multivariate production systems optimization - Stanford University

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concentrations of the phases inside the system are distinct from the rates the phases are crossing the system’s boundaries. Since the phases are not moving in tandem, the phase volumes inside the system cannot be directly inferred from the phase flowrates. The difference between the in situ concentrations inside the system and the concentrations of the phases flowing though the system is known as the holdup phenomenon. Much of the attention of multiphase flow research has been directed at predicting liquid holdup and phase slippage. Now let us define the primary variables used in multiphase flow analysis. Liquid holdup, H L , is defined as the volumetric fraction of liquid residing inside the conduit HL = Volume of liquid Volume of conduit 20 (3.1) This is the true concentration of liquid in the conduit accounting for slippage between the phases. If the boundaries of the system were instantaneously closed and the system were allowed to achieve static equilibrium, this is the fraction of liquid that would settle out. On the contrary, the no-slip liquid holdup, λL , assumes no slippage between the phases and therefor the concentration of the phases in situ is the same as the fractional flow rates through the system Conversely λL = QL QL + QG HG = 1 - HL λG = 1 - λL The slip density and the no-slip density are defined as ρS = ρL HL + ρG HG ρNS = ρL λL + ρG λG Similarly, the slip viscosity and the no-slip viscosity are defined as μS = μ L HL μG HG (3.2) (3.3) (3.4) (3.5) (3.6) (3.7)
μNS = μL λL + μG λG 21 (3.8) The superficial velocity of a phase is the velocity the phase would exhibit if it were the only phase in the system and had access to the entire cross-sectional area of the conduit. The superficial velocity for gas and liquid are VSG = QG A and VSL = QL A and the velocity of the mixture is the sum of the superficial velocities VM = VSG + VSL The superficial phase velocities differ from the true velocity of the phases which are VG = QG A HG and VL = QL A HL Slip velocity is defined as the difference between the true velocities of the phases. VS = VG - VL (3.9) (3.10) (3.11) (3.12) Most multiphase flow correlations only attempt to model binary <strong>systems</strong>. In the context of hydrocarbon <strong>production</strong>, the two phases are typically taken to be gas and liquid. If more than one liquid phase is being considered, such as oil and water, then a weighted average of the liquid parameters is used. where and ρL = ρO fO + ρW fW μL = μO fO + μW fW σL = σO fO + σW fW fO = QO QW + QO fW = QW QW + QO (3.13) (3.14) (3.15) (3.16) (3.17)
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: Chapter 3 VERTICAL MULTIPHASE FLOW
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 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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