multivariate production systems optimization - Stanford University

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educed by an order of magnitude. On the other hand, if a line search indicates that interpolation (a decrease in the step-length ρ) is necessary, the Marquardt parameter is increased by an order of magnitude. If the Marquardt parameter is large, the method imitates the method of steepest descent, which converges slowly but has excellent stability properties. If the Marquardt parameter is small, the method behaves like Newton’s Method, which has good convergence characteristics but is prone to difficulties. Thus, in ill-conditioned regions the Marquardt (1963) modification will imitate the method of steepest descent whereas in wellbehaved areas the method will imitate Newton’s Method. The modification seemingly offers the best of both worlds. The caveat is that the Marquardt (1963) modification can become “stuck” in illconditioned regions if the Marquardt parameter is allowed to increase unfettered. Expressing Equation 6.26 as 1 μ H + I p = - 1 μ g (6.27) it is apparent that as μ → ∞, the step-length becomes infinitesimal. Therefore, when implementing the Marquardt (1963) modification, the Marquardt parameter should always be given an upper bound of some kind and a line search procedure should always be used. 6.2.3 The Greenstadt Modification Greenstadt (1967) proposed a procedure to ensure that the Hessian matrix is positive definite. The Greenstadt (1967) modification involves two steps: 1) All of the negative eigenvalues are replaced with their absolute value; 2) Any small eigenvalues are replaced with infinity. The first step requires spectral decomposition to ascertain the eigenvalues. Although this procedure is very expensive computationally, it avoids the main pitfall of the Marquardt (1963) modification--blindly changing the eigenvalues in lockstep. By taking the absolute value of the negative eigenvalues, the Greenstadt (1967) modification merely reverses the direction of the ascending eigenvectors and forces them to take the same step length in a descent direction. The relative proportions of all the eigenvalues are maintained, the only difference being that they are all moving downhill. 70
The second step of the Greenstadt (1967) modification, replacing small eigenvalues with infinity, is based on the objective function being insensitive to any small eigenvalues. Since the objective function is insensitive to a parameter with a small eigenvalue, this parameter should not be allowed to influence the step-direction. This is precisely what is achieved by replacing the eigenvalue with infinity. Replacing an eigenvalue with infinity is accomplished by replacing the reciprocal of the eigenvalue with zero when inverting the Hessian. Conceptually, the Greenstadt (1967) modification applied to Newton’s Method should result in a robust algorithm that is rapidly convergent and not affected by insensitive parameters; the caveat being that it is very expensive computationally. For information on the performance of the Newton-Greenstadt algorithm applied to petroleum engineering problems, see Barua, Horne, Greenstadt, and Lopez (1989). 6.3 Quasi-Newton Methods Newton’s Method is quadratically convergent when starting with a good initial estimate. However, it is very expensive since the Hessian must be built and solved every iteration, particularly when the Hessian is built with finite difference approximations. The idea behind Quasi-Newton methods is to compromise on the speed of convergence while saving on the expense associated with building and solving the Hessian. Instead of building and solving an exact Hessian every iteration, Quasi-Newton methods attempt to update an approximation of the Hessian. Quasi-Newton theory is based on multidimensional generalizations of the secant method. The object is to build up secant information as the iterations proceed. Suppose at the kth iteration, the Newton step {xk+1 - xk } causes a change in the gradient of {gk+1 - gk }. Then the next Hessian approximation will satisfy the secant passing through these two iterates if xk+1 - xk = Hk+1 gk+1 - gk 71 (6.28) This condition is termed the Quasi-Newton condition and is the design criteria for Quasi- Newton methods. Notice the similarity between this condition and the Newton condition, Equation 6.12. This condition forces the Hessian approximation to exactly match the gradient of the function in the displacement direction, {xk+1 - xk }. For multiple
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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.
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Δ φ SG BG + SO RS BO ρGO S S ρG
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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 and 74: The minimum of a function can occur
Page 75 and 76: F(x + hj ej - hi ei) F(x - hi ei) F
Page 77: 6.2.1 The Method of Steepest Descen
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
Page 125 and 126: • Nonlinear optimization may be u
Page 127 and 128: 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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