Sequential Methods for Coupled Geomechanics and Multiphase Flow

More documents

Recommendations

Info

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

196 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS 6.5.1 Case 6.1—Water injection and oil production in the 1D poroelas- ticity The schematic of this 1D problem is shown in the left plot of Figure 6.2. For Case 6.1, dilation and compaction occur around the injection and production wells, respectively. The water injection rate Qw,inj = 500 kg day −1 is the same as the production rate Qo,prod = 500 kg day −1 . The domain is homogeneous with 15 grid blocks. The length of the domain is Lz = 150 m with grid spacing ∆z = 10 m. We have a constant overburden ¯σ = 2 × 2.125 MPa. A no-displacement boundary condition is maintained at the bottom of the domain. The bulk density of the porous medium is ρb = 2400 kg m −1 . The initial oil pressure is Po,i = 2.125 MPa, where the formation is fully saturated with oil initially. The fluid densities and viscosities are ρo,0 = ρw,0 = 1000 kg m −1 and µo = µw = 1.0 cp, respectively. Permeability is kp = 500 md, porosity is φ0 = 0.3, constrained modulus is Kdr = 1 GPa, and the Biot coefficient is b = 1.0. Capillarity is ignored. A monitoring well is located at the fifth grid block from the top (1, 5). No-flow boundary conditions are applied at the top and bottom. Gravity is neglected. The oil and water compressibilities are similar, co = 3.5 × 10 −9 Pa −1 and cw = 3.0 × 10 −9 Pa −1 , respectively. These values yield τ = 0.95 and τ = 1.05 for single-phase flow, respectively. The residual oil and water saturations are zero. The parameter values for Case 6.1 are also given in Table 6.1. Since the convergence behaviors of the drained and fixed-strain splits depend on the total fluid compressibility from Equations 6.31 and 6.44, water injection can cause non- convergence and instability during simulation. We use a linear relative permeability of phase J, kr,J, with respect to SJ, i.e., where Sr,J is the residual saturation of J phase. kr,J = (SJ − Sr,J), (6.120) The top of Figure 6.3 shows the pressure history at the monitoring well during simulation. The pressure jumps at the initial time because of the excessive overburden and decreases with production. Then the pressure rises because the reservoir is being filled with
6.5. NUMERICAL EXAMPLES WITH THE STAGGERED NEWTON METHODS 197 P d S w 1.4 1.39 1.38 1.37 1.36 1.35 1.34 1.33 Pressure history Drained Fixed−strain Undrained Fully coupled Fixed−stress 1.32 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Pore volume injected t d 1 0.9 0.8 0.7 0.6 0.5 Water saturation profile S w 0.4 0.3 Undrained Fully coupled Fixed−stress 0.2 0 10 20 30 40 50 60 70 80 Vertical distance z (m) Figure 6.3: Top: pressure history at the monitoring well during simulation. Bottom: water saturation profile after simulation. Pd = P/Po,i.
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:
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: 146 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 252 and 253: 226 APPENDIX B. MISCELLANEOUS DERIV
Page 262 and 263: 236 APPENDIX C. CONSTITUTIVE RELATI
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?