# Predictive Modeling of the Uniaxial Compressive Strength of Rocks Using an Artificial Neural Network Approach

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## Abstract

**:**

_{d}) in g/cm

^{3}, Brazilian tensile strength (BTS) in MPa, and wet density (ρ

_{wet}) in g/cm

^{3}. The developed ANN models, M1, M2, and M3, were divided as follows: the overall dataset, 70% training dataset and 30% testing dataset, and 60% training dataset and 40% testing dataset, respectively. In addition, multiple linear regression (MLR) was performed for comparison to the proposed ANN models to verify the accuracy of the predicted values. The performance indices were also calculated by estimating the established models. The predictive performance of the M2 ANN model in terms of the coefficient of determination (R

^{2}), root mean squared error (RMSE), variance accounts for (VAF), and a20-index was 0.831, 0.27672, 0.92, and 0.80, respectively, in the testing dataset, revealing ideal results, thus it was proposed as the best-fit prediction model for UCS of sedimentary rocks at the Thar coalfield, Pakistan, among the models developed in this study. Moreover, by performing a sensitivity analysis, it was determined that BTS was the most influential parameter in predicting UCS.

## 1. Introduction

_{(50)}) in MPa, and ultrasonic (Vp) in m/s, and demonstrated the high performance of the proposed model [21]. The ANN method has proved to be a key method among all intelligent methods and is thus mostly used to solve challenging problems that are reliant on laboratory experimental data because of their high efficiency and ability to learn from inputs [22]. Based on the reliable predictions of ANN methods, some researchers have estimated various mechanical properties of rocks by analyzing the correlation among various physical parameters [23,24]. Yin employed an ANN back-propagation algorithm, which has been considered as the best prediction method based on previous studies [25]. Skentou used hybrid ANN models for predicting UCS of granite rocks with optimal results. Similarly [26], Kaloop developed six hybrid ANN models to predict UCS of different rock types. Based on the performance indicators, such as R

^{2}and RMSE [27], the multivariate adaptive regression splines (MARS) revealed ideal results compared with other models developed in the study. Xiang estimated the in situ rock strength from borehole geophysical logs using ANN models [28]. KÖKEN used different soft computing models including ANN for estimating the fracture toughness of rocks [29]. Table 1 shows previous studies using intelligent methods to predict UCS.

_{d}) in g/cm

^{3}, Brazilian tensile strength (BTS) in MPa, and wet density (ρ

_{wet}) in g/cm

^{3}. A total of 78 sedimentary rock samples, i.e., claystone, sandstone, and siltstone, of each type of core rock were selected from Block IX of the Thar coalfield. For the developed ANN models, the dataset is distributed as follows: model 1 (M1) is the overall dataset, model 2 (M2) consists of 70% as the training dataset and 30% as the testing dataset, and model 3 (M3) consists of 60% as the training dataset and 40% as the testing dataset. Similarly, multiple linear regression (MLR) analyses are performed for comparison to the proposed ANN model to check the accuracy of the predicted values. The performance indices are also calculated by estimating the established models. Furthermore, to determine the effect of each variable on the estimated values of UCS, a sensitivity analysis was performed. The complexity and cost of direct destructive laboratory tests adversely affect the data scarcity problem, making the development of intelligent indirect methods an integral step in attempts to address the problem faced by rock engineering projects. In this study, we apply, for the first time, an intelligent prediction method to predict UCS of sedimentary rocks from Block IX of the Thar coalfield. To the best of the authors’ knowledge, there is no such application of intelligent prediction techniques.

## 2. Materials and Methods

#### 2.1. Building Dataset

_{d}in g/cm

^{3}, BTS in MPa, ρ

_{wet}in g/cm

^{3}, and UCS in MPa, using a universal testing machine (UTM), as shown in Figure 2a,b. Figure 2a,b represent the deformed rock core specimen for UCS and BTS tests, respectively. Table 2 presents the five heads and five tails of the dataset of physical and mechanical parameters. Table 3 shows the minimum, maximum, average, and standard deviation of parameters of rock samples determined in the laboratory.

^{3}), (b) BTS (MPa), (c) wet density (g/cm

^{3}), and (d) UCS (MPa). Figure 4 presents the pairwise plot of the original dataset of different parameters and UCS under this study. Notably, none of the parameters are well-correlated to the UCS, thus all of the parameters are analyzed for UCS prediction. In addition, Figure 4 represents a moderate positive correlation of BTS with UCS; however, the dry density and wet density show a negative correlation with UCS.

#### 2.2. Methods

_{d}(g/cm

^{3}), BTS (MPa), and ρ

_{wet}(g/cm

^{3}). Figure 5 demonstrates the flow chart of the predictive modeling process for UCS. Owing to the small number of resources available for collecting samples, the current study used a limited dataset, that is, 78 samples divided for the established models, including M1, M2, and M3, as presented in Table 4. M1 means the model was trained on the overall dataset, M2 means the model was trained on 70% (55 datasets) of the dataset and tested on 30% (23 datasets) of the dataset, and M3 means the model was trained on 60% (47 datasets) of the dataset and tested on 40% (31 datasets) of the dataset. In addition, Taylor diagram representation was used, which explains a brief qualitative depiction of the best fit of the model to standard deviations and correlations. Moreover, cosine amplitude method (CAM)-based sensitivity analysis was carried out in order to estimate the influence of each input variable on output UCS.

#### 2.2.1. Artificial Neural Network

_{d}, BTS, and ρ

_{wet}, were allocated to an input layer composed of three neurons to predict UCS of the output layer. The ANN models, M1, M2, and M3, were trained, tested, and validated. One hundred epochs were used to train the models and the minimum validation error was considered as a stopping point to prevent overfitting. Figure 7 represents the validation curves for the training performance of the ANN models of UCS. Therefore, model M2 demonstrates the best performance curve of UCS, with validation error equal to 0.14562, which is reached at 0 epochs. Figure 8 illustrates the training scatter plots of predicted UCS against measured UCS, as M1 for overall dataset and as M2 and M3 for the training and testing dataset, respectively.

#### 2.2.2. Multiple Linear Regression

#### 2.2.3. Model Evaluation

^{2}), root mean squared error (RMSE), variance accounts for (VAF), and a20-index of each model, respectively. Table 5 represents the performance indices of the ANN and MLR models for predicting UCS on the overall dataset, training dataset, and testing dataset.

## 3. Prediction and Discussion of Uniaxial Compressive Strength

^{2}, RMSE, VAF, and a20 index were used as performance criteria to examine the final output, to analyze and compare the expected models, and to evaluate the optimal model for data prediction. Model 1 (M1) is the overall dataset, model 2 (M2) consists of 70% as the training dataset and 30% as the testing dataset, and model 3 (M3) consists of 60% as the training dataset and 40% as the testing dataset.

^{2}= 0.793. Based on the M1 predicted outputs, Figure 10a shows the aggregated comparison of predicted versus measured values for UCS. Figure 10b specifies the change in relative error between the measured and predicted values. The MSE value of model M1 achieved is 0.00599. Figure 10c illustrates the error histogram of the established model M1. Here, it can be considered that the distribution of the errors is approximately zero, which is in good agreement with the performance of model M1.

^{2}values of model M2 are 0.834 and 0.831, respectively. According to the M2 estimated results for the training data, Figure 12a displays the aggregated comparison of the predicted against measured values for UCS. Figure 12b shows the change in relative error between the measured and predicted values. The MSE value of model M2 is 0.00002. Figure 12c denotes the error histogram of model M2. It can be seen that the distribution of the errors is almost zero, which indicates that the performance of the proposed model M2 is satisfactory and reliable. Similarly, Figure 12d exhibits the aggregated comparison of the predicted against measured values for UCS of estimated outputs of M3 for the testing data. Figure 12e denotes the change in relative error between the measured and predicted values. The MSE value is achieved as 0.07657. Figure 12f represents the error histogram of model M3. Consequently, it can be seen that the distribution of the errors is nearly zero, which indicates that the performance of the proposed model M2 is acceptable.

^{2}values of model M3 are 0.807 and 0.775 for the training and testing data, respectively. Regarding the estimated results of M3 for the training data, Figure 14a shows the aggregated comparison of the predicted against measured values of UCS. Figure 14b shows the change in relative error between the measured and predicted values. The MSE value of M3 is 0.00015. Figure 14c signifies the error histogram of the developed model M3. Hence, it can be noted that the error distribution approaches zero, which shows that the performance of model M3 is adequate. Likewise, for predictive outputs of M3 for the testing data, Figure 14d reveals the aggregated comparison of the predicted against measured values for UCS. Figure 14e indicates the change in relative error between the measured and predicted values. The MSE value of M3 is 0.04541. Figure 14f presents the error histogram of model M3. Thus, the distribution of the errors is nearly zero, which indicates that the performance of the established model M3 is satisfactory.

^{2}, another very commonly used test is the ANOVA test. In the first case, linear regression was used to determine the relationship between the dependent variable measured UCS and the three independent variables: ${\rho}_{\mathrm{d}}$, BTS, and ${\rho}_{\mathrm{w}\mathrm{e}\mathrm{t}}$. In Table 6, the R

^{2}values of UCS are estimated using different equations of the MLR models, including M1, M2, and M3, for the overall dataset and training and testing data, i.e., 0.187 for M1, 0.292 and 0.066 for M2, and 0.425 and 0.062 for M3, respectively. Therefore, the R

^{2}values of UCS are quite satisfactory in models M1, M2, and M2. Furthermore, the ANOVA test also rejected the null hypothesis at a significance value of p < 0.001.

**Table 5.**Performance indices of the ANN and MLR models for predicting UCS for the overall dataset, training dataset, and testing dataset.

Model | UCS | |||||
---|---|---|---|---|---|---|

R^{2} | RMSE | VAF (%) | a20-index | |||

ANN | M1 | Overall dataset | 0.793 | 0.07739 | 0.96 | 0.95 |

M2 | Train | 0.834 | 0.00484 | 0.99 | 0.99 | |

Test | 0.831 | 0.27672 | 0.92 | 0.80 | ||

M3 | Train | 0.807 | 0.01211 | 0.99 | 0.99 | |

Test | 0.775 | 0.21311 | 0.90 | 0.80 | ||

MLR | M1 | Overall dataset | 0.187 | 6.70404 | 0.98 | 1.07 |

M2 | Train | 0.292 | 3.33067 | 0.77 | 0.80 | |

Test | 0.066 | 1.40950 | 0.81 | 0.99 | ||

M3 | Train | 0.425 | 1.32518 | 0.82 | 1.05 | |

Test | 0.062 | 7.12692 | 0.99 | 0.99 |

**Table 6.**Multiple linear regression analysis for UCS in MPa; ${\rho}_{d}$ (g/cm

^{3}), BTS (MPa), and ${\rho}_{wet}$ (g/cm

^{3}) are the dry density, Brazilian tensile strength, and wet density, respectively.

Model Code | Dataset | Equation | R^{2} |
---|---|---|---|

M1 | Overall | $\mathrm{U}\mathrm{C}\mathrm{S}=1.49-0.93{\rho}_{d}+3.12\mathrm{B}\mathrm{T}\mathrm{S}+0.26{\rho}_{wet}$ | 0.187 |

M2 | Train | $\mathrm{U}\mathrm{C}\mathrm{S}=1.04-1.11{\rho}_{d}+4.35\mathrm{B}\mathrm{T}\mathrm{S}+0.41{\rho}_{wet}$ | 0.292 |

Test | $\mathrm{U}\mathrm{C}\mathrm{S}=7.83-7.24{\rho}_{wet}+4.61{\rho}_{d}+0.80\mathrm{B}\mathrm{T}\mathrm{S}$ | 0.066 | |

M3 | Train | $\mathrm{U}\mathrm{C}\mathrm{S}=0.72-1.80{\rho}_{d}+0.80{\rho}_{wet}+6.17\mathrm{B}\mathrm{T}\mathrm{S}$ | 0.425 |

Test | $\mathrm{U}\mathrm{C}\mathrm{S}=0.59-4.05{\rho}_{wet}+4.90{\rho}_{d}+0.24\mathrm{B}\mathrm{T}\mathrm{S}$ | 0.062 |

#### Taylor Diagram

^{2}, RMSE, and standard deviation of the original and predicted UCS for the M2 and M3 ANN and MLR models for the testing stage. The prediction of ANN model M3 is highly correlated with the original values and, compared with the other developed models, the standard deviation is similar to the original value. Thus, ANN model M2 with R

^{2}= 0.831 is the most suitable for predicting UCS of sedimentary rocks in the Thar coalfield, Pakistan, among the developed models.

^{2}value is highest, the RMSE is lowest, the VAF is at a maximum, and the a20-index is reliable. Therefore, according to Figure 15, ANN model M2 for the testing dataset revealed the optimal results and is proposed as the best-fit prediction model for UCS in this study.

## 4. Sensitivity Analysis

## 5. Conclusions

_{d}, BTS, and ρ

_{wet}as input parameters. The physical and mechanical properties of rock samples were determined in a laboratory in accordance with ISRM and ASTM standards. This study determined the predictive performance of ANN and MLR models by determining the highest R

^{2}, the smallest RMSE, the highest VAF, and a reliable a20-index as follows:

^{2}, RMSE, VAF, and a20-index were 0.793, 0.07739, 0.96, and 0.95, respectively, for M1; 0.834 and 0.831, 0.00484 and 0.27672, 0.99 and 0.92, and 0.99 and 0.80, respectively, for the training and testing dataset of M2; and 0.807 and 0.775, 0.01211 and 0.21311, 0.99 and 0.90, and 0.99 and 0.80, respectively, for the training and testing dataset of M3.

^{2}, RMSE, VAF, and a20-index were 0.187, 6.70404, 0.98, and 1.07, respectively, for M1; 0.292 and 0.066, 3.33067 and 1.40950, 0.77 and 0.81, and 0.80 and 0.99, respectively, for the training and testing dataset of M2; and 0.425 and 0.062, 1.32518 and 7.12692, 0.82 and 0.99, and 1.05 and 0.99, respectively, for the training and testing dataset of M3.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Abbreviation/Symbol | Parameter Name | Abbreviation/Symbol | Parameter Name |

UCS | Uniaxial compressive stregth | R^{2} | Coefficient of determination |

ISRM | International Society of Rock Mechanics | RMSE | Root mean squared error |

ASTM | American Society for Testing and Materials | VAF | Variance accounts for |

ANN | Artificial neural network | µ | Poisson’s ratio |

ANFIS | Adaptive neuro-fuzzy interference system | ρ and r | Density |

PSO | Particle swarm optimization | BTS | Brazilian tensile strength |

GA | Genetic algorithm | SHN | Schmidt hardness |

MARS | multivariate adaptive regression splines | ρ_{wet} | Wet density |

ICA | Imperialist competitive algorithm | N | Porosity |

Is | Point load strength index | I_{s(50)} | Point load index |

Rn | Schmidt hammer rebound number | Vp | P-wave velocity |

BPI | Block punch index | Ab and Wabs | Water absorption |

DD, ρ_{d} | Dry density |

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**Figure 1.**Geological site of the collected rock samples [38].

**Figure 2.**(

**a**) Deformed rock core specimen for Brazilian tensile strength test and (

**b**) deformed rock core specimen for UCS test.

**Figure 3.**Histogram plots of the original dataset in this study: (

**a**) dry density (g/cm

^{3}), (

**b**) BTS (MPa), (

**c**) wet density (g/cm

^{3}), and (

**d**) UCS (MPa).

**Figure 4.**Correlation plot of inputs (dry density (g/cm

^{3}), BTS (MPa), and wet density (g/cm

^{3})) and output (UCS (MPa)) of the original dataset in this study.

**Figure 8.**ANN training scatter plots of predicted UCS against measured UCS for (

**a**) M1, (

**b**) M2, and (

**c**) M3.

**Figure 10.**The demonstration of ANN model M1 for UCS. (

**a**) Model M1 results aggregated with measured UCS. (

**b**) The variation in error between the measured and predicted values. (

**c**) Error histogram.

**Figure 11.**ANN model M2 results for UCS plotted against the measured data for the (

**a**) training and (

**b**) testing data.

**Figure 12.**The demonstration of ANN model M2 for UCS. (

**a**) The model M2 results aggregated with the measured UCS. (

**b**) The variation in error between the measured and predicted values. (

**c**) Error histogram for the training data and (

**d**) model M2 results aggregated with the measured data. (

**e**) The variation in error between the measured and predicted values. (

**f**) Error histogram for the testing data.

**Figure 13.**ANN model M3 results for UCS plotted against the measured data for the (

**a**) training and (

**b**) testing data.

**Figure 14.**The demonstration of ANN model M3 for UCS. (

**a**) Model M3 results aggregated with the measured UCS. (

**b**) The variation in error between the measured and predicted values. (

**c**) Error histogram for the training data and (

**d**) model M3 results aggregated with the measured data. (

**e**) The variation in error between the measured and predicted values. (

**f**) Error histogram for the testing data.

Method | Input | Output | R^{2} | References |
---|---|---|---|---|

ANN | n, Is, µ, ρ, Vp | UCS | 0.97 | (Madhubabu et al., 2016) [1] |

ANN | ρ, n, Vp, Ab | UCS | 0.93 | (Abdi et al., 2018) [4] |

ANN | n, r, Wabs | UCS | 0.92 | (Kamani et al., 2020) [14] |

ANN | Vp, I_{s(50),} BTS | UCS | 0.97 | (Mohamad et al., 2015) [21] |

ANN | Rn, Vp, DD | UCS | 0.82 | (Li et al., 2020) [30] |

ANN | Is, Vp, Rn, n | UCS | 0.93 | (Dehghan et al., 2010) [31] |

ANFIS | BTS, Vp | UCS | 0.60 | (Yesiloglu-Gultekin et al., 2013) [32] |

PSO-BP | DD, MC, Vp, Is_{(50)}, Id_{2} | UCS | 0.999 | (Mohamad et al., 2018) [33] |

ICA-ANN | Rn, Vp, Is_{(50)} | UCS | 0.949 | (Armaghani et al., 2016a) [34] |

ICA-ANN | n, Rn, Vp, Is_{(50)} | UCS | 0.915 | (Armaghani et al., 2016b) [35] |

MLR | n, Is, µ, ρ, Vp | UCS | 0.91 | (Madhubabu et al., 2016) [1] |

MLR | ρ, n, Vp, Ab | UCS | 0.88 | (Abdi et al., 2018) [4] |

MLR | Vp, I_{S(50)}, SHN, BPI | UCS | 0.91 | (Heidari et al., 2018) [36] |

MLR | Id_{2}, Is_{(50)}, N, é | UCS | 0.58 | (Yılmaz et al., 2008) [37] |

Dataset | ρ_{d} (g/cm^{3}) | BTS (MPa) | ρ_{wet} (g/cm^{3}) | UCS (MPa) |
---|---|---|---|---|

1 | 1.91 | 0.305 | 2.13 | 0.404 |

2 | 1.75 | 0.217 | 2.01 | 0.491 |

3 | 1.77 | 0.318 | 2.04 | 0.531 |

4 | 1.78 | 0.271 | 2 | 0.579 |

5 | 1.76 | 0.292 | 2.04 | 0.557 |

… | … | … | … | … |

74 | 1.81 | 0.178 | 2.1 | 0.541 |

75 | 1.84 | 0.189 | 2.11 | 0.476 |

76 | 1.96 | 0.2 | 2.18 | 0.508 |

77 | 1.78 | 0.108 | 2.09 | 0.511 |

78 | 1.84 | 0.138 | 2.09 | 1.415 |

Parameters | ρ_{d} (g/cm^{3}) | BTS (MPa) | ρ_{wet} (g/cm^{3}) | UCS (MPa) |
---|---|---|---|---|

Minimum | 1.22 | 0.023 | 1.63 | 0.304 |

Maximum | 2.12 | 0.627 | 2.3 | 3.55 |

Average | 1.76 | 0.32 | 2.04 | 1.38 |

Standard deviation | 0.22 | 0.13 | 0.15 | 0.98 |

Model Code | Dataset | Dataset Distribution (%) | Total Dataset |
---|---|---|---|

Model 1 (M1) | Overall | 100 | 78 |

Model 2 (M2) | Train | 70 | 55 |

Test | 30 | 23 | |

Model 3 (M3) | Train | 60 | 47 |

Test | 40 | 31 |

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## Share and Cite

**MDPI and ACS Style**

Wei, X.; Shahani, N.M.; Zheng, X. Predictive Modeling of the Uniaxial Compressive Strength of Rocks Using an Artificial Neural Network Approach. *Mathematics* **2023**, *11*, 1650.
https://doi.org/10.3390/math11071650

**AMA Style**

Wei X, Shahani NM, Zheng X. Predictive Modeling of the Uniaxial Compressive Strength of Rocks Using an Artificial Neural Network Approach. *Mathematics*. 2023; 11(7):1650.
https://doi.org/10.3390/math11071650

**Chicago/Turabian Style**

Wei, Xin, Niaz Muhammad Shahani, and Xigui Zheng. 2023. "Predictive Modeling of the Uniaxial Compressive Strength of Rocks Using an Artificial Neural Network Approach" *Mathematics* 11, no. 7: 1650.
https://doi.org/10.3390/math11071650