multivariate production systems optimization - Stanford University

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formulate the problem to solve for the root of a system of nonlinear equations, namely by using Newton’s Method? Simply set up the problem to contain a pressure residual at each node and then use Newton’s Method to solve the equations for the stabilized flow rate. Alternately, you could set up the problem to contain a flow rate residual at each node and then use Newton’s Method to solve the equations for the stabilized pressure at the solution node. Hence, you could use Newton’s Method for two different purposes in the optimization of a production system: to solve for the zero of the function and then again to solve for the zero of the gradient of the function. Several improvements can be made to the current well model. Obviously, it would be interesting to know why the model is nonsmooth with respect to tubing diameter. The cause of this is not yet apparent. In addition, it would be nice to expand the model to include a completion component that models the completion effects (perforations, gravel pack, ....) and also a horizontal flow component that models the pressure loss and phase segregation that occur in surface flowlines between the wellhead and the separator. It would also be interesting to develop a well model based on a compositional equations instead of black oil equations. In addition, one area where the current well model has significant problems is with computational expense. A more efficient manner for solving for the stabilized flow rate should be found. Perhaps the biggest improvement would be to demonstrate the use of different objective criteria in the optimization process. This would clearly demonstrate the inherent advantages of nonlinear optimization over exhaustive iteration. 119
Nomenclature Α Area. B Flow regime boundaries used by Aziz et al (1972) correlation. Bα Formation volume factor of phase α. C Centroid of polytope; Hagedorn and Brown (1965) correlating parameter. CD Discharge coefficient. Cν Cumulative production during time step n. D Diameter; Long-term debt, $; Non-Darcy skin. ΔE Material balance error. Δt Time step, t k - t k-1 . E Outstanding equity, $. ei ith unit vector. EK Kinetic energy component; acceleration term. F Nonlinear function. fα Fugacity of phase α; Fractional flow rate of phase α. fM Moody friction factor. f Partial fugacity. G Cumulative gas production; Mass flux. g Force of gravity; Gradient n-vector of x. gc Gravity constant. Η Hessian matrix. Hα Holdup of phase α; Enthalpy of phase α. h Height of reservoir. hi Finite difference interval of ith dimension. I Identity matrix. i Inflation rate, fraction. K Specific heat ratio, Cp / Cv; Equilibrium ratio; Binary interaction parameter. k Reservoir permeability. krα Relative permeability of phase α. L Flow regime boundary used by Griffith & Wallis (1961), Orkiszewski (1967); Liquid mole fraction. λα No-slip holdup of phase α. M Mobility ratio. m Mass of fluid in the reservoir. 121
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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
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which is valid for steady-state, ra
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Chapter 3 VERTICAL MULTIPHASE FLOW
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μNS = μL λL + μG λG 21 (3.8) T
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-dP = 1 Qα + Q β Mα + Mβ g gC d
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• The term NGV NL 0.38 ND 2.14 is
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HL = 1 - 1 2 1 + VM VS - 1 + VM VS
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A paper by Hazim and Nimat (1989) p
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NV ≤ 18 25 18 < NV < 250 69 NV -0
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NWE Nμ > 0.005 ε = 0.3713 σL D 3
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NLV 100. 10. 1.0 0.1 Bubble Griffit
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V M < 10 δ ≥ -0.065 VM V M > 10
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RETURN ΔZ=.95 Δ Z ΔP=.95 Δ Pnew
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• maintain stable pressure downst
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R P = producing gas-liquid ratio, M
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flowrate is straightforward. Fortun
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Mass Flux Residual G1 - G2 0 Spurio
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segregation. The gas phase must exi
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API, GOR, Bo API Gravity First Stag
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Zi = Xi L + Yi V (5.3) where V and
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aα m = ∑ ∑ YiYj aα ij i j bm
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ψ1 = 2 cos θ 3 - Ω 3 (5.28) ψ2
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Chapter 6 NONLINEAR OPTIMIZATION Th
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IMSL (1987). For a broad overview o
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The minimum of a function can occur
Page 75 and 76: F(x + hj ej - hi ei) F(x - hi ei) F
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
Page 125: • Nonlinear optimization may be u
Page 129 and 130: xe Expansion point. xr Reflection p
Page 131 and 132: Bibliography [1] Achong, I.: “Rev
Page 133 and 134: [23] Charnes, A., and Cooper, W. W.
Page 135 and 136: Techniques,” SPE Paper 12159, pre
Page 137: [67] Strang, G.: Introduction to Ap
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