Sequential Methods for Coupled Geomechanics and Multiphase Flow

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200 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS the error decreases rapidly with iterations. In contrast to pressure, all the schemes shows the same convergence behavior for saturation at td = 0.08. 6.5.2 Case 6.2—Water injection and oil production in 2D with poroelas- ticity Figure 6.2 illustrates a 2D oil production problem. The injection and production wells are located at the right bottom and left top, respectively. The monitoring well is at (3,2) grid block. The input data for the 2D problem are shown in Table 6.2. The dimension of the domain is 100 m × 20 m with 10 × 4 grid blocks under the plane strain mechanical condition. The water injection rate Qw,inj = 5000 kg day −1 is the same as the production rate Qo,prod = 5000 kg day −1 . The domain is homogeneous. The domain has an overburden ¯σ = 3 × 2.125 MPa on the top, no horizontal displacement on the left side, a side burden ¯σh = 2.125 MPa on the right side, and no vertical displacement at the bottom. The bulk density of the porous medium is ρb = 2400 kg m −1 . Initial oil pressure is Po,i = 2.125 MPa, where oil fully saturates the formation. The density, viscosity and compressibility of water are ρw,0 = 1000 kg m −1 , µw = 1.0 cp, and cw = 3.5 × 10 −10 Pa −1 , respectively. The density, viscosity, and compressibility of oil are ρo,0 = 1000 kg m −1 , µo = 1.0 cp, and co = 3.5 × 10 −8 Pa −1 , respectively. The permeability, kp, is 500 md, and the porosity, φ0, is 0.3. Young’s modulus is E = 300 MPa, and Poisson’s ratio is ν = 0.35. The Biot coefficient is b = 1.0. There is no capillarity. No-flow boundary conditions are applied at all sides. Gravity is neglected. The residual oil and water saturations are zero. The oil and water compressibilities differ from each other by two orders of magnitude. It is anticipated that the drained and fixed-strain splits will face instability because of water injection. The relative permeability is the same as Equation 6.120. Figure 6.5 shows the distributions of pressure and water saturation at td = 0.3. As shown in the top of Figure 6.6, the drained and fixed-strain splits deviate from the true solution around td = 0.2 and become unstable. From the pressure behaviors around td = 0.22, the solutions by the drained and fixed-strain splits are not reliable, even though they provide finite pressure values. In contrast to the drained and fixed-strain
6.5. NUMERICAL EXAMPLES WITH THE STAGGERED NEWTON METHODS 201 P d S w 1.9 1.8 1.7 1.6 1.5 1.4 0 1 0.8 0.6 0.4 0.2 0 0 5 5 10 z (m) 10 z (m) 15 15 Pressure distribution (P d ) 20 20 0 0 20 20 40 x (m) Water saturation distribution (S w ) Figure 6.5: Pressure (top) and water saturation (bottom) distributions after simulation. 40 x (m) 60 60 80 80 100 100
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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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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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