multivariate production systems optimization - Stanford University

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

Abstract The objective of this research has been to investigate the effectiveness of nonlinear optimization techniques to optimize the performance of hydrocarbon producing wells. The performance of a production well is a function of several variables. Examples of these variables are tubing size, choke size, and perforation density. Changing any of the variables will alter the performance of the well. There are several ways to optimize the function of well performance. An insensible way to optimize the function of well performance is by exhaustive iteration: optimizing a single variable by trial and error while holding all other variables constant, and repeating the procedure for different variables. This procedure is computationally expensive and slow to converge--especially if the variables are interrelated. A prudent manner to optimize the function of well performance is by numerical optimization, particularly nonlinear optimization. Nonlinear optimization finds the combination of these variables that results in optimum well performance by approximating the function with a quadratic surface. The advantages of nonlinear optimization are many. Using nonlinear optimization techniques, there is no limit to the number of decision variables that can be optimized simultaneously. Moreover, the objective function may be defined in a wide variety of ways. Nonlinear optimization achieves quadratic convergence in determining the optimum well performance and avoids a trial and error solution. Several different optimization methods are investigated in this study: Newton’s Method, modified Newton’s Method with Cholesky factorization, and the polytope heuristic. Significant findings of this study are: the performance of Newton’s Method can be greatly improved by including a line search procedure and a modification to ensure a direction of descent; For nonsmooth functions, the polytope heuristic provides an effective alternative to a derivative-based method; For nonsmooth functions, the finite difference approximations are greatly affected by the size of the finite difference interval. This study found a finite difference interval of one-tenth of the size of the variable to be advisable. iv
Acknowledgements I would like to thank Dr. Roland Horne for providing me with thoughtful guidance, sage advice, and understanding patience. I also extend thanks to Dr. Khalid Aziz for providing sound and judicious counsel and to Dr. Hank Ramey for his generous contributions. I would like to thank Rafael Guzman for the time he invested proofreading several drafts of this manuscript. I would also like to thank Cesar Palagi, Evandro Nacul, Adalberto Rosa, Michael Riley, and Mariyamni Awang for providing a constant source of quality advice. I would like to thank Russell Johns for providing experience in the drafting of this manuscript. I would like to thank Suhail Qadeer and Daulat Mamora for helping me with the computer hardware and David Brock and Chick Wattenbarger for being of assistance with the computer graphics. I would also like to express thanks to Dr. Walter Murray and Sam Eldersveld of the Stanford Optimization Laboratory for their accessibility and helpfulness. I would like to thank Dr. Jawahar Barua for writing a superb thesis that was instrumental in my fully appreciating nonlinear concepts. I would like to thank Gunnar Borthne of the Norwegian Institute of Technology and Godofredo Perez of the University of Tulsa for providing me with computer programs that facilitated my work. I would also like to thank Joseph Gump of Chevron, U.S.A., Gene Kouba of Chevron Exploration and Production Services, and Dr. Rajesh Sachdeva of Simulation Sciences for providing me with a great deal of time and help pertaining to system theory. I would like to thank Dr. Parvis Moin of NASA Ames Research Center for inspiring my work by showing me the beauty and simplicity of numerical methods. I would like to express my sincere gratitude to Dr. Elias Schultzman and the National Science Foundation for providing me with the financial capacity to pursue this research. I would also like to thank the Stanford University Petroleum Research Institute for providing resources and support. Finally, I would like to thank the two most important people in the world, James and Darlene Carroll, mom and dad, for giving me this opportunity. v
Page 1 and 2: MULTIVARIATE PRODUCTION SYSTEMS OPT
Page 3: iii Approved for the Department: Ro
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 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
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segregation. The gas phase must exi
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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
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ψ1 = 2 cos θ 3 - Ω 3 (5.28) ψ2
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Chapter 6 NONLINEAR OPTIMIZATION Th
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IMSL (1987). For a broad overview o
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The minimum of a function can occur
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F(x + hj ej - hi ei) F(x - hi ei) F
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6.2.1 The Method of Steepest Descen
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The second step of the Greenstadt (
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direct search method is simple to u
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Original Triangular Polytope Worst
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Chapter 7 RESULTS Each of the prece
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Figure 7.2: Flow Rate Surface of Si
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more stable than unmodified Newton
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Figure 7.5: Hagedorn and Brown Grad
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Figure 7.7: Aziz, Govier, and Fogar
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Figure 7.9: Orkeszewski Gradient Ma
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The Sachdeva et al. (1986) model wa
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Figure 7.11: Constrained Present Va
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At the end of each time step, the w
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Figure 7.13: Present Value Surface
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Figure 7.15: Profile of Tubing Diam
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Figure 7.17: Close-up of Tubing Dia
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Figure 7.19: First Derivative of Se
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Figure 7.21: First Derivative of Tu
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Figure 7.23: Second Mixed-Partial D
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Figure 7.24: Minimum Eigenvalue of
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Figure 7.26: Curvature of Hessian M
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Figure 7.27: Convergence Path of Un
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Figure 7.29: Convergence Path of Po
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Figure 7.31: Convergence Path of Tr
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• Nonlinear optimization may be u
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Nomenclature Α Area. B Flow regime
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xe Expansion point. xr Reflection p
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Bibliography [1] Achong, I.: “Rev
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[23] Charnes, A., and Cooper, W. W.
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Techniques,” SPE Paper 12159, pre
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[67] Strang, G.: Introduction to Ap
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