Sequential Methods for Coupled Geomechanics and Multiphase Flow

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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.
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SEQUENTIAL METHODS FOR COUPLED GEOM
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I certify that I have read this dis
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splits, are, at best, conditionally
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deeply. There was deep discussion o
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3 Stability of the Drained and Undr
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6.2.3 Undrained split . . . . . . .
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3.7 Left: the Terzaghi problem in 1
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4.1 The distribution of the error a
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5.3 Case 5.2 (1D injection-producti
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5.18 Pressure history at the observ
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2 CHAPTER 1. INTRODUCTION Reservoir
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4 CHAPTER 1. INTRODUCTION problem (
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6 CHAPTER 1. INTRODUCTION undrained
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8 CHAPTER 1. INTRODUCTION where A c
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10 CHAPTER 1. INTRODUCTION Even whe
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12 CHAPTER 1. INTRODUCTION
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14 CHAPTER 2. FORMULATION full form
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16 CHAPTER 2. FORMULATION dE dt =
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18 CHAPTER 2. FORMULATION where the
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20 CHAPTER 2. FORMULATION coupling
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22 CHAPTER 2. FORMULATION where Div
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24 CHAPTER 2. FORMULATION between g
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28 CHAPTER 3. STABILITY OF THE DRAI
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100 CHAPTER 4. CONVERGENCE OF THE D
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Page 240 and 241: 214 APPENDIX A. STABILITY IN SPACE
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246 BIBLIOGRAPHY Aug. Murad M.A. an
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248 BIBLIOGRAPHY ˇZeniˇsek A. 198
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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