Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

76 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS Case 3.5—Fluid production scenario in 2D with elastoplasticity The dimensions of the domain are 50 m×50 m with 5×5 grid blocks under the plane strain mechanical conditions. The domain is homogeneous with an overburden ¯σ = 2.125 MPa, side burden ¯σh = 2.125 MPa on both sides, and no-vertical and no-horizontal displacement boundary at the bottom. The bulk density of the porous medium is ρb = 2400 kg m −1 . Initial fluid pressure is Pi = 2.125 MPa. Fluid density and viscosity are ρf,0 = 1000 kg m −1 and µ = 1.0 cp, respectively. Permeability is kp = 50 md, and porosity is φ0 = 0.3. Young’s modulus is E = 350 MPa, and Poisson’s ratio is ν = 0.35. The Biot coefficient is b = 1.0. For the modified Cam-clay model, the virgin compression index is λ = 0.37, the swell index is κ = 0.054, the critical state slope is Mmcc = 1.4, and the preconsolidation pressure is pco,0 = −1.0 MPa, where tensile stress is positive. The production and observation wells are located at the center of the domain (3,3). The production rate is Qprod = 1000 kg day −1 . A no-flow boundary condition is applied to the domain. There is no gravity in the domain. The input parameters for Case 3.5 are listed in Table 3.4. Figure 3.18 shows that the drained split is not stable in the plastic regime, even though it is stable in the elastic regime. This is because at around td ≈ 0.018, we reach the plastic regime and the coupling strength increases beyond one. On the other hand, the undrained split is stable in the plastic regime. Note that there is a small difference between the undrained-split and the fully coupled results, and this is due to using a single iteration. As the figure indicates, when two iterations are used, the solution from the undrained split is quite close to the fully coupled results. Staggered Newton schemes for Case 3.5 The staggered Newton method has the following solution procedure (Schrefler et al., 1997). 1. Linearize the flow and mechanical problems. 2. Given the linearized coupled system, solve the flow and mechanical problems in a sequential way.
3.7. NUMERICAL EXAMPLES 77 Dimensionless pressure(P/P i ) τ 1 0.9 0.8 0.7 0.6 0.5 (plasticity with hardening) Dr Und(1 iter) Und (2 iter) FC 0.4 0 0.01 0.02 0.03 0.04 Dimensionless time (pore volume produced) 8 7 6 5 4 3 2 1 Variation of coupling strength 0 0 0.01 0.02 0.03 0.04 Dimensionless time (pore volume produced) Figure 3.18: 2D problem with a production well (Case 3.5). Top: pressure history. Bottom: the history of the coupling strength. The model enters the plastic regime at td ≈ 0.018. Beyond this point, the drained split becomes unstable and fails to produce a solution at all.
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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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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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126 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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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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