multivariate production systems optimization - Stanford University

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account for different flow regimes. The more recent correlations of Duns & Ros (1963), Orkiszewski (1967), Aziz et al. (1972), and Beggs & Brill (1973) account for slippage between the phases and provide different algorithms to model different flow regimes. 3.3 The Correlation of Hagedorn and Brown (1965) The version of Equation 3.24 presented by Hagedorn and Brown (1965), in field units, is 144 ΔP ΔZ = ρS + 4 fm Qo + Qw 2 M2 2.9652 x 1011 D5 + ρS 24 ρS Δ VM ΔZ 2gc (3.25) Hagedorn and Brown (1965) adopted an approach of backing out the liquid holdup. After obtaining multiphase flow performance data from an experimental well, the acceleration term and the friction term were solved in the conventional manner and then a value of liquid holdup was calculated to satisfy the observed pressure gradient. Thus the liquid holdup used in the above equation is not a true measure of the volume of the pipe occupied by the liquid but is merely a correlating parameter. To correlate the liquid holdup, Hagedorn and Brown (1965) drew upon the dimensionless groups defined by Ros (1961). These are NVL , the liquid velocity number; NVG , the gas velocity number; ND , the diameter number; and NL , the liquid viscosity number, modified by Hagedorn and Brown (1965) as 4 NLV = 1.938 VSL ρL σL 4 NGV = 1.938 VSG ρL σL (3.26) (3.27) ND = 120.872 D ρL σL (3.28) 4 NL = 0.15726 μL 1 3 ρLσL (3.29) These four numbers are used in conjunction with three graphs published by Hagedorn and Brown in their original paper (1965) to obtain HL as follows: • N L is used with the first graph to obtain the product CN L , where C is a correlating parameter.
• The term NGV NL 0.38 ND 2.14 is used with the second graph to obtain ψ, where ψ is a second correlating parameter. • The term NLV to obtain the term P 0.1 CNL 0.575 NGV PSC ND 25 is used in association with the third graph HL ψ which yields the liquid holdup. These graphs have been tabulated and are presented in Table 3.1. Table 3.1: Correlating Functions of Hagedorn and Brown (1965). GRAPH 1 GRAPH 2 GRAPH 3 N L CN L NGV NL 0.38 ND 2.14 ψ NLV P 0.1 CNL 0.575 NGV PSC ND .002 .0019 .010 1.00 0.2 .04 .005 .0022 .020 1.10 0.5 .09 .010 .0024 .025 1.23 1. .15 .020 .0028 .030 1.40 2. .18 .030 .0033 .035 1.53 5. .25 .060 .0047 .040 1.60 10. .34 .100 .0064 .045 1.65 20. .44 .150 .0080 .050 1.68 50. .65 .200 .0090 .060 1.74 100. .82 .400 .0115 .070 1.78 200. .92 --- --- .080 1.80 300. .96 --- --- .090 1.83 1000. 1.00 Having obtained the correlated value for liquid holdup, the pressure gradient due to hydrostatic head is simply dP g = dZ HH gC ρL HL + ρG HG = g gC ρS HL ψ (3.30) In determining the friction component, Hagedorn and Brown (1965) elected to correlate a two-phase friction factor with a two-phase Reynold's number using the standard Moody diagram. The two-phase Reynold's number, as defined by Hagedorn and Brown (1965), is
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: -dP = 1 Qα + Q β Mα + Mβ g gC d
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 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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