Sequential Methods for Coupled Geomechanics and Multiphase Flow

More documents

Recommendations

Info

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

158 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS From Figure 5.13, the fixed-stress and fully coupled methods are convergent with one iteration, showing that the errors decrease as O(∆t) as the time step size is refined. This also confirms that the fixed-stress and fully coupled methods have O(∆t) error in time. However, the fixed-strain split does not show convergence, but yields zeroth-order accuracy (O(1)). Figures 5.14 and 5.15 show the spatial distributions of pressure and displacement by the fixed-strain and fixed-stress splits, respectively. As the time step size is refined, the fixed-strain split with one iteration does not converge to the true solution but to a different solution (Figure 5.14), whereas the fixed-stress spit with one iteration converges to the true solution (Figure 5.15). The numerical results for Case 5.1 support our a-priori error estimates of the fixed-strain and fixed-stress splits from Equations 5.67 and 5.76. When full iterations are performed for Cases 5.1 and 5.4, the fixed-stress split takes a maximum of two iterations to converge to the solutions from the fully coupled method. Cases 5.2 and 5.4 take one and two iterations, respectively, to match the fully coupled method. These results are consistent with our a-priori estimates of the convergence rate of the fixed-stress split (i.e., G 2 = 0 in Equation 5.77). Note that a few more iterations may be required in the presence of complex boundary conditions, even for elasticity because we do not have accurate estimates of the local Kdr in the flow problem. 5.6.3 Effect of the deviation factor η Since the local Kdr is not known a-priori for complex boundary conditions and production scenarios, we investigate the effect of the deviation factor η on the fixed-stress split. We use two test cases adopting the backward Euler time discretization. Case 5.5 Two dimensional consolidation problem with a constrained boundary (the left picture in Figure 5.16). This is equivalent to a half domain of the Terzaghi problem by symmetry. The true Kdr is K1D dr , and the porous medium is elastic. Case 5.6 Two dimensional consolidation problem with a unconstrained boundary (the right picture in Figure 5.16). The true Kdr is close to K2D dr , and the porous medium is elastic.
5.6. NUMERICAL EXAMPLES 159 Figure 5.16: Two test cases to investigate the effect of different η’s. Case 5.5: consolidation problem with constrained boundary (left picture). Case 5.6: consolidation problem with unconstrained boundary (right picture). For Case 5.5, the dimension of the domain is 3 m×14 m with 3×7 grid blocks under the plane strain mechanical condition. The domain is homogeneous. Grid spacings, ∆x and ∆z are 1 m and 2 m, respectively. The domain has an overburden ¯σ = 2×2.125 MPa on the top, no horizontal displacement boundary on both sides, and no vertical or horizontal displacement at the bottom. The bulk density of the porous medium is ρb = 2400 kg m −1 . Initial fluid pressure is Pi = 2.125 MPa. The 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 = 500 MPa, and Poisson’s ratio is ν = 0.0. The Biot coefficient is b = 1.0. The observation well is located at (2,5). We have a drainage boundary for flow on the top, where the boundary fluid pressure is Pbc = 2.125 MPa. No-flow boundary conditions are applied to both sides and the bottom. Gravity is neglected.
Page 1 and 2:
SEQUENTIAL METHODS FOR COUPLED GEOM
Page 3:
I certify that I have read this dis
Page 6 and 7:
splits, are, at best, conditionally
Page 8 and 9:
deeply. There was deep discussion o
Page 10 and 11:
3 Stability of the Drained and Undr
Page 12 and 13:
6.2.3 Undrained split . . . . . . .
Page 14 and 15:
xiv
Page 16 and 17:
xvi
Page 18 and 19:
3.7 Left: the Terzaghi problem in 1
Page 20 and 21:
4.1 The distribution of the error a
Page 22 and 23:
5.3 Case 5.2 (1D injection-producti
Page 24 and 25:
5.18 Pressure history at the observ
Page 26 and 27:
xxvi
Page 28 and 29:
2 CHAPTER 1. INTRODUCTION Reservoir
Page 30 and 31:
4 CHAPTER 1. INTRODUCTION problem (
Page 32 and 33:
6 CHAPTER 1. INTRODUCTION undrained
Page 34 and 35:
8 CHAPTER 1. INTRODUCTION where A c
Page 36 and 37:
10 CHAPTER 1. INTRODUCTION Even whe
Page 38 and 39:
12 CHAPTER 1. INTRODUCTION
Page 40 and 41:
14 CHAPTER 2. FORMULATION full form
Page 42 and 43:
16 CHAPTER 2. FORMULATION dE dt =
Page 44 and 45:
18 CHAPTER 2. FORMULATION where the
Page 46 and 47:
20 CHAPTER 2. FORMULATION coupling
Page 48 and 49:
22 CHAPTER 2. FORMULATION where Div
Page 50 and 51:
24 CHAPTER 2. FORMULATION between g
Page 52 and 53:
26 CHAPTER 2. FORMULATION
Page 54 and 55:
28 CHAPTER 3. STABILITY OF THE DRAI
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
82 CHAPTER 4. CONVERGENCE OF THE DR
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
100 CHAPTER 4. CONVERGENCE OF THE D
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135: 108 CHAPTER 4. CONVERGENCE OF THE D
Page 144 and 145: 118 CHAPTER 5. FIXED-STRAIN AND FIX
Page 190 and 191: 164 CHAPTER 6. COUPLED MULTIPHASE F
Page 234 and 235:
208 CHAPTER 7. CONCLUSIONS 7.2 Coup
Page 236 and 237:
210 CHAPTER 7. CONCLUSIONS 7.2.3 Ac
Page 238 and 239:
212 CHAPTER 7. CONCLUSIONS
Page 240 and 241:
214 APPENDIX A. STABILITY IN SPACE
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
226 APPENDIX B. MISCELLANEOUS DERIV
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
236 APPENDIX C. CONSTITUTIVE RELATI
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
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
show all

Sequential Methods for Coupled Geomechanics and Multiphase Flow

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?