Sequential Methods for Coupled Geomechanics and Multiphase Flow

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146 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS Dimensionless pressure(P/P i ) Dimensionless pressure(P/P i ) 1 0.95 0.9 0.85 0.8 0.75 η=0.33 η=0.37 Pressure τ=3.33, α=1.0 0.7 0 0.005 0.01 0.015 0.02 0.025 Dimensionless time (pore volume produced) 1 0.95 0.9 0.85 0.8 0.75 η=0.48 η=0.52 Pressure τ=∞, α=1.0 0.7 0 0.005 0.01 0.015 0.02 0.025 Dimensionless time (pore volume produced) Figure 5.6: Stability behaviors of different η’s and τ’s. Top: pressure history at the observation well when τ = 3.33. Bottom: pressure history at the observation well when τ = ∞.
5.6. NUMERICAL EXAMPLES 147 Behavior for high coupling strength In this and the previous chapters, we have shown that, unlike the drained and fixed-strain splits, the undrained and fixed-stress splits are unconditionally stable. We have not ad- dressed, however, the efficiency of the methods. Figure 5.7 shows a comparison between the undrained and fixed-stress splits for Cases 5.1 and 5.2, at very high coupling strength. In these cases, the undrained split requires many iterations to match the fully coupled solution, while the fixed-stress split does so with only one iteration per time step. In particular, we have shown in Chapter 4 that the undrained split is not convergent for incompressibility of the fluid and the solid grains. On the other hand, the fixed-stress split takes only one iteration in Case 5.2 and two iterations in Case 5.1 in order to yield a virtually exact match with the fully coupled method. The flow problem for the first iteration in the Terzaghi problem (Case 5.1) is trivial, since the driving force is instantaneous loading from the mechanical problem. This result implies that the fixed-stress split yields almost the same solution as the fully coupled method with one or two iterations. This behavior is consistent with the fact that the error amplification factor of the fixed-stress split is zero. Case 5.3—The Mandel problem Figure 5.8 indicates that the fixed-strain split (top) is stable for τ < 1, while it is unstable when τ > 1. Similar to the drained split, the fixed-strain split can yield the severe oscillations at early time. Due to the oscillations, the early time solution is not computed properly by the fixed-strain split, even though the late time solution converges to the analytical result. The fixed-stress split, on the other hand, is stable and non-oscillatory under all conditions as shown in Figure 5.8. The Mandel–Cryer effect can be also captured by the fixed-stress split showing good agreement with the analytical solution. For the vertical and horizontal displacements, the fixed-strain split causes severe oscillations even though it is stable (the top of Figures 5.9 and 5.10). But, when the coupling strength is greater than one, the fixed-strain split is unstable (the bottom of Figures 5.9
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SEQUENTIAL METHODS FOR COUPLED GEOM
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splits, are, at best, conditionally
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deeply. There was deep discussion o
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3 Stability of the Drained and Undr
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6.2.3 Undrained split . . . . . . .
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3.7 Left: the Terzaghi problem in 1
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4.1 The distribution of the error a
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5.3 Case 5.2 (1D injection-producti
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5.18 Pressure history at the observ
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2 CHAPTER 1. INTRODUCTION Reservoir
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4 CHAPTER 1. INTRODUCTION problem (
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6 CHAPTER 1. INTRODUCTION undrained
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8 CHAPTER 1. INTRODUCTION where A c
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10 CHAPTER 1. INTRODUCTION Even whe
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12 CHAPTER 1. INTRODUCTION
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14 CHAPTER 2. FORMULATION full form
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16 CHAPTER 2. FORMULATION dE dt =
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18 CHAPTER 2. FORMULATION where the
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20 CHAPTER 2. FORMULATION coupling
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22 CHAPTER 2. FORMULATION where Div
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24 CHAPTER 2. FORMULATION between g
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26 CHAPTER 2. FORMULATION
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28 CHAPTER 3. STABILITY OF THE DRAI
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196 CHAPTER 6. COUPLED MULTIPHASE F
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208 CHAPTER 7. CONCLUSIONS 7.2 Coup
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210 CHAPTER 7. CONCLUSIONS 7.2.3 Ac
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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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