Empirical Estimation of Uniaxial Compressive Strength of Rock: Database of Simple, Multiple, and Artificial Intelligence-Based Regressions

Empirical relationships for estimating Uniaxial Compressive Strength (UCS) of rock from other rock properties are numerous in literature. This is because the laboratory procedure for determination of UCS from compression tests is cumbersome, time consuming, and often considered expensive, especially for small to medium-sized mining engineering projects. However, these empirical models are scattered in literature, making it difficult to access a considerable number of them when there is need to select empirical model for estimation of UCS. This often leads to bias in estimated UCS data as there may be underestimation or overestimation of UCS, because of the site-specific nature of rock properties. Therefore, this study develops large database of empirical relationships between UCS and other rock properties that are reported in literatures. Statistical analysis was performed on the regression equations in the database developed. The typical ranges and mean of data used in developing the regressions, and the range and mean of their R2 values were evaluated and summarised. Most of the regression equations were found to be developed from reasonable quantity of data with moderate to high R2 values. The database can be easily assessed to select appropriate regression equation when there is need to estimate UCS for a specific site.


Introduction
The uniaxial compressive strength (UCS) is a mechanical property of intact rocks that is important in civil and mining engineering works Wang and Aladejare 2016a). Design and stability analysis of underground excavations and other geotechnical structures require the input of data like UCS on the geomechanical behaviour of rocks (Ulusay et al. 1994). Adebayo and Aladejare (2013) explained that UCS of rock has effect on excavation-loading operation of rock fragments. According to Hoek (1977), UCS is a required property when considering a variety of problems encountered during blasting, excavation, and support in engineering works. In addition, UCS is essential for classification of rock masses into different groups for engineering applications, and these classifications are used to determine their suitability for different construction purposes (Sachpazis 1990). For example, UCS is used as input in rock mass classification systems like rock mass rating (RMR) (Bieniawski 1974; Aladejare and Wang 2019a; Aladejare and Idris 2020) and rock mass index (RMi) (Palmstrøm 1996), and in predicting strength parameters of rock masses through Hoek-Brown failure criterion (Hoek et al. 2002). In the probabilistic characterization of Hoek-Brown mi, Aladejare and Wang (2019b) used UCS data in a Bayesian framework to simulate samples of Hoek-Brown m i , which are useful for probability-based estimation of rock mass properties through the Hoek-Brown failure criterion. The UCS also serve as input data when using empirical equations to predict deformation modulus of rock masses (Aladejare and Wang 2019b) and characteristic impedance of rocks (Zhang et al. 2020). All these make UCS an important parameter to most rock and mining engineering designs and analyses. According to a survey reported by Bieniawski (1976), mining engineers request the UCS more often than any other rock material property. From surface to underground mine design and construction, UCS is a key parameter and it is required that UCS be known with certainty to a great extent for engineering analysis.
The guidelines and method for laboratory determination of UCS have been suggested by International Society of Rock Mechanics (ISRM) (Ulusay and Hudson 2007). However, the laboratory determination of UCS is expensive and time consuming. Therefore, for most mining projects, especially small to mediumsized projects, data of UCS are not often available (Aladejare 2016). For this reason, numerous regression equations have been developed in literature for estimation of UCS, when they cannot be directly obtained through laboratory testing (Sachpazis 1990;Gökçeoglu 1996;Chatterjee and Mukhopadhyay 2002;Yılmaz and Sendır 2002;Dincer et al. 2004Dincer et al. , 2008Gokceoglu and Zorlu 2004;Hudyma et al. 2004;Sabatakakis et al. 2008;Tiryaki 2008;Diamantis et al. 2009;Khandelwal and Singh 2009;Moradian and Behnia 2009;Yasar et al. 2010;Mishra and Basu 2012;Khandelwal 2013;Minaeian and Ahangari 2013;Mohamad et al. 2015; Kallu and Roghanchi 2015;Fereidooni 2016;Sharma et al. 2017;Heidari et al. 2018;Aliyu et al. 2019). Results of some physical and mechanical tests have been recommended for indirect estimation of UCS. Numerous studies of empirical equations developed for indirect estimation of UCS in the literature generally include those using physical properties such as Schmidt hardness number, shore hardness, density, water content, porosity, P-wave velocity, S-wave velocity, unit weight, Equotip hardness number (also referred to as Leeb hardness number) and slake durability index, and mechanical properties such as block punch index, Young's modulus, Brazilian tensile strength, and point load strength as inputs (Tugrul and Zarif 1999;Vasarhelyi 2005;Shalabi et al. 2007;Cobanoglu and Celik 2008;Török and Vasarhelyi 2010;Mishra and Basu 2013;Tandon and Gupta 2015;Mohamad et al. 2015;Najibi et al. 2015;Kahraman et al. 2016;Sharma et al. 2017;Uyanik et al. 2019). Simple and multiple regressions are available in literature for estimating UCS from these properties. In the recent past, artificial intelligence has been used to develop models for estimation of UCS, using techniques such as artificial neural network (ANN), support vector machine (SVM), Fuzzy inference system (FIS), genetic programming (GP) and hybrid ANN (Monjezi et al. 2012;Rezaei et al. 2014;Jalali et al. 2017;Aboutaleb et al. 2018;Armaghani et al. 2018;Mohamad et al. 2018;Ren et al. 2019).
With the numerous regression equations available in literature, there is a need to systematically select equations which suit specific sites. Aladejare (2015, 2016a) developed methods for selecting models and estimation of UCS. The Bayesian frameworks developed in studies such as Aladejare (2015, 2016a, b) need empirical equations as input. However, lack of accessibility to a great number of equations is a drawback. This is because when decision is to be made on the regression equation to be used for estimation of UCS, only equations that are readily assessed in literatures are considered. The regression equations developed are scattered in literatures, with no study yet that has systematically compiled them together for use during selection and estimation of UCS of rock. In order to solve this problem, this paper develops a database, which is a global compilation of empirical equations for estimating UCS from physical and mechanical properties of rocks. To provide a global compilation of different forms of regression equations, an extensive review of previous studies is performed to collect and compile information of different regression equations for estimation of the UCS of rock. This study is particularly beneficial for engineering projects when considering any analysis that involves the use of UCS as an input. This is because it serves as the equations bank from which different regression equations can be assessed for selection and their subsequent use for estimation of UCS. Engineering with Computers were used to compile information of regression equations for estimating UCS, ranging from simple to multiple regression and artificial intelligence-based models. The regression equations that are documented in the database only includes those whose data were obtained according to testing procedure standards set by ISRM or American Society for Testing and Materials (ASTM). This ensures that all equations in the database were developed from test results involving consistent sample length to diameter ratio and testing conditions (Aladejare and Wang 2017). Note that only equations developed for rocks are considered in the database, soil and other weathered rocks which behave as soil are not considered in the database. Geo-materials whose equations are included in the database are generally referred to as rock samples in the original literatures. They generally include grade I-III weathered rocks (i.e., ranging from fresh rocks to slightly weathered rock and moderately weathered rocks. Grade IV or above weathered geo-material is generally referred to as soil (e.g. Ehlen 2002; Aladejare and Wang 2017) and are not considered in the database.

Database Development and Description
In the database, there are different types of regression equations ranging from simple to multiple regressions and artificial intelligence-based regressions such as ANN, SVM, FIS, GP, and hybrid ANNs. In addition, there are different modes of equations such as linear, power, exponential, logarithmic and polynomial functions in the database. The equations contained in the database include those developed for estimating UCS from rock properties such as Schmidt hardness number (N), shore hardness (SH), density (q), porosity (n), P-wave velocity (V p ), S-wave velocity (V s ), unit weight (c), equotip hardness number (L D ), slake durability index (I d2 ), block punch index (BPI), Young's modulus (E), Brazilian tensile strength (BTS) and point load strength (Is (50) ). Equations between UCS and other less frequently measured rock properties such as grain size (GS), shape factor (SF), quartz content (Qtz), particle diameter (D), single compressive strength index (SCSI) among others that are available in literature are also included in the database.
For each regression equation, number of data from which it was developed and the correlation coefficient (R 2 ) are documented. The mean (l) of a group of data is calculated as: where h i is a set of rock property data and n t is the total number of rock data present in a group of data. The range and mean of number of data used in equation development and their R 2 for each regression equation are also included in the database.

Simple Regression
Simple regression is a statistical method for studying relationships between two continuous variables, in which one variable is regarded as the predictor or independent variable, and the other variable is regarded as the outcome or dependent variable (Freedman 2009). Assuming two groups of data (Y i ; X ai ); i = 1, …, n, where X ai = (X a1 ,…X an ) is a vector of independent variable and Y i a real-valued dependent variable for the ith observation, a regression equation f is a model that makes a prediction _ Y of Y for a potentially new input vector X a , written as: Simple regression for estimating UCS can take any form such as linear, logarithmic, exponential, power, and polynomial forms (Diamantis et al. 2009;Yasar et al. 2010;Nefeslioghu 2013;Azimian et al. 2014;Kallu and Roghanchi 2015), and the difference in the models is the way that f(X a ) in Eq. (2) is expressed for each regression equation. In this study, the simple regressions are grouped under two headings into those regressions derived from physical properties and those derived from mechanical properties as discussed in the following subsections.

Simple Relationship Between UCS and Physical Properties
Physical tests are generally easier and less expensive to perform, and for this reason many simple regressions are available in literature for estimating UCS from physical properties of rock (Cobanoglu and Celik 2008;Heidari et al. 2018;Aliyu et al. 2019; be observed from Tables 1, 2 , 3, 4, 5, 6, 7, 8 and 9 indicate that not all equations will be suitable for specific site. Having database of regression equations will give mining engineers and other practitioners the opportunity to fairly assess all regression equations before deciding on the regression equations for a specific site. Recent studies in mining and geotechnical engineering have developed model selection approaches to select appropriate model from candidate models (e.g., Aladejare, 2015, 2016a). With many regression equations available in a paper, mining practitioners can subject many regression equations to assessment before deciding on the appropriate regression equation. Table 10 shows the information about the statistics of the regression equations in Tables 1, 2, 3, 4, 5, 6, 7, 8, and 9 that were used to develop the regression equations and the range and mean of their R 2 values. The mean of group data ranges from 24 to 210, while the lowest and highest R 2 values are 0.11 and 0.98, respectively. The quantity of data in a group and R 2 values shows that the equations collated in Tables 1,  2, 3, 4, 5, 6, 7, 8, and 9 may produce satisfactory estimation of UCS when they are used to estimate UCS for deposits of similar rock type.        Genetic rock type codes representing varying spectral absorptions using reflectance spectroscopy   Table 15 shows the information about the statistics of the regression equations in Tables 11, 12, 13 and 14, which includes the range and mean of group of data that were used to develop the regression equations and the range and mean of their R 2 values. The mean of group data ranges from 46 to 150, while the lowest and highest R 2 values are 0.33 and 0.99, respectively. The R 2 value for regression equations using physical properties are higher than those using mechanical properties. This may indicate that the regression equations using physical properties produce low errors when they are used to estimate UCS.

Multiple Regression
Multiple regression is an extension of simple regression. It is used to predict the value of a variable based on the value of two or more other variables. The concept of multiple regression reflects the likelihood that a variable may have relationship with more than one variable. In such case, all the independent variables can be systematically combined to estimate a dependent variable (Aiken et al. 1991). Assuming groups of data (Y i ; X ai …X zi ); where X ai …X zi are vector of independent variables from X a . . .X z , i = 1, …, n Represent Igneous with UCS below and above 20 MPa respectively representing the number of data for each independent variable, and Y i a real-valued dependent variable for the ith observation, a regression equation f is a model that makes a prediction _ Y of Y for a potentially new input vectors X a …X z , written as: Like simple regression, multiple regression for estimating UCS can take any form such as linear, logarithmic, exponential, power, and polynomial forms (Majdi and Rezaei 2013;Cheshomi et al. 2015;Ng et al. 2015;Madhubabu et al. 2016;Armaghani et al. 2018), and the difference in the models will reflect how f(X a …X g ) f X a . . .X z ð Þ in Eq. (3) is expressed for each regression equation.

Artificial Intelligence
Artificial intelligence refers to the simulation of human intelligence in machines that are programmed       UCS and I s(50) in MN/m 2 5.2 Support Vector Machine SVM models are supervised learning models with associated learning algorithms that analyse data used for classification and regression analysis (Aboutaleb et al. 2018). It is an approach of artificial intelligence that enables non-linear mapping of an n-dimensional input space into a higher-dimensional feature space where, for example, a linear classifier can be used. The method can train non-linear models based on the structural risk minimization principle that seeks to minimize an upper bound of the generalization error rather than minimize the empirical error as implemented in other neural networks (Khandelwal et al. 2010). The approach has been used in rock mechanics to estimate UCS (Ceryan 2014;Ren et al. 2019). Table 18 lists some SVM-based estimation of UCS from other rock properties. The analysis of the R 2 of the studies compiled show that SVM models have R 2 value ranging from 0.60 to 0.99.

Fuzzy Inference System
A fuzzy inference system (FIS) is a system that uses fuzzy set theory to map inputs to outputs (Gokceoglu and Zorlu 2004). Fuzzy logic accomplishes machine intelligence by providing a mean for representing and reasoning about human knowledge that is imprecise by nature (Gupta and Kulkami 2013). Fuzzy inference is a method that interprets the values in the input vector and based on some sets of rules, assigns values to the output vector. In fuzzy logic, the truth of any statement becomes a matter of a degree. FIS has been used in rock mechanics to estimate rock properties. Specifically, the technique has been to estimate UCS from other rock properties (Grima and Babuška 1999;Gokceoglu and Zorlu 2004;Karakus and Tutmez 2006;Heidari et al. 2018). Table 19 lists some FIS-based estimation of UCS of different rock types from other rock properties. The analysis of the R 2 of the studies compiled show that FIS models have R 2 values ranging from 0.64 to 0.98.

Genetic Programming
Genetic programming (GP) is a technique of evolving programs, starting from a population of usually random programs, fit for a task by applying operations analogous to natural genetic processes to the population of programs. It is a technique for the automatic generation of computer programs by means of natural selection (Beiki et al. 2013). The GP process starts by creating a large initial population of programs that are random combinations of elements from the problemspecific function sets and terminal sets. Improvements are made possible by stochastic variation of programs and selection according to pre-specified criteria for judging the quality of a solution (Brameier and Banzhaf 2001). GP has been used in rock mechanics for estimating UCS from other properties (Canakci et al. 2009;Armaghani et al. 2018). Table 20 lists some studies where GP has been used to estimate UCS from other rock properties. The statistics of the R 2 values of the models generated for the studies listed in the table shows a range of 0.63-0.97.

Hybrid Artificial Neural Network
ANN has several disadvantages such as long training time, unwanted convergence to local instead of global optimal solution, and large number of parameters (Liou et al. 2009). To overcome these drawbacks, there have been attempts to remedy some of these disadvantages by combining ANN with another algorithm that can take care of a specific problem. Hybrid forms of ANN such as ANFIS, PSO-ANN, ICA-ANN,         Based on the database developed, typical ranges and mean of data used in developing the regressions, and the range and mean of the R 2 values of regressions for estimating UCS from other rock properties were evaluated and summarised. The empirical relationships considered in this study include simple regressions, multiple regressions, and artificial intelligence-based relations for estimating UCS using approaches such as ANN, SVM, FIS, GP, and hybrid ANN like ANFIS, PSO-ANN, ICA-ANN, and GA-ANN. The database of regression equations between UCS and other rock properties provides a systematic and logical assemblage of empirical relations that can be used in mining engineering practice. The relationships between UCS and other rock properties can be assessed to decide on the regression equation to be used for estimation of UCS at a specific site for a rock type. This will eliminate the problem of overestimation or underestimation of rock properties often encountered when regression equations are used to estimate the UCS. In addition, the database will serve as a useful companion to rock characterization approaches developed for mining and geotechnical application, especially when there is need to perform model selection and when quantifying the variability of UCS at a project site. The database will be particularly beneficial at small to medium-sized project sites, where rock properties data are often too sparse and there is need to estimate UCS of rock for mine planning and design purposes. A future study can investigate the possibility of developing an approach to rank the reliability of the regression equations in the database when they are used for estimation of UCS.
Funding Open access funding provided by University of Oulu including Oulu University Hospital.

Declarations
Conflict of interest The authors declare that there are no known conflicts of interest.
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