Sequential Methods for Coupled Geomechanics and Multiphase Flow

More documents

Recommendations

Info

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

132 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS unconditional stability (i.e., B-stability) for the fixed-stress split is achieved when 0.5 ≤ α ≤ 1. 5.5 Error Estimation Stable sequential schemes yield the same solution as the fully coupled method when full iterations are taken and the sequential schemes are convergent. In practice, we take a fixed number of iterations due to limited computational resources. In this situation, first-order accuracy in time is desired. For example, in coupled flow and dynamics, the staggered method can preserve first-order accuracy in time based on Lie’s formula (Armero and Simo, 1992; Chorin et al., 1978; Lapidus, 1981). However, when a fixed number of iterations is performed, typical sequential methods do not guarantee convergence (Turska et al., 1994). The drained split shows non-convergence under a fixed number of iterations for a slightly compressible fluid, whereas the undrained split shows convergence. But, the undrained split is not convergent when both the fluid and the solid grains are incompressible. To determine the convergence properties of the fixed-strain and fixed-stress splits, we employ matrix algebra and spectral analysis as we did in Chapter 3. The linear coupled problem is considered here for simplicity. We use the finite volume and finite element methods for flow and mechanics, respectively, and the backward Euler time discretization. The procedures of the error estimation for the fixed-strain and fixed-stress splits are quite similar to those of the drained and undrained splits used in Chapter 3. The error associated with a sequential method can be decomposed into two terms as e n+1,niter ts ≤ x n+1 t − xn+1 f + = O(∆t) + x n+1 f x n+1 f − xn+1,niter s − xn+1,niter s . (5.55)
5.5. ERROR ESTIMATION 133 Using the fully coupled method, the problem is written as ⎡ F L ⎣ −Lt ⎤⎡ ⎦⎣ K A p ⎤n+1 ⎡ ⎤⎡ ⎦ u L Q − ⎣ ⎦⎣ 0 0 B p ⎤ ⎦ u ⎡ x = ⎣ p ⎤ ⎦, F = Q + ∆tT, u Sequential methods decompose the matrix A into ⎡ F + R 0 ⎣ −Lt ⎤⎡ ⎦⎣ K p ⎤ ⎦ u ⎡ ⎤⎡ L Q − ⎣ ⎦⎣ 0 0 p ⎤n ⎡ ⎦ = u n+1,k+1 ⎣ fp fu n = ⎡ ⎣ fp fu ⎤ ⎦ f ⎡ ⎤⎡ R −L − ⎣ ⎦⎣ 0 0 p ⎤ ⎦ u ⎤ ⎦ n+1 n+1 n+1,k , (5.56) , (5.57) where R is the relaxation matrix in a sequential method. R depends on the specific sequential method used. Subtracting Equation 5.57 from Equation 5.56, the errors of pressure and displacement are written as ⎡ ⎣ efs,p efs,u ⎤ ⎦ n+1,k+1 = = ⎡ F + R 0 ⎣ −Lt ⎤ ⎦ K ⎡ ⎤ ⎡ L Q + ⎣ ⎦ 0 0 ⎡ ⎤⎡ ⎜ R −L ⎝⎣ ⎦ 0 0 ⎤n⎞ −1 ⎛ ⎣ efs,p efs,u ⎦ ⎠ ⎛ ⎣ efs,p efs,u ⎤ ⎦ n+1,k ⎡ (F + R) ⎣ −1 K 0 −1Lt (F + R) −1 K−1 ⎤⎜⎡ ⎤⎡ ⎜ ⎦⎜ R −L ⎜⎣ ⎦⎣ ⎜ ⎝ 0 0 M N efs,p ⎤n+1,k ⎦ efs,u ⎡ ⎤ ⎡ L Q + ⎣ ⎦ ⎣ 0 0 efs,p ⎤n⎞ ⎦ ⎠ . (5.58) efs,u
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 126 and 127: 100 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 208 and 209:
182 CHAPTER 6. COUPLED MULTIPHASE F
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?