Introduction

Deformation modulus of rock mass (Em) is a rock mass property that depicts the stability or deformability of rock structure. It is an important aspect of many mining engineering practices because it is required during designs and analyses of surface and underground structures in rocks (Shen et al. 2012; Aladejare and Idris 2020). In addition, Em is essential during classification of rocks for mining engineering applications. These classifications are frequently used to determine the suitability of rocks for various designs and construction applications (Sachpazis 1990; Aladejare and Wang 2017). In situ field tests such as plate bearing, flatjack, pressure chamber, borehole jacking, and dilatometer tests are used to determine Em (Ulusay and Hudson 2007). However, these in situ tests are expensive, time-consuming, and often difficult to perform. Direct measurements of Em are not often available for most small- to medium-scale mining engineering projects because of the bottlenecks (huge cost and technical difficulties) associated with its in situ determination. As a result of this, several empirical models have been proposed in the literature to indirectly estimate Em from other commonly available and easier-to-estimate rock parameters. Empirical models are also used as inputs in Bayesian framework for probabilistic characterization of Em (Aladejare and Wang 2019).

The different empirical models reported in the literature can be broadly grouped into three categories. The first category uses intact rock properties as input parameters. Commonly used intact rock properties for estimating Em comprise uniaxial compressive strength (UCS) and Youngs’ modulus (Ei) (Agan 2014; Diamantis and Migiros, 2019; Kincal and Koca 2019). The idea behind the use of intact rock properties is that they can be used to approximate rock mass properties since rock mass consists of both intact rock and any associated discontinuities, such as joints, faults, fissures, or bedding planes. However, there has been concern on the reliability of using intact rock properties to estimate rock mass property like deformation modulus. The dimension and natural condition of discontinuities in rock mass are not fully represented in intact rock specimen. This makes it difficult to account for effects of discontinuities in rock mass while estimating Em using models developed from intact rock properties. Hence, the development of the empirical models is the second category, which uses rock mass classification indices as input parameters. Rock mass classification indices commonly used to estimate Em comprises geological strength index (GSI) (Gokceoglu et al 2003; Kayabasi et al. 2003; Isik et al. 2008; Agan 2014), rock mass index (RMi) (Palmström 1995; Palmström and Singh 2001), rock mass rating (RMR) (Bieniawski 1978; Chun et al. 2006; Mohammadi and Rahmannejad 2010; Khabbazi et al. 2013; Nejati et al. 2014; Alemdag et al. 2015; Kavur et al. 2015), rock quality designation (RQD) (Gardner 1987; Zhang and Einstein 2004), and tunnelling quality index (Q) (Barton et al. 1980; Grimstad and Barton 1993; Palmström and Singh, 2001; Kang et al. 2013; Ajalloeian and Mohammadi 2014). The third category uses both intact rock properties and rock mass classification indices as input parameters. In the literature, it is common to find different combinations for estimation of Em such as Ei and RMR (Ramamurthy 2004; Sonmez et al. 2006; Galera et al. 2007; Shen et al. 2012), Ei and RQD (Gardner 1987; Zhang and Einstein 2004), UCS and GSI (Hoek and Brown 1997; Beiki et al. 2010), UCS and RQD (Alemdag et al. 2016), UCS and Q (Barton 1996), Is(50) and RQD (Karaman et al. 2015), and other combinations may include three different inputs, made up of one or two intact properties or rock mass classification indices (Hoek et al. 2002; Beiki et al. 2010; Agan 2014).

The empirical models developed for estimation of Em are in different regression types and functions, scattered in different research articles and sources in the literature. Selecting the appropriate model when there is a need for indirect estimation of Em for a project is often difficult, and practitioners merely resort to using available models without evaluating the reliability of a variety of models. The idea of compiling the empirical models available in the literature was inspired by the UCS regression models compiled by Aladejare et al. (2021). This study aims to develop a similar comprehensive database for empirical models of Em and conduct a reliability check on the models. Zhang (2017) reviewed some empirical models for estimating Em. His efforts majorly centred on empirical models using rock mass classification indices and some intact rock properties as inputs. Empirical models using majorly intact rock properties are not considered in his study. In addition, no study has logically compared the reliability of using different input categories for estimation of Em, with a view to ranking their compatibilities for Em estimation.

To systematically compile different empirical models for estimation of rock mass deformation modulus scattered in the literature, an extensive literature review was performed to collect and compile information on the models reported in the literature. Information on the empirical models such as number and range of data used for their development, coefficient of determination (R2), rock types for which the models are developed, country of origin, condition for use if there is any and the authors of the models are compiled and presented. Then, a comparative analysis was performed using some illustrative project sites to assess the reliability and issues with the use of the models for estimation of rock mass deformation modulus.

Database of empirical models for estimating rock mass deformation modulus

As noted in the previous section, numerous empirical models have been developed and are available for estimation of Em at rock sites. These empirical models present simple, time and cost-effective methods of estimating Em. However, the reliability of the models depends on factors such as the number and quality of data used in their development among others. As will be noted from Tables 1, 2, 3 and 4, more than seventy empirical models have been developed for the estimation of Em in the last few decades. With the numerous models available, rock engineers and practitioners are often faced with the challenge of deciding which model(s) should be used for estimation of Em, and which one provides the most reliable result. It is more difficult because of the different categories of input property that have been used to estimate Em in the literature. Hence, a database of empirical models for estimating rock mass deformation for igneous, sedimentary, and metamorphic rocks was developed from an extensive literature review. A considerable number of research articles from internationally leading rock mechanics and mining engineering journals, covering a period from 1978 to 2019, were used to compile information on simple multiple regression models of deformation modulus for different rock types. The different empirical models in the database are classified based on the category of their input rock property and are presented in the following subsections.

Table 1 Simple regressions between \({E}_{m}\) and intact rock properties
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Table 2 Simple regressions between \({E}_{m}\) and rock mass classification indices
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Table 3 Multiple regressions of \({E}_{m}\) from combination of intact rock property and one or more rock mass classification indices
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Table 4 Multiple regressions of \({E}_{m}\) from combination of three or more input parameters
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Category I: models relating Em with intact rock properties

Table 1 summarizes the available empirical models for estimating deformation modulus using intact rock properties. The table listed the number of data used to develop each model, the range of the data, R2 value, and the rock type from which the equation was developed. Only UCS and Ei are intact rock properties that are reported as input for simple regressions to estimate Em. Most of the models are developed from a limited number of intact rock property data and the range of the data is small.

Category II: models relating Em with rock mass classification indices

Table 2 summarizes the available empirical models for estimating deformation modulus using rock mass classification indices for different rock types. The table shows that GSI, RMi, RMR, RQD, and Q have been used as indices for estimating Em in the literature. The number of models using rock mass classifications indices in the literature exceeds that of intact rock properties. This might be because of the opinion of researchers that since Em is a rock mass property, it is better estimated using rock mass classification indices. Unlike the intact rock properties, rock mass classification indices have conditions for use during the estimation of Em. For instance, the model proposed by Isik et al (2008) for estimating Em using GSI requires that the site RMR > 27, Bieniawski (1978) model using RMR gives negative values of Em when RMR is smaller than 50, Palmström (1995) model for estimating Em using RMi requires that the RMi > 1, while Ajalloeian and Mohammadi (2014) model using Q as input requires that the Q values should be greater than 1 but less than 50. It is evident that the use of rock mass classification indices for estimation of Em requires careful assessment because of conditions that are required to be satisfied for most of the models.

Category III: models relating Em with the combination of intact rock properties and rock mass classification indices

Table 3 summarizes the empirical models for estimating deformation modulus for different rock types using a combination of intact rock properties and rock mass classification indices. The table also includes a model proposed by Agan (2014) which combines two different rock mass classifications indices (RMR and GSI) to estimate Em. It can be noted that Ei and RMR have been mostly combined to estimate Em in the literature. Other notable combinations include Ei and RQD, and UCS and GSI among others. Like models using only rock mass classification indices as inputs, some of the models using combined inputs have conditions for use during estimation of Em. It is interesting to note that the models only have conditions in one of the parameters and not in both input parameters. For instance, the models proposed by Gardner (1987), Ramamurthy (2004), and Zhang and Einstein (2004) have both Ei and RQD as inputs but only have conditions pertaining to RQD, and the model proposed by Hoek and Brown (1997) has UCS and GSI as inputs but has a condition for use in only UCS. In this study, empirical models for estimating Em from a combination of three or more input parameters are presented in Table 4. Most of the models have at least one rock mass classification index as input. Like models using only rock mass classification indices as inputs, some of the models in Table 4 have conditions for use during the estimation of Em.

Illustrative data and performance indicators

To comparatively assess the reliability as well as rank the compatibility of using different input categories to estimate Em, rock data from two sites were used as illustration. Two different performance indicators were employed to compare the estimated Em from different empirical equations with measured Em by Karaman et al. (2015) and Kıncal and Koca (2019). Details of the rock data from the sites and the performance indicators are discussed in subsequent subsections.

Rock data

To compare the reliability of using different input categories for estimation of Em, three categories of models are considered. These comprise models using only intact rock properties as input parameters (Table 1), models using rock mass classification indices as input parameters (Table 2), and models using a combination of intact rock properties and rock mass classification indices as input parameters (Tables 3 and 4). To demonstrate the site-specific nature of the rock, two different sites were used for illustrations. The first site, named as Çambası HEPP site, is located 83 km away from the city of Trabzon in the northeast of Turkey. The lithology of the tunnel route mainly consists of basalt, metabasalt, limestone, dacite, volcanic breccia (Karaman et al. 2015). Table 5 presents the results of field and laboratory studies for the site. The second site is along a 9.3 km length of the İzmir subway route in western Turkey (Kincal and Koca 2019). Table 6 presents the results of field and laboratory studies for the andesites at the railway elevation of the İzmir subway. The measured Em values by Karaman et al. (2015) and Kıncal and Koca (2019) (see Tables 5 and 6) are included for comparison with estimated Em values from empirical models for the sites. The data points in Tables 5 and 6 are thirty-seven and thirty-two, respectively. These amounts of data are considered sufficient to perform statistical analysis. A minimum of thirty data are recommended data points for estimating reliable statistical analysis (Walpole et al 1993; Wang and Aladejare 2015).

Table 5 Properties of different rocks from Çambası HEPP site, Turkey (Karaman et al. 2015)
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Table 6 Properties of andesitic rock materials along the İzmir subway line, Turkey (Kıncal and Koca, 2019)
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Performance indicators for the evaluation of empirical equations

The values of absolute average relative error percentage (AAREP) (Eq. (1)) and variance accounted for (VAF) (Eq. (2)) are adopted in this research as indicators to assess and compare the predictions of Em from different categories of input properties in empirical models:

$$\mathrm{AAREP}=\frac{1}{{\mathrm{n}}_{\mathrm{t}}}{\sum }_{i=1}^{{\mathrm{n}}_{\mathrm{t}}}\left|\frac{{\mathrm{Em}}_{i\mathrm{ mes}}-{\mathrm{Em}}_{i\mathrm{ est}}}{{\mathrm{Em}}_{i\mathrm{ mes}}}\right|\times 100$$
(1)
$$\mathrm{VAF}=\left(1-\frac{Var({\mathrm{Em}}_{\mathrm{mes}}-{\mathrm{Em}}_{\mathrm{est}})}{Var({\mathrm{Em}}_{\mathrm{mes}})}\right)\times 100$$
(2)

where nt is the number of rock data used in analysis; \({\mathrm{Em}}_{i\mathrm{ mes}}\) and \({\mathrm{Em}}_{i\mathrm{ est}}\) are the Em obtained from field testing and empirical estimation, respectively, and \(Var\) denotes variance. AAREP as defined is effectively the measure of the errors associated with the estimation. Therefore, the smaller the AAREP, the more reliable the estimation (Aladejare 2021). VAF generally depicts the extent of deviation between two sets of values. For exact prediction (i.e., when the predicted values do not differ from the measured values), the value of VAF will be 100%. On the other hand, VAF reduces as the predicted values differ from the measured values and may tends to 0%. If the quality of the estimation is extremely poor, the VAF can be negative.

Comparison of Em estimation from different input categories

Estimations of Em from different models

Majority of the empirical models presented in Tables 1, 2, 3 were used to estimate Em from their respective input data. The estimated Em were plotted together with measured Em by Karaman et al. (2015) and Kıncal and Koca (2019) in Figs. 1, 2, 3, 4, 5, 6. For Figs. 1, 2, 3, 4, 5, 6, there are subset figures with varying numbering from a, b, c, and d to show the input parameter for Em estimation in each case. Figures 1, 2, 3 show the measured and predicted values of Em for the first illustrative site (Çambası HEPP site) using input categories I–III, respectively, while Figs. 4, 5, 6 show the measures and predicted values of Em for the second illustrative site (İzmir subway site) using input categories I–III, respectively. For the first site (Figs. 1, 2, 3), some of the models show prediction capacities to some extent. It is obvious that Em estimations from models using intact rock properties differ considerably from the measured Em by Karaman et al. (2015). This is more evident in the Em estimations from the model using UCS as input (see Fig. 1). From visual examinations of Figs. 2 and 3, it can be observed that most of the models using rock mass classification indices and those using a combination of intact rock properties and rock mass classification indices provide Em estimations that are close to the measured Em by Karaman et al. (2015). For the second site (Figs. 4, 5, 6), some of the models also show prediction capacities to some extent. For intact rock properties as input, Ei provides closer estimations to the measured Em. In a similar pattern to the estimations of Em for the first site, the model using UCS provides estimations that differ considerably from the measured Em by Kıncal and Koca (2019) (see Fig. 4). From visual examinations of Fig. 5, it can be observed that most of the models using rock mass classification indices provide Em estimations that are close to the measured Em. In the case of models using rock mass classification indices and intact rock properties, the estimations of Em differ considerably from the measured Em by Kıncal and Koca (2019) (see Fig. 6). This may be because the models use UCS as one of the inputs, and UCS has poor prediction capacity as noted in Figs. 1 and 4. The poor estimation capacity of UCS in this study may also arise from the model that was used with it. Agan (2014) was developed from data of weak rocks. Therefore, the performance of the model may not be convincing when it is used to estimate Em of hard rocks. The model using UCS as input produced non-consistent estimations in two different sites may mean that UCS is not an effective index for estimating Em.

Fig. 1
figure 1

Comparison of Em estimation from models using results of intact rock properties

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Fig. 2
figure 2

Comparison of Em estimation from models using results of rock mass classification indices

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Fig. 3
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Comparison of Em estimation from models using intact rock properties and rock mass classification indices

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Fig. 4
figure 4

Comparison of Em estimation from models using intact rock properties

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Fig. 5
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Comparison of Em estimation from models using rock mass classification indices

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Fig. 6
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Comparison of Em estimation from models using intact rock properties and rock mass classification indices

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Prediction performance of the empirical models from different input categories

AAREP and VAF were used to statistically analyze the performance of each empirical model when estimating Em. Tables 7 and 8 present the results of the analyses for the first and second illustrative sites whose data are presented in Tables 5 and 6, respectively. Comparison of the results shows that the prediction performance of empirical models varies with performance indicators. In this study, the best five prediction performances are identified for each of the performance indicators. In Table 7, the models with the best performances when AAREP is considered are mostly models (4 out of 5) using rock mass classification indices (category II models) while the remaining model is the type using the combination of intact rock properties and rock mass classification indices (category III models). The five models with the best performances have AAREP values ranging from 24.07 to 35.31%. From Table 7, it can be observed that the models using RMR, RMi, and Q as input produced estimations of Em with minimal errors compared to the rest of the models. For VAF, models using rock mass classification indices and those using the combination of intact rock properties and rock mass classification indices (categories II and III models) have the best performances, which range from 59.81 to 72.96%. The two highest VAF values are from models in category III. In Table 8, the models with the best performances when AAREP is considered are mostly models in category III (3 out of 5) while the remaining models (2 out of 5) are category II models, which use GSI as input. The five models with the best performances have AAREP values ranging from 25.20 to 55.15%. From Table 8, it can be observed that the models using GSI and a combination of Ei and RMR, and Ei and RQD as inputs produced estimations of Em with minimal errors compared to the rest of the models. For VAF, models in category I with Ei as the input parameter (2 out of 5), category II with GSI as the input (1 out of 5), and category III with input being Ei and RMR, and Ei and RQD have the best performances, which range from 68.95 to 88.11%. The three highest VAF values are from models in categories II and III. Although two models in category I, which use Ei as input, are among the five models with the best performances, their VAF values are the lowest among the five models. From the statistical analyses of the two different illustrative sites performed in this study, the models with the best prediction performances are the empirical models in category II, which use rock mass classification indices as input. This is followed by models in category III, which use the combination of intact rock properties and rock mass classification indices as input, while the models in category I, which use intact rock properties as input, has the least prediction performance.

Table 7 Prediction performance of empirical models in estimation of Em using data in Table 5 (first site)
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Table 8 Prediction performance of empirical models in estimation of Em using data in Table 6 (second site)
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Limitations

From the analysis in this study, it is obvious that there is no single equation for Em estimation. It is impossible to obtain only one equation applicable to all rock types. The differences among the equations proposed by the researchers may be related to variations in the type and characteristics of the rock and rock mass studied, range of dataset used, and the different regions studied. This study only considers commonly available rock parameters in its analysis, some rock parameters are not considered. Furthermore, data from two sites are used to analyze the performance of the models in this study. Also, different sites will have peculiar geology and rock condition. Therefore, the study may not fully depict rock behavior and the interaction of rock parameters with the empirical models for all geological conditions and rock sites. If possible or necessary, mining engineers and practitioners may perform range analysis of the data of their preferred or selected empirical model to compare with available data at rock site when there is a need to use empirical model for Em estimation.

Conclusions

The deformation modulus of rock mass is a required input for designs and analyses of most mining engineering projects. The field tests for the determination of deformation modulus of rock mass are tedious and expensive. Therefore, mining engineers often estimate deformation modulus based on empirical models and values of intact rock properties as well as rock mass classification indices. Using empirical models for rock mass property comes with the uncertainty caused by the site-specific nature of rock properties, the specific nature of rock mass and the type of input data that are used in the empirical models.

In this paper, empirical models for the estimation of deformation modulus of rock masses published in the literature have been reviewed and compiled. The empirical models were grouped into three categories according to the type of their required input variables. Models using only intact rock properties as input were grouped in category I, those using rock mass classification indices were grouped in category II, while those using the combination of intact rock properties and rock mass classification indices were grouped in category III. The prediction performance of the models was assessed using well-acknowledged published in situ data from two sites. The data for the analyses include 37 data of intact rock, deformation modulus of rock mass and rock mass classification results from Çambası HEPP site, Turkey (Karaman et al 2015) and 32 data of intact rock, deformation modulus of rock mass and rock mass classification results along the İzmir subway line, Turkey (Kıncal and Koca 2019).

Performance indicators such as AAREP and VAF were used in the general assessment of the empirical models considering the data from the two sites adopted in this study. Although notable deviations were observed in the estimations by most of the empirical models, some of the empirical models using rock mass classification index and those using the combination of intact rock properties and rock mass classification indices produced acceptable results. Among the models that use intact rock properties as input, it is observed that a model with uniaxial compressive strength as input does not provide a good approximation of deformation modulus of the rock mass. On the other hand, models with Youngs’ modulus as input performed better. Hence, they can be used to approximate the deformation modulus of rock mass when there is no possibility of performing field mapping for rock mass classification. Rock mass classification indices such as RMR, RMi, Q, and GSI are promising in estimation of deformation modulus of rock mass. This is because rock mass classification indices work on rock masses and can reflect the geo-structures such as joints and other discontinuities contained in rock mass which affect rock mass properties like deformation modulus.

The results of model analysis in this study are typical, and the reliability of an empirical model depends on the number and quality of available data. The results show that values of deformation modulus from different empirical models can be vary significantly. Therefore, it is recommended that the estimation of deformation modulus of rock mass should be based on multiple models, to get possible range of values of deformation modulus of rock mass.