Sequential Methods for Coupled Geomechanics and Multiphase Flow

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124 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS different from those for the undrained split. Interestingly, they are identical to those for the fully coupled method. Remark 5.2. When the mechanical problem has complicated boundary conditions in multiple dimensions, we might not obtain the exact local value of Kdr in the flow problem of Equation 5.20 for the fixed-stress split. Let K est dr be an estimated local Kdr. We define η = Kdr/K est dr as the deviation factor between the true and estimated local drained bulk moduli. Then, following the same procedure of the Von Neumann method with the backward Euler time discretization, we obtain the condition for the linear stability as η ≥ 1 1 − 2 1 , (5.26) τ where one of the amplification factors is negative if η < 1, but all the amplification factors are positive if η > 1. From Equation 5.26, η ≥ 0.5 provides unconditional stability for linear problems. Suppose we use K est dr = K3D dr for incompressible fluid and solid grains when the true Kdr is K1D dr . In this case, τ = ∞ and η = 3(1 − ν)/(1 + ν), where ν is Poisson’s ratio. Since 0 ≤ ν ≤ 0.5, 1.0 ≤ η ≤ 3.0, which gives unconditional stability with monotonic behavior. On the other hand, we obtain η = (1 + ν)/3(1 − ν) where we use K est dr = K1D dr while the true Kdr is K3D dr . In this case, instabilities may occur because 0.33 ≤ η ≤ 1.0 (e.g., η = 0.33 when ν = 0.0), and we may lead to oscillatory behaviors, even though the scheme is stable. Hence, a less stiff K est dr than the true Kdr yields unconditional stability with monotonicity because it guarantees 1.0 ≤ η ≤ 3.0. Thus, it is appropriate to choose K 1D dr as Kest dr ’s for one, two, and three dimensional problems, respectively. , K2D dr , and K3D dr
5.3. CONTRACTIVITY OF THE NONLINEAR CONTINUUM PROBLEM 125 5.3 Contractivity of the Nonlinear Continuum Problem 5.3.1 Non-contractivity of the fixed-strain split The coupled problem is contractive relative to the norms · N and · T , as shown in Chapter 3. We investigate whether the fixed-strain split is contractive, or not. We consider two arbitrary initial conditions as we did in Chapter 3. The fixed-strain split can be written as ⎡ ⎤ ⎡ ⎤ ⎣ dun dpn ⎦ Apsn −→ ⎣ du∗ dpn+1 ⎦ Ausn −→ ⎡ ⎣ dun+1 ⎤ dpn+1 ⎦, where ⎧ A ⎪⎨ p sn : dm ˙ + Div dv = 0, δdεv ˙ = 0, Au sn : Div dσ = 0, dp = 0 ⎪⎩ ⇒ Div dσ ′ = 0. (5.27) Equation 5.27 has homogeneous boundary conditions with no source terms. Since the pressure in the mechanical problem is prescribed, the pressure is not affected by pertur- bations of the initial condition, thus dp = 0. When solving the flow problem, A p sn, we obtain d dχ 2 N dt = ∂dχ2 N : dεe ˙ + ∂dεe ∂dχ2 N · ˙ ∂dχ dξ + ∂dξ 2 N dme ˙ ∂dme = dσ Ω ′ dme : dεe ˙ − dκ · dξ ˙ − M − bdεe,v bdεe,v ˙ ρf,0 + M dme − bεe,v dme ˙ dΩ ρf,0 ρf,0 = dσ : dε ˙ dp + dm ˙ dΩ − dσ ′ : dεp ˙ + dκ · dξ ˙ dΩ = Ω ρf,0 Ω D d p≥0 dσ : dε ˙ −1 − dv · k µdv Ω dm ˙ −1 = −dv · k µdv from Equation 5.27 dΩ − D d p ≤ 0, (5.28) where v ∈ [H (div,Ω)] ndim. Note that we consider maximum plastic dissipation, D d p ≥ 0, when solving the flow problem.
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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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6.2.3 Undrained split . . . . . . .
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3.7 Left: the Terzaghi problem in 1
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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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26 CHAPTER 2. FORMULATION
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28 CHAPTER 3. STABILITY OF THE DRAI
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174 CHAPTER 6. COUPLED MULTIPHASE F
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208 CHAPTER 7. CONCLUSIONS 7.2 Coup
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210 CHAPTER 7. CONCLUSIONS 7.2.3 Ac
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212 CHAPTER 7. CONCLUSIONS
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214 APPENDIX A. STABILITY IN SPACE
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226 APPENDIX B. MISCELLANEOUS DERIV
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236 APPENDIX C. CONSTITUTIVE RELATI
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244 BIBLIOGRAPHY Mech Eng 122: 145-
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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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