Sequential Methods for Coupled Geomechanics and Multiphase Flow

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180 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS From det(B s sn) = 0, we have γe = 0, − Kdr which yields the additional condition for convergence as 6.3.4 Fixed-stress split b 2 1 Mmp + (BoT n o + BwT n w) ∆t h2 , (6.60) 2 (1 − cos θ) τ ≡ b2 Mmp Kdr < 1. (6.61) The fixed-stress split with IMPES predicts the strain change of the flow problem as Equa- tion 6.45. Then, the error equation for flow is hA s,n p + b2 − T n s h KdrBw e k+1 Po,j ∆t e k+1 − 2ek+1 + ek+1 Po,j+1 Po,j Po,j−1 e k Po,j ∆t hAs,n u + ∆t (− ek U j− 1 2 − e k U j+ 1 2 b2 − KdrBw h ) = 0. (6.62) The error equation for mechanics is the same as Equation 6.41. Using the spectral method, the matrix equation is written as where ⎡ B s ss ⎡ ⎣ ˆ ⎤ ⎡ Po ⎦ = ⎣ Û 0 ⎤ ⎦, (6.63) 0 Bs ss = ⎣ As,n p hγe + T n s ∆t b2 h 2 (1 − cos θ) + KdrBw (γe − 1) A s,n u 2i sin θ 2γ From det(B s ss) = 0, we have b2i sin θ 2 γe Kdr h 2 (1 − cos θ)γe γe = 0 , (6.64) ⎤ ⎦.
6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 181 which yields no additional restriction on convergence. 6.4 Stability Analysis via the Energy Method 6.4.1 Constitutive relations for multiphase flow and elastoplasticity We showed the constitutive relations for elastic mechanics and multiphase flow in Chapter 2. Based on Coussy (1995)’s approach, we extend the constitutive relations for elasticity under isothermal conditions to those for poro-elasto-plasticity, which are written as δσ = Cdr : (δε − δεp) − bJδpJ1, (6.65) δm δpJ = MJK −bK(δεv − δεv,p) + − δφK,p , (6.66) ρ where the variation of the elastic fluid content of phase J is δm = ρ J,e K δm − δφJ,p. (6.67) ρ J Repeated indices indicate summation, again. (·)J,e and (·)J,p mean elastic and plastic quantities for fluid phase J, respectively. The plastic porosity of phase J and plastic strain can be related to each other by assuming that ˙ φJ,p = βJ ˙εp,v. Here, we assume that βJ = bJ, similar to single-phase flow, which yields δφJ,p = bJδεp,v. (6.68) For hardening, we can reuse the relation of the hardening variable ξ and force κ for single-phase flow, namely, δκ = −H · δξ. (6.69)
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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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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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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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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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