Introduction

The uniaxial compressive strength (UCS) of intact rocks is an important rock property that is widely used in mining engineering because of its ability to depict the mechanical behaviour of rock. The UCS affects excavation and loading operations in mines (Adebayo and Aladejare 2013), and they are required as input in many mining engineering applications such as Hoek–Brown failure criterion (Hoek et al. 2002; Aladejare and Wang 2019a) and classifications of rock mass such as rock mass rating (RMR) and rock mass index (RMi) (Bieniawski 1974; Palmstrøm 1996; Aladejare and Wang 2019b; Aladejare and Idris 2020), as input parameter for as estimating characteristic impedance (Zhang et al. 2020; Aladejare et al. 2022; Zhang et al. 2023) among other rock properties. This is why most studies on experimental, empirical and numerical methods for estimation of rock strength are centred on UCS (Jamshidi et al 2018a, b; Sharma et al. 2017; Aboutaleb et al. 2018; Armaghani et al. 2018; Uyanik et al. 2019; Aladejare 2020; Aladejare et al. 2021). However, the laboratory tests for measuring the UCS of rocks is difficult, costly and takes time. In addition, extracting cores to meet the specimen requirements for the laboratory tests is not always feasible, because of the nature of some rocks. For instance, it may be difficult to extract acceptable cores from weak, porous, or weathered rocks. In such instances, rock engineers and practitioners frequently utilize empirical methods to estimate UCS of intact rocks from other indices that can be easily obtained through either field or laboratory tests, such as simple regressions for estimation of UCS (Jamshidi et al. 2016; Jamshidi 2022, 2022; Ajalloeian et al. 2020; Aladejare 2020).

There are empirical models for estimating UCS based on unit weight (ϒ), and schmidt hardness rebound (N) (Deere and Miller 1966; Aufmuth 1974; Beverly et al. 1979; Kidybinski 1980). Jajali et al. (2017) and Heidari et al. (2018) developed empirical models for estimating Em based on the N, block punch index (BPI), point load strength (IS(50)), and P-wave velocity (Vp). Tiryaki (2008) developed empirical model based on density (ρ), shore hardness (SH), and cone indenter (CI). Armaghani et al. (2018) developed an empirical formula based on ρ, slake durability Index (Id2), and Brazilian tensile strength (BTS). Çobanoğlu and Çelik (2008), Diamantis et al. (2009), Azimian et al.(2014), and Ng et al. (2015) developed empirical models for estimating UCS based on Is(50) and Vp. Çobanoğlu and Çelik (2008) related UCS to Is(50), Vp, and N. Moradian and Behnia (2009) developed an empirical model based on ρ and Vp. Majdi and Rezaei (2013) developed an empirical model with N, ρ and porosity (n) as input parameters. Sharma et al. (2017) proposed a model which uses Vp, Id2, and ρ as inputs to estimate UCS. Dehghan et al. (2010) developed a model for estimation of UCS based on Is(50), n, Vp, and N. Madhubabu et al. (2016) provided an empirical model using Is(50), n, Poisson’s ratio (ν), ρ, and Vp. Jamshidi et al. (2018a, b) developed two empirical models with one using ρ and Vp as input parameters and the other model using n and Vp as input parameters. Aladejare et al. (2021) developed several databases of empirical equations for estimating UCS from physical and mechanical properties of rocks. They collated several models that were developed using simple regression, multiple regression, and artificial intelligence-based methods in the databases.

Since rocks are heterogeneous in nature and variable because of their formation processes, hence, there is need to properly characterise the randomness for reliable preliminary design investigations and analysis (Sari et al. 2010). When deterministic estimation of UCS is performed, only a unique value is used, usually the average of the input rock property. The deterministic estimations do not consider the various uncertainties associated with UCS and their measurements or estimations. Using empirical equations provides a single output value of UCS for the single value of each input rock property. In addition, the method cannot produce reliable probability distributions of the investigated UCS because empirical models cannot incorporate the quantification of the uncertainties associated with the variables that are used as inputs. These shortcomings often result in underestimation or overestimation of rock UCS, which propagate through design processes resulting in unexpected performance of rock structures.

To overcome the limitations faced by using empirical equations to estimate UCS and the shortcomings resulting from process, Bayesian and machine learning methods have been used to characterize rock property variability and uncertainty (Aboutaleb et al. 2018; Aladejare and Wang 2017a; Aladejare 2020; Aladejare et al. 2020, 2022; Armaghani et al. 2018; Madhubabu et al. 2016; Majdi and Rezaei 2013; Momeni et al. 2015; Wang and Aladejare 2016a, b). Both methods can consider the full range of data concerning the specific random characteristic of UCS. With both methods, the probability distributions, range of values that the variable could take within the range and their relative frequency can be obtained (Fattahi et al. 2019). However, both methods require complex mathematical formulations and great computational skills which makes their adaptation to real-life rock engineering practice to be difficult for practitioners.

To bypass the difficulty often encountered in using the Bayesian and machine learning methods, this study seeks to apply Monte Carlo simulation (MCS) to estimate UCS. MCS is a commonly used sampling technique for stochastic modelling and estimation. MCS has successful applications in different areas of mining and geotechnical engineering, such as mine design, slope stability, foundation design to consider the uncertainty in design parameters for various mining and geotechnical engineering systems (Chiwaye and Stacey 2010; Ghasemi et al. 2010; Fattahi et al. 2013; Wang 2013; Wang and Cao 2014). In addition, Morin and Ficarazzo (2006) used the Kuz–Ram model in MCS to simulate and predict the blasting fragmentation. Aladejare and Wang (2017b) used MCS to model the uncertainty of input parameters in reliability-based design of square pad foundation. Sari et al. (2014) applied MCS to predict the backbreak area resulting from blasting operation. Ghasemi et al. (2012), and Armaghani et al. (2016) applied MCS to model flyrock resulting from production blasting. Fattahi et al. (2013) evaluated the damaged zone around underground excavations using MCS-based neuro-fuzzy clustering. Aladejare and Wang (2018) and Aladejare and Akeju (2020) applied MCS to quantify associated uncertainties in reliability-based designs of rock slopes.

To estimate UCS using MCS in this study, a multiple linear regression (MLR) is used to develop an empirical equation. Then, the developed empirical equation is applied to estimate UCS using MCS. Finally, a hypothesis testing is performed to assess the influence of the input parameters on the estimation of UCS.

The description of the database

This study uses a database comprising a total of 150 data sets obtained from laboratory testing of migmatite blocks collected during site investigation of locations in the Sanandaj-Sirjan zone of Iran (Saedi et al. 2019). The details of the sites including the geographical map and methods used to measure the rock parameters can be found in Saedi et al. (2019). The parameters contained in the database are BPI, BTS, Is(50), vp, and UCS, whose statistical characteristics are given in Table 1.

Table 1 Statistics of inputs and output data set
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Regression analysis

The estimation of UCS is a linear problem which may be influenced by individual parameters. In this section, simple regression analysis is performed to developed model for estimating UCS from BPI, UCS from BTS, UCS from Is(50), and UCS from Vp. The regression equations from the analyses are listed from Eqs. (1)-(4). In addition, the UCS estimation may be influenced by a combination of multiple parameters. The MLR is a widely acceptable approach for developing multiple equations between an output parameter and multiple input parameters. The method has been used severally to proffer solutions to rock mechanics problems. This study uses a similar methodology, inspired by previous studies such as Ghasemi et al. (2012), Sari et al. (2014), Armaghani et al. (2016), Fattahi et al. (2019), and Aladejare (2021). Therefore, MLR analysis was carried out using Microsoft Excel,a and the MLR model is presented in Eq. (5). For the development of simple and multiple regressions, the data in the database was divided into 80% for training the model and 20% for testing the developed models to assess their reliabilities for estimation purposes.

$$\mathrm{UCS }= 5.0468{\text{BPI}}+7.6329$$
(1)
$$\mathrm{UCS }= 8.2924{\text{BTS}}+21.4040$$
(2)
$$\mathrm{UCS }= 9.2781{\text{Is}}_{(50)}+27.7480$$
(3)
$$\mathrm{UCS }= 0.0199{\text{Vp}}+36.303$$
(4)
$$\mathrm{UCS }= -4.1385+1.5995{\text{BPI}}+3.7194{\text{BTS}}+{2.9286{\text{Is}}}_{(50)}+0.0061{\text{Vp}}$$
(5)

To evaluate the reliability of the developed models (i.e., Eqs. (1)-(5)), key performance indicators for regression analysis are used. In this study, five performance indicators are applied, namely determination coefficient (R2), root mean square error (RMSE), absolute average relative error percentage (AAREP), mean absolute error (MAE), and variance accounted for (VAF), which are expressed as below:

$${R}^{2}=1-\frac{{\sum }_{i=1}^{{{\text{n}}}_{{\text{t}}}}{\left({{\text{UCS}}}_{i\mathrm{ m}}-{{\text{UCS}}}_{i\mathrm{ e}}\right)}^{2}}{{\sum }_{i=1}^{{{\text{n}}}_{{\text{t}}}}{\left({{\text{UCS}}}_{i\mathrm{ m}}-{{\text{UCS}}}_{{\text{mean}}}\right)}^{2}}$$
(6)
$${\text{RMSE}}=\sqrt{\frac{1}{{{\text{n}}}_{{\text{t}}}}\sum_{i=1}^{{{\text{n}}}_{{\text{t}}}}{\left({{\text{UCS}}}_{i\mathrm{ m}}-{{\text{UCS}}}_{i\mathrm{ e}}\right)}^{2}}$$
(7)
$${\text{AAREP}}=\frac{1}{{{\text{n}}}_{{\text{t}}}}\sum_{i=1}^{{{\text{n}}}_{{\text{t}}}}\left|\frac{{{\text{UCS}}}_{i\mathrm{ m}}-{{\text{UCS}}}_{i\mathrm{ e}}}{{{\text{UCS}}}_{i\mathrm{ m}}}\right|\times 100$$
(8)
$${\text{MAE}}=\frac{{\sum }_{i=1}^{{{\text{n}}}_{{\text{t}}}}\left|{{\text{UCS}}}_{i\mathrm{ m}}-{{\text{UCS}}}_{i\mathrm{ e}}\right|}{{{\text{n}}}_{{\text{t}}}}$$
(9)
$${\text{VAF}}=\left(1-\frac{{\text{Var}}({{\text{UCS}}}_{{\text{m}}}-{{\text{UCS}}}_{{\text{e}}})}{{\text{Var}}({{\text{UCS}}}_{{\text{m}}})}\right)\times 100$$
(10)

where \({{\text{UCS}}}_{\mathrm{i m}}\) and \({{\text{UCS}}}_{\mathrm{i e}}\) are the UCS values obtained from laboratory testing (i.e., measured UCS) and empirical estimation (i.e., estimated UCS), respectively. \({{\text{UCS}}}_{{\text{mean}}}\) is the mean of the measured UCS, Var is variance, and \({{\text{n}}}_{{\text{t}}}\) is the number of rock data used in analysis.

Table 2 presents the evaluated performances of the developed models for the training data set. From the five statistical analyses performed in this study, the MLR has the best prediction performance than the simple regressions. Considering that the MLR model has the best performance in all the indictors for training data, it can be inferred that the MLR is better than the simple regressions. For the MLR model, there are relatively little errors and high correlations between the measured and estimated UCS data. The results also show that a high proportion of the total variance in the measured UCS values is accounted for by the variance in the estimated UCS values. Therefore, the proposed MLR model may provide better and reliable estimates of UCS values than the simple regressions. In this example, the MLR model benefits from information provided by different input parameters, which are consistent with the site UCS data.

Table 2 Model performance of different regressions for predicting UCS using the training data
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Validation of developed models using independent data

The testing data is used to validate the models proposed in Eqs. (1)-(5). For each input data, UCS data is estimated using the equations. Table 3 presents the estimated UCS and includes the measured UCS, which is used for comparison and validation of the proposed models. The mean from the UCS estimated from MLR model is closer to the measured UCS than those estimated from the simple regression models. Also, the standard deviation of the UCS estimated from MLR model is within the range of the standard deviation of the measured UCS. These indicate that the MLR model performed better than the simple regression. The MLR model benefits if the data of the input parameters is consistent with the UCS measured at site.

Table 3 Uniaxial compressive strength estimated from different models
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To further validate the proposed models, performance indicators are used to evaluate the models. Table 4 present the results of the analyses, using the testing data. The MLR model produced the least error across all the indicators. It also has the highest VAF among the proposed models, which indicates that a high proportion of the total variance in the measured UCS values is accounted for by the variance in the estimated UCS values.

Table 4 Performance analysis of different models for predicting UCS using the testing data
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Stochastic estimation of UCS using Monte Carlo simulation

MCS is implemented to UCS by considering the uncertainty in each of the input parameters. MCS simply perform calculations of empirical model repeatedly using variables that are random and of known or assumed probability distributions (Ang and Tang 2007). The result from each calculation is considered as a sample of the solution of the empirical model, and similar to test sample from a laboratory investigation. MCS methods have gained popularity over the years because they can address complex rock mechanics and mining engineering problems. Among several research studies, Morin and Ficarazzo (2006) used MCS as a tool to predict blasting fragmentation based on the Kuz–Ram model. Similarly, Idris et al. (2013) used MCS to perform probabilistic estimation of rock masses properties in Malmberget mine, Sweden, while Lu et al. (2019) performed MCS-based uncertainty analysis of rock mass quality Q in underground construction. In this paper, the probability distributions of available site information (e.g., statistics of BPI, BTS, Is(50) and Vp) are employed in the MCS to estimate UCS. The calculations are performed repeatedly until a total of nt calculations results are obtained. The results from the calculations describe the statistics and probability distribution of UCS. MCS methods provide logical route to consider and incorporate the uncertainties in input parameters towards the estimation of UCS (Morin and Ficarazzo 2006; Fattahi et al. 2019). This will be particularly helpful in probability-based designs and other reliability analyses where there is need to consider the propagation of uncertainties in design parameters to design of mining systems.

In this study, the main goal of MCS is to consider the uncertainties in the input parameters to accurately quantify the variability in the estimated UCS values. This will increase the applicability of the model and address the limitation of regression models. Note that regression models do not consider the uncertainties in the input parameter and analysis from such data may give misleading performance expectations of mining constructions and structures. The MCS in this study also seeks to reveal the relative contribution of uncertainties of the input parameters to the overall data scatteredness and the range of UCS results (Armaghani et al. 2016; Fattahi et al. 2019). The advantage of MCS over conventional deterministic methods is that while deterministic methods provide fixed estimated values, the MCS uses a large number of estimated values as inputs in the repeated calculations and its output is a range of estimated values. In the MCS, the process takes random values of the input rock properties from the prescribed range and output is calculated using the empirical model developed. This process is repeated many times as needed using the values of input rock properties that are randomly drawn from a prescribed probability distribution. MCS can generate a large amount of data, and Wang (2011) explained that the reliability and confidence of the results from MCS increase with increasing MCS samples numbers. In a typical MCS, a value of nt is set to repeat the simulation process. Through the simulation process, a large number of results (UCS as the output parameter in this study) are obtained as output values. MCS models use the independence nature of random variables, which means that a simulated value for a variable does not affect the simulated value by another variable. This makes it logically possible to incorporate and completely quantify the uncertainty in the estimation process. The obtained results can be used to describe the probability distribution of UCS (Ghasemi et al. 2012). In addition, the simulated samples can be used as inputs in reliability analyses or any probability-based designs involving the use of rock parameter values (Wang 2013; Aladejare and Wang 2017b).

In this study, Microsoft Excel was used to implement MCS for estimation of UCS. The model which performed best (i.e. the MLR equation) and expressed in Eq. 5 was utilized to simulate UCS and to quantify the significance of the input parameters on the estimation of UCS. In the MCS model, a probability distribution was assumed for each of the input parameters based on the analysis of the available data of input parameters (i.e., BPI, BTS, Is(50) and Vp). To avoid bias resulting from assumed probability distribution of input parameters, Kolmogorov–Smirnov (K–S) test is performed in this study for each input parameter. The Kolmogorov–Smirnov (K–S) test is a popular test for analysing and determining the appropriate distribution for a group of data to be used in reliability analysis. The results of K-S tests conducted on the data of the input parameters shows that the best distribution for the parameters are normal distributions, and they are presented in Table 5. To further demonstrate the information of the input parameters as captured in Table 5, the probability distributions of the input parameters applied in MCS are shown in Fig. 1.

Table 5 Probability distribution functions of input parameters
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Fig. 1
figure 1

Histogram showing the distributions of the model inputs used in MCS

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The MCS uses 10000 values of each input parameter to estimate 10000 values of the output parameter. The MCS randomly selects values within the defined distribution of each input parameter. As a result, the variables of input parameters have relationships and hence can significantly affect the output from MCS process. Therefore, the correlation between the variables of input parameters were analysed to improve the MCS model in UCS simulation. Table 6 presents the Spearman’s correlation coefficients between the variables of the input parameters.

The steps involved in the stochastic estimation of UCS are given below:

  1. 1.

    The best-fit distribution functions were determined for each input parameter using MATLAB software.

  2. 2.

    Stochastic estimation of UCS through Eq. (5) using each combination of input values obtained from the assumed probability distribution.

  3. 3.

    Incorporation of the correlation coefficients as presented in Table 6 into the MCS model.

  4. 4.

    To eliminate bias and account for the transformation uncertainty associated with the model developed in Eq. (5), the mean of the MCS-simulated UCS were calculated from a different number of estimations. In all, 15 different numbers of estimations (i.e., 15 different values of nt) were considered to check the convergence of MCS. As listed in Table 7, the starting trial number of estimations is 1000 and with 1000 incremental steps until 15000 estimations. As shown in Table 7, the mean of the MCS samples for the 15 different values of nt are close, and there is good convergence at MCS with 10000 samples.

Table 6 Spearman’s correlation coefficients for model inputs
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Table 7  The mean of the MCS-simulated UCS for 15 different values of nt
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The distribution and frequency of the MCS-simulated UCS samples and their summary of statistics are shown in Fig. 2. The mean of the MCS-simulated UCS is 119.10 MPa, while the mean for the measured UCS is 118.42 MPa (for all datasets). It was found that the normal distribution used in this study is appropriate for estimating UCS. The results show that the MLR model can satisfactorily simulate UCS. In addition, Fig. 3 shows the results of laboratory-measured UCS, MLR-estimated UCS and MCS-simulated UCS. The results show that the MLR-estimated UCS and MCS-simulated UCS values are consistent with the laboratory-measured UCS for all the data points. Generally, the MCS-simulated UCS plots more closely to the laboratory-measured UCS. This shows that the use of the MLR model in MCS further reduces the uncertainty contained in the estimation process and provides UCS values that are consistent with the measured UCS values.

Fig. 2
figure 2

Histogram of MCS simulated UCS along with summary of statistics

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Fig. 3
figure 3

Comparison of laboratory-measured UCS, MLR-estimated UCS, and MCS-simulated UCS

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Hypothesis testing

This section quantifies and statistically compares the effect of uncertainties in the four input parameters on the simulation of UCS. The comparison is performed by using the simulated UCS samples that fall outside the 90” confidence interval of the total simulated samples. The mean \({\mu }_{X}\) of UCS samples outside the 90% confidence interval can be remarkably different from the mean \({\mu }_{0}\) of the total simulated UCS samples. Such instance indicates that the uncertainty in the input parameters significantly impact on the simulated UCS samples. Hypothesis test can be applied to evaluate the statistical difference between \({\mu }_{X}\) and \({\mu }_{0}\). Let H0 denotes a null hypothesis while HA denotes alternative hypothesis, and defined as (Wang et al. 2010):

$$\begin{array}{c}{\text{H}}_0 : {\mu }_{X}={\mu }_{0}\\ \mathrm{H_A }: {\mu }_{X}\ne {\mu }_{0}\end{array}$$
(11)

Let ZH represents a hypothesis test statistic of each input parameter, which is formulated as:

$${{\text{Z}}}_{{\text{H}}}=\frac{{\mu }_{X}-{\mu }_{0}}{{\sigma }_{u}/{n}_{u}}$$
(12)

where \({\sigma }_{u}\) is standard deviation of the parameter and \({n}_{u}\) is the number of simulated UCS samples that fall outside the 90% confidence interval. A relatively large absolute value of ZH indicates that \({\mu }_{X}\) deviates statistically from \({\mu }_{0}\). The higher the absolute value of ZH, the greater the effect of the parameter on the simulated UCS samples, and this is reflected by the significant statistical difference between \({\mu }_{A}\) and \({\mu }_{0}\). Therefore, the absolute value of ZH is adopted in this study as a statistical index to quantify the effect of each input parameter on simulated UCS samples and to rank their effect on simulated UCS samples.

When an empirical model is used to estimate UCS, the estimated UCS is dependent on all input parameters. At 90% confidence interval, the 5% percentile and 95% percentile marks are at about 55 MPa, and 185 MPa respectively. The hypothesis test statistics ZH defined by Eq. (3) is calculated for each input parameter based on the simulated UCS samples that are outside the 90% confidence interval. Figure 4 show the results of the calculations of the absolute value of ZH for the four input parameters considered in this study. The absolute value of ZH ranges from less than 1.5 for P-wave velocity of rock Vp to more than 2 for Brazilian tensile strength of rock BTS. The decreasing order of the ZH absolute values is: BTS, Is(50), BPI, and Vp. This shows that BTS is the most important parameter and its uncertainty has the greatest effects on the simulated UCS followed by Is(50), while the uncertainty of Vp contributes the least to the simulated UCS.

Fig. 4
figure 4

Results of hypothesis test on the system output

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Discussion

The UCS is an important input parameter commonly requested during major mining engineering designs and analyses. International Society of Rock Mechanics (ISRM) (Ulusay and Hudson 2007) has documented the procedures for laboratory measurement of UCS. Although the recommended method is the best technique, it is time and resources consuming. Also, there may be difficulty in sample preparation in some rock types, such as weathered rocks, badly fractured rocks, and soft rocks among others. This study adopted a case study and used four input parameters that have been widely used to estimate UCS in literature to probabilistically model UCS values. Four simple regressions were developed using data of BPI, BTS. Is(50), and Vp. In addition, a MLR model was developed using the four input parameters. 30 data of BPI, BTS. Is(50), and Vp that were not used in the development of the models were applied test the proposed models. The MLR model produced least error and the variance in the estimated UCS values from the MLR model is consistent with that of the measured UCS values. As the MLR model produced the best performance among the five models developed, the MLR model was applied to perform stochastic estimation of UCS values using MCS. Stochastic estimation of UCS using MCS allow the uncertainties in the input parameters to be incorporated. This is important to avoid misleading engineering analysis which can result from underestimation or overestimation of rock properties. Furthermore, a hypothesis testing was applied to extrapolate insights about population of the data used in the analysis. It was used to evaluate the contribution of the four input parameters on UCS estimation.

Conclusions

Below are the conclusions from the study:

  1. 1.

    The performance analyses of the developed MLR model showed that it can estimate UCS in a reliable and accurate manner as can be noted in Tables 3 and 4.

  2. 2.

    The mean of the MCS-simulated UCS values is 119.10 MPa, and the mean of the measured UCS values is 118.42 MPa. Both values are close with small relative difference. In addition, the values and spread of the MCS-simulated UCS values are consistent with the laboratory-measured UCS. This shows that the proposed MCS method can reliably simulate UCS values.

  3. 3.

    Incorporating the MLR model into MCS significantly improves UCS estimation, because it considers the uncertainties in the input parameters.

  4. 4.

    As can be noted in the MLR model developed in Eq. (5), all the input parameters (i.e., BPI, BTS Is(50), and Vp) showed positive correlation with UCS. This is logical considering the physical meaning of the properties of rock that each of the input parameters quantify. Furthermore, the results of the hypothesis testing showed that BTS play the most significant role in the estimation of UCS.

  5. 5.

    The proposed MLR model is robust and can be used to estimate UCS. However, since rock properties are site-specific, the model may have varying performances when applied to other sites/conditions. Hence, there is a need for re-analysis of the presented process to determine the suitability of the proposed model to other sites.