Technical noteA generalized Bayesian approach for prediction of strength and elastic properties of rock
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 qu, deformation modulus of intact rock Ei, point load strength index I) of different rock types (e.g., sedimentary, igneous, metamorphic). Hendron et al. (1970) reported engineering properties (e.g., qu, Ei, rock mass deformation modulus Em) of weak sedimentary rocks (i.e., black and gray Pennsylvanian shales in Illinois, United States). Rowe and Armitage (1984) developed a relationship between qu and Em 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 Em, 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 qu and Ei based on the standard penetration test (SPT). Heap et al. (2020) developed a relationship for prediction of Em of volcanic rocks that is based on the model of Hoek and Diederichs (2006). To predict Ei in Hoek and Diederichs (2006) for volcanic rock, Heap et al. (2020) proposed a relationship between Ei 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 (Em) 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 Ei - qu Bayesian predictive models. Wang and Aladejare (2016), using a Bayesian framework, proposed joint probability distribution functions for Ei - qu 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, Gs, water content, w, porosity, ϕ, rock quality designation, RQD, uniaxial compressive strength, qu, point load strength index, I, Brazilian tensile strength, σt, deformation modulus, Ei, 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 (mi) based on the results of unconfined compression test (i.e., qu) 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 qu 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 qu 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.
Section snippets
Model formulation
Reliability analysis of underground structures requires the use of probabilistic models that can properly quantify the design uncertainty. Following the model formulation proposed by Gardoni et al. (2002), a probabilistic model is formulated for estimation of physical properties of rock:
where Q is a quantity of interest or its suitable transformation into a new space to satisfy the additivity, homoskedasticity and normality assumptions; γ(x, θ) = ∑j=1nθjhj(x) is a function expressed in
Databases
We developed two databases: (i) “training data” and (ii) “testing data.” Training databases are first used for evaluation of the existing predictive models. The training data are then used for calibration of probabilistic model parameters. The proposed models are then evaluated using both training and testing databases. The testing database would allow for an independent evaluation of our proposed models.
Existing models for rock properties
Various investigators have proposed predictive models for rock physical properties (see Labuz et al. 2018 for a review of models for rock strength; see Hoek and Diederichs, 2006 and Gercek, 2007 for review of models for rock elastic properties). The following sections are intended to provide a brief summary of the relevant models which are subsequently evaluated using the entire database (i.e., training and testing data).
Updated models and their evaluation
A general Bayesian framework (Gardoni et al., 2002; Tabandeh et al., 2020) is used to develop probabilistic models for the prediction of the strength and elastic properties of rock. The general form of the likelihood function is given in Eq. (4) and is used to estimate the model parameters in Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15. We use our training databases to estimate the unknown model parameters Θ = (θ, σ) for different rock types for each predictive
Conclusions
In this paper, databases of in situ and laboratory data on rock mass deformation modulus (Em), deformation modulus of intact rock (Ei), unconfined compressive strength (qu), point load index (I), Poisson's ratio (ν), and water content (w) for rocks are developed. The parameters affecting each rock mass property are discussed. The available models for the prediction of these important rock properties are reviewed and evaluated using the databases developed in this study. A Bayesian approach is
Data availability statement
All data, models, and code generated or used during the study appear in the submitted article.
Author statement
Pouyan Asem collected the data, performed analysis and wrote the manuscript. Paolo Gardoni assisted in data reduction, analysis and writing of manuscript.
Declaration of Competing Interest
There are no competing interests for this manuscript.
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