multivariate production systems optimization - Stanford University

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we obtain V, the matrix of eigenvectors, and Λ, the diagonal matrix of eigenvalues. Since the Hessian is symmetric and V is unitary, the inverted Hessian is given as H -1 = V Λ -1 V T (6.15) Expressing the gradient as a linear combination of the eigenvectors, the step-direction is given by g = V α (6.16) p = - H -1 g (6.12) = - V Λ -1 V T V α = - V Λ -1 α = - v1 λ1 v2 λ2 v3 λ3 66 vn λn (6.17) (6.18) α (6.19) Recall from linear algebra that an eigenvalue that is close to zero indicates that the function is close to being singular and therefore will change very little along the corresponding eigenvector. Equation 6.19 indicates that the smaller an eigenvalue is, the larger the stepdirection will be in the direction of the corresponding eigenvector. Therefore, Newton’s Method will take the largest strides in the directions that the function is least sensitive. While this feature is desirable with regard to convergence, it is extremely threatening to the stability of the algorithm if left unchecked. Since Newton’s Method is only a quadratic approximation to the local surface of the objective function, all steps should be somewhat tentative. The primary expense of Newton’s Method is not the matrix solution (Equation 6.12), but the evaluation of the derivatives required for the gradient and the Hessian. If analytic expressions are unavailable for the derivatives, they must be evaluated using finite n difference approximations. For n decision variables, 2 + n 2 second partial derivatives are necessary for the Hessian (recall that the Hessian is symmetric) and n first partial derivatives are necessary for the gradient. If ei is the ith unit vector and hi is the finite difference interval for the ith decision variable (see Figure 6.2), then the elements of the gradient are given by the expression
F(x + hj ej - hi ei) F(x - hi ei) F(x - hj ej - hi ei) gi = F x+hiei - F x-hiei 2hi The diagonal elements of the Hessian are given by Hii = F x+hiei - 2F x + Fx-hiei h i 2 and the off-diagonal elements of the Hessian are given by Hij = F x+hiei+hjej + F x-hiei-hjej - F x-hiei+hjej - F x+hiei-hjej 4hihj hj F(x + hj ej) F(x + hj ej + hi ei) F(x) F(x - hj ej) F(x - hj ej + hi ei) hi 67 F(x + hi ei) Figure 6.2: Discretization Scheme Used for Finite Difference Approximations (6.20) (6.21) (6.22) All of the finite difference approximations are second-order accurate which preserves the second-order accuracy of the quadratic model. If first-order accuracy is acceptable in the mixed partial derivatives, significant savings can be realized by evaluating the off-diagonal elements of the Hessian as Hij = F x+hiei+hjej + F x - Fx+hjej - F x+hiei hihj Notice that after determining the elements of the gradient with Equation 6.20 and the diagonal elements of the Hessian with Equation 6.21, each off-diagonal element can be obtained with Equation 6.23 by a single additional function evaluation as opposed to the four additional function evaluations that would be required by Equation 6.22. (6.23)
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MULTIVARIATE PRODUCTION SYSTEMS OPT
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iii Approved for the Department: Ro
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Acknowledgements I would like to th
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5.2.4 Determine Partial Fugacities
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List of Figures Figure 1.1 Graphica
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Chapter 1 INTRODUCTION The performa
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eservoir model. Kuller and Cummings
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optimization, it is merely the proc
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Chapter 2 RESERVOIR AND INFLOW PERF
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where RS = V GO S 9 V OO S rS = V O
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The source-sink term of Equation 2.
Page 23 and 24: Δ φ SG BG + SO RS BO ρGO S S ρG
Page 25 and 26: 3. Estimate the average reservoir p
Page 27 and 28: which is valid for steady-state, ra
Page 29 and 30: Chapter 3 VERTICAL MULTIPHASE FLOW
Page 31 and 32: μNS = μL λL + μG λG 21 (3.8) T
Page 33 and 34: -dP = 1 Qα + Q β Mα + Mβ g gC d
Page 35 and 36: • The term NGV NL 0.38 ND 2.14 is
Page 37 and 38: HL = 1 - 1 2 1 + VM VS - 1 + VM VS
Page 39 and 40: A paper by Hazim and Nimat (1989) p
Page 41 and 42: NV ≤ 18 25 18 < NV < 250 69 NV -0
Page 43 and 44: NWE Nμ > 0.005 ε = 0.3713 σL D 3
Page 45 and 46: NLV 100. 10. 1.0 0.1 Bubble Griffit
Page 47 and 48: V M < 10 δ ≥ -0.065 VM V M > 10
Page 49 and 50: RETURN ΔZ=.95 Δ Z ΔP=.95 Δ Pnew
Page 51 and 52: • maintain stable pressure downst
Page 53 and 54: R P = producing gas-liquid ratio, M
Page 55 and 56: flowrate is straightforward. Fortun
Page 57 and 58: Mass Flux Residual G1 - G2 0 Spurio
Page 59 and 60: segregation. The gas phase must exi
Page 61 and 62: API, GOR, Bo API Gravity First Stag
Page 63 and 64: Zi = Xi L + Yi V (5.3) where V and
Page 65 and 66: aα m = ∑ ∑ YiYj aα ij i j bm
Page 67 and 68: ψ1 = 2 cos θ 3 - Ω 3 (5.28) ψ2
Page 69 and 70: Chapter 6 NONLINEAR OPTIMIZATION Th
Page 71 and 72: IMSL (1987). For a broad overview o
Page 73: The minimum of a function can occur
Page 77 and 78: 6.2.1 The Method of Steepest Descen
Page 79 and 80: The second step of the Greenstadt (
Page 81 and 82: direct search method is simple to u
Page 83 and 84: Original Triangular Polytope Worst
Page 85 and 86: Chapter 7 RESULTS Each of the prece
Page 87 and 88: Figure 7.2: Flow Rate Surface of Si
Page 89 and 90: more stable than unmodified Newton
Page 91 and 92: Figure 7.5: Hagedorn and Brown Grad
Page 93 and 94: Figure 7.7: Aziz, Govier, and Fogar
Page 95 and 96: Figure 7.9: Orkeszewski Gradient Ma
Page 97 and 98: The Sachdeva et al. (1986) model wa
Page 99 and 100: Figure 7.11: Constrained Present Va
Page 101 and 102: At the end of each time step, the w
Page 103 and 104: Figure 7.13: Present Value Surface
Page 105 and 106: Figure 7.15: Profile of Tubing Diam
Page 107 and 108: Figure 7.17: Close-up of Tubing Dia
Page 109 and 110: Figure 7.19: First Derivative of Se
Page 111 and 112: Figure 7.21: First Derivative of Tu
Page 113 and 114: Figure 7.23: Second Mixed-Partial D
Page 115 and 116: Figure 7.24: Minimum Eigenvalue of
Page 117 and 118: Figure 7.26: Curvature of Hessian M
Page 119 and 120: Figure 7.27: Convergence Path of Un
Page 121 and 122: Figure 7.29: Convergence Path of Po
Page 123 and 124: Figure 7.31: Convergence Path of Tr
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• Nonlinear optimization may be u
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Nomenclature Α Area. B Flow regime
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xe Expansion point. xr Reflection p
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Bibliography [1] Achong, I.: “Rev
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[23] Charnes, A., and Cooper, W. W.
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Techniques,” SPE Paper 12159, pre
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[67] Strang, G.: Introduction to Ap
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