Bayesian methods to treat geotechnical uncertainty in risk‑based design of open pit slopes

Abstract

Common questions asked during the process of mine design are “how much geotechnical information is required for an acceptable design” and “how to measure its confidence”. These are key aspects associated not only with the determination of parameters but more generally with the definition of the geotechnical model for design.The definition of the geotechnical model for slope design is based on four main components including the geological, structural, rock mass and hydrogeological models. Each model is described by different sets of information and parameters and is defined at a scale of interest for the purpose of the analysis of slope behaviour. In the area of slope design in particular, the estimation of geotechnical parameters is normally supported by small data sets, which are evaluated with simple statistical procedures based on frequentist concepts. The geotechnical model defined in this manner lacks a proper measure of its confidence levels, which in turn complicates judging the sufficiency of data and precludes planning the data collection based on strategy at the various stages of project development.The Bayesian approach is an alternative route to the conventional probabilistic methods used in slope design. The approach is based on a particular interpretation of probability and provides a suitable framework to treat uncertainty in the geotechnical model for slope design. Two important features of the approach are the possibility of combining data with subjective information and the ability to quantify the uncertainty of the parameters or models given the available data. The first point is especially relevant in the area of mine slope design considering that subjective information such as expert opinion or engineering judgement is a common element present in the geotechnical design process. The second point provides a contrast with the situation within the frequentist approach where the uncertainty measures apply to the data rather than to the parameters or models, which are the objects of interest to the analyst.The first part of the research focused on reviewing the concepts of uncertainty and probability to derive the arguments supporting the statement that the Bayesian approach offers a better framework for the quantification of uncertainty in the slope design process. The result of this work is illustrated with simple examples and is described in detail in the two papers included as Chapters 3 and 4. The second part of the research was aimed at demonstrating the use of the Bayesian approach for the inference of geotechnical parameters in typical situations encountered during the design of rock slopes. The examples presented in the papers included as Chapters 3 to 6 refer to the rock mass strength parameters of the Hoek-Brown criterion. These examples were used to highlight the advantages of the methodology for the quantification of geotechnical uncertainty.The core procedure of the Bayesian approach for the inference of parameters is the evaluation of the posterior probability function. There are various methods to evaluate this function as described briefly in Chapter 2. However, the specific method used in the research is the Markov Chain Monte Carlo (MCMC) simulation. This method was selected because it can be easily applied by the geotechnical practitioner using existing tools, without relying too much on the use of intricate mathematical procedures. Chapter 2 presents a summary of the principles of this technique and describes the more common MCMC algorithms. Nevertheless, all the analyses included in the thesis were carried out with a powerful MCMC sampler named ‘emcee’, which was developed and is used extensively by the astrophysics community. The sampler, as well as the models presented in the thesis, are coded in the Python programming language.The cases of Bayesian interference of parameters covered by the research include the intact rock strength parameters σci and mi, and the geological strength index (GSI) from the Hoek-Brown strength criterion. The analysis of GSI was based on a correlation commonly used in the design of mine slopes that relates GSI with the rock mass factors block volume (Vb) and joint condition (Jc). Moreover, the research also included the use of the geotechnical parameters inferred with the Bayesian approach for the analysis of the reliability of the slope and the back-analysis of slope failure to illustrate how the observed performance of the slope could be used to update the parameters. The Bayesian analysis involving the stability of the slope require an explicit representation of the slope model that can be incorporated into the posterior function. Therefore, the topic of construction of a surrogate model using the response surface (RS) methodology is also discussed in detail.The research served to identify the main features of the Bayesian methodology that make it a suitable approach for the quantification of the geotechnical uncertainty in the slope design process in mining projects. The examples presented showed the benefits of the approach by contrasting the results with those from conventional frequentist methods.