Sequential Methods for Coupled Geomechanics and Multiphase Flow

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

230 APPENDIX B. MISCELLANEOUS DERIVATIONS Remark D.2. Assume 1D oil-water flow with capillarity with σv ˙ = 0 in order to study the properties of Equations B.12 – B.14 in a numerical sense. Using Equations B.12 – B.14, the governing equations become where N is ⎡ ⎣ Noo Now Nwo Nww ⎡ ⎣ Noo ⎤⎡ Now ⎦⎣ Nwo Nww N ˙ po pw ˙ σv ˙ = Kdr ˙ ⎤ ⎦ = ⎡ εv − b (So ˙ ⎤ ⎦ + b po + Sw ˙ ⎡ pw) bJ ˙ pJ ⎣ φSoco − φ dSw dpco ⎣ So Sw − + So b−φ Ks So φ dSw b−φ + Sw dpco Ks So ⎤ ⎦εv ˙ = ⎡ ⎣ − Div vo + fo − Div vw + fw ⎤ ⎦, (B.16) dSw −bpco ( po ˙ − pw) ˙ , (B.17) dpco ˜ bJ ˙ SJ φ dSw b−φ + So dpco Ks Sw φSwcw − φ dSw b−φ + Sw dpco Ks Sw Then, substituting Equation B.17 into Equation B.16 and using σv ˙ = 0, we obtain ⎛ ⎜⎡ ⎜ ⎜⎣ ⎜ ⎝ Noo ⎤ Now ⎦ Nwo Nww N + b2 Kdr ⎡ ⎣ S2 o SoSw SwSo S 2 w ⎤ ⎦ ˜S − b2 dSw pco Kdr dpco = ⎡ ⎣ − Div vo + fo − Div vw + fw ⎤ W ¯S ⎞ ⎤ ⎦. ⎡ ⎣ So ⎤⎟⎡ −So ⎟ ⎦ ⎟ ⎟⎣ ⎟ Sw −Sw ⎠ ˙ po pw ˙ ⎦. (B.18) For the left terms of Equation B.18, the positive-definiteness of W is not guaranteed because ¯ S is indefinite. This is the case even though N and ˜ S are positive-definite and positive-semidefinite, respectively. The eigenvalues of ¯ S are 0, So − Sw. When |pco| and |dSw/dpco| are large enough such N and ˜ S are ignored, we may face an ill-posed problem. ⎤ ⎦
B.5. δφ FOR SINGLE-PHASE FLOW 231 B.5 δφ for single-phase flow We derive the expression of δφ based on Geertsma (1957)’s work, where p is the fluid tension. Note that p in this thesis is the fluid pressure. We denote by Vp and Vb the pore and bulk volumes, respectively. Then the change of Vp with respect to the movement of the solid skeleton is expressed as which yields δVp Vp = 1 Vp ∂Vp ∂p δVp = ∂Vp ∂p σv − 1 Vp σv δp + ∂Vp ∂σv p p ∂Vp δp + ∂σv 1 Vp δσv, (B.19) ∂Vp ∂σv p (δσv + δp). (B.20) Similarly, the change of Vb with respect to the movement of the solid skeleton is described as which yields δVb Vb = 1 Vb ∂Vb ∂p δVb = ∂Vb ∂p σv − 1 Vb σv δp + ∂Vb ∂σv p p ∂Vb δp + ∂σv 1 Vb δσv, (B.21) ∂Vb ∂σv p (δσv + δp). (B.22) Under the condition that δσv + δp = 0, we obtain for the dilation of the solid grains δVp Vp = δVb Vb = − 1 δp = Ks 1 δσv, (B.23) Ks where Ks is the stiffness of the solid grains. Since the drained bulk modulus Kdr is defined as 1 Vb ∂Vb ∂σv p = 1 , (B.24) Kdr
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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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