Development Of Bayesian Approaches For Uncertainty Quantification In In Situ Stress Estimation

Knowledge of the in situ stress state is crucial for a wide range of rock mechanics applications, but stress estimation at different scales may be unreliable due to various sources of uncertainty. It is therefore critical to be able to probabilistically quantify uncertainty in stress estimation, as it both permits quantitative assessment of the reliability of stress estimations and facilitates application of the more rational probabilistic design in rock engineering. There is also a crucial need for logically incorporating stress information from other sources in order to obtain more reliable stress estimations. This thesis makes contributions of developing novel Bayesian approaches for probabilistic uncertainty quantification in stress estimation at the measurement and local scales and demonstrating how additional stress information can be incorporated for improved stress estimation. The work begins by discussing major sources of uncertainty in stress estimation and why Bayesian approaches are appropriate for uncertainty quantification. Following this, a principal contribution is made by proposing a statistical simulation approach that provides insights into the common question during local stress field estimation: how many stress measurements are required in order to obtain a reliable mean stress estimate? Then, a principal contribution is made by presenting a Bayesian multivariate model that can probabilistically quantify uncertainty in local stress field estimation; importantly this model allows rigorous quantification of uncertainty in principal stresses and formal incorporation of additional stress information through informative priors. Subsequently, a principal contribution is made by presenting a Bayesian regression model that can probabilistically quantify uncertainty in overcoring stress estimation. We demonstrate how this model can overcome two critical limitations of the classical approach to overcoring stress estimation. After this, a principal contribution is made by presenting a powerful Bayesian hierarchical regression model for analysis of multiple overcoring tests that can simultaneously quantify uncertainty in overcoring stress estimation and local stress field estimation. We show that this hierarchical model can give improved stress estimation for each individual overcoring test by borrowing information from the other tests and can account for uncertainties in these individual tests to give a more reasonable estimation of uncertainty in the mean stress.

Rock Mechanics for Natural Resources and Infrastructure Development contains the proceedings of the 14th ISRM International Congress (ISRM 2019, Foz do Iguaçu, Brazil, 13-19 September 2019). Starting in 1966 in Lisbon, Portugal, the International Society for Rock Mechanics and Rock Engineering (ISRM) holds its Congress every four years. At this 14th occasion, the Congress brings together researchers, professors, engineers and students around contemporary themes relevant to rock mechanics and rock engineering. Rock Mechanics for Natural Resources and Infrastructure Development contains 7 Keynote Lectures and 449 papers in ten chapters, covering topics ranging from fundamental research in rock mechanics, laboratory and experimental field studies, and petroleum, mining and civil engineering applications. Also included are the prestigious ISRM Award Lectures, the Leopold Muller Award Lecture by professor Peter K. Kaiser. and the Manuel Rocha Award Lecture by Dr. Quinghua Lei. Rock Mechanics for Natural Resources and Infrastructure Development is a must-read for academics, engineers and students involved in rock mechanics and engineering. Proceedings in Earth and geosciences - Volume 6 The ‘Proceedings in Earth and geosciences’ series contains proceedings of peer-reviewed international conferences dealing in earth and geosciences. The main topics covered by the series include: geotechnical engineering, underground construction, mining, rock mechanics, soil mechanics and hydrogeology.

Modern statistical and machine learning models excel at capturing complex non-linear relationships between outcomes and predictors, resulting in high accuracy. However, the complexity of these models can impede statistical inference and interpretation. This dissertation confronts and tries to overcome the emerging challenges presented by intricate models and big data. One significant challenge involves modeling and statistical inference for zero-inflated semi-continuous data. Thus, in the first part, we develop a flexible Bayesian semi-parametric mixture model for zero-inflated skewed longitudinal data, generating credible intervals for not only the mean but also any quantiles of the parameters and predictions, aiding population inference of skewed data. The model is applied to evaluate how number of binge drinking episodes changes with neuromaturation using the National Consortium on Alcohol and Neuro-Development in Adolescence data. On the other hand, credible or confidence intervals do not directly address a common question: can we identify a subset of predictions or parameters with true values exceeding a specific threshold with confidence? To tackle this, in the second part, we improve upon the inverse set estimation framework that estimates such sets by developing an approach with fewer assumptions and broader applicability to various data settings. We construct an excursion set map with probability guarantee on the North American Regional Climate Change Assessment Program data using the proposed method. Moreover, we use this new method to discover characteristics of in-patients at high risk for severe outcomes using University of California San Diego hospital data. In the third part, we apply this inverse set estimation inference framework to quantify prediction model uncertainty and develop theories and algorithms that ensure non-conservative coverage rates for a single threshold in non-asymptotic settings in regression problems. We demonstrate the effectiveness of the constructed confidence sets for uncertainty quantification and interpretation in both simulate data and PhysioNet sepsis prediction data.

The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers can find data used in the exercises and other supplementary material.

The purpose of this project was to investigate the use of Bayesian methods for the estimation of the reliability of complex systems. The goals were to find methods for dealing with continuous data, rather than simple pass/fail data; to avoid assumptions of specific probability distributions, especially Gaussian, or normal, distributions; to compute not only an estimate of the reliability of the system, but also a measure of the confidence in that estimate; to develop procedures to address time-dependent or aging aspects in such systems, and to use these models and results to derive optimal testing strategies. The system is assumed to be a system of systems, i.e., a system with discrete components that are themselves systems. Furthermore, the system is 'engineered' in the sense that each node is designed to do something and that we have a mathematical description of that process. In the time-dependent case, the assumption is that we have a general, nonlinear, time-dependent function describing the process. The major results of the project are described in this report. In summary, we developed a sophisticated mathematical framework based on modern probability theory and Bayesian analysis. This framework encompasses all aspects of epistemic uncertainty and easily incorporates steady-state and time-dependent systems. Based on Markov chain, Monte Carlo methods, we devised a computational strategy for general probability density estimation in the steady-state case. This enabled us to compute a distribution of the reliability from which many questions, including confidence, could be addressed. We then extended this to the time domain and implemented procedures to estimate the reliability over time, including the use of the method to predict the reliability at a future time. Finally, we used certain aspects of Bayesian decision analysis to create a novel method for determining an optimal testing strategy, e.g., we can estimate the 'best' location to take the next test to minimize the risk of making a wrong decision about the fitness of a system. We conclude this report by proposing additional fruitful areas of research.

This open access book provides an introduction to uncertainty quantification in engineering. Starting with preliminaries on Bayesian statistics and Monte Carlo methods, followed by material on imprecise probabilities, it then focuses on reliability theory and simulation methods for complex systems. The final two chapters discuss various aspects of aerospace engineering, considering stochastic model updating from an imprecise Bayesian perspective, and uncertainty quantification for aerospace flight modelling. Written by experts in the subject, and based on lectures given at the Second Training School of the European Research and Training Network UTOPIAE (Uncertainty Treatment and Optimization in Aerospace Engineering), which took place at Durham University (United Kingdom) from 2 to 6 July 2018, the book offers an essential resource for students as well as scientists and practitioners.

Bayesian analysis is one of the important tools for statistical modelling and inference. Bayesian frameworks and methods have been successfully applied to solve practical problems in reliability and survival analysis, which have a wide range of real world applications in medical and biological sciences, social and economic sciences, and engineering. In the past few decades, significant developments of Bayesian inference have been made by many researchers, and advancements in computational technology and computer performance has laid the groundwork for new opportunities in Bayesian computation for practitioners. Because these theoretical and technological developments introduce new questions and challenges, and increase the complexity of the Bayesian framework, this book brings together experts engaged in groundbreaking research on Bayesian inference and computation to discuss important issues, with emphasis on applications to reliability and survival analysis. Topics covered are timely and have the potential to influence the interacting worlds of biostatistics, engineering, medical sciences, statistics, and more. The included chapters present current methods, theories, and applications in the diverse area of biostatistical analysis. The volume as a whole serves as reference in driving quality global health research.

Throughout his career Dennis Lindley has insisted on thinking things through from first principles and on basing developments on firm, logical foundations. Although his fundamental contributions to Bayesian statistics and decision theory are universally recognised, it is less well known that he arrived at the Bayesian position as a result of seeking to establish a rigorous axiomatic justification for classical statistical procedures.

Almost any informed decision in the oil and gas industry depends on the forecasts of unknown quantities. The challenging part of forecasting is that forecasts are subject to uncertainties that originate from our lack of knowledge of parameters of the subsurface system. Because those parameters are an essential input for building subsurface models that are in turn input to simulators that forecast different quantities such as oil rate, it is important to have a good estimate of such parameters. Engineers rely on measured quantities like field oil rate and pressure to estimate these parameters. The process of using observations to estimate the causal factors is named an inverse problem and is challenging because it is not invertible, non-unique, and often ill-posed. The process of building a full field reservoir model involves a series of inversion processes beginning at the core scale and ending at the field scale. Because the result of each inversion process on a specific scale becomes the input for the subsequent inversion process, uncertainty propagates through scales. Every physical process in each scale is complex in itself, but the interaction of all of them is immensely difficult for the human mind to process. Quantifying and tracking uncertainties through scales is central to incorporating the effect of all uncertainties on the final decision. Traditionally, a deterministic approach is followed where parameters of a single model are modified until the simulation response is sufficiently close to the observed data. The disadvantage of this approach is that the resulting 'history matched' model is unrealistic and consequently unreliable because the applied ad hoc modifications, often, are physically inconsistent. Because the deterministic approach results in a single model, we are not able to compute any statistics to quantify uncertainty. In this dissertation we build and expand on two probabilistic approaches: the Bayesian Causal Analysis (BCA) for all scales where we compute parameter posteriors and the Bayesian Evidential Learning (BEL) protocol to quantify the uncertainty of the decision variable on the field scale. Both BCA and BEL deliver a stochastic perspective to the problem and consequently do not aim to find a single model that best matches the observed data but rather aim to identify a set of models in the vicinity of the observed data. Both approaches build a statistical relationship; in the case of BCA, between the parameters of the model and the desired data variable (output of the simulator such as oil rate versus time) and in the case of BEL between the data variable and forecast variable of interest. The statistical relationship in combination with measured data enables us to predict directly the posterior of the model parameters (BCA) or the posterior of the prediction variable (BEL) that enables us to estimate reliably the associated uncertainty. We begin by quantifying the uncertainty of a core-scale experiment from a lab experiment. Even though parameters believed to be relevant for recovery cannot be directly observed or measured in the lab, we can use the physical response measured in the lab to infer these parameters. In the case of relative permeability, we can measure the oil rate over time for a core flood and use it to infer relative permeability curves. For some reservoirs, parameterized relative permeability models are commonly used, but in other cases with advanced physical processes these models do not suffice due to their complex shape. We quantify the uncertainty of relative permeability curves that originate from an experiment that investigated a voidage-replacement-ratio of less than unity and often exhibits irregular shapes that are difficult to parameterize. We use a novel formulation of the gradual deformation algorithm to compute multiple realistic relative permeability curves that describe the measured lab data. This workflow enables the engineer to quantify the uncertainty of the relative permeabilities that can then be used as an input for the next scale. In other cases where experiments are performed to enhance the understanding of the underlying processes, a posterior distribution of parameters can reduce the risk of misinterpretation because the entire possible range of a parameter is known to the scientist. Next, we give an example of uncertainty quantification of the well scale. Produced water always contains suspended particles. If re-used for injection this water can potentially create a fracture by plugging the formation resulting in the injection pressure exceeding the breakdown pressure. This fracture will continue to grow as we inject water and potentially bypass oil by short circuiting injector/producer pairs. We first introduce a novel definition for risk for the case of a fractured water injector to answer the question: How much water can we inject and not impair oil production? Second, we explore two avenues, namely an optimization and a machine learning technique, that use observed data to reduce previously defined risk and derive posterior probability density functions for all model parameters. Both the optimization as well as the machine learning approach reduce the risk and are able to reduce the uncertainty for all model parameters. We also outline the differences of both strategies and show that the machine learning approach reduces computational time up to 70% and increases the consistency of the parameter posteriors. The posterior parameters of this chapter including the Young's modulus, the minimum horizontal stress, and the Poisson ratio describing the geomechanical model of the subsurface are used as critical input for the reservoir scale. In practice this workflow enables an engineer to propose an injection rate for a newly drilled injector given the risk attitude with the developed risk plot. Furthermore, the workflow helps and directs the engineer to select an appropriate signal, such as the bottom hole pressure, that reduces the uncertainty of the risk. Finally, we reach the field scale to provide the necessary input to make well-informed decisions. On the field scale, we aim to reduce the uncertainty in forecast variables that influences our decision at hand. Contrary to preceding scales, we do not want to compute posterior model parameters on the field scale because they do not influence our decision and also do not comprise the input for a subsequent scale. For that reason, we follow the Bayesian Evidential protocol that focuses on the forecast rather than the parameters that are used to build the models. We extend the formulation of the Bayesian Evidential Learning framework to predict the incremental oil of an enhanced oil recovery project, a polymer flood, and demonstrate those capabilities in a case study. We also develop a novel extension of the Bayesian Evidential Learning protocol to compute the value of information of a polymer pilot. The combination of decision theory and the BEL protocol enables engineers both to reduce the human bias in decision making and to assess the value of a pilot under uncertainty before the information of the pilot is collected. The main advantage of this novel work-flow is the fact that we can generate a significant number of forecasts for each decision branch so that we can derive robust statistics for good decisions. Finally, we also argue and demonstrate that the integration of the uncertainty of the fiscal model can be of paramount importance in making decisions.