleaching. The findings offer both theoretical guidance and practical references for optimizing the processing of low-grade copper sulfide ores.

1. Introduction

Global demand for copper, as a strategic mineral resource, is rising due to the rapid advancement of information technology and low-carbon energy technologies worldwide. Meanwhile, prolonged large-scale mining has led to the depletion of many high-grade mineral resources, shifting attention towards low-grade and complex copper deposits [1,2]. Although traditional copper smelting is effective for high-grade sulfide ores, its adaptability to low-grade ores is limited and involves high energy consumption and environmental emissions. These drawbacks have encouraged the development of less expensive hydrometallurgical alternatives [2]. Heap bioleaching for extracting low-grade ores was first proposed in the 1970s [1]. The core of this process is incorporating microorganisms and using their iron- and sulfur-oxidizing metabolism to promote metal extraction. Heap bioleaching, which is environmentally attractive and cost-effective, has been widely applied in secondary copper sulfide leaching, as these minerals dissolve more readily in the presence of ferric iron generated by microorganisms [2,3]. In contrast, primary sulfide copper ores such as chalcopyrite have dense structures and low dissolution rates at moderate temperatures, which limits their broader industrial application [2,4]. To overcome these challenges, researchers have developed strategies such as biochemical parameter regulation [5,6], catalyst addition [7], and process modeling. More comprehensive reviews by Watling [2] and Saldaña et al. [8] provide useful background on sulfide bioleaching mechanisms and bioleaching modeling, respectively.

In practice, heap bioleaching encompasses key steps such as mining, crushing ore, preparing base pads, stacking, liquid distribution, ventilation, and setting up metal recovery ponds. Figure 1 illustrates the overall bioleaching process. Specifically, crushed ore is stacked onto the prepared base pad. Next, a microbial solution is sprayed from the top, allowing it to percolate through the ore pile. During percolation, microorganisms dissolve metal minerals via mechanisms including molecular diffusion, adsorption, and catalytic oxidation. The leachate flows through the pile base under gravity and enters the collection pond. Leachate from the lean pond is recycled for ore pile spraying. Meanwhile, metals in the rich pond are recovered via solvent extraction-electrowinning [9]. This recycling method enables efficient resource recovery, lowers production costs, and reduces environmental impact.

Heap bioleaching has become a primary method for processing low-grade ores and waste rocks because of its operational simplicity, relatively low cost, and environmental advantages over conventional smelting techniques [2,3,10]. Nevertheless, industrial and pilot leaching operations face persistent challenges, including low or variable recovery, long leaching cycles, acid consumption, uneven liquid distribution, limited oxygen and carbon dioxide supply, and increased operating costs under unfavorable ore or climatic conditions [2,10,11,12]. The performance of a heap is controlled by coupled heap–structure factors (particle size distribution, agglomeration quality, porosity, permeability, and compaction), solution chemistry factors (pH, redox potential, ferric/ferrous ratio, acid, and dissolved oxygen), microbial factors (community composition, abundance, and activity), and operational controls such as irrigation, aeration, rest periods, and solution recycling [2,8,13,14,15,16]. These factors interact nonlinearly, which makes process optimization difficult if each variable is adjusted independently. Thus, developing modeling and control strategies that can accurately represent these coupled effects is a key focus in current heap bioleaching research [9].

Advances in computer software have made numerical simulation an important tool in optimizing heap bioleaching [8,17]. Heap bioleaching involves complex, coupled multi-physical phenomena, including microbial growth and metabolism, ferric/ferrous redox cycling, sulfide mineral dissolution and precipitation, unsaturated fluid flow, gas transport, heat transfer, and evolving pore structures [17,18,19]. Pore-scale models can be constructed digitally using mathematical algorithms based on images or experimental data to recreate the internal structural features of ore beds or particles [20,21,22]. At larger scales, continuum and computational-fluid-dynamics models can evaluate how the irrigation rate, airflow intensity, ore particle size, temperature, and heap geometry influence leaching efficiency [18,19,23]. These models provide a theoretical foundation and guidance for optimization, but their reliability depends on the quality of the kinetic parameters, validation data, and the degree to which laboratory or column-scale assumptions remain valid at the industrial scale [2,8,12].

This review focuses on key factors affecting heap bioleaching efficiency and the numerical simulation approaches used to describe them (Figure 2). Particular attention is paid to the regulation of redox potential, microbial activity, temperature, aeration, and solution flow, as well as the limitations of micro-, meso-, and macro-scale models. We also distinguish evidence derived from laboratory reactors, column tests, pilot heaps, and industrial operations because direct extrapolation between these scales may be misleading without validation. Through this analysis, we address current challenges in bioleaching process control and provide technical references for optimizing the processing of low-grade ores and waste rocks.

2. Key Parameters of Heap Bioleaching and Regulatory Mechanisms

In copper sulfide ore bioleaching systems, interactions between leaching microorganisms and minerals are driven by the synergistic effects of multiple parameters [2,24]. Mineral chemical composition, surface properties, and dissolution characteristics directly affect the thermodynamics and kinetics of leaching. Variations in microbial adsorption selectivity on mineral surfaces influence electron transfer efficiency at the three-phase interface [24,25], affecting leaching outcomes. For process parameters, ore particle size affects mass transfer efficiency and pore connectivity, temperature and pH modulate microbial enzyme activity, and aeration intensity regulates dissolved oxygen and carbon dioxide availability [2,13,15,17].

Each parameter shows a nonlinear relationship with target metal leaching rates. Therefore, analyzing these factors’ coupling mechanisms and optimizing the parameters are essential to improving valuable metal leaching rates and advancing bio-hydrometallurgical technology.

2.1. Key Factors Affecting Heap Bioleaching

The efficiency of heap bioleaching is governed by a complex interplay of multiple factors, which can be systematically categorized into three primary groups. As summarized in

Table 1. Key parameters and key regulatory issues affecting heap bioleaching.

Factors/Regulation	Specific Parameters
Mineral pile physical and solution chemical factors	Mineral type, mineral particle size distribution, particle size, ore dispersion characteristics, porosity, permeability, temperature, pH, redox potential, dissolved oxygen content, carbon dioxide content, iron concentration, other ions and impurities, etc.
Microbial factors	The composition of microbial communities, microbial abundance and activity, etc.
Environmental and controllable factors	Ore pre-treatment, solution redox potential, microbial activity during leaching, permeability of the ore pile, leaching mode, spraying system, ventilation intensity, solvent extraction operation, solution neutralization and purification treatment, etc.

, these include the physical and chemical factors of the mineral pile (e.g., particle size distribution, porosity, permeability, pH, redox potential, and ion concentrations), microbial factors (e.g., community composition, abundance, activity and succession), and a range of key regulatory methods (e.g., ore pre-treatment, aeration, spraying systems, and solution management). This structured classification is consistent with previous reviews of copper sulfide bioleaching and bioleaching modeling [2,8,10] and highlights that both the inherent properties of the heap and operational control strategies must be optimized to enhance bioleaching performance.

2.1.1. Ore Properties

In bioleaching, a thorough preliminary mineralogical analysis of the ore is vital for system efficiency [3], as the results guide the selection of key parameters and process control strategies. Ore particle size and mineral composition [14] interact synergistically, profoundly influencing the leaching process, particularly its efficiency. Ore particle size distribution directly impacts leaching rates and mineral dissolution kinetics [26]. Studies show that ores with loose structures and high permeability promote leachate diffusion and seepage, which markedly improves valuable metal leaching rates. Industrial practices confirm that ore is typically crushed to a particle size of 10–40 mm [26]. Small particles can cause heap blockages, reducing porosity and causing uneven leachate seepage, which leads to local pooling and hypoxia. Conversely, large particles impede leachate penetration, hindering microbial attachment to ore surfaces, which reduces the leaching rate of valuable metals and extends cycles. Additionally, different particle sizes cause variations in heap temperature, humidity, oxygen concentration, and permeability, affecting microbial types and distribution. Thus, optimal ore particle size selection fosters a suitable microbial environment and maintains heap permeability, which are essential to effective leaching [24].

The type and concentration of sulfide minerals, which are important carriers of metal elements, directly influence the complexity of the leaching process. The relative kinetic characteristics of different copper minerals depend on their specific types; simple or complex sulfides often exhibit slow oxidation kinetics, necessitating the use of oxidants to accelerate their oxidation in the leaching system [27]. For instance, in nature, chalcopyrite is typically found coexisting with pyrite, bornite, and gangue minerals [28]. This symbiotic relationship significantly impacts the leaching of chalcopyrite. Pyrite enhances the preferential leaching of chalcopyrite in the chalcopyrite/pyrite system through electrochemical effects, while its oxidative dissolution provides acid and Fe³⁺ ions necessary for the heap leaching system [25]. Furthermore, the heat released during the oxidation process helps maintain the temperature within the heap, thereby promoting the dissolution of copper minerals [29]. However, when the content of gangue minerals, such as quartz (SiO₂) and feldspar, is excessively high, they can occupy a substantial amount of pore space in the ore [30], severely hindering solution permeability and making it difficult for the leachate to contact the surface of the metal sulfides. Clay gangue minerals possess strong water absorption properties and can easily swell in humid environments, further obstructing the pore spaces of the ore [31]. This reduces the permeability of the ore pile and increases the difficulty of metal leaching. These factors not only result in increased energy consumption during production and prolonged leaching cycles but also reduce the economic viability and efficiency of the bioleaching process.

In summary, the particle size of the ore and its mineral composition are closely related and significant factors in the bioleaching process, influencing both the efficiency and economy of leaching from various perspectives. During the initial stages of process design, it is essential to thoroughly analyze the mineral composition of the ore, accurately control the particle size distribution, and fully consider the synergistic effects of different factors. This comprehensive approach enables the development of a scientifically sound optimization strategy, thereby establishing a solid foundation for efficient bioleaching.

2.1.2. Distribution and Activity of Microbial Communities

Due to the close relationship between microbial communities and mineral dissolution, community succession is a crucial factor in determining mineral leaching rates [32]. The distribution and metabolic activity of microbial communities play a decisive role in the bioleaching process. Numerous studies have shown that Acidithiobacillus ferrooxidans and Acidithiobacillus thiooxidans are common acidophilic iron- and sulfur-oxidizing species in copper sulfide bioleaching systems [33]. The spatial distribution of microorganisms on the surface of the ore directly affects their contact efficiency with the mineral matrix; a uniform and dense community structure can catalyze the oxidative dissolution of minerals more efficiently, thereby increasing metal leaching rates. Furthermore, for chalcopyrite, the bioleaching process is influenced by microbial selection, pH, temperature, redox potential, and iron-ion speciation, among which microbial selection is one of the most important factors [32]. Compared with pure culture systems, mixed culture systems often show advantages in copper extraction because iron oxidizers, sulfur oxidizers and heterotrophs can perform complementary metabolic functions [30,33]. Based on their optimal growth temperatures, leaching microorganisms can be classified as psychrophiles (≤20 °C), mesophiles (20 °C < temperature ≤ 40 °C), moderate thermophiles (40 °C < temperature ≤ 60 °C) [34], and extreme thermophiles (>60 °C). Extremely thermoacidophilic archaea [35] such as Sulfurisphaera ohwakuensis [36], Acidianus sp. [29], and Metallosphaera sedula [27] can accelerate chalcopyrite dissolution under suitable laboratory conditions, but their tolerance to slurry shear, pulp density, and changing heap environments remains a barrier to broad industrial implementation [28,36].

Similarly, microbial spatial distribution on ore surfaces directly impacts contact efficiency with the mineral matrix. A homogeneous, dense community structure more effectively catalyzes mineral oxidative dissolution. Moreover, abundant attached microorganisms play a key role in mineral dissolution [25]. Community diversity is also vital [30]. Various functional microorganisms—including iron-oxidizing, sulfur-oxidizing, and heterotrophic bacteria—perform ferrous oxidation, sulfide reduction, and organic matter degradation via synergistic activity. This collectively boosts leaching reaction efficiency. Additionally, microbial metabolic activity directly affects leaching kinetics. Highly active microorganisms rapidly generate essential leaching products and enzymes, like Fe³⁺ and H₂SO₄, accelerating mineral bio-oxidation. However, environmental factors like pH and temperature strongly regulate microbial activity. Competitive or symbiotic microbial relationships also influence functional expression. Therefore, optimizing environmental conditions (e.g., pH adjustment, temperature control, and aeration improvement) and inoculating efficient microbial communities can regulate community distribution and activity. This enhances leaching efficiency and metal recovery in bioleaching.

2.1.3. Environmental Factors

(1)

pH and redox potentials

pH is a crucial parameter that influences the dissolution and passivation of chalcopyrite, as well as microbial activity during bioleaching [37]. The complexity and highly heterogeneous nature of low-grade ores result in significant pH gradients within ore piles [30]. Research has demonstrated that a pH range of approximately 2.0–2.5 is generally favorable for the bioleaching of metal sulfides by acidophilic iron- and sulfur-oxidizing microorganisms, including Acidithiobacillus spp. [38], Leptospirillum spp. [39], Sulfobacillus spp. [40]. and Ferroplasma spp. [41]. However, the bioleaching process of chalcopyrite is characterized by a complex acid–base equilibrium. On one hand, the leaching reaction consumes acid; on the other hand, oxidation by microorganisms generates sulfuric acid. This dynamic equilibrium may lead to a decrease in pH, which can promote the formation of jarosite-type precipitates, reduce soluble iron concentration, and ultimately affect chalcopyrite leaching efficiency [37].

Redox potential is another key controlling factor determined by the ratio of ionic concentrations of Fe³⁺ and Fe²⁺ in solution [42]. Notably, a high redox potential favors the bioleaching of pyrite, while a low redox potential favors the bioleaching of chalcopyrite [31]. Kametani and Aoki (1985) [43] were the first to demonstrate that the leaching rate of chalcopyrite depends on the redox potential of the leaching solution. This finding was further elucidated by Hiroyoshi et al. [44], who revealed that at lower potentials, chalcopyrite can first be reduced to chalcocite, which is more readily oxidized by Fe³⁺ or O₂. This significantly improves the chalcopyrite leaching rate. Consequently, they were able to precisely regulate their leaching system’s potential by applying an external voltage and adding ores that form redox electrode pairs with chalcopyrite to optimize the bioleaching process.

(2)

Forced ventilation

As a crucial tool for enhancing the heap bioleaching process, the development history, mechanism of action, and industrial applications of forced aeration technology hold significant research value. This technology can be traced back to the mid-1990s, when it was developed by research institutions in Chile and Australia concurrently [11]. Forced aeration technology is mainly employed to address the issue of gas transport in the copper sulfide ore leaching system. Under natural convection conditions, air enters through the interface of the ore heap and then forms a vertical airflow by buoyancy. This can inflate the whole interface, but it is often difficult to satisfy the oxygen demand of the leaching system [45]. This problem is mainly due to the microbial metabolic characteristics of the biological heap leaching process. Most leaching microorganisms are aerobic bacteria, and their growth, reproduction, and metabolic activity are highly dependent on the supply of oxygen and carbon dioxide. Importantly, the gas distribution in the heap is affected by multiple factors such as temperature, heap height, aeration intensity, and spraying conditions, and natural aeration is usually insufficient to maintain the optimal gas environment required by microorganisms. Therefore, forced aeration technology was developed and has been successfully applied in the copper and uranium heap bioleaching industries in Peru, the United States, and Chile [46]. Compared with natural convection, forced aeration not only significantly improves the transfer efficiency of oxygen and carbon dioxide but also optimizes the reaction conditions in the heap. At the same time, the airflow not only carries heat through the heap but also promotes mineral dissolution to release more reaction heat [23], thus increasing the leaching rate and metal recovery.

Although forced aeration has demonstrated significant advantages in engineering practice, its theoretical mechanisms and the dynamic response characteristics of the leaching system still require in-depth exploration [46]. For instance, the quantitative relationships between aeration parameters (e.g., airflow rate, pressure, and distribution uniformity) and microbial activity, as well as mineral dissolution kinetics, have not been fully elucidated. Future research should focus on establishing a coupled inflation–reaction model, optimizing process parameters, and exploring intelligent regulation strategies to further enhance the economic viability and environmental sustainability of heap bioleaching. In conclusion, as theoretical research deepens and engineering experience accumulates, forced-gas filling technology will play an increasingly important role in the field of bio-metallurgy, providing robust support for the efficient utilization of mineral resources and sustainable development.

(3)

Temperature

Temperature is a critical factor influencing the bioleaching of sulfide minerals, and its impact on the reaction rate is evident in three main areas: chemical oxidation, biological activity, and mineral dissolution kinetics. In chalcopyrite bioleaching systems, temperature increases within the heap primarily result from the exothermic oxidation of sulfide minerals, with temperatures reaching as high as 55 °C [47]. This temperature condition has a dual effect. On one hand, excessively high temperatures inhibit the physiological activity of mesophilic bacteria [48] and restrict both chemical and biological oxidation reactions. On the other hand, the kinetics of chalcopyrite dissolution are slower at lower temperatures [49], and a moderate increase in temperature can significantly enhance its dissolution. Research indicates a notable difference in chalcopyrite dissolution efficiency across various temperature regimes, with leaching systems that use extremely thermophilic archaea demonstrating superior dissolution kinetics, followed by those employing moderately thermophilic bacteria [37]. Furthermore, excessively high temperatures can hinder the conversion of Fe³⁺ to yellow potassium ferroalum precipitate. The reduction of this precipitate, although it prevents passivation of the mineral surface [50], may disrupt the redox equilibrium of the leaching system. This complex interplay of temperature effects underscores the need for precise control of temperature parameters during the bioleaching process to satisfy the kinetic requirements for mineral dissolution and to maintain an optimal activity range for the leaching microorganisms.

In the process of chalcopyrite bioleaching, temperature, potential, and pH interact with one another [49]. Fe³⁺ serves as a crucial factor in the initial leaching stage, as it controls the oxidation reaction rate of the entire leaching system. In the intermediate stage, when the pH value is low, temperature becomes a significant factor regulating the leaching process. As H⁺ ions are depleted, maintaining an appropriate range of leaching potential becomes a key factor in successful chalcopyrite leaching [37]. Therefore, to achieve efficient bioleaching of chalcopyrite, it is essential to fully consider the synergistic effects of temperature, potential, and pH. By dynamically regulating these critical parameters and optimizing the leaching conditions, leaching efficiency be significantly enhanced, and the stability of the entire process can be ensured. This multi-parameter synergistic strategy effectively addresses challenges associated with leaching efficiency and mineral surface passivation caused by the imbalance of a single parameter. Consequently, it provides a solid theoretical foundation and reliable technical support for the industrial application of copper sulfide ore bioleaching, which holds great scientific significance and offers broad application prospects.

It is important to distinguish the origin and scale of the data used in these discussions. Many temperature, pH, redox and microbial activity relationships reported for chalcopyrite bioleaching were obtained in shake flasks, stirred tanks, small columns, or pilot heaps under controlled conditions [3,14]. These experiments are essential for identifying mechanisms and kinetic parameters, but their results cannot be directly transferred to industrial heaps, where heterogeneity in particle size, permeability, solution channeling, oxygen supply, and self-heating produces strong spatial gradients [3,38]. Therefore, laboratory and column results should be used together with pilot and industrial monitoring data when building or validating numerical models for heap bioleaching.

2.2. Control Mechanism of the Heap Bioleaching Reaction Process

2.2.1. Temperature, Aeration, and Spray Control in the Heap Leaching Process

In the heap leaching of copper sulfide ores, synergistic control over temperature, aeration, and spraying is crucial for achieving efficient leaching. Temperature variations within the heap are influenced by the solution flow rate, airflow, and rate of heat generation from reactions [15]. During heap bioleaching, the heat released via the biological and chemical oxidation of sulfide minerals can enhance the metal-leaching kinetics; however, if local temperatures exceed the optimal growth range of the dominant microorganisms, microbial activity can decrease, and sulfide oxidation can slow down [51]. This phenomenon is particularly important in systems dominated by moderately thermophilic bacteria, whose practical upper growth range is generally below approximately 60° [52]. Petersen and Dixon [53] conducted GEOCOAT™ mini-column leaching experiments combined with heat conservation modelling to evaluate the thermophilic heap leaching of chalcopyrite concentrate. They demonstrated that appropriate control of irrigation, aeration, and heap height could enable high copper extraction and sustain heap temperatures suitable for extreme thermophiles. Although extremely thermophilic archaea demonstrate good leaching performance under laboratory conditions [54], their industrial application remains challenging because of constraints related to environmental adaptation, hydrodynamics, and community stability.

Regarding energy balance and temperature prediction, the temperature dynamics within the heap can be described by an energy balance equation that accounts for heat generated by exothermic reactions, heat transfer via solution flow and aeration, and heat lost to the surroundings:

$${\rho }_b{C}_p{\displaystyle \frac{\partial T}{\partial t}}=\nabla \cdot \left(k_{\mathrm{e}\mathrm{f}\mathrm{f}}\nabla T\right)+Q_{\mathrm{r}\mathrm{x}\mathrm{n}}-{\rho }_l{C}_{p,l}\left(u_l\cdot \nabla T\right)-{\rho }_g{C}_{p,g}\left(u_g\cdot \nabla T\right)$$

(1)

where ${\rho }_b$ is the bulk density of the ore bed (kg/m³); $C_p$ is the effective heat capacity (J/(kg·K)); $k_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective thermal conductivity (W/(m·K)); $Q_{\mathrm{r}\mathrm{x}\mathrm{n}}$ is the volumetric rate of heat generated by sulfide oxidation (W/m³); $u_l$ and $u_g$ are the densities of liquid and gas phases; $C_{p,l}$ and $C_{p,g}$ are their heat capacities, respectively; and ul and ug are the superficial velocities of the leach solution and air, respectively. This equation forms the core of temperature prediction modules in simulation tools like COMSOL 6.12, enabling the evaluation of different aeration and spraying strategies.

Temperature regulation can be achieved by adjusting the aeration intensity and spray rate [52]. The ratio of the spray liquid flow rate to the airflow rate significantly impacts the temperature distribution within the ore pile. Li Hongxu et al. [15] confirmed through experimental simulations that the optimal temperature distribution in the ore pile is achieved when the ratio of the airflow rate to the spray liquid flow rate is 2:3. Furthermore, the spray liquid flow rate not only affects the penetration and distribution of the leach solution but also influences the temperature changes within the heap [12]. Therefore, establishing a well-designed coupled spray–aeration regulation mechanism is essential for maintaining the appropriate temperature range for biological reactions and preventing a decrease in leaching efficiency due to temperature anomalies.

In real-world heap leaching operations, flexibly adjusting the spraying and aeration parameters (manipulated variables) based on the characteristics of the ore and the specific requirements of the heap leaching process is essential to controlling variables (temperature, O₂ concentration) and achieving operational objectives (maximizing recovery). For instance, in forced-ventilated bio-heap leaching, numerical simulation studies conducted by Huang et al. [23] demonstrated that varying combinations of spraying rates and ventilation intensities shifted the mineral dissolution process from being controlled by oxygen transfer to being controlled by the spraying rate under conditions of high ventilation intensity. This finding indicates that differentiated parameter regulation strategies should be implemented at various stages of the leaching process. By thoroughly considering the physical and chemical properties of the ore, as well as microbial activity and reaction kinetics, a multi-parameter synergistic optimization model can be developed, significantly enhancing the stability and metal recovery of the leaching process.

In summary, synergistic control of temperature, aeration, and spraying is essential to optimizing the heap leaching process for copper sulfide ores. Establishing a parameter optimization system that considers the characteristics of the ore and microbial metabolism and achieving a dynamic balance between thermodynamic conditions and reaction kinetics can significantly enhance leaching efficiency and ensure process stability.

2.2.2. Redox Potential Regulation

Complex copper minerals, including chalcopyrite and cuprite oxide, are prone to oxidation during bioleaching; therefore, the extraction of metallic copper can be optimized by controlling the redox potential [32]. The bioleaching of sulfide minerals involves a complex redox reaction process in which sulfide/sulfur serves as an electron donor and iron ions and oxygen function as electron acceptors [12]. Research has demonstrated that a relatively low redox potential in the leaching solution is favorable for chalcopyrite dissolution, whereas a higher potential tends to cause passivation of chalcopyrite [55], which impedes its further dissolution. Consequently, by regulating the redox potential, the formation of intermediates such as porphyry or pyrochlore on the surface of chalcopyrite can be enhanced, thereby accelerating the dissolution of chalcopyrite [56].

In attempting to optimize this process, researchers have found that adding a specific amount of pyrite can significantly enhance the chalcopyrite leaching efficiency. The dissolution of pyrite provides microorganisms with the nutrient Fe²⁺; further, pyrite and chalcopyrite can undergo a redox reaction within the leaching system, effectively regulating the redox potential of the leaching solution. Zheng et al. [57] demonstrated that when the optimal mass ratio of chalcopyrite to pyrite was 1:1, the chalcopyrite leaching rate increased significantly, ranging from 15% to 63%. Electrochemical tests revealed that chalcopyrite and pyrite formed a protocell in this system, resulting in an increase in redox current density and a decrease in resistance. This phenomenon suggests that the addition of pyrite can substantially promote the oxidation rate of chalcopyrite and enhance the rate of electron exchange, thereby accelerating the dissolution of chalcopyrite. Hong et al. [58] further explored the role of pyrite in different mass ratios in regulating the redox potential of chalcopyrite undergoing bioleaching in a system using Leptospirillum ferriphilum. The results showed that when the mass ratio of pyrite to chalcopyrite was 3:1, the redox potential of the leaching solution was effectively maintained at 350–470 mV for a long time. After 30 days of leaching, the copper leaching efficiency was significantly increased to 70%. This indicates that the addition of pyrite can effectively control the redox potential and improve the leaching efficiency of chalcopyrite (Figure 3). In addition, Ferna’ndez-Reyes et al. [59] established a bio-electrochemical system coupling chalcopyrite bio-oxidation, bioproduction of electricity, and cathodic reduction, which further demonstrated that leaching microorganisms are electrically active in the oxidation process.

In conclusion, precisely regulating redox potential is critical for optimizing the chalcopyrite bioleaching process. Research has demonstrated that the introduction of exogenous iron sources, such as pyrite, can effectively regulate the Eh value of the leaching system, thereby significantly enhancing the dissolution kinetics of chalcopyrite and increasing the copper leaching rate. This finding not only clarifies the regulatory mechanisms of the electrochemical dissolution of sulfide minerals but also provides essential guidance for optimizing the industrial heap leaching process. In practice, a precise control strategy based on redox potential values is anticipated to maximize leaching efficiency and optimize production costs, which holds substantial practical significance for advancing the industrial application of bio-hydrometallurgical technology.

2.2.3. Regulation of Microbial Leaching Activity

Acidophilic microorganisms involved in the bioleaching of copper sulfide ores are autotrophic organisms capable of thriving in low-pH inorganic media and tolerating high concentrations of metal ions. Their primary function is to oxidize Fe²⁺ to Fe³⁺ and sulfur to sulfuric acid (H₂SO₄) [60]. These microorganisms continue to push the boundaries of survival in extreme environments, such as thermophilic and acidophilic conditions, through long-term adaptive evolution. However, their environmental adaptation and resistance are highly dependent on the efficient expression of a large number of genes. By integrating the bioleaching environment, microbial community structure, and molecular functional characteristics, we can elucidate the coupling of their environmental adaptation mechanisms and resistance strategies.

In recent years, researchers have leveraged the synergistic effects of mixed strains and the construction of microbial communities to develop highly efficient leaching bacteria, thereby increasing the copper leaching rate. For instance, Liao [61] conducted leaching experiments on chalcopyrite using a moderately thermophilic mixed bacterial culture and discovered that the dominant strains varied at different stages of the leaching process. After 40 days of leaching, the copper leaching rate reached 70.86%. Similarly, Peng et al. [40] performed bioleaching of chalcopyrite using a mixture of strains at a low temperature of 6 °C. Their results indicated that the copper leaching rate in the mixed strain culture system was higher than that observed with a single strain of Sulfobacillus acidophilus. Additionally, Sulfobacillus acidophilus and Sulfobacillus sulfuricus were found to dominate the mixed-strain system.

In a heap bioleaching system, the structure of the microbial community exhibits dynamic succession characteristics. Changes in population composition and metabolic activity can significantly influence leaching efficiency. The dominant species adapt to the leaching system by responding to various environmental factors, such as temperature, pH, and oxygen concentration within the heap (as illustrated in Figure 4). Acosta et al. [16] found that as the height of the heap increased, when the temperature within the heap exceeded 32 °C, the predominant species in the microbial community shifted from mesophilic to moderately thermophilic bacteria. The primary species identified were Leptospirillum ferriphilum and Sulfobacillus thermosulfidooxidans. This indicates that the dynamics of microbial populations in the heap bioleaching process are closely related to environmental factors.

Microorganisms play a crucial role in the bioleaching of copper sulfide ores, and regulating their activity is essential for enhancing leaching efficiency. This can be achieved by optimizing the microbial community structure, leveraging the synergistic effects of mixed strains, and carefully regulating environmental factors (such as temperature, pH, and oxygen concentration) within the heap. This approach not only provides a theoretical foundation for improving the bioleaching of complex copper minerals but also serves as an important reference for practical industrial applications. In particular, in bio-pile leaching, greater leaching efficiency and reduced production costs can be achieved by precisely regulating microbial activity and environmental conditions.

3. Current Status of Heap Leaching Numerical Simulation Studies

Copper ore heap leaching is a complex reaction system characterized by multi-field coupling (thermal–hydro–mechanical–chemical–biological, THMCB) and multi-phase interactions (gas–solid–liquid) [17]. In this system, thermodynamic gradients drive solution permeation, while the mechanical stress field influences the pore structure of the heap. Additionally, chemical oxidation and microbial metabolism jointly govern the kinetics of mineral dissolution. However, the complexity of THMCB coupling mechanisms at the industrial scale has resulted in a lack of universal empirical models to accurately describe the system’s dynamic behavior. Currently, modeling studies in this field rely primarily on deterministic approaches to characterize key physicochemical processes by constructing control equations; however, these approaches are still complicated by significant challenges. First, nonlinearities arising from multi-field coupling complicate equation solving, exemplified by the complex feedback effects between biofilm growth and permeability. Second, the spatial heterogeneity of industrial landfills—such as variations in ore particle size and solution trenching phenomena—renders the calibration of model parameters reliant on high-cost field monitoring data. Therefore, this study was conducted to address these bottlenecks by incorporating data assimilation techniques and machine learning-assisted parameter inversion.

3.1. Leaching Model

A substantial number of studies have analyzed the mechanisms of heap leaching through mathematical models, which can be categorized into macroscale models (ore heap models), mesoscale models (pore-scale models), and microscale models (single-mineral/single-ore models) based on their respective study scales [8,17,62,63] (as illustrated in Figure 5). Traditional mathematical models exhibit limitations in characterizing the complex multiphase reaction processes within heap bioleaching systems. In particular, the dynamic representation of the coupling between solid, liquid, and gas phases—including mass transfer, heat transfer, hydrodynamic behavior, microbial growth, and mineral precipitation—remains challenging [8,17]. To overcome this technical bottleneck, researchers have increasingly opted for high-precision in situ or non-destructive characterization techniques, such as atomic force microscopy [64] (AFM), X-ray tomography [65] (X-ray CT), magnetic resonance imaging [66,67] (MRI), and synchrotron radiation [68]. These techniques are relevant because they provide direct information about pore structure, liquid distribution, gas-–liquid–solid phase contact, mineral surface evolution, and reaction fronts, which are key inputs for model construction and validation. For example, Fagan et al. [22] used MRI to analyze a flowing low-grade copper ore column and generated a three-dimensional phase map that quantitatively identified solid, liquid, and gas positions, voidage, liquid holdup, and interfacial areas. As MRI has already been applied to pore-scale and column-scale analysis of heap leaching systems, it should be acknowledged when discussing high-precision detection methods. Such imaging and spectroscopy approaches can therefore strengthen model validation by linking microscale and mesoscale observations to continuum-scale parameters used in heap simulations.

3.1.1. Microscale Modeling

Microscale models are crucial theoretical tools for interpreting the microscopic mechanisms involved in the heap leaching process. They systematically characterize the coupled mass transfer and reaction processes within ore particles, the dynamic evolution of pore structures, and the interaction mechanisms at the mineral–microbe interface. The classical Shrinking Core Model (SCM) idealizes ore particles as spherical geometries, assuming that the mineral components are uniformly distributed and that the concentration of the primary leaching solution remains constant. This model reveals that the mineral dissolution process is governed by three successive mechanisms: the mass transfer of the leaching agent through the boundary layer of the liquid film, the diffusive mass transfer of reaction products within the porous layer, and the interfacial chemical reaction occurring on the surface of the unreacted core [21,69,70,71,72], as illustrated in Equation (2) [70]. Despite the fact that the ore morphology and leaching environment undergo dynamic changes during real-world leaching processes, these models still provide an important basis for predicting mineral reaction rates. In recent years, microscale models have been continuously developed and improved, such as the composite model established by Ranjbar et al. [72]. By coupling the particle size distribution (PSD) and reaction kinetics, it uses the Rosin–Rammler function to describe the particle distribution and combines the chemical control with the diffusion control mechanism, thereby significantly improving the accuracy of predicting the leaching behaviour of chalcopyrite. Current trends in research trend indicate that the microscale model is transitioning from ideal assumptions to actual conditions. By integrating the multi-physical field processes of mass transfer, reaction and structure evolution, and combining advanced characterization techniques, it can deepen our understanding of the leaching mechanism, providing theoretical support for the optimization of macroscopic heap leaching processes.

$${\displaystyle \frac{3K_1{C}_{\mathrm{b}}}{{\rho }_{\mathrm{s}}R}}t=X_{\mathrm{c}}$$

(2)

$${\displaystyle \frac{6D_{\mathrm{e}}{C}_{\mathrm{b}}}{{\rho }_{\mathrm{s}}{R}^2}}t=\left[1-3(1-X_{\mathrm{c}}{)}^{{\displaystyle \frac{2}{3}}}+2(1-X_{\mathrm{c}})\right]$$

(3)

$${\displaystyle \frac{k_{\mathrm{s}}{C}_{\mathrm{b}}}{{\rho }_{\mathrm{s}}R}}t=\left[1-(1-X_{\mathrm{c}}{)}^{{\displaystyle \frac{1}{3}}}\right]$$

(4)

where $X_{\mathrm{c}}$ represents the conversion rate of the mineral particles, ${\rho }_{\mathrm{s}}$ indicates the molar density of the ore particles, $R$ is the initial radius of the ore particles, t denotes time, $C_{\mathrm{b}}$ refers to the concentration of the external constant leach solution, $K_1$ is the mass transfer coefficient, $D_{\mathrm{e}}$ denotes the diffusion coefficient, and $k_{\mathrm{s}}$ is the surface reaction rate coefficient.

3.1.2. Mesoscale Modeling

A mesoscopic scale heap leaching model is a digital characterization of the pore structure of the ore heap, established using 3D CT reconstruction technology. This type of model can accurately depict the pore distribution and connectivity properties within the heap. Researchers, both in China and abroad, have conducted numerical simulation studies based on CT reconstructions of porous media in the fields of petroleum, chemical engineering, and materials science. However, most existing models are limited to simulating a single leaching flow, mass transfer, or heat transfer process, making it challenging to fully capture the multi-field coupling effects occurring in real-world heap leaching processes [70]. The numerical modeling of multi-field coupling in the mineral heap leaching process remains in the exploratory stage and is primarily constrained by the complexity of multi-physical field interactions in porous media and the nonlinear characteristics of the mineral dissolution kinetics. Presently, three primary equations are utilized for modeling fluid flow in porous media: Darcy’s law, the Richards equation, and the Navier–Stokes–Brinkman equation (as shown in Equations (5)–(7)) [21]. The applicability of these equations depends on the pore scale and flow characteristics.

$$Q=K_{s}A\nabla H={\displaystyle \frac{K_{s}A}{\rho g}}\nabla p$$

(5)

$${\displaystyle \frac{\partial \theta }{\partial t}}=-\nabla ·\left[K\left(\theta \right)\nabla h\left(\theta \right)\right]+{\displaystyle \frac{\partial K\left(\theta \right)}{\partial z}}$$

(6)

$$\nabla \cdot \left(\mu \left(\nabla u+(\nabla u{)}^T\right)-\nabla p=0\right.$$

(7)

where $Q$ is the fluid volume flow rate (m³·s⁻¹), $K_s$ is the hydraulic conductivity coefficient of the saturated medium (m·s⁻¹), $A$ is the cross-sectional area of the porous medium (m²), $\nabla H$ denotes the fluid hydraulic gradient (m·m⁻¹), $\nabla p$ denotes the fluid pressure gradient (Pa·m⁻¹), $\rho$ is the fluid density (kg·m⁻³), $\theta$ is the volumetric water content (m³·m⁻³), $K\left(\theta \right)$ is the capillary conductivity coefficient of an unsaturated medium (m·s⁻¹), $h\left(\theta \right)$ is the height of fluid suction due to capillary action (m), $\mu$ is the hydrodynamic viscosity (kg·m⁻¹·s⁻¹), $u$ is the fluid flow rate (m·s⁻¹), $T$ is matrix transpose, and $p$ is the fluid pressure (Pa).

3.1.3. Macroscale Modeling

Macroscale heap leaching models typically treat the entire ore heap as a homogeneous porous medium, assuming that the material properties of the heap—such as the ore density, porosity, permeability, and thermal conductivity—are isotropic. These models also presume that minerals are uniformly distributed throughout the heap. Changes in the morphology and physical properties of the ore throughout the leaching process are generally not considered. Instead, the average ore grain size is used to approximate the ore gradation across the heap, and a shrinking core model is utilized to predict the leaching reaction rate [73]. However, this assumption may diverge from real-world conditions encountered during heap leaching, as the particle size distribution, mineral composition, and pore structure of the ore change dynamically throughout the leaching process. These changes, in turn, influence the leaching efficiency and distribution of the leaching solution. Therefore, although the macroscale model offers certain advantages in simplifying calculations and facilitating preliminary design, it is limited in its ability to accurately simulate and optimize the heap leaching process. The heap leaching model is primarily used to describe the equilibrium of solution, material, and heat flow within the entire leaching system, with its fundamental governing equations encompassing gas flow, liquid–phase mass transfer, and heat transfer, as illustrated in Equations (8) and (9) [74].

$${\displaystyle \frac{\partial }{\partial t}}\left({\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{\nu }_{\mathrm{g}}\right)+{\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}\nabla \cdot {\nu }_{\mathrm{g}}=-{\epsilon }_{\mathrm{g}}\nabla p+{\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}\Delta {\nu }_{\mathrm{g}}-{\displaystyle \frac{{\epsilon }_{\mathrm{g}}^2{\mu }_{\mathrm{g}}}{K}}{\nu }_{\mathrm{g}},$$

(8)

$${\epsilon }_l{\displaystyle \frac{\partial \left(C_i\right)}{\partial t}}=D_l{\epsilon }_l\Delta C_i-q_l\cdot \nabla C_i+R_i$$

(9)

$$C_{\mathrm{p},\mathrm{B}}{\rho }_{\mathrm{B}}{\displaystyle \frac{\partial T}{\partial t}}=k_{\mathrm{B}}\Delta T-{\displaystyle \frac{\partial H_{\mathrm{L}}}{\partial y}}-\left({\displaystyle \frac{\partial H_{\mathrm{g}}}{\partial x}}+{\displaystyle \frac{\partial H_{\mathrm{g}}}{\partial y}}\right)+\left(-\Delta H_{\mathrm{R}}\right)R_{\mathrm{c}\mathrm{h}}$$

(10)

where $v_g$ is the gas flow rate (m·s⁻¹), ${\epsilon }_g$ is the air void ratio (gas phase volume fraction), ${\rho }_g$ is the air density (kg·m³), P is the gas pressure (Pa), ${\mu }_g$ is the aerodynamic viscosity (Pa·s), K is the bed permeability (m²), $C_i$ is the concentration of the ith component (mol·m⁻³), ${\epsilon }_l$ is the liquid phase volume fraction (the proportion of the liquid phase to the total volume of the heap leaching system, $q_l$ is the Darcy flow rate of the liquid phase (m·s⁻¹), and satisfies $q_l$ = ${\epsilon }_l{v}_l$, where $v_l$ is the apparent velocity vector, $R_i$ is the diffusion coefficient of oxygen in the liquid phase (mol·m⁻³·s⁻¹), $D_l$ is the diffusion coefficient of oxygen in the liquid phase (m²·s⁻¹), “-“ denotes exothermic, $C_{\mathrm{p},\mathrm{B}}$ is the average heat capacity (J·kg⁻¹·K⁻¹), $k_{\mathrm{B}}$ is the heat transfer coefficient (W·m⁻¹·K⁻¹), $H_{\mathrm{L}}$ is the liquid-phase enthalpy on the potential area (J·m⁻²), $H_{\mathrm{g}}$ is the gas-phase enthalpy per unit area (J·m⁻²), $\Delta H_{\mathrm{R}}$ is the average heat of reaction (J·mol⁻¹), and $R_{\mathrm{c}\mathrm{h}}$ is the chemical reaction rate constant (mol·m⁻³·s⁻¹).

3.1.4. Neural Network Model

An artificial neural network (ANN) is a data-driven modeling tool inspired by the functions of artificial nervous systems and based on interconnected neurons [75]. The advantages of neural networks lie in their ability to handle multiple inputs and multiple outputs, enabling parallel processing of data and self-learning capabilities. Currently, the backpropagation (BP) neural network and the radial basis function (RBF) network are the two most mature and widely applied networks in the field. Taking the BP neural network as an example, the neural network structure consists of an input layer, a hidden layer, and an output layer (as shown in Figure 6). After providing learning samples to the input neurons, the activation values of the neurons propagate from the input layer through each hidden layer to the output layer. Finally, the network’s response to the input is expressed at each neuron in the output layer. According to the direction that reduces the error between the network output and the actual output sample, the signal is propagated backward through each hidden layer back to the input layer, thereby gradually correcting the connection weights. The formula is as follows:

$$u_k={\displaystyle \sum _{j=1}^P} w_{kj}{x}_j,v_k=u_k-{\theta }_k,y_k=\phi \left({\nu }_k\right)$$

(11)

where $x_1$, $x_2$, …, $x_p$ are the input signals; $w_1$, $w_2$, …, $w_{kp}$ are the weights of neuron k; $u_k$ is the result of the linear combination; ${\theta }_k$ is the threshold; $\phi (·)$ is the activation function; and $y_k$ is the output of neuron k.

Multi-scale modeling of the heap leaching process has produced a systematic theoretical framework that provides an important basis for revealing the leaching mechanism and optimizing the process. At the micro scale, the shrinking core model and its improved method were gradually extended from the ideal scenario to real-world leaching conditions by coupling the mass transfer reaction process and particle distribution characteristics. The mesoscopic scale model based on CT reconstruction technology has been used to accurately characterize pore structure, but it still requires further work regarding the multi-physical-field coupling simulation. The macro scale model uses a homogenization assumption to simplify the calculation, but it cannot accurately describe the dynamic evolution of the ore pile. These existing models remain insufficient to characterize key processes such as the microbe–mineral interface, passivation layer formation mechanism, and multi-component competitive leaching. With the help of an advanced deep learning algorithm, the neural network model significantly simplifies the complex and cumbersome experimental process, thus greatly reducing the commercial cost. However, this model’s training process relies on high-quality field data to ensure accuracy and reliability. Future research should develop multi-scale coupling methods and combine computational fluid dynamics, multi-physical-field modeling, machine learning, and other technologies to build more accurate prediction models to promote the optimization and innovation of heap leaching.

Among whole-heap-modeling tools, HeapSim deserves particular attention because it was developed to integrate multiple interacting heap processes rather than treating flow, heat, reactions, and biology in isolation. Ogbonna and Dixon [76] described HeapSim as a model that accounts for mineral kinetics, particle-level effects, bacterial growth, oxidation and adsorption, gas absorption, pore diffusion, bulk advection, gas balance, and heat conservation. Bouffard [77] subsequently applied the HeapSim model to the Pueblo Viejo ore deposit and used it to evaluate heap height, aeration rate, irrigation rate, and microbial inoculation strategies. These studies illustrate one of the earliest and most explicit efforts to couple the major heap-scale variables required for design and operation, providing a useful reference point for later THMCB and digital-twin modeling frameworks.

4. Common Methods for Numerical Simulation of Heap Leach Models

To provide a clear overview of the numerical simulation methods available for heap leaching,

Table 2. Comparison of numerical simulation methods for heap-leaching models.

Method	Type	Advantage	Limitations	Typical Applications
Finite Element Analysis (FEA)	Physics-based	Handles complex geometries; supports multi-physics coupling, including thermal, hydraulic, and mechanical processes; high accuracy for pore-scale simulation	Computationally intensive; requires detailed material properties; mesh quality significantly affects results	Pore-scale flow simulation; stress–strain analysis; coupled thermo-hydro-mechanical problems
Computational Fluid Dynamics (CFD)	Physics-based	Simulates complex fluid flow, heat/mass transfer, and reactions; adapts to varying boundary conditions; provides detailed spatial–temporal distributions	High computational cost; requires validation with experimental data; sensitive to turbulence models and closure assumptions	Industrial-scale heap flow prediction; unsaturated flow in porous media; gas–liquid transport
Finite Volume Method (FVM)	Physics-based	Conservative formulation for mass, momentum, and energy; robust for fluid-dominant problems; well-suited to large-scale domains	Less flexible for complex geometries than FEA; mesh generation can be challenging	Solution transport in heaps; coupled chemical–biological processes; industrial-scale simulations such as PHOENICS
Artificial Neural Networks (ANN)	Data-driven	Captures nonlinear relationships without explicit physical equations; fast prediction once trained; handles noisy data	Requires large training datasets; limited extrapolation capability; “black box” nature limits interpretability	Leaching rate prediction; process optimization; real-time control surrogate models

presents a comparative summary of the key approaches, including Finite Element Analysis (FEA), Computational Fluid Dynamics (CFD), Finite Volume Method (FVM), and Artificial Neural Networks (ANNs). The table distinguishes between physics-based models (FEA, CFD, and FVM) and data-driven models (ANNs), highlighting their respective advantages, limitations, and typical applications. This comparison serves as a roadmap for selecting appropriate simulation strategies based on specific research or industrial needs.

The selection of an appropriate method depends on the specific objectives (e.g., mechanistic understanding vs. predictive optimization), available data, computational resources, and required accuracy. Furthermore, all simulation methods must be rigorously validated against experimental measurements, with careful quantification of errors and uncertainties. These validation strategies should be directly linked to the physical parameters discussed in Section 1 (e.g., temperature, pH, particle size, redox potential) and the mathematical models presented in Section 2 (e.g., mass/energy balances, microbial kinetics).

4.1. Finite Element Analysis Methods

Researchers have employed finite element analysis (FEA) to explore the effects of ore particle size, particle density, and pore structure on solution flow and leaching efficiency during heap leaching, highlighting the critical roles of these factors. FEA can accurately simulate fluid flow behavior at the pore scale, offering a multi-scale perspective that ranges from micro to macro for the study. However, as leaching progresses, the bulk medium of the heap undergoes significant structural changes due to particle transport, deposition, chemical dissolution, precipitation, and microbial activity (as shown in Figure 7). Therefore, it is essential to integrate microscopic pore structural parameters with macroscopic seepage characteristics to more comprehensively reflect the dynamic features of the heap leaching process.

Regarding error, uncertainty, and validation, the accuracy of FEA and FVM simulations depends strongly on several factors: (i) mesh resolution and quality, (ii) constitutive relationships for material properties (e.g., permeability as a function of porosity), (iii) boundary and initial conditions, and (iv) numerical solver settings. Systematic mesh refinement studies should be conducted to ensure mesh-independent solutions. Validation against experimental data—such as column leach tests or pilot-scale heap measurements—is essential. Common validation metrics include the coefficient of determination (R²), root mean square error (RMSE), and mean absolute percentage error (MAPE). For example, a simulated copper extraction curve should be compared with measured data from column tests, with acceptable error margins typically within ±10%–15% for industrial applications. Uncertainty quantification can be performed using methods such as Monte Carlo simulation or sensitivity analysis to assess how variations in input parameters (e.g., reaction rate constants, thermal conductivity) affect model outputs. These uncertainties must be propagated through the model to provide confidence intervals for predictions.

Using multiscale modeling combined with digital image processing techniques, Maghsoudy et al. [21] revealed an inverse relationship between fluid velocity and porosity. They found that pressure distribution is influenced by the arrangement of particles. For the first time, heap leaching field images were utilized for pore scale, enabling the correlation of parameters from laboratory to industrial scales. The authors reported validation against experimental measurements, with discrepancies attributed to image resolution limitations and assumptions regarding particle shape uniformity.

Meanwhile, numerous researchers have employed computational fluid dynamics (CFD) techniques to predict flow and leaching behavior in heaps. It is important to distinguish CFD as a general approach that can be implemented using different numerical discretization schemes, primarily FVM and, to a lesser extent, FEA. The finite volume method (FVM) is particularly well-suited to fluid-dominant problems due to its conservative formulation, which ensures that mass, momentum, and energy are conserved locally. Simulation combining CFD and FVM can be defined as an analysis of a system involving mass transfer, heat transfer, fluid flow, and chemical reactions based on numerical calculations [78]. CFD has the advantage of simulating complex physical phenomena and adapting to varying environmental conditions. This capability facilitates detailed design, diagnostics, and optimization analysis of the heap leaching process.

For example, C.R. Bennett et al. [18] developed a copper heap leaching model based on CFD-FVM techniques, with which they accurately simulated the mineral dissolution reaction through column leach test data validation in combination with the shrinking core model, in addition to describing the precipitate species, bacterial action, and the mode of flow of unsaturated fluids and gases in the porous medium. The authors reported good agreement between simulated and experimental copper recovery curves (R² > 0.95), though uncertainties in the prediction of intermediate species concentrations persisted due to limited kinetic data.

McBride et al. [19,79] developed a model for the oxidation of gold–silver–copper by means of CFD software, creating a comprehensive model applicable to gold–silver–copper oxide ore heap leaching systems and developing a code of equations for solving industrial heap leaching problems that comprehensively reflects the complex processes in porous media. Their validation effort included a comparison with pilot-scale heap data, with errors in metal recovery predictions ranging from 5% to 12% depending on the ore type and operating conditions.

In addition, M. Yaghobi Moghaddam et al. [78] provided reasonable and accurate results for heap leaching using the numerical finite volume method and the CFD model PHOENICS, showing that chemical and biological processes have a significant effect on the solution transport mechanism and leaching efficiency. The study emphasized the importance of validating kinetic parameters (linked to Section 2.2.3 on microbial activity) against independent column test data before increasing to heap scale.

The physical parameters discussed in Section 2—including temperature (Section 2.2.1), redox potential (Section 2.2.2), and microbial activity (Section 2.2.3)—serve as critical inputs and boundary conditions for FEA/CFD/FVM models. For instance, the temperature-dependent microbial growth kinetics described in Section 2 (Equation (5)) must be incorporated into the reaction source terms in the mass balance equations solved via CFD. Similarly, the redox potential regulation mechanisms discussed in Section 2.2.2 inform the electrochemical boundary conditions for species transport models. Validation of these simulation results against experimental data provides a crucial feedback loop for refining the mathematical models presented in Section 2.

On this basis, utilizing simulation software, researchers have investigated the response mechanisms of leaching systems under multi-phase and multi-factor conditions. They established a mathematical model encompassing pore development, solute migration, heat transfer, and gas convection within the leaching system. By employing multi-field coupling software, they assessed the influence of various factors, including porosity, solution flow rate, temperature, and gas concentration. Furthermore, they proposed optimization and control measures for the microfabricated leaching environment to enhance the efficiency of leaching for multiple complex minerals [80].

4.2. Artificial Neural Network Method

Artificial neural networks (ANNs) represent an effective computational and simulation technology within the broader category of data-driven models. Unlike physics-based models (FEA, CFD, and FVM) that solve fundamental conservation equations, ANNs learn patterns directly from data without requiring explicit physical equations. The concept of neural networks stems from neuroscience and involve a variety of intricately interconnected neurons that form a neural system capable of learning for modeling and simulation [81]. Over the past two decades, the hydrometallurgical industry has increasingly adopted simulation and modeling techniques to assist in production decision-making. This approach has facilitated process control and optimization, enabling the acquisition of valuable data, such as accurate ore particle size measurements, improved metal recovery, and enhanced predictive capabilities [82].

Compared to physics-based models, ANNs present unique challenges with respect to error and uncertainty quantification. Key sources of error include (i) training data quality and quantity—insufficient or unrepresentative data leads to poor generalization; (ii) network architecture selection (number of layers, neurons, activation functions); (iii) overfitting, where the model learns noise rather than underlying patterns; and (iv) extrapolation beyond the range of training data, which is highly unreliable. Standard validation practices for ANNs include.

• Data partitioning—splitting available data into training (typically 60%–70%), validation (15%–20%), and testing (15%–20%) sets.
• Cross-validation—k-fold cross-validation to assess model stability.

Performance metrics—R², RMSE, MAPE, and Akaike Information Criterion (AIC) for model comparison.

3.

Uncertainty quantification—Bayesian neural networks or Monte Carlo dropout can provide prediction intervals rather than point estimates.

It is critical to recognize that ANNs cannot reliably predict conditions outside the domain of their training data. Therefore, extrapolation for novel ore types or significantly different operating conditions should be avoided without additional experimental data.

In recent years, researchers have introduced methods such as grey prediction models and artificial neural networks (ANNs) to more accurately predict the metal leaching rate during heap leaching. These methods have been employed to assess the leachability and economic viability of metal-bearing mines. Jiang Huichun et al. [83] established a grey prediction model (GM) based on grey prediction theory and verified its accuracy in predicting the leaching rate in column leaching tests through residual analysis and mean square ratio tests. The authors reported prediction errors within ±5% for the tested conditions, though the model’s applicability to other ore types remains to be validated.

Hoseinian et al. [84] utilized an ANN and its Genetic Algorithm Neural Network (GANN) to optimize the column leaching of copper oxide, effectively addressing complex nonlinear relationships and enhancing prediction accuracy. The study reported that the GANN model achieved an R² of 0.96 on the test dataset, outperforming standard ANN (R² = 0.89) and multiple linear regression (R² = 0.72) approaches. Uncertainty analysis revealed that the most influential parameters were the solution pH and flow rate, consistent with the physical understanding presented in Section 2.1.

Flores and Leiva [85] demonstrated that ANNs can accurately predict fine-scale features, such as solution flow trajectories and seepage velocity fields in the ore heap, using real data from a copper mine in northern Chile. Their results provide robust support for optimizing the heap leaching process. The study emphasized the importance of validation against independent field measurements, with reported errors in velocity predictions ranging from 8% to 15% depending on spatial location within the heap.

In addition to laboratory and column data, machine learning has also been applied directly to industrial bioleaching datasets. Demergasso et al. [86] developed a decision support system for the heap bioleaching process at Minera Escondida using industrial records that included mineralogical and chemical descriptors, operating variables, metallurgical performance indicators, acid consumption, temperature, and microbial information obtained through real-time PCR. Their approach used data-mining and machine learning tools to build rules and recommendations associated with copper recovery, oxidation test behavior, and bacterial activity. This example is relevant to the present review because it shows that microbial community composition, abundance, and functional activity can be incorporated as predictive variables rather treated as only qualitative background information.

On the subject of integrating physics-based and data-driven models, a promising recent development is the hybridization of physics-based models (FEA/CFD/FVM) with data-driven models (ANN). These hybrid approaches—sometimes termed physics-informed neural networks (PINNs)—leverage the strengths of both paradigms, combining the physical consistency and extrapolation capability of mechanistic models with the pattern recognition and fast prediction capabilities of ANNs. For example, a CFD model can generate synthetic training data for an ANN surrogate model, which can then be used for real-time process optimization. Conversely, experimental data can be used to calibrate uncertain parameters in a physics-based model through ANN-assisted inverse modeling. This integration directly addresses the limitation of purely data-driven models (limited extrapolation) and pure physics-based models (computational expense).

Regarding physical parameters (Section 1) and models (Section 2), the input variables for ANN models in heap leaching typically include the physical parameters discussed in Section 1, such as the temperature, pH, redox potential, particle size, and solution flow rate. The target outputs often include metal recovery, leaching rate constants, or final copper extraction. The mathematical relationships learned by the ANN should ideally be consistent with the kinetic models presented in Section 2 (e.g., Monod kinetics, shrinking core model). Discrepancies between ANN predictions and mechanistic model outputs can help identify missing physics or incorrect assumptions in the mechanistic framework, providing a valuable tool for model refinement.

5. Suggestions and Reflections on Heap Leaching Simulation

Amid the ongoing decline in copper sulfide ore grades, heap bioleaching technology has emerged as a pivotal direction in hydrometallurgy, owing to its substantial environmental and economic benefits. Heap bioleaching constitutes a complex integrated process, encompassing chemical reactions such as acid dissolution, Fe³⁺ oxidation of sulfide minerals, and oxidation reactions involving dissolved oxygen, along with electrochemical processes; microbial processes including adsorption, growth, and catalysis; mass transfer processes comprising oxygen transport in the ore heap via air diffusion and natural convection, as well as diffusion of aqueous species within ore particles; and heat transfer processes arising from endothermic and exothermic reactions, leading to heating or cooling of the ore heap and environmental heat exchange. The overall process involves numerous variables, presenting substantial challenges for mathematical simulation and automated control.

Employing COMSOL for simulating the entire bio-heap leaching process allows the ore heap bottom to be assumed as rectangular, with cross-sections perpendicular to the long edge forming isosceles trapezoids, and all cross-sections being uniform. Based on these assumptions, the ore heap can be developed into an operable two-dimensional mathematical model. Utilizing the COMSOL Subsurface Flow module enables the calculation of four key variables at any point (X, Y) in the two-dimensional model: airflow velocity, gas-phase oxygen concentration, temperature, and leaching rate at a specified time. The four equations for the key variables are provided below.

• Leaching rate model for bacterial leaching (Cu, Ni, Fe, etc.):

$${\displaystyle \frac{d\alpha }{dt}}={\displaystyle \frac{\beta }{{\rho }_b{G}_0}}X{V}_M{\displaystyle \frac{C_I}{K_M+C_I}}$$

(12)

2.

The flow velocity of gas through the ore heap can be expressed using Darcy’s law:

$${\epsilon }_g{\upsilon }_g=q_g=-{\displaystyle \frac{K{K}_{rg}}{{\mu }_g}}\left(\nabla P-{\rho }_{g}g\right)$$

(13)

3.

Mass balance for gaseous oxygen:

$${\epsilon }_g{D}_g\left({\displaystyle \frac{{\partial }^2{C}_g}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{C}_g}{\partial y^2}}\right)-{\epsilon }_g\left({\displaystyle \frac{\partial \left({\upsilon }_g{C}_g\right)}{\partial x}}+{\displaystyle \frac{\partial \left({\upsilon }_g{C}_g\right)}{\partial y}}\right)={\displaystyle \frac{{\rho }_B{G}^{\mathsf{\Theta }}}{{\sigma }_1}}{\displaystyle \frac{d\alpha }{dt}}$$

(14)

4.

Heat balance:

$$K_B\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)-{\displaystyle \frac{\partial h_1}{\partial y}}-\left({\displaystyle \frac{\partial h_g}{\partial x}}+{\displaystyle \frac{\partial h_g}{\partial y}}\right)=\left(-\Delta H_R\right){\displaystyle \frac{{\rho }_B{G}^{\mathsf{\Theta }}}{{\sigma }_1}}{\displaystyle \frac{d\alpha }{dt}}$$

(15)

The aforementioned equations are transformed into a dimensionless system and discretized using the finite difference method, yielding a set of nonlinear algebraic equations. This system is solved via the successive relaxation iteration method. First, the gas density distribution is assumed to obtain estimates of the airflow function and air flow velocity through the ore bed. Then, the distributions of oxygen concentration and temperature are solved for. Finally, the obtained oxygen concentration and temperature values are used to compute gas density values, iteratively solving for the airflow function and air velocity. This process is repeated until convergence is achieved.

6. Future Perspectives and Challenges

With the annual decline in copper sulfide ore grades, heap bioleaching has emerged as an important direction in hydrometallurgy owing to its environmental and economic benefits [1,2,3,10]. However, traditional heap leaching control strategies rely primarily on single-factor optimization, which hinders a comprehensive assessment of multi-factor interactions [2,8]. Key parameters—including ore particle size distribution, pore structure, solution pH, temperature, microbial activity, and oxygen transfer efficiency—collectively influence the leaching process [2,8,13,15,17]. The complex interrelationships among these factors render traditional methods inadequate for regulating leaching rates, maintaining microbial activity, and controlling passivation layer formation. Therefore, advanced control methods, such as multi-parameter optimization through numerical simulation, industrial data assimilation, and intelligent systems, are needed to enable more reliable prediction and dynamic regulation of heap bioleaching [8,86].

Based on a comparative analysis of modeling approaches, several conclusions can be drawn regarding the current state of knowledge. First, while the key controlled variables (temperature, redox potential, pH, and oxygen concentration) and manipulated variables (aeration rate, spray intensity, and additive dosage) have been clearly identified, the empirical relationships linking these variables to operational objectives have been established from a mixture of laboratory reactors, column studies, pilot heaps, and selected industrial datasets [2,8,12,15,16,86]. Therefore, the scale and origin of data must be stated explicitly when parameters are transferred into numerical models. For example, the airflow-to-spray ratio of 2:3 reported by Li Hongxu et al. [15] is an experimental simulation result for a specific ore and test configuration rather than a universal design rule for all industrial heaps. Similarly, redox potential regulation via pyrite addition has demonstrated significant improvements in chalcopyrite bioleaching under controlled experimental conditions [57,58], but field-scale performance depends on ore mineralogy, permeability, aeration, and microbial community dynamics. Second, physics-based models such as finite element analysis, computational fluid dynamics, and finite volume methods can simulate pore- and column-scale phenomena, including fluid flow, heat transfer, mass transport and chemical reactions, when validated against experimental data [18,19,78,79]. Third, data-driven models, particularly artificial neural networks and decision-support systems, can capture nonlinear relationships in large operational datasets, but their reliability depends on representative training data and validation against independent field measurements [84,85,86]. Fourth, microbial community dynamics follow temperature- and chemistry-dependent succession patterns, and industrial heap studies show that such microbial information should be included in predictive models where data are available [16,86].

Despite significant progress, several critical gaps persist between current modeling capabilities and industrial requirements. The first gap is the disconnect between laboratory-scale kinetics and industrial-scale performance: the vast majority of kinetic parameters are derived from well-mixed laboratory reactors under controlled conditions but are routinely applied to heterogeneous heap environments without rigorous validation. Further, the extrapolation error from column tests (meters) to industrial heaps (decameters) remains poorly quantified. The second gap is the inadequate representation of spatial heterogeneity: most existing models assume homogeneous or simplified heterogeneous properties, yet real heaps exhibit multi-scale heterogeneity from particle-scale composition variations to heap-scale temperature gradients and compaction zones, and current models cannot reliably predict the formation and evolution of preferential flow paths that dramatically affect leaching efficiency. The third gap is the limited integration of microbial community dynamics into transport–reaction models: microbial kinetics are often represented by Monod-type equations, but these formulations do not capture community succession, interspecies interactions, or adaptive responses to environmental stressors such as temperature spikes and metal toxicity, and the functional redundancy within microbial consortia—where multiple species can perform the same biochemical function—is rarely considered, despite its significant implications for process robustness. The fourth gap is the absence of standardized validation protocols and uncertainty quantification: there is no established framework for validating heap leaching models against industrial data, and validation metrics, acceptable error thresholds, and uncertainty propagation methods vary widely across studies, preventing meaningful comparison between different modeling approaches and hindering regulatory acceptance of simulation results for process design. The fifth gap is that real-time control and digital twin implementation remain aspirational: despite frequent mentions of digital twins in the literature, no fully functional digital twin has been developed for industrial heap bioleaching, with major barriers including the lack of reliable in situ sensors for key variables such as microbial activity and passivation layer thickness, the computational expense of multi-physics models precluding real-time simulation, and the absence of robust data assimilation frameworks to update model states from measurements.

With this review, we make three distinctive contributions to the field of heap bioleaching modeling. First, we establish a formal process control framework that explicitly distinguishes between controlled variables, manipulated variables, and operational objectives, providing a structured basis for control system design and enabling direct comparison between different modeling approaches in terms of their suitability for control applications. Second, we provide a critical comparison of modeling approaches—finite element analysis, computational fluid dynamics, finite volume method, and artificial neural networks—systematically evaluating each method based on its advantages, limitations, and typical applications, thereby offering actionable guidance for model selection and clearly articulating the distinction between physics-based models, data-driven models, and emerging hybrid approaches. Third, it we identify five specific knowledge gaps and propose prioritized research directions with associated time horizons and feasibility assessments, moving beyond generic recommendations to provide a concrete roadmap for future research.

Based on the gap analysis described above, prioritizing certain research directions is recommended. The first direction, physics–machine learning integration, requires exploiting the complementary strengths of physics-based and data-driven models through hybrid approaches. Physics-informed neural networks should be developed that incorporate mass and energy conservation laws into the loss function, ensuring physical consistency while leveraging data for parameter calibration; these can be used to solve inverse problems to estimate spatially varying parameters from sparse measurements, a task that is computationally prohibitive with traditional methods. Surrogate models trained on CFD or FVM simulations can reduce the computation time from hours to milliseconds, enabling real-time prediction for control applications. Transfer learning should be explored to leverage laboratory-scale data for industrial-scale predictions to directly address the extrapolation challenge. The second direction, real-time control strategies, requires moving beyond single-factor optimization to integrated multi-variable control. Model predictive control frameworks should use reduced-order models to optimize manipulated variables in real time based on measured or inferred controlled variables, with control objectives explicitly including both recovery maximization and cost minimization parameters, such as energy consumption and reagent usage. Low-cost, robust sensors must be developed for key variables for which reliable measurement technologies are currently lacking, with priority targets including indicators of in situ microbial activity, passivation layer thickness, and solution flow distribution. Data assimilation frameworks such as ensemble Kalman filters should continuously update model states from sensor measurements, enabling adaptive control that responds to unforeseen disturbances like weather events or ore variability. The third direction, industrial scaling criteria and validation standards, requires bridging the gap between academic research and industrial adoption through standardized validation protocols that include minimum data reporting requirements for model validation studies, accepted error metrics and tolerance bands such as RMSE below 10% for copper recovery at pilot scale, as well as guidelines for uncertainty quantification and propagation. Dimensionless scaling parameters should be developed to characterize the transition from laboratory to industrial behavior—including Biot numbers for heat transfer, Damköhler numbers for reaction-transport regimes, and Péclet numbers for dispersion—with regime maps based on these parameters predicting when laboratory-scale kinetics are directly scalable versus when meso-scale phenomena such as channeling predominate. Benchmark datasets from industrial operations should be published, with anonymity where needed, to enable model intercomparison through a community challenge approach similar to those in climate modeling or fluid dynamics. The fourth direction, multi-scale modeling integration, requires addressing spatial heterogeneity through hierarchical modeling frameworks that couple pore-scale simulations of fluid–mineral–microbe interactions to continuum-scale heap models using homogenization techniques such as volume averaging or asymptotic expansion for mathematical rigor, combined with stochastic representations of heterogeneity like random fields for porosity and permeability with uncertainty quantification, where ensemble simulations provide probabilistic predictions of recovery rather than deterministic point estimates to better inform decision-making under uncertainty.

In conclusion, heap bioleaching of low-grade copper sulfide ores is a mature technology at the intersection of hydrometallurgy, microbiology, and transport phenomena. Numerical simulation has advanced from descriptive modeling to prediction at the laboratory and pilot scales; however, the incorporation of industrial-scale digital twins and realization of real-time control remain incomplete. The path forward requires integrated approaches that combine the physical rigor of conservation laws with the pattern-recognition capabilities of machine learning, validated against industrial data using standardized protocols [8,86]. Although numerical simulation techniques have mitigated some shortcomings of traditional experimental methods, significant challenges remain in practical industrial applications, including the complexity of bioreaction kinetics, the spatial heterogeneity of the heap leaching system, the coupling mechanisms of reaction–transport processes, and data acquisition and processing challenges. By addressing the gaps identified in this review—particularly the disconnect between laboratory kinetics and heap performance, the inadequate representation of spatial heterogeneity, the limited incorporation of microbial community composition and activity, and the absence of real-time control frameworks—the field can move toward intelligent, efficient, and sustainable bioleaching for the strategic recovery of mineral resources. The integration of artificial neural networks for analyzing metabolic composition and community regulation, the development of kinetic models for multi-phase interfacial reactions, the implementation of multi-dimensional data acquisition systems, and the eventual establishment of digital-twin platforms will collectively support intelligent decision-support systems for low-grade copper ore development.