Sequential Methods for Coupled Geomechanics and Multiphase Flow

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splits, are, at best, conditionally stable. Moreover, their stability limit depends on the coupling strength only and is independent of time step size. On the other hand, the derived a-priori estimates indicates that the undrained and fixed-stress splits are unconditionally stable regardless of the coupling strength. All the results have been verified by performing numerical simulations for several test cases. To analyze the convergence rates of the various coupling algorithms, we use matrix and spectral methods. The drained and fixed-strain splits can suffer from non-convergence, even when they are stable. On the other hand, the undrained split yields first-order accuracy in time for a compressible fluid, but it exhibits slow convergence rates for high coupling strength and suffers from non-convergence for purely incompressible systems (solid grains and fluid). The fixed-stress split shows first-order accuracy in time regardless of the fluid type and coupling strength, and it yields a less stiff mechanical problem. Furthermore, the fixed-stress split requires only a few iterations to converge, even for very difficult problems with strong coupling. The stability and convergence behaviors of the four sequential methods for coupled multiphase flow and geomechanics are also analyzed using spectral and energy methods. The formulation for the flow part can be either fully implicit, or IMPES (IMplicit Pressure, Explicit Saturations). The derived a-priori estimates for the four sequential methods are similar to their single-phase counterparts. That is, the undrained and fixed-stress splits show unconditional stability, and the fixed-stress split exhibits faster convergence rates compared with the other sequential methods. Therefore, we strongly recommend the fixed-stress split with backward Euler time integration, a finite-volume scheme for flow, and a finite-element discretization for mechanics. vi
Acknowledgements This thesis is made by many people who helped me with considerable effort as follows. My deepest appreciation goes to my advisor, Prof. Hamdi A. Tchelepi in the department of energy resources engineering at Stanford. I have worked for him since 2005. For four years, I have overcome a lot of difficulty on research with which I would have been lost. Whenever I faced a problem, he always encouraged me with warm advice. He guided me to the right track, showing me valuable and critical suggestions and providing me with the freedom for various ideas. This was a great opportunity and privilege for me to work for Prof. Hamdi A. Tchelepi. Prof. Ruben Juanes in the department of civil and environmental engineering at MIT deserves my sincere appreciation from the bottom of my heart. I have worked for him for five years since 2004 when I took the first step in the U.S. He first motivated me for coupled flow and geomechanics as the Ph.D research topic. After he moved to MIT, he has still advised me with enthusiasm all the time inviting me or visiting Stanford. I would like to express my heartfelt thanks to Prof. Ronaldo I. Borja in the department of civil and environmental engineering at Stanford. His classes on computational geomechanics and plasticity have become critical resources for this thesis. Also, he listened to my research challenges and gave me fruitful advice on coupled flow and geomechanics at a turning point of my research. My sincere thanks go to other members of my Ph.D committee, Prof. Mark D. Zoback and Prof. Khalid Aziz, who provided me with valuable comments. I would like to thank Prof. Ignacio Romero in the Universidad Politecnica de Madrid vii
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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214 APPENDIX A. STABILITY IN SPACE
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226 APPENDIX B. MISCELLANEOUS DERIV
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244 BIBLIOGRAPHY Mech Eng 122: 145-
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248 BIBLIOGRAPHY ˇZeniˇsek A. 198
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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