Sequential Methods for Coupled Geomechanics and Multiphase Flow

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$Drilling-induced tensile wall-fractures - Stanford University$

194 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS ≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫ = − Using Equations 6.94 and 6.116, Equation 6.92 becomes Since dp n+α J α∆dε n : dσ ′n+α dΩ = 0. (6.116) dΣ n+1 2 E − dΣn 2 E + (2α − 1) dΣ n+1 − dΣ n 2 ≤ 0. (6.117) E = 0, Equation 6.103 provides dp n+1 2 M − dpn 2 M = − (2α − 1) dp n+1 − dp n 2 . (6.118) M Then Equations 6.117 and 6.118 yield dχm n+1 2 Nm − dχm n 2 = dζm n+1 2 Nm Tm − dζm n 2 Tm = dΣ n+1 2 E − dΣn 2 E + dp n+1 2 M − dpn 2 M dΣ ≤ −(2α − 1) n+1 − dΣ n 2 E + dp n+1 − dp n 2 , (6.119) M from which B-stability is achieved if 0.5 ≤ α ≤ 1 when we solve the mechanical problem, A u,m ss . Therefore, from Equations 6.114 and 6.119, the fixed-stress split has B-stability if 0.5 ≤ α ≤ 1.
6.5. NUMERICAL EXAMPLES WITH THE STAGGERED NEWTON METHODS 195 6.5 Numerical Examples with the staggered Newton methods We test the four sequential methods based on a staggered Newton approach. We use one and two dimensional oil production problems with water injection and overburden. Case 6.1 Water injection and oil production in a 1D poroelastic medium. The driving force is due to injection, production, and an overburden (the left picture in Figure 6.2). Case 6.2 Water injection and oil production in a 2D poroelastic medium. The compaction and dilation of the reservoir occur due to production of oil and injection of water, respectively (the right picture in Figure 6.2). Figure 6.2: Water injection and oil production. Left: coupled multiphase flow and geomechanics in a 1D poroelastic medium with overburden. Right: coupled multiphase flow and geomechanics in a 2D poroelastic medium with overburden and side burden
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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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xiv
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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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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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