Sequential Methods for Coupled Geomechanics and Multiphase Flow

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226 APPENDIX B. MISCELLANEOUS DERIVATIONS P co Schematic capillary pressure P co 0 1 S w P cg Schematic capillary pressure P cg 0 1 S g Figure B.1: Typical shapes of capillary pressure regarding wetting and non-wetting phases (e.g., Sw and Sg, respectively). pco = po − pw, and pcg = pg − po. The capillary pressure curves are drawn based on the work of Lenhard and Parker (1987) Note from Chapter 2 that N −1 = M and both N and M are symmetric. Then we obtain p t Np = p 2 oNoo + p 2 wNww + p 2 gNgg + 2Nowpopw + 2Nogpopg + 2Nwgpwpg = φSocop 2 o + φSwcwp 2 w + φSgcgp 2 g − φ dSw dpco 2 po − 2popw + p 2 w +φ dSg 2 po − 2popg + p dpcg 2 b − φ 2 g + Sop Ks 2 o + S 2 wp 2 w + S 2 gp 2 g +2SoSwpopw + 2SoSgpopg + 2SwSgpwpg) = φ Socop 2 o + Swcwp 2 w + Sgcgp 2 dSw g − φ (po − pw) dpco 2 + φ dSg (po − pg) dpcg 2 + b − φ (Sopo + Swpw + Sgpg) Ks 2 ≥ 0, (B.3) where the equality is only satisfied with p = 0. Hence, N is positive-definite, from which M is also positive-definite.
B.2. ACCURACY OF THE FULLY COUPLED METHOD 227 B.2 Accuracy of the Fully Coupled Method The matrix form of the differential governing equations is written as ⎡ ⎣ K −Lt 0 T ⎤⎡ ⎦ Z ⎣ utr ptr xtr ⎤ ⎡ ⎤⎡ 0 0 ⎦ − ⎣ ⎦ L Q B x = [u,p] t , ⎣ ˙utr ˙ptr ⎤ ⎦ = ⎡ ⎣ hu hp ⎤ h ⎦, (B.4) where h is the source-term vector. Consistency of the fully coupled method can be shown as follows. The matrix form of the difference equations for the fully coupled method yields ⎡ ⎣ K −Lt L F ⎤⎡ ⎦ A ⎣ utr ptr ⎤ ⎦ xtr n+1 ⎡ ⎤⎡ 0 0 − ⎣ ⎦⎣ L Q B utr ⎤ ⎦ ptr Rn+1 = 1 ∆tru,rp t , n = ⎡ ⎣ fu fp ⎤ ⎦ f n+1 + ∆tR n+1 , (B.5) where R n+1 is the truncation-error vector for the mechanical (ru) and flow (rp) problems. hu = fu and ∆thp = fp. Using a Taylor expansion, x n tr = x n+1 tr tr − ∆t∂xn+1 ∂t Combining Equations B.5 and B.6, we have Ax n+1 tr which yields B ∂xn+1 tr ∂t − B x n+1 tr tr − ∆t∂xn+1 ∂t 1 + 2 ∆t2∂2 x n+1 tr ∂t2 + O(∆t3 ) · · · . (B.6) 1 + 2 ∆t2∂2 x n+1 tr ∂t2 + O(∆t3 ) 1 1 + (A − B)xn+1 tr − B ∆t 2 ∆t∂2 x n+1 tr ∂t2 + O(∆t2 ) = f n+1 + ∆tR n+1 , (B.7) = 1 ∆t fn+1 + R n+1 . (B.8)
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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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20 CHAPTER 2. FORMULATION coupling
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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