Sequential Methods for Coupled Geomechanics and Multiphase Flow

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182 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS 6.4.2 Contractivity of multiphase flow Similar to single-phase flow, we introduce the norm based on the complementary Helmholtz free energy to study the contractivity of coupled mechanics and multiphase flow, namely, ζm 2 Tm 1 ′ −1 = σ : C 2 dr Ω σ′ + κ · H −1 κ + pJNJKpK dΩ, (6.70) Tm := ζm := (σ ′ , κ,p) ∈ S × R nint × R np : σ ′ ij ∈ L 2 (Ω), κi ∈ L 2 (Ω), pJ ∈ L 2 (Ω) , (6.71) where np is the number of fluid phases. p = {pJ}. N = {NJK} and M = {MJK}, where M = N −1 , as shown in Chapter 2. Let (u0, p0, ξ 0) and (ũ0, ˜p0, ˜ ξ0) be two arbitrary initial conditions. Let (u, p, ξ) and (ũ, ˜p, ˜ ξ) be the corresponding solutions, yielding (σ ′ , m, κ, εp) and (˜σ ′ , ˜m, ˜κ, ˜εp), respectively, where m = {mJ}. Denote the difference between the two solutions by d(·) = (·) − ˜ (·). Assume the corresponding solutions from two arbitrary initial conditions are close enough, such that they honor the incremental form of the constitutive relations which is given by dσ = Cdr : (dε − dεp) − bJdpJ1, (6.72) dm dpJ = MJK −bK(dεv − dεv,p) + − dφK,p , (6.73) ρ dκ = −H · dξ. (6.74) Then Equations 6.72 – 6.74 yield K
6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 183 dζm 2 Tm 1 = 2 = 1 2 + Ω ′ −1 dσ : Cdr dσ′ + dκ · H −1 dκ + dpJNJKdpK dΩ, (dεe : Cdrdεe + dξ · Hdξ Ω dm − bJdεe,v ρ J,e MJK dm ρ K,e − bKdεe,v = dχm 2 , (6.75) Nm where we define the norm of χm Nm as χm 2 Nm 1 = (εe : Cdrεe + ξ · Hξ 2 Ω m + − bJεe,v ρ J,e MJK m ρ K,e − bKεe,v Nm := χ := (εe, ξ, mJ,e) ∈ S × R nint × R np : εeij ∈ L2 (Ω), dΩ, dΩ, (6.76) ξi ∈ L 2 (Ω), mJ,e ∈ L 2 (Ω) , (6.77) which originates from the Helmholtz free energy (Coussy, 1995). Since the corresponding solutions from two arbitrary initial conditions satisfy the govern- ing equations and boundary conditions, the fully coupled method yields from Equation 6.1 ⎡ ⎣ dun dp n J ⎤ ⎦ Am fc −→ ⎡ ⎣ dun+1 dp n+1 J ⎤ ⎦, where A m fc : ⎧ ⎪⎨ Div dσ = 0, ⎪⎩ dmJ ˙ + Div dwJ = 0, (6.78) where maximum plastic dissipation is assumed for elasto-plasticity. Note that homogeneous boundary conditions are obtained. Then the coupled problem is contractive relative to the norms, · Nm and · Tm . Specifically,
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SEQUENTIAL METHODS FOR COUPLED GEOM
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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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