Sequential Methods for Coupled Geomechanics and Multiphase Flow

More documents

Recommendations

Info

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

48 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS where E contains the origin (0,0), and fY is the yield surface, which is assumed to be a convex function. Σ = (σ ′ , κ) is a generalized effective stress constrained to lie within the elastic domain. The bilinear form ≪ ·, · ≫ is defined as ≪ Σ, Π ≫ = σ ′ C −1 dr π′ + κ · H −1 η dΩ , (3.72) which forms the norm as 2 Σ 2 E =≪ Σ, Σ ≫. The top of Figure 3.6 shows the associative flow rule based on maximum plastic dissipation. To show the contractivity of the coupled problem, let (u0, p0, ξ 0) and (ũ0, ˜p0, ˜ ξ0) be two arbitrary initial conditions and let (u, p, ξ) and (ũ, ˜p, ˜ ξ) be the corresponding solutions, yielding (σ ′ , m, κ, εp) and (˜σ ′ , ˜m, ˜κ, ˜εp), respectively. By subtracting the two solutions, we obtain ⎡ ⎣ dun dp n ⎤ ⎦ Afc ⎡ −→ ⎣ dun+1 dp n+1 ⎤ ⎧ ⎪⎨ Div dσ = 0, ⎦, where Afc : ⎪⎩ dm ˙ + Div dw = 0, (3.73) where d(·) = (·) − ˜ (·). Equation 3.73 has homogeneous boundary conditions with no source terms. From Equations 3.64, 3.65, and 3.69, dσ, dm, and dκ can be expressed as dσ = Cdr : (dε − dεp) − bdp1, (3.74) dm ρf,0 = bdεv + 1 dp, (3.75) M dκ = −H · dξ, (3.76) where dφp = bdεp,v is used in Equation 3.75. From Equation 3.70, we obtain Ω Ω (˜σ ′ − σ ′ ) : εp ˙ + (˜κ − κ) · ˙ ξ dΩ ≤ 0 , Choosing (π, η) = (˜σ ′ , ˜κ), (3.77) (σ ′ − ˜σ ′ ) :˙˜εp + (κ − ˜κ) ·˙˜ξ dΩ ≤ 0 , Choosing (π, η) = (σ ′ , κ). (3.78)
3.5. CONTRACTIVITY OF THE NONLINEAR CONTINUUM PROBLEM 49 Figure 3.6: Top: the associative flow rule under the assumption of maximum plastic work (Coussy, 1995). Bottom: geometric interpretation of the algorithmic plastic dissipation, which corresponds to the associative flow rule (Simo and Hughes, 1998).
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 124 and 125:
98 CHAPTER 4. CONVERGENCE OF THE DR
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:
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

Create successful ePaper yourself

Delete template?

Save as template?