multivariate production systems optimization - Stanford University

List of Figures
Figure 1.1 Graphical Examples of Nodal Analysis.................................................. 4
Figure 1.2 Schematic of a Well Model. ............................................................... 4
Figure 2.1 Phase Behavior Assumptions of Reservoir Material Balance Procedure............. 9
Figure 3.1 Flow Regimes of Aziz, Govier, and Fogarasi (1972).................................. 28
Figure 3.2 Flow-Pattern Map Proposed by Aziz, Govier, and Fogarasi (1972)................. 29
Figure 3.3 Flow-Pattern Map Proposed by Orkiszewski (1967). ................................. 35
Figure 3.4 Pressure Traverse Flow Diagram......................................................... 39
Figure 4.1 Positive Choke.............................................................................. 41
Figure 4.2 Mass Flow Rate versus Pressure Ratio. ................................................. 43
Figure 4.3 Real and Spurious Roots of the Mass Flux Residual................................... 48
Figure 5.1 Schematic of Horizontal Separator. ...................................................... 49
Figure 5.2 Stage Separation. ........................................................................... 50
Figure 5.3 Effect of Separator Pressure in a Two-Stage Separation Process..................... 52
Figure 6.1 A Maximum and a Minimum.............................................................. 62
Figure 6.2 Discretization Scheme Used for Finite Difference Approximations .................. 67
Figure 6.3 Two-Dimensional Polytope. .............................................................. 75
Figure 7.1 Flow Rate Surface of Single-Phase Model, 3-D........................................ 78
Figure 7.2 Flow Rate Surface of Single-Phase Model, 2-D........................................ 79
Figure 7.3 Convergence Path of Single-Phase Model............................................... 80
Figure 7.4 Hagedorn and Brown Gradient Map, 2-D............................................... 82
Figure 7.5 Hagedorn and Brown Gradient Map, 3-D............................................... 83
Figure 7.6 Aziz, Govier, and Fogarasi Gradient Map, 2-D. ....................................... 84
Figure 7.7 Aziz, Govier, and Fogarasi Gradient Map, 3-D. ....................................... 85
Figure 7.8 Orkiszewski Gradient Map, 2-D.......................................................... 86
Figure 7.9 Orkiszewski Gradient Map, 3-D.......................................................... 87
Figure 7.10 Present Value Surface of Choke Diameter vs. Tubing Diameter...................... 90
Figure 7.11 Constrained Present Value Surface of Choke Diameter vs. Tubing Diameter....... 91
Figure 7.12 Revenue Stream is Discounted from Center of Time Step............................. 93
Figure 7.13 Present Value Surface of Separator Pressure vs. Tubing Diameter................... 95
Figure 7.14 Rough Features of Present Value Surface. .............................................. 96
Figure 7.15 Profile of Tubing Diameter for Constant Separator Pressure.......................... 97
Figure 7.16 Close-up of Tubing Diameter Profile for Constant Separator Pressure. ............. 98
ix