Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

218 APPENDIX A. STABILITY IN SPACE z/L z 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Case A.1. The Terzaghi problem Numerical c v t/h 2 =0.4 Analytic c v t/h 2 =0.4 Numerical c v t/h 2 =0.0001 Analytic c v t/h 2 =0.0001 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 p d Figure A.2: Case A.1 Stability of pressure at early time for Terzaghi’s problem. Top: pressure distributions at early and late time by the nodal based finite element method, where linear interpolation is used for pressure and displacement (Wan, 2002). Bottom: pressure distributions at early and late time by the combined finite-volume and finiteelement approach.
A.3. NUMERICAL EXAMPLES 219 Table A.1: Input data for Case A.1 Property Value Permeability (kp) 100 md Porosity (φ0) 0.3 Drained modulus (Kdr) 6 GPa Bulk density (ρb) 2400 kg m −3 Fluid density (ρf,0) 1000 kg m −3 Fluid viscosity (µ) 1.0 cp Initial pressure (Pi) 2.125 MPa Boundary pressure (Pbc) 2.125 MPa Overburden (¯σ) 2×2.125 MPa Grid spacing (∆z) 2 m is clear that the instability propagates from the drainage boundary to the interior domain. However, the combined finite-volume for flow and finite-elements for mechanics provide stable and accurate pressures at early time that match the analytical solution, as shown in the bottom of Figure A.2. For the combined finite-volume and finite-element method, Figure 4.4 shows that the fully coupled method yields first-order accuracy in time. A.3.2 Case A.2—Two dimensional plain strain strip footing consolidation problem At the top layer, the left part has an overburden, ¯σ = 2 × 2.125 MPa, and the right part is a the drainage boundary. The domain is 160 m × 70 m with 16 × 7 grid blocks with a plane strain mechanical problem. The domain is homogeneous. The overburden at the drainage boundary is ¯σ = 2.125 MPa. No-horizontal displacement boundary conditions are used on both sides and no-vertical displacement boundary condition is used at the bottom for mechanics. The bulk density of the porous medium is ρb = 2400 kg m −1 . Initial fluid pressure is Pi = 2.125 MPa. The fluid density and viscosity are ρf,0 = 1000 kg m −1 and µ = 1.0 cp, respectively. The fluid compressibility is cf = 3.5 × 10 −8 Pa −1 . Permeability is kp = 50 md, and porosity is φ0 = 0.3. Young’s modulus is E = 300 MPa, and Poisson’s ratio is ν = 0.0. The Biot coefficient is b = 1.0. A no-flow boundary condition is applied to the both sides, the bottom, and the left side of the top. No gravity is applied. The values
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SEQUENTIAL METHODS FOR COUPLED GEOM
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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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3.7 Left: the Terzaghi problem in 1
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4.1 The distribution of the error a
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2 CHAPTER 1. INTRODUCTION Reservoir
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6 CHAPTER 1. INTRODUCTION undrained
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14 CHAPTER 2. FORMULATION full form
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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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28 CHAPTER 3. STABILITY OF THE DRAI
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82 CHAPTER 4. CONVERGENCE OF THE DR
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164 CHAPTER 6. COUPLED MULTIPHASE F
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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