multivariate production systems optimization - Stanford University

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

employed, significant enhancements can be made to the polytope algorithm on a case-bycase basis. 76
Chapter 7 RESULTS Each of the preceding chapters has expounded the theory of a single facet of the well model. This chapter will discuss how this study integrated each of these individual components into a functional production system model and how the production system model was then optimized. 7.1 Model Development 7.1.1 Developing a Well Model Prototype To test out the general concept of using numerical techniques to optimize a well model, a simplified well model was initially developed. The simplified model was a single-phase, dry-gas model that consisted of three distinct components: a generalized inflow performance relationship to model the reservoir deliverability; the Cullender and Smith (1956) correlation to model the flow in the vertical tubing; and the Weymouth equation to model the flow through a horizontal flowline. The well components were coupled together and optimized with Newton’s Method. The decision variables of tubing inside-diameter and flowline inside-diameter were used to optimize the objective criterion of instantaneous flow rate (see Figures 7.1 and 7.2). Due to the simplified nature of the model, the surface was unbounded--the larger the inside-diameters, the larger the flowrate. The insidediameters were constrained to be a maximum of five inches each. Figure 7.3 shows the convergence path for two different starting points, both of which converge to the maximum inside-diameters allowed. Having proven the general concept of production optimization on a prototype, a more ambitious model-development program was launched. The new well model was to have significant improvements over the initial model. The new model was to account for multiphase flow in the reservoir, in the tubing, and in the surface facilities. The reservoir component of the new model was to account for the reservoir performance over time. The new model was to model flow through the surface choke, have an optimization scheme 77
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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
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 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: Original Triangular Polytope Worst
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
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
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[67] Strang, G.: Introduction to Ap
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