Sequential Methods for Coupled Geomechanics and Multiphase Flow

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202 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS P d S w 1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36 1.34 1.32 Drained Fixed−strain Undrained Fully coupled Fixed−stress Pressure history 1.3 0 0.05 0.1 0.15 0.2 0.25 0.3 Pore volume injected t d 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Water saturation profile S w (Top layer) Undrained Fully coupled Fixed−stress 0 0 20 40 60 80 100 Horizontal distance x (m) Figure 6.6: Top: pressure history at the monitoring well during simulation. Bottom: water saturation profile at the bottom layer after simulation.
6.5. NUMERICAL EXAMPLES WITH THE STAGGERED NEWTON METHODS 203 Table 6.2: Input data for Case 6.2 Property Value Permeability (kp) 500 md Porosity (φ0) 0.3 Young modulus (E) 300 MPa Poisson ratio (ν) 0.35 Biot coefficient (b) 1.0 Bulk density (ρb) 2400 kg m −3 Oil compressibility (co) 3.5×10 −8 Pa −1 Oil density (ρo,0) 1000 kg m −3 Oil viscosity (µo) 1.0 cp Water compressibility (cw) 3.5×10 −10 Pa −1 Water density (ρw,0) 1000 kg m −3 Water viscosity (µw) 1.0 cp Initial oil pressure (Po,i) 2.125 MPa Side burden 2.125 MPa Overburden (¯σ) 3×2.125 MPa Grid spacing (∆x) 10 m Grid spacing (∆z) 5 m Grid 10 × 4 Water injection rate 5000 kg day −1 Total production rate 5000 kg day −1 splits, the undrained and fixed-stress splits yield stable and accurate solutions. The bottom of Figure 6.6 shows the water saturation profile for the bottom layer at td = 0.3. The high water saturation reduces the total fluid compressibility, and leads to violating the convergence conditions of the drained and fixed-strain splits. To analyze the convergence of each scheme, we investigate the behaviors as a function of iterations, as shown in Figure 6.7. We choose the first time step for convergence behaviors because the effect of instantaneous loading is considerable for the first time step. The drained and fixed-strain splits show severe oscillations and slow convergence. The undrained split is monotonic and yields better convergence. The fixed-stress split is monotonic and yields the best convergence behavior. There is a large difference between the fixed-stress and fully coupled solutions for the first iteration of the first time step because the fixed- stress split does not capture the instantaneous loading. But, in the second iteration, the
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SEQUENTIAL METHODS FOR COUPLED GEOM
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I certify that I have read this dis
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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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82 CHAPTER 4. CONVERGENCE OF THE DR
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100 CHAPTER 4. CONVERGENCE OF THE D
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118 CHAPTER 5. FIXED-STRAIN AND FIX
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Page 240 and 241: 214 APPENDIX A. STABILITY IN SPACE
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Page 262 and 263: 236 APPENDIX C. CONSTITUTIVE RELATI
Page 270 and 271: 244 BIBLIOGRAPHY Mech Eng 122: 145-
Page 272 and 273: 246 BIBLIOGRAPHY Aug. Murad M.A. an
Page 274: 248 BIBLIOGRAPHY ˇZeniˇsek A. 198
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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