STATISTICAL LEARNING AND INFERENCE OF SUBSURFACE PROPERTIES UNDER COMPLEX GEOLOGICAL UNCERTAINTY WITH SEISMIC DATA A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Anshuman Pradhan December 2020 © 2020 by Anshuman Pradhan. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/sg756xs1514 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Tapan Mukerji, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Biondo Biondi I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Jef Caers Approved for the Stanford University Committee on Graduate Studies. Stacey F. Bent, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract Attempting to characterize, image or quantify the subsurface using geophysical data for exploration and development of earth resources presents interesting and unique challenges. Subsurface heterogeneities are the result of abstract paleo geological events, exhibiting variability that is spatially complex and existent across multiple scales. This leads to significantly high-dimensional inverse problems under complex geological uncertainty, which are computationally challenging to solve with conventional geophysical and statistical inference methods. In this dissertation, we discuss these challenges within the context of subsurface property estimation from seismic data. We discuss three specific seismic estimation problems and propose methods from statistical learning and inference to tackle these challenges. The first problem we address is that of incorporating constraints from geological history of a basin into seismic estimation of P-wave velocity and pore pressure. In particular, our approach relies on linking velocity models to the basin modeling outputs of porosity, mineral volume fractions, and pore pressure through rock-physics models. We account for geologic uncertainty by defining prior probability distributions uncertain basin modeling parameters. We have developed an approximate Bayesian inference framework that uses migration velocity analysis in conjunction with well and drilling data for updating velocity and pore pressure uncertainty. We apply our methodology in 2D to a real field case from the Gulf of Mexico. We demonstrate that our methodology allows for building a geologic and physical model space for velocity and pore-pressure prediction with reduced uncertainty. In the second problem, we investigate the applicability of deep learning models for conditioning reservoir facies models, parameterized by geologically realistic geostatistical models such as training-image based and object-based models, to seismic iv data. In our proposed approach, end-to-end discriminative learning with convolutional neural networks (CNNs) is employed to directly learn the conditional distribution of model parameters given seismic data. The training dataset for the learning problem is derived by defining and sampling prior distributions on uncertain parameters and using physical forward model simulations. We apply our methodology to a 2D synthetic example and a 3D real case study of seismic facies estimation. Our synthetic experiments indicate that CNNs are able to almost perfectly predict the complex geological features, as encapsulated in the prior model, consistently with seismic data. For real case applications, we propose a methodology of prior falsification for ensuring the consistency of specified subjective prior distributions with real data. We found modeling of additive noise, accounting for modeling imperfections and presence of noise in the data, to be useful in ensuring that a CNN, trained on synthetic simulations, makes reliable predictions on real data. In the final problem, we present a framework that enables estimation of low- dimensional sub-resolution reservoir properties directly from seismic data, without requiring the solution of a high dimensional seismic inverse problem. Our workflow is based on the Bayesian evidential learning approach and exploits learning the direct relation between seismic data and reservoir properties to efficiently estimate reservoir properties. The theoretical framework we develop allows incorporation of non-linear statistical models for seismic estimation problems. Uncertainty quantification is performed with approximate Bayesian computation. With the help of a synthetic example of estimation of reservoir net-to-gross and average fluid saturations in sub- resolution thin sand reservoir, several nuances are foregrounded regarding the applicability of unsupervised and supervised learning methods for seismic estimation problems. Finally, we demonstrate the efficacy of our approach by estimating posterior uncertainty of reservoir net-to-gross in sub-resolution thin sand reservoir from an offshore delta dataset using pre-stack seismic data. v Acknowledgements As I was penning the final words of this Ph.D. dissertation, I could not help but recall a beautiful quote from poet Robert Frost: “My object in living is to unite my avocation and my vocation, as my two eyes make one in sight”. I realized that during the six years that I have spent working on this Ph.D., I actually did make my “vocation and avocation one in sight”. Beginning my Ph.D. with a purely geophysical background, I did not have the slightest idea that one day I would feel so passionately about subjects such as Bayesian inference, machine learning, geological modeling and uncertainty quantification. Learning and incorporating ideas from these scientific and mathematical domains in my research has been somewhat of an uphill task, but one that I have thoroughly enjoyed. And as it is with accomplishing all challenging tasks, completion of this Ph.D. would not have possible without the guidance, help and support of many. First and foremost, I would like to heartfully thank my advisor Tapan Mukerji for the outstanding guidance and mentorship he has constantly provided me with. Tapan has meticulously guided me during the numerous ‘tricky situations’ I have found myself in with my research. Tapan has always motivated me to question my thinking and explore ideas beyond my zone of comfort. Tapan has inspired me to take so many excellent classes at Stanford. Tapan has patiently listened to my personal problems and shown his understanding and constant support. Tapan has helped me grow tremendously as a scientist, as a researcher and as a person. I could not have hoped for a better Ph.D. advisor than Tapan. I would also like to thank my Ph.D. defense committee members: Jef Caers, Biondo Biondi, Gary Mavko, Allegra Hosford Scheirer and Louis Durlofsky. They have been instrumental in the development of several ideas and insights that went into writing vi this dissertation. I am grateful to Jef for helping me think about uncertainty quantification in a way that I had never done before. Learning about seismic imaging and rock physics from Biondo and Gary was a true pleasure. Discussions on geology with Allegra (as well as the dinners at her lovely place) will be something that I will sorely miss. During my Ph.D., I had the opportunity to collaborate with many research groups at the Stanford school of Earth Sciences and, as a result, network with several outstanding researchers and students. For the engaging research discussions and helpful technical collaborations, I would like to thank (1) Celine, David, Lijing, Alex (B&M), Riyad, Liang, Lewis, Geet and Ogy from SCERF group; (2) Nader, Noelle, Ken, Les, Laura, Tanvi, Best, Zack, Anatoly, Marcelo from BPSM group; (3) Humberto, Abdullah, Priyanka, Ankush from SRB group; and (4) Huy from SEP group. “Uniting my avocation and my vocation” would definitely have been a daunting task without the love and support of the valuable friends I made while at Stanford. I consider myself truly blessed to have been surrounded by amazing friends. Thank you, Vishal, Mustafa, Noe, Jihoon, Wen, Halldora and Erik for all the fun times, for staying with me till the very end of most ‘Friday Social’s and for being there whenever I needed you. Special appreciation and love go to my group of ‘Can Bee’ friends that I made towards the end of my Ph.D. Rayan, Rustam, Josue, Sergei, Tyler, Albert, Anna, Dalila and Margariete; you all ensured I made through the dreariest of times, smiling and cheerful! No word can ever capture the true extent of the contribution that my parents, Jayashree Pradhan and Sanjeev Kumar Pradhan, have made towards completion of this Ph.D. Right from my childhood, they have motivated me for academic excellence and inspired me to always do better than myself. Their motivation, support, care and unconditional love has always