Copyright by Xue Li 2012

Transcription

1 Copyright by Xue Li 2012

2 The Thesis Committee for Xue Li Certifies that this is the approved version of the following thesis: A Collection of Case Studies for Verification of Reservoir Simulators APPROVED BY SUPERVISING COMMITTEE: Supervisor: Kamy Sepehrnoori M. Hosein Kalaei

3 A Collection of Case Studies for Verification of Reservoir Simulators by Xue Li, B.S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2012

4 Dedication To my parents and my friends

5 Acknowledgements It is my honor to be a student in the Petroleum Engineering and Geosystem Department at The University of Texas at Austin. I would like to express my sincere gratitude to my supervisor Professor Kamy Sepehrnoori for his guidance, care and understanding. I am very grateful for joining his research group and for everything I learned from him. I really appreciate Dr. Mohammad Hosein Kalaei for his continuous and various guidance in my research. He was helpful in providing technical instructions and insight on UTCOMP, CMG, and valuable suggestions to this thesis. I would also like to thank Dr. Abdoljalil Varavei for his consistent assistance with GPAS and CMG. In addition, I am thankful for Dr. Chowdhury Mamun for his time and great help in revising my thesis. The members of our research group were very kind and helpful in providing suggestions to my research. I wish to thank Dr. Francisco Marcondes and Bruno Ramon Batista Fernandes for their help on UTCOMP_IMPSAT. Thanks also go to Luiz Otavio Schmall dos Santos for his help on GPAS_COATS. I would like to thank Hamid Reza Lashgari and Mohsen Taghavifar for providing data for my research. I want to extend my thank-you to Mojtaba Ghasemi Doroh, Wei Yu, Ali Moinfar, Ali Goudarzi, and Mahdi Haddad for sharing their knowledge with me. Finally I would like to show a deep gratitude to my parents who brought me to this world; I really appreciate their care, support, and understanding throughout my life. I also want to thank my Chinese friends in this department. v

6 Abstract A Collection of Case Studies for Verification of Reservoir Simulators Xue Li, M.S.E. The University of Texas at Austin, 2012 Supervisor: Kamy Sepehrnoori A variety of oil recovery improvement techniques has been developed and applied to the productive life of an oil reservoir. Reservoir simulators have a definitely established role in helping to identify the opportunity and select the most suitable techniques to optimum improvement in reservoir productivity. This is significantly important for those reservoirs whose operating and development costs are relatively expensive, because numerical modeling helps simulate the increased oil productivity process and evaluates the performance without undertaking trials in field. Moreover, rapid development in modeling provides engineers diverse choices. Hence the need for complete and comprehensive case studies is increasing. This study will show the different characteristics of in-house (UTCOMP and GPAS) and commercial simulators and also can validate implementation and development of models in the future. The purpose of this thesis is to present a series of case studies with analytical solutions, in addition to a series of more complicated field cases studies with no exact solution, to verify and test the functionality and efficiency of various simulators. These vi

7 case studies are performed with three reservoir simulators, including UTCOMP, GPAS, and CMG. UTCOMP and GPAS were both developed at the Center for Petroleum and Geosystem Engineering at The University of Texas at Austin and CMG is a commercial reservoir simulator developed by Computer Modelling Group Ltd. These simulators are first applied to twenty case studies with exact solutions. The simulation results are compared with exact solutions to examine the mathematical formulations and ensure the correctness of program coding. Then, ten more complicated field-scale case studies are performed. These case studies vary in difficulty and complexity, often featuring heterogeneity, larger number of components and wells, and very fine gridblocks. vii

8 Table of Contents Table of Contents... viii List of Tables... xiv List of Figures... xviii Chapter 1: Introduction... 1 Chapter 2: Description of the Reservoir Simulators UTCOMP UTCOMP IMPSAT GPAS GPAS COATS CMG-GEM CMG-STARS SENSOR Chapter 3: Simulation Case Studies Introduction Case Studies Case Study 1: One-Dimensional Incompressible Flow with Horizontal Displacement Case Study 2: One-Dimensional Incompressible Flow with Vertical Displacement Case Study 3: One-Dimensional Compressible flow Case Study 4: Two-dimensional Compressible Flow Case Study 5: One-Dimensional Capillary End Effect viii

9 3.2.6 Case Study 6: One-Dimensional Convection-Diffusion Equation Case Study 7: Two-Dimensional Transverse Dispersion Case Study 8: Tracer Flow in a Five-Spot Well Pattern Case Study 9: One-dimensional Waterflooding in X Direction without Capillary Pressure Case Study 10: One-Dimensional Waterflooding in X Direction with Capillary Pressure Case Study 11: One-Dimensional Waterflooding in Z Directions without Capillary Pressure Case Study 12: Miscible WAG Displacement with Secondary Displacements and Low-WAG injection Case Study 13: Miscible WAG Displacement with secondary displacements and high-wag injection Case Study 14: Miscible WAG Displacement with tertiary displacements and low-wag injection Case Study 15: Miscible WAG Displacement with tertiary displacements and high-wag injection Case Study 16: Miscible WAG Displacement with tertiary displacements and water-free solvent injection Case Study 17: Dietz Displacement with miscible displacement and lowlongitudinal dispersivity Case Study 18: Dietz Displacement with miscible displacement and highlongitudinal dispersivity Case Study 19: Dietz Displacement with immiscible displacement and no longitudinal dispersivity ix

10 Case Study 20: Two-Dimensional Convection-Diffusion Equation Case Study 21: Three Dimensional Waterflooding Case Study 22: Three-Dimensional Gas and Solvent Injection with Three Hydrocarbon Components in Reservoir Case Study 23: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Reservoir Case Study 24: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Large Reservoir Case Study 25: Two-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir with Twenty Oil Components in the Reservoir Case Study 26: Three-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir Case Study 27: Scenario Two of SPE Fifth Comparative Solution Project Case Study 28: Gas Injection Involving second Hydrocarbon Phase Generation Case Study 29: Primary, Secondary, and Tertiary Production Process in a Highly Heterogeneous Reservoir Case Study 30: Waterflooding in a Highly Heterogeneous Reservoir with 800,000 Gridblock and 16 Production/Injection Wells Chapter 4: Results Introduction Comparative Results of Case Studies Case Study 1: One-Dimensional Incompressible Flow with Horizontal Displacement x

11 4.2.2 Case Study 2: One-Dimensional Incompressible Flow with Vertical Displacement Case Study 3: One-Dimensional Compressible flow Case Study 4: Two-dimensional Compressible Flow Case Study 5: One-Dimensional Capillary End Effect Case Study 6: One-Dimensional Convection-Diffusion Equation Case Study 7: Two-Dimensional Transverse Dispersion Case Study 8: Tracer Flow in a Five-Spot Well Pattern Case Study 9: One-dimensional Waterflooding in X Direction without Capillary Pressure Case Study 10: One-Dimensional Waterflooding in X Direction with Capillary Pressure Case Study 11: One-Dimensional Waterflooding in Z Directions without Capillary Pressure Case Study 12: Miscible WAG Displacement with Secondary Displacements and Low-WAG injection Case Study 13: Miscible WAG Displacement with secondary displacements and high-wag injection Case Study 14: Miscible WAG Displacement with tertiary displacements and low-wag injection Case Study 15: Miscible WAG Displacement with tertiary displacements and high-wag injection Case Study 16: Miscible WAG Displacement with tertiary displacements and water-free solvent injection xi

12 Case Study 17: Dietz Displacement with miscible displacement and lowlongitudinal dispersivity Case Study 18: Dietz Displacement with miscible displacement and highlongitudinal dispersivity Case Study 19: Dietz Displacement with immiscible displacement and no longitudinal dispersivity Case Study 20: Two-Dimensional Convection-Diffusion Equation Case Study 21: Three Dimensional Waterflooding Case Study 22: Three-Dimensional Gas and Solvent Injection with Three Hydrocarbon Components in Reservoir Case Study 23: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Reservoir Case Study 24: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Large Reservoir Case Study 25: Two-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir Case Study 26: Three-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir Case Study 27: Scenario Two of SPE Fifth Comparative Solution Project Case Study 28: Gas Injection Involving second Hydrocarbon Phase Generation Case Study 29: Primary, Secondary, and Tertiary Production Process in a Highly Heterogeneous Reservoir Case Study 30: Waterflooding in a Highly Heterogeneous Reservoir with 800,000 Gridblock and 16 Production/Injection Wells xii

13 Chapter 5: Summary, Conclusions and Recommendations Summary Conclusions Recommendations Appendix: MATLAB Program for Analytical Solutions of First Twenty Case Studies in Chapter Nomenclature References xiii

14 List of Tables Table 3.1: Reservoir d fluid property for Case Study Table 3.2: Well operation conditions for Case Study Table 3.3: Relative permeability parameters for Case Study Table 3.4: Reservoir and fluid property for Case Study Table 3.5: Relative permeability parameters for Case Study Table 3.6: Well operation conditions for Case Study Table 3.7: Component Properties for Case Study Table 3.8: Reservoir and fluid property for Case Study Table 3.9: Relative permeability parameters for Case Study Table 3.10: Well operation conditions for Case Study Table 3.11: Reservoir and fluid property for Case Study Table 3.12: Relative permeability parameters for Table 3.13: Capillary pressure parameters for Case Study Table 3.14: Well operation conditions for Case Study Table 3.15: Reservoir and fluid property for Case Study Table 3.16: Relative permeability parameters for Case Study Table 3.17: Well operation conditions for Case Study Table 3.18: Reservoir and fluid property for Case Study Table 3.19: Relative permeability parameters for Case Study Table 3.20: Well operation conditions for Case Study Table 3.21: Reservoir and fluid property for Case Study Table 3.22: Relative permeability parameters for Case Study Table 3.23: Well operation conditions for Case Study Table 3.24: Reservoir and fluid property for Case Study Table 3.25: Relative permeability parameters for Case Study Table 3.26: Well operation conditions for Case Study Table 3.27: Capillary pressure parameters for Case Study xiv

15 Table 3.28: Reservoir and fluid property for Case Study Table 3.29: Relative permeability parameters for Case Study Table 3.30: Well operation conditions for Case Study Table 3.31: Well operation conditions for Case Study Table 3.32: Well operation conditions for Case Study Table 3.33: Well operation conditions for Case Study Table 3.34: Well operation conditions for Case Study Table 3.35: Reservoir and fluid property for Case Study Table 3.36: Relative permeability parameters for Case Study Table 3.37: Well operation conditions for Case Study Table 3.38: Input data for Case Study Table 3.39: Reservoir and fluid property for Case Study Table 3.40: Relative permeability parameters for Case Study Table 3.41: Well operation conditions for Case Study Table 3.42: Reservoir and fluid property for Case Study Table 3.43: Relative permeability parameters for Case Study Table 3.44: Well operation conditions for Case Study Table 3.45: Component Properties for Case Study Table 3.46: Reservoir and fluid property for Case Study Table 3.47: Relative permeability parameters for Case Study Table 3.48: Well operation conditions for Case Study Table 3.49: Component Properties for Case Study Table 3.50: Reservoir and fluid property for Case Study Table 3.51: Relative permeability parameters for Case Study Table 3.52: Well operation conditions for Case Study Table 3.53: Component Properties for Case Study Table 3.54: Binary coefficients for Case Study Table 3.55: Reservoir and fluid property for Case Study xv

16 Table 3.56: Relative permeability parameters for Case Study Table 3.57: Well operation conditions for Case Study Table 3.58: Component Properties for Case Study Table 3.59: Binary coefficients for Case Study Table 3.60: Input data for Case Study Table 3.61: Reservoir and fluid property for Case Study Table 3.62: Relative permeability parameters for Case Study Table 3.63: Well operation conditions for Case Study Table 3.64: Component Properties for Case Study Table 3.65: Binary coefficients for Case Study Table 3.66: Input data for Case Study Table 3.67: Reservoir and fluid property for Case Study Table 3.68: Relative permeability and capillary pressure data for Case Study Table 3.69: Well operation conditions for Case Study Table 3.70: Reservoir data by layers for Case Study Table 3.71: Component Properties for Case Study Table 3.72: Binary coefficients for Case Study Table 3.73: Reservoir and fluid property for Case Study Table 3.74: Relative permeability parameters for Case Study Table 3.75: Well operation conditions for Case Study Table 3.76: Component Properties for Case Study Table 3.77: Binary coefficients for Case Study Table 3.78: Reservoir and fluid property for Case Study Table 3.79: Relative permeability parameters for Case Study Table 3.80: Well operation conditions for Case Study Table 3.81: Component Properties for Case Study Table 3.82: Binary coefficients for Case Study Table 3.83: Reservoir and fluid property for Case Study xvi

17 Table 3.84: Relative permeability parameters for Case Study Table 3.85: Well operation conditions for Case Study Table 3.86: Component Properties for Case Study Table 3.87: Binary coefficients for Case Study xvii

18 List of Figures Figure 3.1: Schematic of one-dimensional reservoir with incompressible fluid displacement in the x-direction Figure 3.2: Schematic of one-dimensional reservoir with incompressible fluid displacement in the z-direction Figure 3.3: Schematic of one-dimensional reservoir with compressible fluid displacement in the x-direction Figure 3.4: Schematic of two-dimensional reservoir with compressible fluid displacement Figure 3.5: Schematic of one-dimensional convection and diffusion reservoir Figure 3.6: The relative permeability curve for water and oil flow of Case Study Figure 3.7: Schematic of one-dimensional for convection and diffusion problem Figure 3.8: Schematic of reservoir geometry for two-dimensional transverse dispersion test Figure 3.9: Schematic of one quarter of Five-spot Well Pattern Figure 3.10: Schematic of one-dimensional waterflooding reservoir Figure 3.11: Schematic of reservoir and the well locations for one-dimensional waterflooding in the z-direction with no capillary pressure Figure 3.12: Schematic of miscible WAG displacement reservoir and well locations Figure 3.13: Fractional flow curves for water/oil and water/solvent flow Figure 3.14: Schematic of Dietz Displacement reservoir and well locations Figure 3.15: Schematic of Dietz Displacement reservoir and well locations Figure 3.16: Schematic of Two-Dimensional Convection-Diffusion numerical model for Case Study Figure 3.17: Schematic of Two-Dimensional Convection-Diffusion analytical solution for Case Study Figure 3.18: The relative permeability curve for water and oil flow of Case Study xviii

19 Figure 3.19: The relative permeability curve for oil and gas flow of Case Study Figure 3.20: The relative permeability curve for water and oil flow of Case Study Figure 3.21: The relative permeability curve for oil and gas flow of Case Study Figure 3.22: Schematic of Reservoir and well locations for Case Study Figure 3.23: The relative permeability curve for water and oil flow of Case Study Figure 3.24: The relative permeability curve for liquid and gas flow of Case Study Figure 3.25: Permeability distribution of Case Study Figure 3.26: Porosity distribution of Case Study Figure 3.27: The relative permeability curve for water and oil flow of Case Study Figure 3.28: The relative permeability curve for liquid and gas flow of Case Study Figure 3.29: Permeability distribution of Case Study Figure 3.30: Porosity distribution of Case Study Figure 3.31: The relative permeability curve for water and oil flow of Case Study Figure 3.32: The relative permeability curve for liquid and gas flow of Case Study Figure 3.33: Two-dimensional well locations of Case Study Figure 3.34: Three-dimensional well locations of Case Study Figure 3.35: Permeability distribution of Case Study Figure 3.36: Porosity distribution of Case Study Figure 3.37: Depth of cell top distribution of Case Study Figure 4. 1: Comparison of the pressure drop profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG for Case Study Figure 4. 2: Comparison of the pressure drop profile of analytical solution with that of UTCOMP, GPAS, GPAS_COATS and CMG for Case Study Figure 4. 3: Comparison of the pressure profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG at t D =0.157 for Case Study xix

20 Figure 4. 4: Pressure profile of the analytical solution of dimensionless pressure versus dimensionless distance at different dimensionless time (t D = 0.1, 0.16, 0.21, 0.27, 0.33, 0.39, 0.44, 0.5, 0.56, 0.61 and 0.67) for Case Study Figure 4. 5: Pressure profile of the analytical solution of real pressure versus dimensionless distance at different real time (t=3.18, 5, 6.82, 8.63, 10.45, 12.26, 14.08, 15.89, 17.71, 19.52, Day) for Case Study Figure 4. 6: Comparison of the pressure profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG at y=840ft and t=365days for Case Study Figure 4. 7: Comparison of the water saturation profiles of the analytical solution with that of the simulation results of UTCOMP and CMG for Case Study Figure 4. 8: Comparison of the dimensionless concentration profile of the analytical solution with that of the simulation results of UTCOMP, GPAS_COATS, and CMG when peclet number is 200 at 0.5 pore volume for Case Study Figure 4. 9: Comparison of UTCOMP simulation result of the dimensionless concentration profile with peclet number varying at 0.5 pore volume using third-order TVD method for Case Study Figure 4. 10: Comparison of UTCOMP simulation result of the dimensionless concentration profile with different dispersion control method when Peclet number is 1000 at 0.5 pore volume for Case Study Figure 4. 11: Comparison of the normalized concentration profile of the analytical solution with that of the simulation results of UTCOM and CMG at x D = for transverse dispersivity of and longitudinal dispersivity of 0.02 for Case Study Figure 4. 12: Comparison of the normalized concentration profile of the analytical solution with that of the simulation results of UTCOM and CMG at x D = for transverse dispersivity of 0.02 and longitudinal dispersivity of 0.02 for Case Study xx

21 Figure 4. 13: Comparison of the normalized effluent tracer concentration of the analytical solution with that of the simulation results of UTCOMP and CMG for Case Study Figure 4. 14: Comparison of the normalized effluent tracer concentration of the analytical solution with that of the simulation results of UTCOMP and CMG for variable order of numerical dispersion control methods Figure 4. 15: Comparison of the water saturation profiles of Buckley-Leverett solution and the simulation result of UTCOMP, GPAS and CMG at 0.2 pore volume injected using one-point upstream weighting for Case Study Figure 4. 16: Comparison of the water saturation profile of the Buckley-Levereett solution, the Terwilliger solution, and the simulation results of UTCMOP and CMG at 0.2 pore volume injected using the third-order TVD method with TVD for Case Study Figure 4. 17: Comparison of the water saturation profile of the Buckley-Levereet solution and the simulation results of UTCOMP, GPAS, and CMG at 0.2 pore volume injected using one-point upstream weighting for Case Study Figure 4. 18: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.6 pore volume injected for Case Study Figure 4. 19: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.4 pore volume injected for Case Study Figure 4. 20: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.25 pore volume injected for Case Study Figure 4. 21: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.3 pore volume injected for Case Study Figure 4. 22: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.25 pore volume injected for Case Study Figure 4. 23: Profiles of 0.5 solvent concentration of UTCOMP simulation for Case Study 17 with a xxi

22 Figure 4. 24: Profiles of 0.5 solvent concentration of CMG simulation for Case Study 17 with a longitudinal dispersivity of 1 ft Figure 4. 25: Profiles of 0.5 solvent concentration of UTCOMP simulation for Case Study 18 with a longitudinal dispersivity of 2 ft Figure 4. 26: Profiles of 0.5 solvent concentration of CMG simulation for Case Study 18 with a longitudinal dispersivity of 2 ft Figure 4. 27: Profiles of water saturation of UTCOMP simulation for Case Study Figure 4. 28: Profiles of water saturation of CMG simulation for Case Study Figure 4. 29: Profiles of water saturation of GPAS_COATS simulation for Case Study Figure 4. 30:Three-dimensional view of normalized concentration of analytical solution at 2, 20, 40, 80, 100, and 150 days for Case Study Figure 4. 31: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 2 days and z=0 for Case Study Figure 4. 32: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS and CMG for Case Study Figure 4. 33: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS and CMG for Case Study Figure 4. 34: Comparison of water production rate of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS, and CMG for Case Study Figure 4. 35: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study Figure 4. 36: Water saturation distribution of UTCOMP using one-point upstream weighting at 1000 days for Case Study Figure 4. 37: Oil saturation distribution of UTCOMP using one-point upstream weighting at 1000 days for Case Study xxii

23 Figure 4. 38: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study Figure 4. 39: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study Figure 4. 40: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study Figure 4. 41: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study Figure 4. 42: Oil saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Figure 4. 43: Gas saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Figure 4. 44: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 45: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 46: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 47: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study Figure 4. 48: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Figure 4. 49: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Figure 4. 50: Comparison of average pressure of UTCOMP and CMG for Case Study Figure 4. 51: Comparison of oil production rate of UTCOMP and CMG for Case Study xxiii

24 Figure 4. 52: Comparison of gas production rate of UTCOMP and CMG for Case Study Figure 4. 53: Oil saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Figure 4. 54: Gas saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Figure 4. 55: Comparison of average pressure of UTCOMP and CMG for Case Study Figure 4. 56: Comparison of oil production rate of UTCOMP and CMG for Case Study Figure 4. 57: Comparison of gas production rate of UTCOMP and CMG for Case Study Figure 4. 58: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Figure 4. 59: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Figure 4. 60: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 61: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 62: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study Figure 4. 63: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study Figure 4. 64: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Figure 4. 65: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study xxiv

25 Figure 4. 66: Comparison of cumulative oil production of UTCOMP, CMG, and SENSOR for Case Study Figure 4. 67: Comparison of cumulative oil production vs. cumulative water injection of UTCOMP, CMG, and SENSOR for Case Study Figure 4. 68: Comparison of producing gas-oil ratio of UTCOMP, CMG, and SENSOR for Case Study Figure 4. 69: Comparison of producing water cut of UTCOMP, CMG, and SENSOR for Case Study Figure 4. 70: Comparison of average reservoir pressure of UTCOMP, CMG, and SENSOR for Case Study Figure 4. 71: UTCOMP simulation result of second hydrocarbon phase saturation using one-point upstream weighting at 100, 1000, 2100, 2850, 3500, and 4550 days for Case Study Figure 4. 72: Comparison of average reservoir pressure of UTCOMP and CMG for Case Study Figure 4. 73: Comparison of oil production rate of UTCOMP and CMG for Case Study Figure 4. 74: Comparison of gas production rate of UTCOMP and CMG for Case Study Figure 4. 75: Comparison of average pressure of UTCOMP and CMG for Case Study Figure 4. 76: Comparison of oil production rate of UTCOMP and CMG for Case Study Figure 4. 77: Comparison of gas production rate of UTCOMP and CMG for Case Study Figure 4. 78: Comparison of water production rate of UTCOMP and CMG for Case Study Figure 4. 79: Average reservoir pressure of UTCOMP for Case Study xxv

26 Figure 4. 80: Oil production rate of UTCOMP for Case Study Figure 4. 81: Gas production rate of UTCOMP for Case Study xxvi

27 Chapter 1: Introduction The life of an oil reservoir will mainly go through three stages where all kinds of production techniques are performed to maintain the oil production rate at the maximum possible level. These three distinct stages consist of primary, secondary, and tertiary recovery. During the primary recovery stage, oil is produced by a number of natural mechanisms. These natural mechanisms range from expansion of gas in the gas-cap or initially dissolved in the crude oil, gravity force drainage in dip reservoirs, to waterdriven process in the natural aquifer. Primary recovery typically yields a small amount of oil recovery from the total oil capacity. Because of the large amount of oil remaining after primary production in oil producing reservoirs, secondary recovery mechanisms are employed. After the primary recovery stage, the reservoir pressure decreases and there is no sufficient initial energy to drive oil to the surface. The external energy is applied to the reservoir in the form of injecting fluids to increase the reservoir pressure. Common secondary recovery techniques are water flooding and sometimes gas flooding. In these two methods, aqueous or gaseous fluid is injected at one or several points of the reservoir toward the producers. Hence, oil is displaced from pores and driven ahead of the water/oil or gas/oil front. Because oil is generally immiscible in water, waterflooding is incapable of displacing oil with very high viscosity or oil trapped in the small pores because of capillary pressure between the interface of the water and hydrocarbons. The secondary recovery method is able to improve the reservoir productivity up to 30% of the total oil capacity. 1

28 The tertiary recovery, or Enhanced Oil Recovery (EOR), introduces methods to increase the mobility of the residual/trapped oil by decreasing the viscosity of oil or by decreasing the interfacial tension. EOR is achieved by injecting fluids such as miscible gas, steam, polymer solution, and surfactant solution. Compared to primary and secondary recoveries, the tertiary recovery enables accessing up to more than half of the original oil reserves. Some of the common EOR processes are discussed below. Gas injection is the most commonly used technique in the EOR. It does not only help maintain the reservoir pressure, but also reduces oil viscosity as gas mixes with oil. The gas used usually contains carbon dioxide, nitrogen, hydrocarbon gases. T he gas injection efficiency relies on phase behavior of the injected gas and oil displaced, reservoir temperature, and pressure. In high pressure reservoirs with light oil, for instance, CO 2 can be miscible with crude oil and improve oil recovery by dissolving, and/or swelling in oil and reducing the oil viscosity. A large amount of injected CO 2 is produced with oil; it thus can be re-injected in a cyclic injection mode. In the case of low pressure, CO 2 may not be miscible or only be partially miscible with the oil. Other gases instead of CO 2 can be used as well, such as compressed nitrogen and hydrocarbon gases. Chemical EOR involves mixing of various chemical agents with the injected water (polymer, surfactant, and alkaline solution). In polymer flooding, the viscosity of injected water is increased by added polymer. Polymer flooding aims to decrease the mobility ratio between displacing and displaced fluid, leading to a more efficient displacement of viscous oil. In addition to the beneficial effect of water viscosity, surfactant/polymer (SP) flooding aids recovery by significantly lowering the interfacial tension between aqueous and oleic phases. The trapped residual oil can be displaced because of low interfacial tension. In alkaline-surfactant-polymer (ASP) flooding, 2

29 alkaline solution reacts with acids in crude oil to generate surfactants or pseudosurfactant (soap). Alkaline solution also helps impede surfactant retention on the surface of the rock. Thermal EOR, on the other hand, heats the reservoir oil to lower the oil viscosity, thus making the oil easier to flow. Thermal EOR includes steam injection, hot water injection, and in-situ combustion. In steam flooding, oil is heated to expand and evaporate in the steam zone, causing oil viscosity reduction. In order to achieve high efficiency in this method, steam injection has to be cyclical or continuous. In-situ combustion or fire-flooding involves injection of air or oxygen. In addition to oil viscosity reduction, large hydrocarbon molecules will be cracked and vaporized; lighter hydrocarbons play the role of miscible displacement. The purpose of this thesis is to present a series of case studies with exact solutions, as well as a range of more complicated field case studies with no exact solution, to test the functionality and efficiency of various simulators. For this purpose, we use three compositional simulators to simulate the cases. These compositional reservoir simulators include UTCOMP (Chang, 1990), GPAS (Wang et al. 1997; Wang et al. 1999) and CMG (CMG User s Guide, 2010). UTCOMP and GPAS were both developed at the Center for Petroleum and Geosystem Engineering at The University of Texas at Austin and CMG is a commercial reservoir simulator developed by Computer Modelling Group Ltd. UTCOMP is an isothermal, three-dimensional, equation of state (EOS) compositional reservoir simulator with IMPES (implicit pressure and explicit phase saturations and compositions) formulation. It is capable of simulating a variety of important enhanced oil recovery processes, such as immiscible and miscible gas flooding. 3

30 GPAS is a t hree-dimensional multicomponent, multiphase, fully implicit compositional simulator. It includes a cubic equation-of-state model for the hydrocarbon phase behavior. The hydrocarbon phase behavior is calculated by the Peng-Robinson Equation of State (PRES). Most of the runs are performed using CMG s advanced general EOS compositional modules called GEM. GEM was developed to simulate compositional effects of reservoir fluid for the treatment of primary, secondary, and enhanced oil recovery processes. The rest of runs are made with CMG s new generation advanced processes reservoir modules called STARS. This was designed to simulate thermal flood injection and combustion, as well as many types of chemical additive processes. Some of the applications include chemical/polymer flooding, and steam injection. This thesis presents a total of thirty case studies. These cases vary from small size homogeneous reservoirs to highly heterogeneous large size reservoirs. In Chapter 2, we introduce the simulators that are used in the case studies. These simulators, as mentioned earlier, are UTCOMP, GPAS, and CMG. Compositional simulation modules use Peng- Robinson EOS for phase equilibrium calculations. Description of the problems is given in Chapter 3. T he first twenty case studies provide the exact analytical solution and numerical simulation solution. MATLAB codes for analytical solutions are available in Appendix A. The exact solutions have attempted not only to highlight different aspects of these simulators, but also provide a step to validate two reservoir simulators, UTCOMP and GPAS. The remaining ten case studies include complicated field scenarios with simulation results. All relevant variables for each case have been well defined. Tables of input data and detailed mathematical equations for the analytical solution, as well as the process of simulation, are provided. The simulation cases for testing include first-contact miscible displacement, water flooding, and multi-contact miscible gas displacement in 4

31 one, two, and three dimensions. Chapter 4 s hows the simulations and some analytical solutions for the case studies described in Chapter 3. Differences in these results are discussed. Discussion of results for each case study involves comparison of simulation results and comparison between analytical and simulation results. Comparison of simulation results shows different characteristics, performances, and application categories among various reservoir simulators. This comparison can also be helpful in the development of new models and in optimizing the performance of exiting reservoir simulators. A summary of results, conclusions, and some recommendations for the future work is presented in Chapter 5. Having introduced the research project, it is crucial to mention why this project is important, so to speak. First, this project deals with two kinds of problems: one kind whose analytical solution is known and thus given; the other kind whose analytical solution, far from known, cannot perhaps ever be known, other than solution by numerical simulation means. Hence, the first kind offers readers, researchers, and practitioners embarked on de veloping new simulators or adding innovations and improvements into existing ones the opportunity to test and verify new models whose exact solutions are already provided or can be obtained via dimensionless analysis or lumping of relevant parameters. (Analogy is that of solving mass and momentum transport problems via exact forms and solutions of heat diffusion problems.) In this framework, this research obviously is of immense help and potential. The second kind of problems undoubtedly is more complex, elaborate, and thus time-consuming. Such problems enable us to probe further the varied features (say, flash computations), governing physics, numerical demands of size and numerical schemes, and processes (say, different stages of enhanced oil recovery), otherwise not amenable to 5

32 or possible with other simulators. This at the same time opens the window to our in-house simulators: UTCOMP and GPAS, particularly in regard to what capabilities and innovations are already present or being pursued at the backdrop limitations of various other simulators. In this regard, this thesis has both lofty objective and purpose. Second, regardless of type of problems studied and whether analytical solutions available or not, an essential ingredient of reservoir studies and enhanced oil recovery processes is envisioning what can be studied with present or future resources. For example, with all analytical solutions at hand, can we imagine new processes or features fitting the analytical frameworks? In other words, can we visualize new processes where a few parameters will change or be added and still maintain similar nature of governing equations? We may essentially perturb solution or process and still employ similar analytical solutions. Almost the same spirit imbues problems whose analytical solutions are currently elusive but their numerical solutions are possible. Take for instance, in certain enhanced oil recovery studies more than five hydrocarbon or non-hydrocarbon phases may be envisioned; can we study such eventualities? Yes, all UTCOMP has to do is enlarge its repertoire of flash calculations and calculation schemes (using a reduced set of canonical variables, for example). Thus, both kinds of problems provide a framework to envision future strides. 6

33 Chapter 2: Description of the Reservoir Simulators In this chapter a bref description of the simulators used in this study is presented. 2.1 UTCOMP UTCOMP, developed at the Center for Petroleum and Geosystems Engineering in The University of Texas at Austin, is an isothermal, three-dimensional, equation-of-state, implicit pressure and explicit phase saturations and compositions (IMPES), compositional reservoir simulator capable of addressing a v ariety of enhanced oil recovery processes, such as immiscible and miscible gas flooding. The formulation is based on the volume-balanced approach (Acs et al. 1985) with some modifications. A detailed description of the UTCOMP can be found in the work of Chang (1990). The solution of UTCOMP is analogous to IMPES (the grid block pressure is solved implicitly whereas the component mole rather than phase saturation is calculated explicitly). UTCOMP can model up to four-phase flow behavior. They are aqueous phase, an oleic phase, a gaseous phase, and a second nonaqeous liquid phase. The aqueous phase is entirely water and hydrocarbon components can be soluble in the aqueous phase. The hydrocarbon phase behavior is modeled using both the Peng- Robinson (PR) EOS (Peng and Robinson 1976) and a modified version of the Redlich- Kwong (RK) EOS (Turek et al. 1984). A volume-shift parameter option, based on the work of Jhaveri and Youngren (1988), is designed to adjust hydrocarbon-phase density calculations. UTCOMP is capable of modeling tracers, surfactant, foam, and polymer effects. UTCOMP includes a number of advanced features as following: 7

34 Rigorous and simplified flash calculations (including three-phase flashcalculation capability) K-value option for phase-behavior calculations Higher-order total variation diminishing (TVD) finite-difference method Full physical-dispersion tensor including molecular diffusion Variable-width cross-section option (similar to radial coordinate) Vertical or horizontal well Tracer-flood capability Polymer-flood capability Dilute-surfactant option Gas-foam-flood capability Asphaltene precipitation model CO2 sequestration in aquifers A third-order finite-difference method is employed by UTCOMP to reduce numerical dispersion and for grid orientation control (Chang et al., 1990). Two versions of this third-order method have been implemented so that cell Peclet numbers of at least 1000 can be used without oscillations. The stability and accuracy of this third-order scheme have been dramatically improved by adding a flux limiter that constitutes the method of total variation diminishing (TVD) and the changing the time integration from first-order to a higher-order correct method (Liu et al., 1994). Physical dispersion is simulated using the full dispersion tensor, and the elements of the dispersion tensor emerge from molecular diffusion and mechanical dispersion. 8

35 Relative permeability, interfacial tension, and capillary pressure are included. The relative permeability and capillary pressure can be dependent on interfacial tension, through the concept of capillary number. The interfacial tension between hydrocarbon phases is computed using the MacLeod-Sugden correlation (Reid et al., 1977). Water viscosity is constant and hydrocarbon viscosity is calculated using the Lohrenz, et al. (1964) correlation. Equation (2.1) describes component mole balance n N p i Vb Vb ξjλjxij ( Pj γ j D) + φξ j SK j ij xij qi = 0 for i= 1, 2,, nc. (2.1) t j= 1 Phase-Equilibrium Relationship After solving the conservation equations for component moles in a gridblock, the phase-equilibrium calculations are required to calculate the number, amounts, and composition of all equilibrium phases. The equilibrium solution must be checked with three kinds of constraints, molar-balance constraint, chemical potentials, and Gibbs free energy. The chemical potential for each component must be the same in all phases and the total Gibbs free energy must be minimum at constant temperature and pressure. Equation (2.2) defines equality of component fugacities among all phases, which come from the derivative of the total Gibbs free energy with respect to the independent variables. j r f f = 0 ( i = 1, 2,, n ; j = 1, 2,, n 1). (2.2) i i c p The phase composition constraint is the following: 9

36 n c xij 1 = 0 ( j = 1, 2,, np ). (2.3) i= 1 Equation (2.4) is used to calculate the phase mole fractions of two hydrocarbon phases and is implicitly used in the solution of the fugacity equation, Equation (2.2), i= 1 ( Ki 1) v( K ) n c zi = 0, (2.4) 1+ 1 i where v is the ratio of moles of gas to total moles. Volume Constraint The volume constraint states that the total fluid volumes fully occupy the pore volume in each of the cells: = 0, (2.5) n n c p Ni Lv j j Vp i= 1 j= 1 where L j is a ratio of moles in phase j to the total number of moles in the mixture, v j is the molar volume of phase j, and v p is the pore volume of a grid block. Pressure Equation The grid block pressure equation is solved implicitly and it satisfies the condition that the pore volume should be filled completely by the total fluid volume V( PN, ) = V( P) t p, (2.6) where the fluid is assumed to be a function of pressure and total number of moles of each component and the pore volume are related to pressure only. 10

37 We differentiate both volumes with respect to time and use the chain rule to expand both terms with respect to their independent variables. After rearrangement and substitution of Equation (2.2) into the resultant equations, it gives the following final expression for pressure equation at a s pecified time t under the assumption of slightly compressible formation: nc + 1 np 0 Vt P Vp cf Vb Vti k rj j xij P P λξ t i= 1 j= 1 n + 1 n n + 1 n n + 1 = V V k x P D + V V S K x + V q c p c p c λ ξ ( 2 γ ) φξ b ti rj j ij c j j b ti j j ij ti i i= 1 j= 1 i= 1 j= 1 i= 1. (2.7) 2.2 UTCOMP IMPSAT Watts (1986) formulation method was implemented into the UTCOMP simulator as an alternative approach to the existing solution scheme based on Acs et al. s (1985) formulation. The purpose for this implementation was to improve the computational efficiency of UTCOMP simulator by using a more stable formulation by allowing UTCOMP to employ larger time steps in simulations. Unlike Acs et al. using an implicit pressure/explicit concentration (IMPEC) approach (Acs et al. is formulated), Watts constructed the formulation on the basis of an implicit pressure/implicit saturation (IMSAT) scheme. The phase velocities and the mass balance equations for components are determined from calculated pressure and saturations. A new saturation equation is solved implicitly only for the IMPSAT formulation. The following equation is the final saturation equation in terms of total velocity: 11

38 V P ( SV p ) = t P N t Nc + 1 N p Np qk Vb V k xkjξ + j f j ( vt + λmk( gρj D gρm D Pcmo + Pcjo )) + k= 1 j= 1 m= 1 Vb. (2.8) 2.3 GPAS General Purpose Adaptive Simulator (GPAS) is a fully implicit, three dimensional, multiphase, and multicomponent compositional simulator developed at the Center for Petroleum and Geosystems Engineering in The University of Texas at Austin. GPAS has an additional capability of parallel processing and is extended to model chemical oil recovery processes, which allows much larger-scale simulation. More detail can be obtained from the work of Wang et al. (1997, 1999) EOS Compositional Model Phase behavior of oleic phase is determined fully implicitly from the Peng- Robinson EOS. Physical properties of hydrocarbons are calculated based on t heir compositions; physical properties of water are dependent on chemical species present and these properties are calculated separately. The mass balance equation can be expressed in terms of moles per unit time for a hydrocarbon component i: kk V N V x P D q = 0, (2.9) n p rj ( ϕ ) ξ ( γ ) b i b j ij j j i t j= 2 µ j where N i is the number of moles of component i per pore volume and is defined by N n p = ξ Sx. (2.10) i j j ij j= 2 For the water phase, there is an additional material balance equation as following: 12

39 kk V N V p D q rw ( ϕ ) ξ ( γ ) = 0 b w b w w w w t µ w. (2.11) The equality of component fugacities constraint is as follows: f o i g f = 0. i (2.12) Mole fraction constraints: n c xij = 0 for j = 2... np. (2.13) i= 1 Volume constraint is defined by Nw N N o g + + 1= 0 or So + Sg + Sw 1= 0. (2.14) ξ ξ ξ w o g Number of hydrocarbon phases and their compositions can be determined from the EOS. With one-point upstream weighting method for the transmissibility terms, the massbalance, Equation (2.9), is discretized in a fully implicit approximation. This results a system of nonlinear equations that must be solved iteratively. Newton s method is used to solve the nonlinear equations using an approximated set of linear equations. The linear equations underscore the Jacobian matrix generated. Elements of the Jacobia matrix are the derivatives of the governing equations with respect to the independent variables. Phase equilibrium is used to update the number, the amounts, and the compositions of the phases in equilibrium. Newton s method is used for above calculations that are repeated until residual tolerances meet the convergence criteria. Computational Framework 13

40 In order to handle the simulation task with parallel processing, GPAS runs under the framework called Integrated Parallel Accurate Reservoir Simulator (IPARS) (Parashar et al., 1997). The framework supports three-dimensional, multiphase, multispecies plus an immobile solid rock phase with adsorbing components and separates the physical model development from parallel processing, solvers, and other auxiliary functions. The IPARS framework handles many tasks including input/output to solvers, message passing between processors, memory allocation, and parallel computation. The primary source of the solvers used in the framework comes from the software package called PETSc (Portable Extensible Toolkit for Scientific Computation). Chemical Model The chemical module has been successfully implemented in GPAS with chemical species such as tracers, polymer, surfactant, and electrolytes, which occupy negligible volume and do not affect the EOS model governing equations. The microemulsion phase (the surfactant-rich aqueous phase) with the presence of oil is considered a pseudocomponent with the same composition as the oil phase in the EOS model. Volumetric concentrations of surfactant, oil, and water are used as coordinates on a ternary diagram and the phase behavior is based on W insor, 1954; Reed and Healy, 1977; Nelson and Pope, 1978; Prouvost et al., 1985; and Camilleri et al., Although salinity, alcohol, and divalent cations affect the microemulsion phase behavior significantly, GPAS models the phase behavior as tough dependent on salinity only. 2.4 GPAS COATS Coats (1980) formulation is an implicit, three-dimensional, three-phase flow compositional simulation model under viscous, gravity, as well as capillary forces. The 14

41 purpose for the implementation of the coats formulation in GPAS is to enhance the efficiency as well as the robustness of most numerical compositional modeling problems. Coats approach is capable of providing consistent and smoother hydrocarbon compositions and properties at near a critical point. Their applications range from depletion or cycling of volatile oil and gas condensate to MCM process. Coats model uses Soave-Redlich-Kwong (1972) and Peng-Robinson (1976) equation of state for phase equilibrium and property calculations (hydrocarbon densities, fugacities, and K-values). Hydrocarbon viscosities are determined from the Lohrenz et al. (1964) correlation; relative permeability and capillary pressure are dependent on saturations. Assumptions in Coats formulation are instantaneous equilibrium between gas and oil phases in any grid block and mutual insolubility of water and hydrocarbon components. While diffusion is neglected in the original publication, in the GPAS implementation, it is added to guarantee reliability when miscible processes are modeled. There are N c +1 mass balances for N c hydrocarbon components (Equation (2.9)) and water (Equation (2.11)) associated with the constraint equations (Equation (2.12) through Equation (2.14)). Coats (1980) chose the following unknown quanlities as primary variables, pressure (p), saturations of oil and gas (S o and S g ), and N c -2 mole fractions of the gas phase (y i for i=3 N c ). The unknowns require a simultaneous solution of N c +1 material balance equations for each grid block in a fully implicit manner. Like Fussell and Fussel (1979), Coats formulation checks the stability of each single-phase grid block by comparing the calculated grid block pressure with the fluid saturation pressure. If the grid block pressure is equal or smaller than the saturation pressure, the grid block changes from one to two hydrocarbon phases. This method is proven to be not robust for many situations, such as above the cricondentherm, or close to the critical 15

42 point. On account of this, in GPAS, we use Michelsen s tangent plane procedure (Michelsen, 1982). 2.5 CMG-GEM GEM (GEM user s Guide, 2010) is CMG's advanced general equation-of-state compositional simulator. GEM is capable of compositional simulations of a variety of large and difficult problems using state-of-the-art algorithms and computers for several improved oil recovery processes. The main features of GEM include dual porosity, CO 2, miscible gases, volatile oil, gas condensate, horizontal wells, well management, and complex phase behavior. Some of the main features of GEM are listed in the following Formulations Three options are provided for time stepping: IMPES, fully implicit, and adaptive modes. Properties Phase equilibrium compositions and densities of hydrocarbon phases are evaluated using either the Peng-Robinson or the Soave-Redlich-Kwong equation of state. Complex reservoirs Many types of grids can be used to describe the reservoir, such as Cartesian, Cylindrical, refined grids of both Cartesian and Hybrid type, variable thickness/variable depth type, as well as corner point type, either with or without user-controlled Faulting. 16

43 Geomechanical model In some simulation process, the producing formation is sensitive and dynamical to changes in applied stresses. This phenomenon will be considered in the geomechanical model, such as plastic deformation, shear dilatancy, and compaction drive in cyclic injection/production strategies, injection induced fracturing, as well as near-well formation failure and sand co-production. Well A comprehensive well control facility and an extensive list of constraints are available. Matrix solution method AIMSOL provides GEM a state-of-the art linear solution routine based on incomplete Gaussian Elimination to generalized minimal residual (GMRES) iteration and preconditioners such as adaptive implicit Jacobian matrices. 2.6 CMG-STARS STARS (STARS user s Guide, 2011) is CMG's new generation advanced process equation-of-state compositional reservoir simulator which was developed to simulate steam flood, steam cycling, steam-with-additives, dry and wet combustion, along with many types of chemical additive processes, using a wide range of grid and porosity models in both field and laboratory scales. The applications in STARS include chemical/polymer flooding, thermal applications, steam injection, horizontal wells, dual porosity/permeability, directional permeabilities, flexible grids and fireflood. Some of the novel features of STARS are the following: 17

44 Dispersed component including foam The concept of dispersed components provides a unifying point of view in modeling of polymers, gels, fines, emulsions, and foam. Adaptive implicit formulation STARS can be run in either fully implicit or adaptive implicit formulations. Discretized wellbore STARS discretize wellbore flow and solve the resulting coupled wellbore/reservoir flow to handle the problem from the horizontal well technology. Fully implicit wells In order to avoid convergence problems for wells with multiple completions in highly stratified reservoir, block variables are solved fully implicit for the blocks where the well is completed. Aquifer models Geomechanical model Matrix solution method Local cartesian Flexible grid system Naturally fractured reservoirs 2.7 SENSOR SENSOR (SENSOR manual, 2011) which stands for System for Efficient Numerical Simulation of Oil Recovery, is a three-dimensional reservoir simulator designed to optimize and predict oil and gas recovery processes via simulation of compositional and black-oil fluid flow in single porosity, dual porosity, and dual 18

45 permeability petroleum reservoirs. SENSOR offers IMPES and IMPLICIT formulations. Its three linear solvers are reduced bandwidth direct, Orthomin preconditioned by Nested Factorization, and Orthomin preconditioned by ILU with red-black and residual constraint options. Sensor utilizes Peng-Robinson and Soave-Redlich-Kwong EOS to calculate fluid properties with optional shift actors. The main applications available in SENSOR can be summarized as follows: Three kinds of grid type can be used: first is conventional, seven point orthogonal Cartesian grid; second is cylindrical coordinate system; third is any grid (eg. Corner-point, refined, unstructured, or hybrid). Black oil PVT Compositional PVT Multiple reservoirs and multiple PVT Relative permeability and capillary pressure treatment Compaction Initialization (Equilibration) Dual porosity systems for fractured reservoirs Coal bed methane Implicit well treatment Platforms (Gathering Centers) Tracers Regions Stable step logic 19

46 Dynamic dimensioning Active-block storage and CPU 20

47 Chapter 3: Simulation Case Studies 3.1 Introduction The purpose of this chapter is to offer verification of the mathematical formulations, such as pressure equation, boundary conditions, volume derivatives in the pressure equation, and physical property models in various reservoir simulators. In addition, case studies stated in this chapter are necessary to test the characteristic and capability of the simulators. This chapter presents the description of thirty case studies, including problem statement, simulation process, equations used in analytical solution, calculations for numerical simulation input data, input data for numerical simulation, and the schematic of reservoir and well locations. There are twenty case studies with exact analytical solutions. These case studies are one-dimensional incompressible and compressible flows, capillary end-effects, one-dimensional and two-dimensional convection-diffusion equations, transverse dispersion flow, five-spot well pattern tracer injection, one-dimensional waterflooding with and without capillary pressure, miscible WAG displacements, and Dietz displacement (miscible and immiscible). Some of these validation problems are taken from Chang s dissertation (1990), but they are run with different methods, in terms of both analytical solution and simulation using different simulators. Simulation results are of course compared with the exact solutions, which are important for checking mathematical equations and codings in the simulators. When single phase, we actually do not require relative permeability data; however, all simulators -UTCOMP, CMG, and GPAS- need relative permeability information to process the input files. Relative permeability curves (or functions) must be provided, in order for the simulation to be carried out. The rest of case studies are more complicated field studies, varying from homogeneous reservoir with coarse grid blocks to a highly 21

48 heterogeneous reservoir with 800,000 grid blocks. Field cases that we mentioned include three dimensional waterfloding, three dimensional gas and solvent injection (three, six, and twenty hydrocarbon components), SPE Fifth Comparative Solution Project (Killough and Kossack, 1987), gas injection involving second hydrocarbon phase generation, complex recovery processes in a highly heterogeneous reservoir (primary, secondary, and tertiary), waterflooing in a highly heterogeneous reservoir with 800,000 gridblocks and 16 production/injection wells. These case studies are important for testing the compositional and three-dimensional capability, phase behavior calculations as in threephase flash calculation (gas, oil, second nonaqueous liquid) capability, EOR processes, and heterogeneity properties. Simulation results from different simulators are compared, to study different characteristics of varied numerical reservoir simulators. Result of variable simulation solutions and analytical solutions are presented in Chapter 4. Analytical solutions for the validation of case studies by MATLAB are available in Appendix. 3.2 Case Studies In the following sections, all case studies are described Case Study 1: One-Dimensional Incompressible Flow with Horizontal Displacement This case study is made to establish the validity of one-dimensional, incompressible single-phase flow with Darcy s law for a one-dimensional reservoir with a length of 2 ft and a cross-sectional area of 0.01 ft 2. Absolute permeability is 500 md and porosity is 0.2. Reservoir and fluid property data are shown in Table 3.1. Well operation conditions are given in Table 3.2. The initial pressure of this reservoir is 2000 psi. The schematic of reservoir and well locations are shown in Figure

49 Figure 3.1: Schematic of one-dimensional reservoir with incompressible fluid displacement in the x- direction. There is a production well at the right edge of the reservoir. The production well is produced at a constant bottom-hole pressure of 2000 psi. There is an injector at the left edge of the reservoir where a single-phase fluid with a density of 44.7 lb/ft 3 and a viscosity of cp is injected at a rate of 0.04 ft 3 /day. In order to reach the purpose of fluid injection, the property of this fluid is imposed on w ater. Water viscosity, mass density, and molar density are set for cp, 44.7 lb/ft 3, and lb-mole/ft 3, respectively. In this case study, we use Corey s model (Corey, 1986) for generation of water/oil relative permeability data as presented in the following: water: n k = k ( S) w, rw rw (3.1) oil: n k = k (1 S) o, ro ro (3.2) where, Sw Swr S = 1 S S or wr, (3.3) where k rw and k ro are water and oil end point relative permeability, respectively. S is normalized water saturation. Also n w is water relative permeability exponent while n o is oil relative permeability exponent. Relative permeability parameters for Corey s correlation are given in Table 3.3. Water is the only fluid in the reservoir during the simulation process (i.e. the initial water saturation is 100%); therefore, the effect of 23

50 relative permeability is negligible. Darcy s law, which is Equation (3.4), is used to calculate the pressure gradient for this case study. This case is a s teady-state process (time independent). Darcy s law is given by ka dp Q = µ dl, (3.4) where, Q: injection flow rate, bbl/d l: distance in the direction of flow, ft µ: viscosity, cp k: absolute permeability, Darcy A: cross sectional area of flow path, ft 2 P: fluid pressure, psi 24

51 Table 3.1: Reservoir and fluid property for Case Study 1. Grid blocks dimension in x, y, and z directions Length (ft) 2 Width (ft) 0.1 Thickness (ft) 0.1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Water viscosity (cp) Water density (lb/ft 3 ) 44.7 Water compressibility (psi -1 ) 0.0 Initial water saturation (fraction) 1.0 Initial reservoir pressure (psi) 2000 Table 3.2: Well operation conditions for Case Study 1. Fluid injection rate (ft 3 /day) 0.04 Production well bottom-hole pressure (psi) 2000 Maximum time (Day) 0.01 Table 3.3: Relative permeability parameters for Case Study 1. Water Oil Endpoint phase relative permeability Relative permeability Exponent 1 2 Residual saturation

52 3.2.2 Case Study 2: One-Dimensional Incompressible Flow with Vertical Displacement This case study is used to test the gravity term in the flow equation. This case has the same fluid and reservoir properties as case study 1. The only difference is that the reservoir is built in a vertical mode. Hence, the grid block dimension is The fluid is injected from the top of the reservoir. The schematic of reservoir and well locations are shown in Figure 3.2. Figure 3.2: Schematic of one-dimensional reservoir with incompressible fluid displacement in the z- direction. The analytical solution is computed for a vertical injector with the same reservoir and fluid property as case study 1. Because the fluid is under the effect of gravity, a gravity term is added to Equation (3.4), which consequently becomes Equation (3.5). This is a steady-state case study. Darcy s law for inclined flow is as following ka dp dz Q = ( ± γ ), (3.5) µ dl dl 26

53 where, z: distance in the vertical direction, ft γ: specific gravity of the fluid, dimensionless l: distance in the direction of flow, ft 0.433: pressure gradient of water, psi/ft Since injection is performed from the top of the reservoir, we can simplify Equation (3.5) to ka dp Q = ( γ ). (3.6) µ dl Case Study 3: One-Dimensional Compressible flow This case study is generated to validate the conservation equation for the linear flow of a single-phase slightly compressible fluid in a homogeneous and isotropic porous medium. For a one-dimensional compressible flow, reservoir and fluid property data are listed in Table 3.4; relative permeability data are shown in Table 3.5; well operation conditions are shown in Table 3.6; and component properties are given in Table 3.7. The reservoir is divided into 100 grid blocks. There is one production well at the left edge of reservoir, which is shown in Figure 3.3. Figure 3.3: Schematic of one-dimensional reservoir with compressible fluid displacement in the x- direction. 27

54 The well constraint is constant well bottom-hole pressure of 1900 psi. Corey s model for generation of water/oil relative permeability data is used. Relative permeability parameters for Corey s correlation are given in Table 3.5. There is one fluid flow in the simulation process; so the effect of relative permeability is negligible. The pressure profile of analytical and simulation solutions are compared at dimensionless days (5 days). Conservation equation is used to calculate one-dimensional flow of a single-phase slightly compressible fluid in a homogeneous and isotropic porous medium (Ahmed, 2006): 2 2 P φµ ct P P = = 2 x k t, (3.7) where c t is total compressibility and x represents distance. The analytical solution for Equation (3.7) is 2 2 PD( xd, td ) = exp( γntd) sin( γnxd), (3.8) γ where, P Pe PD = P P x t D D i e n= 1 n, (3.9) x =, (3.10) L kt =, (3.11) 2 φµc L t γ n 1 = (2 n 1) π. (3.12) 2 for the boundary conditions of P x,0 = P P = 1 (3.13) ( ) i D 28

55 ( ) P 0, t = P P = 0. (3.14) e D Table 3.4: Reservoir and fluid property for Case Study 3. Grid blocks dimension in x, y, and z directions Length (ft) 2000 Width (ft) 10 Thickness (ft) 10 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) Reservoir temperature ( F) 200 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Water viscosity (cp) 1.0 Water density (lb/ft 3 ) 62.4 Water compressibility (psi -1 ) 0.0 Fluid compressibility (psi -1 ) Fluid viscosity (cp) Initial water saturation (fraction) 0.2 Residual water saturation 0.2 Initial reservoir pressure (psi) 2000 Reservoir fluid initial composition (mole fraction) n-c 10 H Table 3.5: Relative permeability parameters for Case Study 3. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation

56 Table 3.6: Well operation conditions for Case Study 3. Production well bottom-hole pressure (psi) 1900 Maximum time (Day) 5 Table 3.7: Component Properties for Case Study 3. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) n-c 10 H Case Study 4: Two-dimensional Compressible Flow This case study is used to model the pressure distribution in a rectangular compressible reservoir. The reservoir is assumed to be homogeneous with no-flow boundaries. The schematic of reservoir and well location are shown in Figure 3.4. Figure 3.4: Schematic of two-dimensional reservoir with compressible fluid displacement. An injection well is located at the center of this rectangular porous medium with grid dimension. The fluid properties of this case are the same as those in case study 3. In UTCOMP simulation, an oil component, n-c 10 H 22, is directly injected into the reservoir. While in the simulation of GPAS, GPAS_COATS and CMG, in order to accomplish the oil injection, the property of injected fluid (density, viscosity and fluid compressibility) is imposed on water. Water mass density is the same as that of the oil 30

57 component (n-c 10 H 22 ) in the reservoir condition, water viscosity is cp, and water compressibility is psi -1. Water is the only movable fluid in the simulation process, so the initial water saturation in the input file of GPAS, GPAS_COATS and CMG is 100%. Reservoir and fluid property data are listed in Table 3.8; relative permeability parameters for Corey s correlation in Table 3.9; and, well operation conditions in Table The pressure profile of analytical and simulation solutions are plotted at 365 days ( pore volume) and y D equaling 0.42 (y=840 ft). The linear partial differential equation represented the pressure distribution for this case study is shown as Equation (3.15) (Hovanessian, 1961). There is a single source or sink flow with a rate of Q. The relevant parameters are marked in Figure P P = + α P + βqlq 2 2 (, ), (3.15) x y t where, φ ct µ α = , k Boµ β = , k x, y: space coordinates, ft 1, q: location of source or sink, ft P: pressure at (x, y), psi ϕ: porosity, fraction c t : total compressibility of fluids and rock structure, psi -1 µ: viscosity, cp k: permeability, md t: time, day 31

58 B o : oil formation volume factor, reservoir bbl/stb Q: specific production (+) or injection (-) rate, STB/D-ft of thickness The solution of Equation (3.15) is applied to the no-flow boundary condition and to an initial constant pressure distribution of P i. Equation (3.16) can be used to determine the pressure for any specified location at any given time except at the location of the line source (where it diverges to infinity) 2 2 t 1 π m mπl mπx exp cos cos 2 t α m= 1 π ( m / a ) α a a a 2 2 β Q 1 π n nπq nπy P( xyt,, ) = Pi exp t cos cos ab n= 1 π ( n / b ) α b b b π m n mπl nπq mπx nπy exp + t cos cos cos cos n= 1 m= 1 π ( m / a + n / b ) α a b a b a b, (3.16) where, P i : uniform initial pressure distribution, psi m, n : summation indices, integers a, b : dimensions of the rectangular section, ft 32

59 Table 3.8: Reservoir and fluid property for Case Study 4. Grid blocks dimension in x, y, and z directions Length (ft) 2000 Width (ft) 2000 Thickness (ft) 1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) Reservoir temperature ( F) 200 Permeability in x-direction (md) 1.5 Permeability in y-direction (md) 1.5 Permeability in z-direction (md) 1.5 Initial water saturation 0.2 Residual water saturation 0.2 Water compressibility (psi -1 ) 0.0 Fluid viscosity (cp) Fluid compressibility (psi -1 ) Initial reservoir pressure (psi) 2000 Reservoir fluid initial composition (mole fraction) n-c 10 H Table 3.9: Relative permeability parameters for Case Study 4. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.10: Well operation conditions for Case Study 4. Total injection rate (ft 3 /day) 8.3 Production well bottom-hole pressure (psi) 2000 Maximum time (D)

60 3.2.5 Case Study 5: One-Dimensional Capillary End Effect This case study is established to validate modeling of outflow boundary condition in the simulators. Capillary-end effect is caused by the discontinuity of the phase pressure across the boundary. Capillary pressure suddenly changes from a finite value in the core to a value close to zero outside the core. The saturation of the wetting phase must increase to the value corresponding to zero capillary pressure. This causes a w etting saturation gradient near the end of the core (Willhite, 1986). In this case study, there is a one-dimensional reservoir which is shown in Figure 3.5. Figure 3.5: Schematic of one-dimensional convection and diffusion reservoir. We examine the saturation distribution caused by the capillary-end effect. The finite-difference solution of the nonlinear partial differential equation induces the truncation error which can smear an otherwise sharp saturation front as if additional physical dispersion is present. The smearing of front, caused by truncating Taylor's series, is called numerical dispersion. In order to minimize the numerical dispersion, a third-order total variation diminishing (TVD) finite-difference method was introduced (Liu et al., 1994). This method is used in UTCOMP while one-point upstream weighting scheme is used in CMG_STARS. Reservoir and fluid property data are listed in Table Figure 3.6 depicts the relative permeability curve. Corey s correlation is selected to calculate relative permeability with parameters given in Table Capillary pressure parameters are given in Table And Table 3.14 contains well operation conditions. The analytical solution of the saturation profile along a one-dimensional, homogeneous 34

61 porous medium was introduced by Richardson et al. (1952). The following equations listed below are used in the analytical solution. Equation (3.17) and Equation (3.18) are Darcy s law applied to the water and oil phase, respectively. qwµ wdl dpw =, (3.17) kk A rw qoµ odl dpo =. (3.18) kk A ro Equation (3.19) expresses in differential form that the capillary pressure relating pressures in the two phases: dp = dp dp. c o w (3.19) By substituting Equation (3.17) and (3.18) into Equation (3.19), the following differential equation can be derived: dsw 1 qwµ w qoµ o 1 dpc 1 = ( ) ( ). (3.20) dl kk kk A ds rw ro w Equation (3.21) is the capillary function between water and oil (UTCOMP Technical Documentation, 2003) φ E P (1 ) pc c = Cpcσ wo S. (3.21) k y By differentiating Equation (3.21) with respect to water saturation, the following equation can be resulted: dpc φ Epc 1 1 = Cpcσ wo Epc (1 S) (1 Swr Sor ). (3.22) ds k w y The following differential equation can be used to determine ds w /dl for water saturation by substituting Equation (3.22) and Equation (3.3) into Equation (3.20), 35

62 dsw 1 qwµ w qoµ 1 o φ Sw S wr Epc 1 1 = ( ) Cpcσ wo Epc ( 1 ) (1 Swr Sor ) dl kkr w kkr o A ky 1 Sor Swr 1. (3.23) Then (ds w /dl) -1 is plotted as a function of water saturation. By integrating this relation, the saturation profile as a function of the distance from the outflow end is obtained. where, q w : water flow rate, bbl/d µ w : water viscosity, cp P w : pressure in water phase, psi k rw : water relative permeability, dimensionless k: absolute permeability, Darcy A: cross sectional area, L: distance from outlet, ft 2 ft P o : pressure in oil phase, psi q o : oil flow rate, bbl/day µ o : oil viscosity, cp k ro : oil relative permeability, dimensionless k y : permeability in the y-direction, mili Darcy (md) P c : capillary pressure between water and oil, psi S: normalized water saturation (Equation (3.3)) S w : water saturation, fraction S wr : residual water saturation, fraction S or : residual oil saturation, fraction C pc : constant of the capillary pressure function, psi md dyne cm σ wo : interfacial tension between water and oil, dynes/cm 36

63 ϕ: porosity, fraction E pc : exponent of capillary pressure function, dimensionless Table 3.11: Reservoir and fluid property for Case Study 5. Grid blocks dimension in x, y, and z directions Length (ft) 2 Width (ft) 0.1 Thickness (ft) 0.1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Oil viscosity (cp) 20 Water viscosity (cp) 1.0 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) 0.0 Initial water saturation (fraction) 0.2 Initial reservoir pressure (psi) 2000 Reservoir fluid initial composition (mole fraction) n-c 10 H Table 3.12: Relative permeability parameters for Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation

64 Table 3.13: Capillary pressure parameters for Case Study 5. pc ( ( / )) C psi md dyne cm 6.78 E pc 2 σ wo (dynes/cm) 42 Table 3.14: Well operation conditions for Case Study 5. Water injection rate (lb-mole/day) Oil injection rate (lb-mole /day) Production well bottom-hole pressure (psi) 2000 Maximum dimensionless time (pore volume) 1.2 Relative permeability water relative permeability Oil relative permeability Water saturation Figure 3.6: The relative permeability curve for water and oil flow of Case Study Case Study 6: One-Dimensional Convection-Diffusion Equation This case study is a o ne-dimensional miscible flow for testing the conservation equations with longitudinal dispersion. The schematic of reservoir and well locations are shown in Figure

65 Figure 3.7: Schematic of one-dimensional for convection and diffusion problem. The length of the reservoir is 40 ft and its cross sectional area is 0.01 ft 2. The absolute permeability is 500 md and the porosity is 0.2. The properties of the injected fluid are identical to those of the fluid in the reservoir. The courant number, Equation (3.32), are the same for all simulations with the constant Δt equaling Day. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Well operation conditions are given in Table In CMG simulation, resident fluid initially contains 0.5 molality of NaCL salt. The NaCL concentration of the injected fluid is 5 molality. In UTCOMP simulation, tracer option is activated and the injected tracer concentration is 2500 ppm. Simulation of convection diffusion equation with Peclet number (N pe : the ratio of the dispersive transport to the convective transport of a particle) of 200 and 100 g rid blocks with courant number of 0.05 are considered. UTCOMP uses third-order TVD finite-difference method while CMG-GEM uses two-point fluxes under the control of a Total Variation Limiting flux limiter (CMG modules do not have the third-order TVD). In this case study, dispersion and convection are the two main transport mechanisms in porous media. In order to analyze how the diffusion and convection affect the salt concentration profile, three values of Peclet number are used in UTCOMP simulations, which are 50, 200, and1000. Physical dispersion comprises of molecular diffusion and mechanical dispersion. Molecular diffusion means particles moves from high concentration regions to low 39

66 concentration regions. Mechanical dispersion is caused by the convection, due to the variations in magnitude and direction of local velocity. Numerical dispersion refers to the smearing of front caused by the truncation error. Many techniques are developed to reduce numerical dispersion, such as third-order TVD method. In this study, we compensate the physical dispersion with varying numerical dispersion control in UTCOMP simulations. Two dispersion controls are used: third-order TVD method and the other is one-point upstream weighting. All results are compared at 0.5 pore volume. The convection-diffusion equation (CDE) describes the conservation of the displacing component with mass concentration of C (Lake, 1989) as 2 C C φ + u φk C L = 0 2 t x x, (3.24) where u is bulk fluid velocity and K L is longitudinal dispersion coefficient. Equation (3.24) assumes incompressible fluid and rock, ideal mixing, and a single phase at unit saturation. Rewriting Equation (3.24) in the dimensionless form gives 2 CD CD 1 CD + = 0 2, (3.25) t x N x D D Pe D where dimensionless parameters are defined as C Co CD =, C C t D x D N J ut =, φl Pe x =, L L o ul =, φk (3.26) (3.27) (3.28) (3.29) 40

67 with C D : dimensionless concentration C o : initial concentration C J : injection concentration t D : dimensionless distance N pe : Peclet number Analytical solution can be calculated for one-dimensional CDE with the following boundary conditions and initial condition on C D (x D, t D ): CD( xd,0) = 0, xd 0 C D( xd, td) = 0, td 0 CD(0, td) = 1, td 0 (3.30) The exact analytical solution (Lake, 1989) for the above conditions is xdnpe 1 NPe xd td e NPe xd + td CD = erfc( ) + erfc( ), (3.31) 2 t 2 2 t 2 D D In this case study, the time-step size has been expressed in a d imensionless form of Courant number (C) as q t C =, x y zφ (3.32) where q/ φ x y z refers to its maximum value over all grid blocks. For one-dimensional flow, the longitudinal dispersion coefficient K L is given by K ( vd L p ) β = C1+ C2, (3.33) D D o o where C 1, C 2, and β are properties of the permeable medium and the flow regime. D o is the effective binary molecular diffusion coefficient between the miscible displacing and displaced fluids and D p is average particle diameter. 41

68 If the interstitial velocity is greater than about 3 cm/day, the local mixing term in equation (3.34) dominates the first term then, D ( vd o p ) β Kl = + C2 Do αl v, (3.34) φf D where, α L : the longitudinal dispersivity, ft v: interstitial velocity, cm/day o Thus the Peclet number (Equation (3.29)) now becomes independent of velocity as L N pe =. (3.35) α L Table 3.15: Reservoir and fluid property for Case Study 6. Grid blocks dimension in x, y, and z directions Length (ft) 40 Width (ft) 0.1 Thickness (ft) 0.1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Fluid viscosity (cp) Fluid density (lb/ ft 3 ) 44.7 Initial fluid saturation 1.0 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 )

69 Table 3.16: Relative permeability parameters for Case Study 6. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.17: Well operation conditions for Case Study 6. Fluid injection rate (ft 3 /day) 0.04 Production well bottom-hole pressure (psi) 2000 Maximum time (pore volume) 0.5 In order to make sure Equation (3.35) can be used to calculate the Peclet number in this case study, the value of interstitial velocity should be checked. According to the data given in Table 3.15 and Table 3.17, the interstitial velocity is u q 0.04 scf / day ν = 20 ft / day cm / day φ = Aφ = 0.1ft 0.1ft 0.2 = =. (3.36) Since cm/day>3 cm/day, Equation (3.35) can be applied in this simulation Case Study 7: Two-Dimensional Transverse Dispersion This case study is a two-dimensional miscible flow to validate simulation of fluid flow with longitudinal and transverse dispersion. There is an analytical solution introduced by Giordano et al. (1985). The solution applies to a steady-state tracer concentration profile for a unit-step input of tracer into the center of the inlet face of an infinitely long, rectangular porous medium. The reservoir geometry is shown in Figure

70 Figure 3.8: Schematic of reservoir geometry for two-dimensional transverse dispersion test. The reservoir is 2 ft by 2 ft and tracer is injected at the interval of 0.3 ft. Fresh water is injected at the interval of 0.7ft at the same time. The injection rate of tracer and fresh water are proportional to the grid block width where the injector is located. Lower part of the reservoir is simulated because of the symmetry of the reservoir. Two cases with different transverse Peclet number (100 and 1000) are tested with a co nstant longitudinal Peclet number of 100. There are forty grid blocks in the x-direction and ten grid blocks in the y-direction. The concentration profile for the analytical solution and the simulation solution by UTCOMP and CMG are compared along x D = , when the simulation is steadystate. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table 3.18 and Table 3.20 contains well operation conditions. Since this is a single phase and steady state case, the relative permeability and the injection rate of tracer and fresh water will not affect the concentration profile when the simulation approaches steady state, as long as the injection rate of tracer and fresh water are in proportion to the grid block width. The following equation is used for the analytical solution (Giordano et al., 1985): 44

71 C = 2ξ + o (1 A) η 4sin(2 πvξ ) cos(2 πvξ) exp o 2Na, l, (3.37) v= 1 πv(1 + A) where, 2 2 = a, l a, t, A N N v π N = a L, N a, l L/ 2 a, t= al t / W, η = x/ L, ξ = y/ W, The tracer concentration is normalized by using Equation (3.26). 45

72 Table 3.18: Reservoir and fluid property for Case Study 7. Grid blocks dimension in x, y, and z directions Length (ft) 2 Half width (ft) 1 Thickness (ft) Tracer injection interval (ft) 0.3 Fresh water injection interval (ft) 0.7 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Water viscosity (cp) 1.0 Water molar density (lb-mole/ft 3 ) Initial water saturation 1.0 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Reservoir fluid initial composition (mole fraction) n-c 10 H Longitudinal dispersivity (ft) 0.02 Transverse 1 dispersivity (ft) 0.02 Transverse 2 dispersivity (ft) Table 3.18: Relative permeability parameters for Case Study 7. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation

73 Table 3.19: Well operation conditions for Case Study 7. UTCOMP CMG Tracer injection rate 30 (lb-mole/day) 1.54 (STB/day) Tracer concentration 2500 (ppm) 5 (molality) Production well bottom-hole pressure (psi) Maximum time (pore volume) Case Study 8: Tracer Flow in a Five-Spot Well Pattern This case study is made to study modeling of two-dimensional miscible flow with longitudinal and transverse dispersion for a five-spot well pattern. An equation is derived by Abbaszadeh-Dehghani and Brigham which predicts effluent tracer concentration from a homogeneous five-spot well pattern for a slug of tracer injected into the patterns (Abbaszadeh-Dehghani and Brigham, 1984). Unit mobility ratio in the development of this analytical solution is assumed. The schematic of reservoir and well locations are shown in Figure 3.9. Figure 3.9: Schematic of one quarter of Five-spot Well Pattern. Reservoir and fluid property data are listed in Table Relative permeability parameters are used for Corey s correlation presented in Table 3.22 and Table 3.23 contains well operation conditions. The five-spot well pattern is symmetric, so onequarter of this well pattern is numerically modeled with grid block dimension. 47

74 A 0.02 pore volume of tracer is injected into the reservoir, followed by a chasing fluid (water) to displace resident fluid through the formation. The tracer slug will be distributed among the reservoir with longitudinal dispersivity of 0.66 f t and transverse dispersivity of ft. To control the numerical dispersion, third-order TVD method is used in UTCOMP while two-point fluxes under the control of a Total Variation Limiting flux limiter is used in CMG-GEM. In order to evaluate how much the order of numerical dispersion will affect the simulation result, a series of simulations is repeated with the variable order of numerical dispersion control method in UTCOMP and CMG. The concentration equation applied in the analytical solution for the homogeneous fivespot well pattern is as KmK m a 2 exp ( V pdbt ( ψ ) V pd ) 4 KmK ( ) ( m) C = dψ, (3.38) D '2 π ( ) ( ) ' 4 2 π Y ( ψ) αl 2 π π 0 Y ( ψ) where, 2 η = tan ( ψ), (3.39) m = 0.5, (3.40) ( ) K ' ( m) K m = =. (3.41) The pore volume injected into the system at the time of breakthrough of streamline ψ is π 2 VpDbt = (1 + η) K(1 η ). (3.42) ' 4 KmKm ( ) ( ) The mixing-line integral for five-spot pattern is defined as the following: 3 tdt 2 Y = (1 + η). (3.43) ( t + 1)( t + η )( t + η) 0 48

75 Substituting the constant into Equation (3.44), effluent concentration for five-spot well pattern is as π a 2 exp ( ( ) ) 4 V pdbt ψ V pd Y ( ψ) α L CD = dψ. (3.44) Y ( ψ ) 0 The dimensionless variable for this case are defined as C CD = (Dimensionless effluent tracer concentration). (3.45) a CF o r α L Tracer slug volume injected into the pattern in terms of fraction of pattern PV is VTr F = r Aφ hs, w (3.46) where, ψ: stream function or value of streamline, angle K (m): complementary complete elliptic integrals of the first kind K (m): incomplementary elliptic integrals of the first kind m: parameter of elliptic integral V pdbt : breakthrough pore volume or breakthrough areal sweep efficiency of a pattern, dimensionless V pdbt (ψ): pore volume of displacing fluid injected at breakthrough of streamline, ψ, dimensionless Y: mixing-line integral C D : dimensionless effluent tracer concentration, dimensionless a: distance between like wells, ft α L : longitudinal dispersivity, ft 49

76 V pd : pore volume injected into the five-spot pattern, dimensionless C : effluent tracer concentration from a homogeneous pattern at V pd, fraction C o : initial tracer concentration, fraction F r : tracer slug volume injected into the pattern in terms of fraction of pattern PV, fraction V Tr : total volume of tracer slug injected into the pattern, ft 3 A: cross sectional area of flow path, ft 2 ϕ: porosity, fraction h: thickness, ft S w : water saturation, fraction 50

77 Table 3.20: Reservoir and fluid property for Case Study 8. Grid blocks dimension in x, y, and z directions Length (ft) 165 Width (ft) 165 Thickness (ft) 1.0 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Water viscosity (cp) Water molar density (lb-mole/ft 3 ) Initial water saturation 1.0 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Reservoir fluid initial composition n-c 10 H (mole fraction) Longitudinal dispersivity (ft) 0.66 Transverse dispersivity (ft) Table 3.21: Relative permeability parameters for Case Study 8. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.22: Well operation conditions for Case Study 8 UTCOMP CMG Water injection rate (lb-mole/day) (STB/day) Production well bottom-hole pressure (psi) Tracer injected time (pore volume)

78 3.2.9 Case Study 9: One-dimensional Waterflooding in X Direction without Capillary Pressure This case study is a o ne-dimensional incompressible waterflooding simulation problem in x-direction without capillary pressure. The analytical solution is the classical Buckley-Leverett problem (Buckley and Leverett, 1942). The schematic of reservoir and well locations are shown in Figure Figure 3.10: Schematic of one-dimensional waterflooding reservoir. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Well operation conditions are given in Table One-point upstream-weighting is used in UTCOMP, GPAS, and CMG simulation. Water is injected at a rate of 0.1 ft 3 /day. There are 500 grid block in x- direction. The following equations listed below are used in the analytical solution. Equation (3.47) is the fractional flow expression for water kkro A Pc 1 + ( ± ρg sin α) qµ o x fw =, (3.47) kroµ w 1+ k µ where, rw f w : water fractional flow, dimensionless k: absolute permeability, m 2 k ro : oil relative permeability, dimensionless o k rw : water relative permeability, dimensionless µ o : oil viscosity, Pa s 52

79 µ w : water viscosity, Pa s α : reservoir dip angle A: cross sectional area of flow path, m 2 P c : capillary pressure between water and oil, atm g: gravity constant, 9.81 m/s 2 ρ : density difference between water and oil, kg/m 3 By neglecting capillary pressure between oil and water (P c ) and for a reservoir with a dip angle of zero, water fractional flow is as 1 fw = kroµ. w 1+ k µ rw o (3.48) And Equation (3.49) is the Buckley-Leverett theory for two-phase flow: dx u dfw =. (3.49) dt φ ds Sw w This is the velocity of a front of constant water saturation. Then integrate this equation and impose the initial condition that at t=0 and x=0: u df w xs = t w s. w φ dsw (3.50) Here are some dimensionless variables used in analytical solution: Dimensional distance and time used in the analytical solution are x xd =, L t D = qt V ALφ = V, p (3.51) (3.52) where V p is pore volume. Hence the Beckley-Leverett equation can be written in dimensionless form of 53

80 x D df w = td. dsw (3.53) Equation (3.54) is the expression of velocity of shock front x q fwf fwi q f = = w. (3.54) t Aφ S S Aφ S wf wi w The dimensionless form of the above equation is xd fw v S = =. (3.55) w t S D w The water front saturation can be solved, if the velocity of the shock and the velocity of the continuous saturation front are equal fw df = w. (3.56) Sw ds w S = S w wf The water saturation profile at a certain pore volume can be calculated by using Equation (3.53) and plugging the derivative of the water fractional flow curve at a corresponding water saturation point. 54

81 Table 3.23: Reservoir and fluid property for Case Study 9. Grid blocks dimension in x, y, and z directions Length (ft) 2 Width (ft) 0.1 Thickness (ft) 0.1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Oil viscosity (cp) 20 Water viscosity (cp) 1.0 Water density (lb /ft 3 ) 62.4 Initial water saturation 0.2 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Reservoir fluid initial composition (mole fraction) n-c 10 H Table 3.24: Relative permeability parameters for Case Study 9. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.25: Well operation conditions for Case Study 9. Water injection rate (ft 3 /day) 0.1 Production well bottom-hole pressure (psi) 2000 Maximum dimensionless time (pore volume)

82 Case Study 10: One-Dimensional Waterflooding in X Direction with Capillary Pressure This case study is a o ne-dimensional incompressible waterflooding simulation problem in x-direction with capillary pressure. This case study is almost the same as case study 9, the differences are that the capillary pressure is considered and numerical solution uses one hundred and sixty grid blocks with third-order TVD method. The capillary pressure parameters are shown in Table The schematic of reservoir and well locations are the same as case study 9, which are shown in Figure Terwilliger et al. (1951) provide an analytical solution of one-dimensional immiscible flow with capillary pressure. In his theory, there are two zones for an immiscible flow with capillary pressure. One is stabilized zone, which refers to a portion of the saturation distribution curve maintaining the same shape from a point until the time of water breakthrough. The remainder of the saturation profile is the non-stabilized zone. The equations used in the analytical solution are divided into two parts, stabilized zone and unstabilized zone. In the stabilized zone, all saturation points move forward at the same rate, thus the saturation distribution curve keeps the same shape until water breakthrough. Equation (3.47) is the fractional flow function describing fluid flow behavior of water and oil. Neglecting dip angle, water fractional equation becomes: kkro A Pc 1+ qµ o x fw =. (3.57) kroµ w 1+ k µ rw o where, A: cross-sectional area of flow path, cm 2 56

83 k: absolute permeability, Darcy x: distance in x-direction, cm μ: viscosity, cp q: total flow rate, cm 3 /s P c : capillary pressure, atm Because capillary pressure is a function of water saturation, chain rule of differentiation can be used to replace dp c /dx: Pc Pc Sw =. (3.58) x S x w Then Equation (3.57) becomes kkro A Pc S w 1+ qµ o Sw x fw =. (3.59) kroµ w 1+ k µ rw o The water fractional curve of the stabilized zone is the straight line portion of a f w vs S w curve constructed by Equation (3.48). The straight line is a tangent line to this curve and starts from the equilibrium water saturation at a point where f w =0. Now we read Sw and fw values from straight line portion. Then we substitute Sw and fw in fractional flow Equation (3.59) with these new S w and f w values. kkro A Pc S w 1+ qµ o Sw x fw_ new =. (3.60) kroµ w 1+ k µ rw o Now we solve for S w / x in Equation (3.60) at corresponding water saturations. The length and shape of this stabilized zone is given by 57

84 L = S w x dsw, (3.61) S Swir w where, x : distance in x-direction, cm L: distance in the x-direction (cm) from irreducible water saturation (S wir ) to any water saturation (S w ) This method does not provide the lateral position of the stabilized zone. Hence, the lateral location of stabilized zone is obtained by matching the analytical solution with UTCOMP simulation. The unstabilized zone can be calculated by setting Equation (3.58) to zero. Then, the water fractional curve of the non-stabilized zone is Equation (3.48), which establishes fw as a f unction of Sw. The values of f w / S w in Equation (3.49) can be determined by differentiating Equation (3.48). Thus Equation (3.49) relates the velocity of each point of saturation with f w / S w. The saturation profile in this zone can be calculated, if we use Equation (3.53). Table 3.26: Capillary pressure parameters for Case Study 10. pc ( ( / )) C psi md dyne cm 3 E pc 2 σ wo (dynes/cm) Case Study 11: One-Dimensional Waterflooding in Z Directions without Capillary Pressure This case study is a o ne-dimensional incompressible waterflooding simulation problem in z-direction without capillary pressure. This case study is almost the same as case study 9, except that the reservoir is vertical and the water is injected from top of the 58

85 reservoir. Oil is drained downward by gravity and produced from the bottom. The grid block dimension is The schematic of reservoir and well locations are shown in Figure Figure 3.11: Schematic of reservoir and the well locations for one-dimensional waterflooding in the z- direction with no capillary pressure. The analytical solution is applied by Beckley-Leverett theory. The fractional flow of water can be calculated using the Equation (3.47) by neglecting capillary pressure and substituting a dip angle of 90. Because the water is injected from the top of the vertical reservoir, positive sign in Equation (3.47) is taken. As we mentioned previously, Corey s model parameters for relative permeability are the same with case study 9, which are given in Table kkro A 1+ ρ g sin 90 qµ o fw = kroµ w 1+ k µ ( ( )) rw o. (3.62) Equation (3.56) is used to solve for the water saturation of front as following: 59

86 fw df = w S ds = w w Sw Swf. Water saturation profile can be found by substituting the derivative of Equation (3.62) into the dimensionless Beckley-Leverett Equation (3.53) as x D = (f w /ds w )t D. Δρ is chosen as g/cm Case Study 12: Miscible WAG Displacement with Secondary Displacements and Low-WAG injection This case study is a one-dimensional fractional flow to displace oil by a miscible solvent in the presence of an immiscible aqueous phase. Walsh and Lake (1989) derived the analytical solution for this problem. In order to examine effects of simultaneous water-solvent injection and the performance of various simulators, five case studies (12 through 16) are run with the following reservoir geometry and well locations (Figure 3.12). Figure 3.12: Schematic of miscible WAG displacement reservoir and well locations. The five case studies have almost the same simulation process, except for the initial water saturation and injection conditions. Reservoir and fluid property data of case study 12 are listed in Table Relative permeability parameters for Corey s correlation are given in Table Also Table 3.30 contains well operation conditions. In the simulation process, water and solvent are injected at the same time with total 60

87 injection rate of ft 3 /day. The solvent is totally dissolved in oleic phase with no trapped oil. The viscosity of water and solvent mixture is calculated using linear mixing rule. Third-order TVD method is used in UTCOMP and one-point upstream-weighting is used in simulations by CMG_STARS (one-point upstream-weighting is the only method in CMG_STARS). The comparison of analytical solution and simulation result of UTCOMP, GPAS, and CMG is carried out up to 0.6 pore volume fluid (water and solvent) injected. Figure 3.13 shows the fractional flow curves for water/oil and water/solvent. Water Fractional Flow Water/Oil Water/Solvent Water Saturation Figure 3.13: Fractional flow curves for water/oil and water/solvent flow. The water-solvent fractional flow curves form the water-oil curve is constructed by replacing the oil viscosity by the solvent viscosity in Equation (3.48). 61

88 Initial water saturation (S wi ) is uniform and no solvent is present initially (t D = 0). The injected condition (x D =0) is two-phase mixture of solvent and water specified by the water fractional flow (f wj ) at the injection end. WR fwj =. (3.63) 1 + W R The water fractional flow is then related to the water-solvent ratio. Where, W R is the ratio of water solvent injected simultaneously; both volumes are in reservoir volumes. The velocity of a continuous variation in concentration is given by f w vcw =, Sw xd (3.64) v Cs ( Cso fo ) ( C S ) = so o xd. (3.65) The velocity of an abrupt saturation change or a shock is given by fw v Cw =, S w (3.66) v Cs ( Cso fo ) ( C S ) =, (3.67) so o where, Cso is volume fraction of solvent in the oleic phase and Δ represents change between the upstream and the downstream value. The solvent travels through the medium with a miscible wave of constant velocity expressed by fw 1 ( 1 Csw ) vc =, (3.68) Sw { 1 SOM ( 1 CsT ) ( 1 Csw )} where, 62

89 C sw : volume fraction of solvent in the aqueous phase S OM : miscible-flood residual hydrocarbon saturation C st : volume fraction of solvent in the residual phase f w : water fractional flow S w : water saturation This equation is slope of a straight line through the point of ( {[1-S OM (1-C ST )]/(1-C SW )}, 1/(1-C SW ) ) to the point on the solvent-water fractional flow curve. This equation involves the effects of solvent water solubility, trapped oil saturation, and solvent partitioning in the trapped oil saturation. If the solvent is completely dissolved in oleic phase and no trapped oil present, the previous equation can be simplified as following: fw 1 vc =. S 1 w (3.69) The followings are the calculations and equations used in numerical simulation process of case study 12 through 16. Simulation by UTCOMP Because water compressibility is taken as zero in this case, the volume of water at surface condition and reservoir condition is almost the same. Volume of CO2 changes a lot; so CO 2 volume factor should be known from Vreservoir ρs tan dard Bo = =. (3.70) V ρ s tan dard reservoir Density of CO2 at standard condition and reservoir condition can be determined from the literature and output file of simulation. ρs tan dard lb cuft 3 3 Bo = = = ft ρ lb cuft scf. (3.71) reservoir 63

90 Total injection rate is ft D at reservoir condition; fwj also refers the water injection ratio at reservoir condition. The surface injection rate of water and CO2 are calculated as follows: For f wj = 0.3, 3 3 qw reservoir ft day ft = = SinceC w = 0, tan = scf day, w s dard 3 3 qco 2 reservoir ft day ft = = Since, B = ft scf, o qco s tan dard ft day ft scf Mscf day = =, tan = scf day and w s dard q co2 s dard 4 = Mscf tan will be used as the surface injection rate for water and CO 2 in UTCOMP (IQTYPE=4). For f wj = 0.7, 3 3 qw reservoir ft day ft = = SinceC w = 0, tan = scf day, w s dard 3 3 qco reservoir ft day ft day = =. 3 3 Since, B = ft scf, o qco s tan dard ft day ft scf Mscf day = =, 64

91 s tan dard = scf day and q co2 s dard 4 = Mscf tan will be used as the surface injection rate for water and CO 2 in UTCOMP (IQTYPE=4). Simulation by CMG: In the beginning, we can set up the compressibility of all components in CMG as zero or very close to zero; so all the components are incompressible. By this way, the injection rate of condition. The flow rate of injection rate of water and solvent. For f wj = 0.3, qw@ s tan dard = scf day, ft D ay is both under the reservoir condition and standard ft D ay can be directly used as the total surface q = ft day = scf day. 3 solvent@ s tan dard For f wj = 0.7, qw@ s tan dard = scf day, q = ft day = scf day. 3 solvent@ s tan dard The above values are water and solvent surface injection rate used in GMG 65

92 Table 3.27: Reservoir and fluid property for Case Study 12. Grid blocks dimension in x, y, and z directions Length (ft) 10 Width (ft) 1 Thickness (ft) 1 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Solvent viscosity (cp) 0.1 Oil viscosity (cp) 1 Water viscosity (cp) 0.5 Water density (lb /ft 3 ) 62.4 Initial water saturation 0.2 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Reservoir fluid initial composition n-c 10 H (mole fraction) Longitudinal dispersivity (ft) 0.04 Transverse dispersivity (ft) 0 Table 3.28: Relative permeability parameters for Case Study 12. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation

93 Table 3.29: Well operation conditions for Case Study 12. Total injection rate (ft 3 /day) Water fractional flow (f wj ) 0.3 Production well bottom-hole pressure (psi) 2000 Maximum dimensionless time (pore volume) Case Study 13: Miscible WAG Displacement with secondary displacements and high-wag injection This case is almost the same as case study 12, except that water fractional flow injection increased to 0.7. Hence, well operation conditions are presented in Table The comparison of analytical solution and simulation results is compared at 0.4 pore volume. Table 3.30: Well operation conditions for Case Study 13. Water fractional flow (f wj ) 0.7 Maximum dimensionless time (pore volume) Case Study 14: Miscible WAG Displacement with tertiary displacements and low-wag injection This case is almost the same as case study 12, except that initial water saturation increased to Table 3.31 shows well operation conditions. The comparison of analytical solution and simulation results is made at 0.25 pore volume. Table 3.31: Well operation conditions for Case Study 14. Initial Water Saturation (fraction) 0.65 Maximum time (pore volume)

94 Case Study 15: Miscible WAG Displacement with tertiary displacements and high-wag injection This case is almost the same as case study 12, except that water fractional flow injection increased to 0.7 and the initial water saturation is changed to Well operation conditions are given in Table The comparison of analytical solution and simulation results is made at 0.3 pore volume. Table 3.32: Well operation conditions for Case Study 15. Initial Water Saturation (fraction) 0.65 Water fractional flow (f wj ) 0.7 Maximum time (pore volume) Case Study 16: Miscible WAG Displacement with tertiary displacements and water-free solvent injection This case is almost the same as case study 12, except that the initial water saturation increases to 0.65 and the water fractional flow injection decreased to 0. Well operation conditions are presented in Table The comparison of the analytical solution and simulation results is at 0.25 pore volume. Table 3.33: Well operation conditions for Case Study 16. Initial Water Saturation (fraction) 0.65 Water fractional flow (f wj ) 0 Maximum time (pore volume)

95 Case Study 17: Dietz Displacement with miscible displacement and lowlongitudinal dispersivity This case study is a two-dimensional miscible displacement in a dipping reservoir with low-longitudinal dispersivity, which can be used to validate the dipping reservoir option and gravity term with the analytical solution of Dietz displacement (Dietz, 1953). Dietz displacement refers to a stable displacement under segregated flow conditions, the angle between the fluid interface and the direction of flow will reach steady state at a constant value throughout the flooding. The displacement angle will be affected by reservoir dip angle (Figure 3.14), viscous and gravity forces acting on the fluids. α =30 Figure 3.14: Schematic of Dietz Displacement reservoir and well locations. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table And Table 3.37 contains well operation conditions. Solvent is injected at the top and displaced oil moves downward at rate of 10 ft 3 /day using grid blocks. The concentration front is smeared by a longitudinal dispersivity of 1ft, but the velocity of the 0.5 solvent concentration front is not changed with the longitudinal dispersivity value. The position of 0.5 s olvent concentration is determined for a series of pore volumes injected, which is 0.1, 0.2, 0.3, 0.4, 0.5, and

96 The following equations are used in case study 17 and case study 18. Dietz stability equation for solvent displacement of oil downward becomes: 1 M e tan β = + tanα, (3.72) MN cosα e ge where α is reservoir dip angle and β is interface angle. End-point gravity number for oil-gas is defined as k Ak ro ( ρo ρs) g N ge =. (3.73) µ Q o The end point mobility ratio is defined as k rs k ro M e =, (3.74) µ s µ o where, k ro : end point relative permeability, dimensionless k rs : end point solvent relative permeability, dimensionless µ o : oil viscosity, Pa s µ s : solvent viscosity, Pa s A: cross sectional area of flow path, m 2 k: absolute permeability, m 2 ρ o : oil density, kg/m 3 ρ s : solvent density, kg/m 3 g : gravity, 9.81 m/s 2 Q: injection rate, m 3 /day Because the mobility ratio for this case is 1, Dietz stability Equation (3.72) can be simplified to tan β = tanα = 30. (3.75) 70

97 Table 3.34: Reservoir and fluid property for Case Study 17. Grid blocks dimension in x, y, and z directions Length (ft) 100 Width (ft) 10 Thickness (ft) 10 Dip Angle θ x (degree) 30 Dip Angle θ y (degree) 0 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Solvent viscosity (cp) 1 Oil viscosity (cp) 1 Water viscosity (cp) 0.5 Water density (lb /ft 3 ) 62.4 Initial water saturation 0.2 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Fluid specific density difference 0.46 Reservoir fluid initial composition (mole fraction) n-c 10 H Longitudinal dispersivity (ft) 1 Transverse dispersivity (ft) 0 Table 3.35: Relative permeability parameters for Case Study 17. Water Oil Endpoint phase relative permeability Relative permeability Exponent Residual saturation

98 Table 3.36: Well operation conditions for Case Study 17. Total injection rate (ft 3 /day) 10 Production well bottom-hole pressure (psi) 2000 Maximum dimensionless time (pore volume) Case Study 18: Dietz Displacement with miscible displacement and highlongitudinal dispersivity In order to verify that longitudinal dispersivity will not affect the interface angle, a run is made based on case study 17. This case study is a t wo-dimensional miscible displacement in a dipping reservoir with high longitudinal dispersivity. Input data in Table 3.35 through Table 3.37 are also applied in this case, except that grid block dimension and longitudinal dispersivity are changed to the values given in Table The position of 0.5 solvent concentration is determined at the same pore volumes injected in case study 17. Table 3.37: Input data for Case Study 18. Grid block dimension in x, y, and z directions Longitudinal dispersivity (ft) Case Study 19: Dietz Displacement with immiscible displacement and no longitudinal dispersivity This case study is a t wo-dimensional immiscible displacement in a dipping reservoir without longitudinal dispersivity. In this case, Dietz theory is tested in immiscible condition. As shown in Figure 3.15, the reservoir dip angle is

99 α = -30 Figure 3.15: Schematic of Dietz Displacement reservoir and well locations. Water injected at the bottom of the tilted reservoir is at a rate of 4 ft 3 /day and oil is displaced upward. The grid block dimension is There is no longitudinal or transverse dispersivities in this case. The position of water saturation is found at 0.1, 0.2, 0.3, and 0.4 pore volume injected. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Table 3.40 contains well operation conditions. Dietz stability equation for immiscible displacement is the same as Equation (3.72), but the definition of end-point gravity number and end- point mobility ratio are different. End-point gravity number for oil and water is defined as k Ak ro ( ρw ρo) g N ge =. (3.76) µ Q o End-point mobility ratio for oil and water system is k rw k ro M e =, (3.77) µ w µ o where μ w is water viscosity in Pa s and ρ w is water density in kg/m 3 73

100 Table 3.38: Reservoir and fluid property for Case Study 19. Grid blocks dimension in x, y, and z directions Length (ft) 100 Width (ft) 10 Thickness (ft) 10 Dip Angle θ x (degree) -30 Dip Angle θ y (degree) 0 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 500 Permeability in y-direction (md) 500 Permeability in z-direction (md) 500 Solvent viscosity (cp) 1 Oil viscosity (cp) 1 Water viscosity (cp) 0.5 Water density (lb /ft 3 ) 62.4 Initial water saturation 0.2 Initial reservoir pressure (psi) 2000 Water compressibility (psi -1 ) 0.0 Fluid specific density difference 0.35 Reservoir fluid initial composition n-c 10 H (mole fraction) Longitudinal dispersivity (ft) 0 Transverse dispersivity (ft) 0 Table 3.39: Relative permeability parameters for Case Study 19. Water Oil Endpoint phase Relative permeability Relative permeability Exponent Residual saturation

101 Table 3.40: Well operation conditions for Case Study 19. Total injection rate (ft 3 /day) 4 Production well bottom-hole pressure (psi) 2000 Maximum dimensionless time (pore volume) Case Study 20: Two-Dimensional Convection-Diffusion Equation Convection-Diffusion mass-transfer is one of the most important EOR mechanisms in solvent-based on EOR techniques. In this case study, in order to validate the formulation of the conservation equations, numerical solution of two-dimensional miscible flow with longitudinal and transverse dispersion is compared with the analytical solution of the two-dimensional convection-diffusion equation. Figure 3.16 shows the schematic of a two-dimensional first-contact miscible displacement process in its initial stage. Figure 3.16: Schematic of Two-Dimensional Convection-Diffusion numerical model for Case Study 20. The homogenous reservoir considered is almost oil saturated, except for a small amount of solvent present in the black rectangular area A in Figure 3.16 while oil concentration is zero in this area. Oil is uniformly injected along the left edge into the 75

102 reservoir (v y =0). The solvent is diluted and drained forward by viscous forces and produced along the right edge of the reservoir. The injection rate is the same as the production rate. The injected oil has properties identical to those of the resident oil and solvent, in terms of critical properties. The longitudinal and transverse dispersivities are 1.0 ft and 0.5 ft, respectively, in the simulation model. In order to achieve the purpose of velocity in y-direction equaling to zero, in the simulations, oil is injected into the reservoir at the x-z cross section (Figure 3.16) with a rate of 0.5 bbl/day at reservoir condition in CMG and MSCF/day at standard condition in UTCOMP. The oil injected rate of UTCOMP is obtained from the output of CMG, for the purpose of result comparison with the same input information. The block address for solvent source (SOL) is (10, 1, 50) in x, y, and z directions, if we define the positive direction of z axis up. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Table 3.44 contains well operation conditions and Table 3.45 contains component properties. Figure 3.17: Schematic of Two-Dimensional Convection-Diffusion analytical solution for Case Study

103 Figure 3.17 presents the schematic of analytical solution. The point of initial solvent source (SOL) is the origin, where x=0 and z=0. The comparative results of normalized concentration (C/C o ) of the analytical solution and simulations by UTCOMP and CMG are obtained at 2 da ys and z equaling to zero. Two-dimensional convectiondiffusion mass-transfer equation, under the assumption of incompressible fluid, constant velocity and dispersion coefficients, and infinite medium, can be formulated in the following form (Cleary and Ungs, 1978): 2 2 C C C C C = K L K 2 T v 2 x v z, (3.78) t x z x z where K L and K T represent longitudinal and transverse dispersion coefficients, respectively. 2 vx KL = αtv+ ( αl αt), (3.79) v K 2 vz = α v+ ( α α ), (3.80) v T T L T q v =, (3.81) Aφ where v is average velocity (interstitial velocity) and α L and α T are longitudinal and transverse dispersivity, respectively. Initially, the solvent is only in the black rectangular area in Figure Thus, the boundary conditions (BC) and initial conditions (IC) can be expressed as Co x1 x x2; z1 z z2 C( xz,,0) = 0 other x & z (3.82) C x = x = (3.83) 77

104 C z = z = (3.84) The solution of Equation (3.78) with above BC and IC is Ci x x1 vt x x x2 vt x z z1 v z t z z2 v z t C ( x, z, t) = erf erf erf erf (3.85) 4 ( 4Dt) ( 4 ) ( 4 ) 2 DT ( 4 ) 2 D L Dt L Dt T Dt T T With the assumption of uniform flow in the simulation model, Equation (3.85) can be simplified as C i x x1 vt x x2 vt z z 1 z z 2 C ( x, z, t) = erf erf erf erf ( 4Dt L ) ( 4Dt L ) ( 4Dt T ) ( 4Dt T ). (3.86) Though we use third-order TVD method of numerical dispersion control, the numerical longitudinal dispersivity could not be avoided and ignored in this case study. Equation (3.87) and Equation (3.88) are used to determine numerical longitudinal and transverse dispersivities from Fanchi (1983) and Lantz (1971) models. In order to match the analytical solution, numerical dispersivity and input dispersivity are summed up as total dispersivity in analytical solution (Equation (3.89) and Equation (3.90)). x 1 α L _ num = = 0.5 ft, (3.87) 2 2 α =, (3.88) T _ num 0 α = α + α =, (3.89) L _ AnalyticalSolution L _ input L _ num 1.5 ft α = α + α =. (3.90) T _ AnalyticalSolution T _ input T _ num 0.5 ft 78

105 Table 3.41: Reservoir and fluid property for Case Study 20. Grid blocks dimension in x, y, and z directions Length (ft) 50 Width (ft) 0.5 Thickness (ft) 50 Porosity (fraction) 0.38 Rock compressibility (psi -1 ) Reservoir temperature ( F) 60 Permeability in x-direction (md) 50 Permeability in y-direction (md) 50 Permeability in z-direction (md) 50 Solvent viscosity (cp) 1.0 Oil viscosity (cp) 1.0 Water viscosity (cp) 1.0 Water density (lb /ft 3 ) 62.4 Initial water saturation 0.0 Initial reservoir pressure (psi) 4500 Water compressibility (psi -1 ) Block address in x, y, and z UTCOMP (10,1,51) directions of solvent source (SOL) CMG (10,1,50) Longitudinal dispersivity (ft) 1.0 Transverse dispersivity (ft) 0.5 Table 3.42: Relative permeability parameters for Case Study 20. Water Oil Endpoint phase Relative permeability 1 1 Relative permeability Exponent 1 1 Residual saturation Table 3.43: Well operation conditions for Case Study 20. UTCOMP CMG Total injection rate (MSCF/day) 0.5 (bbl/day) Production well 4500 (psi) 0.5 (bbl/day) Maximum time (Day)

106 Table 3.44: Component Properties for Case Study 20. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) OIL SOL Case Study 21: Three Dimensional Waterflooding This case study is a t hree-dimensional incompressible waterflooding simulation problem with a horizontal homogeneous reservoir. A reservoir with a size of ft 3 is used for a water-flooding case study with 8000 grid blocks. Water is injected at a constant injection rate of 3500 STB/day. The production well is produced at a constant bottom-hole pressure of 200 psi. Reservoir and fluid property data are listed in Table In this case study, Corey s model is employed in this case study to generate water/oil and gas/oil relative permeability data. Relative permeability parameters for Corey s correlation are given in Table Table 3.47 contains well operation conditions. Table 3.49 shows the component properties. The relative permeability curves for water/oil and oil/gas are shown in Figure 3.18 and Figure

107 Table 3.45: Reservoir and fluid property for Case Study 21. Grid blocks dimension in x, y, and z directions Length (ft) 1600 Width (ft) 1600 Thickness (ft) 50 Porosity (fraction) 0.2 Rock compressibility (psi -1 ) 0.0 Reservoir temperature ( F) 60 Permeability in x-direction (md) 1000 Permeability in y-direction (md) 1000 Permeability in z-direction (md) 100 Water viscosity (cp) 1.0 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) 0.0 Initial water saturation (fraction) 0.2 Initial reservoir pressure (psi) 200 Reservoir fluid initial composition (mole fraction) C Table 3.46: Relative permeability parameters for Case Study 21. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.47: Well operation conditions for Case Study 21. Water injection rate (STB/day) 3500 Production well bottom-hole pressure (psi) 200 Maximum time (Day) 6000 Table 3.48: Component Properties for Case Study 21. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) C

108 Relative permeability water relative permeability Oil relative permeability Water saturation Figure 3.18: The relative permeability curve for water and oil flow of Case Study 21. Relative permeability Oil relative permeability Gas relative permeability Gas saturation Figure 3.19: The relative permeability curve for oil and gas flow of Case Study

109 Case Study 22: Three-Dimensional Gas and Solvent Injection with Three Hydrocarbon Components in Reservoir This case study is a three-dimensional gas and solvent injection simulation problem with three oil components in the reservoir. The reservoir with a size of ft 3 is used for a three-component reservoir fluid mixture over 8000 grid blocks. CO 2 and methane are injected at a constant injection rate of 20 MMSCF/day. The production well is produced at a constant bottom-hole pressure of 3000 psi. Reservoir and fluid property data are listed in Table Corey s model is chosen to evaluate relative permeability with given parameters in Table Table 3.52 contains well operation conditions. Table 3.52 shows the component properties. All binary coefficients are zeros except those given in Table The relative permeability curves for water/oil and oil/gas are also shown in Figure 3.20 and Figure Table 3.49: Reservoir and fluid property for Case Study 22. Grid blocks dimension in x, y, and z directions Length (ft) 1600 Width (ft) 1600 Thickness (ft) 200 Porosity (fraction) 0.3 Rock compressibility (psi -1 ) Reservoir temperature ( F) 80 Permeability in x-direction (md) 1000 Permeability in y-direction (md) 1000 Permeability in z-direction (md) 100 Water viscosity (cp) 0.8 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) Initial water saturation (fraction) 0.25 Initial reservoir pressure (psi) 3000 Reservoir fluid initial CO composition C (mole fraction) NC

110 Table 3.50: Relative permeability parameters for Case Study 22. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.51: Well operation conditions for Case Study 22. Injected gas/solvent CO composition (mole fraction) C Gas inject rate (MMSCF/day) 20 Production well bottom-hole pressure (psi) 3000 Maximum time (Day) Table 3.52: Component Properties for Case Study 22. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) CO C NC Table 3.53: Binary coefficients for Case Study 22. CO 2 and C CO 2 and NC 16 84

111 water relative permeability Oil relative permeability Relative permeability Water saturation Figure 3.20: The relative permeability curve for water and oil flow of Case Study 22. Relative permeability Oil relative permeability Gas relative permeability Gas saturation Figure 3.21: The relative permeability curve for oil and gas flow of Case Study

112 Case Study 23: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Reservoir This case study is a three-dimensional gas and solvent injection simulation problem with six oil components in the reservoir. The reservoir with a size of ft 3 is used for a six component reservoir fluid mixture over 9,600 grid blocks. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Table 3.56 contains well operation conditions. Table 3.57 shows the component properties. All binary coefficients are zeros except those given in Table A mixture of methane and propane is injected at the rate of 5 MMSCF/day at the same time. The production well is produced at a constant bottom-hole pressure of 3100 psi. 86

113 Table 3.54: Reservoir and fluid property for Case Study 23. Grid blocks dimension in x, y, and z directions Length (ft) 3000 Width (ft) 3000 Thickness (ft) 300 Porosity (fraction) 0.3 Rock compressibility (psi -1 ) Reservoir temperature ( F) 150 Permeability in x-direction (md) 100 Permeability in y-direction (md) 100 Permeability in z-direction (md) 100 Water viscosity (cp) 1.0 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) Initial water saturation (fraction) 0.25 Initial reservoir pressure (psi) 3100 C C Reservoir fluid initial C composition C (mole fraction) C C Table 3.55: Relative permeability parameters for Case Study 23. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation

114 Table 3.56: Well operation conditions for Case Study 23. Injected gas/solvent C composition (mole fraction) C Gas inject rate (MMSCF/day) 5 Production well bottom-hole pressure (psi) 3100 Maximum time (Day) 9000 Table 3.57: Component Properties for Case Study 23. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) C C C C C C Table 3.58: Binary coefficients for Case Study 23. C 1 and C C 1 and C 15 C 1 and C Case Study 24: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Large Reservoir This case is almost the same as case study 23, except that this is a larger reservoir size with a higher number of grid blocks of 33,750 and is run for 10,000 days. The different input data from case study 23 are listed Table Input information from Table 3.55 to Table 3.59 are also used for this case, except those parameters in Table 3.60 are changed to new values. 88

115 Table 3.59: Input data for Case Study 24. Grid blocks dimension in x, y, and z directions Length (ft) 5625 Width (ft) 5625 Maximum time (Day) Case Study 25: Two-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir with Twenty Oil Components in the Reservoir The reservoir with a size of ft 3 is used for a twenty component reservoir fluid mixture over 1600 grid blocks. A mixture of methane and CO 2 is injected at a co nstant bottom-hole pressure of 2900 psi. The production well produces at the constant bottom-hole pressure of 2400 psi. Reservoir and fluid property data are listed in Table Since there is not enough information for the 20 components, component 7 to 20 have the same properties. In this case study, we use Corey s model for generation of water/oil and gas/oil relative permeability data. Relative permeability parameters for Corey s correlation are given in Table Table 3.63 contains well operation conditions. Table 3.64 shows the component properties. All binary coefficients are zeros except those given in Table

116 Table 3.60: Reservoir and fluid property for Case Study 25. Grid blocks dimension in x, y, and z directions Length (ft) 2000 Width (ft) 2000 Thickness (ft) 20 Porosity (fraction) 0.25 Rock compressibility (psi -1 ) Reservoir temperature ( F) 260 Permeability in x-direction (md) 100 Permeability in y-direction (md) 100 Permeability in z-direction (md) 10 Water viscosity (cp) 0.79 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) Initial water saturation (fraction) 0.25 Initial reservoir pressure (psi) 2850 CO C Reservoir fluid initial C composition C (mole fraction) C C Component 7-10 C Component C Table 3.61: Relative permeability parameters for Case Study 25. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation

117 Table 3.62: Well operation conditions for Case Study 25. Injected gas/solvent composition (mole fraction) CO C C C Constant bottom hole injection pressure (psi) 2900 Production well bottom-hole pressure (psi) 2400 Maximum time (Day) 6000 Table 3.63: Component Properties for Case Study 25. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) CO C C C C C C Table 3.64: Binary coefficients for Case Study 25. CO 2 and C 1 CO 2 and C 2-3 CO 2 and C CO 2 and C CO 2 and C CO 2 and C

118 Case Study 26: Three-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir This case study is almost the same as case study 25, except that this is a threedimensional reservoir with 33,750 grid blocks. The data given in Table 3.66 are used in this case study and the rest data can be drawn from Table 3.61 through Table Table 3.65: Input data for Case Study 26. Grid blocks dimension in x, y, and z directions Length (ft) 3750 Width (ft) 3750 Height (ft) Case Study 27: Scenario Two of SPE Fifth Comparative Solution Project The SPE Fifth Comparative Solution Projection (Killough and Kossack, 1987) is designed to test the abilities of compositional simulators to model the water alternating gas (WAG) injection process into a volatile oil reservoir. Three scenarios are generated for this comparative project. In this case study, scenario two is chosen. The compositional fluid description is a six component Peng-Robison (PR) characterization. A threedimensional reservoir with is used for this simulation case. The injection well is located at grid block (1, 1, 1) and the production well is located at grid block (7, 7, 3). Both wells are located in the center of the grid. Figure 3.22 shows the schematic of reservoir and well locations. 92

119 Figure 3.22: Schematic of Reservoir and well locations for Case Study 27. The producer is constrained to produce at a maximum oil rate of 12,000 STB/day with the minimum bottom-hole pressure of 3000 psi. The injection well is constrained to injection at a co nstant rate of 20 MMSCF/day for gas and 45MSTB/day for water and with a maximum bottom-hole pressure of 4500 psi. WAG injection starts initially on a standard three month WAG cycle. A limiting GOR of 10 MCF/STB and a WOR limit of 5 STB/STB are used to shut-in the well. The simulation should be run for 20 years. Reservoir and fluid property data are listed in Table The relative permeability data are shown in Table Table 3.68 contains well operation conditions. Reservoir data by layers are shown in Table Table 3.71 shows the component properties. All binary coefficients are zeros except those given in Table The relative permeability curves for water/oil and oil/gas are shown in Figure 3.23 and Figure In UTCOMP input file, there is an option to use different formulations for calculating fluid s relative permeability curve. This option is controlled by an index called IPERM. In the current case study, IPERM is chosen to be 8 to match the relative permeability data used in SPE fifth comparative case study (Killough and Kossack, 1987). Also, by selecting this index to be 8, capillary pressure curves for oil-water and oil-gas will be automatically calculated using Equation 93

120 (3.91) and Equation (3.92). The Fluids relative permeability formulation for this index is from Equation (3.93) to Equation (3.97). Pcow Pcog ( S ) = exp (3.91) ( S ) exp =. (3.92) k rw S1 0.2 = (3.93) k rg S = (3.94) k 0.7 S row = 2. (3.95) k 0.65 S rog = ( )( ) ( ) ro row rw rog rg rw rg. (3.96) k = k + k k + k k + k. (3.97) 94

121 Table 3.66: Reservoir and fluid property for Case Study 27. Grid blocks dimension in x, y, and z directions Length (ft) 500 Width (ft) 500 Porosity (fraction) 0.3 Rock compressibility (psi -1 ) Reservoir temperature ( F) 160 Water viscosity (cp) 0.7 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) Initial water saturation (fraction) 0.2 Reservoir fluid initial composition (mole fraction) C C C C C C Table 3.67: Relative permeability and capillary pressure data for Case Study 27. S w K rw k row P cow S liq K rliq k rg P cgo

122 Table 3.68: Well operation conditions for Case Study 27. Injected gas/solvent composition (mole fraction) C C C Gas inject rate (MMSCF/day) 20 Water injection rate (STB/day) Maximum injection bottom hole pressure (psi) 4500 Oil production rate (STB/day)) Minimum bottom-hole pressure (psi) 3000 Reference Depth (ft) 8400 Wellbore radius (ft) 0.25 Maximum time (Day) 7300 Layer Table 3.69: Reservoir data by layers for Case Study 27. Horizontal permeability (md) Vertical Permeability (md) Thickness (ft) Elevation (ft) Initial Pressure (psi) Table 3.70: Component Properties for Case Study 27. Component P ci (psi) T ci ( R) C ritz ω i W ti (lb/lb-mole) C C C C C C Table 3.71: Binary coefficients for Case Study 27. C 1 and C 15 C 1 and C C 3 and C 15 C 3 and C

123 Relative permeability Water relative permeability Oil relative permeability Water saturation Figure 3.23: The relative permeability curve for water and oil flow of Case Study 27. Relative permeability Liquid relative permeability Gas relative permeability Liquid saturation Figure 3.24: The relative permeability curve for liquid and gas flow of Case Study

124 Case Study 28: Gas Injection Involving second Hydrocarbon Phase Generation This case study is a three-phase hydrocarbon flash calculation simulation problem with a three-dimensional heterogeneous reservoir. A reservoir with a size of ft 3 is used to test the simulators for three-phase flash calculation over 100,000 grid blocks. The object of this case is to test three hydrocarbon phases simulation by UTCOMP and CMG. However, CMG does not have the capacity to conduct three hydrocarbon phases calculation; only have a model called WINPROP which can simply detect three hydrocarbon phases. The Dykstra-Parsons coefficient (V DP ) of permeability is about 0.5. Figure 3.25 and Figure 3.26 give the 3D view of permeability and porosity distribution throughout reservoir. A mixture of seven fluid components is injected at constant bottom-hole pressure of 1250 psi. The production well is produced at the constant bottom-hole pressure of 1100 psi. Reservoir and fluid property data are listed in Table Relative permeability parameters for Corey s correlation are given in Table Table 3.75 contains well operation conditions. Table 3.75 shows component properties. All binary coefficients are zeros except those given in Table The relative permeability curves for water/oil and oil/gas are given in Figure 3.27 and Figure

125 Figure 3.25: Permeability distribution of Case Study 28. Figure 3.26: Porosity distribution of Case Study

126 Table 3.72: Reservoir and fluid property for Case Study 28. Grid blocks dimension in x, y, and z directions Length (ft) 2500 Width (ft) 2500 Thickness (ft) 200 Porosity (fraction) Data file Rock compressibility (psi -1 ) Reservoir temperature ( F) 105 Permeability in x-direction (md) Data file Permeability in y-direction (md) Data file Permeability in z-direction (md) 10 Water viscosity (cp) 0.79 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) 0.0 Initial water saturation (fraction) 0.25 Initial reservoir pressure (psi) 1100 CO C Reservoir fluid initial C composition C (mole fraction) C C C Table 3.73: Relative permeability parameters for Case Study 28. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation

127 Table 3.74: Well operation conditions for Case Study 28. Injected gas/solvent composition (mole fraction) CO C C 2-3 C 4-6 C 7-15 C C Constant bottom hole pressure injection (psi) 1250 Production well bottom-hole pressure (psi) 1100 Maximum time (Day) 4560 Table 3.75: Component Properties for Case Study 28. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) CO C C C C C C Table 3.76: Binary coefficients for Case Study 28. CO 2 and C 1 CO 2 and C 2-3 CO 2 and C CO 2 and C 7-15 CO 2 and C CO 2 and C

128 Relative permeability water relative permeability Oil relative permeability Water saturation Figure 3.27: The relative permeability curve for water and oil flow of Case Study 28. Relative permeability Oil relative permeability Gas relative permeability Gas saturation Figure 3.28: The relative permeability curve for liquid and gas flow of Case Study

129 Case Study 29: Primary, Secondary, and Tertiary Production Process in a Highly Heterogeneous Reservoir This case study is a series of oil recovery processes with a three-dimensional highly heterogeneous reservoir. A reservoir with a size of ft 3 is used to simulate the whole production process including primary production, water flooding, and WAG. The reservoir is divided into 200,000 grid blocks. The V DP of permeability is 0.9. Figure 3.29 and Figure 3.30 give the full view of permeability and porosity distribution through reservoir. For the first 200 days, there is only one well producing at a constant bottom-hole pressure of 3000 psi until the oil production rate is almost zero. From 200 days to 2010 days, water flooding is adopted as a secondary recovery method until WOR exceeds 90%. Between 2010 days and 2410 da ys, water flooding is replaced by WAG process. The WAG is performed on a 100-day cycle to improve recovery further. The gas injection is first on for 100 days, and then alternates with water flooding. The WAG process carries on for 2 cycles and stops at 2410 days. The well constraint of water or gas injection is constant well bottom-hole pressure of 4500 psi. Reservoir and fluid property data are listed in Table The absolute permeability data and porosity data are presented in various data files. Relative permeability parameters for Corey s correlation are given in Table Table 3.80 contains well operation conditions. Table 3.81 shows the component properties. All binary coefficients are zeros except those given in Table The relative permeability curves for water/oil and oil/gas are presented in Figure 3.31 and Figure

130 Figure 3.29: Permeability distribution of Case Study 29. Figure 3.30: Porosity distribution of Case Study

131 Table 3.77: Reservoir and fluid property for Case Study 29. Grid blocks dimension in x, y, and z directions Length (ft) 3500 Width (ft) 3500 Thickness (ft) 100 Porosity (fraction) Data file Rock compressibility (psi -1 ) Reservoir temperature ( F) 160 Permeability in x-direction (md) Permeability in y-direction (md) Data file Data file Permeability in z-direction (md) 10 Water viscosity (cp) 0.7 Water density (lb/ ft 3 ) 62.4 Water compressibility (psi -1 ) Initial water saturation (fraction) 0.2 Initial reservoir pressure (psi) 4000 C C Reservoir fluid initial C composition C (mole fraction) C C Table 3.78: Relative permeability parameters for Case Study 29. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation

132 Table 3.79: Well operation conditions for Case Study 29. Injected gas/solvent composition (mole fraction) C C C Injection well bottom-hole pressure (psi) 4500 Production well bottom-hole pressure (psi) 3000 Maximum time (Day) 2410 Table 3.80: Component Properties for Case Study 29. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) C C C C C C Table 3.81: Binary coefficients for Case Study 29. C 1 and C C 1 and C 20 C 3 and C C 3 and C

133 Relative permeability water relative permeability Oil relative permeability Water saturation Figure 3.31: The relative permeability curve for water and oil flow of Case Study 29. Relative permeability Oil relative permeability Gas relative permeability Gas saturation Figure 3.32: The relative permeability curve for liquid and gas flow of Case Study Case Study 30: Waterflooding in a Highly Heterogeneous Reservoir with 800,000 Gridblock and 16 Production/Injection Wells This case study is a three-dimensional highly heterogeneous reservoir. A reservoir with a size of ft 3 is used to test the simulators for a six component 107

134 reservoir fluid mixture with 800,000 grid blocks and 16 injection/production wells. The type of pattern employed in this case study is seven-spot well patterns as illustrated in Figure 3.33 and Figure The V DP of permeability and porosity is Figure 3.35, Figure 3.36 and Figure 3.37 depict the full view of permeability and porosity distribution through reservoir and depth of cell top distribution. Reservoir and fluid property data are listed in Table In this case study, we use Corey s model for generation of water/oil and gas/oil relative permeability data. Relative permeability parameters for Corey s correlation are given in Table Table 3.84 contains well operation conditions. Table 3.85 shows the component properties. All binary coefficients are zeros except those given in Table The reservoir is subjected to waterflooding. Figure 3.33: Two-dimensional well locations of Case Study

135 Figure 3.34: Three-dimensional well locations of Case Study 30. Figure 3.35: Permeability distribution of Case Study

136 Figure 3.36: Porosity distribution of Case Study 30. Figure 3.37: Depth of cell top distribution of Case Study

137 Table 3.82: Reservoir and fluid property for Case Study 30. Grid blocks dimension in x, y, and z directions Length (ft) 8000 Width (ft) 8000 Thickness (ft) 50 Porosity (fraction) Data file Rock compressibility (psi -1 ) Reservoir temperature ( F) 150 Permeability in x-direction (md) Permeability in y-direction (md) Data file Data file Permeability in z-direction (md) 100 Water viscosity (cp) 1.0 Water density (lb/ ft 3 ) Water compressibility (psi -1 ) Initial water saturation (fraction) 0.25 Initial reservoir pressure (psi) 3100 C C Reservoir fluid initial C composition C (mole fraction) C C Table 3.83: Relative permeability parameters for Case Study 30. Water Oil Gas Endpoint phase relative permeability Relative permeability Exponent Residual saturation Table 3.84: Well operation conditions for Case Study 30. Water injection rate (STB/day) 3500 Production well bottom-hole pressure (psi) 3100 Maximum time (Day)

138 Table 3.85: Component Properties for Case Study 30. Component P ci (psi) T ci ( R) V ci (ft 3 /lb-mole) ω i W ti (lb/lb-mole) C C C C C C Table 3.86: Binary coefficients for Case Study 30. C 1 and C C 1 and C C 1 and C

139 Chapter 4: Results 4.1 Introduction This chapter aims at providing a benchmark data-base, which can be used to reflect on the performance of in-house reservoir simulators (UTCOMP and GPAS) and commercial simulator (CMG). Furthermore, it may also help validate simulation results via comparisons with analytical solutions or the solution by commercial simulator CMG with regard to both existing and any future model developments and implementations. All the case studies that we conducted in this chapter have been elaborated and furnished in Chapter 3. This chapter primarily gives comparison of results. These comparisons of various case studies shed light on the reservoir simulators with regard to phase saturation (its profile or its 3D distribution), reservoir pressure profile, concentration profile, effluent concentration, full range of heterogeneity properties (depth of top cell, permeability, and porosity distribution), average pressure and production rate histories, gas-oil ratio (GOR), water-cut, and time-step selection during simulation process, among other factors. To this end, analytical solutions and varied numerical solutions of twenty validation cases are addressed. Moreover, comparative results obtained by different simulators for more complex field studies are provided as well. Obviously not all simulators could be used for this study: GPAS, because it is still under development and many functions required for running some of these cases are still unavailable at present (thus they could not be run with GPAS); CMG-GEM, because it does not possess threephase flash procedures. It is to be noted that the case study 28 is the second scenario of SPE Fifth Comparative Solution Project (Killough and Kossack, 1987). SPE Comparative Solution Project is a series of comparative solution projects initiated by the 113

140 Society of Petroleum Engineers (SPE). Case study 28 i s the only case having the comparative solution with SENSOR issued by Coats Engineering website. Materials presented in this chapter are important for various reasons. Comparison with analytical solution provides the framework for determining accuracy of simulations of both exact (that is identical) problems (here solutions by analytical and by simulation ought to be almost identical) and perturbed problems (here analytical solution will only offer and highlight the trend). As alluded to in Chapter 1, s tudying analytical solution gives us the opportunity to envision new problems, whose solution can be by both analytical and numerical simulation or simply by one of the two. For either case, we would have reasonable solutions. Furthermore, these analytical and simulated solutions, covered throughout the chapter, offer researchers the chance and ease to verify and justify their own problems, not exactly ours and yet similar. This surely gives them ample confidence since our solutions are always almost impeccably correct. 4.2 Comparative Results of Case Studies The following section shows various comparisons of analytical solutions and different simulators Case Study 1: One-Dimensional Incompressible Flow with Horizontal Displacement Results are shown in Figure 4.1. In this figure, the simulation result from all simulators matches the analytical result very well. 114

141 Pressure Drop (Psi), (P-Pinitial) Analytical Solution UTCOMP CMG GPAS Dimensionless Distance in X-Direction, (l/l) Figure 4. 1: Comparison of the pressure drop profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG for Case Study Case Study 2: One-Dimensional Incompressible Flow with Vertical Displacement Results are shown in Figure 4.2, the agreement between simulation results of UTCOMP, GPAS, and GPAS_COATS with analytical solution is excellent. There is a small discrepancy between the solution of CMG and the analytical solution. The results of UTCOMP, GPAS, GPAS_COATS and CMG, have the same slope. 115

142 Pressure Drop (Psi), (P-Pinitial) Analytical Solution CMG UTCOMP GPAS Dimensionless Distance in Z-Direction, (l/l) Figure 4. 2: Comparison of the pressure drop profile of analytical solution with that of UTCOMP, GPAS, GPAS_COATS and CMG for Case Study Case Study 3: One-Dimensional Compressible flow Results of analytical solution and the simulators match very well, as shown in Figure 4.3. The analytical solution is run for several other days both for dimensionless and real pressure profiles. Results are shown in Figure 4.4 and Figure

143 Pressure (Psi) td=0.157 Analytical Solution GPAS CMG UTCOMP Dimensionless Distance Figure 4. 3: Comparison of the pressure profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG at t D =0.157 for Case Study 3. Dimensionless Pressure td=0.1 td=0.16 td=0.21 td=0.27 td=0.33 rtd=0.39 td=0.44 td=0.5 td=0.56 td=0.61 td= Dimensionless Distance Figure 4. 4: Pressure profile of the analytical solution of dimensionless pressure versus dimensionless distance at different dimensionless time (t D = 0.1, 0.16, 0.21, 0.27, 0.33, 0.39, 0.44, 0.5, 0.56, 0.61 and 0.67) for Case Study

144 2000 t=3.18day t=5day t=6.82day t=8.63day t=10.45day t=12.26day t=14.08day t=15.89day t=17.71day Pressure (Psi) Dimensionless Distance Figure 4. 5: Pressure profile of the analytical solution of real pressure versus dimensionless distance at different real time (t=3.18, 5, 6.82, 8.63, 10.45, 12.26, 14.08, 15.89, 17.71, 19.52, Day) for Case Study Case Study 4: Two-dimensional Compressible Flow The analytical and simulation results at 365 days ( pore volume) and y D equaling 0.42 (y=840 ft) are shown in Figure 4.6. The agreement between the analytical and the simulation results is very good. 118

145 Analytical Solution CMG UTCOMP GPAS Pressure (Psi) Distacne in X-Direction (ft) Figure 4. 6: Comparison of the pressure profile of the analytical solution with that of the simulation results of UTCOMP, GPAS, GPAS_COATS and CMG at y=840ft and t=365days for Case Study Case Study 5: One-Dimensional Capillary End Effect Result is given in Figure 4.7. The agreement between the analytical solution and the simulation result of UTCOMP is excellent. The curve shows that there is no capillary end-effect in the simulation result of CMG_STARS. 119

146 Water Saturation Analytical Solution UTCOMP CMG Dimensionless Distance Figure 4. 7: Comparison of the water saturation profiles of the analytical solution with that of the simulation results of UTCOMP and CMG for Case Study Case Study 6: One-Dimensional Convection-Diffusion Equation The analytical solution and the numerical solutions are compared at 0.5 por e volume and for a Peclet number of 200. T he comparison is shown in Figure 4.8. The results of UTCOMP, GPAS_COATS, and CMG match the analytical solution very well. Figure 4.9 shows concentration profiles of UTCOMP at 0.5 pore volume with the Peclet number varying. As Peclet number increases the front diffuses less, which means that convective transport dominates dispersive mixing. The diffusion effect is less important than that of convection. Figure 4.10 shows how different numerical dispersion controls influence the concentration profile when Peclet number is 1000 at 0.5 pore volume. In the simulation result of UTCOMP, the result with third-order total variation diminishing (TVD) finite-difference method matches better with analytical solution (Figure 4.10). The solution using one-point upstream weighting displays a lot of numerical dispersion. 120

147 Dimensionless Concentration Analytical Solution UTCOMP GPAS_COATS CMG Dimensionless Distance Figure 4. 8: Comparison of the dimensionless concentration profile of the analytical solution with that of the simulation results of UTCOMP, GPAS_COATS, and CMG when peclet number is 200 at 0.5 pore volume for Case Study 6. Npe=50 Npe=200 Npe=1000 Dimensionless Concentration Dimensionless Distance Figure 4. 9: Comparison of UTCOMP simulation result of the dimensionless concentration profile with peclet number varying at 0.5 pore volume using third-order TVD method for Case Study

148 Dimensionless Concentration Analytical Solution One-point Upstream Weighting Third-order TVD Method Dimensionless Distance Figure 4. 10: Comparison of UTCOMP simulation result of the dimensionless concentration profile with different dispersion control method when Peclet number is 1000 at 0.5 pore volume for Case Study Case Study 7: Two-Dimensional Transverse Dispersion Figure 4.11 and Figure 4.12 show the concentration profile comparison result of analytical solution and simulation solution for UTCOMP and CMG at X D = with different transverse dispersivity. In Figure 4.11, UTCOMP matches well with analytical solution except for a small area at the front. There is a big difference between analytical solution and the simulation result of CMG with both one-point and two-point upstream weighting. There are two fluctuations in CMG results at around 0.5 to 0.6 dimensionless distance, with two-point upstream weighting and the normalized tracer concentration is even larger than 1 at those fluctuations. In Figure 4.12, the agreement between the analytical and the numerical solutions of UTCOMP is excellent. There is still big discrepancy between analytical and simulation result of CMG with both one-point 122

149 upstream and two-point upstream weighting. UTCOMP uses third-order TVD method while CMG-GEM uses two-point upstream weighting under the control of a Total Variation Limiting flux limiter (TVL) (CMG modules do not have the third-order TVD method). Normalized Tracer Concentration Analytical Solution (al=0.02 at=0.002) UTCOMP Third-order TVD Method (al=0.02 at=0.002) CMG One-point Upstream Weighting (al=0.02 at=0.002) CMG Two-point Upstream Weighting with TVL (al=0.02 at=0.002) Dimensionless Distance in Y-Direction Figure 4. 11: Comparison of the normalized concentration profile of the analytical solution with that of the simulation results of UTCOM and CMG at x D = for transverse dispersivity of and longitudinal dispersivity of 0.02 for Case Study

150 Normalized Tracer Concentration Analytical Solution (al=0.02 at=0.02) UTCOMP Third-order TVD Method (al=0.02 at=0.02) CMG One-point Upstream Weighting (al=0.02 at=0.02) CMG Two-point Upstream Weighting with TVL (al=0.02 at=0.02) Dimensionless Distance in Y-Direction Figure 4. 12: Comparison of the normalized concentration profile of the analytical solution with that of the simulation results of UTCOM and CMG at x D = for transverse dispersivity of 0.02 and longitudinal dispersivity of 0.02 for Case Study Case Study 8: Tracer Flow in a Five-Spot Well Pattern Figure 4.13 shows comparison of effluent tracer concentrations obtained from the analytical solution, UTCOMP, and CMG simulation results. The comparison indicates that the simulation solution by UTCOMP matches well with the analytical solution, while CMG gives much lower maximum tracer concentration and slightly later breakthrough time than the analytical solution. The comparison in Figure 4.14 illustrates that UTCOMP with two-point upstream weighting yields the best simulation result and the accuracy decreases with the decreasing of order of numerical dispersion control. The CMG with one-point upstream weighting gives better result than that by two-point upstream weighting with TVD. 124

151 Normalized Effluent Tracer Concentration Analytical Solution UTCOMP CMG Pore Volumes Injected Figure 4. 13: Comparison of the normalized effluent tracer concentration of the analytical solution with that of the simulation results of UTCOMP and CMG for Case Study 8. Normalized Effluent Tracer Concentration Analytical Solution UTCOMP One-point Upstream Weighting UTCOMP Two-point Upstream Weighting CMG Two-point Upstream Weighting CMG One-point Upstream Weighting Pore Volume Injected Figure 4. 14: Comparison of the normalized effluent tracer concentration of the analytical solution with that of the simulation results of UTCOMP and CMG for variable order of numerical dispersion control methods. 125

152 4.2.9 Case Study 9: One-dimensional Waterflooding in X Direction without Capillary Pressure A comparison of analytical solution and the simulated result of UTCOMP, GPAS, and CMG is made at 0.2 pore volume. Figure 4.15 shows comparison result at 0.2 pore volume. Overall, the simulation results from UTCOMP, GPAS, and CMG match well with the analytical solution, except for the slightly smearing water front. Water Saturation Analytical Solution UTCOMP CMG GPAS Dimensionless Distance in X-Direciton Figure 4. 15: Comparison of the water saturation profiles of Buckley-Leverett solution and the simulation result of UTCOMP, GPAS and CMG at 0.2 pore volume injected using one-point upstream weighting for Case Study Case Study 10: One-Dimensional Waterflooding in X Direction with Capillary Pressure Figure 4.16 shows numerical solution at 0.2 pore volume injected using one-point upstream weighting, along with the analytical solutions of Buckley-Leveret problem and Terwilliger et al. (1951). The results of UTCOMP and CMG match well with the 126

153 analytical solution of Terwilliger et al. The water front of Terwilliger et al. is not sharp anymore compared to the Buckley-Leveret solution. Water Saturation Terwilliger Solution CMG Buckley-Leverett Solution UTCOMP Dimensionless Distance in X-Direction Figure 4. 16: Comparison of the water saturation profile of the Buckley-Levereett solution, the Terwilliger solution, and the simulation results of UTCMOP and CMG at 0.2 pore volume injected using the third-order TVD method with TVD for Case Study Case Study 11: One-Dimensional Waterflooding in Z Directions without Capillary Pressure A comparison of analytical solution and simulation result of UTCOMP, GPAS, and CMG is made at 0.2 pore volume with five hundred grid blocks in z-direction, which is shown in Figure The simulation result matches well with the analytical solution, except that the simulation front is little bit smeared compared to that of analytical solution. 127

154 Analytical Solution CMG GPAS UTCOMP Water Saturation Dimensionless Distance in Z-Direction Figure 4. 17: Comparison of the water saturation profile of the Buckley-Levereet solution and the simulation results of UTCOMP, GPAS, and CMG at 0.2 pore volume injected using one-point upstream weighting for Case Study Case Study 12: Miscible WAG Displacement with Secondary Displacements and Low-WAG injection Figure 4.18 shows the comparison result. The comparison indicates a good match between analytical solution and simulation results. 128

155 Analytical Solution CMG UTCOMP Oil Saturation Dimensionless Distance Figure 4. 18: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.6 pore volume injected for Case Study Case Study 13: Miscible WAG Displacement with secondary displacements and high-wag injection The results of various simulators along with the analytical solution are shown in Figure As the water fractional flow f wj is increased to 0.7, t here are two shock fronts. The later shock front of numerical solution smears a lot, compared to the early shock front. 129

156 Oil Saturation Analytical Solution CMG UTCOMP Dimensionless Distance Figure 4. 19: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.4 pore volume injected for Case Study Case Study 14: Miscible WAG Displacement with tertiary displacements and low-wag injection With increasing initial water saturation to 0.65, there are three shock fronts in this case. As shown in Figure 4.20, the simulation result shows more smearing, and the numerical shock front in the middle completely disappeared. UTCOMP offers more accurate result at the front compared to CMG. 130

157 Oil Saturation CMG Analytical Solution UTCOMP Dimensionless Distance Figure 4. 20: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.25 pore volume injected for Case Study Case Study 15: Miscible WAG Displacement with tertiary displacements and high-wag injection As shown in Figure 4.21, the simulation result matches better with the analytical solution compared to result of case study 14. There are two fronts in this case. The numerical shock front on the left in Figure 4.21 smeared more than that of right one. 131

158 Analytical Solution CMG UTCOMP Oil Saturation Dimensionless Distance Figure 4. 21: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.3 pore volume injected for Case Study Case Study 16: Miscible WAG Displacement with tertiary displacements and water-free solvent injection There are two shock fronts in this case. As shown in Figure 4.22, the simulation result smears more than that of case study 14. Only one shock front can be found in the simulation results for UTCOMP and CMG, however UTCOMP gives better result compared to CMG. 132

159 Oil Saturation Analytical Solution CMG UTCOMP Dimensionless Distance Figure 4. 22: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 0.25 pore volume injected for Case Study Case Study 17: Dietz Displacement with miscible displacement and lowlongitudinal dispersivity The simulation result of UTCOMP and CMG are shown in Figure 4.23 and Figure The stabilized interface angle is 30, which is the same as calculated from the analytical solution. 133

160 Distance in Z-Direction (ft) 0.1PV 0.2PV 0.3PV 0.4PV 0.5PV 0.6PV Distance in X-Direction (ft) Figure 4. 23: Profiles of 0.5 solvent concentration of UTCOMP simulation for Case Study 17 with a longitudinal dispersivity of 1 ft. 0.1PV 0.2PV 0.3PV 0.4PV 0.5PV 0.6PV 1.25 Distance in Z-Direction (ft) Distance in X-Direction (ft) Figure 4. 24: Profiles of 0.5 solvent concentration of CMG simulation for Case Study 17 with a longitudinal dispersivity of 1 ft. 134

161 Case Study 18: Dietz Displacement with miscible displacement and highlongitudinal dispersivity The simulation result of UTCOMP and CMG are given in Figure 4.25 and Figure The stabilized interface angle is still 30, these results show that the longitudinal dispersivity will not alter the interface angle and the velocity profile of 0.5 concentration front. Distance in Z-Direction (ft) 0.1PV 0.2PV 0.3PV 0.4PV 0.5PV 0.6PV Distance in X-Direction (ft) Figure 4. 25: Profiles of 0.5 solvent concentration of UTCOMP simulation for Case Study 18 with a longitudinal dispersivity of 2 ft. 135

162 PV 0.2PV 0.3PV 0.4PV 0.5PV 0.6PV Distance in Z-Direction (ft) Distance in X-Direction (ft) Figure 4. 26: Profiles of 0.5 solvent concentration of CMG simulation for Case Study 18 with a longitudinal dispersivity of 2 ft Case Study 19: Dietz Displacement with immiscible displacement and no longitudinal dispersivity The simulation result of UTCOMP, GPAS_COATS, and CMG are given in Figure 4.27, Figure 4.28, and Figure

163 Distance in Z-Direction (ft) PV 0.2PV 0.3PV 0.4PV Distance in X-Direction (ft) Figure 4. 27: Profiles of water saturation of UTCOMP simulation for Case Study 19. Distance in Z-Direction (ft) 0.1PV 0.2PV 0.3PV 0.4PV Distance in X-Direction (ft) Figure 4. 28: Profiles of water saturation of CMG simulation for Case Study

164 Distance in Z-Direction (ft) PV 0.2PV 0.3PV 0.4PV Distance in X-Direction (ft) Figure 4. 29: Profiles of water saturation of GPAS_COATS simulation for Case Study Case Study 20: Two-Dimensional Convection-Diffusion Equation Figure 4.30 depicts the solvent-concentration movement with a range of time. Initially the source point of solvent is at the origin. As time progresses, solvent is drained along the x-axis by injected gas and dispersed in both x- and y- directions. At about 100 days, solvent begins to breakthrough and graduately disappear. Figure 4.31 shows the comparative results of the analytical solution and simulations by UTCOMP and CMG. Overall, there is a good agreement between analytical solution and numerical results, but both UTCOMP and CMG predict slightly higher concentration around x equals to zero. 138

165 Figure 4. 30:Three-dimensional view of normalized concentration of analytical solution at 2, 20, 40, 80, 100, and 150 days for Case Study

166 Normalized Concentration (C/Co) Analytical Solution CMG UTCOMP Distance in x-drection (ft) Figure 4. 31: Comparison of the analytical solution with the simulation results of UTCOMP and CMG at 2 days and z=0 for Case Study Case Study 21: Three Dimensional Waterflooding Figure 4.32 shows comparison of average reservoir pressure over time for UTCOMP, UTCOMP_IMSPAT, GPAS, GPAS_COATS, and CMG. Oil and water production histories are shown in Figure 4.33 and Figure 4.34, respectively. As seen in Figure 4.32, average pressure for all simulators behaved almost the same, except that the average pressure predicted by GPAS_COATS is a little bit lower than that of other simulators at the early time. As shown in Figure 4.33 and Figure 4.34, there is good agreement between the results of the simulators. Figure 4.35 shows comparison of timestep during the simulation time between UTCOMP and UTCOMP_IMPSAT. The timestep of UTCOMP_IMPSAT increases to the maximum time of 50 da y at around 3000 days and stays on the maximum time-step to the end of simulation. UTCOMP can reach the maximum time-step of 10 days, but it will reduce the time-step immediately after it 140

167 gets to the maximum time and oscillates heavily between maximum and minimum timesteps. By increasing the maximum time-step in UTCOMP, the run will fail or the simulation result oscillates too much. Figure 4.36 and Figure 4.37 give the water and oil distribution graphs of UTCOMP at 1000 days for one-point upstream weighting. Average Pressure (Psi) CMG UTCOMP GPAS GPAS_COATS UTCOMP_IMPSAT Time (Day) Figure 4. 32: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS and CMG for Case Study

168 Oil Production Rate (STB/Day) CMG UTCOMP GPAS GPAS_COATS UTCOMP_IMPSAT Time (Day) Figure 4. 33: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS and CMG for Case Study 21. Water Production Rate (STB/Day) CMG UTCOMP GPAS GPAS_COATS UTCOMP_IMPSAT Time (Day) Figure 4. 34: Comparison of water production rate of UTCOMP, UTCOMP_IMPSAT, GPAS, GPAS_COATS, and CMG for Case Study

169 60 50 IMPSAT UTCOMP Time Step (Day) Time (Day) Figure 4. 35: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study 21. Figure 4. 36: Water saturation distribution of UTCOMP using one-point upstream weighting at 1000 days for Case Study

170 Figure 4. 37: Oil saturation distribution of UTCOMP using one-point upstream weighting at 1000 days for Case Study Case Study 22: Three-Dimensional Gas and Solvent Injection with Three Hydrocarbon Components in Reservoir Figure 4.38 shows comparison of average reservoir pressure versus time for UTCOMP, UTCOMP_IMPSAT, GPAS_COATS and CMG. Figure 4.38 indicates that the average pressure predicted by GPAS_COATS is slightly higher than that of other simulators around 1000 days (breaktrough time). It matches well with other simulators from 1000 days to the end of simulation. CMG, UTCOMP and UTCOMP_IMPSAT have great agreement with each other. Figure 4.39 and Figure 4.40 give result of oil and gas production rates, respectively. There is good agreement among UTCOMP, UTCOMP_IMPSAT, and CMG. Figure 4.41 shows comparison of time-step during the simulation time of UTCOMP and UTCOMP_IMPSAT. This figure illustrates that the trend of the result for UTCOMP is almost the same as UTCOMP_IMPSAT, except that 144

171 UTCOMP_IMPSAT has bigger oscillations between about 1000 d ays and 4500 da ys. Figure 4.42 and Figure 4.43 present oil and gas saturation distribution graphs of UTCOMP using one-point upstream weighting at days. Average Reservoir Pressure (Psi) CMG GPAS_COATS UTCOMP UTCOMP_IMPSAT Time (Day) Figure 4. 38: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study

172 Oil Production Rate (STB/Day) CMG UTCOMP GPAS_COATS UTCOMP_IMPSAT Time (Day) Figure 4. 39: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study 22. Gas Production Rate (MMSCF/Day) CMG UTCOMP GPAS_COATS UTCOMP_IMPSAT Time (Day) Figure 4. 40: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, GPAS_COATS, and CMG for Case Study

173 IMPSAT UTCOMP Time Step (Day) Time (Day) Figure 4. 41: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study 22. Figure 4. 42: Oil saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study

174 Figure 4. 43: Gas saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Case Study 23: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Reservoir Figure 4.44 shows comparison of average reservoir pressure versus time for UTCOMP, UTCOMP_IMPSAT, and CMG. Figure 4.44 indicates that there is good agreement between UTCOMP and UTCOMP_IMPSAT. Figure 4.45 displays oil production rate. In this figure, the result of UTCOMP, UTCOMP_IMPSAT, and CMG match each other very well until about 5800 d ays. After about 5800 days, CMG gives lower values than that by UTCOMP and UTCOMP_IMPSAT, but they have the same trend. Figure 4.46 shows gas production rate. The gas production rate starts to increase at 4500 days; it is because the injected gas begins to produce. After about 5700 days, the 148

175 front of gas flooding arrives at the producer; gas production rate almost does not increase. Figure 4.47 gives the time-step history during simulation. The time-step of UTCOMP_IMPSAT is almost the same with UTCOMP. But UTCOMP_IMPSAT has less oscillations between 3000 t o 5000 da ys, when reaches the maximum time-step. Figure 4.48 and Figure 4.49 show oil and gas saturation distribution graphs of UTCOMP at 3000 days using one-point upstream weighting. Average Reservoir Pressure (Psi) CMG UTCOMP_IMPSAT UTCOMP Time (Day) Figure 4. 44: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study

176 Oil Production Rate (STB/Day) CMG UTCOMP_IMPSAT UTCOMP Time (Day) Figure 4. 45: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study23. Gas Production Rate (MMSCF/day) CMG UTCOMP_IMPSAT UTCOMP TIME (Day) Figure 4. 46: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study

177 6 5 UTCOMP IMPSAT Time Step (Day) Time (Day) Figure 4. 47: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study 23. Figure 4. 48: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study

178 Figure 4. 49: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Case Study 24: Three-Dimensional Gas and Solvent Injection with Six Hydrocarbon Components in Large Reservoir Figure 4.50 shows comparison of average reservoir pressure versus time for UTCOMP and CMG. Figure 4.51 and Figure 4.52 show oil and gas production rates, respectively. There is an excellent agreement between UTCOMP and CMG. Figure 4.53 and Figure 4.54 give oil and gas saturation distribution graphs of UTCOMP at days with one-point upstream weighting. 152

179 Average Reservoir Pressure (Psi) CMG UTCOMP Time (Day) Figure 4. 50: Comparison of average pressure of UTCOMP and CMG for Case Study 24. Oil Production Rate (STB/Day) CMG UTCOMP Time (Day) Figure 4. 51: Comparison of oil production rate of UTCOMP and CMG for Case Study

180 Gas Production Rate (MMSCF/Day) CMG UTCOMP Time (Day) Figure 4. 52: Comparison of gas production rate of UTCOMP and CMG for Case Study 24. Figure 4. 53: Oil saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study

181 Figure 4. 54: Gas saturation distribution of UTCOMP using one-point upstream weighting at days for Case Study Case Study 25: Two-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir Figure 4.55 shows comparison of average reservoir pressure versus time for UTCOMP and CMG. Results of UTCOMP and CMG match each other very well until about 1900 days. After 1900 days, results of UTCOMP are higher than that of CMG, but they have the same trend. Figure 4.56 shows oil production rate. There is a difference between results of UTCOMP and CMG. Gas production history is shown in Figure Figure 4.58 and Figure 4.59 show oil and gas saturation distribution graphs of UTCOMP using one-point upstream weighting at 3000 days. 155

182 Average Reservoir Pressure (Psi) CMG UTCOMP Time (Day) Figure 4. 55: Comparison of average pressure of UTCOMP and CMG for Case Study 25. Oil Production Rate (STB/Day) CMG UTCOMP Time (Day) Figure 4. 56: Comparison of oil production rate of UTCOMP and CMG for Case Study

183 Gas Production Rate (MMSCF/Day) CMG UTCOMP Time (Day) Figure 4. 57: Comparison of gas production rate of UTCOMP and CMG for Case Study 25. Figure 4. 58: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study

184 Figure 4. 59: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Case Study 26: Three-Dimensional Gas and Solvent Injection with Twenty Hydrocarbon Components in Reservoir Figure 4.60 shows comparison of average reservoir pressure versus time for UTCOMP, UTCOMP_IMPSAT, and CMG. Figure 4.60 indicates that there is good agreement between UTCOMP and UTCOMP_IMPSAT. Results of CMG, UTCOMP and UTCOMP_IMPSAT are comparable. Figure 4.61 displays oil production rate. In this figure, there is a difference among the results of CMG and UTCOMP, UTCOMP_IMPSAT. CMG gives higher estimates than by UTCOMP and UTCOMP_IMPSAT, but they maintain the same trend to the end of simulation. It is suspected that the difference between the results in case studies 25 and 26 is due to the use of different flash calculation algorithms; however we did not investigate this issue further. Figure 4.62 shows gas production rate. Gas production rate increases, after 158

185 yielding a stable rate for about 2600 da ys. This is because the injected gas reaches the producer. Figure 4.63 gives the time-step during simulation by UTCOMP and UTCOMP_IMPSAT. Results of UTCOMP and UTCOMP_IMPSAT are almost the same, and they keep oscillating from the beginning to the end. The time-step of UTCOMP_IMPSAT doesn t either exceed that of UTCOMP or stay at the maximum time-step for a while, such as the results in case study 21. Figure 4.64 and Figure 4.65 present oil and gas saturation distribution graphs of UTCOMP at 3000 days for one-point upstream weighting. Average Reservoir Pressure (Psi) CMG UTCOMP_IMPSAT UTCOMP Time (Day) Figure 4. 60: Comparison of average pressure of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study

186 Oil Production Rate (STB/Day) CMG UTCOMP_IMPSAT UTCOMP Time (Day) Figure 4. 61: Comparison of oil production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study 26. Gas Production Rate (MMSCF/Day) CMG UTCOMP_IMPSAT UTCOMP Time (Day) Figure 4. 62: Comparison of gas production rate of UTCOMP, UTCOMP_IMPSAT, and CMG for Case Study

187 2.5 IMPSAT UTCOMP Time Step (Day) Time (Day) Figure 4. 63: Comparison of time-step of UTCOMP and UTCOMP_IMPSAT for Case Study 26. Figure 4. 64: Oil saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study

188 Figure 4. 65: Gas saturation distribution of UTCOMP using one-point upstream weighting at 3000 days for Case Study Case Study 27: Scenario Two of SPE Fifth Comparative Solution Project Result of scenario two is shown in Figure 4.66 through Figure In general, UTCOMP results are comparable with CMG and SENSOR (COATS, 1992). In Figure 4.66, UTCOMP gives slightly higher cumulative oil production. Figure 4.67 shows cumulative oil production versus cumulative water injection; UTCOMP shows slightly lower result than by CMG and SENSOR. UTCOMP gives higher gas oil ratio after 12 years, but the total gas-oil ratio is comparable to CMG and SENSOR in Figure UTCOMP reaches the limiting GOR of 10 MCF/STB in about 15 years. In Figure 4.69, UTCOMP and CMG have almost zero water cut before the well is shut in, while the water cut of SENSOR suddenly increases at year 16. All simulation results do not get to the limiting WOR of 5 STB/STB before the well is shut in. Figure 4.70 shows average 162

189 reservoir pressure. UTCOMP gives lower values than by CMG and SENOR before year 6, while the result graduately increases and exceeds CMG and SENSOR. Cumulative Oil Production (MSTB) UTCOMP CMG SENSOR Time(Years) Figure 4. 66: Comparison of cumulative oil production of UTCOMP, CMG, and SENSOR for Case Study

190 Cumulative Oil Production (MSTB) UTCOMP CMG SENSOR Cumulative Water Injection (MSTB) Figure 4. 67: Comparison of cumulative oil production vs. cumulative water injection of UTCOMP, CMG, and SENSOR for Case Study 27. UTCOMP CMG SENSOR Time(Years) Gas Oil Ratio (MSCF/STB) Figure 4. 68: Comparison of producing gas-oil ratio of UTCOMP, CMG, and SENSOR for Case Study

191 Water Cut (%) UTCOMP CMG SENSOR Time(Years) Figure 4. 69: Comparison of producing water cut of UTCOMP, CMG, and SENSOR for Case Study 27. Average Reservoir Pressure (Psi) UTCOMP CMG SENSOR Time (Years) Figure 4. 70: Comparison of average reservoir pressure of UTCOMP, CMG, and SENSOR for Case Study

192 Case Study 28: Gas Injection Involving second Hydrocarbon Phase Generation This model has PVT data generating three hydrocarbon phases (gas, oil, and a second liquid) around 1280 psi and below about 105 F. CMG-GEM cannot simulate three hydrocarbon phases; it runs poorly and eventually fails before 4560 days. However, UTCOMP has the ability to simulate three hydrocarbon phases. Figure 4.71 gives UTCOMP simulation of second liquid saturation as time progresses. Because of heterogeneity in permeability and porosity, the front of second liquid saturation does not move with the same velocity level. Second liquid saturation moves faster along higher permeability and porosity paths. Figure 4.72 through Figure 4.74 present comparison of UTCOMP and CMG results with one-point upstream weighting. 166

193 100 days 2850 days 1000 days 3500 days 2100 days 4550 days Figure 4. 71: UTCOMP simulation result of second hydrocarbon phase saturation using one-point upstream weighting at 100, 1000, 2100, 2850, 3500, and 4550 days for Case Study