Impact of pyrite microscopic features on shale mechanical properties: a machine learning and simulation study

Abstract

Multi-scale mechanical research has become an effective approach for analyzing the behavior of complex materials, enabling the prediction of macroscopic properties from microscale characteristics. Pyrite, a key mineral affecting the mechanical properties of shale, has attracted considerable attention due to its brittleness and distinctive microstructure. However, the specific effects of pyrite’s microscopic features on shale mechanics have not been systematically investigated. In this study, Python-based machine learning techniques are employed to establish an automated framework for quantitatively analyzing pyrite’s microscopic characteristics. The influence of pyrite content, particle size, shape factor, and distribution on shale mechanical properties and fracture network formation is examined using a random forest regression algorithm and numerical simulations. The results show that pyrite content exerts the strongest influence on shale mechanics, followed by shape factor, while particle size and distribution mode have comparatively weaker effects. Compressive strength and fracture behavior are particularly sensitive to elevated pyrite content and shape factor. All four factors (pyrite content, particle size, shape factor, and particle distribution) affect mechanical properties through stress concentration, with shape factor additionally governing particle interlocking. Distribution mode further modulates the mechanical response by influencing the formation of force chain networks. Numerical simulations reveal that fracture network development is optimized when pyrite content is 3.00% and the shape factor is 0.81, thereby enhancing the fracturing effect. This study provides theoretical support for hydraulic fracturing in shale reservoirs and introduces a novel perspective on the role of pyrite’s microscopic characteristics in governing shale mechanical behavior.

Graphical abstract

Keywords

pyrite microstructure / machine learning / numerical simulations / mechanical properties / fracture networks

1 Introduction

Unconventional natural gas resources, including coalbed methane, shale gas, and tight sandstone gas, are widely distributed across the globe. Achieving their commercial development requires large-scale reservoir modification. The effectiveness of such development is strongly dependent on the mechanical properties of the reservoirs, such as Young’s modulus and Poisson’s ratio (Li et al., 2021; Liao et al., 2024). The factors influencing these mechanical properties can be broadly categorized into macroscopic and microscopic factors. Macroscopic factors primarily include mineral composition, temperature, pressure, and other reservoir-scale characteristics. Microscopic factors, in contrast, mainly involve the development of pore and fracture systems, the storage characteristics of rock components, and other features at the microstructural scale (Li et al., 2018a; Yu et al., 2022; Liu et al., 2023a; Li et al., 2024).

Shale is composed of diverse inorganic minerals, including quartz, feldspar, calcite, dolomite, pyrite, and clay minerals, as well as organic matter and pore space (Ougier-Simonin et al., 2016; Saif et al., 2017a). It is widely recognized that brittle minerals such as quartz and feldspar increase Young’s modulus and reduce Poisson’s ratio, thereby promoting fracturing. In contrast, plastic minerals such as clays decrease Young’s modulus and increase Poisson’s ratio, making fracturing less favorable. Ahmed et al. (2019) investigated the role of montmorillonite swelling in shale fracturing instability by integrating LST (linear swelling test), pore pressure testing, SEM (scanning electron microscope), and XRD (X-ray diffraction). Liu et al. (2020) applied triaxial compression tests and XRD mineral analysis to examine the influence of mineral composition on the mechanical properties and fracturing behavior of organic-rich shales. More recently, Lin et al. (2024) explored the relationship between fracture propagation during stimulation and the distribution of shale minerals, organic matter, and pores through hydraulic fracturing experiments combined with SEM and FIB-SEM (focused ion beam scanning electron microscope) analyses.

From a macroscopic perspective, studies on the mechanical properties of unconventional reservoirs often neglect their inherent heterogeneity, typically assuming the rock to be homogeneous, continuous, and isotropic. In reality, the microstructures of unconventional oil and gas reservoirs are highly heterogeneous. It is therefore essential to investigate the evolution of mechanical properties and behavior at the microscopic scale. Taking shale as an example, quartz is one of its most significant mineral constituents and forms through multiple processes. Quartz produced by different formation pathways exhibits distinct microstructural characteristics, which in turn affect the mechanical properties of shale. Ye et al. (2023) demonstrated, through finite element simulations and scanning electron microscopy, that quartz content, grain size, and distribution strongly influence shale mechanics, with authigenic microcrystalline quartz notably enhancing compressive strength and brittleness. Liang et al. (2023) applied molecular dynamics simulations to examine the micromechanical behavior of quartz and found that its high brittleness and elasticity substantially improve shale’s fracture formation capacity, thereby optimizing hydraulic fracturing. Zhang et al. (2024), using X-ray diffraction, scanning electron microscopy, and elemental analyses, showed that increasing biogenic quartz enhances shale brittleness and fracturing, whereas higher proportions of detrital and authigenic quartz reduce brittleness and hinder fracture development. Collectively, these studies highlight that mineralogical composition at the microscale exerts a critical influence on shale’s mechanical behavior and fracturing potential.

Although quartz is typically the most abundant mineral in shale and has therefore been extensively studied, other minerals present in smaller proportions—such as pyrite—can also exert a significant influence on reservoir properties. Pyrite is widely distributed in unconventional oil and gas reservoirs and forms through multiple processes. However, detailed investigations into its complex microstructural characteristics and their effects on reservoir mechanical behavior remain limited. Despite its relatively low abundance in shale, pyrite, with its high Young’s modulus and low Poisson’s ratio, has been shown to strongly influence shale brittleness (Li, 2022). Moreover, the role of microstructure in reservoir reformability becomes increasingly pronounced when mineral content is low (Li et al., 2022). Based on these insights, we hypothesize that the microstructural characteristics of pyrite—including content, particle size, shape factor, and distribution—have a significant and quantifiable impact on the mechanical properties and fracture behavior of shale. This study aims to systematically evaluate these effects and elucidate the underlying mechanisms.

Previous research has reported various influences of pyrite on shale mechanics. Wu et al. (2020) used digital rock modeling combined with the finite element method (FEM) and demonstrated that increasing pyrite content markedly enhances the elastic modulus of shale. He et al. (2022) investigated shales from the Cambrian Niutitang Formation using NMR (nuclear magnetic resonance), TOC (total organic carbon) measurements, and XRF (X-ray fluorescence spectroscopy) analyses, showing that higher pyrite content increases compressive strength while promoting crack initiation and propagation. Liu et al. (2023b) applied FIB-SEM imaging and 3D numerical simulations to analyze pyrite distribution, revealing that pyrite boundaries act as primary sites for fracture initiation and extension. Li et al. (2025b) examined the effect of pyrite on fracture behavior and anisotropic strength through uniaxial compression tests and CT (computed tomography) imaging, integrated with PFC (particle flow code) simulations, and found that high-density minerals such as pyrite can guide fracture propagation and generate rough fracture surfaces.

Unsurprisingly, numerical simulation has become an increasingly valuable tool for investigating rock fracture behavior (Mohammadnejad et al., 2021). However, many existing models rely on simplified or assumed geometric parameters for pyrite rather than those obtained from high-resolution imaging. Accurate quantification of mineral microstructures is essential for reliable numerical modeling. Machine learning algorithms provide a novel means of extracting information and performing analytical classification from microscopic images, thereby improving modeling accuracy and ensuring that results more closely reflect real reservoir conditions (Holm et al., 2020; Vranjes-Wessely et al., 2021). Wu et al. (2019) developed a machine-learning-based method for SEM image segmentation that efficiently identifies pores, organic matter, matrix, and pyrite in organic-rich shales. Davletshin et al. (2021) employed convolutional neural networks to automatically recognize framboidal pyrite in SEM images, revealing diameters predominantly ranging from 1.2 to 15 μm. Li et al. (2025a) applied deep learning segmentation to accurately identify 5% pyrite in shale, assign its mechanical properties in 2D FEM modeling, and predict the elastic modulus based on real mineral composition. While these studies demonstrate the potential of integrating image analysis with mechanical modeling, such approaches have rarely been implemented in a fully coupled manner. To date, few studies have systematically combined machine-learning-based SEM quantification with numerical simulation to examine how real-world geometric characteristics of pyrite affect shale’s mechanical behavior. Addressing this gap forms the foundation of the present study.

In this study, we focus on pyrite in shale and quantitatively characterize its microstructure using machine learning. After extracting microstructural parameters, including perimeter, area, minimum enclosing circle diameter, and centroid coordinates of pyrite particles, we construct a shale microstructural model with RFPA-2D software. This model is then used to investigate the mechanisms through which pyrite microstructure influences the mechanical properties of shale. The methods and findings presented herein provide a valuable reference for evaluating the mechanical behavior of similar unconventional reservoirs.

2 Development of a python framework for pyrite characterization in shale SEM images

Scanning electron microscopy (SEM) is one of the primary techniques for investigating the microstructure of minerals in rocks (Vos et al., 2014; Saif et al., 2017b). SEM images (Fig. 1) reveal notable variations in the content, grain size, shape, and distribution of pyrite within the shale samples. To evaluate the influence of pyrite’s microstructural characteristics on the mechanical properties of shale, it is necessary to quantify these parameters and construct a corresponding microscopic model of shale. The application of programming software substantially improves the speed, reproducibility, and flexibility of such analyses compared with manual parameter extraction and model construction (Campbell et al., 2018; Bangaru et al., 2022). Among available tools, Python has become one of the most widely used programming languages owing to its efficiency in data processing (Richert, 2013; Ketkar and Santana, 2017).

This study presents a framework for constructing a microscopic model of shale with varying pyrite characteristic factors using the NumPy, SciPy, and Matplotlib libraries in Python, following a control variables approach. The framework comprises four main modules: field-of-view determination, parameter extraction, parameter specification, and image rendering. The detailed workflow is illustrated in Fig. 2.

2.1 Field of view determination

For parameter extraction, variations in SEM image resolution may lead to misleading results. Therefore, determining the optimal representative elementary area (REA) through statistical analysis of images at different resolutions is a critical step for subsequent analysis (Peng et al., 2012). Following the method proposed by Medina et al. (2022) for evaluating porosity REA size, a set of SEM images enriched with pyrite was selected. From the standard SEM image (1.98 mm × 2.78 mm), sub-images of varying sizes and random positions were extracted. The sub-images were measured in pixels, with side lengths ranging from 150 to 500 pixels in increments of 50 pixels. For each sub-image, the percentage of area occupied by pyrite was calculated. The results show that pyrite content values gradually converge to the total pyrite content of the entire image as the sub-image area increases. Accordingly, an area of 0.1089 mm² is identified as the representative elementary area (REA) that best reflects pyrite enrichment in shale (Fig. 3).

2.2 Parameter extraction

This section outlines the extraction of four key parameters from SEM images: pyrite content, particle size, shape factor, and distribution mode. The shape factor is a dimensionless metric that quantifies how closely a pyrite particle’s shape approximates a perfect circle, thereby reflecting its degree of roundness or angularity. The distribution mode refers to the spatial dispersion of pyrite particles in shale, measured by the largest eigenvalue of the covariance matrix of particle centroids.

A total of 74 REA SEM images were processed and converted to grayscale using OpenCV (cv2). Histogram analysis indicated that the grayscale values corresponding to pyrite typically ranged between 200 and 255. Within this range, a series of threshold values was tested to assess segmentation quality, and a threshold of 220 consistently yielded the most effective visual separation of pyrite particles from the surrounding matrix (Fig. 4). To ensure consistency and reliability in morphological parameter extraction, this threshold was adopted for subsequent binarization.

External contours were then detected using the cv2.findContours function, and the area of each contour was calculated with cv2.contourArea. Regions smaller than 15 pixels were excluded to eliminate noise, ensuring that only valid pyrite particles were retained. Each identified particle was assigned a unique label, and its morphological parameters were extracted. Finally, all data were compiled and exported into an Excel file for further analysis.

1) Pyrite content

In this study, pyrite content was quantified as the ratio of the total area of segmented pyrite particles to the area of the original image. The extraction results indicate that pyrite content ranges from 0.47% to 4.85%.

2) Particle size

In this study, the diameter of the minimum enclosing circle for each labeled contour was calculated using the cv2.minEnclosingCircle function, representing the maximum particle size of individual pyrite particles. The results indicate that the maximum particle size of pyrite ranges from 4.21 μm to 57.07 μm.

3) Shape factor (Circ.)

Wadell (1933) first proposed the concept of roundness to describe particle angularity. In this study, the shape factor of pyrite, denoted as Circ., is calculated using the roundness formula (Eq. (1)):

(1)\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}.=4\text{π}\times \frac{Area}{{Perim}^2},

where Area represents the area of a single pyrite particle, calculated using cv2.contourArea, and Perim denotes the perimeter of a single pyrite particle, calculated using cv2.arcLength. The results indicate that the shape factor of pyrite ranges from 0.03 to 0.94.

4) Distribution mode

Parker and Asencio (2009) demonstrated that the degree of dispersion can be used to analyze the distribution of particles in a two-dimensional image. Snijders (1981) proposed that this dispersion can be characterized by variance. Drozyner (1981) further argued that the eigenvector corresponding to the largest eigenvalue of the covariance matrix indicates the direction of greatest variance. Accordingly, this study employs the maximum eigenvalue of the covariance matrix to characterize the distribution mode of pyrite in shale SEM images. Given a set of mutually independent pyrite data sets X_1,X_2,\dots ,X_n\in {\mathbb{R}}^m with sample size n, the covariance matrix \mathit{\Sigma } is expressed as Eq. (2), and its eigen decomposition is defined in Eq. (3) (Johnstone and Paul, 2018).

(2)\mathit{\Sigma }=n^{-1}\sum _{i=1}^n\left(X_I-\overline{X}\right){\left(X_I-\overline{X}\right)}^{\prime },

(3)\mathit{\Sigma }={\lambda }_1{\nu }_1{\nu }_1^{\prime }+\cdots +{\lambda }_m{\nu }_m{\nu }_m^{\prime }=V\mathrm{\Lambda }{V}^{\prime }.

The columns of the orthogonal matrix represent the eigenvectors, while the diagonal elements of the diagonal matrix \mathrm{\Lambda }correspond to the eigenvalues arranged in descending order. The coordinates of individual pyrite particles were calculated using cv2.moments. The results indicate that the distribution mode of pyrite ranges from 1952.61 to 593912.20.

2.3 Parameter determination

Cluster analysis in unsupervised learning is widely used to identify the underlying structure of data and reduce its complexity (Kao et al., 2008). Through this process, patterns can be revealed, similarities detected, and complex data sets efficiently grouped, thereby providing a clearer perspective for subsequent analysis. Among available methods, Gaussian mixture clustering is one of the most established approaches (Liu et al., 2021). It is based on the Gaussian probability density function and consists of multiple Gaussian distributions (Jia et al., 2019). In a DDD-dimensional space, a Gaussian mixture model with K components is defined by Eq. (4).

(4)\{\begin{array}{l}p\left(x|\text{Uptheta}\right)={\displaystyle \sum _{k=1}^K{\alpha }_k\text{Uppsi}\left(x|{\mu }_k,{\text{Upsigma}}_k\right)}\\ \text{Uppsi}\left(x|{\mu }_k,{\text{Upsigma}}_k\right)={\displaystyle \frac{1}{\left(2\text{π}\right){\displaystyle \frac{D}{2}}{\left|{\text{Upsigma}}_k\right|}^{\frac{1}{2}}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{1}{2}}{\left(x-{\mu }_k\right)}^{\mathrm{T}}\sum _k^{-1}\left(x-{\mu }_k\right)\right]\\ {\displaystyle \sum _k^K{\alpha }_k=1,{\alpha }_k\ge 0}\end{array},

where \text{Uptheta}={\left\{{\alpha }_k,{\mu }_k,{\text{Upsigma}}_k\right\}}_{k=1}^K represents the set of model parameters; x denotes the individual elements in the data set; K is the number of clusters; {\alpha }_k is the weight of the kth Gaussian component; and {\mu }_kand {\text{Upsigma}}_k are the mean vector and covariance matrix of the kth Gaussian component, respectively. The maximum likelihood estimation method, known as the EM algorithm (Qiu et al., 2019), is typically used to obtain the optimal parameters Θ of the Gaussian mixture model (GMM) by iteratively performing the expectation step (E-step) and maximization step (M-step) until convergence.

The Bayesian Information Criterion (BIC) is commonly applied to determine the number of clusters in Gaussian mixture models (Yang et al., 2019). It is based on selecting the lowest point on the BIC curve with respect to the number of clusters as the optimal cluster number (Gogebakan, 2021). In this study, clustering of each pyrite parameter was performed automatically using the Gaussian Mixture function in Python (Fig. 5). The clustering analysis classified pyrite content into 0.47%, 1.84%, 2.42%, 3.00%, and 4.85%; particle size into 4.21 μm, 9.50 μm, 18.38 μm, 32.16 μm, and 57.07 μm; shape factor into 0.20, 0.40, 0.61, and 0.81; and distribution mode into 38232.1, 93464.9, 166027.0, 267707.3, and 383454.3.

2.4 Image rendering

The adaptive quadtree is a recursion-based spatial decomposition technique widely applied in image segmentation, data indexing, and collision detection (Huo et al., 2019; Yuan et al., 2019). In this study, quadtrees were employed for mesh subdivision during image plotting to enhance model generation efficiency and reduce computational costs. Specifically, four plotting modules—triangle, rhombus, right-angled trapezoid, and ellipse—were created using the Matplotlib library, with each module corresponding to progressively increasing values of the shape factor. The centroids of pyrite particles in each model were generated randomly, with a minimum distance of 5 pixels between pyrite particles and the edges of the representative elementary area (REA) to prevent distortion during meshing. Additionally, a minimum interparticle distance of 10 pixels was maintained to avoid overlap. The adaptive quadtree technique was then applied to recursively subdivide the mesh, minimizing the number of generated cells while ensuring model accuracy. This approach significantly reduced both computational cost and memory consumption. The adaptive subdivision method enabled a more accurate representation of pyrite morphology and distribution in the microscopic model while improving computational efficiency (Liang and Borthwick, 2009). Based on the control variables method, 19 model groups were finally generated using the parameter values of the factors identified above (Fig. 6).

3 Numerical simulation of shale mechanical properties using HFGMC and RFPA

The High Fidelity Generalized Method of Cells (HFGMC) is an algorithm developed from the Method of Cells (MOC) and the Generalized Method of Cells (GMC). It can accurately simulate both micro-level stress and strain fields and the macro-level constitutive response of multiphase composites subjected to multiaxial loading (Aboudi, 1989; Paley and Aboudi, 1992; Aboudi et al., 2002). The method is well suited for modeling material responses under complex loading conditions, based on the assumption that repetitively distributed sub-cells, regardless of their arrangement, conform to the framework of the homogenization method (Fig. 7; Aboudi, 2004).

In this study, numerical simulations were performed using microscopic models of shale with varying pyrite characteristics. The simulations employed both the High Fidelity Generalized Method of Cells (HFGMC) and Rock Failure Process Analysis (RFPA) (Fig. 8). During parameter assignment, all finite elements corresponding to pyrite were treated as a unified entity, while the remaining components were considered as the substrate. Assuming that the mechanical properties follow a Weibull distribution, the relationship between macro- and micro-parameters was established using the probability density function (PDF) (Eq. (5)), which was then used to determine the micro-level values (Zhu and Tang, 2004; Li et al., 2022).

(5)f\left(u\right)=\frac{m}{u_0}{\left(\frac{u}{u_0}\right)}^{m-1}\mathrm{e}\mathrm{x}\mathrm{p}{\left(-\frac{u}{u_0}\right)}^m.

In this context, u represents the mechanical parameters, such as Young’s modulus and compressive strength. The parameter m denotes material homogeneity, where larger values of m correspond to greater homogeneity, with m\ge 1.

The mechanical parameters for the shale matrix and pyrite phases were determined at the outset of the modeling process. The model was first assumed to be homogeneous for micromechanical uniaxial compression simulations, and the resulting values were converted to macroscopic mechanical parameters using Eq. (5). These parameters were then iteratively refined by comparing them with empirical values reported in the literature, employing a trial-and-error approach until the simulated results aligned with the characteristic properties of shale and pyrite. Specifically, the homogeneity coefficient of pyrite was set to 15, the microscopic Young’s modulus to 3 × 10⁵ MPa, and the uniaxial compressive strength to 250 MPa. For the shale matrix, the homogeneity coefficient was set to 3, the microscopic Young’s modulus to 6.3 × 10⁴ MPa, and the uniaxial compressive strength to 194 MPa. This process ensured that all assigned values were both reasonable and representative for the numerical simulations.

4 Results and discussion

The mechanical parameters were obtained primarily from the stress–strain curve (Fig. 9), which captures the complete damage process of the rock and characterizes its mechanical properties before and after failure (Rahimzadeh Kivi et al., 2018; Li, 2022).

In the initial stage, the presence of pre-existing cracks produces a nonlinear response in the stress–strain curve. As microcracks close, the material enters the elastic phase, represented by a constant slope corresponding to Young’s modulus. At the end of this linear segment, the curve begins to bulge, marking the onset of the plastic strain phase, which extends until the peak stress at point D, corresponding to compressive strength. At peak stress, the rock ruptures, and the stress decreases at a rate defined by modulus M, eventually stabilizing at the residual stress level F (Eberhardt et al., 1999; Hou et al., 2019; Cheng et al., 2024). Previous studies have shown that higher compressive strength, Young’s modulus, and drop modulus are associated with increased brittleness (Chen et al., 2019). In this study, natural fractures are not considered, and the shale model stress–strain curve begins at segment A–B. In rock mechanics, compressive strength is not only a key parameter for evaluating mechanical behavior but also an important indicator for predicting fracture initiation (Huang et al., 2012). Accordingly, this paper focuses on the mechanisms by which pyrite parameters affect variations in shale mechanical properties, with particular emphasis on compressive strength.

4.1 Single-factor analysis of pyrite effects on shale mechanical properties

4.1.1 Pyrite content

Research has shown that shale compressive strength first increases and then decreases with rising pyrite content (Fig. 10). In composite materials mechanics, it is well established that higher stress concentration reduces compressive strength (Khechai et al., 2018; Liu et al., 2018). Accordingly, the effect of pyrite content on compressive strength can be divided into two distinct phases.

In the initial phase, when pyrite content is below 2.42%, stress within the material primarily concentrates around individual particles. These localized stresses substantially exceed the average stress level, thereby facilitating the initiation of damage and cracks. As pyrite content increases, the particles become more uniformly distributed within the matrix, reducing stress concentration and enhancing compressive strength. This behavior is consistent with established findings in materials mechanics, which emphasize that a uniform particle distribution significantly improves the strength of composite materials (Lauke, 2008).

In the second phase, as pyrite content increases from 2.42% to 4.85%, the spacing between particles gradually decreases. When the distance between adjacent particles falls below twice the particle diameter (2d), the stress distribution zones begin to overlap, markedly intensifying stress concentration and thereby reducing compressive strength. Sun et al. (2005) demonstrated that the stress concentration factor rises significantly once particle spacing is less than 2d. Under the assumption of a periodic particle arrangement within the matrix, a spacing of 2d corresponds to an elliptical particle content of approximately 2.8%, at which point compressive strength reaches its maximum. Thus, peak compressive strength occurs when pyrite content ranges from 2.42% to 3%, beyond which compressive strength declines with further increases in pyrite content.

Analysis of the stress–strain curves reveals that the variations in Young’s modulus, drop modulus, and Poisson’s ratio closely parallel the trends in compressive strength. The maximum Young’s modulus is observed at a pyrite content of 1.84%. Furthermore, within the range of 1.84% to 2.42% pyrite content, the changes in these parameters remain relatively minor.

4.1.2 Particle size

Research has shown that the mechanical properties of the shale model decrease with increasing pyrite particle size, followed by stabilization (Fig. 11). Larger particle sizes result in more uneven stress distribution within the material, particularly at particle–matrix interfaces, where pronounced stress concentrations promote localized failure, crack initiation, and propagation, thereby reducing compressive strength. This observation is consistent with previous studies (Huang and Li, 2005; Zhong et al., 2014; Liu et al., 2018) and conventional statistical theory (TST), which indicates that the uniaxial compressive strength of rock materials decreases exponentially with increasing grain size (Wang et al., 2019). In addition, acoustic emission (AE) technology effectively monitors strain energy release during rock rupture. Elastic waves generated under stress loading, resulting from microcrack initiation and propagation, provide insight into the material’s damage process (Saeedifar and Zarouchas, 2020; Luo et al., 2022). AE monitoring of five shale models revealed a slight increase of 3.31% in signal count as particle size increased from 18.38 μm to 57.07 μm. This indicates that, within this range, the number of microcracks remains relatively constant and compressive strength stabilizes. Thus, beyond an optimal particle size, further increases exert minimal influence on microcrack development, and compressive strength remains unchanged despite persistent stress concentrations (Lauke, 2008).

4.1.3 Shape factor (Circ.)

Research has shown that shale compressive strength decreases initially and then increases with rising shape factor (Fig. 12). This behavior results from the combined influences of stress concentration and mechanical embedding (Hatefi et al., 2024). In composite materials, irregularly shaped particles typically sustain higher loads than spherical ones, as particle corners induce more pronounced stress concentrations (Liu et al., 2018). The embedding effect is commonly explained by frictional resistance within the Mohr–Coulomb strength framework, which posits that the overall material strength improves by restricting particle movements such as misalignment, slippage, and rotation (Liu et al., 2023c).

For triangular particles (Circ. = 0.2), stress concentration primarily occurs at the sharp corners, where interparticle stresses connect, thereby reducing the overall stress concentration. The pointed tips and shorter bottom edges of these particles provide greater resistance, restricting rotation and movement, which in turn enhances the shale’s compressive strength. By contrast, rhombic particles (Circ. = 0.4) concentrate stresses at their acute corners. Owing to their geometric symmetry, they are more prone to rotation and sliding, resulting in lower compressive strength. Right-angled trapezoidal particles (Circ. = 0.61) exhibit stress concentration at the right angles. Their asymmetry limits particle mobility, thereby improving compressive strength compared with symmetric shapes (Prashant et al., 2020). Elliptical particles (Circ. = 0.81), lacking sharp edges, experience reduced stress concentration, which enhances compressive strength. However, their smooth geometry facilitates rotation under compression, partially compromising the material’s overall stability (Zhou et al., 2021; Ali et al., 2023). Furthermore, Young’s modulus exhibits a trend consistent with compressive strength, whereas Poisson’s ratio remains largely stable, indicating that the elastic deformation properties are relatively unaffected by particle shape.

4.1.4 Distribution mode

Research indicates that as the distribution of pyrite particles in the shale model becomes more dispersed, the material’s mechanical properties exhibit only minor fluctuations (Fig. 13). Force chains, defined as linear or quasi-linear arrays of particles formed through contact interactions involving normal and shear forces, play a key role in load transmission within particulate systems. Structurally, these chains align with the principal stress direction and are laterally supported by the surrounding matrix. Such an arrangement stabilizes the overall structure and markedly enhances the material’s resistance to deformation (Tordesillas and Muthuswamy, 2009; Shi et al., 2024).

Data analysis shows that the material exhibits higher compressive strength when the distribution mode is below 93464.9, corresponding to a dense particle arrangement with an average spacing of approximately 90 μm. Under these conditions, shorter force chains, together with the surrounding matrix, provide sufficient lateral support to pyrite particles, thereby forming a stable force-chain network that enhances compressive strength. In contrast, as the distribution mode increases to 166027 and 267707.3, compressive strength displays a downward trend. For elliptical particles with an aspect ratio of approximately 2.0, the critical force chain length was found to span 9–10 particles, equivalent to about 164.6 μm in this model (Guo, 2012).

As the distribution mode increases, the frequency of particle spacings exceeding the critical length becomes higher. Under these conditions, the surrounding matrix is insufficient to provide adequate support for the elongated force chains formed by the dispersed particles. This inadequacy causes the force chains to fracture under stress, leading to the destabilization of the network and a reduction in compressive strength (McBeck et al., 2019; Jiang and Liu, 2023). When the distribution mode reaches 383454.3, however, the generally discrete particle arrangement occasionally gives rise to localized dense clusters. These clusters facilitate the formation of stable force-chain networks, thereby producing a modest increase in the overall compressive strength.

4.2 Multifactorial analysis of shale properties using random forest regression

4.2.1 Assessing pyrite’s impact on shale mechanics using random forest regression

Random Forest is an ensemble learning algorithm built on the bagging principle, designed to improve predictive accuracy by incorporating stochasticity into the training of individual decision trees (Breiman, 2001; Liaw and Wiener, 2002). In this study, the microstructural properties of pyrite are expressed by the data matrix X_i\left(k\right), while the mechanical parameters of shale—particularly compressive strength—are represented by the reference matrix Y_j\left(k\right). The Random Forest algorithm generates M regression trees, each constructed by sampling subsets of the training data set and randomly selecting features at each node. The final model prediction is obtained by averaging the outputs of all regression trees, thereby reducing variance and enhancing robustness (Rodriguez-Galiano et al., 2015; Wang et al., 2016; Li et al., 2018b).

In situations where the data set is limited, out-of-bag (OOB) samples are employed for internal cross-validation. Typically, two-thirds of the data are randomly selected to train each tree, while the remaining one-third are used to evaluate model performance (Svetnik et al., 2003). Model accuracy was assessed using the Root Mean Square Error (RMSE) as the primary metric (Eq. (6)). As RMSE approaches zero, the model’s predictions become increasingly consistent with the actual values, reflecting higher regression accuracy (Li et al., 2018b). However, due to the limited data set of 74 SEM images, external validation could not be performed. Instead, OOB samples provided an effective means of internal validation. This limitation is acknowledged, and future work will focus on acquiring additional data to enable external validation and further evaluation of model generalizability:

(6)\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}{n}}},

where n denotes the number of samples; y_i represents the actual value; \widehat{y_i} the predicted value; and \overline{y_i} the mean of the response variable.

In this study, Python was employed to develop a random forest regression model, with hyperparameters optimized via RandomizedSearchCV to minimize the Root Mean Square Error (RMSE). The optimal configuration included 399 trees, a maximum depth of 20, a minimum of three samples required for node splitting, at least one sample per leaf, a minimum splitting gain of 0.01, no feature sub-sampling, and a full sample ratio of 1.0. Based on this model, the influence of each pyrite characteristic was evaluated using the weighted product of splitting gain and the number of test samples. The results reveal that pyrite content exerts the strongest influence on shale compressive strength, followed by the shape factor, while particle size and distribution mode have comparatively weaker effects (Fig. 14). These findings are consistent with the stress concentration mechanisms discussed in Section 4.1: variations in pyrite content redistribute stress within the rock matrix, thereby controlling fracture initiation, whereas the shape factor governs stress concentration and transmission around individual particles, predominantly influencing fracture propagation. Collectively, pyrite content and shape factor jointly regulate the formation and complexity of fracture networks in shale.

Model performance was assessed using %RMSE, defined as the ratio of RMSE to the mean of the observed values, following the criteria proposed by Li et al. (2013). The training RMSE was 3.34 (%RMSE = 8%), reflecting excellent model performance, whereas the test RMSE was 6.32 (%RMSE = 16%), indicating good predictive capability. Collectively, these results demonstrate that the model exhibits both robustness and reliability.

4.2.2 Combined effects of pyrite content and shape factor on shale behavior

Integrating the results of the weighting and single-factor analyses, 12 models with varying pyrite content and particle shapes were constructed and subjected to RFPA-based uniaxial compression simulations. The corresponding variations in mechanical parameters are presented in Fig. 15.

An increase in pyrite content exerts a pronounced influence on the mechanical properties of shale, and this effect varies with particle shape factor. At a shape factor of 0.20, the compressive strength decreases by approximately 3.95% as pyrite content rises from 0.47% to 1.84%, and by about 8.04% when the content reaches 3.00%. By contrast, at a shape factor of 0.81, compressive strength decreases by roughly 4.21% between 0.47% and 1.84% pyrite content, and by as much as 26.96% at 3.00%. These results demonstrate that the magnitude of strength reduction is substantially amplified at higher shape factors, indicating that shale mechanical properties are increasingly sensitive to pyrite content variations as particle shape approaches more elongated geometries.

An increase in shape factor, under varying pyrite contents, generally results in a reduction of shale mechanical properties. For example, at a pyrite content of 0.47%, Young’s modulus decreases by 5.07% as the roundness increases from 0.20 to 0.40. This reduction continues with an additional 2.07% decrease at a roundness of 0.61, and a further 2.62% decrease at 0.81. At a higher pyrite content of 3.00%, the decline in Young’s modulus becomes more pronounced, decreasing by 26.34% as roundness rises from 0.20 to 0.40, followed by an 11.29% decrease at 0.61, and a further 15.33% decrease at 0.81. These results demonstrate that the influence of shape factor is strongly amplified at elevated pyrite contents, indicating that shale mechanical properties are increasingly sensitive to particle geometry under higher mineral concentrations.

Nonetheless, variations in Poisson’s ratio remain minimal across different shape factors and pyrite contents, suggesting that the balance between lateral and axial strains in shale is relatively stable. In contrast, the elastic modulus exhibits a more complex response to these factors, which can be attributed to stress redistribution under varying structural conditions. Taken together, the results demonstrate that both pyrite content and shape factor exert a significant influence on shale mechanical behavior, with higher values of either parameter leading to more pronounced changes in material properties.

4.3 Evaluating pyrite’s role in shale fracturability and fracture network development

The commercial development of shale reservoirs largely relies on hydraulic fracturing techniques that employ volume stimulation. The fracture networks generated in brittle rocks through this process are essential for enhancing gas production and recovery rates. Thus, the capacity to establish a well-developed fracture network serves as a critical indicator of the effectiveness of shale gas development strategies (Guo et al., 2015). Brittleness, a fundamental mechanical property of rocks, is commonly quantified by the brittleness index (BI). Numerous studies have confirmed that BI plays a pivotal role in controlling fracture formation in shale (Hou et al., 2014; Zhang et al., 2016; Li et al., 2018c; Yang et al., 2023). Mineralogical composition is the primary factor influencing rock brittleness (Yang et al., 2020). Among the constituent minerals, pyrite—characterized by its intrinsic brittleness—exerts a significant effect on BI (Yasin et al., 2021). To investigate this influence, 16 representative model groups were constructed in this study to examine the impact of pyrite on the mechanical properties of shale. The brittleness index was determined using the energy balance analysis method applied to stress–strain curves. Subsequently, BI values were integrated with compressive strength and Poisson’s ratio data to evaluate the morphology and distribution of microfractures within the models following fracturing (Fig. 16).

By comparing the particle size cases of 0.81 and 0.61 in Fig. 16(d), it is evident that the cracks in the 0.81 case are both more numerous and more widely distributed throughout the model, whereas those in the 0.61 case are concentrated in a localized region. A similar trend is observed when comparing the 3.00% and 0.47% pyrite content cases in Fig. 16(a), suggesting that as the brittleness index increases, the plastic deformation capacity of shale decreases. This reduction in plasticity lowers resistance to crack initiation, resulting in a greater number of initial cracks and a more extensive fracture network. In Fig. 16(b), a comparison between the particle size cases of 0.2 and 0.4 reveals that, although the crack areas and counts are comparable, the cracks in the 0.4 case are significantly longer than those in the 0.2 case. This indicates that higher compressive strength suppresses crack propagation, thereby restricting crack extension and weakening connectivity between microcracks, which ultimately reduces the overall fracture network. A similar phenomenon is seen in Fig. 16(c), where, despite originating from similar initiation points, cracks in 0.4 particle size case are longer and more interconnected, while some cracks in the 0.2 case remain unconnected. The influence of Poisson’s ratio further highlights the complexity of crack behavior. A higher Poisson’s ratio not only increases the difficulty of crack initiation but also slows crack propagation. Under uniaxial compression, where longitudinal deformation remains constant, larger transverse deformation facilitates directional changes in crack propagation. This effect is clearly illustrated in Fig. 16(a) by comparing 0.47% and 2.42% pyrite content cases: cracks in the latter are shorter, more easily redirected, and often exhibit a distinct “V-shaped” morphology. Overall, when pyrite content reaches 3.00% and the shape factor approaches 0.81, cracks in shale are longer, more widely distributed, and more prone to forming interconnected networks, thereby producing an enhanced fracturing effect. This enhancement primarily arises from localized stress concentrations induced by pyrite particles, which strongly promote the initiation and branching of microcracks and contribute to more complex and interconnected fracture systems. Although pyrite–matrix interfaces and grain boundaries likely act as preferential pathways for crack initiation and propagation, the present numerical model did not explicitly account for interfacial strength due to software limitations. It is important to note that the practical application of these theoretical insights—such as optimizing hydraulic fracturing parameters including proppant volume, fluid injection rate, and stage spacing—requires further integration with field-scale data, including in situ fracture development patterns and operational conditions. Such integration lies beyond the scope of the current study but represents an essential direction for future research.

5 Conclusions

This study quantitatively analyzed the microscopic characteristics of pyrite in shale—including content, particle size, shape factor, and distribution mode—using scanning electron microscopy combined with Gaussian hybrid clustering. Numerical simulations were then conducted to examine how variations in pyrite influence the mechanical properties of shale, providing a scientific basis for evaluating the mechanical behavior of comparable unconventional reservoirs. Results demonstrate that pyrite content and shape exert a significant influence on shale mechanical behavior and on the initiation and propagation of microcracks. These findings provide geological guidance for identifying favorable hydraulic fracturing zones and for optimizing fracturing parameters in unconventional reservoirs. However, the practical application of these insights requires integration with field-scale data and validation across diverse geological settings, which will be the focus of future research. The main conclusions are summarized as follows.

1) A framework was established for the automatic extraction and quantitative characterization of pyrite content, particle size, shape factor, and distribution mode, based on threshold segmentation and Gaussian hybrid clustering of scanning electron microscope images. Furthermore, 19 shale models with varying pyrite micro-characteristics were generated using an adaptive quadtree structure.

2) Numerical simulations conducted with the high-precision cytogenetic method and RFPA-2D software reveal that pyrite content, particle size, shape factor, and distribution mode exert significant influences on shale mechanical properties. Among them, pyrite content exhibits nonlinear relationships with mechanical parameters, particle size shows a negative correlation, the shape factor exerts complex effects, and the distribution mode presents fluctuating impacts.

3) The impact of pyrite micro-characteristics on shale mechanical properties is primarily governed by stress concentration. Variations in pyrite content, maximum particle size, shape factor, and distribution mode intensify localized stress concentration, facilitating crack initiation and propagation, thereby reducing compressive strength. At the same time, differences in particle shape enhance inter-particle friction through mechanical embedding, which limits misalignment and slippage, ultimately strengthening the material. Moreover, particle distribution regulates the development and stability of force chains, exerting further control over shale compressive strength.

4) Random forest regression and numerical simulations indicate that pyrite content has the strongest influence on shale mechanical properties, followed by the shape factor, while maximum particle size and distribution mode exert comparatively weaker effects. Shale mechanical behavior is particularly sensitive to high pyrite content and elevated shape factors. Furthermore, fracture propagation simulations show that when pyrite content reaches ~3.00% and the shape factor is ~0.81—under comparable particle size and distribution conditions—fracture networks become more extensive and interconnected, thereby enhancing fracturing effectiveness.

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