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NUMERICAL SIMULATION OF PORE-SCALE HETEROGENEITY AND ITS EFFECTS ON ELASTIC, ELECTRICAL AND TRANSPORT PROPERTIES A DISSERTATION SUBMITTED TO THE DEPARTMENT OF GEOPHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Ratnanabha Sain November 2010 iv Abstract “Asato mā sad gamaya Tamaso mā jyotir gamaya” (Sanskrit) meaning: From ignorance, lead me to truth; From darkness, lead me to light; This dissertation describes numerical experiments quantifying the influence of pore-scale heterogeneities and their evolution on macroscopic elastic, electrical and transport properties of porous media. We design, implement and test a computational recipe to construct granular packs and consolidated microstructures replicating geological processes and to estimate the link between process-to-property trends. This computational recipe includes five constructors: a Granular Dynamics (GD) simulation, an Event Driven Molecular Dynamics (EDMD) simulation and three computational diagenetic schemes; and four property estimators based on GD for elastic, finite-elements (FE) for elastic and electrical conductivity, and Lattice- Boltzmann method (LBM) for flow property simulations. v Our implementation of GD simulation is capable of constructing realistic, frictional, jammed sphere packs under isotropic and uniaxial stress states. The link between microstructural properties in these packs, like porosity and coordination number (average number of contacts per grain), and stress states (due to compaction) is non-unique and depends on assemblage process and inter-granular friction. Stable jammed packs having similar internal stress and coordination number (CN) can exist at a range of porosities (38-42%) based on how fast they are assembled or compressed. Similarly, lower inter-grain friction during assemblage creates packs with higher coordination number and lower porosity at the same stress. Further, the heterogeneities in coordination number, spatial arrangement of contacts, the contact forces and internal stresses evolve with compaction non-linearly. These pore-scale heterogeneities impact effective elastic moduli, calculated by using infinitesimal perturbation method. Simulated stress-strain relationships and pressure-dependent elastic moduli for random granular packs show excellent match with laboratory experiments, unlike theoretical models based on Effective Medium Theory (EMT). We elaborately discuss the reasons why Effective Medium Theory (EMT) fails to correctly predict pressure-dependent elastic moduli, stress-strain relationships and stress-ratios (in uniaxial compaction) of granular packs or unconsolidated sediments. We specifically show that the unrealistic assumption of homogeneity in disordered packs and subsequent use of continuum elasticity-based homogeneous strain theory creates non-physical packs, which is why EMT fails. In the absence of a rigorous theory which can quantitatively account for heterogeneity in random granular packs, we propose relaxation corrections to amend EMT elastic moduli predictions. These pressure-dependent and compaction-dependent (isotropic or uniaxial) correction factors are rigorously estimated using GD simulation without non-physical approximations. Further, these correction factors heuristically represent the pressure- dependent heterogeneity and are also applicable for amending predictions of theoretical cementation models, which are conventionally used for granular packs. For predicting stress-ratios in uniaxial compaction scenario, we show the vi inappropriateness of linear elasticity-based equations, which use elastic constants only and do not account for dissipative losses like grain sliding. We further implement and test a computational recipe to construct consolidated microstructures based on different geological scenarios, like sorting, compaction, cementation types and cement materials. Our diagenetic trends of elastic, electrical and transport properties show excellent match with laboratory experiments on core plugs. This shows the feasibility of implementing a full-scale computational-rock- physics-based laboratory to construct and estimate properties based on geological processes. However, the elastic property estimator (FE simulation) shows limitations of finite resolution while computing elastic properties of unconsolidated sediments and fluid-saturated microstructures. vii viii Acknowledgement My PhD would not have ―Piled-higher-and-Deeper‖ (© JC) without the inspiration, support and benevolence of several people. This work was financially supported by Stanford Rock Physics and Borehole Geophysics (SRB) project and its affiliates, for which I am truly indebted. I acknowledge my qualifying exam committee—Gary Mavko, Tapan Mukerji, Greg Beroza and Wei Cai—for finding my research proposal worthy of consideration, and my reading committee—Gary, Tapan, Jack Dvorkin and Mark Zoback—for accepting this work as fully adequate in scope and quality for PhD. I owe a lot of gratitude to my adviser, Gary Mavko. I have truly cherished countless discussions with him: technical, professional and personal. Throughout my 5+ years at Stanford, I have wondered the depths of his scientific insight, knowledge, ix and ideas. He has helped me see physics on his yellow notepads and through his amazing illustrations, specially the one with rubber balls and Hawaiian sand. Thanks for accepting me as your student, inspiring my thoughts to bloom, teaching me how to present my talks, and allowing my weird schedule! Truly, I have not met a professor with a better sense of humor and (stunning) photographic skills. Thanks to my co-adviser, Tapan Mukerji, for introducing me to diverse subjects including this dissertation subject. Ever since my first acquaintance with Stanford, he has guided and advised me regarding scientific research, courses (he possibly knows all Stanford courses), simulations, books, market investments and lately (sea)food. Contrary to my fears, his transition from the small cubicle in Mitchell basement to 3X- sized room in Green basement only increased the length of our discussions! Often, these discussions extended to our walks from Main Quad to Stanford Avenue. The bulk of what we do at SRB is inspired by Amos Nur—our father figure. His work and ideas carry the vision of understanding physics in uncertain conditions. Thanks Amos! Thanks to Mark Zoback and Jack Dvorkin for your excellent recommendations. I am grateful to Tiziana Vanorio for showing me how real laboratory experiments are done: they helped me better understand my simulations. I would also like to thank Mike Payne, Chris Finn and Ezequiel Gonzalez for the wonderful mentorship during my industry internships, which made me appreciate the practical industry problems. A special note of thanks to Ronny Hofmann and Jim Spencer for their incredible suggestions regarding my research. I am grateful to Trilok N. Singh for encouraging me during undergraduate research. Suggestions from Dawn Burgess about editing the thesis were really helpful. I thank Dennis Michael, Bob Clapp and Stanford CEES for access and help with the computational cluster. There are certain people at Stanford who visibly and invisibly make our stay smoother: SRB and department administrators. I thank Fuad Nijm and Tara Ilich for helping me enormously in terms of administrative hassles. Discussions with both of them have ranged from serious financial and technical problems to banter about habits, cuisines and cultures. Thanks to Susan Phillips-Moskowitz and Lauren Nelson as well. x I am grateful to all my colleagues for sharing time during my stay at Stanford: Ezequiel, Kyle, Kevin, Jeetu, Franklin, Kaushik, Tanima, Richa, Cinzia, Carmen, Hiroki, Piyapa, Ramil, Indrajit, Abhishek, Danica, Stephanie, Adam Tew, Kenichi, Adam Allan, Amrita, Dario, Yu and Xi. I would like to specially acknowledge Youngseuk Keehm and Fabian Krzikalla for our discussions on computational methods. Special thanks to my lunch-n-coffee-mates: Madhur (partner-in-crime for photoshoots) and Nishank, who have been excellent friends. Thanks to Anu and Mita for helping me on various occassions. I sincerely thank Tanima for lots of things, especially her sisterly care during my surgery. I won‘t forget those long—mostly meaningful—discussions with Kaushik on the patio (sometimes at wee hours of night)! Lest I forget, I should acknowledge my very special friends: Madhav, Amit, Saurabh, Durga, Sandeep, Sreekar, Saurav, Jayank, Indira and Harry. These people provided me enough strength and distractions, so that I could stay put through my grad life. Friendship with Madhav was forged on Day One at Stanford and evolved through uncountable arguments, cooking and sports (someday I will beat you). Amit, Saurabh and Durga were like my family for the past 3 years. Thanks, mates! Lastly, any amount of acknowledgement would be insufficient for my extended family: especially my grandparents, my parents and my would-be-in-a-month-wife and her mother! My daddy, Naba Kumar, and my ma, Ratna, have continuously provided encouragement for everything I have done until now, and this PhD is their dream. While my childhood dream was to be an accountant, my accountant Daddy‘s dream was to be a scientist—and so we swapped dreams! Ma has always ensured appropriate (boundary) conditions during my real experiments with life since early childhood. My other-Ma, Purnima, has enquired that I eat-and-sleep properly during most strenuous times. My
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