The Pennsylvania State University
The Graduate School
John and Willie Leone Family Department of Energy and Mineral Engineering
INTEGRATION OF NUMERICAL AND MACHINE LEARNING PROTOCOLS FOR COUPLED RESERVOIR-WELLBORE MODELS: A STUDY FOR GAS LIFT OPTIMIZATION
A Dissertation in
Energy and Mineral Engineering
by
Venkataramana Balamurugan Srikanth Putcha
2017 Venkataramana Balamurugan Srikanth Putcha
Submitted in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
December 2017
ii The dissertation of Venkataramana Balamurugan Srikanth Putcha was reviewed and approved* by the following:
Turgay Ertekin Professor Emeritus of Petroleum and Natural Gas Engineering Chair of Committee Dissertation Advisor
Eugene Morgan Assistant Professor of Petroleum and Natural Gas Engineering
Kamesh Madduri Assistant Professor of Computer Science and Engineering
Hamid Emami-Meybodi Assistant Professor of Petroleum and Natural Gas Engineering
Sridhar Anandakrishnan Professor of Geosciences
Luis F. Ayala H. Professor of Petroleum and Natural Gas Engineering Associate Department Head for Graduate Education
*Signatures are on file in the Graduate School
iii ABSTRACT
As the reservoir pressure declines with time, many of the wells do not have adequate bottom- hole pressure to carry the fluids to the surface. Under such circumstances, artificial lift mechanisms must be employed. Amongst various artificial lift mechanisms, a significant proportion of wells utilize the gas-lift mechanism, which is an extension of the natural flow. In gas-lift implementation, high pressure gas is injected into the wellbore through a valve, where injected gas supports production by altering the composition and reducing the density, and increasing the velocity of the produced fluids.
In order to design a gas-lift system, a study of the inflow performance of the fluid from the reservoir into the wellbore, combined with the outflow performance of the fluids from the bottom of the wellbore to the surface is necessary. For this purpose, existing technologies for optimization of gas-lift systems predominantly use empirical correlations in order to reduce the computational overhead. These systems use a single-equation based inflow performance relations and black-oil outflow performance correlations that have restricted applicability in systems where the fluid composition varies spatially and temporally. The contemporary protocols consider the oil flow rate, water cut and formation gas-liquid ratio and well productivity index at a given instant of time to calculate the optimal quantity of gas lift injection. Due to this methodology, the effects of pressure decline and subsequent variations in well performance are not adequately captured. This results in a solution which determines the maximum liquid flow rate expected for a given gas lift injection rate only for the instantaneous period at which the study has been performed. This optimal gas lift injection rate may or may not provide the maximum total output of oil over the producing life of the well.
iv As a first step, a compositional coupled numerical reservoir and wellbore hydraulics models has been developed as a part of this work. These hard-computing tools simulate the variations in composition, pressure and production profiles of a gas lift well and its associated reservoir from inception to abandonment. One more advantage of this method is that it can predict the future performance of a well with or without the details of well production history. This capability can be useful when gas lift is introduced in a well immediately after its completion post a drilling or a work-over job.
Soft computing tools have gained popularity in the petroleum industry due to their speed, simplicity, wide range of applicability, capacity to identify patterns and ability to provide inverse solutions. The fully numerical coupled reservoir-wellbore simulator developed is computationally expensive. In order to develop a faster system, firstly, an ANN based wellbore hydraulics tool is developed and coupled with the numerical reservoir simulator. The data utilized for training the
ANN tool was generated using the numerical wellbore hydraulics tool.
Both the numerical and ANN wellbore hydraulics models were validated against cases from the field and another compositional numerical model from the literature. The average relative deviation with respect to field data was observed to be 2.2% and 2.4% respectively for the ANN and numerical wellbore hydraulics model, respectively. When compared against another compositional numerical model, the average relative deviation for the ANN based model was observed to be between 3.3% and 7.1%, while it was between 2.3% and 8.1% for the numerical model developed in this work. While the ANN based wellbore hydraulics model maintained the accuracy of the numerical model, it outperformed its counterpart the numerical model, by four orders of magnitude in terms of speed-up.
The ANN based wellbore model was also coupled with the numerical reservoir simulator. This resultant model which involves a coupled numerical-ANN system is faster than the fully numerical coupled system by about 160 times. This coupled tool was used to generate a gas lift v database of cumulative oil production of a well with various reservoir and wellbore operating conditions under a range of operating gas lift injection depths and flow rates. This database was used to develop an ANN based gas lift model that is capable of generating performance curves plotting total oil produced during the producing life of a well as a function of gas lift injection rate. Blind testing of the ANN gas lift model showed an average absolute error of 16.6 % with respect to the predictions of the coupled numerical-ANN reservoir wellbore model. This fully
ANN based gas lift model provided a speed-up by four orders of magnitude with respect to the coupled numerical-ANN based model. Hence, a fast, robust and versatile model has been developed for maximizing total primary oil recovery using gas lift optimization through integration of numerical and neuro-simulation.
vi TABLE OF CONTENTS
List of Figures ...... ix
List of Tables ...... xii
Nomenclature ...... xvi
Acknowledgements ...... xix
Chapter 1 Introduction ...... 1
Chapter 2 Literature Survey ...... 5
2.1. Gas lift technology ...... 5 2.2. Gas lift optimization ...... 6 2.3. Inflow performance relationship (IPR): ...... 8 2.4. Outflow performance relationship: ...... 10 2.5. Machine learning methods in simulation and artificial lift: ...... 12
Chapter 3 Problem statement and objectives ...... 13
Chapter 4 Methodology ...... 16
4.1. Hard computation tools: ...... 17 4.1.1. Compositional numerical reservoir model ...... 17 4.1.2. Compositional numerical wellbore hydraulics model ...... 17 4.1.3. Coupled numerical reservoir-wellbore simulator ...... 18 4.1.4. Fully numerical gas lift performance tool ...... 18 4.2. Soft computation tools ...... 18 4.2.1. ANN based wellbore hydraulics model ...... 18 4.2.2. Fully ANN based gas lift performance model ...... 18 4.3. Coupled hard and soft computation tools ...... 19 4.3.1. Coupled numerical reservoir- ANN based wellbore model ...... 19 4.3.2. Coupled numerical - ANN based gas lift performance model ...... 19
Chapter 5 Development of a numerical reservoir model ...... 20
5.1. Primary equations and variables ...... 20 5.2. Auxiliary equations: ...... 21 5.3. Description of the flow geometry: ...... 22 5.4. Finite difference equations for the reservoir model: ...... 24 5.5. Initial and boundary conditions: ...... 26 5.6. Newton-Raphson protocol for linearizing equations ...... 28 5.7. Computational protocol followed: ...... 29 5.8. Volumetric flow rates of phases ...... 31 5.9. Three-phase relative permeability model ...... 32
Chapter 6 Numerical wellbore model ...... 33 vii 6.1. Primary equations and variables...... 34 6.2. Auxiliary equations: ...... 35 6.3. The Drift-flux model: ...... 36 6.3.1 Bubbly flow: ...... 37 6.3.2. Slug flow: ...... 38 6.3.3. Churn flow: ...... 38 6.3.4. Annular flow: ...... 39 6.4. Transition smoothening: ...... 39 6.5. Finite difference equations for wellbore: ...... 39 6.6. Initial and boundary conditions: ...... 42 6.7. Newton-Raphson protocol for linearizing equations ...... 44
Chapter 7 Nuero-simulation of wellbore hydraulics ...... 46
7.1. ANN based wellbore model ...... 46 7.2. Data generation and pre-processing ...... 46 7.3. ANN architecture selection and training ...... 49 7.4. Modular ANN toolbox ...... 49 7.5. Computational performance comparison of the numerical model and neuro- simulation ...... 54
Chapter 8 Validation of the standalone models ...... 55
8.1. Standalone numerical compositional reservoir simulator ...... 55 8.2. Standalone compositional wellbore simulator ...... 62 8.3. Summary ...... 73
Chapter 9 Coupled reservoir-wellbore simulator ...... 75
9.1. Coupling mechanism for a fully numerical model ...... 75 9.2. Coupling mechanism for a numerical reservoir-ANN based wellbore model ...... 77 9.3. Comparison with a commercial simulator ...... 78 9.3.1. Single-phase liquid case ...... 78 9.3.2. Single-phase gas case ...... 80 9.3.3. Two-phase case ...... 82
Chapter 10 Integrated gas lift simulation and optimization ...... 86
10.1. Model 1- Fully numerical system ...... 86 10.1.1. Implementation of gas-lift in the wellbore and modification of boundary conditions ...... 86 10.2. Model 2- Coupled numerical-ANN system ...... 87 10.3. Model 3- Fully ANN based system ...... 88 10.3.1. Data generation and pre-processing for the ANN based gas lift model (GL-ANN) ...... 88 10.3.2. Architecture selection and training of the GL-ANN ...... 94 10.4. Comparisons of the gas lift models ...... 96 10.4.1. Case GL1: Performance comparison between all three gas lift models ...... 96 viii 10.4.2. Case GL2: Performance comparison between numerical-ANN coupled and fully ANN based gas lift models ...... 102 10.5. Gas lift case studies ...... 108 10.5.1. Gas lift case study 1 ...... 109 10.5.2. Gas lift case study 2-Field case ...... 113 10.6. Monte Carlo simulation ...... 122
Chapter 11 Conclusions and future work ...... 127
11.1. Summary ...... 127 11.2. Conclusion ...... 129 11.3. Future work ...... 131
References ...... 132
Appendix A- Critical properties of components ...... 136
Appendix B- Tabulated results for Section 10.4.2.1- case GL 2b ...... 137
ix LIST OF FIGURES
Figure 2-1. Schematic of a typical gas lift system ...... 5
Figure 2-2. Schematic of a typical gas lift performance curve ...... 7
Figure 2-3. Gas lift performance curve in monetary units ...... 7
Figure 2-4. Inflow performance relationship described by Vogel (1966) ...... 9
Figure 3-1. Plot to determine operating flow conditions...... 13
Figure 4-1. Schematic of the methodology followed ...... 16
Figure 5-1. System of equations for a cell in the discretized reservoir model ...... 28
Figure 5-2. Schematic for the procedure used in the reservoir simulator ...... 30
Figure 6-1. Visual representation of various flow regimes ...... 37
Figure 6-2. Schematic of the discretized wellbore model ...... 40
Figure 6-3. System of equations for a cell in the discretized wellbore hydraulics model ...... 44
Figure 6-4. Schematic representation of the computational protocol used in numerical wellbore simulation ...... 45
Figure 7-1. Histogram of error counts for various wellbore ANN modules ...... 52
Figure 7-2. Cross plots of prediction versus targets for various wellbore ANN modules ...... 53
Figure 8-1. Schematic of the reservoir simulation ...... 55
Figure 8-2. Oil flow rate comparison – Case 1 ...... 56
Figure 8-3. Oil flow rate comparison – Case2 ...... 57
Figure 8-4. Gas flow rate comparison – Case2 ...... 58
Figure 8-5. Case 1: Comparison of wellbore models developed in this research with field data and mechanistic models ...... 64
Figure 8-6. Black-oil case: Comparison of wellbore models developed in with Pourafshary et al. (2009) ...... 66
Figure 8-7. Volatile-oil case: Comparison of wellbore models developed with Pourafshary et al. (2009) ...... 68
Figure 8-8. Gas condensate case: Comparison of wellbore models developed in the current work with the model by Pourafshary et al. (2009) ...... 70 x Figure 8-9. Three-phase flow case: Comparison of wellbore models developed in this work with Pourafshary et al. (2009) ...... 71
Figure 9-1. Single-phase liquid case: Comparison with a commercial numerical simulator ... 79
Figure 9-2. Single-phase gas case: Comparison with a commercial numerical simulator ...... 81
Figure 9-3. Two-phase case: Comparison of cumulative oil production with commercial numerical simulator ...... 83
Figure 9-4. Two-phase case: Comparison of cumulative gas production with commercial numerical simulator ...... 84
Figure 10-1. Histogram of error counts for the ANN based gas lift model ...... 95
Figure 10-2. Cross plots of prediction versus target oil recovery factor for the ANN based gas lift model ...... 95
Figure 10-3(a). Gas lift performance curve predicted by models 1-3 for case GL1 ...... 99
Figure 10-3(b). Derivative of the gas lift performance curve predicted by models 1-3 for case GL1 ...... 99
Figure 10-4. Gas lift performance curve with varying gas lift injection point ...... 103
Figure 10-5. Case GL2b-Gas lift performance curve for Fluid A ...... 105
Figure 10-6. Case GL2b-Gas lift performance curve for Fluid B ...... 105
Figure 10-7. Case GL2b-Gas lift performance curve for Fluid C ...... 106
Figure 10-8. Case GL2b-Gas lift performance curve for Fluid D ...... 106
Figure 10-9. Case GL2b-Gas lift performance curve for Fluid E ...... 106
Figure 10-10. Case GL2b-Gas lift performance curve for Fluid F ...... 107
Figure 10-11. Case GL2b-Gas lift performance curve for Fluid G ...... 107
Figure 10-12. Case GL2b-Gas lift performance curve for Fluid H ...... 107
Figure 10-13. Gas lift case study 1: gas lift performance curve for standalone wellbore model ...... 111
Figure 10-14. Gas lift case study 1: gas lift monetary performance curve for standalone wellbore model ...... 112
Figure 10-15. Gas lift case study 1: slope of gas lift monetary performance curve for standalone wellbore model ...... 112 xi Figure 10-16. Gas lift case study 2: gas lift performance curve for standalone wellbore model ...... 116
Figure 10-17. Gas lift case study 2: gas lift performance curve for a well R3-3 for total production period ...... 118
Figure 10-18. A high-resolution gas lift performance plot generated using the gas lift ANN ...... 118
Figure 10-19. Gas lift case study 2: Gas lift monetary performance curve for an integrated gas lift system...... 119
Figure 10-20. Gas lift case study 2: Slope of gas lift monetary performance curve to locate economic optimum ...... 120
Figure 10-21. Cumulative distribution function for optimum gas lift injection rate ...... 124
Figure 10-22. Cumulative distribution function for the total oil produced ...... 125
xii LIST OF TABLES
Table 6-1. Hasan and Kabir (2007a) drift-flux model parameters ...... 37
Table 7-1. Summary of the input parameters used in the ANN based wellbore model ...... 48
Table 7-2. Architecture description and blind testing results of the wellbore ANN modules ...... 51
Table 8-1. Inputs for comparison of commercial reservoir simulator with current work - Case 1 ...... 56
Table.8-2. Results for comparison of commercial reservoir simulator with current work- Case 1 ...... 56
Table.8-3. Inputs for comparison of commercial reservoir simulator with current work - Case 2 ...... 57
Table.8-4. Results for comparison of commercial reservoir simulator with current work - Case 2 ...... 57
Table 8-5. Input for transient-well test analysis-current work ...... 58
Table 8-6. Comparison of current work with transient-well test analysis with current work ...... 59
Table 8-7. Input for pseudo steady state-well test analysis- current work ...... 59
Table 8-8. Comparison of current work with pseudo steady state- well test analysis- current work ...... 60
Table 8-9. Input for transient-well test analysis-commercial simulator ...... 60
Table 8-10. Comparison of current work with transient-well test analysis with commercial simulator ...... 60
Table 8-11. Input for pseudo steady state-well test analysis- commercial simulator ...... 61
Table 8-12. Comparison of current work with pseudo steady state- well test analysis- commercial simulator ...... 61
Table 8-13. Case 1: Black-oil production field data ...... 63
Table 8-14. Case 1: Fluid composition-calculated for field data ...... 63
Table 8-15. Case 1: Results- comparison of developed ANN and numerical wellbore models with field data and mechanistic models ...... 63
Table 8-16. Case 2: Wellbore description ...... 65 xiii Table 8-17. Case 2: Fluid composition input ...... 65
Table 8-18. Case 2: Results- comparison for black-oil fluid ...... 65
Table 8-19. Case 3: Wellbore description ...... 67
Table 8-20. Case 3: Fluid composition input ...... 67
Table 8-21. Case 3: Results- comparison for Volatile-oil fluid ...... 67
Table 8-22. Case 4: Wellbore description ...... 68
Table 8-23. Case 4: Fluid composition input ...... 69
Table 8-24. Case 4: Results- comparison for gas condensate fluid ...... 69
Table 8-25. Case 5: Wellbore description ...... 70
Table 8-26. Case 5: Hydrocarbon composition input ...... 71
Table 8-27. Case 5: Results- comparison for three-phase system ...... 71
Table 8-28. Summary of the observed deviations for wellbore comparison cases 1 through 5 ...... 72
Table 9-1. Single phase liquid case: System inputs ...... 78
Table 9-2. Single-phase liquid case: Results ...... 79
Table 9-3. Single-phase gas case: System inputs ...... 80
Table 9-4. Single-phase gas case: Results ...... 81
Table 9-5. Two-phase case: System inputs ...... 82
Table 9-6. Two-phase case: Results ...... 83
Table 10-1. Summary of inputs for the ANN based gas lift model (GL-ANN) ...... 90
Table 10-1. Summary of inputs for the ANN based gas lift model (GL-ANN) (continued) ... 91
Table 10-2. Architecture description and blind testing results of the GL-ANN ...... 94
Table 10-3. Case GL1-Fluid molar composition for comparison study between gas lift models 1-3 ...... 97
Table 10-4. Case GL1- reservoir and well inputs for comparison study between gas lift models 1-3 ...... 97 xiv Table 10-5. Case GL1-comparison of predicted total oil produced by the system by gas lift models 1-3 ...... 98
Table 10-6. Case GL1-comparison of incremental total oil produced per unit change in gas lift injection rate ...... 100
Table 10-7. Case GL1: Computational time comparison for models 1-3 ...... 101
Table 10-8. Case GL2a-comparison of gas lift performance with varying gas lift injection depth ...... 102
Table 10-9. Case GL2b- reservoir and well inputs for comparison study at various API gravities ...... 104
Table 10-10. Case GL2b- Lift gas molar composition ...... 104
Table 10-11. Case GL2b- API gravity and OGIP: OOIP ratio of the fluids considered ...... 105
Table 10-12. Case GL2b- Comparison of Model 2 and Model 3 for various reservoir fluids considered ...... 108
Table 10-13. Well#1 description for gas lift case study 1 as provided by Redden et al. (1974) ...... 109
Table 10-14. Hydrocarbon composition of fluids for gas lift case study 1 ...... 110
Table 10-15. Well R3-3 description for gas lift case study 2 as provided by Abdel-Waly et al. (1996) ...... 113
Table 10-16. Hydrocarbon composition of fluids for gas lift case study 2 ...... 114
Table 10-17. Comparison of pressure predictions for well R3-3 from Abdel-Waly et al. (1996) ...... 114
Table 10-18. Gas lift case 2: gas lift performance comparison- standalone wellbore models ...... 116
Table 10-19. Gas lift case 2-Input reservoir parameters considered for coupled reservoir- well gas lift simulation ...... 117
Table 10-20. Gas lift case 2- gas lift performance for the production period of well R3-3 (Abdel-Waly et al., 1996)...... 117
Table 10-21. Gas lift case 2 summary: benefits of using integrated gas lift optimization approach ...... 121
Table 10-22. Summary of the range of variation of reservoir parameters ...... 122
Table 10-23. Probability values for the estimates of optimum gas lift injection rate ...... 123 xv Table 10-24. Probability values for the estimates of maximum total oil produced ...... 123
Table 10-25. Probability values for the estimates of maximum total oil produced ...... 125
xvi NOMENCLATURE A = Area of cross-section (ft2) Co = Distribution Coefficient (dimensionless) d = Tubing internal diameter (ft) f = Fugacity (psia) ftp = Friction factor (dimensionless) FBHP = Flowing bottom-hole pressure (psia) g = Acceleration due to gravity (ft/s2) H = Hold-up (dimensionless) K = Thermodynamic equilibrium constant (dimensionless) k = Absolute permeability (mD) kr = Permeability in the radial direction (mD) kθ = Permeability in the angular direction (mD) kz = Permeability in the vertical direction (mD) krg = Three-phase relative permeability of gas kro = Three-phase relative permeability of oil krw = Three-phase relative permeability of water krog = Two-phase relative permeability of oil-gas krow = Two-phase relative permeability of oil-water M = Molecular weight (lb/lb-moles) Mginj = Molecular weight of the injected lift gas (lb/lb-moles) nphase = Mole fraction of a phase (dimensionless) nc = Total number of hydrocarbon components (dimensionless) nwb = Number of discretized well blocks (dimensionless) N = Molar flow rate of a component (lb-moles/s) P = Pressure (psia) Pchi = Parachor of a component Pcow = Capillary pressure in oil-water contact (psia) Pcgo = Capillary pressure in gas-oil contact (psia) Pi = Initial reservoir pressure (psia) Pr = Wellhead specified pressure (psia) Pr = Average reservoir pressure (psia) Pw = Pressure of a well-block (psia) PI = Productivity index of a well (STB/day/psi) qo = volumetric flow rate of oil (STB/day) qg = volumetric flow rate of gas (SCF/day) qginj = Gas lift injection rate at standard conditions (MSCF/day) qw = volumetric flow rate of water (STB/day) qo = Absolute open flow potential of oil (STB/day) r = Radius (ft) R = Residual of a discretized equation (lb-moles/ft3s for continuity equations, psia/ft for momentum equations) S = Saturation of a phase (dimensionless) SBHP = Static bottom-hole pressure (psia) Swc = Critical water saturation xvii STB = Stock Tank barrels SCF = Standard Cubic Feet t = Time (seconds for wellbore description, days for reservoir description) T = Temperature (°F) v = Velocity (ft/s) vsl = Superficial velocity of the liquid phase (ft/s) vsg = Superficial velocity of the gas phase (ft/s) V = Volume (ft3) x = Mole fraction of a component in the oleic phase (dimensionless) y = Mole fraction of a component in the gas phase (dimensionless) z = Global mole fraction of a component (dimensionless) zc = Global mole fraction of a component c in the reservoir or wellfeed zginjc = Global mole fraction of a component c in lift gas = Depth/Elevation/ of a well/ Thickness of reservoir block (ft)
Greek symbols
Δ = Change in a variable (dimensionless) φ = Porosity (dimensionless) Φ = Potential (psia) θ = Angular displacement along the radial direction (radians) ρ = Density (lb/ft3) ρ ginj = Density of the injected lift gas (lb/SCF) µ = Viscosity (cp for reservoir and lb-moles/ft-s for wellbore) σgo = Interfacial tension at the gas-oil contact (dynes/cm)
Subscripts b = Bulk c = A given component label d = Drift g = Gas i = ith block of a discretized wellbore in the vertical direction ginj = Gas lift injection gas l = Liquid (oil and water) m = Mixture nwb = Wellhead block o = Oil r = Radial block index sc = Standard conditions sf = Sandface conditions sp = Specified condition- At the wellhead tot = Total w = Water Z = Vertical block index xviii θ = Angular block index
Superscripts k = Iteration number for a given time step n = Time step number
Unit scale
M = Thousand units MM = Million units B = Billion units T = Trillion units
xix
ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor Dr. Turgay Ertekin for his constant guidance and support throughout my life at Penn State. His dedicated efforts towards excellence in education and research at Penn State has positively impacted the life of several generations of students, and has inspired me to strive for consistent improvement as a person, student and researcher. I would also like to thank my PhD committee members Dr. Eugene Morgan, Dr.
Kamesh Madduri, Dr. Hamid Emami-Meybodi, Dr. Sridhar Anandakrishnan and Dr. Luis Ayala for their time and valuable insights. The Department of Energy and Mineral Engineering has been the backbone of my work, supporting me with excellent infrastructure and a commendable set of people. The warmth and help provided by Judi, Jaime, Carole, Bob and Missy made me feel at home.
Coming to the United States and pursuing research has been a dream fulfilled through the love, affection and unconditional support provided my father Satya Prasad, my mother
Annapurna and my sister Sreevarsha. They made me what I am today.
I have been lucky to find friends in Vaibhav, Nirjhor, Sarath, Gireesh, Madhu and Sachin who have helped me understand myself, my courses and research better. I am thankful to Kelvin, Jian,
Nitesh, Aniruddh, Manik, Aditi Khadilkar, Aditi Bhat, Anahita and Dhvani for being exemplary student researchers and friends who have helped to expand my horizons of thinking and made me push myself a step further. I am glad that I met Sandhya, Janahvi, Arun, Rimsha, Manasi and several other friends who have made life at Penn State filled with happiness. It is always the collective efforts of countless people, their faiths, criticisms and appreciations that go into an achievement like this, and I will always be obliged to each one of that. 1
Chapter 1
Introduction
Petroleum is one of the most important energy sources in the world currently, if not the most.
Apart from being an energy source petroleum also acts as a raw material for various industries in order to produce chemical products such as solvents, pharmaceuticals, fertilizers, plastics, pesticides, polymers etc. With the world population still growing, the demand for energy and chemicals is only increasing. In this situation, every country strives its best to ascertain its energy security through a consistent supply of oil.
According to the World Bank report (2008), National Oil Companies (NOCs) control 90% of the world’s oil reserves and 75% of the oil production. A majority of the fields belonging to this production comes from mature fields. With many of the oil fields having depleted with time, the operating wells do not have sufficient pressure at the bottom to drive the fluids to the surface.
Under such cases, the wells need support through the means of artificial lift systems. Such a requirement only increases when the reservoir is depleted of gas, or under situations where the oil is very heavy and also in systems where the water cut is high.
There are several artificial lift mechanisms, primary modes being gas lift, rod pump, electrical submersible pump (ESP), and progressive cavity pump (PCP) and jet pump. Of all these mechanisms, gas lift is the most versatile mode which can be used in deep wells and high temperature environments unlike rod pump and PCP systems; An ESP is optimal only in high oil flow rate conditions and cannot operate in high gas-oil ratio (GOR) systems or wells having sand production issues. Gas lift is also much more energy efficient when compared to a Jet pump and 2 requires less area at the well site. Due to these advantages, gas lift is one of the most popular modes of artificial lift used in the world, especially in offshore systems. (Brown, 1984)
Since gas lift is basically an extension of natural flow of a well, wells are commonly provided gas lift support even before they cease to flow in order to improve the performance of the well.
With the technology of wireline retrievable gas lift valves (GLVs) being available, the cost of maintenance and repair of a gas lift well is much lower, as compared to other modes which usually require a work-over operation. Although the replacement of a damaged GLV may not require a work-over, a faulty calculation of the operating GLV depth may require a work-over operation as it involves the running-in of a gas lift mandrel along with the tubing string. Hence, it is very important to design the placement of operating gas lift mandrel at the right depth, which can accommodate gas lift operation for a well under various conditions based on the reservoir performance.
The placement of the operating GLV (bottom-most gas lift valve) is dependent on the estimated flow rates and a designated lift gas injection rate. This part is the most crucial step of the design as the operating valve is the only valve which is expected to stay open to allow gas to flow from annulus to production tubing. The remaining GLVs open only during kick-off to unload a static fluid column. While the position of the operating GLV is a part of the gas lift design and optimization process, the placement of the remaining GLVs is considered a static problem, in this process the static fluid pressure gradients and the opening and closing pressures of gas lift valves are considered as a part of the design.
There are two kinds of gas lift operation mechanisms: Continuous and intermittent. For continuous gas lift operation, it is common to place an orifice gas lift valve, which stays open at all times, except under the action of back-pressure exceeding annulus pressure. This is the common mechanism used in wells which are producing above 10 STB/day of oil. Wells which produce below 10 STB/day (IOGCC, 2012) of oil are considered to be stripper wells, in such case 3 the intermittent gas lift mechanism is used. The current work focuses on optimizing continuous gas lift systems.
Current methods for gas lift optimization involve the representation of reservoir performance through an Inflow performance relation curve, which is an equation defining the producing oil flow rate for an operating flowing bottom-hole pressure (FBHP) at a given point of time. These
Inflow performance relations currently available in the literature are extremely simplified systems which may not accurately account for the variations in the reservoir performance due to compositional changes and future behavior of the reservoir. Further description of the Inflow performance relations is provided in the literature review section.
Using a black oil flow correlation for the outflow performance is the currently popular mode, which when used along with an Inflow performance relationship provides the operating flow rate at different operating flowing bottom-hole pressures. This operating flow rate is used as an initial basis for designing a gas lift system. For a given oil production rate, when an operating gas lift valve is placed at a particular depth and lift gas is injected at a particular rate, the bottom-hole pressure varies according to these gas lift parameters, which in turn recursively varies the operating flow rate. Thus, the operating gas lift injection rate, the operating production rates are obtained iteratively. The rates production performance predicted by this method pertains to an instantaneous measurement as it assumes that the IPR will hold true for a later stage of production as well. However, it has been documented that the IPR varies along with depletion (Vogel, 1968).
For a private oil operator, the primary target lies in getting the maximum economic value from the oil resource during any given period of time. While, for an NOC, the primary target is to maximize the total oil recovery from a given resource in the long term. The economic planning of such organizations is based on this target rather than to vary its production strategy primarily according to market forces. This necessity to maximize the net oil recovery over the production life of a well is only accentuated for NOCs belonging to a country which has high oil 4 consumption and is largely a net importer of oil. This is because a country which has a high domestic consumption of oil will prefer to secure its energy interests through its local resources rather than through imports.
To accommodate for this long-term strategy of NOCs, an alternative approach is proposed which is more inclusive and evaluates performance of the gas lift well in terms of total oil recovered from a well during primary recovery. This requires a development of new method for gas lift optimization based on the production modeling of a well and its associated reservoir during the producing life of the well.
The current work proposes the use of a coupled reservoir-wellbore compositional numerical system which will account for all the variations in the reservoir described above, in order to replace the currently used correlations. The popularity of the currently used system is based on its speed. Using a coupled reservoir-wellbore simulator to generate results for a gas lift design takes a much longer time as compared to using a single equation based IPR coupled with a black oil outflow performance correlation.
Hence, in order to address the issue of speed, integration of numerical and ANN based models is proposed to developed a large database of gas lift design configurations generated. These models are expected to be fast and as they are based on the accurate results generated by the coupled compositional numerical simulator it is expected the resultant tools are fast, versatile and accurate.
5 Chapter 2
Literature Survey
2.1. Gas lift technology
A schematic of a typical modern gas lift system is presented in Figure 2-1. Gas is injected from the surface through the casing annulus and it enters from the casing annulus to the tubing through the gas lift valve. In 1920s, air was pumped through a simple U-tube channel for lifting wellbore fluids, since then numerous advancements have followed, from usage of other gases for lifting, to installation of pressure operated gas lift valves (GLVs), to changes in the gas lift design procedure. This evolution has been described by Osuji (1994). A schematic of a typical modern gas-lift system is presented in Figure 2-1. Gas is injected from the surface through the casing annulus and it enters from the casing annulus to the tubing through the gas-lift valve.
Figure 2-1. Schematic of a typical gas lift system (source: http://petrowiki.org/File:Vol4_Page_522_Image_0001.png)
6 During 1930s to 1940s, gas lift using GLV’s had begun, but the work done on analyzing the details of flow performance in a holistic manner by linking both inflow and outflow performance was very limited. Gilbert (1954) first proposed a method of well analysis technique involving the study of oil production rates versus flowing bottom-hole pressure, along with the study of two phase vertical flow performance. The work also described the basic gas lift design procedure on the basis of pressure gradients in the tubing and annulus, without including the pressure drops taking place across the GLV.
Kirkpatrick (1959), provided a detailed description of working mechanism of each component of gas lift system including GLV mechanics. Pittman (1982) came up with a detailed work on gas lift design and performance, explaining the procedure for optimizing design of gas lift. Mach, et al. (1983) described the effect of valve spacing, differential pressure, and optimal gas liquid ratio in gas lift design. Mukherjee and Brown (1986) presented a sensitivity analysis for improvement of gas lift performance through fine tuning valve port sizes, lift gas injection rate, for modifying and troubleshooting gas lift systems according to changing flow patterns.
2.2. Gas lift optimization
Redden et al. (1974) described a method to optimize the gas lift injection rate for a standalone well using a constant productivity index model for inflow performance and the Orkiszewski
(1967) correlation for outflow performance. The performance of the gas lift system was measured in terms of the resulting fluid production rate and also the cost benefit. Figure 2-2 describes a typical gas lift performance curve and Figure 2-3 display the gas lift performance in terms of monetary units for conditions where the gas availability is either unlimited or limited. These points of monetary optima are chosen based on the operating costs of a given operator. 7
Figure 2-2. Schematic of a typical gas lift performance curve (source: Redden et al.,1974)
Figure 2-3. Gas lift performance curve in monetary units (source: Redden et al.,1974)
Abdel-Waly et al. (1996) applied a single well gas lift optimization model to the wells of
Ramadan field in the Gulf of Suez. In this work, the gas lift well performance was compared with various outflow performance correlations and it was determined that the Beggs and Brill (1973) and the Hagedorn and Brown (1965) models were described to be best representative of the 8 wellbore hydraulics in the field. These outflow performance models were combined with an
Inflow performance relation to develop gas lift performance curves depicting the improvement in liquid production with gas lift injection rate for each individual well.
Dutta-Roy et al. (1997) developed a gas lift optimization model for a network of wells, considering the effect of separator back pressure and the pressure losses in the network of flow lines. Salazar-Mendoza (2006) developed a new method for representing gas lift optimal allocation curves for a system of six wells when all the wells are flowing simultaneously.
Rashid et al. (2012) compared the various methods available for gas-lift design and optimization. Among the methods dealing with single-well gas lift optimization the performance curve analysis, nodal analysis and curve-based models are fast and simple, however, these models are either standalone well models or pseudo-steady state models. In comparison, a coupled simulation is detailed and includes transient effects. Its main drawback is the high computational cost, which has been addressed in this work through the development of an ANN wellbore simulator. Wang et al. (2008) and Lu et al. (2012) have developed numerical algorithms for field scale gas lift optimization. Pinto et al. (2015) and Rasouli et al. (2015) have used genetic algorithms for field scale gas lift optimization. These studies are focused on gas lift allocation to each well, considering the back pressure from the surface pipeline network and separators.
In order to optimize the design of a gas lift system, a study of the inflow performance of the fluid from the reservoir to the bottom of the wellbore, combined with the outflow performance of the fluids from the bottom of the wellbore to the surface is necessary. These models are described subsequently.
2.3. Inflow performance relationship (IPR):
Before 1966, a constant productivity index based relationship was used to relate oil production rate to flowing bottom-hole pressure in the form of a resultant straight line. This straight line IPR 9 is only valid when gas is not evolved during production as is still popularly used in cases where the reservoir is operating above bubble point pressure. Vogel (1966) generated a series of plots between the fractions of well flow rate with respect to the maximum flow rate versus the fractions of flowing bottom-hole pressure with respect to average reservoir pressures at various conditions
of recovery, using a numerical model, as shown in Figure 2-2.
), ),
r
/P
wf
Bottomwell hole (P pressure Fractionreservoir pressure of
Producing rate (qo/(qo)max), Fraction of maximum
Figure 2-4. Inflow performance relationship described by Vogel (1966) (Source: Vogel, 1966) As it can be observed in Figure 2-4 the inflow performance of a reservoir varies temporally at different stages of production for a given reservoir even for a black-oil formulation. Vogel’s work was based on Weller’s work on solution gas drive reservoirs (1966) and assumed that the oil and gas compositions do not change. Vogel came up with a single equation to represent the inflow performance relationship (IPR):
푞 푃 푃 2 표 = 1 − 0.2 ( 푤푓 ) − 0.8 ( 푤푓 ) (2.1) 푞표푚푎푥 푃푟 푃푟
10 However, Vogel’s relationship was found to be less accurate at later stages of depletion, as more gas evolved out of oil. Fetkovich (1973) proposed a new relation, based on multi-rate tests:
2 푛 푞표 푃푤푓 = 1 − ( 2 ) (2.2) 푞표푚푎푥 푃푟
This model requires a multi-rate test to determine the backpressure exponent (n) and assumes oil mobility to be a linear function of average reservoir pressure. Klins and Majcher (1992) arrived at a new IPR correlation, by considering the variation in bubble point as pressure declines.
Wiggins (1993) proposed an IPR expression for three phase flow. Gallice and Wiggins (2004) performed a comparison of various two phase IPRs and concluded: “Because of depletion effects, one IPR method may be reliable at one reservoir pressure but unreliable at another. This may be caused by changes in reservoir parameters with time that can lead to changes in reservoir flow properties. Once again, this suggests the use of multiple IPR methods to estimate well performance.” Khasanov and Krasnov (2012) proposed an IPR which is derived by using an oil pseudo pressure function based on oil mobility at a given average reservoir pressure.
Most of the above IPR methods are applicable for two phase conditions and have pressure or water/gas saturation based constraints. Although it is possible to develop a composite IPR combining different IPR curves, these methods represent the reservoir performance only at a given stage of reservoir depletion. A reservoir simulator is necessary to capture the effects of variable inflow performance under the effects of depletion.
2.4. Outflow performance relationship:
Multi-phase flow in wellbores is a complex phenomenon which involves pressure losses due to gravitational, acceleration and friction effects. The nature and extent of the pressure loss varies with various flow regimes. Several models are available in the literature for describing wellbore 11 hydraulics. Firstly, empirical models were developed to describe the pressure drop under different flow regimes for two phase flow in a vertical wellbore, such as Duns et al. (1963), Hagedorn et al.
(1965), Orkiszewski (1967) and Beggs et al. (1973). These models have an applicability pertaining to a certain ranges of tubing internal diameter, gas-liquid ratio, oil API gravity and water cut. Later, Hasan et al. (1988), Ansari et al. (1994) and Petalas et al. (1998) developed mechanistic models. Brown (1977-1984) and Takács (2005) had described and summarized the applicability and constraints of these empirical and mechanistic models.
All of the above-mentioned models are black-oil based. The use of compositional wellbore models has recently gained the attention of the petroleum and reservoir engineering community, although a majority of the wellbore models described in the literature are black-oil (Shahamiri et al. (2015)). Pourafshary et al. (2009) developed a compositional wellbore model and provided a comparison between black-oil and compositional models at various hydrocarbon compositions where it was shown that black-oil models were restricted in their accuracy under conditions where the oil and gas compositions vary. Cao (2002), developed an isothermal compositional wellbore simulator which is fully coupled with a compositional reservoir simulator. Bahonar et al.
(2011a) developed a transient coupled reservoir-wellbore model for gas well testing and Bahonar et al. (2011b) developed and described an unsteady state heat transfer model in wells with steam injection, considering the effects of heat flow through annulus and cementing. Shirdel (2012) developed a thermal compositional numerical wellbore model for damage prediction and remediation. Livescu et al. (2008) developed a thermal, compositional, multi-segmented wellbore model based on a drift-flux formulation, which is coupled with a reservoir model. The main advantage of the drift-flux formulation is its simplicity, continuity and differentiability. Xiong
(2014) developed a standalone thermal compositional wellbore simulator based on the drift-flux models developed by Shi et al. (2005) and Hasan and Kabir (2007a). In this current work, the 12 drift-flux model developed by Hasan and Kabir (2007a) has been used for representing various flow regimes in multi-phase numerical wellbore simulation.
2.5. Machine learning methods in simulation and artificial lift:
Numerical compositional wellbore simulation is a computationally expensive process
(Pourafshary et al. (2009)) and various constraints need to be imposed on the system which may lead to time-step cuts, especially when there is a phase change near the wellbore boundaries
(Shahamiri et al. (2015)). Gaganis et al. (2012) have pointed out that close to 50% of the computational time in a compositional simulation is spent on flash calculations and they have used machine learning methods to perform flash calculation and phase stability determination to reduce the computational load. An ANN approach is used in the current work to develop a compositional wellbore simulation tool which is fast and robust. Previously, Osman et al. (2005) and Mohammadpoor et al. (2010) have developed ANN tools for predicting bottom-hole pressures in flowing oil wells. These tools were modelled upon 206 datasets from fields of middle-east and 167 datasets from fields of Iran, respectively. The range of API gravity in the datasets of Osman et al. (2005) was 30-37.
Rashidi et al. (2010) used genetic algorithms for determining an optimal tubing size, gas lift injection depth and gas lift injection rate simultaneously. In this work, they used a constant productivity index model to represent the reservoir performance and developed a model which considers maximum liquid flow rate at an instant as the objective function. Ranjan et al. (2015) discussed the application of a feed-forward ANN for gas lift optimization using data from Indian fields. In this model, they used a constant productivity index, a constant static bottom-hole pressure, flowing bottom-hole pressure and separator pressure and temperature, choke size for determining the gas injection rate and oil production rate for a well.
13 Chapter 3 Problem statement and objectives
A summary of the current practices for single well gas lift optimization can be found in the gas lift manual by Takács (2005). The basis for these models lies in the determination of operating point at various gas lift injection rates. The operating point is the point of intersection between the inflow and outflow performance curves as shown in Figure 3-1. It is to be noted that outflow performance curve A is a representation of a tubing system in which the well is not flowing, as the inflow and outflow performance curves are not intersection. The well is made to flow through introduction of gas lift which results in outflow performance curve B, which has a lower flowing bottom-hole pressure as compared to outflow performance curve A. A vertical line dropped from the operating point on the x-axis represents the operating production rate. Under further improved gas lift conditions, the outflow performance curve C can be observed, where the operating flowing bottom-hole pressure is lower and the operating production rate is higher.
Figure 3-1. Plot to determine operating flow conditions. (Source: http://petrowiki.org/File%3AVol4_Page_030_Image_0001.png)
14
A majority of the existing models in the literature use the same basis as shown in Figure 3-1 with an objective maximizing the oil or liquid production rate at a given point of time. None of the existing methodologies for gas lift optimization are focused on maximizing the cumulative oil production over the life of a well. Such a study requires to capture the variations in the performance of an integrated reservoir-wellbore system with varying pressure, production and fluid composition profiles seen along the life of a well as the associated reservoir undergoes depletion. This current work proposes to develop a gas lift optimization model for maximizing the cumulative oil production based on compositional formulations through integration of numerical simulation and machine learning methodologies.
The objectives of the current study can be summarized as:
To develop a three-phase, compositional numerical reservoir simulator, which will be used to
represent inflow performance under various reservoir conditions.
To develop a three-phase, compositional numerical wellbore hydraulics model, which can
simulate flow of oil, gas and water in vertical wells under non-isothermal conditions, with or
without gas lift.
To develop an ANN based wellbore hydraulics model trained using the data generated by the
numerical wellbore hydraulics model. This ANN based wellbore model is expected to
provide fast and accurate predictions of flowing bottom-hole pressure for a provided well
operation and inflow conditions from the reservoir.
To develop coupled reservoir-wellbore simulation models. The first model will consist of a
fully numerical model, while the second model will consist of a numerical reservoir simulator
coupled with the ANN based wellbore model.
To generate a database of well performance with various gas lift specifications using the
coupled reservoir-wellbore models. 15
To develop an ANN based tool for predicting the cumulative oil produced from a well from its
inception till abandonment under various gas lift specifications, in order to choose optimal
gas lift operating conditions.
To perform Monte Carlo simulations to account for the uncertainties in the reservoir
parameters and obtain probabilities for the estimate of gas lift injection rate and the total oil
produced from a well.
16 Chapter 4 Methodology
The proposed study involves the use of hard computing tools to generate databases to create an ANN based tools capable of predicting gas lift performance. We can divide the work into two broad categories: the development of hard and soft computational tools. A schematic for the proposed methodology is shown in Figure 4-1.
Hard Computation Tools
Numerical reservoir Numerical wellbore model Coupling hydraulics model
Coupling Generation of wellbore hydraulics database Gas lift simulation, Numerical optimization reservoir model Training, Testing coupled with ANN and Validation and database Coupling generation based wellbore hydraulics model
ANN based wellbore Training, Testing hydraulics model and Validation
ANN based gas lift simulation and optimization tool
Soft Computation Tools
Figure 4-1. Schematic of the methodology followed
17 4.1. Hard computation tools:
As a part of hard computational tools, the following numerical models have been developed:
4.1.1. Compositional numerical reservoir model
A three-phase, single well, compositional, isothermal numerical reservoir simulator with three-dimensional radial cylindrical coordinates is being used for modelling the reservoir for a single well system. The model has the option of introducing a multilayered well, so that contribution from different layers of reservoir can be tracked.
The compositional reservoir simulator is used to represent the inflow performance for each reservoir system, at different stages of depletion. The numerical equations required for building this simulator are primarily based on the incorporation of Darcy’s law in the continuity equation for a porous medium. Other important concepts such as phase equilibrium, relative permeability, and effect of capillary forces have an important role as well, and have been incorporated in the development of the reservoir simulator.
4.1.2. Compositional numerical wellbore hydraulics model
A one dimensional, non-isothermal wellbore simulator has been developed. Pressure loss in vertical flow through tubing can be broadly divided into acceleration, gravitational and frictional losses which are represented through a momentum equation, however, the effect of multiphase flow behavior in various flow regimes makes the system complex. To account for these complexities, a drift flux model based on the work of Hasan and Kabir (2007a) has been employed. The wellbore hydraulics simulator is utilized for representing an outflow performance curve, which defines the flow rate of oil out of the well to the surface corresponding to a flowing bottom-hole pressure. 18 4.1.3. Coupled numerical reservoir-wellbore simulator
The boundary conditions of the wellbore and the reservoir simulators have been coupled to provide a consistent result. The molar flow rate produced by the numerical reservoir simulator acts as an input for the numerical wellbore hydraulics model. The stabilized flowing bottom-hole pressure calculated by the wellbore hydraulics model acts as an input for each iteration of the sand-face pressure specified reservoir simulator.
4.1.4. Fully numerical gas lift performance tool
The coupled numerical wellbore-reservoir simulator developed is capable of simulating a system with various options for gas lift injection rates, gas compositions, operating GLV depth and time of gas lift injection. However, this tool is computationally taxing and hence soft computational tools have been developed which serve the same purpose with better speed while maintaining a good level of accuracy.
4.2. Soft computation tools
As a part of soft computational tools, the following ANN based models have been developed:
4.2.1. ANN based wellbore hydraulics model
An ANN based wellbore hydraulics model has been developed which takes wellbore construction parameters and fluid compositional parameters as an input and predicts the flowing bottom-hole pressure as an output. This ANN based tool provides a speed-up of four orders of magnitude while showing an average error of 1-6% with respect to the numerical wellbore hydraulics model.
4.2.2. Fully ANN based gas lift performance model
An ANN based gas lift performance model has been developed. This model can predict the cumulative oil produced by a gas lift well based on a compositional reservoir and wellbore 19 description, for varying gas lift injection rates, gas lift composition, operating GLV depths and the start time of gas lift injection.
4.3. Coupled hard and soft computation tools
4.3.1. Coupled numerical reservoir- ANN based wellbore model
The fully numerical reservoir-wellbore model takes hours to days of run time for simulating the life of a gas lift well depending on the computational platform available. Since the wellbore hydraulics model is called at each iteration of the reservoir simulator to receive the sand-face pressure specification, the ANN based wellbore hydraulics model has been coupled with the numerical reservoir simulator. This model works in a very similar way as compared to the fully numerical coupled reservoir-wellbore model. The compositional molar flow rate output of the reservoir simulator is used as the input for the ANN based wellbore hydraulics model, which in turn predicts the corresponding flowing bottom-hole pressure to be used by the numerical reservoir simulator for the subsequent iteration. This model takes between minutes to a few hours of run time on average to simulate the results of the production life of a gas lift well.
4.3.2. Coupled numerical - ANN based gas lift performance model
The developed coupled numerical reservoir-ANN based wellbore model is capable of generating a large database of gas lift well performance under variations of well, reservoir, fluid composition and gas lift specifications. This coupled numerical-ANN model can also be used for generating a gas lift performance curve and perform sensitivity analysis of a gas lift system.
In summary, in this study, three different models have been developed for simulating and optimizing a gas lift system: The fully-numerical model, the coupled numerical-ANN based model and the fully ANN based model.
20 Chapter 5 Development of a numerical reservoir model
In this section, the differential equations and the discretized finite difference equations describing the compositional reservoir model are displayed and the procedure used to solve the equations is explained.
Considering an isothermal compositional reservoir system with three operating phases: oleic, gaseous and aqueous, the model is built on a three-dimensional radial cylindrical geometry with a single well.
Assumptions:
1. Reservoir is isothermal and the temperature is uniform throughout the reservoir.
2. There is only primary (matrix) porosity.
3. Flow in porous medium is governed by Darcy’s law and all the assumptions governing
Darcy’s law apply.
4. The water component exists only in the aqueous phase.
5. Hydrocarbon components exist only in oleic and gaseous phases and not in aqueous
phase.
5.1. Primary equations and variables
The Primary equations describing the reservoir model are:
1. Hydrocarbon continuity equation incorporating Darcy’s law (nc equations):
1 휕 푟푥푐휌표푘푟푘푟표 휕Φo 푟푦푐휌푔푘푟푘푟푔 휕Φg 1 휕 푥푐휌표푘휃푘푟표 휕Φo [ + ] + 2 [ + 푟 휕푟 휇표푀표 휕푟 휇푔푀푔 휕푟 푟 휕휃 휇표푀표 휕휃 푦 휌 푘 푘 휕Φ 휕 푥 휌 푘 푘 휕Φ 푦 휌 푘 푘 휕Φ 푐 푔 휃 푟푔 g] + [ 푐 표 푧 푟표 o + 푐 푔 푧 푟푔 g] = 휇푔푀푔 휕휃 휕푧 휇표푀표 휕푧 휇푔푀푔 휕푧 1 휕 푆 푥 휌 푆푔푦푐휌푔 [ 휙 ( 표 푐 표 + )] (5.1) 5.615 휕푡 푀표 푀푔
21
2. Water continuity equation incorporating Darcy’s law (one equation):
1 휕 푟휌푤푘푟푘푟푤 휕Φ푤 1 휕 휌푤푘휃푘푟푤 휕Φ푤 휕 휌푤푘푧푘푟푤 휕Φ푤 [ ] + 2 [ ] + [ ] = 푟 휕푟 휇푤푀푤 휕푟 푟 휕휃 휇푤푀푤 휕휃 휕푧 휇푤푀푤 휕푧 1 휕 푆 휌 [ 휙 ( 푤 푤)] (5.2) 5.615 휕푡 푀푤
Total primary equations: nc+1
Considering nc hydrocarbon components and one water component, the primary unknowns involved:
1. The global mole fraction of nc-1 hydrocarbon components(zc): nc-1 unknowns
2. The water saturation (Sw): one unknown
3. The oil pressure in the reservoir (Po): one unknown
Total primary unknowns: nc+1 5.2. Auxiliary equations:
The auxiliary equations involve:
1. The overall mole fraction constraint:
∑ 푧푐 − 1 = 0 (5.3)
2. The thermodynamic equilibrium relations:
푦푐 = 퐾푐푥푐 (5.4)
푧푐 = 푦푐푛표 + 푥푐(1 − 푛표) (5.5)
푧푐 푥푐 = (5.6) 푛표 + (1 − 푛표)퐾푐
푓푔 = 푓표 (5.7)
22 휌 푛 ( 표 표) (1 − 푆 ) 푀 푤 푆 = 표 (5.8) 표 휌 푛 휌 푛 ( 표 표) + ( 푔 푔) 푀표 푀푔
3. Saturation constraints:
푆표 + 푆푔 + 푆푤 = 1 (5.9)
4. Capillary pressure relationships:
푃푐표푤 = 푃표 − 푃푤 (5.10)
푃푐푔표 = 푃푔 − 푃표 (5.11)
5.3. Description of the flow geometry:
The Radial-cylindrical Three-Dimensional reservoir system has been developed following the work of Abou-Kassem and Farouq Ali (2006).
Considering a cylindrical reservoir whose outer bounder of radius is re (ft) and wellbore radius is rw (ft). For the reservoir to be discretized, the grid is logarithmically spaced. The counters i,j,k represent blocks along the radial, angular and vertical directions respectively. For the total number of blocks along the radial, angular and vertical direction to be nr, nt , nz respectively, we define:
1. Logarithmic ratio constant, 1 푟푒 푛푟 훼lg = ( ) (5.12) 푟푤
2. Radius of the first block, 훼lg log푒 훼lg 푟1 = [ ] 푟푤 (5.13) 훼lg −1 23
3. Radius to the ith point, 푖−1 푟푖 = (훼lg ) 푟1 (5.14)
4. The bulk volume of each block,
for i 2 (훼2 − 1) 1 lg 2 (5.15푎) 푉푏푖,푗,푘 = 2 2 푟푖 ( ∆휃푗) ∆푧푖,푗,푘 훼lg log푒 훼lg 2 for i=nr, log 훼 2 1 푒 lg 2 2 2 2 (5.15푏) 푉푏푖,푗,푘 = {1 − [ ] (훼lg − 1)/(훼lg log푒 훼lg ) }푟푒 ( ∆휃푗) ∆푧푛푟,푗,푘 훼lg − 1 2 For defining the fixed part of the transmissibilities pertaining to the geometry of the system, the following geometric factors have been defined: 1. for i>1 ∆휃푗 퐺푟 1 = 푖−2,푗,푘 {log푒 [훼lg log푒 훼lg /(훼lg − 1)]/(∆푧푖,푗,푘푘푟푖,푗,푘) + log푒[(훼lg − 1)/ log푒(훼lg )]/(∆푧푖−1,푗,푘푘푟푖−1,푗,푘) } (5.16푎) 2. for i=1, 푘푟1,푗,푘ℎ1,푗,푘 퐺 1 = ∆휃 푟푖− ,푗,푘 푗 2 푟1,푗,푘 (5.16푏) (log푒 〖( ))〗 푟푤 24 3. for i ∆휃푗 퐺푟 1 = (5.17푎) 푖+ ,푗,푘 훼 훼lg − 1 2 lg log푒 〖 ] 훼lg − 1 log푒〖(훼lg)] log푒 [훼lg ) + log푒 〗 (∆푧 푘 〗 ∆푧푖,푗,푘푘푟푖,푗,푘 푖+1,푗,푘 푟푖+1,푗,푘 { [ } 4. for i=nr, 퐺푟 1 = 0 (5.17푏) 푖+ ,푗,푘 2 5. for all i,j,k 2 log푒 훼lg 퐺 1 = (5.18) 휃푖,푗± ,푘 ∆휃푗 ∆휃푗±1 2 + ∆푧 푘 ∆푧 푘 푖,푗,푘 휃푖,푗,푘 푖,푗±1,푘 휃푖,푗±1,푘 6. for all i,j 푉푏 2( 푖,푗,푘) ∆푧푖,푗,푘 퐺 1 = (5.19) 푧푖,푗,푘± ∆ 푧푖,푗,푘 ∆ 푧푖,푗,푘±1 2 + 푘푧푖,푗,푘 푘푧푖,푗,푘±1 7. for k=1, top-most layer 퐺 1 = 0 (5.20) 푧푖,푗,1− 2 8. for k=nz, bottom-most layer 1 ( ) 퐺푧푖,푗,푛 + = 0 5.21 푧 2 5.4. Finite difference equations for the reservoir model: Since an analytical solution for the equations (6.1) through (6.3) is not available, a numerical solution using the finite difference method is sought. A second order central finite difference 25 scheme is used for the spatial derivative and a backward finite difference scheme is used for the temporal derivative and the system is solved in a fully implicit manner. For such a system, the primary equations are discretized as such: Hydrocarbon continuity equation: 퐾푟표푥푐휌표 푅 = [퐺 1 ( ) (Φ − Φ ) ] − 푐푖 푟푖+ ,푗,푘 1 o푖+1,푗,푘 o푖,푗,푘 2 휇표푀표 푖+ ,푗,푘 2 퐾푟표푥푐휌표 − [퐺 1 ( ) (Φ − Φ ) ] + 푟푖− ,푗,푘 1 o푖,푗,푘 o푖−1,푗,푘 2 휇표푀표 푖− ,푗,푘 2 퐾푟푔푦푐휌푔 + [퐺푟 1 ( ) (Φg − Φ푔 ) ] − 푖+ ,푗,푘 휇 푀 1 푖+1,푗,푘 푖,푗,푘 2 푔 푔 푖+ ,푗,푘 2 퐾푟푔푦푐휌푔 − [퐺푟 1 ( ) (Φg − Φg ) ] + 푖− ,푗,푘 휇 푀 1 푖,푗,푘 푖−1 ,푗,푘 2 푔 푔 푖− ,푗,푘 2 퐾푟표푥푐휌표 + [퐺 1 ( ) (Φ − Φ ) ] − 휃푖,푗+ ,푘 1 o푖,푗+1,푘 o푖,푗,푘 2 휇표푀표 푖,푗+ ,푘 2 퐾푟표푥푐휌표 − [퐺 1 ( ) (Φ − Φ ) ] + 휃푖,푗− ,푘 1 o푖,푗,푘 o푖,푗−1 ,푘 2 휇표푀표 푖,푗− ,푘 2 퐾푟푔푦푐휌푔 + [퐺휃 1 ( ) (Φg − Φ푔 ) ] − 푖,푗+ ,푘 휇 푀 1 푖,푗+1,푘 푖,푗,푘 2 푔 푔 푖,푗+ ,푘 2 퐾푟푔푦푐휌푔 − [퐺휃 1 ( ) (Φg − Φg ) ] + 푖,푗− ,푘 휇 푀 1 푖,푗,푘 푖,푗−1,푘 2 푔 푔 푖,푗− ,푘 2 퐾푟표푥푐휌표 + [퐺 1 ( ) (Φ − Φ ) ] − 푧푖,푗,푘+ 1 o푖,푗,푘+1 o푖,푗,푘 2 휇표푀표 푖,푗,푘+ 2 퐾푟표푥푐휌표 − [퐺 1 ( ) (Φ − Φ ) ] + 푧푖,푗,푘− 1 o푖,푗,푘 o푖,푗,푘−1 2 휇표푀표 푖,푗,푘− 2 퐾푟푔푦푐휌푔 + [퐺푧 1 ( ) (Φg − Φ푔 ) ] − 푖,푗,푘+ 휇 푀 1 푖,푗,푘+1 푖,푗,푘 2 푔 푔 푖,푗,푘+ 2 퐾푟푔푦푐휌푔 − [퐺푧 1 ( ) (Φg − Φg ) ] (5.22) 푖,푗,푘− 휇 푀 1 푖,푗,푘 푖,푗,푘−1 2 푔 푔 푖,푗,푘− 2 26 Water continuity equation: 퐾푟푤휌푤 푅 = [퐺 1 ( ) (Φ − Φ ) ] − 푐푖 푟푖+ ,푗,푘 1 w푖+1,푗,푘 w푖,푗,푘 2 휇푤푀푤 푖+ ,푗,푘 2 퐾푟푤휌푤 − [퐺 1 ( ) (Φ − Φ ) ] + 푟푖− ,푗,푘 1 w푖,푗,푘 w푖−1,푗,푘 2 휇푤푀푤 푖− ,푗,푘 2 퐾푟푤휌푤 + [퐺 1 ( ) (Φ − Φ ) ] − 휃푖,푗+ ,푘 1 w푖,푗+1,푘 w푖,푗,푘 2 휇푤푀푤 푖,푗+ ,푘 2 퐾푟푤푤 − [퐺 1 ( ) (Φ − Φ ) ] + 휃푖,푗− ,푘 1 w푖,푗,푘 w푖,푗−1 ,푘 2 휇푤푀푤 푖,푗− ,푘 2 퐾푟푤푤 + [퐺 1 ( ) (Φ − Φ ) ] − 푧푖,푗,푘+ 1 w푖,푗,푘+1 w푖,푗,푘 2 휇푤푀푤 푖,푗,푘+ 2 퐾푟푤푤 − [퐺 1 ( ) (Φ − Φ ) ] (5.23) 푧푖,푗,푘− 1 w푖,푗,푘 w푖,푗,푘−1 2 휇푤푀푤 푖,푗,푘− 2 The fluid densities, viscosities and molecular weights are obtained at the interface through arithmetic averaging and the more non-linear terms such as the relative permeability and the phase mole fractions are obtained through single point upstream weighting. 5.5. Initial and boundary conditions: Initial conditions: The reservoir simulator is initialized with a uniform assignment of the primary variables. 0 ( ) 푃표푖 = 푃푖푛푖푡푖푎푙 5.24 0 ( ) 푧푐푖 = 푧푖푛푖푡푖푎푙 5.25 0 ( ) 푆푤푖 = 푆푤푖푛푖푡푖푎푙 5.26 27 Boundary conditions: The outer boundaries of the reservoir in both the radial and vertical direction are no flow boundaries for which the geometric transmissibility factors are zero. For the inner boundary in the radial direction, at r = rw, 휕푃 푁푐 휇표푀표 푟 표 = − 표 (5.27) 휕푟 2휋푘푟푘푟표푥푐휌표ℎ For a sand-face pressure specified system, Equation (5.27) is simplified as: 푘푟표푥푐휌표 푁푐 = −퐺 1 ( ) (Φo − 푃푠푓) (5.28) 표 푟 − ,푗,푘 1 푖,푗,푘 1 2 휇표푀표 1− ,푗,푘 2 푘 푦 휌 푟푔 푐 푔 ( ) 푁푐푔 = −퐺 1 ( ) (Φg − 푃푠푓) 5.29 푟1− ,푗,푘 휇 푀 1 푖,푗,푘 2 푔 푔 1− ,푗,푘 2 푘푟푤휌푤 푁푤 = −퐺 1 ( ) (Φw − 푃푠푓) (5.30) 푟 − ,푗,푘 1 푖,푗,푘 1 2 휇푤푀푤 1− ,푗,푘 2 푁푐 = 푁푐표 + 푁푐푔 (5.31) For an oil molar component flowrate specified system, 푁푐 푃 = Φ − 표 (5.32) 푠푓 o푖,푗,푘 푘 푥 휌 퐺 ( 푟표 푐 표) 1 1 푟1− ,푗,푘 휇표푀표 1− ,푗,푘 2 2 For a gas molar component flowrate specified system, 푁푐푔 푃푠푓 = Φg − (5.33) 푖,푗,푘 푘푟푔푦푐휌푔 퐺 1 ( ) 푟 − ,푗,푘 1 2 휇푔푀푔 1 1−2,푗,푘 28 For a water molar component flowrate specified system, 푁 푃 = Φ − 푤 (5.34) 푠푓 w푖,푗,푘 퐾 휌 퐺 ( 푟푤 푤) 1 1 푟1− ,푗,푘 휇푤푀푤 1− ,푗,푘 2 2 It is to be noted that for Equations (5.28) through (5.30), the units of molar flow rate are in (lb- moles/day) (RB/RCF). Where, RB and RCF are reservoir barrels and reservoir cubic feet respectively. 5.6. Newton-Raphson protocol for linearizing equations As it can be observed, our reservoir equations are not linear in nature and hence the Newton- Raphson protocol for a multivariable system is utilized to linearize the system. The system of equations is represented in the form JX=R, where J represents the Jacobian matrix, X represents the vector of the primary unknown variables and R represent the residual vector. The reservoir system of equations using four hydrocarbon components for a single block is represented in Figure 5-1. (푛+1) 휕푅 휕푅 휕푅 휕푅 휕푅 (푘) ( 푛푐) ( 푛푐) ( 푛푐) ( 푛푐) ( 푛푐) 휕푧1 휕푧2 휕푧3 휕푃표 휕푆푤 (푛+1) (푛+1) 휕푅2 휕푅2 휕푅2 휕푅2 휕푅2 ( ) ( ) ( ) ( ) ( ) ∆푧 (푘+1) −푅 (푘+1) 1 1 휕푧1 휕푧2 휕푧3 휕푃표 휕푆푤 ∆푧 −푅 2 2 휕푅3 휕푅3 휕푅3 휕푅3 휕푅3 ∆푧 −푅 ( ) ( ) ( ) ( ) ( ) 3 = 3 휕푧1 휕푧2 휕푧3 휕푃표 휕푆푤 ∆푃표 −푅푛푐 휕푅1 휕푅1 ∆푆푤 −푅푤 휕푅1 휕푅1 휕푅1 ( ) ( ) [ ] ( ) ( ) ( ) 휕푃 휕푆 [ ]푖 푖 휕푧 휕푧 휕푧 표 푤 1 2 3 휕푅푤 휕푅푤 0 0 0 ( ) ( ) [ 휕푃표 휕푆푤 ]푖 Figure 5-1. System of equations for a cell in the discretized reservoir model 29 The derivative terms in the Jacobian can be calculated using a numerical differentiation scheme. This system of equations above is solved using a linear solver. In the C++ based implementation GMRES developed by Youcef (1986) linear solver is used, while in the MATLAB© implementation, the linsolve command is used. The change in the primary unknowns obtained through solving for X=R/J is added to their respective variables from the previous iteration to get an improved estimate. This process is iteratively repeated for a time step until the maximum value of the residual is less than 0.01 (lb- moles/day) (RB/RCF). The entire procedure followed has been displayed in the form of a flow chart in Figure 5-2 . It is to be noted that in Jacobian matrix, the position of the residuals of continuity equations of the first and last hydrocarbon components have been interchanged in order to ensure diagonal dominance as described by Ayala (2004). 5.7. Computational protocol followed: The computational protocol followed in the operation of the numerical reservoir simulator is displayed in Figure 5-2. While the simulation is performed to calculate the performance of a reservoir of a given period, it is typical to divide the total simulation run days into several time steps. Under constraining conditions, where the numerical solution is not achieved within ten iterations, the time step size is reduced as described in the Figure 5-2. The program is terminated when the target time period initially defined has been achieved. ∗ MATLAB© is a registered trademark of The MathWorks, Inc 30 Initialize the system describing the pressure, water saturation and global composition using the conditions described in section 5.5 Calculate fluid properties by performing Peng-Robinson(1978) Equation of State (PREOS) based flash calculations and using correlations for phase viscosities at each block for a given pressure and composition, use table look-up and interpolation for calculating phase relative permeabilities and capillary pressures ∆푡 = 푡1 Calculate the residuals using the equations in section 5.6 and construct a Jacobian matrix to solve for the system of equations 5.35 ∆푡 = ∆푡/4 Update primary variables No Yes Is any of the following conditions true? : Is the number of iterations less than 10? Is Is any of the primary variable value maximum negative or imaginary or not a number? residual<0.01? Is the matrix singular? Is the mole fraction of any component less than 0 or greater than 1? Yes Proceed to next time step No Has the total run time reached target time? Yes Terminate Figure 5-2. Schematic for the procedure used in the reservoir simulator 31 5.8. Volumetric flow rates of phases The molar flow rates produced from the reservoir is converted to volumetric flow rates in reservoir conditions using the following equations: Oil flow rate in reservoir conditions, 푙푏 푅퐵 푙푏푚표푙푒푠 푅퐵 푀표 ( ) 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 (5.35) 표푖푙 푐표 푙푏 푑푎푦 푑푎푦 푅퐶퐹 휌 ( ) 표 푅퐶퐹 Gas flow rate in reservoir conditions, 푙푏 푅퐶퐹 푙푏푚표푙푒푠 푅퐵 푀푔 ( ) 푅퐶퐹 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 × 5.615 ( ) (5.36) 푔푎푠 푐푔 푙푏 푑푎푦 푑푎푦 푅퐶퐹 휌 ( ) 푅퐵 푔 푅퐶퐹 Water flow rate in reservoir conditions, 푙푏 푅퐵 푙푏푚표푙푒푠 푅퐵 푀푤 ( ) 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 (5.37) 푤푎푡푒푟 푤 푙푏 푑푎푦 푑푎푦 푅퐶퐹 휌 ( ) 푤 푅퐶퐹 In order to obtain the flow rates of various phases in surface conditions, the molar flow rate ratios of hydrocarbon components are taken to obtain the composition of the hydrocarbon mixture at surface conditions and a flash calculation is performed at standard conditions ( 14.7 psia pressure, 60 F Temperature). Based on the fluid properties obtained at standard conditions (subscript ‘sc’), the surface flow rates are calculated using: Oil flow rate in standard conditions, 푙푏 푆푇퐵 푙푏푚표푙푒푠 푆푇퐵 푀표푠푐 ( ) 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 (5.38) 표푠푐 푐표푠푐 푙푏 푑푎푦 푑푎푦 푆퐶퐹 휌 ( ) 표푠푐 푆퐶퐹 Gas flow rate in standard conditions, 푙푏 푆퐶퐹 푙푏푚표푙푒푠 푆푇퐵 푀푔푠푐 ( ) 푆퐶퐹 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 × 5.615 ( ) (5.39) 푔푠푐 푐푔푠푐 푙푏 푑푎푦 푑푎푦 푆퐶퐹 휌 ( ) 푆푇퐵 푔푠푐 푆퐶퐹 32 Water properties at surface conditions are obtained using table look-up and the water flow rate at standard conditions is given by: 푙푏 푆푇퐵 푙푏푚표푙푒푠 푆푇퐵 푀푤 ( ) 푞 ( ) = 푁 ( ) ( ) × 푙푏푚표푙푒푠 (5.40) 푤푠푐 푤 푙푏 푑푎푦 푑푎푦 푆퐶퐹 휌 ( ) 푤푠푐 푆퐶퐹 5.9. Three-phase relative permeability model The Stone’s [1973] three phase relative permeability model has been used to obtain the oil relative permeability, Kro: 푘푟표푤(푆푤) 푘푟표푔(푆푔) 푘푟표 = 푘푟표푤(푆푤푐) [( + 푘푟푤(푆푤)) + ( + 퐾푟푔(푆푔)) − (푘푟푤(푆푤) + 푘푟푔(푆푔))] (5.41) 푘푟표푤(푆푤푐) 푘푟표푤(푆푤푐) The single phase and two-phase water and gas relative permeabilities used in Equation (5.41) are obtained through interpolation on table look-up. The numerical reservoir model has been developed as described in Chapter 5. As a subsequent step, the numerical wellbore hydraulics model is developed as shown in Chapter 6. The numerical reservoir simulator is coupled with a wellbore hydraulics model to model the performance of a well over a period of time. The description of this coupling mechanism is described in Chapter 9. 33 Chapter 6 Numerical wellbore model In this section, the differential equations and the discretized finite difference equations describing the wellbore model are displayed and the procedure used to solve the equations is explained. Considering an Isothermal vertical wellbore system with two operating phases: liquid and gas, with the oleic and the aqueous phases lumped into a liquid phase. The oil and water phase velocity is hence assumed to be equal to the liquid phase velocity vl. Assumptions: 1. Flow is unidimensional along the vertical axis and velocity components along the radial and angular directions of a cylindrical system are considered negligible. 2. Flow is primarily modeled for cocurrent upward flow, although the drift-flux model has the capability to account for countercurrent and downward flow as well. 3. Pressure of all phases are equal and capillary effects are negligible. 4. Frictional effects are calculated using mixture properties and friction between the gas and liquid phases is currently considered negligible, although it can be incorporated later. 5. The temperature of the wellbore is provided as an input and the heat exchange between fluids is considered as negligible. 6. The water component exists only in the liquid phase. 7. Hydrocarbon components exist only in oleic and gaseous phases and not in the aqueous phase. 8. The production feed from the reservoir flows only in the tubing of the well and not the annulus. 34 6.1. Primary equations and variables The primary equations describing the wellbore system in oilfield units are: 1. The hydrocarbon components molar balance (Continuity equations, nc equations): 휕 휌푔퐻푔푦푐 휌 퐻 푥 휕 푣푔휌푔퐻푔푦푐 푣 휌 퐻 푥 푁 − [( ) + ( 표 표 푐)] + [( ) + ( 푙 표 표 푐)] + ( 푐) = 0 (6.1) 휕푡 푀푔 푀표 휕푍 푀푔 푀표 푉푏 2. The water component molar balance (continuity equation, one equation): 휕 휌 퐻 휕 푣 휌 퐻 푁 − [( 푤 푤)] + [( 푙 푤 푤)] + ( 푤) = 0 (6.2) 휕푡 푀푤 휕푍 푀푤 푉푏 3. The mixture momentum equation (one equation): 푤 2 휕푃 휕 휕 푓푡푝휌푚푣푚 4634.6 + 휌 푔 + (휌 푣 ) + (휌 푣2 ) + ( ) = 0 (6.3) 휕푧 푚 휕푡 푚 푚 휕푍 푚 푚 2푑 Total primary equations: nc+2 Considering nc hydrocarbon components and one water component, the primary unknowns involved: 1. The global mole fraction of nc-1 hydrocarbon components(zc): nc-1 unknowns 2. The water holdup (Hw): one unknown 3. The pressure in the well (Pw) : one unknown 4. The Mixture velocity (Vm) : one unknown Total primary unknowns: nc+2 In a non-isothermal system, the temperature of the system adds as a primary variable and the energy balance equation will add an additional primary equation to balance the system. 35 6.2. Auxiliary equations: The auxiliary equations include: 1. The overall mole fraction constraint: ∑ 푧푐 − 1 = 0 (6.4) 2. The thermodynamic equilibrium relations: 푦푐 = 퐾푐푥푐 (6.5) 푧푐 = 푦푐푛표 + 푥푐(1 − 푛표) (6.6) 푧푐 푥푐 = (6.7) 푛표 + (1 − 푛표)퐾푐 푓푔 = 푓표 (6.8) 휌 푛 ( 표 표) (1 − 퐻 ) 푀 푤 퐻 = 표 (6.9) 표 휌 푛 휌 푛 ( 표 표) + ( 푔 푔) 푀표 푀푔 3. Hold-up constraints: 퐻푙 + 퐻푔 = 1 (6.10) 퐻푙 = 퐻표 + 퐻푤 (6.11) 4. Drift flux equations: 푣푔 = 퐶표푣푚 + 푣푑 (6.12) 푓푡 푣 ( ) = 퐻 푣 (6.13) 푠푔 푠 푔 푔 36 푓푡 푣 ( ) = 퐻 푣 (6.14) 푠푙 푠 푙 푙 푣푚 = 푣푠푙 + 푣푠푔 (6.15) All intensive mixture properties, including density, viscosity have been calculated using a volumetric averaging of phase properties. The description of the terms Co and vd (ft/s) in the drift flux equations will be explained in a next subsection describing the drift flux model. 6.3. The Drift-flux model: In two phase vertical pipe flow, the velocity of the gas is higher than that of liquid due to the effects of slippage and drift. Also, the velocity of the fluid is higher in the center of the pipe as compared to the pipe walls. These effects are represented in terms of the flow parameter, Co and the drift velocity, vd. During vertical two-phase flow, various configurations involving combinations of liquid and gas hold-ups and velocities are represented by various flow regimes such as bubbly flow (dispersed bubble flow), slug flow, churn flow and annular flow. The visual representation of these flow regimes has been shown in Figure 6-1. The Hasan and Kabir (2007a) drift-flux model has been used in the current work. The main type of flow which is being modelled in the current work is upward-cocurrent flow of liquid and gas. For this flow condition, the flow parameter, Co and the drift velocity, vd have been defined as explained below in Table 6-1. 37 Figure 6-1. Visual representation of various flow regimes (reproduced from http://www.drbratland.com/PipeFlow2/chapter1.html) Table 6-1. Hasan and Kabir (2007a) drift-flux model parameters Profile parameter Co Flow Drift regime Upward Cocurrent Countercurrent Downward velocity Vd Bubbly 1.2 2 1.2 푣푑푏 Slug 1.2 1.2 1.12 푣̅̅푑̅ Churn 1.15 1.15 1.12 푣̅̅푑̅ Annular 1 1 1 0 6.3.1 Bubbly flow: In bubbly flow, the drift velocity is defined by the bubble rise velocity, vdb. where, in field units: 1 4 푓푡 휌푙 − 휌푔 푣푑푏 ( ) = 0.3315 휎푔표 (푔 2 ) (6.16) 푠 휌푙 38 6.3.2. Slug flow: During slug flow, there is a Taylor bubble velocity, vdt introduced along with the bubble rise velocity. In field units: 푓푡 푣 ( ) = 0.1 푔 (휌 − 휌 ) /휌 (6.17) 푑푡 푠 √ 푙 푔 푙 For transition between bubbly to slug flow, a new velocity term, vgb is introduced: For upward concurrent flow in a vertical system: 푓푡 푣 ( ) = 0.43 푣 + 0.36푣 (6.18) 푔푏 푠 푠푙 푑푏 when the gas hold-up exceeds 0.25 and the gas superficial velocity, vsg exceeds vgb , the flow transitions from bubbly flow to slug flow. The drift velocity, vd in slug flow regime is defined as: 0.1푣 0.1푣 푓푡 − 푔푏 − 푔푏 푣 ( ) = 푣̅̅̅ = 푣 (1 − 푒 푣푠푔−푣푔푏) + 푣 (푒 푣푠푔−푣푔푏 ) (6.19) 푑 푠 푑 푑푏 푑푡 6.3.3. Churn flow: When the mixture velocity is high, the slugs break and a churn flow is established. This happens when the velocity vms (ft/s) is greater than mixture velocity, vm (ft/s). The slug velocity, vms is defined by the equation described by Shoham (1982): 0.4 0.6 1.2 푓푡푝 휌푙 0.4휎푔표 푣푠푔 2푣푚푠 ( ) ( ) [√ ] = 0.725 + 4.15√ (6.20) 2푑 휎푔표 푔(휌푙 − 휌푔) 푣푚 For transition from bubbly flow to churn flow, vsg >vsl. 39 퐶표푣푠푙 + 푣푑푏 푣푔푏 = (6.21) 4 − 퐶표 The drift velocity in the churn flow is defined using equation (6.19). 6.3.4. Annular flow: The transition from churn to annular flow is defined when vsg > vgc , where : 0.25 푔휎 (휌 − 휌 ) 푔 푙 푔 푣푔푐 = 3.1 [ 2 ] (6.22) 휌푔 6.4. Transition smoothening: To have a transition smoothing, for churn flow: 푣 푣 −0.1 푚푠 −0.1 푚푠 푣 −푣 푣 −푣 퐶표 = 1.2 [1 − 푒 푚 푚푠 ] + 1.15 [푒 푚 푚푠 ] (6.23) For annular flow: 푣 푣 −0.1 푔푐 −0.1 푔푐 푣푠푔−푣푔푐 푣푠푔−푣푔푐 퐶표 = 1.15 [1 − 푒 ] + 1.0 [푒 ] (6.24) It is to be noted that in all the equations above, wherever the interfacial tension has been used, it is represented in dynes/cm and calculated using the Weinaug and Katz (1943) equation: 1 푛푐 4 휌푙푥푐 휌푔푦푐 휎푔표 = ∑ 푃푐ℎ푖[ − ] #(6.24) 62.4푀표 62.4푀푔 푖=1 6.5. Finite difference equations for wellbore: An analytical solution for the Equations (6.1) through (6.3) is not available, hence a numerical solution using the finite difference method is sought. Considering a One-Dimensional vertical system operating only along the z-direction, we discretize the wellbore into grid spacing 40 containing blocks numbered in an increasing order as we go from the bottom of the well to the surface as shown in Figure 6-2. Figure 6-2. Schematic of the discretized wellbore model A second order central finite difference scheme is used for the spatial derivative and a backward finite difference scheme is used for the temporal derivative and the system is solved in a fully implicit manner. For such a system, the primary equations are discretized as such: 41 Hydrocarbon continuity equation: (푛+1) (n+1) 푣푔 1휌푔 1퐻푔 1푦푐 1 푣푙 1휌표 1퐻표 1푥푐 1 푖+ 푖+ 푖+ 푖+ 푖+ 푖+ 푖+ 푖+ [( 2 2 2 2) + ( 2 2 2 2) ] 푀푔 1 푀표 1 푖+ 푖+ 2 2 ∆z (푛+1) (n+1) 푣푔 1휌푔 1 퐻푔 1푦푐 1 푣푙 1휌표 1퐻표 1푥푐 1 푖− 푖− 푖− 푖− 푖− 푖− 푖− 푖− [( 2 2 2 2) + ( 2 2 2 2) ] 푀푔 1 푀표 1 푖− 푖− 2 2 − ∆z 푣 휌 퐻 푦 (푛+1) 푣 휌 퐻 푥 (n+1) [( 푔푖 푔푖 푔푖 푐푖) + ( 푙푖 표푖 표푖 푐푖) ] 푀푔 푀표 + 푖 푖 ∆푡 푣 휌 퐻 푦 (푛) 푣 휌 퐻 푥 (푛) [( 푔푖 푔푖 푔푖 푐푖) + ( 푙푖 표푖 표푖 푐푖) ] 푀 푀 푛+1 푔푖 표푖 푁푐푖 − − = 푅푐푖 (6.25) ∆푡 푉푏푖 Water continuity equation: (푛+1) (푛+1) 푣푙 1휌푤 1퐻푤 1 푣푙 1휌푤 1퐻푤 1 푖+ 푖+ 푖+ 푖− 푖− 푖− ( 2 2 2 ) − ( 2 2 2) 푀푤 1 푀푤 1 푖+ 푖− 2 2 + ∆푧 (푛+1) (푛) 푣푙푖휌푤푖퐻푤푖 푣푙푖휌푤퐻푤푖 [( ) − ( ) ] ( ) 푀 푀 푁 푛+1 푤푖 푤푖 푤푖 + − = 푅푤푖 ∆푡 푉푏푖 (6.26) Mixture momentum equation: (푛+1) 2 2 (휌 1푣 1 − 휌 1푣 1) 푤 푤 (푛+1) 푚푖+ 푚푖+ 푚푖− 푚푖− 4634.6(푃푖+1 − 푃푖 ) (푛+1) 2 2 2 2 + (휌푚 푔) + ∆푧 푖 ∆푧 (푛+1) (푛+1) (푛) 푓푡푝푖휌푚푖푣푚푖 (휌푚푖푣푚푖) − (휌푚푖푣푚푖) + ( ) + = 푅푚푖 (6.27) 2푑푖푛 ∆푡 42 1 For any entry in Equations (6.25) through (6.27) with subscripts ± , for example 휌, 2 휌푖 + 휌(푖±1) 휌 1 = (6.28) 푖± 2 2 6.6. Initial and boundary conditions: Initial conditions: In this wellbore model a pressure specification at the wellhead, Psp and a molar flow rate specification at the bottom of the well, N is specified. The initial pressure along the well is calculated using the hydrostatic gradient. The initial density of the fluid used to calculate the hydrostatic gradient is obtained by performing a flash calculation using reservoir source/sink term’s composition and the pressure at the wellhead. Hence, 푤0 푃푖 = 푃푠푝 + 휌푚푠푝(푍푤푒푙푙ℎ푒푎푑 − 푍푖) (6.29) 0 ( ) 푧푐푖 = 퐼푛푖푡푖푎푙 푟푒푠푒푟푣표푖푟 푐표푚푝표푠푖푡푖표푛 6.30 0 ( ) 퐻푤푖 = 퐼푛푖푡푖푎푙 푤푎푡푒푟 푠푎푡푢푟푎푡푖표푛 푖푛 푟푒푠푒푟푣표푖푟 6.31 The initial composition of the fluids in the well is uniform and is equal to the molar ratio of the source/sink terms coming from the reservoir. For the drift-flux implementation an initial value of superficial velocity for liquid and gas is required. This is obtained using the equations: 푡표푡푎푙 푚표푙푎푟 푓푒푒푑 × 푔푎푠 푚표푙푒 푓푟푎푐푡푖표푛 × 푔푎푠 푚표푙푒푐푢푙푎푟 푤푒푖푔ℎ푡 푛 × 푀 푔푖 푔푖 ( ) 푣푠푔푖 = = 푓푒푒푑 × 6.32 푃푖푝푒 푎푟푒푎 표푓 푐푟표푠푠 푠푒푐푡푖표푛 × 푔푎푠 ℎ표푙푑 − 푢푝 퐴푖 × 퐻푔푖 푡표푡푎푙 푚표푙푎푟 푓푒푒푑 × 푙푖푞 푚표푙푒 푓푟푎푐푡푖표푛 × 푙푖푞 푚표푙푒푐푢푙푎푟 푤푒푖푔ℎ푡 푛 × 푀 푙푖 푙푖 ( ) 푣푠푙푖 = = 푓푒푒푑 × 6.33 푃푖푝푒 푎푟푒푎 표푓 푐푟표푠푠 푠푒푐푡푖표푛 × 푙푖푞푢푖푑 ℎ표푙푑 − 푢푝 퐴푖 × 퐻푙푖 43 The initial gas and liquid hold-ups are calculated using the equations (6.9) through (6.11) after a flash calculation has been performed. The liquid mole fraction and molecular weights are calculated using molar weighted averaging of oil and water properties. Boundary Conditions: The work of Shirdel (2012) has been referred to for this part. Two reflection blocks are used at the top and bottom of the well. The boundary at the image block in the bottom serves as a no-flow boundary and the image block at the top serves as a constant flow boundary. At the bottom (block#1 – no flow boundary): 푤푛+1 푤푛+1 푤푛+1 푃1 = 2푃2 − 푃3 (6.34) 푛+1 푛+1 푛+1 푧푐 1 = 2푧푐 2 − 푧푐 3 (6.35) 휌푛+1 푛+1 푛+1 푚 2 푣푚 1 = −푣푚 2( 푛+1 ) 휌푚 1 (6.36) 푛+1 푛+1 푛+1 퐻푤 1 = 2퐻푤 2 − 퐻푤 3 (6.37) At the top (block#nwb– constant flow boundary): 푤푛+1 푛+1 푤푛+1 푃푛푤푏 = 2푃푠푝 − 푃푛푤푏−1 (6.38) 푛+1 푛+1 푛+1 푧푐 푛푤푏 = 2푧푐 푛푤푏−1 − 푧푐 푛푤푏−2 (6.39) 푛+1 푛+1 푛+1 푣푚 푛푤푏 = 2푣푚 푛푤푏−1 − 푣푚 푛푤푏−2 (6.40) 44 푛+1 푛+1 푛+1 퐻푤 푛푤푏 = 2퐻푤 푛푤푏−1 − 퐻푤 푛푤푏−2 (6.41) 6.7. Newton-Raphson protocol for linearizing equations As it can be observed, our wellbore equations are not linear in nature and hence the Newton- Raphson protocol for a multivariable system is used to linearize the system. The system of equations is represented in the form JX=R, where J represents the Jacobian matrix, X represents the vector of the primary unknown variables and R represent the residual vector. The wellbore system of equations using four hydrocarbon components for a single block is represented in Figure 6-3: (푛+1) 휕푅1 휕푅1 휕푅1 휕푅1 휕푅1 휕푅1 (푘) ( ) ( ) ( ) ( 푤) ( ) ( ) 휕푧1 휕푧2 휕푧3 휕푃 휕푣푚 휕퐻푤 휕푅2 휕푅2 휕푅2 휕푅2 휕푅2 휕푅2 ( ) ( ) ( ) ( ) ( ) ( ) (푛+1) (푛+1) 휕푧 휕푧 휕푧 휕푃푤 휕푣 휕퐻 1 2 3 푚 푤 ∆푧1 (푘+1) −푅1 (푘+1) 휕푅 휕푅 휕푅 휕푅 휕푅 휕푅 3 3 3 3 3 3 ∆푧2 −푅2 ( ) ( ) ( ) ( 푤) ( ) ( ) 휕푧1 휕푧2 휕푧3 휕푃 휕푣푚 휕퐻푤 ∆푧3 −푅3 푤 = 휕푅 휕푅 휕푅 ∆푃 −푅푛푐 ( 푛푐) ( 푛푐) ( 푛푐) 휕푅푛푐 휕푅푛푐 휕푅푛푐 푤 ∆푣 −푅 ( ) ( ) ( ) 휕푃 휕푣푚 휕퐻푤 푚 푚 휕푧1 휕푧2 휕푧3 [∆퐻 ] [−푅 ] 휕푅푚 휕푅푚 휕푅푚 푤 푖 푤 푖 휕푅푚 휕푅푚 휕푅푚 ( 푤 ) ( ) ( ) ( ) ( ) ( ) 휕푃 휕푣푚 휕퐻푤 휕푧1 휕푧2 휕푧3 휕푅푤 휕푅푤 휕푅푤 0 0 0 ( 푤) ( ) ( ) [ 휕푃 휕푣푚 휕퐻푤 ]푖 Figure 6-3. System of equations for a cell in the discretized wellbore hydraulics model The derivative terms in the Jacobian are calculated using numerical differentiation. This system of equations above is solved using a linear solver. In the C++ based implementation GMRES is used, while, in the MATLAB implementation, the linsolve command is used. The change in the primary unknowns obtained through solving for X=R/J is added to their respective variables from the previous iteration to get an improved estimate. This process is iteratively 45 repeated for a time step until the maximum value of the residual is less than 0.01 psi. The entire procedure followed has been displayed in the form of a flow chart in figure 6-4. Figure 6-4. Schematic representation of the computational protocol used in numerical wellbore simulation 46 Chapter 7 Nuero-simulation of wellbore hydraulics 7.1. ANN based wellbore model The numerical wellbore hydraulics model is computationally expensive and takes a significant time, ranging from minutes to hours to predict the pressure distribution of the stabilized system. The deeper a given well is, the simulation grid is divided into more number of blocks in order to control the numerical dispersion. The complexity of the computation increases with an increase in the number of blocks as more flash calculations are to be performed and also because a larger matrix has to be inverted at each iteration. Hence, an alternative model which can simulate this process quickly while maintaining the accuracy was sought. Artificial Neural Networks as classification and regression tools are able to solve complex problems in a short time and hence their applicability in this junction is explored. 7.2. Data generation and pre-processing The numerical wellbore model can be operated using any number of components (it is typical to use less than 45 components). However, for training an ANN, it was considered necessary to define a fixed number of components. For such purposes, the total number of components had been set to seven. The first five components are fixed: Methane, Ethane, n-Propane, iso-Butane and iso-Pentane, the last two components are pseudo-components C6+ and C20+, which represent a lumping of the mixture of components between C6 through C19 and C20 through C45 respectively. 47 The numerical wellbore simulator was used to generate the data for training, validation and testing of the ANN. As a first step, the numerical model was validated against cases from literature and after a good agreement was observed, the model was run under various specifications. The ranges of the various input parameters are described in Table 7-1. The parameters, molar feed into the well, wellhead pressure, tubing internal diameter, wellhead temperature, well temperature gradient, initial water mole fraction in well feed, water specific gravity, water viscosity, total depth of the well, pipe roughness and one component each between C6 through C19 and C20 through C45 respectively were selected at random. The rest of the inputs, physical properties such as hydrocarbon phase densities, viscosities and black-oil properties such as gas-oil ratio and gas-Liquid ratio, oil and gas flow rates were obtained through a flash calculation for the molar compositional feed at standard conditions. A total of almost 80,000 data sets have been generated. As it can be observed in Table 7-1, input number 5 represents the depth as a fraction of the total depth for the point at which we intend to predict the wellbore pressure. After obtaining raw pressure data from each wellbore simulation, the pressures were interpolated at depths representing each tenths of the total depth of a well. This process is performed to ensure that wells with greater depth do not encounter abnormally excessive data points. After the data generation, all of the input and output parameters with a numerical range greater than 1,000 were reduced to a logarithm to place the data on similar scale before normalization process. After this step, the entire set of inputs is normalized to values between 0 and 1. Hence, the output predicted by the ANN is denormalized and subjected to an exponentiation to the base 10 in order to obtain the final pressures. 48 Table 7-1. Summary of the input parameters used in the ANN based wellbore model Category Inputs Units Min Max Wellhead pressure psia 100 2000 Tubing diameter inches 1 4 Pipe roughness ft 1.00E-04 2.00E-03 Total Depth ft 200 16000 Fraction of total depth at the point of prediction fraction 0 1 Temperature at the point of prediction °F 60 340 Well construction Well Wellhead temperature °F 60 180 Well temperature gradient °F/ft 0.005 0.02 Molar feed into the well lb moles/s 0.00125 0.5 Initial water mole fraction in well feed fraction 0 1 Water Cut percentage 0 98.7 Oil viscosity lb-moles/ft-s 1.7E-05 1.5E-03 Gas viscosity lb-moles/ft-s 3.4E-06 9.9E-05 Water viscosity lb-moles/ft-s 4.7E-04 8.1E-04 oil properties oil - Oil specific gravity fraction 0.49 0.94 Gas specific gravity fraction 0.47 6.93 Water specific gravity fraction 1 1.05 Oil flow rate STB/day 10 32767 Gas flow rate MMSCF/day 0 15.6 Fluid flow and Black and flow Fluid Water flow rate STB/day 0 2180 Gas-Oil Ratio SCF/STB 0 168068 Gas-Liquid Ratio SCF/STB 0 94868 Feed -global mole fraction of C1 fraction 0.20 0.94 Feed -global mole fraction of C2 fraction 2.4E-05 0.59 Feed -global mole fraction of C3 fraction 1.9E-06 0.56 Feed -global mole fraction of C4 fraction 9.3E-06 0.65 Feed -global mole fraction of C5 fraction 6.5E-07 0.54 Feed -global mole fraction of C6+ fraction 2.9E-06 0.62 Composition of feed of Composition Feed -global mole fraction of C20+ fraction 1.7E-05 0.63 Critical temperature of C6+ °R 914 1409 Critical temperature of C20+ °R 1428 1724 Critical pressure of C6+ psia 211 477 Critical pressure of C20+ psia 105 203 Accentricity factor of C6+ unitless 0.28 0.82 Accentricity factor of C20+ unitless 0.86 1.33 Molecular weight of C6+ lb/lb-moles 86 275 Molecular weight of C20+ lb/lb-moles 291 539 Volume shift parameter of C6+ unitless -0.059 0.142 Volume shift parameter of C20+ unitless 0.139 0.358 Critical volume of C6+ cubic ft./lb-mole 5.5 16.5 Critical volume of C20+ cubic ft./lb-mole 17.2 31.3 Thermodynamic properties of C6+ and C20+ and C6+ of properties Thermodynamic Parachor of C6+ unitless 250.1 710.5 Parachor of C20+ unitless 742.2 1090.4 49 7.3. ANN architecture selection and training A MATLAB©* script had been developed to train, validate and test feedforward ANNs with various different architectures and select the model that provides the least error on the data which was isolated for testing. The scheme of this script is similar to that described by Sun-Ertekin (2015). The model has 43 input parameters and one output. A random selection is performed for the assignment of each architectural parameter. For each ANN, 1,000 different architectures have been tried and blind tested. The data from training testing and validation is randomly subdivided in the ratio of 80:10:10, respectively. The training has been performed using the scaled conjugate gradient method due to its high speed, robustness and lower memory requirement (Rajput (2012)). The number of epochs was set to a high value of 10,000 and the validation fail parameter is set to 1,000 counts. The error from each epoch was reduced by updating the neuron weights using a back-propagation method. The training was stopped if either the maximum number of epochs are reached or the validation fail count is approached. The architecture and the performance statistics of each ANN are described in Table 7-2. 7.4. Modular ANN toolbox The ANN toolbox is divided into modules. An appropriate module can be selected for a given wellbore hydraulics problem depending on the conditions of operation. Modules ANN-1A and ANN-2A are developed to predict the pressure profile at various points along the depth of a well. Modules ANN-1B through ANN-5B are developed to predict the pressure at the deepest point in ∗ MATLAB© is a registered trademark of The MathWorks, Inc 50 the section of the well under consideration. These modules are assembled to provide the flowing bottom-hole pressure values to the sand-face pressure specified reservoir simulator in coupled reservoir-wellbore simulation. ANN modules for pressure prediction at various points along the entire depth of a well: 1. ANN-1A: Modeled on wells from a depth of 2,400 ft to 6,800 ft. 2. ANN-2A: Modeled on wells from a depth of 7,200 ft to 14,400 ft. ANN modules for pressure prediction at the bottom-most depth of a well section: 3. ANN-1B: Modeled on wells from a depth of 2,400 ft to 16,000 ft. 4. ANN-2B: Modeled on wells from a depth of 200 ft to 2,200 ft, with a wellhead pressure lower than 1,500 psia and molar flow rate operating between 0.00125-0.01 lb-moles/s. 5. ANN-3B: Modeled on wells from a depth of 200 ft to 2,200 ft, with a wellhead pressure between 1,500 to 6,000 psia and molar flow rate operating between 0.00125-0.01 lb- moles/s. 6. ANN-4B: Modeled on wells from a depth of 200 ft to 2,200 ft, with a wellhead pressure lower than 1,500 psia and molar flow rate operating between 0.01-0.5 lb-moles/s. 7. ANN-5B: Modeled on wells from a depth of 200 ft to 2,200 ft, with a wellhead pressure between 1,500 to 6,000 psia and molar flow rate operating between 0.01-0.5 lb-moles/s. ANN-2B through ANN-5B are developed primarily to simulate the pressures in the bottom portion of the wellbore below the depth of operating gas-lift valve. For determining the flowing bottom-hole pressure for a well with gas-lift, the pressure at the point of gas-lift injection is determined using ANN-1B and this pressure is used as the wellhead pressure for the bottom section of the well. The maximum depth for this bottom section is chosen to be 2,200 ft as it is common to place the operating gas lift valve a depth of 600 ft to 2,000 ft above the production interval. The details of the architectures and the blind testing results of each ANN module is 51 summarized in Table 7-2. It is to be noted that ‘logsig’ and ‘tansig’ refer to the logistic sigmoid and the tangential sigmoid transfer functions, respectively. Table 7-2. Architecture description and blind testing results of the wellbore ANN modules Number of Neurons in Number of test Mean R-Squared Module Transfer functions hidden layers hidden layers samples error (%) value ANN-1A 1 [13] Tansig 329 7.87 0.97 ANN-2A 2 [18,13] Tansig, Tansig 1527 5.84 0.985 ANN-1B 4 [19,22,19,13] Logsig, Logsig, Logsig, Tansig 1397 3.59 0.989 ANN-2B 3 [19,12,15] Tansig, Tansig, Logsig 521 2.66 0.993 ANN-3B 2 [16,14] Tansig, Tansig 3076 1 0.997 ANN-4B 3 [9,10,5] Tansig, Tansig, Logsig 1426 3.11 0.98 ANN-5B 4 [10,12,20,11] Tansig, Tansig, Logsig, Logsig 1414 1.12 0.979 The frequency of error counts can be observed through the histograms for each ANN module as displayed in Figure 7-1. It can be observed that the proportion of observations which have less than 0-10% error is much higher in the ANN modules 1B through 5B as compared to 1A through 2A. This displays the improvement in performance and justifies the development and usage of ANN modules for predicting the flowing bottom-hole pressure exclusively at the deepest point as opposed to a model which predicts the pressure along the entire length of the wellbore for coupled reservoir-wellbore simulation. Furthermore, the target bottom-hole pressures provided by the numerical simulator are compared to the predicted bottom-hole pressures by the ANN in cross plots from Figure 7-2. To display the extent of agreement between the target and predicted values, the R-squared values and the line of least square fit are included for each cross plot. It is to be noted that ANNs are most accurate in the range of variables under which they are trained and the accuracy may not be reliable beyond the range of training. 52 Figure 7-1. Histogram of error counts for various wellbore ANN modules 53 Figure 7-2. Cross plots of prediction versus targets for various wellbore ANN modules 54 7.5. Computational performance comparison of the numerical model and neuro-simulation The numerical wellbore model was run for 24 hours with the input variables randomly chosen in a protocol similar to the one followed to generate data for ANN training. The numerical model generated 91 results in this time span. These 91 results were re-simulated on the same computer using the ANN model. Although the ANN is capable of taking inputs simultaneously in the form of a 43×91 matrix and predict a result as a 1×91 vector, an iterative loop was used to run the Neuro-simulation. This was done in order to account for the ANN call time which will be a factor in each iteration of the coupled reservoir-wellbore simulation. It was observed that the total time taken by the ANN toolbox was 2.13 seconds to re-simulate the 91 runs. Since the numerical model was run for 86,400 seconds, the proposed ANN model outperformed the numerical model speed-wise by more than 40,000 times. The ANN model does not encounter any stability issues which may lead to time-step cuts or failure of the program. This speed and robustness of the ANN model provides a significant advantage in coupled reservoir-wellbore simulation studies. This comparison was performed on a computer with a 2.20 GHz processor and 8 GB RAM. 55 Chapter 8 Validation of the standalone models 8.1. Standalone numerical compositional reservoir simulator A three-dimensional, three-phase, single well, compositional, isothermal numerical reservoir simulator has been developed using a radial cylindrical coordinate system. The development and description of this tool is described in chapter 5. The simulator has been validated by comparing the results with the results produced by the compositional commercial numerical simulator (CMG-GEMS©*). The simulation was performed using fifteen grid blocks per layer in radial direction, and two layers in the depth direction at a constant reservoir temperature of 160 F. Figure 8-1 shows the tilted view of the reservoir model being considered. Figure 8-1. Schematic of the reservoir simulation Case1: A reservoir above bubble point, using a fluid initially made up of 99.96% Decane and 0.01% each of Propane, n-Butane, Hexane and Decane with inputs as shown in Table 8-1. ∗ CMG-GEMS© is a registered trademark of Computer Modelling Group Ltd. 56 Table 8-1. Inputs for comparison of commercial reservoir simulator with current work -Case 1 Initial pressure(psia) 5000 Permeability in radial direction (mD) 10 Permeability in depth direction (mD) 1 porosity 0.23 Wellbore radius (ft) 0.25 Radius of the reservoir (ft) 10000 Sand-face pressure(psia) 3500 Thickness (ft) 2x10 The results for a constant sandface pressure specified well are shown in Table 8-2 and Figure 8-2: Table.8-2. Results for comparison of commercial reservoir simulator with current work-Case 1 Commercial Current work Pressure reading difference per block (psia) simulator Average Relative error Relative (%) error (%) Time Oil flow rate-reservoir conditions (bbl/day) Maximum Average (days) 4 19.6 11.2 0.3 597.5 580.7 2.9 64 11.9 8.6 0.2 510.8 493.3 3.5 364 8.1 6.9 0.2 459.1 441.7 3.9 724 12.0 8.9 0.3 422.7 408.0 3.6 700.0 600.0 500.0 reservoir reservoir - 400.0 300.0 Commercial simulator 200.0 Current conditions(bbl/day) Oil flow flow Oil rate 100.0 work 0.0 0 100 200 300 400 500 600 700 800 Time (days) Figure 8-2. Oil flow rate comparison – Case 1 57 Case2: For a reservoir below bubble point, using a fluid with 60% Methane and 10% each of Propane, n-Butane, Hexane and Decane with inputs as shown in Table 8-3 Table.8-3. Inputs for comparison of commercial reservoir simulator with current work -Case 2 Initial pressure(psia) 2100 Permeability in radial direction (mD) 10 Permeability in depth direction (mD) 1 porosity 0.23 Wellbore radius (ft) 0.25 Radius of the reservoir (ft) 10000 Sand-face pressure(psia) 1100 Thickness (ft) 2x10 The results for a constant sandface pressure specified well are shown in Table 8-4 and Figures 8-3 and 8-4, respectively: Table.8-4. Results for comparison of commercial reservoir simulator with current work -Case 2 Commercial Current Commercial Current Pressure reading per block-absolute Average simulator work simulator work difference between commercial relative Relative Relative simulator and current work(psia) error Oil flow rate- reservoir error Gas flow rate-reservoir error conditions conditions Time Maximum Average (%) (bbl/day) (%) (ft3/day) (%) (days) 5 18.5 10.2 0.6 65.9 62.4 5.3 38802.1 40574.7 -4.6 60 15.6 9.9 0.6 57.8 55.6 3.8 32778.3 33728.3 -2.9 360 12.7 8.4 0.5 55.0 53.0 3.6 29499.1 30046.1 -1.9 690 12.5 8.5 0.5 56.7 54.3 4.2 28475.1 28966.5 -1.7 80.0 70.0 60.0 Reservoir Reservoir 50.0 - 40.0 30.0 Commercial simulator 20.0 conditions(bbl/day) Current work oil flow rate 10.0 0.0 0 100 200 300 400 500 600 700 800 Time (days) Figure 8-3. Oil flow rate comparison – Case2 58 50000.0 45000.0 40000.0 35000.0 Reservoir Reservoir - 30000.0 25000.0 20000.0 Commercial 15000.0 simulator 10000.0 current work 5000.0 Gas Gas flow rate conditions(Cubic ft./day) 0.0 0 100 200 300 400 500 600 700 800 Time (days) Figure 8-4. Gas flow rate comparison – Case2 The comparison with the commercial simulator yielded a relative error of about 1% for average of block pressures and 4% for flow rates. The current work was also compared with analytical solutions from well test analysis theory, under transient conditions and pseudo steady state conditions. For this purpose, the reservoir fluid from case 1 of the commercial simulator validation was used, with the entire system above bubble point pressure. The evaluation is performed by comparing the sand-face pressure of a well at different times for a constant production rate. For a case of transient conditions, the inputs used are as described in Table 8-5 and results have been shown in Table 8-6. Transient Conditions Table 8-5. Input for transient-well test analysis-current work Initial pressure (psia) 2000 Permeability (mD) 0.01 Porosity 0.23 Oil viscosity(cP) 0.396 Total compressibility (psia -1) 7.95E-6 Radius-well (ft) 0.25 Reservoir radius(ft) 10000 Oil flow rate-reservoir conditions (bbl/d) 1.06 Reservoir thickness(ft) 1x100 59 Table 8-6. Comparison of current work with transient-well test analysis with current work Sand-face pressure (psia) calculated for a flow rate specified well Time Current Relative Time (days) Well-test difference (hours) work error (%) 10.0 240.0 1705.2 1693.1 12.1 0.7 190.0 4560.0 1617.9 1605.9 12.0 0.7 381.0 9144.0 1596.2 1585.2 11.0 0.7 The comparison of the current work with well test analysis under transient conditions gives a relative error of 0.7% for the sand-face pressure calculated. The sensitivity of the well test solution with respect to the fluid parameters is quite significant. Hence, the fluid properties used are obtained through volumetric weighted averaging. As it is quite probable that a gas lift system is operating in a mature reservoir operating under pseudo steady state conditions, a comparison is performed for pseudo steady state well-test analysis, the inputs and results of a sample case are displayed in Tables 8-7 and 8-8, respectively: Pseudo steady state conditions Table 8-7. Input for pseudo steady state-well test analysis- current work Initial pressure (psia) 2000 Permeability (mD) 10 Porosity 0.23 Oil viscosity(cP) 0.39 Total compressibility (psia -1) 8.275E-6 Radius-well (ft) 0.25 Reservoir radius(ft) 1000 Oil flow rate-reservoir 254 conditions (bbl/d) Reservoir thickness(ft) 1x100 60 Table 8-8. Comparison of current work with pseudo steady state- well test analysis-current work Sand-face pressure (psia) calculated for a flow rate specified well Time Current Relative error Time (days) Well-test difference (hours) work (%) 10.0 240.0 1868.0 1870.6 2.6 0.1 190.0 4560.0 1435.9 1441.4 5.5 0.4 381.0 9144.0 1006.5 985.9 -20.6 2.1 For pseudo steady state conditions, the relative error ranges between 0.1 - 2 %. Based on results in comparison with both a commercial simulator and well test analysis, the reservoir simulator has been considered to be validated. However, to provide a reference, a well test has been conducted with the same reservoir using a commercial simulator. Table 8-9 provides the inputs of the comparison of the commercial simulator with transient state well test analysis solution. Table 8-10 provides the results of the transient state comparison of the commercial simulator with the well test solution. Table 8-9. Input for transient-well test analysis-commercial simulator Initial pressure (psia) 2000 Permeability (mD) 0.01 Porosity 0.23 Oil viscosity(cP) 0.3806 Total compressibility (psia -1) 7.94E-6 Radius-well (ft) 0.25 Reservoir radius(ft) 10000 Oil flow rate-reservoir conditions (bbl/d) 1.06 Reservoir thickness(ft) 1x100 Table 8-10. Comparison of current work with transient-well test analysis with commercial simulator Sand-face pressure (psia) calculated for a flow rate specified well Time Commercial Relative Time (days) Well-test difference (hours) simulator error (%) 10.0 240.0 1710.2 1703.9 6.3 0.4 190.0 4560.0 1628.1 1620.0 8.1 0.5 381.0 9144.0 1607.7 1600.2 7.5 0.5 61 Similarly, a comparison is conducted between the commercial simulator and the well-test solution under pseudo-steady state conditions. The inputs for this test are provided in Table 8-11 and the corresponding results are summarized in Table 8-12. Table 8-11. Input for pseudo steady state-well test analysis- commercial simulator Initial pressure (psia) 2000 Permeability (mD) 10 Porosity 0.23 Oil viscosity(cP) 0.3734 Total compressibility (psia -1) 8.35E-6 Radius-well (ft) 0.25 Reservoir radius(ft) 1000 Oil flow rate-reservoir 254 conditions (bbl/d) Reservoir thickness(ft) 1x100 Table 8-12. Comparison of current work with pseudo steady state- well test analysis-commercial simulator Sand-face pressure (psia) calculated for a flow rate specified well Relative Time Time Commercial Well-test difference error (days) (hours) simulator (%) 10.0 240.0 1834.1 1875.2 41.1 2.2 190.0 4560.0 1407.6 1447.5 39.9 2.8 381.0 9144.0 985.1 993.6 8.5 0.9 It is to be noted in Table 8-9 and Table 8-11 that the viscosity and compressibility of the input data for the comparison of the well-test analysis solution with the commercial simulator varies from the inputs for the current work shown in Table 8-5 and Table 8-7, respectively. This difference is due to the variation in the results of the flash calculations in the current work when compared to the commercial simulator. However, as the commercial simulator has been compared with the well-test analysis solution based on the averaged fluid properties estimated through the flash calculations generated by the commercial simulator, this difference between the 62 flash results of commercial simulator and current work should not make a difference to this study. It can be observed that the commercial simulator has an error of 0.4.-0.5% under transient conditions and 0.9-2.8% under pseudo steady state conditions from Table 8-10 and Table 8-12, respectively. This result is comparable to the results of the error of predictions from current work versus well test analysis based solution. Since, there are several factors which are unknown in the functioning of the commercial simulator, such as, the numerical tolerances and the grid type (whether body centered or mesh centered), the difference in the predictions of the commercial simulator and current work cannot be further reduced. However, with good levels of agreement between the current works, the analytical solution and current work provide encouragement to proceed forward. 8.2. Standalone compositional wellbore simulator The wellbore hydraulics model has been developed using two approaches: (1) Numerical (2) ANN. These models are validated using data found in the literature. Five different cases have been displayed here. In Case 1, a black-oil example had been chosen from Hasan et al. (2002). The data provided pertains to a field example which has also been compared with Hasan and Kabir (1992) and Ansari et al. (1994) mechanistic models. In Cases 2-5 the models developed in this work have been tested with the data from the numerical compositional simulator developed by Pourafshary et al. (2009). These models include fluids systems which contain black-oil, volatile oil, gas condensate and a three-phase system, respectively. Case 1-Field data: In this case, the black-oil flow properties of a well from field data are provided in the literature, hence a composition which reproduces the API gravity, gas gravity and oil flow rate upon performing a flash calculation using the Peng-Robinson equation of state at surface conditions has been chosen. A seven-component system selected to test the results using 63 the ANN model along with the numerical model developed as a part of this study. The total depth of the well has been provided to be 5,151 ft, the tubing diameter is 2.99 inches and the wellhead pressure is 505 psig. The black-oil production data and fluid composition details are described in Tables 8-13 and 8-14, respectively. The results of this study are displayed in Table 8-15 and Figure 8-5, respectively. Table 8-13. Case 1: Black-oil production field data API Gravity (°API) 23 Gas-Oil Ratio (SCF/STB) 450 Oil production rate (STB) 1,140 Gas Gravity 0.8 Table 8-14. Case 1: Fluid composition-calculated for field data Hydrocarbon Mole fraction (Calculated) Methane 0.449 Ethane 0.071 Propane 0.0543 i-Butane 0.0435 i-Pentane 0.0014 C6+ (C8) 0.0013 C20+ (C35) 0.3678 Table 8-15. Case 1: Results- comparison of developed ANN and numerical wellbore models with field data and mechanistic models Pressure (psig) Relative deviation (%) Pressure (psig) Relative deviation (%) Depth Field data Hasan and Ansari Field data vs Field data vs Field Field data (ft) ANN Numerical vs Kabir et al. Hasan and Ansari et al. data vs ANN Numerical (1992) (1994) Kabir (1992) (1994) 0 505 505 505 0.0 0.0 505 505 0.0 0.0 400 582 595 587 0.9 1.3 593 586 1.0 0.2 650 634 655 647 2.0 1.2 654 641 1.1 0.9 1150 753 781 777 3.1 0.5 781 758 0.5 2.4 1650 889 917 920 3.4 0.3 918 885 0.2 3.8 2150 1042 1062 1074 3.0 1.1 1063 1021 1.0 4.9 2650 1208 1215 1237 2.4 1.8 1212 1165 2.0 5.8 3150 1384 1373 1407 1.6 2.4 1369 1316 2.7 6.5 3650 1568 1537 1582 0.9 2.8 1530 1473 3.3 6.9 4150 1756 1706 1850 5.1 7.8 1695 1634 8.4 11.7 4650 1945 1878 1960 0.8 4.2 1864 1799 4.9 8.2 5151 2135 2052 2105 1.4 2.5 2034 1968 3.4 6.5 Average 2.2 2.4 Average 2.6 5.3 64 Pressure (Psig) 0 500 1000 1500 2000 2500 0 Field ANN 1000 Numerical 2000 Hasan-Kabir (1992) Ansari et al. (1994) 3000 Depth(ft) 4000 5000 6000 Figure 8-5. Case 1: Comparison of wellbore models developed in this research with field data and mechanistic models It can be observed in Table 8-15 that the average relative deviation of the pressure profile predicted by the ANN and the numerical wellbores models with respect to field data is 2.2% and 2.4%, respectively. The mechanistic models developed by Hasan and Kabir (1992) and Ansari et al. (1994) show an average relative deviation of 2.6 % and 5.3 %, respectively with respect to field data. Case 2: A comparison was made for the wellbore models developed in this paper using black- oil fluid system with a numerical simulator from Pourafshary et al. (2009). For this case, the molar flow rates provided in the literature was 1,500 lb-moles/day. The molar composition of the well-feed was provided and the black-oil properties were obtained through a flash calculation performed at standard conditions. The description of the wellbore system in consideration is provided in Tables 8-16 and 8-17, respectively. The results are displayed in Table 8-18 and Figure 8-10, respectively. 65 Table 8-16. Case 2: Wellbore description Total Depth (ft) 5000 Grid spacing (ft) 250 Wellhead Pressure (psia) 300 Tubing diameter (inches) 1.5 Wellhead Temperature (°F) 80 Well Bottom-hole Temperature (°F) 150 Pipe roughness (ft) 6E-4 Table 8-17. Case 2: Fluid composition input Pourafshary et al. (2009) and ANN model current work-Numerical model Component Mole fraction Component Mole fraction Methane 0.3 Methane 0.3 Propane 0.12 Ethane 1E-6 i-Butane 0.12 Propane 0.12 i-Pentane 0.12 i-Butane 0.12 Heptane 0.17 i-Pentane 0.12 Octane 0.17 C6+ 0.34 -2E-6 C20+ 1E-6 Table 8-18. Case 2: Results- comparison for black-oil fluid Pressure (psia) Relative deviation (%) Depth (ft) Pourafshary et Pourafshary et al. (2009) Pourafshary et al. (2009) ANN Numerical al. (2009) * vs ANN vs Numerical 0 300 300 300 0.0 0.0 500 374 359 375 0.2 4.4 1000 429 421 450 4.7 6.5 1500 494 488 525 6.0 7.1 2000 570 562 600 5.1 6.4 2500 660 644 696 5.1 7.5 3000 764 735 780 2.0 5.8 3500 881 836 890 1.0 6.1 4000 1009 947 1000 0.9 5.3 4500 1146 1072 1105 3.7 3.0 5000 1286 1202 1230 4.5 2.3 Average 3.3 5.4 * Numerical data was obtained from the plot in Pourafshary et al. (2009) using a web plot digitizer 66 Figure 8-6. Black-oil case: Comparison of wellbore models developed in with Pourafshary et al. (2009) The fluid composition used as an input for the ANN is slightly modified from the numerical system as it can be observed in Table 8-17. The Heptane and Octane components are lumped together and represented as C6+. Also, the Ethane and C20+ mole fraction is taken to be 10-6 to account for the absence of these components in method compatible with the ANN input. Accordingly, to compensate for these changes the mole fraction of the C6+ component has been reduced by 2×10-6. Such a scheme is followed in cases 3 through 5 as well. It can be observed in Table 8 and Figure 7 that the pressures profile described by the numerical simulator and ANN show an average relative deviation of 5.4% and 3.3% from the values provided by Pourafshary et al. (2009). Case 3: A wellbore with volatile oil fluid system was simulated and compared with literature data. The molar flow rate from this system was provided to be 1,500 lb-moles/day. The well description and the fluid composition are provided in Tables 8-19 and 8-20, respectively. 67 Table 8-19. Case 3: Wellbore description Total Depth (ft) 8000 Grid spacing (ft) 250 Wellhead Pressure (psia) 1000 Tubing diameter (inches) 1.5 Wellhead Temperature (°F) 150 Well Bottom-hole Temperature (°F) 195 Pipe roughness (ft) 6E-4 Table 8-20. Case 3: Fluid composition input Pourafshary et al. (2009) and ANN model current work-Numerical model Component Mole fraction Component Mole fraction Methane 0.55 Methane 0.55 Propane 0.1 Ethane 1E-6 i-Butane 0.1 Propane 0.1 i-Pentane 0.1 i-Butane 0.1 Heptane 0.075 i-Pentane 0.1 Octane 0.075 C6+ 0.15 -2E-6 C20+ 1E-6 Table 8-21. Case 3: Results- comparison for Volatile-oil fluid Pressure (psia) Relative deviation (%) Depth (ft) Pourafshary et Pourafshary et al. (2009) vs Pourafshary et al. (2009) ANN Numerical al. (2009) * ANN vs Numerical 0 1000 1000 1000 0.0 0.0 800 1066 1103 1115 4.4 1.1 1600 1162 1209 1239 6.2 2.4 2400 1264 1319 1354 6.7 2.6 3200 1372 1433 1475 7.0 2.8 4000 1488 1553 1602 7.1 3.1 4800 1612 1677 1723 6.4 2.6 5600 1746 1808 1879 7.1 3.8 6400 1891 1944 1998 5.4 2.7 7200 2048 2086 2123 3.5 1.7 8000 2218 2234 2230 0.5 0.2 Average 5.4 2.3 * Numerical data was obtained from the plot in Pourafshary et al. (2009) using a web plot digitizer For the volatile-oil case, the average relative deviation between the bottom-hole pressures predicted by the ANN and numerical models in comparison to Pourafshary et al. (2009) is 5.4% and 2.3%, respectively. This can be observed in Table 8-22 and Figure 8-7, respectively. The 68 protocols used for obtaining black-oil production and fluid properties is same as that for Case 2. Similarly, the hydrocarbon compositions are adapted for the ANN input as described in Case 2. Figure 8-7. Volatile-oil case: Comparison of wellbore models developed with Pourafshary et al. (2009) Case 4: A gas condensate producing wellbore system was simulated using the ANN and numerical simulators and compared with Pourafshary et al. (2009). The molar flow rate from this system was provided to be 1,500 lb-moles/day. The well description and the fluid composition are provided in Tables 8-22 and 8-23, respectively. Table 8-22. Case 4: Wellbore description Total Depth (ft) 12000 Grid spacing (ft) 250 Wellhead Pressure (psia) 1500 Tubing diameter (inches) 1.5 Wellhead Temperature (°F) 90 Well Bottom-hole Temperature (°F) 170 Pipe roughness (ft) 6E-4 69 Table 8-23. Case 4: Fluid composition input Pourafshary et al. (2009) and ANN model current work-Numerical model Component Mole fraction Component Mole fraction Methane 0.8 Methane 0.8 Propane 0.04 Ethane 1E-6 i-Butane 0.04 Propane 0.04 i-Pentane 0.04 i-Butane 0.04 Heptane 0.04 i-Pentane 0.04 Octane 0.04 C6+ 0.08 -2E-6 C20+ 1E-6 Table 8-24. Case 4: Results- comparison for gas condensate fluid Pressure (psia) Relative deviation (%) Depth (ft) Pourafshary et Pourafshary et al. (2009) Pourafshary et al. (2009) ANN Numerical al. (2009) * vs ANN vs Numerical 0 1500 1500 1500 0.0 0.0 1200 1575 1654 1657 5.0 0.2 2400 1706 1808 1804 5.4 0.2 3600 1849 1964 1998 7.4 1.7 4800 2004 2124 2198 8.8 3.3 6000 2172 2289 2404 9.7 4.8 7200 2354 2460 2601 9.5 5.4 8400 2552 2636 2803 8.9 5.9 9600 2766 2819 3001 7.8 6.1 10800 2997 3009 3118 3.9 3.5 12000 3244 3213 3209 1.1 0.1 Average 6.7 3.1 * Numerical data was obtained from the plot in Pourafshary et al. (2009) using a web plot digitizer From Table 8-24 and Figure 8-8, it can be noted that the average deviation of the ANN and the numerical wellbore models developed in this work with respect to Pourafshary et al. (2009) gas condensate case is 6.8% and 3.2%, respectively. 70 Figure 8-8. Gas condensate case: Comparison of wellbore models developed in the current work with the model by Pourafshary et al. (2009) Case 5: After performing comparisons with various two-phase cases, the wellbore models were compared with a three-phase system from the literature. The wellbore description and the fluid compositions has been provided in Tables 8-25 and 8-26 respectively. Table 8-25. Case 5: Wellbore description Total Depth (ft) 10000 Grid spacing (ft) 250 Wellhead Pressure (psia) 500 Tubing diameter (inches) 1.5 Wellhead Temperature (°F) 90 Well Bottom-hole Temperature (°F) 170 Pipe roughness (ft) 6E-4 Hydrocarbon flow rate (lb-moles/day) 2500 Water flow rate (STB/day) 300 71 Table 8-26. Case 5: Hydrocarbon composition input Pourafshary et al. (2009) and ANN model current work-Numerical model Component Mole fraction Component Mole fraction Methane 0.78 Methane 0.78 Propane 0.08 Ethane 1E-6 i-Butane 0.05 Propane 0.08 i-Pentane 0.05 i-Butane 0.05 Heptane 0.02 i-Pentane 0.05 Octane 0.02 C6+ 0.04 -2E-6 C20+ 1E-6 Table 8-27. Case 5: Results- comparison for three-phase system Pressure (psia) Relative deviation (%) Depth (ft) Pourafshary et Pourafshary et al. (2009) Pourafshary et al. (2009) ANN Numerical al. (2009) * vs ANN vs Numerical 0 500 500 500 0.0 0.0 1000 780 707 653 19.4 8.2 2000 915 895 810 13.0 10.5 3000 1063 1078 983 8.1 9.7 4000 1222 1261 1145 6.7 10.1 5000 1390 1449 1300 6.9 11.4 6000 1569 1642 1508 4.1 8.9 7000 1760 1839 1690 4.1 8.8 8000 1965 2035 1899 3.5 7.2 9000 2187 2218 2122 3.1 4.5 10000 2431 2414 2379 2.2 1.5 Average 7.1 8.1 * Numerical data was obtained from the plot in Pourafshary et al. (2009) using a web plot digitizer Figure 8-9. Three-phase flow case: Comparison of wellbore models developed in this work with Pourafshary et al. (2009) 72 From Table 8-27 and Figure 8-9, it can be observed that the ANN and numerical wellbore models show an average relative deviation of 7.1% and 8.1% respectively in comparison with Pourafshary et al. (2009) for a three-phase system. A summary of the relative deviations of the cases is displayed in Table 8-28. The average relative deviations of pressures predicted along the length of the wellbore for the cases recorded varies in the order of 2.2-7.1 % and 2.3-8.1% for the ANN and the numerical model, respectively. The relative deviation observed in the pressure readings at the deepest point in the well lies in the order of 0.5-4.5% and 0.1-2.5% for the ANN and numerical simulator, respectively. For a coupled reservoir-wellbore simulation, the pressure at the deepest point is of greater importance as this parameter is a boundary condition for the reservoir simulator. Table 8-28. Summary of the observed deviations for wellbore comparison cases 1 through 5 Wellbore Average relative deviation (%) Relative deviation at the deepest point (%) comparison case ANN Numerical ANN Numerical Case 1 2.2 2.4 1.4 2.5 Case 2 3.3 5.4 4.5 2.3 Case 3 5.4 2.3 0.5 0.2 Case 4 6.7 3.1 1.1 0.1 Case 5 7.1 8.1 2.2 1.5 It is to be noted that the numerical model developed by Pourafshary et al. (2009) has some differences when compared to the model developed in this work. Firstly, the current numerical model solves the system of equations containing the transient mass balance equations and mixture momentum balance equation simultaneously for all the well blocks, whereas the Pourafshary et al. (2009) model uses a marching algorithm, the compositions are first solved for using a flash calculation and mass balance equation, following this, the block pressure is calculated using the steady state mixture momentum equation and subsequently the block temperature is calculated by solving the energy balance equation. This procedure is applied block by block. Secondly, the 73 current work uses Hasan and Kabir (2007a) drift-flux model for determining the operating flow regime in order to obtain a relationship between phase velocities and hold-ups, whereas the Pourafshary et al. (2009) model uses the mechanistic models developed by Ansari et al. (1994) and Hasan and Kabir (2007b) for flow regime determination. Apart from these differences, the flash calculation protocol involved in this work is an equilibrium flash calculation, whereas the Pourafshary et al. uses an approach where a portion of the liquid phase is not in equilibrium with the gas phase due to slip velocity, during the flash calculation. These differences in the numerical models can account for the discrepancies observed. 8.3. Summary 1. A numerical compositional reservoir simulator which is isothermal and multiphase in formulation has been developed as described in chapter 5 and validated against a commercial simulator for conditions operating above and below bubble point pressure. The test results gave an average error of less than 1% for pressure comparisons and 2-5 % for flow rate comparisons, as summarized in Tables. 8-2 and 8-4. 2. The developed reservoir simulator has also been tested against an analytical model: well test analysis for both transient and pseudo steady state conditions and has provided an error of 0.1-2 %. As summarized in Tables. 8-6 and 8-8. A correspondingly similar comparison of the well test analysis with commercial simulator shows an error of 0.4- 2.8% as shown in Tables. 8-10 and 8-12, respectively. 3. A numerical transient wellbore simulator which is non-isothermal, compositional and multiphase in formulation has been developed as described in Chapter 6. This numerical wellbore simulator has been used to generate data for developing an ANN wellbore simulator as described in Chapter 7. Both the numerical and ANN wellbore simulators have been validated against sources from literature, including field data, a black-oil 74 model and also against another compositional numerical simulator. Each of these tests have contained a variety of data with well depths ranging from 5000 ft to 12000 ft, wellhead pressure varying from 500 psi to 1500 psi, well temperatures varying from 90 F to 230 F, fluid systems comprising of black-oil, volatile oil and gas-condensate systems with methane percentage varying from 30% to 80%. 4. Among these tests the average error of the ANN wellbore simulator was 0.5-7.1 % and the numerical wellbore simulator was 0.1- 8.1% as summarized in Table 8-24. 5. The methane mole percentages of the fluids in the volatile oil and the gas condensate case was 55% and 80% respectively. This is similar to the mixture fluid composition in a gas- lift well. Hence, it can be considered that the developed wellbore models are capable of simulating wellbore hydraulics in a gas-lift well. It has been shown that the proposed ANN tool is capable of speeding-up the solution process by four-orders of magnitude when compared against the numerical models developed in this work. It can be concluded that the ANN wellbore model can be coupled with the reservoir simulator for fast, robust and accurate coupled simulation results. This integrated simulation model involving numerical and ANN based approaches can been used for simulating and optimizing the performance of gas-lift wells over their life time. 75 Chapter 9 Coupled reservoir-wellbore simulator The standalone reservoir and wellbore models are coupled to produce a model in which the wellbore hydraulics affect the reservoir output and vice-versa. This coupling has been performed in two different ways: 1. Fully numerical model: The numerical reservoir model is coupled with the numerical wellbore hydraulics model 2. Coupled numerical-ANN model: The numerical reservoir model is coupled with the ANN based wellbore hydraulics model 9.1. Coupling mechanism for a fully numerical model The operating mechanism of the coupled model is described here: Initialization: The reservoir section of the model is initialized as defined in Section 5.5. In case of a fully numerical model, the wellbore section is initialized using the initial reservoir fluid composition as described in Section 6.6. The wellhead pressure and temperature, the temperature gradient of the wellbore, the initial reservoir pressure, the initial reservoir composition and the initial water saturation are specified as inputs. Boundary conditions: The reservoir is operated with a sandface pressure specification as defined in section 5.5 and the numerical wellbore model is operated in a wellhead pressure specification on the top and the molar flow rate specification at the bottom of the well. The initial pressure at the bottom of the well is considered to be the sandface pressure for the first iteration of the first time step of the reservoir simulator. This sandface pressure is calculated by assuming a gravitational gradient over the depth of a wellbore filled by a fluid whose composition is identical to that of the initial reservoir composition. 76 First reservoir simulator operation: The reservoir operates based on the initial sandface pressure specification. The reservoir gives a molar flow rate as an output in the first reservoir iteration in the first reservoir time step. First wellbore simulator operation: The obtained molar flow rate from the reservoir is provided as a source term for the wellbore simulator which now operates and runs until a time at which the pressure difference in the wellbore simulator is less than 0.05 psi between five consecutive time steps. This ensures that the well has reached a steady state. Coupled operation: Once steady state is achieved in the wellbore simulation, the wellbore pressure at a given depth pertaining to the reservoir sandface is recorded and is provided as an input for the next iteration of the reservoir simulator. The reservoir simulator uses the sandface pressure to generate the molar flow rate in the subsequent iteration, which is once again provided to the wellbore source term. Under unstable conditions, the time step size of the reservoir simulator is reduced and upon sustained instability, the sandface pressure of the previous time step is used. It is to be noted that the reservoir time step size is in days, while the wellbore time step size is in seconds. Using the procedure as described above, a coupling on an iteration level has been established between the wellbore and reservoir simulators which is not affected by the disparity in the time step sizes. The mole fraction of each component ci entering the wellbore from the reservoir is calculated by using: 푁푐 (푟푒푠푒푟푣표푖푟) 푧 (푤푒푙푙푏표푟푒) = 푖 (9.1) 푐푖 ∑푛푐 1 푁푐푖 where, nc is the total number of hydrocarbon components and Nci is the molar flow rate of each component ci. 77 It is to be noted that the molar flow rate of the source/sink term in the reservoir simulator is in (lbmoles/day) (RB/RCF), which is converted to lbmoles/s for the wellbore simulator by multiplying with a factor of 5.615/86,400 (RCF/RB) (days/s). 푙푏 − 푚표푙푒푠 푙푏 − 푚표푙푒푠 푅퐵 5.615 푅퐶퐹 푑푎푦푠 푁 (푤푒푙푙푏표푟푒) ( ) = 푁 (푟푒푠푒푟푣표푖푟) ( ) ( ) × ( ) ( ) (9.2) 푐1 푠 푐1 푑푎푦 푅퐶퐹 86400 푅퐵 푠 9.2. Coupling mechanism for a numerical reservoir-ANN based wellbore model The coupling mechanism for a numerical reservoir-ANN based wellbore model is similar to the fully numerical model described in Section 9.1. In a numerical reservoir-ANN based wellbore model, the wellbore hydraulics part of the coupling utilizes an ANN based model and hence there is no iterative routine involved. Initialization: The initialization of the reservoir system is similar to that of the fully numerical model. The ANN based wellbore model has no explicit initial step as it involves a non- iterative operating procedure. Boundary conditions: The reservoir system has a sand-face pressure specified boundary condition, which is obtained from the results predicted by the ANN based wellbore model. The ANN based wellbore model takes the molar flow rate and composition of the sink from the reservoir simulator as inputs. First iteration of the coupled system in initial state: As the reservoir has no flow under the initial conditions, while the ANN based wellbore simulator cannot operate under static conditions, the system can be initialized using two options. The first option involves assuming a small molar flow rate of 0.0001 lb-moles/s with a composition same as the initial reservoir composition entering the bottom of the wellbore from the formation. This acts as the flow rate specification for the ANN based wellbore hydraulics model. The ANN calculates the corresponding flowing bottom-hole pressure (as described in Section 7.2) which is used 78 by the reservoir simulator. Under unstable conditions, the time step size of the reservoir simulator is reduced and upon sustained instability, the sandface pressure of the previous time step is used. For the subsequent iterations, the molar composition and flow rate required for the ANN wellbore is obtained as an output from the reservoir simulator defined by Equations (9.1) and (9.2), respectively. 9.3. Comparison with a commercial simulator Both the fully numerical model and the numerical-ANN coupled model have been compared with CMG-GEMS ©*. In this commercial simulator, the reservoir simulator is based on a compositional formulation and the wellbore hydraulics model is based on the black-oil correlation developed by Aziz et al. (1972). 9.3.1. Single-phase liquid case Since the commercial simulator uses a wellbore model which is based on a black-oil formulation, to neutralize the effects of varying composition in a wellbore system, a comparison is performed with a single-phase liquid system. The inputs of the reservoir, wellbore and fluid system have been provided in Table 9-1. Table 9-1. Single phase liquid case: System inputs Reservoir outer radius (ft) 1000 Component Mole fraction Wellbore radius (ft) 1 Methane 0.2 Radial permeability (mD) 5 Ethane 0.01 Porosity 0.1 Propane 0.01 Formation thickness (ft) 60 i-Butane 0.01 Total depth (ft) 2000 i-Pentane 0.01 Wellhead pressure (psi) 900 C6+(C14) 0.2 Wellhead temperature (°F) 60 C20+(C38) 0.56 Reservoir temperature (°F) 100 Initial reservoir pressure (psia) 2500 Initial water saturation 0 ∗ CMG-GEMS© is a registered trademark of Computer Modelling Group Ltd. 79 The coupled simulation is run for a time of 205 days, until the oil flow rate drops to an abandonment rate of 10 STB/day. The cumulative oil produced is compared with the commercial simulator with the productions from the fully numerical model and the numerical-ANN coupled model. The comparison can be seen in Table 9-2 and Figure 9-1. Table 9-2. Single-phase liquid case: Results Time Cumulative oil production (MSTB) (days) Fully numerical Numerical-ANN coupled Commercial simulator 1 0.2 0.2 0.2 2 0.4 0.4 0.3 5 0.9 0.9 0.8 8 1.3 1.3 1.2 17 2.6 2.6 2.3 25 3.5 3.5 3.2 45 5.4 5.4 4.9 65 6.8 6.8 6.3 85 7.9 7.9 7.3 105 8.7 8.7 8.2 125 9.3 9.3 8.8 145 9.8 9.8 9.4 165 10.2 10.2 9.8 185 10.4 10.4 10.1 205 10.7 10.7 10.4 12.0 10.0 8.0 6.0 Fully numerical (MSTB) Num-ANN coupled 4.0 Commercial simulator 2.0 Cumulative Cumulative oilproduction 0.0 0 50 100 150 200 250 Time (days) Figure 9-1. Single-phase liquid case: Comparison with a commercial numerical simulator 80 The difference in the cumulative oil production at the end of the run time was observed to be 2.9 % between the fully numerical system and the commercial simulator and 2.8 % between the numerical-ANN coupled system and the commercial simulator, respectively. The difference between fully numerical and numerical-ANN coupled simulator is less than 0.1%. 9.3.2. Single-phase gas case After a single-phase liquid case, to test the coupled reservoir-wellbore system in the other extreme condition of phase behavior, a comparison is performed under single phase gas conditions. The inputs for the reservoir, wellbore and fluid system for this test are as provided in Table 9-3. Table 9-3. Single-phase gas case: System inputs Reservoir outer radius (ft) 1000 Component Mole fraction Wellbore radius (ft) 1 Methane 0.91 Radial permeability (mD) 5 Ethane 0.04 Porosity 0.1 Propane 0.01 Formation thickness (ft) 60 i-Butane 0.01 Total depth (ft) 2000 i-Pentane 0.01 Wellhead pressure (psi) 200 C6+(C14) 0.01 Wellhead temperature (°F) 80 C20+(C38) 0.01 Reservoir temperature (°F) 110 Initial reservoir pressure (psia) 1000 Initial water saturation 0 The results of the comparison are provided in the Table 9-4 and Figure 9-2. The difference between the fully numerical coupled simulator and the commercial simulator was observed to be 3.5% in terms of the total gas produced at the end of the simulation time. Whereas, when the numerical-ANN coupled reservoir-wellbore model is compared with the commercial simulator, the difference in the cumulative gas produced at the end of the simulation run was found to be 4.0%. The difference between fully numerical and numerical-ANN coupled simulator is 0.6%. 81 Table 9-4. Single-phase gas case: Results Time Cumulative gas production (MMSCF) Numerical -ANN Commercial (days) Fully numerical coupled simulator 1 6 6 5 5 25 25 23 11 51 52 49 25 109 110 103 40 163 165 154 45 182 184 172 65 246 248 233 85 303 305 288 105 354 356 338 125 400 403 383 145 441 444 424 165 479 482 461 185 514 517 495 205 545 548 527 600 500 400 300 200 Fully numerical 100 Num-ANN coupled Commercial simulator Cumulative Cumulative production gas (MMSCF) 0 0 50 100 150 200 250 Time (days) Figure 9-2. Single-phase gas case: Comparison with a commercial numerical simulator 82 Both the single-phase liquid and gas case have been compared with the commercial simulator using a fully-implicit formulation. It was observed that the commercial simulator had multiple reservoir time step cuts as compared to the numerical simulator in this work. The total number time steps taken by the fully numerical and the numerical-ANN coupled models for the single- phase liquid and gas cases was 20. For a given single phase simulation the commercial simulator took 28 reservoir time steps. It is also to be noted that in this work the reservoir and wellbore models have been coupled at a reservoir iteration level. The coupling scheme of the commercial simulator may differ from this. 9.3.3. Two-phase case Having tested the coupled reservoir-wellbore systems developed in this work with a commercial simulator for a single-phase oil and a single-phase gas case, respectively, the system was subsequently tested for a two-phase case. For this, a system is considered in which both the reservoir and wellbore are maintained below bubble point pressure throughout the simulation. The inputs for the reservoir, wellbore and fluid system for this case are provided in Table 9-5. Table 9-5. Two-phase case: System inputs Reservoir outer radius (ft) 1000 Component Mole fraction Wellbore radius (ft) 1 Methane 0.3 Radial permeability (mD) 5 Ethane 0.06 Porosity 0.1 Propane 0.02 Formation thickness (ft) 60 i-Butane 0.01 Total depth (ft) 2000 i-Pentane 0.01 Wellhead pressure (psi) 500 C6+(C14) 0.2 Wellhead temperature (°F) 120 C20+(C38) 0.4 Reservoir temperature (°F) 150 Initial reservoir pressure (psia) 2500 Initial water saturation 0 83 The results of the cumulative oil and gas production comparison from the two-phase case is summarized in Table 9-6, Figure 9-3 and Figure 9-4, respectively. Table 9-6. Two-phase case: Results Time Cumulative oil production (STB) Cumulative gas production (MMSCF) Fully Numerical-ANN Commercial Fully Numerical-ANN Commercial (days) numerical coupled simulator numerical coupled simulator 1 0.5 0.5 0.4 0.2 0.2 0.1 5 2.1 2.1 1.9 0.7 0.6 0.4 25 8.8 8.8 7.9 2.9 3.6 1.8 45 13.6 13.6 13.3 6.0 5.4 3.6 55 16.0 16.0 15.9 7.1 6.2 4.7 75 20.6 20.7 21.0 9.2 8.2 6.9 95 25.3 25.3 26.0 11.8 10.8 9.2 115 29.8 29.8 30.9 14.8 13.8 11.5 135 34.1 34.1 35.8 18.4 17.3 14.0 155 38.3 38.2 40.6 22.3 21.1 16.7 175 42.3 42.2 45.4 26.5 25.3 19.6 195 46.2 46.0 50.2 31.0 29.7 22.8 215 49.9 49.7 55.0 35.7 34.3 26.3 60.0 50.0 40.0 30.0 Fully numerical Num-ANN coupled 20.0 Commercial simulator 10.0 Cumulative Cumulative oilproduction (MSTB) 0.0 0 50 100 150 200 250 Time (days) Figure 9-3. Two-phase case: Comparison of cumulative oil production with commercial numerical simulator 84 40.0 35.0 30.0 25.0 20.0 15.0 Fully numerical 10.0 Num-ANN coupled 5.0 Commercial Cumulative Cumulative production gas (MMSCF) simulator 0.0 0 50 100 150 200 250 Time (days) Figure 9-4. Two-phase case: Comparison of cumulative gas production with commercial numerical simulator It can be observed from Table 9-6 that the difference between the cumulative oil production predicted at the end of simulation time by the fully numerical simulator and commercial simulator is 9.1%. The difference between the cumulative oil production predicted between numerical- ANN coupled model and commercial simulator is 9.6%. For the cumulative gas prediction comparison, the difference between the fully numerical and commercial simulator is 35.6%, while the difference between the numerical-ANN coupled simulator and the commercial simulator is 30.2%. The difference between the fully numerical simulator and the numerical- ANN coupled model is 0.5% and 4% for cumulative oil and gas production respectively. The differences observed in the results predicted by the coupled models developed in this work and the commercial simulator may primarily be due to the black-oil formulation of the commercial simulator’s wellbore model. Considering that the standalone numerical reservoir simulator has shown better agreement with the same commercial simulator as shown in section 8.1, this reason seems more probable. In a compositional wellbore model, phase separation is accompanied with transfer of lighter components into the gaseous phase. Further, with the effect 85 of slippage, the resultant composition of the liquid and gas phases within a wellbore is significantly different from a black-oil based wellbore hydraulics system (Pourafshary,2007). It is expected that the lower oil production and higher gas production of the coupled models developed in this work when compared to the commercial simulator is due to these compositional wellbore hydraulics effects. Having observed a fair agreement with the commercial simulator for a coupled system under single phase liquid and single-phase gas flow respectively, the coupled model models developed in this work have been considered to be validated. Hence, the integrated gas lift simulation and optimization using the developed tools is taken up as the next step. 86 Chapter 10 Integrated gas lift simulation and optimization As it has been shown in literature survey (Section 2.2), currently there is no known work which considers the gas lift optimization problem in terms of total oil production from the well over its production life. The aim of this work is to study the gas lift optimization model from this particular perspective. For this purpose, three models of tools have been developed for gas lift optimization: 10.1. Model 1- Fully numerical system In this model, the fully numerical reservoir-wellbore simulator is used for representing the performance of a gas lift well along with its associated reservoir. The total oil production obtained as a function of varying gas lift parameters is studied. 10.1.1. Implementation of gas-lift in the wellbore and modification of boundary conditions The implementation of gas-lift in the numerical wellbore model section of the coupled model is performed by dividing the parts above and below the gas injection port into two separate simulations. For the top section containing the part above the operating gas-lift valve, for a specified quantity of gas injection flow rate specified at standard conditions, the molar flow rate of injection is calculated as: 푆퐶퐹 푙푏 푞 ( ) × 휌 ( ) 푙푏 − 푚표푙푒푠 푔푖푛푗 푑푎푦 푔푖푛푗 푆퐶퐹 푀표푙푎푟 푔푎푠 푙푖푓푡 푖푛푗푒푐푡푖표푛 푟푎푡푒 푁 ( ) = (10.1) 푔푖푛푗 푠 푙푏 푠 86400 ( ) × 푀 ( ) 푑푎푦 푔푖푛푗 푙푏 − 푚표푙푒푠 The density and the molecular weight of the injection gas are obtained by using the Peng- Robinson Equation of State (PREOS) (Robinson et. al., 1978). The total molar feed Ntot into the section above the point where the gas lift is injected is given by: 87 푙푏 − 푚표푙푒푠 푙푏 − 푚표푙푒푠 푙푏 − 푚표푙푒푠 푙푏 − 푚표푙푒푠 푁 ( ) = 푁 ( ) + 푁 ( ) + 푁 ( ) (10.2) 푡표푡 푠 푔푖푛푗 푠 푤 푠 푐 푠 푀표푙푒 푓푟푎푐푡푖표푛 표푓 푎 푔푖푣푒푛 푐표푚푝표푛푒푛푡 푐1 푖푛 푡ℎ푒 푡표푡푎푙 푓푒푒푑 푖푛푡표 푡ℎ푒 푤푒푙푙푏표푟푒, (푁푐 × 푧푐1) + (푁푔푖푛푗 × 푧푔푖푛푗푐1) 푧푐1 = (10.3) (푁푡표푡 − 푁푤푎푡푒푟) The temperature at the gas lift injection point is obtained through the specified temperature gradient of the well. The bottom-hole pressure predicted by the top section and the temperature at the bottom of that section is used as the wellhead pressure and wellhead temperature for the bottom section of the well. The molar flow rate and the composition of the feed from the reservoir is used for the bottom section. The start time of the gas lift injection, the depth of gas lift injection and the gas lift injection rate is specified by the user. 10.2. Model 2- Coupled numerical-ANN system In this model, the coupled numerical reservoir- ANN based wellbore simulator is used for simulating the performance of a gas lift well along with its associated reservoir. The total oil production of the well obtained with different gas lift parameters is recorded and analyzed. For the implementation of gas lift, the wellbore system is split into two portions. For the section of the well above the gas lift injection point, the ANN-1B from the wellbore ANN module is used for all wells which have a depth greater than 2,200 ft. ANN-2B through ANN-5B have been developed specifically to predict the pressure for the portion of the well below the point of gas lift injection in cases where the gas lift injection takes place at a height between 200 ft to 2,200 ft above the perforation. In cases where the difference in true vertical depth of the point of gas lift injection from the perforation is more than 2,200 ft, ANN-1B is used to predict the pressure at the section of the well below the gas lift injection point. 88 The bottom-hole pressure predicted by the ANN depicting the top section of the well (section above gas lift injection point) acts as the well head pressure specification for the ANN representing the section of the well below the gas lift injection point. The molar flow rates, gas compositions and boundary conditions of Model 2 is same as Model 1 as described in section 10.1.1. The gas lift injection rate, depth and start time are specified by the user. Model 2 has been developed for a faster simulation compared to Model 1 as it is described in the case studies section. 10.3. Model 3- Fully ANN based system In this model, an ANN has been developed for predicting fast and effective results for determining the total oil production of the well with variable gas lift parameters. The data for training, validation and blind testing is generated using Model 2. The ANN model reservoir, wellbore, fluid PVT properties and gas lift parameters as an input and predicts the oil recovery factor as an output. Since, the original oil in place for a given reservoir is constant, the total oil produced from the well, Qo is obtained using equation (10.4): 푄표(푆푇퐵) = 푂푖푙 푟푒푐표푣푒푟푦 푓푎푐푡표푟 × 푂푟푖푔푖푛푎푙 표푖푙 푖푛 푝푙푎푐푒(푆푇퐵) (10.4) 10.3.1. Data generation and pre-processing for the ANN based gas lift model (GL-ANN) The total number of inputs of the ANN based gas lift model are 66 and there is one output predicted. The training data for the ANN is generated by selecting values between a fixed range for each parameter using a random number generator. The details of the input parameters and their respective ranges has been displayed in the Table 10-1.The reservoir external radius, the wellbore radius, the reservoir porosity, radial permeability, formation thickness, initial reservoir pressure, reservoir temperature, wellhead specified pressure, wellhead temperature, total depth of the well, tubing internal diameter, pipe roughness, gas lift injection depth, gas lift start time, gas 89 lift injection rate , the molar composition of the reservoir fluid and the gas lift fluid are provided as inputs to the system. The hydrocarbon components involved in the GL-ANN are similar to the ANN based wellbore model. The mole fraction of the C6+ and C20+ components in the lift gas have been set to 1× 10-6, this is to ensure that no heavy oil components are present in the lift gas. The compositional fluid mole fractions for the reservoir and the lift gas were variably generated through a random number generator such that the sum of the mole fractions equals one. The corresponding black-oil production and volumetric parameters were calculated using a flash calculation based on Peng-Robinson Equation of State (Robinson et al., 1978). The range of the reservoir parameters has been based on an assumption that the drainage area of the reservoir is only influenced by a single well. The range of the drainage area corresponding to the reservoir boundaries defined in Table 10-1 are 43 acres to 103 acres. The petrophysical property ranges of the reservoir are chosen based on the characteristics of a typical conventional oil reservoir. 90 Table 10-1. Summary of inputs for the ANN based gas lift model (GL-ANN) Category Parameter Unit Min Max Reservoir external boundary radius ft 800 1200 Well radius ft 0.5 1.25 Initial reservoir pressure psi 1390 7906 Bubble point pressure of reservoir psi 101 4991 Permeability in the radial direction mD 2 15 Formation thickness ft 10 100 Reservoir geometry Porosity fraction 0.03 0.15 Total depth of the well ft 4000 14000 ry t Tubing internal diameter in 1 4 Wellhead pressure psi 100 1000 Wellhead temperature °F 60 140 Well bottom-hole /Reservoir temperature °F 83 414 Wellboregeome Pipe roughness ft 1.00E-04 2.00E-03 Initial water saturation fraction 0 0.7 Initial oil saturation fraction 0 1 Initial gas saturation fraction 0 0.9 Initial oil density (reservoir conditions) lb/cubic ft 17.2 55.8 Initial gas density (reservoir conditions) lb/cubic ft 4.7 55.8 Initial water density (reservoir conditions) lb/cubic ft 61.8 64.9 Initial oil viscosity at standard conditions cp 0.65 2.2 Initial gas viscosity at standard conditions cp 0.01 0.32 Initial water viscosity at standard conditions cp 0.5 1.3 API gravity of oil- initial reservoir conditions lb/cubic ft 19.7 45 Gas gravity - initial reservoir conditions 1/psi 0.62 2.88 Reservoirproperties fluid Oil density- Standard conditions lb/cubic ft 50 58.4 Oil compressibility-Initial conditions SCF/STB 2.10E-09 1.00E-05 Original oil in place (OOIP) MSTB 15 7523 Original gas in place (OGIP) MMSCF 0 11919 Original water in place (OWIP) MSTB 1 42562 Gas injection depth as a fraction of total depth fraction 0.83 0.99 Gas lift injection rate MSCF/d 0 3000 Density of the injected gas lb/cubic ft 0.029 0.068 Start time of gas lift injection days 0 5847 Lift gas -global mole fraction of C1 fraction 0.77 0.92 Lift gas -global mole fraction of C2 fraction 0 0.2 Lift gas -global mole fraction of C3 fraction 0 0.2 Gas lift Gas properties Lift gas -global mole fraction of C4 fraction 0 0.2 Lift gas -global mole fraction of C5 fraction 0 0.19 91 Table 10-1. Summary of inputs for the ANN based gas lift model (GL-ANN) (continued) Category Parameter Unit Min Max Initial feed into well -global mole fraction of C1 fraction 0 0.8 Initial feed into well -global mole fraction of C2 fraction 0 0.73 Initial feed into well -global mole fraction of C3 fraction 0 0.64 Initial feed into well -global mole fraction of C4 fraction 0 0.61 properties Initial feed into well -global mole fraction of C5 fraction 0 0.71 Initial feed well Initial feed into well -global mole fraction of C6+ fraction 0 0.63 Initial feed into well -global mole fraction of C20+ fraction 0 0.73 Critical Temperature of C6+ °R 913.5 1409.2 Critical Temperature of C20+ °R 1427.9 1724 Critical Pressure of C6+ psi 211.1 477.2 Critical Pressure of C20+ psi 105 203.3 Accentricity factor of C6+ unitless 0.28 0.82 Accentricity factor of C20+ unitless 0.86 1.33 Molecular weight of C6+ lb/lb-mole 86 275 Molecular weight of C20+ lb/lb-mole 291 539 Volume shift parameter of C6+ unitless -0.06 0.14 Volume shift parameter of C20+ unitless 0.14 0.36 Critical volume of C6+ cubic ft/lb 5.5 16.5 Critical volume of C20+ cubic ft/lb 17.2 31.3 Parachor of C6+ unitless 250.1 710.5 Thermodynamic properties of C6+ and C6+ ThermodynamicC20+ properties of Parachor of C20+ unitless 742.2 1090.4 Geometric transmissibility: Radial permeability× unitless 3.1 222.8 formation thickness/log (reservoir radius/well radius) Oil Transmissibility: Initial oil density / (Initial oil lb-moles/ (cubic 0.1 1.7 viscosity × Initial oil molecular weight) ft-cp) OGIP/OOIP SCF/STB 0.1 95120 OGIP/(OOIP+OWIP) SCF/STB 0 21008 mixture fluid density - initial reservoir conditions lb/cubic ft 0.8 63.3 Net pressure drawdown -Initial conditions psi 0 6738 Secondary parameters Net initial pressure drawdown × geometric MSTB/d 0 1048 transmissibility ×Fluid transmissibility For each case, the coupled reservoir-wellbore system (Model 2) was run with a provided gas lift depth, injection rate and start time, respectively. This simulation is run until an abandonment condition is reached. The abandonment condition for this study has been defined as state when the oil production rate below 10 STB/day has be achieved. At this condition, the program is 92 terminated and the total oil produced from the well is recorded. The gas lift specifications are kept constant throughout the run in a specified case. The fluid flow model in this study includes both multi-phase flow in porous media and in a wellbore, the governing equations associated with the flow processes differ as explained in chapter 5 and chapter 6, respectively. To account for these complexities, a set of secondary input parameters have been provided to the ANN. These include: 3. Geometric transmissibility factor of the reservoir system: This parameter represents the contribution of the reservoir radial permeability, the reservoir thickness, the reservoir external radius and the wellbore radius towards the productivity index of the well. This parameter has been defined as: 푅푎푑푖푎푙 푝푒푟푚푒푎푏푖푙푖푡푦(푚퐷) × 푛푒푡 푝푎푦(푓푡) 퐺푒표푚푒푡푟푖푐 푡푟푎푛푠푚푖푠푠푖푏푖푙푡푦 푓푎푐푡표푟 = (10.5) 푟푒푠푒푟푣표푖푟 푒푥푡푒푟푛푎푙 푟푎푑푖푢푠(푓푡) log ( ) 10 푤푒푙푙푏표푟푒 푟푎푑푖푢푠(푓푡) 4. Oil transmissibility: This parameter takes into account, the contribution of the fluid properties to the well productivity index. The fluid properties considered as an input for the ANN pertain to the initial reservoir conditions. The oil transmissibility is defined as: 푙푏 표푖푙 푑푒푛푠푖푡푦 ( ) 푓푡3 푂푖푙 푡푟푎푛푠푚푖푠푖푏푖푙푖푡푦 푓푎푐푡표푟 = (10.6) 푙푏푚표푙푒푠 표푖푙 푣푖푠푐표푠푖푡푦 (푐푝) × 표푖푙 푚표푙푒푐푢푙푎푟 푤푒푖푔ℎ푡 ( ) 푙푏 5. Ratio of the original gas in place to the original oil in place: The user may or may not have any production data available at the initiation of the coupled reservoir-wellbore system, the ratio of the original gas in place to the original oil in place is taken as an input parameter to provide as estimate of the gas-oil ratio of the system. 6. Ratio of the original gas in place to the original liquid in place: This parameter is included to take into account, the effect of the water present in the system and an estimate of the 93 gas-liquid ratio of the reservoir. These parameters can provide a weightage to the fluid properties in terms of whether the mixture fluid in the reservoir will behave more similar to gas, oil or water phase. Also, a system with greater gas-oil/ gas-liquid ratio may require less gas lift support as the production of the fluid may be driven primarily by the relatively low density and low viscosity of the reservoir fluids. 7. The net pressure drawdown in the system: The fluid production of the system can be considered to be dependent on two primary parameters: The transmissibility and the pressure drop across the system. Since the flowing bottom-hole pressure of the system under initial and final conditions is unknown to the user, the hydrostatic pressure drop across the system is calculated using mixture density, total depth of the well, wellhead pressure. 푙푏 푓푡3 [휌 ( ) × (푂푂퐼푃 + 푂푊퐼푃)(푆푇퐵) × 5.615 ( )] 푙푏 푙푠푐 푓푡3 푆푇퐵 푀푖푥푡푢푟푒 푑푒푛푠푖푡푦, 휌푚푖푥 ( ) = (10.7) 푓푡3 푙푏 푆퐶퐹 + [휌푔푠푐 ( ) × 푂퐺퐼푃 ( )] { 푓푡3 푆푇퐵 } 푊푒푙푙 푡표푡푎푙 푑푒푝푡ℎ 푁푒푡 푝푟푒푠푠푢푟푒 푑푟푎푤푑표푤푛 (푝푠푖) = 푃 − (푃 + 휌 × ) (10.8) 푖 푠푝 푚푖푥 144 8. Total oil productivity: The product of the net pressure drawdown to the geometric and oil transmissibilities gives us the total oil productivity. This parameter provides an estimate of the system’s total oil production potential. 푛푒푡 푝푟푒푠푠푢푟푒 푑푟푎푤푑표푤푛 (푝푠푖) 푆푇퐵 × 푔푒표푚푒푡푟푖푐 푡푟푎푛푠푚푖푠푠푖푏푖푙푖푡푦 푇표푡푎푙 표푖푙 푝푟표푑푢푐푡푖푣푖푡푦 ( ) = (10.9) 푑푎푦 푙푏 − 푚표푙푒푠 × 표푖푙 푡푟푎푛푠푚푖푠푠푖푏푙푖푡푦 ( ) { 푓푡3 − 푐푝 } The above described input parameters were subjected to pre-processing. All the parameters with ranges greater than the third order were subjected to a natural logarithm and subsequently, all the parameters were normalized between the values 0 to 1 prior to training. 94 10.3.2. Architecture selection and training of the GL-ANN Based on the scheme mentioned in section 10.3.1, a total of 9,000 cases were generated using the gas lift Model 2. This data was divided in a ratio of 80:10:10 for training, validation and blind testing of the GL-ANN. The training has been performed using the scaled conjugate gradient. The number of epochs was set to 10,000 and the validation fail parameter is set to 1,000 counts. The error from each epoch was reduced by updating the neuron weights using a back-propagation method. The training was stopped if either the maximum number of epochs are reached or the validation fail count is approached. A MATLAB©* script had been developed to train, validate and test feedforward ANNs with various architectures and to select the model that provides the least error on blind testing data similar to the ANN based wellbore model. The scheme of this ANN training and selection script is similar to that described by Sun-Ertekin (2015). The details of the architecture and performance of the ANN is described in Table 10-2. Table 10-2. Architecture description and blind testing results of the GL-ANN Number of hidden layers 3 Neurons per hidden layer [14.9,9] Transfer functions Tansig,Logsig,Tansig Number of test samples 1440 mean error (%) 15.9 R-squared value 0.93 ∗ MATLAB© is a registered trademark of The MathWorks, Inc 95 Histogram of error counts: GL-ANN 500 450 400 350 s t n u 300 o c f o 250 r e b m 200 u N 150 100 50 0 0 5 10 15 20 25 30 35 40 45 50 absolute percentage error (%) Figure 10-1. Histogram of error counts for the ANN based gas lift model Figure 10-2. Cross plots of prediction versus target oil recovery factor for the ANN based gas lift model A histogram of the error counts can be seen in Figure 10-1 and a cross plot between the predicted and the target oil recovery factors can be observed in Figure 10-2. The blind testing 96 mean error for this ANN model is 15.9 % and the R-squared value for the cross plot between predicted and target oil recovery factors is 0.93. The outliers causing a higher than mean error as displayed in Figure 10-1 and Figure 10-2, respectively are pertaining to systems with light oils and high initial gas oil ratio. Further description about cases with high errors is explained in Section 10.4.2. It is to be noted that ANNs are most accurate in the range of variables under which they are trained and the accuracy may not be reliable beyond the range of training. 10.4. Comparisons of the gas lift models The three models developed for gas lift have the same purpose. The Model 2 and Model 3 have been developed to enhance the speed of solution. In this section, the results predicted and the capabilities of the above-mentioned models is tested. 10.4.1. Case GL1: Performance comparison between all three gas lift models A gas lift system has been tested under variable gas lift injection rates using the three different models described in sections 10.1 through 10.3. Gas lift is introduced in a well at the initial time of the production of a well and sustained till abandonment time. The hydrocarbon molar composition of the reservoir-wellbore system in the initial conditions and that of the lift gas have been summarized in Table 10-3. The input reservoir and well properties have been summarized in Table 10-4. 97 Table 10-3. Case GL1-Fluid molar composition for comparison study between gas lift models 1-3 Molar composition of hydrocarbon components Initial reservoir-wellbore fluid Lift gas Methane 0.399 Methane 0.910 Ethane 0.101 Ethane 0.012 Propane 0.104 Propane 0.030 i-Butane 0.016 i-Butane 0.038 i-Pentane 0.190 i-Pentane 0.010 C6+(C18) 0.045 C6+ 1.00E-06 C20+(C37) 0.145 C20+ 1.00E-06 Table 10-4. Case GL1- reservoir and well inputs for comparison study between gas lift models 1-3 Reservoir external radius (ft) 1068 Wellbore radius (ft) 1.03 Radial permeability (mD) 11.16 Porosity 0.101 Formation thickness (ft) 44 Total Depth (ft) 4000 Gas injection depth (ft) 3600 Wellhead pressure (psi) 778 Wellhead temperature (°F) 61 Reservoir Temperature (°F) 2424 Initial Reservoir Pressure (psia) 3465 Initial Water saturation 0.236 Lift gas gravity 0.664 Gas lift start time (days) 0.2 Original oil in place (MMSTB) 1.24 Original gas in place (BCF) 1.49 Original water in place (MMSTB) 3.76 The gas lift injection rate is varied from 100 MSCFD to 3 MMSCFD and the total oil production of the system from initiation till abandonment is recorded and summarized in Table 10-5. It can be noted from Table 10-5, that the total oil production predicted by the Model 2: coupled numerical-ANN model is showing an average difference of 5.07 % when compared to 98 the Model 1: fully numerical system. The Model 3: the full ANN based gas lift system shows an average difference of 1.40% when compared to Model 1: the fully numerical system. In comparison between the Model 2: coupled numerical-ANN model and Model 3: fully ANN based model there is an average difference of 6.1%. Table 10-5. Case GL1-comparison of predicted total oil produced by the system by gas lift models 1-3 Total oil produced Difference Difference Lift gas Fully Coupled Fully Fully injection Fully ANN Numerical Num-ANN numerical vs numerical vs rate Num-ANN Fully ANN MSCFD MSTB MSTB % MSTB % 100 60.23 60.2 0.04 59.99 0.40 500 68.04 70.6 3.76 66.72 1.94 1000 71.27 73.92 3.73 69.82 2.03 1300 71.89 74.85 4.13 70.48 1.96 1500 72.15 75.27 4.33 70.71 1.99 1600 72.23 75.37 4.35 70.79 2.00 1700 72.23 75.64 4.72 70.85 1.92 1800 72.25 75.74 4.82 70.89 1.88 1900 72.05 76.00 5.48 70.92 1.57 2000 71.96 76.02 5.64 70.93 1.43 2100 71.92 76.12 5.85 70.94 1.36 2200 71.81 76.21 6.13 70.94 1.21 2600 71.38 76.38 7.00 70.87 0.72 2800 71.27 76.75 7.69 70.81 0.65 3000 70.71 76.67 8.43 70.72 0.01 Average 5.07 1.40 A gas lift performance curve predicted by Models 1-3 has been plotted in Figure 10-3(a) and the performance of the well in response to a unit change gas lift injection has been shown in Table 10-6. The optimum quantity of the gas lift is dependent upon the definition of the user. It can be observed in Figure 10-3(b) and in Table 10-6 that the incremental gain in oil production 99 per unit gas lift injection is highest at low gas lift injection rates, while it declines at higher gas lift injection rates. 80.00 75.00 70.00 65.00 Fully Num 60.00 Num-ANN Coupled Totaloil produced (MSTB) 55.00 Fully ANN 50.00 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-3(a). Gas lift performance curve predicted by models 1-3 for case GL1 30.0 25.0 20.0 Fully num Num-ANN coupled 15.0 Fully ANN 10.0 5.0 injectionrate (STB/MSCFD) 0.0 Incremental Incremental oilproduced per unit lift gas 500 1000 1500 2000 2500 3000 3500 -5.0 Gas lift injection rate (MSCFD) Figure 10-3(b). Derivative of the gas lift performance curve predicted by models 1-3 for case GL1 100 Table 10-6. Case GL1-comparison of incremental total oil produced per unit change in gas lift injection rate Incremental oil produced per unit incremental gas injection Coupled Fully Lift gas injection rate Numerical- Fully ANN Numerical ANN MSCFD STB/MSCFD STB/MSCFD STB/MSCFD 500 19.5 26.0 16.8 1000 6.4 6.6 6.2 1300 2.1 3.1 2.2 1500 1.3 2.1 1.2 1600 0.9 1.1 0.8 1700 0.0 2.7 0.6 1800 0.2 0.9 0.4 1900 -2.0 2.6 0.3 2000 -0.9 0.3 0.2 2100 -0.5 1.0 0.1 2200 -1.1 0.8 0.0 2600 -1.1 0.4 -0.2 2800 -0.6 1.8 -0.3 3000 -2.8 -0.4 -0.5 From Figure 10-3(b) and Table 10-6, it is observed that the incremental gain in oil per unit increment of gas injection at 1500 MSCFD of gas lift injection rate is 1.3 STB/MSCFD, 2.1 MSCFD and 1.2 MSCFD for the fully numerical, coupled numerical-ANN and the fully ANN based models, respectively. The corresponding total oil produced at 1500 MSCFD of gas lift injection rate is 72.15 MSTB, 75.27 MSTB and 70.71 MSTB as predicted by the fully numerical, coupled numerical-ANN and the fully ANN based models, respectively. It can be observed from Table 10-5 that the difference in predicted total oil production provided by the three models in study is between 2-6 % at a gas lift injection rate of 1500 MSCFD. The point where the incremental gain in oil per unit increment in gas is zero according to the prediction of the fully numerical model is 1700 MSCFD of gas lift injection rate. The gain in oil between 1500 MSCFD 101 and 1700 MSCFD of gas lift injection is 80 STB, 370 STB, and 140 STB corresponding to an incremental gas usage of 200 MSCFD over a period of 527 days. Hence, the operator can take a decision based on the economics of operation to establish the rate of gas lift injection. Above the gas lift injection rate of 1500 MSCFD as the incremental gain in oil per unit increment of gas lift injection is only decreasing, the gas lift injection rate of 1500 MSCFD can be considered to be potentially optimal, with a margin of 200 MSCFD. One of the primary purposes for which different models were developed as a part of this work was to obtain improvement in speed of solution from each subsequent model. In Table 10-7, a time comparison has been shown for the case GL1 using the three different models. Table 10-7. Case GL1: Computational time comparison for models 1-3 Computational time per day of simulation (s) Gas lift Well Fully Numerical- Injection abandonment Numerical- ANN Coupled - Fully ANN- Model 3 rate(MSCFD) time(days) Model 1 Model 2 100 427.29 381.9 2.8 1.98E-04 500 497.29 501.0 2.7 1.98E-04 1000 547.29 469.8 2.4 1.98E-04 1500 527.29 476.5 2.9 1.98E-04 1700 527.29 484.9 2.9 1.98E-04 2000 517.29 469.9 2.9 1.98E-04 2500 517.29 441.1 2.8 1.98E-04 2800 537.29 413.3 2.8 1.98E-04 3000 527.29 465.8 3 1.98E-04 Average 456.0 2.8 1.98E-04 Based on the results from the study from case GL1, it can be concluded that use of ANN based models for gas lift simulation and optimization has provided a speed-up of three to six order of magnitude as compared to the fully numerical model while maintaining its accuracy. 102 10.4.2. Case GL2: Performance comparison between numerical-ANN coupled and fully ANN based gas lift models 10.4.2.1. Case GL2a-Performance comparison with variation in depth of gas lift injection Parametric studies have been performed on the performance of the numerical-ANN coupled gas lift model: Model 2 and the fully ANN based gas lift model: Model 3. The first study involves a study of models with variation in gas lift injection depth. The inputs for the system are the same as shown in Table 10-2 and Table 10-3, with the exception of the gas lift injection depth, which is varied for each run. The gas lift injection rate is fixed at 2 MMSCFD. The results have been summarized in Table 10-8 and Figure 10-4. Table 10-8. Case GL2a-comparison of gas lift performance with varying gas lift injection depth Total Oil Produced (MSTB) Gas lift Numerical-ANN Fully ANN- Difference injection depth Coupled- Model 2 Model 3 (%) (ft) 3900 78.9 75.6 4.2 3800 77.9 74.0 5.0 3700 77.0 72.4 5.9 3600 76.7 70.7 6.7 3500 75.1 69.4 7.6 3400 74.1 67.9 8.3 Average 6.30 103 84.40 79.40 74.40 69.40 Num-ANN Coupled 64.40 Fully ANN Total Oil Produced(MSTB) 59.40 54.40 3300 3400 3500 3600 3700 3800 3900 4000 Gas lift injection depth (ft) Figure 10-4. Gas lift performance curve with varying gas lift injection point While the average difference between Model 2 and Model 3 for this study are observed to be 6.3%, the trends shown by both the models display that the total oil production is improved by placing the operating gas lift valve at a deeper location. This result is in line with conventional knowledge that gas lift injection at a deeper location leads to a lighter column of fluid in the wellbore resulting in lower flowing bottom-hole pressure. Maintenance of lower flowing bottom- hole pressure results in lesser back-pressure on the reservoir leading to higher fluid production. 10.4.2.1. Case GL2b- Performance comparison with variation in Oil API gravity In this study, a comparison was performed on gas lift Model 2 and Model 3 based on variation of the variation in fluid composition of the reservoir. A total of eight different cases consisting of a heavy oil, a medium oil, a medium-light oil and a light oil with corresponding variation in the 104 ratio of original gas in place to original oil in place were considered. All the reservoir, wellbore and gas lift parameters which are fixed for this case study are summarized in Table 10-9 and the lift gas composition which is also constant for this case study is presented in Table 10-10. Table 10-9. Case GL2b- reservoir and well inputs for comparison study at various API gravities Reservoir outer radius (ft) 895 Wellbore radius (ft) 0.966 Radial permeability (mD) 8.6 Porosity 0.131 Formation thickness (ft) 30 Total Depth (ft) 8300 Gas injection depth (ft) 7500 Wellhead pressure (psi) 760 Wellhead temperature (°F) 132 Reservoir Temperature (°F) 177 Initial Reservoir Pressure (psia) 3465 Initial Water saturation 0.301 Lift gas gravity 0.65 Gas lift start time(days) 0.2 Table 10-10. Case GL2b- Lift gas molar composition Methane 0.91 Ethane 0.012 Propane 0.03 i-Butane 0.038 i-Pentane 0.01 C6+ 1.00E-06 C20+ 1.00E-06 The oil API gravity and ratio of original gas in place to original oil in place of the eight fluids considered in this study has been summarized in Table 10-11. 105 Table 10-11. Case GL2b- API gravity and OGIP: OOIP ratio of the fluids considered OGIP: OOIP Oil Category API gravity (SCF/STB) Fluid A Heavy 19.7 1260 Fluid B Medium 26.5 1280 Fluid C Medium-Light 36.0 2020 Fluid D Light 42.0 2130 Fluid E Heavy 20.5 5300 Fluid F Medium 26.6 5600 Fluid G Medium-Light 36.0 3240 Fluid H Light 43.0 6260 The gas lift performance curves for Fluid A through Fluid H are displayed in Figures 10-5 through Figure 10-12, respectively. 12.0 10.0 8.0 6.0 Coupled Numerical-ANN 4.0 Fully ANN 2.0 Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-5. Case GL2b-Gas lift performance curve for Fluid A 40.0 35.0 30.0 25.0 20.0 Coupled Numerical- 15.0 ANN 10.0 Fully ANN 5.0 0.0 0 500 1000 1500 2000 2500 3000 3500 Totaloil produced (MSTB) Gas lift injection rate (MSCFD) Figure 10-6. Case GL2b-Gas lift performance curve for Fluid B 106 60.0 50.0 40.0 30.0 Coupled Numerical- 20.0 ANN Fully ANN 10.0 0.0 Totaloil produced (MSTB) 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-7. Case GL2b-Gas lift performance curve for Fluid C 40.0 35.0 30.0 25.0 20.0 15.0 Coupled Numerical-ANN 10.0 5.0 Fully ANN Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-8. Case GL2b-Gas lift performance curve for Fluid D 4.5 4.0 3.5 3.0 2.5 2.0 1.5 Coupled Numerical- ANN 1.0 Fully ANN 0.5 Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-9. Case GL2b-Gas lift performance curve for Fluid E 107 7.0 6.0 5.0 4.0 3.0 Coupled Numerical- 2.0 ANN Fully ANN 1.0 Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-10. Case GL2b-Gas lift performance curve for Fluid F 80.0 70.0 60.0 50.0 40.0 30.0 Coupled Numerical- 20.0 ANN Fully ANN 10.0 Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-11. Case GL2b-Gas lift performance curve for Fluid G 120.0 100.0 80.0 60.0 Coupled Numerical- 40.0 ANN 20.0 Fully ANN Totaloil produced (MSTB) 0.0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-12. Case GL2b-Gas lift performance curve for Fluid H 108 Table 10-12. Case GL2b- Comparison of Model 2 and Model 3 for various reservoir fluids considered Fluid case Average difference (%) Fluid A 4.6 Fluid B 4.4 Fluid C 5.5 Fluid D 3.5 Fluid E 9.8 Fluid F 5.8 Fluid G 16.5 Fluid H 13.1 The difference in the predictions of the Model 3: fully ANN based and Model 2: numerical- ANN coupled gas lift models is displayed in Table 10-12. It can be observed that the gas lift Model 3 shows an average difference of less than 6% when compared to Model 2 for cases with OGIP to OOIP ratio less than 2200 SCF/STB. The average difference increases to about 5.8% - 16.5% for cases with higher OGIP to OOIP ratio. This difference is pronounced in cases of medium-light to light oils with high content of gas in reservoir. This effect can be observed in the zone with less than 100 MSCFD of gas injection rate. This may be due to the nature of these reservoirs which facilities a high volume of fluid production without the requirement of gas lift. Also, the thermodynamics of the lighter oils are closer to that of volatile oil, causing a deviation in behavior leading to an increase in error. Considering that gas lift may not be required with medium-light and light oil reservoirs with high volume of gas in place, this variation can be considered to be of less significance. Hence, the Model 3: fully ANN based gas lift model can be considered to have been validated for various oil classes and gas lift configurations when compared to Model 2: coupled numerical-ANN based gas lift model. 10.5. Gas lift case studies Having performed internal checks, the gas lift models are now tested for case studies from literature. In this section, two cases are considered. The first case involves a study based on a 109 hypothetical gas lift system which is being optimized based of the economics of operation involved. The second case involves a well from Suez field on which a study was performed to optimize the gas lift system. 10.5.1. Gas lift case study 1 In this study, a standalone gas lift well with a constant productivity index inflow performance has been optimized based with an objective function of matching the expenditure on the gas lift with the revenue from incremental oil produced. This well#1 from the paper published Redden et al. (1974) is considered for this study. The details of the well are provided in Table 10-13. It is to be noted that the optimization is based on the oil production rate at the provided conditions and not for the total oil produced over the production life of the well. The economics of the optimization are based on the prevailing costs of gas and the oil price in 1973, which were $1 per STB of oil and $0.25 /MSCF of natural gas. The composition of the produced fluids and lift gas has been used such that it represents the fluid and production properties provided. The composition used for this cased study is summarized in Table 10-14. Table 10-13. Well#1 description for gas lift case study 1 as provided by Redden et al. (1974) Oil API gravity 35 Tubing inside diameter(in) 2.992 Depth of producing interval (ft) 5500 Depth of operating gas lift valve(ft) 5000 Static bottom-hole pressure (psig) 1400 producing wellhead pressure (psig) 100 Gas specific gravity 0.8 Value of oil ($/bbl) 1 Cost of gas-lift gas ($/MSCF) 0.25 Productivity Index(STB/psi-day) 1.12 Water cut 0 Formation Gas-Oil ratio 500 110 Table 10-14. Hydrocarbon composition of fluids for gas lift case study 1 Molar composition of hydrocarbon components Produced fluids Lift gas Methane 0.359 Methane 0.78 Ethane 0.099 Ethane 0.06 Propane 0.032 Propane 0.06 i-Butane 0.013 i-Butane 0.077 i-Pentane 0.04 i-Pentane 0.023 C6+(C17) 0.4038 C6+ 1.00E-06 C20+(C27) 0.0532 C20+ 1.00E-06 The procedure followed for this optimization process described by Redden et al. (1974) involves: 1. Select a gas lift injection rate. 2. Assume an oil flow rate, derive the gas and water flow rates through the provided gas- liquid ratio and water cut. 3. Calculate the flowing bottom-hole pressure using a wellbore hydraulics model. Redden et al. (1974) have used the Orkiszewski (1967) correlation for this purpose. 4. Calculate the oil production rate using: 푞푙 = {푃퐼 × (푆퐵퐻푃 − 퐹퐵퐻푃)} (10.4) 5. Update the oil flow rate and iterate until the oil flow rate assume equals the oil flow rate provided by equation (10.4). 6. Change the gas lift injection rate and go through steps 2 through 5. Perform this step several times until a maximum gas lift injection rate is achieved. 7. Generate a gas lift performance curve based on the oil production rate for a given gas injection rate. 111 8. Calculate the revenue earned through the oil produced at each flow rate of oil and the cost of lift gas at each step calculated in step 7. 9. Generate a curve of oil revenue versus gas cost. Redden et al. (1974) defined the point at which slope is equal to 1 as the optimal point of gas injection. The have considered the point at which $1 worth of incremental oil is produced per $1 spent on gas. The procedure has been replicated in this study using the standalone ANN wellbore model. The gas lift performance curve, the gas lift monetary performance curve and a plot of slope of gas lift monetary performance curve are shown in Figure 10-13, Figure 10-14 and Figure 10-15, respectively. 1060 1040 1020 1000 980 960 Oil production Oil rate (STB/day) 940 920 0 100 200 300 400 500 600 Gas lift injection rate(MSCFD) Figure 10-13. Gas lift case study 1: gas lift performance curve for standalone wellbore model 112 1060 1040 1020 1000 980 960 Revenue Revenue Oil from ($/day) 940 920 0 20 40 60 80 100 120 140 cost of gas lift ($/day) Figure 10-14. Gas lift case study 1: gas lift monetary performance curve for standalone wellbore model 2.5 2 1.5 curve 1 0.5 Slopeof monetary lift gas performance 0 0 100 200 300 400 500 600 Gas lift injection rate(MSCFD) Figure 10-15. Gas lift case study 1: slope of gas lift monetary performance curve for standalone wellbore model From Figure 10-15 it is observed that the economic optimal point of gas lift injection is at an injection rate of 271 MSCFD where the slope of the monetary gas lift performance curve is 1. 113 The corresponding oil production rate is 1009.7 STB/day. The results for this case provided by Redden et al. (1974) specify the optimal gas flow rate to be 258 MSCFD and the corresponding oil flow rate at 1013 STB/day. 10.5.2. Gas lift case study 2-Field case In this case study, the gas lift optimization study performed by Abdel-Waly et al. (1996). A well R3-3 from the Nubia reservoir, Ramadan field in the Gulf of Suez is chosen for this case study. This well is chosen from the two wells provided in the paper because this well is nearly vertical, with an average inclination of 6.1° below point of gas injection and 4.1° above point of gas injection, while the second well described in the paper has an inclination of 17.6° and has a water cut of 90%. For all calculations in this case study the well R3-3 is considered vertical. The input parameters as provided in the paper are described in Table 10-15. The composition of the produced fluid and lift gas is calculated on the basis of the black-oil properties provided. The composition used for this study is summarized in Table 10-16. Table 10-15. Well R3-3 description for gas lift case study 2 as provided by Abdel-Waly et al. (1996) Oil API gravity 32 Tubing inside diameter(in) 2.992 Total vertical depth of well (ft) 11053 Vertical depth of operating gas lift valve(ft) 6476 producing wellhead pressure (psia) 149 Gas specific gravity 0.9245 Wellhead Temperature (°F) 89 Temperature at the bottom of the well (°F) 288 Temperature at the point of gas injection (°F) 237 Water cut (%) 18 Liquid production rate (STB/day) 1140 Formation Gas-Oil ratio 356 Gas lift injection rate (MSCFD) 3620 114 Table 10-16. Hydrocarbon composition of fluids for gas lift case study 2 Molar composition of hydrocarbon components Produced fluids Lift gas Methane 0.305 Methane 0.77 Ethane 0.095 Ethane 0.018 Propane 0.052 Propane 0.021 i-Butane 0.033 i-Butane 0.021 i-Pentane 0.074 i-Pentane 0.169998 C6+(C17) 0.149 C6+ 1.00E-06 C20+(C27) 0.292 C20+ 1.00E-06 Based on the inputs provided as a first step, the pressure at the point of gas injection and the bottom of the well is calculated. The calculation is performed using the wellbore model developed in this work through the same procedure as described for the standalone wellbore system in section 10.5.1. The results are summarized in Table 10-17. Table 10-17. Comparison of pressure predictions for well R3-3 from Abdel-Waly et al. (1996) ANN Numerical Hagedorn Beggs based Field wellbore Error Error and Error and Error Depth wellbore data hydraulics (%) (%) Brown (%) Brill (%) hydraulics model (1965) (1973) model gas lift injection 774 753 2.8 759 2.0 718 7.2 863 11.5 point at 6476 ft Total depth at 1946 2080 6.6 2033 4.3 1967 1.1 2135 9.7 11053 ft Average 4.7 3.1 4.2 10.6 It was described by Abdel-Waly et al. (1996) that the wellbore hydraulics data from the field matched the best with Hagedorn and Brown (1965) and Beggs and Brill (1973) wellbore correlations. 115 The reservoir parameters and the well productivity index for this case was not provided in this paper. However, from Macary and Desouky (2001) the reservoir pressure of the Nubia reservoir was found to be 4700 psia in the year 1996. Also, from Ahmed and Waller (1997) the details of a few other parameters describing another well R-33 in the Nubia C reservoir were found. From this data, it was gathered that the water saturation of the reservoir was about 13-17%, the porosity was between 11-14% and the reservoir thickness multiplied by drainage area was between 900- 1900 Acre-ft. Using the reservoir pressure of 4700 psi and a flowing bottom-hole pressure of 1946 psi and the provided liquid flow rate of 1140 STB/day, a productivity index of 0.41 STB/day/psi was obtained. Further, based on the well construction considering that the total depth of the well is 11053 ft and the gas lift injection is taking place at 6476 ft, the operating valve depth of the well can be moved further deep. It was described by Abdel-Waly et al. (1996) that as a part of gas lift optimization the operating gas injection depth was proposed to be moved to a depth of 9,984 ft. Further in the paper by Abdel-Waly et al. (1996), a gas lift performance comparison has been performed using a wellbore vertical flow correlation. A study has been performed in the current work to make a comparison of gas lift performance with the Beggs and Brill (1973) and Hagedorn and Brown (1965) correlation, as these models were chosen as the best fit for the wells which were studied by Abdel-Waly et al. (1996). The results of this study based on a standalone wellbore model coupled with a productivity index specification are summarized in Table 10-18 and Figure 10-16. 116 Table 10-18. Gas lift case 2: gas lift performance comparison- standalone wellbore models Liquid production rate (STB/day) Hagedorn Gas lift Current and Beggs and injection rate work Brown Brill (1965) (MSCFD) (1965) 0 860 743 348 380 1394 1273 973 1380 1494 1485 1203 3950 1411 1458 1241 4470 1390 1441 1228 The liquid production rate at the bottom-hole survey conditions for well R3-3 was described to be 1140 STB//day by Abdel-Waly et al.(1996). An improvement of 354 STB/day of liquid and correspondingly of 290 STB/day of oil is predicted by the model from current work by injecting gas at a rate of 1380 MSCFD through an operating gas lift valve at a depth of 9984 ft. The Hagedorn and Brown (1965) model and the Beggs and Brill model (1973) model predict an improvement of 345 STB/day and 7 STB/day of liquid, respectively using the estimated reservoir pressure and well productivity index. 1600 1400 1200 1000 800 Current work 600 Hagedorn and 400 Brown (1965) Beggs and Brill Liquid flow Liquidflow rate (STB/day) 200 (1973) 0 0 1000 2000 3000 4000 5000 Gas lift injection rate (MSCFD) Figure 10-16. Gas lift case study 2: gas lift performance curve for standalone wellbore model 117 After gas lift performance has been studied using standalone wellbore models, the performance of the well is studied in conjunction with its associated reservoir. Hence, the performance of well R3-3 under gas lift is made using Model 2 and Model 3 of the gas lift tools described in this chapter. The reservoir parameters are set as described in Table 10-19. The reservoir simulator is initiated such that a productivity index of 0.4139 STB/day/psi is observed in the initial run and the water cut of the production liquids is 18%. Gas lift is introduced into the well at the very beginning of the run. Table 10-19. Gas lift case 2-Input reservoir parameters considered for coupled reservoir-well gas lift simulation Reservoir external radius(ft) 1000 Wellbore radius (ft) 0.75 Initial reservoir pressure (psia) 4700 Permeability (mD) 2.91 Reservoir thickness (ft) 10 Porosity 0.125 Initial water saturation 0.1735 The results of the gas lift study considering the total production period of the well using gas lift Models 2 and 3, respectively have been summarized in Table 10-20 and Figure 10-17. Table 10-20. Gas lift case 2- gas lift performance for the production period of well R3-3 (Abdel-Waly et al., 1996) Total oil produced (MSTB) Gas lift Total injection rate production (MSCFD) time (days) Coupled numerical-ANN: Model 2 Fully ANN: Model 3 10 1067.3 35 50 40 1167.29 45 61 100 1067.29 68 71 400 1737.29 70 74 700 1767.29 71 73 1000 1737.29 71 72 2000 1707.29 70 70 3000 1647.29 67 68 118 80 70 60 50 40 30 Coupled numerical- 20 ANN Totaloil production (MSTB) Fully ANN 10 0 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-17. Gas lift case study 2: gas lift performance curve for a well R3-3 for total production period From the Figure 10-17 it can be observed that the point of gas lift injection after which the total oil produced declines is close to 400 MSCFD. The optimal gas lift injection rate can be more precisely located as 310 MSCFD producing a maximum total oil of 74.3 MSTB. This is obtained from higher resolution plot as shown in Figure 10-18, generated using the gas lift ANN. 80 75 70 65 60 Totaloil produced (MSTB) 55 50 0 500 1000 1500 2000 2500 3000 3500 Gas lift injection rate (MSCFD) Figure 10-18. A high-resolution gas lift performance plot generated using the gas lift ANN 119 Hence, when compared to the models pertaining to optimization based on standalone wellbore systems which provided a maximum oil flow rate point at 1380 MSCFD, the integrated reservoir- wellbore based gas lift performance model produces a solution which is less by a rate of 1070 MSCFD. The total production time has been recorded from the results of the numerical-ANN coupled model. From this, the average production time is calculated to be 1487.3 days based on the data points in Table 10-20. Based on the U.S. Energy Information Agency data (2017), the WTI oil price from May 1st 2017 is $ 48.83 and the price of natural gas is $ 3.15/MSCF. An economic analysis is performed assuming that these prices stay constant, using the average production time of a well to be 1487.3 days. The high-resolution data generated by the gas lift ANN is used to generate a monetary gas lift performance curve (Figure 10-19) for the producing life of a well. 4000 3500 3000 2500 2000 1500 1000 Total revenue Totalrevenue from oil(Thousand USD) 500 0 0 2000 4000 6000 8000 10000 12000 14000 16000 Total gas lift cost (Thousand USD) Figure 10-19. Gas lift case study 2: Gas lift monetary performance curve for an integrated gas lift system 120 The definition of the economic optimum is not rigid and can be determined by the asset manager. For this case, on the basis of the work by Redden et al. (1974) the economic optimum is defined to be a point where one dollar worth of incremental oil is produced per dollar expense on gas lift injection. To determine the economic optimum, the slope of the gas lift monetary performance curve is plotted against the gas lift injection rate in Figure 10-20. 6 5 4 3 2 1 0 0 100 200 300 400 500 600 Slopeof lift gas monetaryperformance curve -1 Gas lift injection rate (MSCFD) Figure 10-20. Gas lift case study 2: Slope of gas lift monetary performance curve to locate economic optimum It is observed in Figure 10-20 that the point of economic optimum is found at a gas lift injection rate of 97 MSCFD. The corresponding total oil production for the gas lift injection rate can be obtained to be 70.7 MSCFD from Figure 10-18. It is to be noted that these figures have been obtained by assuming that the cost of gas lift injection is equal to the cost of natural gas. Typically, an oil field can use its own produced gas for gas lift. Under such circumstances, the operational cost incurred for gas lift injection mainly pertain to the gas compression costs, which will typically be less than the natural gas price. On the other hand, in cases where lift gas has to be purchased by the owner/operator of the oil field, the total cost of gas lift injection will exceed 121 the cost of natural gas as the compression and processing costs may have to be added to the price of the natural gas. Table 10-21 summarizes the benefits of integrated gas lift optimization over the methods involving a standalone wellbore model in combination with an IPR, based on the assumptions described in this section. Table 10-21. Gas lift case 2 summary: benefits of using integrated gas lift optimization approach Standalone Integrated gas lift- Integrated gas lift - wellbore-IPR Monetary performance performance model method model Predicted optimum gas lift injection rate 1380 310 97 (MSCFD) Estimated total oil 71.4 74.3 70.7 production (MSTB) Total oil revenue 3.5 3.6 3.5 (Million USD) Total gas expense 6.5 1.5 0.5 (Million USD) Net revenue -3.0 2.2 3.0 (Million USD) From this study, it can be inferred that the gas lift injection rate of 1380 MSCFD is only suitable for short-term purposes, where one is trying to maximize the production from a well instantaneously. When planning for the long term, allocation of such high quantity of lift gas does not lead to substantially incremental oil or revenue. Hence it is suggested that the excess gas may be allocated to another well which is facing a deficit of gas lift and plan to allocate 310 MSCFD gas lift injection rate for maximizing the total oil production to get 74.3 MSTB of total oil production. Alternatively, based on a basic economic evaluation an operator may allocate 97 MSCFD of gas lift injection rate to obtain the highest net revenue. 122 10.6. Monte Carlo simulation The estimates of the reservoir parameters associated with a well such as the permeability, thickness, porosity, water saturation, drainage radius carry certain uncertainty. In this case study shown in section 10.5.2, the values for the above mentioned reservoir parameters are summarized in Table 10-19. These values were determined to account for an estimated initial productivity index of 0.4139 STB/day/psi and an initial water cut of 18% to match the production performance of well R3-3 as described by Abdel-Waly et al. (1996). To account for the uncertainties in the reservoir parameters, a Monte Carlo analysis is performed in this section. An estimate of probable values for the optimum gas lift injection rate and the total oil produced from the well is obtained through this analysis. The scheme followed for the Monte Carlo simulation is similar to the procedure described by Murtha et al. (1994). The range of the variation in each reservoir parameter in this study is as displayed in Table 10-22. Table 10-22. Summary of the range of variation of reservoir parameters Reservoir parameter min max Reservoir external radius (ft) 900 1100 Reservoir thickness (ft) 8 28 porosity 0.1 0.15 Initial water saturation 0.1 0.2 radial permeability (mD) 2 6 The reservoir parameters were varied and a set of 3000 samples were generated for each parameter, respectively using a random normal distribution. Using the ANN based gas lift model, the optimum gas lift injection rate and the corresponding total oil produced was calculated for each sample, respectively. The well construction, the reservoir and gas lift fluid parameters are described in Tables 10-15 and 10-16, respectively. The cumulative distribution function for optimum gas lift injection rate and corresponding total oil recovered was plotted with 3000 samples of each reservoir parameter mentioned in the 123 Table 10-22 varied individually, while keeping all the other input variables constant. Subsequently, 3000 samples were generated with all the parameters mentioned in Table 10-22 being varied simultaneously and the corresponding cumulative distribution function was calculated as described by Murtha et al. (1994). The P10, P50 and P90 values for the optimal gas lift injection rate and for the total oil produced is summarized in Tables 10-23 and 10-24, respectively. The P10, P50 and P90 represent cases in which 10%, 50% and 90% of the percentage of the total number of cases lie below the provided value. For example, in Table 10- 23, P50 value of 302 MSCFD indicates that out of the sample of 3000 cases of variable reservoir thickness , 50% of the cases have resulted in an optimum gas lift injection rate of less than 302 MSCFD. Hence, if it is assumed that the reservoir thickness is the only uncertain parameter, the probability that we may have to inject gas lift at an injection rate at less than 302 MSCFD is 50%. Table 10-23. Probability values for the estimates of optimum gas lift injection rate Optimum gas lift injection rate (MSCFD) Variable parameter P10 P50 P90 Reservoir external radius (ft) 285 314 344 Reservoir thickness (ft) 294 302 311 porosity 275 314 354 Initial water saturation 313 319 325 radial permeability (mD) 314 335 356 All the above parameters 256 321 386 Table 10-24. Probability values for the estimates of maximum total oil produced Maximum total oil produced (MSTB) Variable parameter P10 P50 P90 Reservoir external radius (ft) 73 74.3 75.5 Reservoir thickness (ft) 74.7 75.4 76.2 porosity 72.9 74.3 75.7 Initial water saturation 74 75.4 76.9 radial permeability (mD) 74.2 74.3 74.4 All the above parameters 74.1 76.7 79.3 124 A Monte Carlo simulation was performed with 3000 random normal samples, by varying all the parameters as described in Table 10-22. A scatter plot of the cumulative distribution function for optimal gas injection rate and maximum total oil produced in displayed in Figures 10-21 and 10- 22, respectively. It can be observed from these figures that based on the uncertainties of the reservoir parameters the value of the optimum gas lift injection rate lies between 200 MSCFD to 500 MSCFD, however, only 10% of the cases have a value less than 256 MSCFD and similarly 50% and 90% of the cases have an optimum gas lift injection rate less than 321 MSCFD and 386 MSCFD, respectively. Correspondingly, the range of the expected maximum total oil produced from a well is between 69 to 85 MSTB. The P10, P50 and P90 values for the maximum total oil produced is 74.1, 76.7 and 79.3 MSTB, respectively. Hence, a probabilistic estimate can be made using this study, to account for the uncertainties in reservoir parameters. The decision of injecting gas lift at a certain rate can be decided on the basis of the estimated total oil production and economics of operation. Figure 10-21. Cumulative distribution function for optimum gas lift injection rate 125 Figure 10-22. Cumulative distribution function for the total oil produced Another result of this study was to observe the variation in optimum gas lift injection rate and the maximum oil production rate, respectively, corresponding to the variation in each individual reservoir parameter. This variation is summarized in Table 10-25. Table 10-25. Probability values for the estimates of maximum total oil produced fraction of Normalised Normalised Optimum gas lift Maximum total oil range of optimum gas lift maximum total Varied parameter injection- variation produced- variation parameter injection- oil produced- (%) (%) varied variation (%) variation (%) Reservoir external radius (ft) 18.79 3.36 0.50 37.58 6.73 Reservoir thickness (ft) 5.63 1.99 0.22 25.33 8.95 porosity 25.16 3.77 0.42 60.38 9.04 Initial water saturation 3.76 3.85 0.14 26.33 26.92 radial permeability (mD) 12.54 0.27 0.31 40.75 0.87 The percentage variation of optimum gas lift injection rate is obtained by getting the P10 to P90 range as a percentage of the P50 value. This calculated variation is normalized by dividing its value with the fraction of the range in the variation of the given reservoir parameter with respect to the range of the reservoir parameter in the ANN. For example, the range of initial water 126 saturation in this study is 0.1(0.1 to 0.2), while the range of this parameter in the ANN training data is 0.7 (0 to 0.7). Hence, the fraction of range of initial water saturation in this study is 0.14. Through the normalized variation of a given variable, we get an estimate of the sensitivity of the target variable with respect to the variation in the reservoir parameter for this particular case. The optimum gas lift injection rate is observed to be more sensitive to variations in reservoir parameters listed in Table 10-25, compared to the variation in maximum total oil produced. As described earlier, 3,000 samples of a given reservoir parameter were generated for each case. A gas lift optimization curve was generated for each of these reservoir specifications using 3000 different gas lift injection rates varied between 1 MSCFD to 3,000 MSCFD. It was observed that the time taken to generate each cumulative distribution function was about 300 s. Such high resolution parametric analysis and Monte Carlo simulations as shown in this section were possible due to the speed provided the ANN based model. ‘ 127 Chapter 11 Conclusions and future work 11.1. Summary Firstly, a numerical reservoir simulator was developed. The developed model is compositional, three-phase, isothermal and fully-implicit in formulation. This model has been developed using a three-dimensional radial-cylindrical system with a single well at the center. The PVT properties of hydrocarbons are determined through an equilibrium flash calculation based on Peng-Robinson equation of state (1978). The simulator can be operated on a sandface pressure specification as well as under molar flow rate specified boundary condition. This tool is capable of operating as single-phase: oil or gas, two-phase: oil and gas, three phase-oil, gas and water flow systems. The reservoir simulator was tested against a commercial numerical simulator and also against an analytical well-testing solution. As shown in section 8.1, the numerical reservoir simulator has been validated for single and multi-phase flow under transient and pseudo-steady state conditions. A numerical wellbore hydraulics model was developed. This model is compositional, multi- phase and non-isothermal. This model is based on the numerical solution of the continuity and mixture momentum equations based on a finite difference scheme implemented on a one- dimensional radial cylindrical system. The definition of the flow regime and the difference in gas and liquid phase velocities is represented using the Hasan and Kabir (2007a) drift-flux model. The wellhead pressure, the molar flow rate of the reservoir feed, the wellhead temperature and the temperature gradient are specified by the user. This numerical wellbore model was used to generate data for training an ANN based wellbore hydraulics model. 128 A fast and robust ANN based wellbore hydraulics tool was developed. This model is capable of predicting the flowing-bottom hole pressure of a vertical tubing and takes both compositional and black-oil flow properties into consideration. This model can take a compositional input of seven hydrocarbon components. Both the numerical and ANN based wellbore hydraulics models have been validated against field data provided by Hasan and Kabir (2002) under natural flowing conditions and also against field data provided by Abdel-Waly et al. (1996) under gas lift conditions. These wellbore hydraulics model can work with black-oil, volatile oil, gas-condensate and three phase systems and have been validated against another compositional numerical simulator from the literature developed by Pourafshary et al. (2009). The ANN based wellbore hydraulics model has a speed-up of four orders of magnitude compared to the corresponding numerical wellbore hydraulics model. The corresponding numerical model is not at the state of the art in terms of parallelization, other mathematical and computational advancements. However, considering that the ANN requires only a few matrix multiplication operations which bypasses the usage of iterative flash and solver routines, the ANN based model is expected to be significantly faster when compared to a corresponding state of the art numerical model. The mean error between the ANN based and numerical wellbore hydraulics model is 1-8 %. The ANN based wellbore model is capable of generating high-resolution gas lift performance and gas lift monetary performance curves as shown in gas lift case study 1. The standalone ANN based wellbore model was used to perform an economic optimization of a gas lift well based on the current gas lift optimization methodology described by Redden et al. (1974). Three tools were developed for integrated gas lift simulation and optimization. These include: a coupled numerical reservoir-numerical wellbore system, a numerical reservoir-ANN based wellbore coupled system and a fully ANN based gas lift model. The numerical-ANN based 129 wellbore coupled system was used to generate data for training, testing and validation of the ANN based gas lift model. These models are capable of predicting the total oil produced by a well during its production life under various gas lift specifications such as gas lift injection rate, lift gas composition, gas lift start time and gas lift injection depth. Monte Carlo simulations were performed using the ANN based gas lift model. The P10,P50 and P90 values were obtained for the optimum gas lift injection rate and the maximum total oil produced by using 3000 samples of reservoir external radius, reservoir thickness, porosity, initial water saturation and radial permeability. The operator can obtain accelerated results using the ANN based gas lift model for developing integrated gas lift performance curves and to perform Monte Carlo type analysis. Subsequently, the operator can utilize the fully numerical and the numerical-ANN coupled models for intermittent verification purposes or for full-fledged usage based on the time constraints. 11.2. Conclusion A new method for integrated gas lift optimization has been developed based on numerical and ANN based tools. The tools developed as a part of this work facilitate fast and accurate modelling of a gas lift operation coupled with a reservoir. As Rashid, et al. (2012) summarized, the prevalence of nodal analysis and use of inflow and outflow performance curves is due to its speed and simplicity as compared to a coupled simulation. The coupled simulation includes the transient effects of reservoir-wellbore interaction; however, it is computationally expensive. Addressing this particular issue has been one of the primary objectives of this work. Among the tools developed for integrated gas lift optimization, the ANN based gas lift model provides a speed-up of four to six orders of magnitude compared to the fully numerical model and the numerical-ANN coupled model. This tool can be used for obtaining a fast, high-resolution 130 estimate of the optimum gas lift injection rate for maximizing the total oil produced by a well. For obtaining the results for the total production time, the liquid and gas production volumes, the variation in production profile with time, the numerical-ANN coupled model can be used as an alternative to fully numerical coupled gas lift simulation. The use of this integrated gas lift optimization methodology can be particularly useful to some of the NOCs whose primary objective is to maximize the ultimate oil recovery from a well, as opposed to the other operators who prefer to modify the production rates according to the market variations. It has been displayed in the gas lift case study 2 that following an integrated approach also facilitates optimal allocation of gas lift resources and generates revenue through the reduced usage of gas lift in the long term. Also, the gas lift quantity cut off from a given well can be reassigned to another well which displays a greater incremental oil production for a unit increment in gas lift injection rate. Another application of this integrated approach can be in newly drilled or worked-over wells where the production data is not available. Current gas lift optimization methods primarily are dependent on production data whereas the tools developed in this work can predict the production estimate from a well based on given reservoir specifications and arrive at an optimal gas lift specification. The production data obtained from the initial flow tests performed on the well is used as a basis for estimating the gas lift injection rate. As the flow behavior of a well undergoes a decline, the initial estimate of the productivity index of the well, the production rates, the optimal gas lift quantity to be injected do not hold true with time. Hence, it can be concluded that use of an integrated approach for gas lift optimization offers several advantages in the long term when compared to the use of inflow and outflow curves. The integration of hard and soft computing tools developed as a part of this work facilitates fast and accurate solutions for such an approach removing the currently existing constraints. 131 11.3. Future work The current work may be extended in the following ways: Development of a fast model for integrated field optimization of gas lift system containing multiple wells, pipelines, separators and compressors. In such a system, the parameters to be optimized will include the separator pressure and the gas lift allocation to each well. The production rate from each well at a given point of time should be controlled such that the ultimate oil recovery from the field is maximized, while minimizing the gas usage. Such a study can explore the usage of ANNs for global sensitivity analysis. An inverse model of an ANN can be developed to predicting the global optimum based on the study. Development of a model for long term optimization of a single-well and/or field scale gas lift systems based on economics of operation and gas supply constraints. Development of a model for long term gas lift optimization for a well with an associated water drive reservoir. In such a system, the reservoir pressure does not decline but the water cut increases with time. Development of a model for integrated gas lift optimization of a hydraulically fractured well with a dual-porosity, dual-permeability based reservoir model. A study involving the usage of machine learning models for estimating and mitigating the effects of paraffin deposition and other flow assurance problems in gas lift wells: Restricted flow due to the presence of paraffin based or calcite based obstructions leads to back-pressure at the sandface. This problem can be aggravated by the cooling caused by gas expansion at a gas lift valve, leading to sub-optimal performance. A machine learning protocol which can recommend appropriate gas lift composition and injection rate which can mitigate the flow assurance issues will make a significant impact. 132 References Abdel-Waly, A. A., Darwish, T. A., Nagger, M., Darwish, T. A., Osman Salama, A., El-Naggar, M..Nagger. (1996). Study optimizes gas lift in Gulf of Suez Field. Oil and Gas Journal, 94(26), 38–44. Abou-Kassem, Jamal H; Farouq Ali, S. M; Islam, Rafiqul, (2006) Petroleum reservoir simulation: a basic approach Gulf Publishing Company Ahmed Tarek (2010), Chapter 1 - Fundamentals of Reservoir Fluid Behavior, In Reservoir Engineering Handbook (Fourth Edition), Gulf Professional Publishing, Boston, 2010, Pages 1-28 Ahmed, A. H., & Waller, H. N. (1997). An Application of Risk Assessment in the Development of a Complex Mature Oil Reservoir. Society of Petroleum Engineers. doi:10.2118/37779-MS Ali S.M. Farouq(1981). A Comprehensive Wellbore Steam/Water Flow model for Steam Injection Wells. J Can Pet Technology 21(2): 82-88. JCPT Paper No. 82-02-04. doi: 10.2118/82-0204. PAlrumah, M., Startzman, R., Schechter, D., & Ibrahim, M. (2005). Predicting Well Inflow Performance in Solution Gas Drive Reservoir by Neural Network. Petroleum Society of Canada. doi:10.2118/2005-106 Ansari, A.M., Sylvester, N.D., Sarica, C., Shoham, O., and Brill, J. P. (1994): “A Comprehensive Mechanistic Model for Upward Two-Phase Flow in Wellbores,” SPE Production & Facilities Journal, 1994, 143-151. Ayala, L.F. (2004). Compositional Modeling of Naturally-Fractured Gas-Condensate Reservoirs in Multi-mechanistic Flow Domains. PhD Dissertation, The Pennsylvania State University, University Park, Pennsylvania. Bahonar, M., Azaiez, J., & Chen, Z. (2011a). Transient Nonisothermal Fully Coupled Wellbore / Reservoir Model for Gas-Well Testing , Part 1 : Modelling. Journal of Canadian Petroleum Technology, (October), 37–50. https://doi.org/10.2118/149618-PA Bahonar, M., Azaiez, J., & Chen, Z. J. (2011b). Transient Nonisothermal Fully Coupled Wellbore/Reservoir Model for Gas-Well Testing, Part 2: Applications. Society of Petroleum Engineers. doi:10.2118/149618-PA Baldwin, J. L., Otte, D. N., & Whealtley, C. L. (1989). Computer Emulation of Human Mental Processes: Application of Neural Network Simulators to Problems in Well Log Interpretation. Society of Petroleum Engineers. doi:10.2118/19619-MS Beggs, H. D., Brill, J. P. (1973) A Study of Two-Phase Flow in Inclined Pipes. Journal of Petroleum Technology, May 1973, 607-617. Brown, K. E. (1977-1984). The technology of artificial lift methods. Tulsa: PPC Books Cao, H.(2002) Development of Techniques for General Purpose Simulators, Ph.D dissertation, Stanford University Center for energy economics. (2008). A CITIZEN ’ S GUIDE TO NATIONAL OIL COMPANIES: TECHNICAL REPORT- PART A. Austin: The world bank group. Cullender, M.H. and Smith, R.V. (1956). Practical Solution of Gas-Flow Equations for Wells and Pipelines with Large Temperature Gradients. Trans., AIME 207: 281 D.B. Robinson, D.Y. Peng, (1978) The characterization of the heptanes and heavier fractions for the GPA Peng–Robinson programs, Gas processors association, Research report RR-28, 1978 (Booklet only sold by the Gas Processors Association, GPA). Duns, H. Jr., Ros, N. C. J. (1963) Vertical Flow of Gas and Liquid Mixtures in Wells. Proceeding of the Sixth World Petroleum Congress, Frankfurt, June 1963, Section II, Paper 22-PD6. Dutta-Roy, K., & Kattapuram, J. (1997). A New Approach to Gas-Lift Allocation Optimization. Society of Petroleum Engineers. doi:10.2118/38333-MS Ertekin T, Abou-Kassem JH, King GR. (2001) Basic applied reservoir simulation. Richardson, Tex: Society of Petroleum Engineers; 2001 F.Rashidi, E.Khamehchi, H.Rasouli, Rashidi, F., Khamehchi, E., & Rasouli, H. (2010). Oil Field Optimization Based on Gas Lift Optimization. 20th European Symposium on Computer Aided Process Engineering – ESCAPE20, (January 2016). https://doi.org/10.13140/2.1.2799.6969 133 Fancher, G.H., Jr., and Brown, K.E.(1963), "Prediction of Pressure Gradients for Multiphase Flow in Tubing," SPE Journal, March 1963, pp. 59-69. Fetkovich, M. J. (1973). The Isochronal Testing of Oil Wells. Society of Petroleum Engineers. doi:10.2118/4529-MS Gaganis, V., & Varotsis, N. (2012). Machine Learning Methods to Speed up Compositional Reservoir Simulation. Society of Petroleum Engineers. doi:10.2118/154505-MS Gallice, F., & Wiggins, M. L. (2004). A Comparison of Two-Phase Inflow Performance Relationships. Society of Petroleum Engineers. doi:10.2118/88445-PA Gilbert, W. E. (1954). Flowing and Gas-lift well Performance. American Petroleum Institute. Hagedorn, A. R., Brown, K. E. (1965) Experimental Study of Pressure Gradients Occurring During Continuous Two- Phase Flow in Small Diameter Vertical Conduits. Journal of Petroleum Technology, April 1965, 475-484. Hasan, A.R., and Kabir, C.S(1988).: “A Study of Multiphase Flow Behavior in Vertical Wells,” SPE Production Engineering Journal, May 1988, 263–272. Hasan, A. R., & Kabir, C. S. (1992). Two-phase flow in vertical and inclined annuli. International Journal of Multiphase Flow, 18(2), 279–293. https://doi.org/10.1016/0301-9322(92)90089-Y Hasan, A.R. and Kabir, C.S (2002). Fluid Flow and Heat Transfer in Vertical Wellbores. Richardson, Texas: Textbook Series, SPE. Hasan, A. R., Kabir, C. S., & Sayarpour, M. (2007a). A Basic Approach to Wellbore Two-Phase Flow Modeling. Society of Petroleum Engineers. doi:10.2118/109868-MS Hasan, A. R., & Kabir, C. S. (2007b). A Simple Model for Annular Two-Phase Flow in Wellbores. Society of Petroleum Engineers. doi:10.2118/95523-PA Interstate Oil and Gas Compact Commision (2012). Marginal wells: Fuel for Economic Growth, 2012 Report Kirkpatrick, C. V. (1959). Advances in Gas-lift Technology. American Petroleum Institute. Klins, M. A., & Majcher, M. W. (1992). Inflow Performance Relationships for Damaged or Improved Wells Producing Under Solution-Gas Drive. Society of Petroleum Engineers. doi:10.2118/19852-PA Krasnov, V., Khabibullin, R., Musabirov, T., & Khasanov, M. (2012). Self Consistent Approach to Construct Inflow Performance Relationship for Oil Well. Society of Petroleum Engineers. doi:10.2118/160782-MS Livescu, S., Durlofsky, L. J., Aziz, K., & Ginestra, J.-C. (2008). Application of a New Fully-Coupled Thermal Multiphase Wellbore Flow Model. Society of Petroleum Engineers. doi:10.2118/113215-MS Lu, Q., & Fleming, G. C. (2012). Gas Lift Optimization Using Proxy Functions in Reservoir Simulation. Society of Petroleum Engineers. doi:10.2118/140935-PA Mach, J. M., Proano, E. A., Mukherjee, H., & Brown, K. E. (1983). A New Concept in Continuous-Flow Gas-Lift Design. Society of Petroleum Engineers. doi:10.2118/8026-PA Mohammadpoor, M., Shahbazi, K., Torabi, F., & Qazvini Firouz, A. R. (2010). A New Methodology for Prediction of Bottomhole Flowing Pressure in Vertical Multiphase Flow in Iranian Oil Fields Using Artificial Neural Networks (ANNs). Society of Petroleum Engineers. doi:10.2118/139147-MS Mukherjee, H., & Brown, K. E. (1986). Improve Your gas-lift Design. Society of Petroleum Engineers. doi:10.2118/14053-MS Murtha, J. A. et al. (1994). Incorporating historical data into monte carlo simulation.SPE Computer Applications, 6(02):11–17. Ocanto, L., & Rojas, A. (2001). Artificial-Lift Systems Pattern Recognition Using Neural Networks. Society of Petroleum Engineers. doi:10.2118/69405-MS Orkiszewski, J. (1967) Predicting Two-Phase Pressure Drops in Vertical Pipe. Journal of Petroleum Technology, June 1967, 829-838. Osman, E.-S. A., Ayoub, M. A., & Aggour, M. A. (2005). An Artificial Neural Network Model for Predicting Bottom- hole Flowing Pressure in Vertical Multiphase Flow. Society of Petroleum Engineers. doi:10.2118/93632-MS Osuji, L. C. (1994). Review of Advances in gas-lift Operations. Society of Petroleum Engineers. 134 Petalas, N., & Aziz, K. (1998). A Mechanistic Model For Multiphase Flow In Pipes. Petroleum Society of Canada. doi:10.2118/98-39 Pinto, M. A. S., Gildin, E., & Schiozer, D. J. (2015). Short-Term and Long-Term Optimizations for Reservoir Management with Intelligent Wells. Society of Petroleum Engineers. doi:10.2118/177255-MS Pittman, R. W. (1982). gas-lift Design and Performance. Society of Petroleum Engineers. doi:10.2118/9981-MS Pourafshary P (2007). A Coupled Wellbore/Reservoir Simulator to Model Multiphase Flow and Temperature Distribution, Ph.D dissertation, University of Texas-Austin. Pourafshary, A. Varavei, K. Sepehrnoori, A. Podio (2009), A compositional wellbore/reservoir simulator to model multiphase flow and temperature distribution, Journal of Petroleum Science and Engineering, Volume 69, Issues 1– 2, November 2009, Pages 40-52, ISSN 0920-4105, http://dx.doi.org/10.1016/j.petrol.2009.02.012 Rajput.V. (2012). A production performance prediction and field development design tool for coalbed methane reservoirs, Masters Thesis, The Pennsylvania State University, University Park, Pennsylvania. Ranjan, A., Verma, S., & Singh, Y. (2015). gas-lift Optimization using Artificial Neural Network. Society of Petroleum Engineers. doi:10.2118/172610-MS Rashid, K., Bailey, W., & Couët, B. (2012). A survey of methods for gas-lift optimization. Modelling and Simulation in Engineering, doi:http://dx.doi.org.ezaccess.libraries.psu.edu/10.1155/2012/516807 F.Rashidi, E.Khamehchi, H.Rasouli, Rashidi, F., Khamehchi, E., & Rasouli, H. (2010). Oil Field Optimization Based on Gas Lift Optimization. 20th European Symposium on Computer Aided Process Engineering – ESCAPE20, (January 2016). https://doi.org/10.13140/2.1.2799.6969 Rasouli, H., Rashidi, F., Karimi, B., & Khamehchi, E. (2015). A Surrogate Integrated Production Modeling Approach to Long-Term Gas-Lift Allocation Optimization. Chemical Engineering Communications, 202(5), 647-654. doi:10.1080/00986445.2013.863186 Redden, J. D., Sherman, T. A. G., & Blann, J. R. (1974). Optimizing Gas-Lift Systems. Society of Petroleum Engineers. doi:10.2118/5150-MS Salazar-Mendoza, R. (2006). A New Representative Curves for Gas-Lift Optimization. Society of Petroleum Engineers. doi:10.2118/104069-MS Saputelli, L., Malki, H., Canelon, J., & Nikolaou, M. (2002). A Critical Overview of Artificial Neural Network Applications in the Context of Continuous Oil Field Optimization. Society of Petroleum Engineers. doi:10.2118/77703-MS Semenova, A. P., Livescu, S., Durlofsky, L. J., & Aziz, K. (2010). Modeling of Multisegmented Thermal Wells in Reservoir Simulation. Society of Petroleum Engineers. doi:10.2118/130371-MS Shahamiri, A., Heidari, M., Buchanan, W. L., & Nghiem, L. X. (2015). A CFD Compositional Wellbore Model for Thermal Reservoir Simulators. Society of Petroleum Engineers. doi:10.2118/175056-MS Shi, Hi., Holmes, J.A., Diaz, L.R., Durlofsky, L.J., Aziz, K., (2005). Drift-flux Parameters for Three-Phase Steady-State Flow in Wellbores, SPE Paper 89836. SPE J 10(2), 130-137. Shirdel, M. (2012). Development of a Coupled Wellbore/Reservoir Simulator for Damage Prediction and Remediation. Society of Petroleum Engineers. doi:10.2118/160914-STU Shirdel, M., & Sepehrnoori, K. (2012). Development of a Transient Mechanistic Two-Phase Flow Model for Wellbores. Society of Petroleum Engineers. doi:10.2118/142224-PA Shoham, O.(1982) Flow Pattern Transition and Characterization in Gas–Liquid Two-Phase Flow in Inclined Pipes. PhD Dissertation, U. of Tel Aviv, Tel Aviv, Israel Stone, T. W., Bennett, J., Law, D. H.-S., & Holmes, J. A. (2002). Thermal Simulation With Multisegment Wells. Society of Petroleum Engineers. doi:10.2118/78131-PA Stone, T.W., Edmunds, N.R., Kristoff, B.J., (1989). A Comprehensive Wellbore/Reservoir Simulator. SPE Paper 18419, presented at the SPE Symposium on Reservoir Simulation, Houston, TX, Feb. 6-8, 1989. Sun, Q., & Ertekin, T. (2015). The Development of Artificial-neural-network-based Universal Proxies to Study Steam Assisted Gravity Drainage (SAGD) and Cyclic Steam Stimulation (CSS) Processes. Society of Petroleum Engineers. doi:10.2118/174074-MS 135 Takács, G. (2005). gas-lift manual. Tulsa, Okla.: PennWell. Vicente, R., Sarica, C., & Ertekin, T. (2000). A Numerical Model Coupling Reservoir and Horizontal Well Flow Dynamics: Transient Behavior of Single-Phase Liquid and Gas Flow. Society of Petroleum Engineers. doi:10.2118/65508-MS Vogel, J. V. (1968). Inflow Performance Relationships for Solution-Gas Drive Wells. Society of Petroleum Engineers. doi:10.2118/1476-PA Wang, P., & Litvak, M. L. (2008). Gas Lift Optimization for Long-Term Reservoir Simulations. Society of Petroleum Engineers. doi:10.2118/90506-PA Weinaug, C.F. and Katz, D.L. (1943). Surface Tensions of Methane-Propane Mixtures. Ind. Eng. Chem. 35 (2): 239-246 Weller, W. T. (1966). Reservoir Performance During Two-Phase Flow. Society of Petroleum Engineers. doi:10.2118/1334-PA Wiggins, M. L. (1993). Generalized Inflow Performance Relationships for Three-Phase Flow. Society of Petroleum Engineers. doi:10.2118/25458-MS Xiong.W (2014), Development of a Standalone Thermal Wellbore Simulator, M.S.Thesis, University of Alberta Youcef Saad and Martin H Schultz. (1986). GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Computer. 7, 3 (July 1986), 856-869. DOI=http://dx.doi.org/10.1137/0907058 Zuber, N. and Findlay, (1965), J.A. Average Volumetric Concentration in Two-Phase Flow Systems. J. Heat Transfer, Trans. ASME 87,453. 136 Appendix A- Critical properties of components Volume Critical Critical Accentric Molecular Critical Component shift Parachor pressure temperature factor weight volume parameter atm K lb/lb-mole (l/mol) CO2 72.8 304.2 0.225 44.01 -0.082 0.094 78 N2 33.5 126.2 0.040 28.013 -0.193 0.090 41 CH4 45.4 190.6 0.008 16.043 -0.160 0.099 77 C2H6 48.2 305.4 0.098 30.07 -0.113 0.148 108 C3H8 41.9 369.8 0.152 44.097 -0.086 0.203 150.3 IC4 36.0 408.1 0.176 58.124 -0.084 0.263 181.5 IC5 33.4 460.4 0.227 72.151 -0.061 0.306 225 C6 32.5 507.5 0.275 86 -0.059 0.344 250.1 C7 31.0 543.2 0.308 96 -0.019 0.381 278.4 C8 29.1 570.5 0.351 107 -0.002 0.421 309.0 C9 26.9 598.5 0.391 121 0.013 0.471 347.2 C10 25.0 622.1 0.444 134 0.031 0.521 381.9 C11 23.2 643.6 0.477 147 0.055 0.574 415.9 C12 21.6 663.9 0.522 161 0.070 0.626 451.6 C13 20.4 682.4 0.560 175 0.077 0.674 486.5 C14 19.3 700.7 0.605 190 0.084 0.723 522.9 C15 18.3 718.6 0.651 206 0.091 0.777 560.6 C16 17.2 734.5 0.684 222 0.104 0.835 597.2 C17 16.4 749.2 0.729 237 0.111 0.884 630.5 C18 15.7 760.5 0.757 251 0.115 0.930 660.7 C19 15.1 771 0.790 263 0.123 0.973 685.9 C20 14.4 782.9 0.816 275 0.142 1.027 710.5 C21 13.8 793.3 0.858 291 0.139 1.073 742.2 C22 13.3 804.4 0.879 300 0.166 1.126 759.6 C23 12.8 814 0.916 312 0.173 1.151 782.2 C24 12.4 823.2 0.940 324 0.184 1.170 804.2 C25 11.8 832.7 0.965 337 0.199 1.202 827.3 C26 11.5 841.2 0.993 349 0.206 1.240 847.9 C27 11.1 849.6 1.017 360 0.217 1.279 866.3 C28 10.8 857.7 1.042 372 0.227 1.323 885.7 C29 10.5 864.3 1.063 382 0.234 1.356 901.4 C30 10.1 872.53 1.082 394 0.247 1.404 919.7 C31 9.9 880 1.103 404 0.255 1.438 934.4 C32 9.6 887.3 1.123 415 0.266 1.480 950.2 C33 9.4 893.9 1.143 426 0.271 1.516 965.3 C34 9.1 900 1.162 437 0.280 1.558 980.0 C35 8.9 905.9 1.179 445 0.288 1.589 990.3 C36 8.7 912.1 1.196 456 0.297 1.632 1004.0 C37 8.5 917.3 1.213 464 0.301 1.657 1013.7 C38 8.3 923.4 1.227 475 0.311 1.702 1026.5 C39 8.1 928.2 1.245 484 0.316 1.734 1036.6 C40 7.9 934.3 1.259 495 0.326 1.780 1048.4 C41 7.8 938.5 1.273 502 0.331 1.806 1055.7 C42 7.6 942.8 1.289 512 0.337 1.844 1065.6 C43 7.5 947.6 1.302 521 0.342 1.878 1074.3 C44 7.3 953.7 1.316 531 0.354 1.928 1083.4 C45 7.1 957.8 1.330 539 0.358 1.955 1090.4 137 Appendix B- Tabulated results for Section 10.4.2.1- case GL 2b Fluid A: API 19.7 OGIP/OOIP-1260 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical- Fully ANN- (MSCFD) (%) ANN Model 2 Model 3 1 8.7 8.7 0.2 5 8.8 8.3 5.8 10 8.8 8.0 9.1 50 9.0 8.8 2.3 100 9.2 9.3 1.2 500 10.0 10.1 0.7 1000 10.2 10.4 1.6 2000 9.8 10.5 7.1 3000 9.1 10.5 13.5 Average 4.61 Fluid B: API 26.5 OGIP/OOIP-1280 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical- Fully ANN- (MSCFD) (%) ANN Model 2 Model 3 1 27.0 25.0 8.1 5 27.1 25.2 7.6 10 27.2 25.7 6.0 50 28.0 28.4 1.5 100 29.4 30.3 3.0 500 36.0 34.4 4.7 1000 36.2 34.6 4.4 2000 34.7 33.9 2.3 3000 32.4 33.0 1.9 Average 4.40 138 Fluid C: API 36 OGIP/OOIP-2020 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical-ANN Fully ANN- (MSCFD) (%) Model 2 Model 3 1 42.4 34.9 21.7 5 42.5 36.3 17.2 10 42.6 37.4 13.9 50 43.0 40.0 7.6 100 43.8 42.4 3.2 500 48.3 47.0 2.8 1000 48.6 46.4 4.6 2000 46.9 44.8 4.6 3000 44.1 43.3 1.7 Average 5.49 Fluid D: API 42 OGIP/OOIP- 2130 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical-ANN Fully ANN- (MSCFD) (%) Model 2 Model 3 1 34.0 29.5 15.1 5 34.0 30.4 12.0 10 34.1 30.6 11.4 50 34.2 31.8 7.5 100 34.6 33.8 2.2 500 35.8 36.2 1.0 1000 35.9 35.9 0.0 2000 35.0 35.1 0.5 3000 33.9 34.4 1.6 Average 3.45 139 Fluid E: API 20.5 OGIP/OOIP-5300 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical- Fully ANN- (MSCFD) (%) ANN Model 2 Model 3 1 3.8 3.2 17.4 5 3.8 3.0 27.9 10 3.8 3.0 25.7 50 3.8 3.6 6.0 100 3.7 3.8 0.5 500 3.8 3.9 1.2 1000 3.9 3.8 2.1 2000 3.9 3.8 2.8 3000 3.9 3.7 4.7 Average 9.81 Fluid F: API 26.6 OGIP/OOIP-5600 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical- Fully ANN- (MSCFD) (%) ANN Model 2 Model 3 1 5.8 5.2 11.0 5 5.8 5.7 1.7 10 5.8 5.9 1.5 50 5.8 6.2 5.9 100 5.8 6.1 5.1 500 5.8 5.8 1.2 1000 5.6 5.6 0.8 2000 5.7 5.2 9.6 3000 5.7 5.0 15.0 Average 5.75 140 Fluid G: API 36 OGIP/OOIP-3240 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical-ANN Fully ANN- (MSCFD) (%) Model 2 Model 3 1 58.7 48.5 21.1 5 58.8 46.8 25.5 10 59.0 47.7 23.7 50 60.0 51.5 16.5 100 61.5 54.3 13.2 500 70.8 61.5 15.1 1000 72.3 61.9 16.8 2000 70.2 60.5 16.0 3000 66.9 58.8 13.8 Average 16.45 Fluid H: API 43 OGIP/OOIP- 6260 SCF/STB Gas lift rate Total oil produced (MSTB) Difference Coupled Numerical-ANN Fully ANN- (MSCFD) (%) Model 2 Model 3 1 83.9 56.1 49.7 5 84.1 59.2 42.0 10 83.8 61.5 36.3 50 84.9 73.0 16.3 100 86.6 81.4 6.3 500 95.3 92.5 3.1 1000 97.5 90.9 7.3 2000 94.9 86.0 10.3 3000 91.2 81.2 12.3 Average 13.11 Vita Venkataramana Balamurugan Srikanth Putcha Venkataramana Putcha hails from a Telugu speaking family in India and was raised in several parts of India. He obtained a Bachelors of Technology degree in Chemical Engineering (2009) from National Institute of Technology, Trichy, India. Subsequently, he worked as a Senior Executive at SRF limited and later as an Assistant Executive Engineer (Production) at Oil and Natural Gas Corporation Limited. He has pursued his PhD at the Penn State University from 2013 to 2017 under the guidance of Dr. Turgay Ertekin.