multivariate production systems optimization - Stanford University

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3.4.4 Annular-Mist Flow Regime For modeling the annular-mist flow regime, Aziz et al. (1972) used the procedure developed by Duns and Ros (1963). Duns and Ros assumed that the high gas velocity of the annular-mist region would allow no slippage to occur between the phases. The mixture density used to calculate the density component is, therefore, the no-slip density, ρNS . The expression for the density component is dP = dZ HH g gC ρNS = g gC ρL λL + ρG λG 32 (3.70) The friction component for the annular-mist region is based solely on the gas phase and is given by dP dZ F = fM 2 ρG VSG 2 gC D where the Moody friction factor is based on the Reynold’s number of the gas Re = ρG VSG D μG Duns and Ros (1963) gave special treatment to the manner that the relative roughness was determined. They discovered that the pipe roughness was altered by the thin layer of liquid on the wall of the pipe. Two variables are used to characterize this effect. The first is a form of the Weber number NWE = ρG 2 VSG ε σL and the second is dimensionless number based on liquid viscosity Nμ = μL ρL σL ε (3.71) (3.72) (3.73) (3.74) Duns and Ros (1963) proposed the following relationship to model the relative roughness: NWE Nμ ≤ 0.005 ε = 0.0749 σL D 2 ρG VSG D (3.75)
NWE Nμ > 0.005 ε = 0.3713 σL D 33 NWE Nμ 2 ρG VSG D 0.302 The value of the relative roughness is constrained to be no less than the actual relative roughness of the pipe and no more than 0.5. (3.76) For values of relative roughness greater than 0.05, the limits of the Moody diagram are approached. Duns and Ros (1963) proposed the following relation for the moody friction factor for values of relative roughness greater than 0.05: fM = 4 1 4 log 0.27 ε 2 D + 0.067 ε D We can account for the acceleration component by defining E K as The total pressure gradient is then given by EK = VM VSG ρNS gC P dP = dZ MIST dP dZ HH + dP dZ F 1 - EK As mentioned earlier, treating the kinetic energy term in this fashion requires care as the term will become indeterminant as E K approaches one. 3.5 The Correlation of Orkiszewski (1967): 1.73 (3.77) (3.78) (3.79) The work of Orkiszewski (1967) is a composite product: after testing the available correlations for their accuracy, he selected the flow correlations he considered to be the most accurate. These include the Griffith and Wallis (1961) method for bubble flow, the Duns and Ros (1963) method for annular-mist flow and transition flow, and a method he proposed for slug flow. The Griffith and Wallis (1961) method was previously discussed with the Hagedorn and Brown (1965) correlation and the Duns and Ros (1963) method was previously discussed with the Aziz et al. (1972) correlation. Discussion of these methods will not be repeated here.
Page 1 and 2: MULTIVARIATE PRODUCTION SYSTEMS OPT
Page 3 and 4: iii Approved for the Department: Ro
Page 5 and 6: Acknowledgements I would like to th
Page 7 and 8: 5.2.4 Determine Partial Fugacities
Page 9 and 10: List of Figures Figure 1.1 Graphica
Page 11 and 12: Chapter 1 INTRODUCTION The performa
Page 13 and 14: eservoir model. Kuller and Cummings
Page 15 and 16: optimization, it is merely the proc
Page 17 and 18: Chapter 2 RESERVOIR AND INFLOW PERF
Page 19 and 20: where RS = V GO S 9 V OO S rS = V O
Page 21 and 22: 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: NV ≤ 18 25 18 < NV < 250 69 NV -0
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 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
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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