Sequential Methods for Coupled Geomechanics and Multiphase Flow

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4.1 The distribution of the error amplification factors of the drained (top) and undrained (bottom) splits with respect to pressure diffusivity χ and fre- quency. The coupling strength τ is 0.05. . . . . . . . . . . . . . . . . . . . . 97 4.2 The difference of the magnitude of the error amplification factors between the drained and undrained splits. . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Case 4.1: the Terzaghi problem in one dimension (left). Case 4.2: the consolidation problem in two dimensions (right). . . . . . . . . . . . . . . . . . 99 4.4 Convergence analysis of Case 4.1: pressure (top) and displacement (bottom). The coupling strength τ = b 2 M/Kdr, is 0.95. FC, Dr, and Und indicate the fully coupled, drained split, and undrained split methods, respectively. ∆td = 4cv∆t/(Lz) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Non-convergence of the drained split with one iteration for Case 4.1: pressure (top) and displacement (bottom). . . . . . . . . . . . . . . . . . . . . . . . . 101 4.6 Convergence of the undrained split with one iteration for Case 4.1: pressure (top) and displacement (bottom). . . . . . . . . . . . . . . . . . . . . . . . . 102 4.7 Comparison between more iterations and refined time step size in the drained split for Case 4.1: pressure (top) and displacement (bottom). The coupling strength is close to one (τ = 0.95). . . . . . . . . . . . . . . . . . . . . . . . 103 4.8 Convergence analysis of pressure (top) and displacement (bottom) for a nearly incompressible fluid, where τ = 9.5 × 10 4 and cf = 3.5 × 10 −13 Pa −1 . The staggered method is used. The undrained split is not convergent, showing almost zeroth-order accuracy. The fully coupled method shows first-order accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.9 Spatial distributions of pressure (top) and displacement (bottom) for a nearly incompressible fluid. The undrained split produces large errors, not converg- ing to the true solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 xx
4.10 Convergence analysis of pressure (top) and displacement (bottom) for an incompressible fluid, where τ = 9.5×10 11 ∼ = ∞ and cf = 3.5×10 −20 Pa −1 ∼ = 0. The undrained split is not convergent, showing zeroth-order accuracy. But, the fully coupled method shows first-order accuracy. . . . . . . . . . . . . . 107 4.11 Spatial distributions of pressure (top) and displacement (bottom) for an incompressible fluid. The undrained split does not converge to the true solutions. There is no pressure jump and displacement by the overburden at early time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.12 Convergence analysis of Case 4.2 on pressure (top) and displacement (bottom). Pd = P/Pi , ud = u/ √ LxLz, and ∆td = 4∆tcv/L 2 x, where cv is the consolidation coefficient along the horizontal direction (Abousleiman et al., 1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.13 Non-convergence of the drained split with one iteration for Case 4.2: pressure (top) and displacement (bottom). ux is the horizontal displacement. . . . . 112 4.14 Comparison between more iterations and refined time step size in the drained split for Case 4.2: pressure (top) and displacement (bottom). . . . . . . . . 113 4.15 Convergence of the undrained split with one iteration for Case 4.2: pressure (top) and displacement (bottom). . . . . . . . . . . . . . . . . . . . . . . . . 114 4.16 Comparison of the rate of convergence at low (top) and high χ (bottom) for Case 4.1. The coupling strength τ is 0.95. . . . . . . . . . . . . . . . . . . . 115 4.17 Comparison of the rate of convergence at low (top) and high χ (bottom) for Case 4.2. The coupling strength τ is 0.77. . . . . . . . . . . . . . . . . . . . 116 5.1 Iteratively coupled methods for flow and geomechanics. Left: fixed-strain split. Right: fixed-stress split. . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Case 5.1 (the Terzaghi problem). Evolution of the dimensionless pressure as a function of dimensionless time. Shown are the results for the fully coupled method, the fixed-strain, and fixed-stress splits. Top: coupling strength τ = 0.83. Bottom: coupling strength τ = 1.21. . . . . . . . . . . . . . . . . . . . 141 xxi
Page 1 and 2: SEQUENTIAL METHODS FOR COUPLED GEOM
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82 CHAPTER 4. CONVERGENCE OF THE DR
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118 CHAPTER 5. FIXED-STRAIN AND FIX
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164 CHAPTER 6. COUPLED MULTIPHASE F
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208 CHAPTER 7. CONCLUSIONS 7.2 Coup
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212 CHAPTER 7. CONCLUSIONS
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214 APPENDIX A. STABILITY IN SPACE
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226 APPENDIX B. MISCELLANEOUS DERIV
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236 APPENDIX C. CONSTITUTIVE RELATI
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244 BIBLIOGRAPHY Mech Eng 122: 145-
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246 BIBLIOGRAPHY Aug. Murad M.A. an
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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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