multivariate production systems optimization - Stanford University

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

amifications as accurately as possible and therefore a criterion of net present value should be used. 7.3 Results 7.3.1 The Surface of the Well Model Once the well model was completed and the objective criterion was decided upon, a surface map was generated of the objective criterion as a function of two decision variables. Namely, the present value of the <strong>production</strong> stream was plotted as a function of the separator pressure and the tubing diameter (see Figure 7.13). The surface appears to be a textbook example of an <strong>optimization</strong> surface: nice, smooth contour lines bounding the extreme value on all sides. However, closer inspection of the surface reveals a different story. As shown in Figure 7.14, looking at a closer level reveals a surface that is very rough and ill-behaved. The rough features were masked by the graphics software in Figure 7.13 but are plainly visible in Figure 7.14. A better understanding of the surface can be obtained by examining several unidimensional profiles. Figures 7.15 through 7.18 show four different profiles of the surface. The first three figures keep the separator pressure constant while varying the tubing diameter. It is clearly visible that the surface is very rough and discontinuous along this dimension. Figure 7.17 shows the amount of noise present at the maximum of the surface. Figure 7.18 is a profile of the other dimension. This profile was obtained by holding the tubing diameter constant and varying the separator pressure. The figure shows that in this dimension, the surface is a very smooth and continuous function. Thus, the surface is rough as a function of tubing diameter and smooth as a function of separator pressure. This conclusion about the surface obtained from the profiles is reinforced by investigating the derivatives of the function. Figures 7.19 and 7.20 show the first and second derivatives of separator pressure. These derivatives clearly exhibit that the surface is very smooth as a function of separator pressure (be advised that the two graphs are shown from two different viewing angles). The smoothness of the function with respect to separator pressure is very much in contrast to the roughness of the function with respect to tubing diameter. Figures 7. 21 and 7.22 show the first and second derivatives of tubing 94
Figure 7.13: Present Value Surface of Separator Pressure vs. Tubing Diameter. 95
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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
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 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: At the end of each time step, the w
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
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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