PREFACE

"I think that when we know that we actually do live in uncertainty, then we ought to admit it; it is of great value to realize that we do not know the answers to different questions. This attitude of mind - this attitude of uncertainty - is vital to the scientist, and it is this attitude of mind which the student must first acquire"

Richard P. Feynman, Noble Laureate in Physics, 1965

This book offers five substantial case studies on decision making under uncertainty for subsurface systems. The strategies and workflows designed for these case studies are based on a Bayesian philosophy, tuned specifically to the particularities of the subsurface realm. Models are large and complex; data are heterogeneous in nature; decisions need to address conflicting objectives; the subsurface medium is created by geological processes that are not always well understood; and expertise of a large variety of scientific and engineering disciplines need to be synthesized.

There is no doubt that we live in an uncertain time. With growing population, resources such as energy, materials, water, and food will become increasingly critical in their exploitation. The subsurface offers many such resources, important to the survival of humankind. Drinking water from groundwater systems is gaining in importance, as aquifers are natural purifiers and can store large volumes. However, the groundwater system is fragile, subject to contamination from agriculture practices and industries. Before renewables become the dominant energy sources, oil and gas will remain a significant resource in the next few decades. Geothermal energy both deep (power) and shallow (heating) can contribute substantially to alleviating reliance on fossil fuels. Mining minerals used for batteries will aid in addressing intermittency of certain renewables, but mining practices will need to address environmental concerns.

Companies and governmental entities involved in the extraction of these resources face considerable financial risk because of the difficulty in accessing the poorly understood subsurface and the cost of engineering facilities. Decisions regarding exploration methods, drilling, extraction methods, and data-gathering campaigns often need to balance conflicting objectives: resource versus environmental impact, risk versus return. This can be truly addressed only if one accepts uncertainty as integral part of the decision game. A decision based on a deterministic answer when uncertainty is prevailing is simply a poor decision, regardless of the outcome. Decisions and uncertainty are part of one puzzle; one does not come before the other.

Uncertainty on key decision variables such as volumes, rates of extraction, time of extraction, spatiotemporal variation on fluid movements needs to be quantified. Uncertainty quantification, in this book shortened to UQ, requires a complex balancing of several fields of expertise such as geological sciences, geophysics, data science, computer science, and decision analysis. We gladly admit that we do not have a single best solution to UQ. The aim of this book is to provide the reader with a principled approach, meaning a set of actions motivated by a mathematical philosophy based on axioms, definitions, and algorithms that are well understood, repeatable, and reproducible, as well as a software to reproduce the results of this book. We consider uncertainty not simply to be some posterior analysis but a synthesized discipline steeped in scientific ideas that are still evolving. Ten chapters provide insight into our way of thinking on UQ.

Chapter 1 introduces the five case studies: an oil reservoir in Libya, a groundwater system in Denmark, a geothermal source for heating buildings in Belgium, a contaminated aquifer system in Colorado, and an unconventional hydrocarbon resource in Texas. In each case study, we introduce the formulation of the decision problem, the types of data used, and the complexity of the modeling problem. Common to all these cases is that the decision problem involves simple questions: Where do we drill? How much is there? How do we extract? What data to gather? The models involved on the other hand are complex and high dimensional, the forward simulators time-consuming. The case studies set the stage.

Chapter 2 introduces the reader to some basic notions in decision analysis. Decision analysis is a science, with its own axioms, definitions, and heuristics. Properly formulating the decision problem, defining the key decision variables, the data used to quantify these, and the objectives of the decision maker are integral to such decision analysis. Value of information is introduced as a formal framework to assess the value of data before acquiring it.

Chapter 3 provides an overview of the various data science methods that are relevant to UQ problems in the subsurface. Representing the subsurface requires a high-dimensional model parametrization. To make UQ problems manageable, some form of dimension reduction is needed. In addition, we focus on several methods of regression such as Gaussian process regression and CART (classification and regression trees) that are useful for statistical learning and development of statistical proxy models. Monte Carlo is covered extensively as this is instrumental to UQ. Methods such as importance sampling and sequential importance resampling are discussed. Lastly, we present the extension of Monte Carlo to Markov chain Monte Carlo and bootstrap; both are methods to address uncertainty and confidence.

Chapter 4 is dedicated to sensitivity analysis (SA). Although SA could be part of Chapter 3, because of its significance to UQ, we dedicate a single chapter to it. Our emphasis will be on global SA and more specifically Monte Carlo-based SA since this family of methods (Sobol', regionalized sensitivity analysis, CART) provides key insight into understanding what model variables most impact data and prediction variables.

Chapter 5 introduces the philosophy behind Bayesian methods: Bayesianism. We provide a historical context to why Bayes has become one of the leading paradigms to UQ, having evolved from other paradigms such as induction, deduction, and falsification. The most important contribution of Thomas Bayes is the notion of the prior distribution. This notion is critical to UQ in the subsurface, simply because of the poorly understood geological medium that drives uncertainty. The chapter, therefore, ends with a discussion on the nature of prior distributions in the geosciences, how one can think about them and how they can be established from physical, rather than statistical principles.

Chapter 6 then extends on Chapter 5 by discussion on the role of prior distribution in inverse problems. We provide a brief overview of both deterministic and stochastic inversion. The emphasis lies on how quantification of geological heterogeneity (e.g., using geostatistics) can be used as prior models to solve inverse problems, within a Bayesian framework.

Chapter 7 is perhaps the most novel technical contribution of this book. This chapter covers a collection of methods termed Bayesian evidential learning (BEL). Previous chapters indicated that one of the major challenges in UQ is model realism (geological) as well as deal with large computing times in forward models related to data and prediction responses. In this chapter, we present several methods of statistical learning, where Monte Carlo is used to generate a training set of data and prediction variables. This Monte Carlo approach requires the specification of a prior distribution on the model variables. We show how learning the multivariate distribution of data and prediction variables allows for predictions based on data without complex model inversions.

Chapter 8 presents various strategies addressing the decision problem of the various case studies introduced in Chapter 1. The aim is not to provide the best possible method but to outline choices in methods and strategies in combination to solve real-world problems. These strategies rely on materials presented in Chapters 2-7.

Chapter 9 provides a discussion of the various software components that are necessary for the implementation of the different UQ strategies presented in the book. We discuss some of the challenges faced when using existing software packages as well as provide an overview of the companion code for this book.

Chapter 10 concludes this book by means of seven questions that formulate important challenges that when addressed may move the field of UQ forward in impactful ways.

We want to thank several people who made important contributions to this book, directly and indirectly. This book would not have been possible without the continued support of the Stanford Center for Reservoir Forecasting. The unrestricted funding provided over the last 30 years has aided us in working on case studies as well as fundamental research that focuses on synthesis in addition to many technical contributions in geostatistics, geophysics, data science, and others. We would also like to thank our esteemed colleagues at Stanford University and elsewhere, who have been involved in many years of discussion around this topic. In particular, we would like to thank Tapan Mukerji (Energy Resources Engineering & Geophysics), who has been instrumental in educating us on decision analysis as well as on the geophysical aspects of this book. Kate Maher (Earth System Science) provided important insights into the modeling of the case study on uranium contamination. We thank the members of the Ensemble project funded by...