Sequential Methods for Coupled Geomechanics and Multiphase Flow

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92 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS errors of the form ek Uj = γk eei(j)θ eU, ˆ ek Vj = γk eei(j)θ eV ˆ , and ek Pj = γk eei(j)θ eP, ˆ we obtain ⎡ ⎢ ⎣ γe −∆tγe 0 ∆tKdr ρbh 2 2(1 − cos θ)γe γe ∆tb θ ρbh2i sin 2 0 ∆tMb θ h 2i sin 2γe γe + ∆tMkp µh2 B 2(1 − cos θ)γe ∗ dr ⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣ eU ˆ eV ˆ eP ˆ ⎤ ⎥ ⎦ = ⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ ⎦ 0 . (4.37) From det(B∗ dr ) = 0, the error amplification factors of the coupled flow and dynamics for the drained split are given by γe = 0, − Equation 4.38 indicates that ∆t 2 Mb 2 ρbh2 2(1 − cos θ) 1 + ∆t2Kdr ρbh2 2(1 − cos θ) 1 + ∆tMkp µh2 . (4.38) 2(1 − cos θ) lim ∆t→0 max |γe| = 0. (4.39) When we follow a similar procedure for the drained split in coupled flow and statics (Equations 4.5 – 4.7) and add vk to the unknowns, convergence is obtained for the drained split of the coupled flow and dynamics from Equations 4.1 and 4.39 because lim e n+1 = 0. As explained by Armero and Simo (1992), when we use the staggered ∆t→0 fs method (i.e., one iteration), convergence and first-order accuracy can be directly estimated by Lie’s formula (Chorin et al., 1978; Lapidus, 1981), written as lim n→∞ [exp(A1t/n)exp(A2t/n)] n = exp[(A1 + A2)t] , (4.40) because Equation 4.29 is split by the drained split as ˙ξ = A1ξ + s1, ˙ ξ = A2ξ + s2, (4.41)
4.2. SPECTRAL ANALYSIS 93 where ⎛ ⎜ A1ξ = ⎜ ⎝ vk 1 ρb Div(Cdr : ε − bp1) 0 ⎞ ⎛ ⎟ ⎠ , s1 ⎜ = ⎜ ⎝ 0 1 ρb Div bp01 + g 0 ⎞ ⎟ ⎟, (4.42) ⎠ ⎛ ⎜ A2ξ = ⎜ ⎝ 0 0 M Div kp µ Gradp − bM Grads ⎞ vk : 1 ⎛ ⎟ ⎟, ⎠ (4.43) ⎜ s2 = ⎜ ⎝ 0 0 −M Div kp µ ρfg ⎞ + f ⎟ ⎟. ⎠ (4.44) Equation 4.40 implies that the product formula of the operator splitting converges to the original operator as the time step size goes to zero. Thus, the drained split of the coupled flow and dynamics is convergent, whereas the drained split of the coupled flow and statics is not convergent, especially close to the stability limit. Remark 4.5. Similar to coupled flow and statics in Remark 4.4, max|γe| ≤ 1 for all θ in Equation 4.38 provides a necessary condition for stability in the coupled flow and dynamics. Consider the undrained limit corresponding to kp = 0 in order to compare with the stability estimate of thermoelasticity in Armero and Simo (1992). Then, the stability condition becomes (see Appendix B.2) 2as 2 ∆t (τ − 1) ≤ 1, (4.45) h where a 2 s = Kdr ρb , and as is the speed of sound (Hughes, 1987). Equation 4.45 has a similar form to that of Armero and Simo (1992). The slight difference between the two is due to the different space and time discretization. In contrast to coupled flow and statics, we
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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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4.1 The distribution of the error a
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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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142 CHAPTER 5. FIXED-STRAIN AND FIX
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164 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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