The rock’s mechanical characteristics, including uniaxial compressive strength [UCS (MPa)] and elastic modulus [E (GPa)], play a pivotal role in the planning and design of relevant projects in mining and civil engineering. The success of both underground and surface mining endeavors greatly relies on a comprehensive understanding of various rock characteristics, with UCS and E being pivotal components of rock mechanical assessment. Ensuring the accuracy and precision of UCS and E measurements is fundamental in the design and execution of any mining engineering project. The development of machine learning (ML) intelligent indirect methods for studying subsurface structures based on limited data holds the potential to save time and cost while ensuring structural stability. This study carries significant socio-economic advantages and stands as an essential contributor to sustainable development. Furthermore, it specifically focuses on Block IX of the Thar Coalfield in Pakistan, with the primary objective of assessing the stability of the surrounding rock during underground excavation. This effort aims to prevent any disturbance to overlying aquifers and the ground surface due to underground mining while minimizing adverse environmental impacts. The study offers an in-depth exploration of rock deformation and characterization induced by changes in the stress field. “Various forms of rock deformation behavior have been scrutinized by researchers (Zhao et al., 2017; Rahimi and Nygaard, 2018; Davarpanah et al., 2019; Xiong et al., 2019), and the methods for estimating rock strength and deformation encompass both destructive and non-destructive techniques”. As per the recommended standards of the International Society for Rock Mechanics (ISRM) and the American Society for Testing and Materials (ASTM), the direct estimation of UCS and E through laboratory-based destructive testing is recognized as a difficult, time-consuming, and expensive endeavor, especially when working with delicate, internally fractured, thin, or highly foliated rock samples (Jing et al., 2021). Consequently, exploring indirect methods to assess UCS and E, such as rock index tests or predictive approaches based on ML-based intelligent methods.

The investigation of the rocks’ mechanical characteristics is integral to the extraction of energy resources and forms the fundamental basis for their safe exploitation. The importance of rock mechanics extends to the advancement of natural resource extraction, encompassing the safeguarding of energy reserves, such as petroleum products (oil, coal, and natural gas), and the preservation of the surrounding geological environment. Furthermore, waste disposal and hydroelectric energy projects necessitate a deeper exploration of rocks and soils, mandating comprehensive research into the mechanical characteristics of rocks (National Research Council, 1978; Demirdag et al., 2010). Various numerical techniques, such as peridynamic models, general particle dynamics, and uniaxial compression experiments on granite rock samples, have been employed to investigate the fracture behavior of brittle materials featuring preexisting cracks. These methods offer valuable insights into the complex mechanisms governing fracture (Zhou et al., 2014; Zhou et al., 2015; Wang et al., 2016; Wang et al., 2017; Wang et al., 2018; Zhou et al., 2019). The UCS and E of rocks play pivotal roles in addressing issues related to rock mechanics and the design of coal mining operations (Török and Vásárhelyi, 2010; Hakan and Kanik, 2012; Jahed Armaghani et al., 2015a; Armaghani et al., 2016).

“Typically, two common methods, namely, static and dynamic, are utilized to determine E. Static E is often derived by analyzing the stress-strain curve up to 50% of the maximum strength of the rock core sample. Conversely, dynamic E is determined by considering the rock’s density and the velocities of compressional and shear waves”. The disparity between static and dynamic E is a well-researched aspect of rock engineering (Brotons et al., 2016). It is generally observed that the dynamic E tends to be slightly higher than the static E, as observed by various researchers (Zhang, 2006; Kolesnikov, 2009). The dynamic-to-static E ratio has been reported in the range of 1–20 (Wang, 2000).

Different authors have devised predictive models to mitigate these challenges, employing various ML approaches (Ozcelik et al., 2013; Abdi et al., 2018; Yang et al., 2020; Cao et al., 2021; Duan et al., 2021; Harandizadeh and Armaghani, 2021; Pham et al., 2021; Armaghani et al., 2022). This departure from the direct utilization of tests prescribed by international standards is primarily due to the perceived drawbacks of those tests, including their time-consuming, costly, and unreliable nature (Jahed Armaghani et al., 2016; Jamshidi et al., 2016). “The evolution of ML has been strongly influenced by the emergence of novel learning algorithms, theoretical advancements, as well as the continuous enhancement of online data resources and high-speed computing capabilities” (Jordan and Mitchell, 2015). Although these models excel in addressing complex problems promptly and effectively, they predominantly focus on unraveling intricate relationships among variables for goal estimation, rather than offering insights into the associations between predictors and output values (Chelgani et al., 2016). Abdi and Taheri-Garavand (2020) developed the ANFIS (adaptive neuro-fuzzy inference system) algorithm to estimate the UCS of a sandstone consisting of 136 data points. ANFIS proved to be highly accurate based on the evaluation of the model by R ², RMSE and VAF. Ceryan and Samui (2020) predicted the volcanic rocks’ UCS using extreme learning machine (ELM) and minimum probabilistic machine regression (MPMR), and also incorporated the least square support vector machine (LS-SVM) model to compare the performance of the model. The results showed that ELM and MPMR gave better results than LS-SVM. Aboutaleb et al. (2018) used simple regression analysis (SRA), MRA, ANN (artificial neural networks), and support vector regression (SVR) to predict the UCS of carbonate rock. It was found that the SVR model was more accurate than the other models. Ceryan et al. (2018) used various ML models, namely, FIS (fuzzy inference system), ANN, and LV-SVM. So, the LV-SVM model was the best in predicting UCS. Ghasemi et al. (2018) employed the model tree method to estimate the UCS of carbonate rocks and demonstrated the high performance of the method. For predicting the UCS of travertine rocks, (Barzegar et al., 2020) used RF, M5 model trees, and multiple adaptive regression splines (MARS). In addition, they built an ANN model based on an ensemble committee to correlate the results of the implemented models. The results showed that the MARS model outperformed the other models (Barzegar et al., 2020). Zhong et al. (2021) predicted UCS employing the XGBoost model with highly accurate results. Yesiloglu-Gultekin and Gokceoglu (2022) developed NLMR, ANN, and ANFIS models using 137 data points (including unit weight, porosity, and sound velocity) as input features to indirectly estimate the UCS of basalt. The ANN was successful based on the performance metrics of R ², RMSE, VAF, and a20-index. Predicting rock strength is an efficient alternative technique to direct estimation. Diamantis and Moussas (2021) applied multiple regression and ANNs to estimate the UCS of peridotites collected from central Greece. Based on their results, the proposed ANNs were found to be the most effective instead of multiple regression. Mai et al. (2021) employed RF, one of the most powerful ML models to predict concrete strength using GBFS, and concluded that RF is the most powerful prediction tool with R ² = 0.97, which is recommended for engineers to reduce the cost of experiments. The mechanical properties of rock, specifically UCS are considered a key parameter that plays an important role in the design of any rock engineering structure and energy resource recovery and development. Therefore, an accurate estimation of it is essential. UCS was predicted using RF by Matin et al. (2018). For comparison, multivariate regression (MVR) and generalized regression neural network (GRNN) were used for the prediction. According to their results, RF yielded more satisfactory conclusions than MVR and GRNN. Wang et al. (2020) determined the UCS of rocks indirectly by developing RF as a prediction tool using two indirect input features, namely, Schmidt hammer rebound values (L-type) and V_p. Thus, the applied RF model has high accuracy and suggests that the predicted UCS values can be better applied in the fields of rock mechanics and engineering geology. Gupta and Natarajan (2021) developed new intelligent prediction models, namely, RF, ELM, LSSVR, and primal least squares dual SVR (PLSTSVR) proposed as density-weighted least squares twin support vector regression (PDWLSTSVR) to predict the UCS of rock samples. The model’s efficiency was estimated identically at the testing dataset of 47 samples out of overall 179 samples. Consequently, PDWLSTSVR performed well with high accuracy compared to other studied models such as RF, ELM, LSSVR, and PLSTSVR. The UCS of rocks was modeled using soft computing methods such as MLPNN (multilayer perceptron neural network), M5 model tree, and ELM by Gül et al. (2021). The MLPNN model performed excellently with an R ² of 0.9982. Sampath et al. (2019) utilized advanced soft computing models, especially ANFIS and ANN, to effectively predict the strength alterations. Similarly, Abdi et al. (2018) introduced ANN and MRA for predictive modeling of E. Their study incorporated input variables, including porosity, dry density, P-wave velocity, and water absorption. The findings demonstrated that the ANN model outperformed the MRA model. Ghasemi et al. (2018) employed a model tree-based method to evaluate the E of carbonate rocks. Their findings indicated that the studied technique yielded the best predictive results. Shahani et al. (2021) utilized a novel XGBoost algorithm. The applied model, XGBoost achieved a high level of accuracy in predicting E. Furthermore, Shahani et al. (2022a) developed six ML models such as “LightGBM, SVM, Catboost, GBRT, RF, and XGBoost” to estimate E of the Thar Coalfield. Thus, the XGBoost model showed better results than the other models. Umrao et al. (2018) studied the ANFIS method to determine the strength and E of non-homogeneous sedimentary rocks. The anticipated ANFIS model exhibited excellent predictive capabilities. Davarpanah et al. (2020) established both linear and nonlinear relationships between static and dynamic deformation parameters in various types of rocks. Their research revealed a strong correlation between these parameters. For predicting E of CO₂-rich coals, Guha Roy and Singh (2018) employed ANN, ANFIS, and MR techniques. The findings indicated that both ANN and ANFIS have outdone the MR model. Jahed Armaghani et al. (2015b) conducted a study where they predicted the E of rocks, comparing the ANFIS against MRA and ANN. The consequences demonstrated that ANFIS exhibited superior performance compared to MRA and ANN. Singh et al. (2012) introduced the ANFIS architecture as a method for predicting rock E. Cao et al. (2022) adopted an innovative approach by combining XGBoost and the Firefly Algorithm (FA) in supervised ML to predict E. The results showed that this novel method was effective. Yang et al. (2019) used a Bayesian method to predict intact granite’s E, and the model produced suitable predictions. Rastegarnia et al. (2018) predicted the mechanical characteristics of sedimentary rocks, especially UCS and E, using ANN with R ² of 0.99 and 0.97, respectively. Singh et al. (2017) assessed a range of geomechanical parameters, with a specific focus on the parameter E using a combination of MRVA and ANFIS methods. As a result, the ANFIS model yielded a significantly more accurate.

Considering the limitations in existing literature and conventional prediction methods, a single model often proves to be insufficiently adaptable and inclusive, leading to suboptimal solutions in complex scenarios, with varying performance outcomes dependent on input features. To our knowledge, previous research has addressed complex and unpredictable engineering situations without leveraging intelligent prediction methods, specifically in the context of the Thar Coalfield. There is a dearth of research focused on predicting the UCS and E, and the comprehensive exploration of model selection and application in UCS and E prediction remains uncharted territory. To address this gap, this study employs a hybrid ML-based model that amalgamates multiple models to counterbalance the limitations of a single-model approach, substantially enhancing the accuracy of predictive results. In this study, hybrid PSO with tree-based models, such as gradient boosting regressor (GBR), light gradient boosting machine (LightGBM), random forest (RF), and extreme gradient boosting (XGBoost) including wet density (WD) in g/cm³, moisture in %, dry density (DD) in g/cm³, and (BTS) in MPa and shore hardness (SH) as input features. The dataset used in this study was collected from Block IX of the Thar Coalfield in Pakistan. A training-testing split of 80% and 20%, respectively, was implemented on a dataset consisting of 122 samples. To optimize the performance of the developed models, a repetitive hyperparameters tuning and cross-validation method is employed. Furthermore, SHAP (Shapley Additive exPlanations) ML analysis was conducted to identify the influence of each input feature on the predicted UCS and E. This research represents the first application of a hybrid model to predict UCS and E at Block IX of the Thar Coalfield in Pakistan. Figure 1 illustrates the flowchart of the research methodology used in this study.

In this study, 122 stratigraphic rock samples including siltstone, claystone, sandstone, and coal were collected from Block IX of Thar Coalfield using the borehole coring method. These rock samples underwent careful preparation and splitting procedures following the guidelines set forth by the ISRM (International Society for Rock Mechanics) (Brown, 2007) and the ASTM (American Society for Testing and Materials) (ASTM Committee D-18 on Soil and Rock, 2013) to keep consistent core dimensions and geometric features. Subsequent investigational tests were conducted on these rock samples at the laboratory, to determine their physico-mechanical properties. Key properties assessed included wet density (WD), moisture, dry density (DD), Brazilian tensile strength (BTS), shore hardness (SH), elastic modulus (E), and uniaxial compressive strength (UCS). Previously (Shahani et al., 2022a), we used 106 datasets and 4 input parameters with a single predictive model. However, this study used more datasets and input parameters and a hybrid optimization prediction model to improve the accuracy of the data. Table 1 defines the statistical distribution of the original dataset used in this study.

The UCS test carried out following ISRM standards, involved the use of a uniaxial testing machine (UTM) on standardized core samples with NX dimensions, featuring a diameter of 54 mm, and applied a loading rate of 0.5 MPa/s. This test was performed to ascertain the UCS and E of the rock samples. Additionally, to evaluate the BTS of the rock samples, Brazilian tests were performed using the same UTM apparatus.

The Seaborn module in Python was used for visualizing the original dataset in this study. Specifically, Figure 3 presents three-dimensional (3D) surface plots illustrating the relationships between input parameters and the output variables UCS and E.

Additionally, Figure 4 shows a correlation heatmap for the complete dataset. In Figure 4, there are positive correlations between BTS and moisture, as well as SH and E with output UCS, while WD and DD exhibit negative correlations. Similarly, for output E, Figure 4 reveals positive correlations with BTS, SH, and UCS, and negative correlations with WD and DD. The representation of moisture does not exhibit a correlation with the E. RStudio software was employed for the formation of Figure 4, and it is important to note that UCS and E were considered as interrelated input parameters in the analysis.

In developing countries like Pakistan, the use of large-scale conventional tests to determine rock mechanical characteristics, namely, UCS and E is often impractical. Consequently, there is a growing imperative to design intelligent predictive algorithms using ML techniques to address the challenge of data scarcity, which is the focus of this study. This study employs hybrid PSO with tree-based models, including GBR, LightGBM, RF, and XGBoost. The study aims to forecast the UCS and E of rock samples collected from Block-IX of the Thar Coalfield in Pakistan. A concise overview of the developed method is given below.

Grid Search Cross-Validation is a method used for tuning hyperparameters (Bergstra and Bengio, 2012). This method allows for a search within a specified range of hyperparameters and identifies the expected hyperparameter combinations that yield the best results in terms of evaluation metrics such as R ², MAE, and RMSE. In the Python programming language, the GridSearchCV() function is typically used to implement this strategy, and it is readily available in the scikit-learn library. This method essentially computes Cross-Validation scores for all possible combinations within a specific range of hyperparameters.

This study, as shown in Figure 8, conducted 10-fold Cross-Validation to comprehensively assess the performance of hyperparameter combinations. GridSearchCV() not only allows us to discover the optimal hyperparameter combinations but also provides performance metrics for these combinations.

The optimal selection of hyperparameters, specifically max_depth and n_estimators, plays a crucial role in constructing a robust model for predicting UCS and E. max_depth determines the maximum depth of the decision tree, while n_estimators signifies the total number of trees in the forest. Given the computational cost associated with hyperparameter tuning, we employed Python’s range function to efficiently explore the parameter space. For max_depth, values between 4 and 10 are considered, taking into account the computational resources required for selection. Similarly, a range of 10 for n_estimators is utilized, spanning from 10 to 300. The learning rate was kept at its default value. Figure 9; Table 2 illustrate the visualization plot for hyperparameter optimization of the developed models: (a) UCS and (b) E.

Different performance matrices such as R ² (Shahani et al., 2022b; Wei et al., 2023), MAE (Willmott, 1982), and RMSE (Shahani et al., 2022c) have been used by different scholars to evaluate high-accuracy ML models. This suggests that the higher the R ² value and the smaller the MAE and RMSE, the more the model is considered the best predictive model when predicting UCS and E from their original values. In this study, R ², MAE, and RMSE are utilized to assess the relationship between the original and predicted values of UCS and E. R 2 = ∑ i = 1 n S o − S o ¯ S p − S p ¯ ∑ i = 1 n S o − S o ¯ 2 ( S p − S p ¯ 2 (6) M A E = 1 N ∑ i = 1 n | S 0 − S p | (7) R M S E = ∑ i = 1 n S o − S p 2 n (8) where, S o ¯ and S p ¯ represents are the mean values of original and predicted UCS and E, S o and S p are the original and predicted values of UCS and E, respectively.

In conventional rock and mining engineering, accurate estimation of essential parameters such as UCS and E is critical for reliable design and analysis. However, conventional methods, often based on linear regression approaches, face limitations due to the inherent non-linearity of rock parameters. The hybrid PSO with tree-based prediction models proposed in this study offers significant improvements in parameter estimation.

For instance, in the context of Block IX of the Thar Coalfield in Pakistan, where the application of the longwall top coal caving (LTCC) method has been proposed (Shahani et al., 2019; Shahani et al., 2020), real-time estimation of UCS and E can play a crucial role. This estimation directly influences the customization and modification of LTCC design and operations, ensuring safe and cost-effective mining practices. The accurate determination of rock mechanical parameters, particularly UCS and E, is essential for designing mining structures and earth surface profiles, thereby contributing to the safety, economic viability, and sustainability of mining operations.

Although numerous basic studies have been conducted to determine the mechanical properties of rocks at Thar Coalfield, the requirement for more robust models and more diverse datasets is crucial to developing reliable information on the impacts of the consideration of mechanical characteristics of rock, which will be very useful for mine planning and design. ML-based prediction targets making models that after learning from specific training datasets can make accurate predictions on concealed data that has never been introduced to the model, i.e., a model that can be generalized by Chollet in 2018 (Chollet, 2021). Therefore, the main ML-based engineering applications of this study are outlined below:

(1) ML Indirect Techniques for Designing and Excavating Underground Structures: The study introduces ML as a tool for designing and excavating underground structures, leveraging limited data for cost-effective and stable mining structure development. This has technical, economic, and social implications, aligning with sustainability objectives.

The prediction of UCS and E using hybrid PSO with tree-based models not only contributes to the stability of underground mine roadway excavation but also addresses uncertainties associated with sparse datasets. This approach empowers mining and rock engineers to assess the level of uncertainty surrounding predictions, ensuring a more informed decision-making process for the safe and continuous operation of mining activities at Block IX of the Thar Coalfield.