A generalized Bayesian approach for prediction of strength and elastic properties of rock

Highlights

• Data on rock E_m, E_i, I, ν, and w

• The parameters affecting each rock mass property are discussed

• The available models for the prediction of these important rock properties are evaluated

• A Bayesian approach is used to develop probabilistic models for the prediction of rock engineering properties

• The accuracy of the model prediction improves using rock-specific relationships

Abstract

Rock mass elastic and strength properties are needed for calculation of deformation and determination of stability of underground structures. Most available models for prediction of rock mass properties are site-specific, are deterministic, and cannot properly propagate uncertainty in reliability analysis of underground structures. A generalized Bayesian approach is used to develop probabilistic predictive models for the rock mass properties. A set of training and testing databases for rock mass deformation modulus, unconfined compressive strength, and Poisson's ratio based on point load strength index, water content, and geological strength index are also developed. Evaluation of the existing models using our databases show rather large mean absolute percentage errors. Our models, calibrated using training databases, (i) are probabilistic and include model parameter statistics needed for uncertainty propagation in reliability analysis, (ii) are rock-specific, (iii) show smaller prediction errors compared to existing models, and (iv) can be updated as new information become available.

Introduction

Behavior of underground structures is governed by the site-specific physical charateristics of rock (Goodman, 1989; Hoek and Brown, 2019), and the determination of such properties has been of much interest to geophysicists (e.g., Zisman, 1933; Hoek and Brown, 1980; Al-Ajmi and Zimmerman, 2005, Labuz et al., 2018). Rock properties, however, are not routinely available during the preliminary design phase of most underground structures and thus are evaluated using predictive models (e.g., Wuerker, 1956; Deere and Miller, 1966; Rosenbaum et al., 1997; Hoek and Diederichs, 2006; Feng and Jimenez, 2014; Ng et al., 2015; Bertuzzi et al., 2016; Hoek and Brown, 2019; Sun et al., 2019; Salmi and Sellers, 2021; Asem et al., 2021). Uncertainty in model prediction stems from the selected model form, inherent variability in rock masses, and measurement errors. Most deterministic models do not consider these errors. Different researchers have used a variety of methods to mitigate the limitations of deterministic models (e.g., Napa-García et al., 2018; Pandit and Babu, 2018; Zhang et al., 2018; Huang et al., 2019; Shadab Far et al., 2019; Mohammed et al., 2020; Tabandeh et al., 2020; Wang et al., 2020; Xu et al., 2020a; Xu et al., 2020b; Li et al., 2021; Pan et al., 2021; Xiao et al., 2021; Zhao et al., 2021).

Zisman (1933) reported measurements of Young's modulus and Poisson's ratio for granite, norite, gneiss, diabase, marble, limestone and sandstone. Wuerker (1956) developed “annotated” tables to summarize the typical values of physical properties of various rocks (e.g., andesite, basalt, diabase, dolomite, gabbro, gneiss, granite, limestone, marble, sandstone, slate). Deere and Miller (1966) developed a database for physical properties (e.g., unconfined compressive strength q_u, deformation modulus of intact rock E_i, point load strength index I) of different rock types (e.g., sedimentary, igneous, metamorphic). Hendron et al. (1970) reported engineering properties (e.g., q_u, E_i, rock mass deformation modulus E_m) of weak sedimentary rocks (i.e., black and gray Pennsylvanian shales in Illinois, United States). Rowe and Armitage (1984) developed a relationship between q_u and E_m which they back-calculated from in situ rock socket and plate load tests in sandstone, mudstone, shale, siltstone, chalk and andesite. Hoek and Diederichs (2006) developed a relationship between the geological strength (GSI) and E_m, which was mainly based on data reported by Chern et al. (2004) for sedimentary, igneous and metamorphic rocks in China and Taiwan. Asem (2018) provided a summary and evaluation of the existing models and their applicability to predict the properties of weak rock masses (sedimentary, igneous, metamorphic). Asem (2020) proposed a framework for the prediction of q_u and E_i based on the standard penetration test (SPT). Heap et al. (2020) developed a relationship for prediction of E_m of volcanic rocks that is based on the model of Hoek and Diederichs (2006). To predict E_i in Hoek and Diederichs (2006) for volcanic rock, Heap et al. (2020) proposed a relationship between E_i and porosity (ϕ). Other investigators have proposed similar models (see Table 1).

This discussion indicates that most of available models (i) are site-specific (e.g., Hendron et al., 1970; Ng et al., 2015; Sun et al., 2019; Azarafza et al., 2020; Buyer et al., 2020; Salmi and Sellers, 2021), (ii) are deterministic - the uncertainty in the model parameters has not been reported - (e.g., Wuerker, 1956, Deere and Miller, 1966, Hendron et al., 1970, Rowe and Armitage, 1984, Hoek and Diederichs, 2006, Azarafza et al., 2020, also see Table 1), (iii) do not distinguish rock type (e.g., Rowe and Armitage, 1984; Hoek and Diederichs, 2006), and (iv) focus on properties of intact rock (see Table 1); because a rock mass typically consists of relatively intact rock blocks separated by discontinuity surfaces, such models may not adequately represent the rock mass behavior especially at shallow depths where discontinuity spacing is typically smaller than the size of underground structures.

Rosenbaum et al. (1997) developed a Bayesian approach for predicting rock lithology based on analyses of surface deposits and applied their proposed framework to an underground facility at Aspo Hard Rock Laboratory in Sweden. Miranda et al. (2009) used a Bayesian framework for updating information on deformation modulus (E_m) of granite at an underground facility at Venda Nova, Portugal as new data became available. Using tests on igneous, metamorphic and sedimentary rocks, Feng and Jimenez (2014) developed rock-specific E_i - q_u Bayesian predictive models. Wang and Aladejare (2016), using a Bayesian framework, proposed joint probability distribution functions for E_i - q_u relationships for sandstone from Ekbatan dam site, Iran. Aladejare and Wang (2017) developed a database of sedimentary, igneous and metamorphic rock properties (e.g., bulk density, ρ, specific gravity, G_s, water content, w, porosity, ϕ, rock quality designation, RQD, uniaxial compressive strength, q_u, point load strength index, I, Brazilian tensile strength, σ_t, deformation modulus, E_i, Poisson's ration, ν) to evaluate typical ranges of intact rock and rock mass variability but did not propose a probabilistic model. Aladejare and Wang (2019) developed a Bayesian framework for characterization of Hoek-Brown material constant (m_i) based on the results of unconfined compression test (i.e., q_u) and used a limited dataset for granite from Forsmark, Sweden to illustrate their approach. Guevara-Lopez et al. (2019) and Asem et al. (2019) developed probabilistic models for q_u based on corrected point load strength index for a diameter of 50 mm (I₅₀) for sedimentary, metamorphic and igneous rocks. Aladejare et al. (2020) developed probabilistic models for q_u based on punch tests (e.g., point load strength index, I₅₀) using a Bayesian updating method for sandstone from Khammam, India but did not report the statistics of model parameters.

This discussion indicates that (i) most of the existing probabilistic models or frameworks have been applied to site-specific data and not to a universal database, (ii) most models are developed for intact rock properties, (iii) models are not rock-specific and (iv) the statistics (e.g., standard variation, σ, correlation coefficient) for model parameters are often not reported.

In the subsequent sections, two databases for physical properties of sedimentary, ignesous and metamorphic rocks are introduced. These databases consist of “training” and “testing” subgroup. The relationship between rock strength and elastic properties with different indices of rock mass is discussed. These databases and a rigorous Bayesian methodology are used to develop rock-specific probabilistic models for prediction of intact rock and rock mass properties.