Sequential Methods for Coupled Geomechanics and Multiphase Flow

More documents

Recommendations

Info

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

52 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS the drained split can be expressed as ⎡ ⎣ dun dp n ⎤ ⎦ Au dr −→ ⎡ ⎣ dun+1 dp n ⎤ ⎦ Ap dr −→ ⎡ ⎣ dun+1 ⎤ dp n+1 ⎦, where ⎧ ⎪⎨ ⎪⎩ A u dr : Div dσ = 0, δ(dp) = 0, A p dr : ˙ dm + Div dw = 0, dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0, (3.86) which has homogeneous boundary conditions with no source terms. Since ˙ε, εp, ˙ and ˙ ξ are prescribed, they are not affected by the perturbation of the initial condition, yielding dε ˙ = 0, dεp ˙ = 0, and ˙ dξ = 0. When we solve the mechanical problem Au dr split, we have, dΨ d dt = = = Ω Ω dσ : ˙ dε + dp ρf,0 dm ˙ dΩ − dσ ′ : dεp ˙ + κ · ˙ dξ dΩ Ω D d p dσ : dε ˙ dp + b ρf,0 ˙ dεv dΩ − D d p (from δ(dp) = 0) dp by the drained b Ω ρf,0 ˙ dεvdΩ − D d p ≤ 0 (from Div dσ = 0) . (3.87) Equation 3.87 proves that the drained split is not contractive when we solve the mechanical problem. This non-contractivity of the drained split has been pointed out by Armero and Simo (1992) and Armero (1999). Then, when we solve the flow problem A p dr by the drained split, we have dΨ d dt = Ω dσ : dε ˙ dp + dm ˙ dΩ − dσ ρf,0 Ω ′ : dεp ˙ + dκ · ˙ dξ dΩ = −dp Div(dv)dΩ .. . dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0 Ω = − dv · µk −1 dvdΩ ≤ 0. (3.88) Ω Dp
3.6. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 53 ⎡ For the undrained split, we have ⎣ dun dp n ⎤ −→ ⎡ dp ∗ ⎤ −→ ⎡ ⎦ Au ud ⎣ dun+1 ⎦ Ap ud ⎣ dun+1 ⎤ dp n+1 ⎦, where ⎧ ⎪⎨ ⎪⎩ A u ud : Div dσ = 0, δdm = 0, A p ud : ˙ dm + Div dw = 0, dε ˙ = 0, dεp ˙ = 0, dξ ˙ = 0, (3.89) which has homogeneous boundary conditions with no source terms. When we solve the mechanical problem A u ud dΨ d dt = Ω by the undrained split, we have dσ : ˙ dε + dp ρf,0 dm ˙ dΩ − dσ ′ : dεp ˙ + dκ · ˙ dξ dΩ Ω D d p≥0 = dσ : ˙ d dεdΩ − Dp Ω = −D d p ≤ 0. (3.90) Note that the operator for flow in the undrained split is the same as that in the drained split, A p ud is the same as Ap dr . Thus, the undrained split satisfies the contractivity condition for both mechanics and flow. 3.6 Discrete Stability of the Nonlinear Problem As shown in the previous section (Armero, 1999), the undrained split honors the dissipative properties of the coupled problem. But, we need to find which algorithm (i.e. time inte- gration technique) provides unconditional stability - so called B-stability. By B-stability we mean the discrete counterpart of contractivity (Simo, 1991; Simo and Govindjee, 1991; Simo and Hughes, 1998), i.e., dχ n+1 N ≤ dχ n N ∀n. (3.91)
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 108 and 109: 82 CHAPTER 4. CONVERGENCE OF THE DR
Page 126 and 127: 100 CHAPTER 4. CONVERGENCE OF THE D
Page 128 and 129:
102 CHAPTER 4. CONVERGENCE OF THE D
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
118 CHAPTER 5. FIXED-STRAIN AND FIX
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
164 CHAPTER 6. COUPLED MULTIPHASE F
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
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?