Elastic wave propagation and scattering in anisotropic fractured media

In this thesis, we develop several methods for the delineation and analysis of anisotropic fractures in the subsurface. Our principal goal is to develop methods to use seismic data to infer fracture properties. In turn, we relate this information to the fluid flow properties of anisotropically fractured media. The first problem is to examine media with cracks oriented over a range of directions rather than begin perfectly aligned. We use this model of crack distributions to examine both velocity and permeability variations in an anisotropically fractured rock. The analysis begins by considering a material under a uniaxial stress, as cracks in a rock mass subjected to a uniaxial stress will be preferentially closed depending on the angle between the fracture normal vectors and the direction of the applied stress. If the prestress fracture orientation distribution is isotropic, the effective elastic properties of such a material after application of the stress are then transversely isotropic due to the overall alignment of the cracks still open. Velocity measurements in multiple directions are used to invert for the probability density function describing orientations of crack normals in such a rock. The information on fracture distribution obtained from the velocity inversion allows an estimation of the anisotropic permeability of the fractured rock system. Permeability estimates are based on the number of cracks open in each direction. This approach yields a prediction of permeability as a function of the angle from the uniaxial stress axis. The inversion for crack orientation is applied to ultrasonic velocity measurements on Barre granite, and permeability predictions for this sample are presented. The inversion results are good and reproduce velocity measurements well, and the permeability predictions show some of the expected trends. Initial comparisons of the predictions with available permeability data, however, show deviations suggesting that further information on partial crack closure and connectivity of cracks should be included into the permeability model. After considering the more general range of crack orientations considered in the inversion procedure, we analyze the behavior of elastic waves on encountering a fractured region which is too small for velocity variations to become apparent, and therefore too small for observations of shear wave splitting. This new problem is solved through the study of the scattering of elastic waves from isolated fracture zones. When the scattering zone is much smaller than the wavelength of an incident plane wave and has a relatively small difference in elastic properties from the surrounding background medium, the Born approximation allows an estimate of the radiation pattern of elastic wave Rayleigh scattering of both compressional and shear waves due to a perturbation of any combination of the 21 independent elastic constants. Examination of radiation patterns for incident shear and compressional waves shows that the shear waves are the most sensitive to the alignment of fractures in anisotropic zones. As the polarization of the incident S-wave ranges from perpendicular to parallel to the fractures, the amplitude of the scattered waves goes to zero. The calculation of scattered wavefields is extended to larger regions of inhomogeneity by application of a Ray-Born technique. This approach applies ray methods to the computation of Green's tensors for the background medium and uses the Born approximation to determine the scattered wavefield from each volume element within a discretized model of heterogeneity. Comparisons of Ray-Born results to the complete solution for scattering from an elastic sphere show that that method works fairly well for wavelengths on the order of five times larger than the scale lengths typical of the heterogeneity, but then breaks down to the failure of the Born approximation. With this restriction in mind, the method is applied to a hypothetical layered earth model containing a thin, laterally extensive fracture zone. The results confirm that scattering from shear waves will give unique information on fracture orientation even for this extended zone. On the other hand, compressional waves are more useful in inference of nature of the fluid filling the cracks. Modeling of scattered waves in VSP data from the Lardarello geothermal field in Italy demonstrates the applicability of the method and suggests that at least in this locality, anisotropic fracturing is not responsible for the observations. Analysis of the Fresnel zones affecting reflections from the thin fracture zones responsible for the scattering allows a delineation of regions of more intense fracturing, information of importance for the development of geothermal resources. Thesis Advisor: M. Nafi Toksiz Title: Professor of Geophysics Acknowledgements This thesis has benefited from the influence of a number of people. My advisor, Nafi Toks6z, has provided a research environment at the Earth Resources Laboratory with support and direction encouraging me to take my research interests into a variety of areas while contributing his own insights into the nature of rock physics and seismic data. In addition, his prodding helped me to gain the confidence to publish my work. Close work with Ari Ben-Menahem has also been an invaluable aspect of my education. His enthusiasm for and understanding of mathematical geophysics has thoroughly strengthened my understanding of wave propagation. I also learned from him the joy of truly beginning to understand subtle aspects of this field, not to mention the insights of his analyses of World Cup soccer matches! Many conversations with Arthur Cheng have guided my study of the physics of fractured rock and other aspects of the strategy of science at ERL. Roger Turpening has always been supportive and, as usual, has pushed all students in an effort to drive us into examining "real data". Of course, the opportunity he provided for me to visit the Michigan reef experimental site was also very valuable in helping me to understand how unrealiable the collection of this "real data" can be! Wafik Beydoun has always been encouraging and helpful in regard to the implementation and use of Ray-Born methods, while Fausto Batini and Gildes Omnes both provided data and helpful suggestions in the study of the VSP data from the Badia 1A well in Lardarello. Interaction with a number of other staff and faculty members in the Department of Earth, Atmospheric and Planetary Sciences has also been very beneficial. Some of these are Greg Duckworth and Ted Madden, who served as advisors in my early years, Vernon Cormier, and Batakrishna Mandal. ERL administrative and computer staff have also provided a great deal of help in the practical side of getting through MIT. I appreciate the help of Sara Brydges, Naida Buckingham, Liz Henderson, Jane Maloof, Al Taylor and Sue Turbak. One of the most important sides of an education is the interaction with other students. My officemates Lisa Block and Delaine Thompsen have been friendly and encouraging through my time here, although they didn't always appreciate my efforts at air conditioning the office in winter. Arcangelo Sena has always been eager to help with mathematical problems and discussions of some very difficult problems in our various collaborations, and he was a great traveling companion on our trips to Venezuela and to Israel. Ted Charrette was very helpful in his capacity as an nCUBE employee at helping me implement my computer codes on the nCUBE parallel computer, which made the efficient development of the Ray-Born code possible. I have also benefited greatly from working with and enjoying the company of a number of other students, including Chris Bradley, Bob Cicerone, Karl Ellefsen, Jack Foley, Joe Matarese, Jeff Meredith, Michael Prange, Ed Reiter, and Craig Schultz. On a more personal note, I am indebted to my parents and brother for their support and encouragement throughout my education. Even when things were most difficult and stressful, they continued to be a very positive influence. Many friends from the MIT Baptist Student Fellowship and Metropolitan Baptist Church have helped me to maintain a balanced perspective on life as a whole during my stay at MIT. I am especially grateful for the positive spirit of chaplain Betsy Draper and Rev. David Draper ("No P.N.! "), who helped me see how to pursue an understanding of the truly important things in life. Missy Tackett has also been supportive and encouraging as I struggled with this thesis, as were Keith Eady and Steve Walker. This work has been supported by the Reservoir Delineation and Full Waveform Acoustic Logging Consortia at the Earth Resources Laboratory, a National Science Foundation Graduate Fellowship, an nCUBE Fellowship, and the Defense Advanced Research Project Agency through contract #F 19628-90-K-0057 administered by the Air Force Geophysics Laboratory.