Mathematical Modelling and Numerical Simulation of Carbonated ...

Mathematical Modelling and Numerical Simulation of Carbonated Water Injection for Enhanced Oil Recovery and CO2 Storage

By Jalal Foroozesh BSc, MSc Submitted for the degree of Doctor of Philosophy in Petroleum Engineering

Heriot-Watt University

Institute of Petroleum Engineering School of Energy, Geoscience, Infrastructure and Society

July 2015

The copyright in this thesis is owned by the author. Any quotation from the thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information.

Abstract Numerical simulation of carbonated water injection (CWI) as an EOR and CO2 storage technique is studied in this thesis. When carbonated water (CO2 saturated water) contacts oil during injection into oil reservoirs, because of higher solubility of CO2 in hydrocarbons compared to water, CO2 will migrate from water into oil phase. Therefore, oil mobility, and in turn oil recovery, will increase. In addition, CO2 can also be stored securely in reservoir during CWI. The compositional simulation approach should be used for simulation of CWI in order to capture the mechanisms and the changes of composition happening during CWI process. However, the conventional compositional approach is based on the assumption of instantaneous thermodynamic equilibrium. That is, it assumes that the CO2 is transferred and distributed between oil and water phases very fast such that the thermodynamic equilibrium state is reached instantaneously. However, the CWI coreflood experiments presented in the literature show that during CWI, the CO2 transfer between water and oil phases happens slowly and therefore, the assumption of instantaneous equilibrium is not valid during the simulation of CWI coreflood experiments. As a result, the available compositional simulators cannot simulate CWI coreflood experiments correctly. Hence, in this thesis, a new compositional simulator is developed, in which the assumption of instantaneous equilibrium is relaxed by including the kinetics of mass transfer. To evaluate the performance of the developed simulator and to explore its generic capability, two different sets of CWI coreflood experiments performed in a water-wet and a mixed-wet (aged) sandstone core are selected from the literature. These coreflood experiments are simulated and studied in detail including the role of oil swelling and wettability alteration during CWI process. The simulator can predict the production profiles of oil, water and CO2; the CO2 storage profile; the differential pressure across the core and the CO2 concentration in oil and water phases. The impacts of dispersion, injection rate and carbonation level on the performance of CWI process are investigated using the developed simulator. The simulator shows that the dispersion effect on oil production is minimal here during the coreflood experiments. It is also shown that at low injection rates and high carbonation levels, higher oil recovery will be obtained by CWI. In addition, at low injection rates, more CO2 can be stored in core during the coreflood experiments with a lower and delayed CO2 production at the core outlet. Moreover, the compositional simulator of ECLIPSE300 (E300) is used to simulate the CWI coreflood experiments and its capability is compared to the capability i

of the developed simulator. E300 over predicts the oil recovery of CWI coreflood experiments due to the assumption of instantaneous equilibrium made by ECLIPSE 300. A dimensionless number so-called equilibrium number (Ne) is introduced and it is shown that at a specific range of Ne values, the assumption of instantaneous equilibrium made by E300 is acceptable. Accordingly, it is shown that at reservoir-scale, the system will reach the equilibrium state and therefore E300 can be used to simulate the CWI process at reservoir-scale. Based on this, finally, the reservoir-scale simulation of CWI is studied employing the ECLIPSE300 simulator. The impacts of some influential parameters on CWI performance are investigated using the results of reservoir-scale simulation.

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Dedication

I wish to dedicate my thesis:

To my parents, Abdollah and Mahtab for all their support, encouragement and especially for their unconditional loves and prayers.

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Acknowledgements I would like to express my deep gratitude to my first supervisor, Prof. Mahmoud Jamiolahmady (Jami) for his excellent technical guidance, encouragement, support and patience throughout my thesis work. This thesis could never be completed without his constructive criticism and help.

I also would like to express my sincere appreciation to my second supervisor, Prof. Mehran Sohrabi for his encouragement and outstanding technical support and comments.

The financial support of Heriot-Watt University during my study through a full scholarship is highly appreciated. This thesis is the outcome of the joint industry project of ‘Carbonated Water Injection’ at Heriot-Watt University which was equally sponsored by the UK Department of Energy and Climate(DECC), Total, Petrobras, ADCO, BG Group and Galp Energia. Their support is gratefully acknowledged.

Thanks to Dr Alireza Emadi, Dr Masoud Riazi, Dr Nor Idah, Mr Mojtaba Seyyedi, Mr Rasoul Nazari Moghaddam, Mr Amir Farzaneh for their time for discussion and providing valuable comments.

My last (but not least) and special deepest thanks are due to all members of my family, Abdollah and Mahtab (my parents), Javad and Jamal (my brothers), Zohreh, Zahra and Ziba (my sisters) for their encouragement and support.

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ACADEMIC REGISTRY Research Thesis Submission

Name:

JALAL FOROOZESH

School/PGI:

Institute of Petroleum Engineering / School of EGIS

Version: (i.e. First, Resubmission, Final)

Final

Degree Sought (Award and Subject area)

PhD in Petroleum Engineering

Declaration In accordance with the appropriate regulations I hereby submit my thesis and I declare that: 1)

the thesis embodies the results of my own work and has been composed by myself

2)

where appropriate, I have made acknowledgement of the work of others and have made reference to work carried out in collaboration with other persons

3)

the thesis is the correct version of the thesis for submission and is the same version as any electronic versions submitted*.

4)

my thesis for the award referred to, deposited in the Heriot-Watt University Library, should be made available for loan or photocopying and be available via the Institutional Repository, subject to such conditions as the Librarian may require

5)

I understand that as a student of the University I am required to abide by the Regulations of the University and to conform to its discipline.

*

Please note that it is the responsibility of the candidate to ensure that the correct version of the thesis is submitted. Signature of Candidate:

Date:

Submission Submitted By (name in capitals):

JALAL FOROOZESH

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Table of Contents INTRODUCTION ................................................................................................................. 1 1-1. EXPERIMENTAL STUDY OF CWI ................................................................................................................. 2 1-1.1. Micro-Model Tests ..................................................................................................................... 2 1-1.2. Coreflood Experiments .............................................................................................................. 5 1-1.3. Sand-packed Flooding Experiments ........................................................................................... 7 1-2. SIMULATION STUDY OF CWI .................................................................................................................. 14 MODEL STRUCTURE ..........................................................................................................21 2-1. GOVERNING EQUATIONS ....................................................................................................................... 22 2-2. INITIAL AND BOUNDARY CONDITIONS ...................................................................................................... 24 2-3. SOLUTION TECHNIQUE .......................................................................................................................... 24 2-4. DENSITY MODELS ................................................................................................................................ 27 2-4.1. Oil-CO2 Density ........................................................................................................................ 27 2-4.2. Water-CO2 Density .................................................................................................................. 31 2-5. VISCOSITY MODELS .............................................................................................................................. 33 2-5.1. Oil-CO2 Viscosity ...................................................................................................................... 33 2-5.2. Water-CO2 Viscosity ................................................................................................................ 35 2-6. MASS TRANSFER TERM ......................................................................................................................... 36 2-7. MOLECULAR DIFFUSION ........................................................................................................................ 37 2-8. DISPERSION ........................................................................................................................................ 40 2-9. RELATIVE PERMEABILITY RELATIONSHIP .................................................................................................... 42 2-10. CAPILLARY PRESSURE .......................................................................................................................... 42 2-11. HISTORY MATCHING USING GENETIC ALGORITHM BASED OPTIMISATION ..................................................... 43 2-11.1. The Structure of Genetic Algorithm-Based Optimiser ........................................................... 43 SIMULATION OF CWI IN A WATER-WET CORE ...................................................................48 3-1. THE WI AND CWI EXPERIMENTS ............................................................................................................ 48 3-2. SIMULATION OF THE WATER INJECTION EXPERIMENT .................................................................................. 53 3-2.1. Water-Oil Relative Permeability Curve .................................................................................... 53 3-2.2. Capillary Pressure .................................................................................................................... 62 3-3. SIMULATION OF CWI ........................................................................................................................... 65 3-3.1. Compositional Simulation........................................................................................................ 65 3-3.2. Black-oil Simulation ................................................................................................................. 72 3-4. EFFECT OF MIXTURE FLUID PROPERTIES ON THE SIMULATION OF CWI ........................................................... 74 3-4.1. Effect of Mixture Density Model on the Simulation of CWI ..................................................... 74 3-4.2. Effect of Mixture Viscosity Model on the Simulation of CWI ................................................... 80 3-5. CO2 PRODUCTION PROFILE.................................................................................................................... 85 3-5.1. Effect of MTC on the CO2 Production Profile ........................................................................... 87 3-5.2. Effect of Dispersion Coefficient on the CO2 Production Profile ................................................ 90 3-5.3. Effect of Injection Rate on the CO2 Production Profile............................................................. 93 3-6. OIL PRODUCTION PROFILE ..................................................................................................................... 95 3-6.1. Effect of Dispersion Coefficient on the Oil Production Profile ................................................. 95 3-6.2. Effect of Injection Rate on the Oil Production Profile .............................................................. 97 3-6.3. Effect of Carbonation Level on the Oil Production Profile ..................................................... 100 3-7. CO2 STORAGE ................................................................................................................................... 101 3-8. COMPOSITIONAL SIMULATION OF CWI USING ECLIPSE300 (E300) .......................................................... 104 3-9. SUMMARY AND CONCLUSIONS ............................................................................................................. 112 SIMULATION OF CWI IN A MIXED-WET CORE ..................................................................117

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4-1. THE WI AND CWI EXPERIMENTS .......................................................................................................... 117 4-2. SIMULATION OF THE WATER INJECTION EXPERIMENT ................................................................................ 122 4-2.1. Water-Oil Relative Permeability Curve .................................................................................. 122 4-2.2. Capillary Pressure .................................................................................................................. 127 4-3. SIMULATION OF CWI ......................................................................................................................... 131 4-3.1. Compositional Simulation...................................................................................................... 131 4-3.2. Black-oil Simulation ............................................................................................................... 157 4-4. CO2 PRODUCTION PROFILE.................................................................................................................. 160 4-4.1. Effect of MTC on the CO2 Production Profile ......................................................................... 163 4-4.2. Effect of Dispersion Coefficient on the CO2 Production Profile .............................................. 164 4-4.3. Effect of Injection Rate on the CO2 Production Profile........................................................... 165 4-5. OIL PRODUCTION PROFILE ................................................................................................................... 167 4-5.1. Effect of Dispersion Coefficient on the Oil Production Profile ............................................... 167 4-5.2. Effect of Injection Rate on the Oil Production Profile ............................................................ 169 4-5.3. Effect of Carbonation Level on the Oil Production Profile ..................................................... 170 4-6. CO2 STORAGE ................................................................................................................................... 171 4-6.1. Effect of MTC on the CO2 Storage.......................................................................................... 173 4-6.2. Effect of Injection Rate on the CO2 Storage ........................................................................... 174 4-7. COMPOSITIONAL SIMULATION OF CWI USING ECLIPSE300 (E300) .......................................................... 175 4-8. SUMMARY AND CONCLUSIONS ............................................................................................................. 182 RESERVOIR-SCALE SIMULATION OF CWI .........................................................................185 5-1. DIMENSIONLESS NUMBER ANALYSIS ...................................................................................................... 186 5-1.1. Capillary Number (Nc) ........................................................................................................... 186 5-1.2. Equilibrium Number (Ne) ....................................................................................................... 188 5-2. MODEL DESCRIPTION........................................................................................................................ 193 5-3. SIMULATION RESULTS ......................................................................................................................... 199 5-3.1. Secondary WI and CWI .......................................................................................................... 199 5-3.2. Effect of KVKH ratio .............................................................................................................. 204 5-3.3. Initial Water Saturation Effect .............................................................................................. 206 5-4. TERTIARY CWI .................................................................................................................................. 209 5-5. PERFORMANCE OF THE DEVELOPED SIMULATOR FOR A LARGE-SCALE MODEL ................................................ 212 5-6. SUMMARY AND CONCLUSIONS ............................................................................................................. 215 SUMMARY AND CONCLUSIONS ......................................................................................218 6-1. SUMMARY AND CONCLUSIONS ............................................................................................................. 218 6-2. RECOMMENDATIONS .......................................................................................................................... 226 APPENDIX A: DETAILS OF EQUATIONS AND SOLUTION TECHNIQUE ...................................................228 REFERENCES .......................................................................................................................................238

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List of Tables Table 1-1: Summary of micro-model experiments. ..................................................................................... 9 Table 1-2: Summary of coreflood experiments. ......................................................................................... 10 Table 1-3: Summary of sand-packed experiments. .................................................................................... 11 Table 2-1: Density of decane-CO2 mixture measured at P=138 atm and T=40 0C (Song et al. 2012). ....... 29 Table 2-2 : Thermodynamic properties of components (Yaws 2014, Nourozieh et al. 2010). ................... 31 Table 2-3: Density of water-CO2 mixture predicted by EOS as a function of CO2 calculated at P=136.1 atm, T=38 0C. ..................................................................................................................................... 32 Table 2-4: Viscosity of decane-CO2 mixture measured at test conditions (Cullick and Mathis 1984). ...... 35 Table 2-5: Diffusion coefficient of CO2 in decane as a function of pressure at T=38 0C (Renner 1988). .... 39 Table 3-1: Core properties. ........................................................................................................................ 49 Table 3-2: Fluid properties [Ref: NIST 2014]. ............................................................................................. 49 Table 3-3: The operational conditions of the coreflood experiments. ...................................................... 50 Table 3-4: Initial uncertainty range of parameters used in GA. ................................................................. 54 Table 3-5: Optimal Corey parameters of water-oil relative permeability, obtained using E100 linked to the GA-based optimizer as well as assigned komax and swc values, WI experiment............................ 56 Table 3-6: Optimal Corey and Brook-Corey coefficients of water-oil relative permeability and capillary pressure curves obtained by the optimiser linked to E100. .............................................................. 62 Table 3-7: Optimal values of Corey parameters and MTC, obtained using devlpoed simulator linked to the GA-based optimiser compared to manual tuning,CWI experiment. ........................................... 68 Table 4-1: Core properties. ...................................................................................................................... 118 Table 4-2: Fluid properties [Ref: NIST 2014]. ........................................................................................... 118 Table 4-3: The operational conditions of the coreflood experiments. .................................................... 119 Table 4-4: Optimal Corey parameters of water-oil relative permeability, obtained using the developed simulator under the black-oil mode linked to the GA-based optimiser as well as assigned komax and swc values, WI experiment. ...................................................................................................... 122 Table 4-5: The data needed for calculation of pressure drop ratio. .......................................................... 127 Table 4-6: Initial uncertainty range of parameters used in GA. ............................................................... 128 Table 4-7: Optimal Corey and modified Brook-Corey coefficients of water-oil relative permeability and capillary pressure curves obtained by the optimiser linked to the E100. ....................................... 129 Table 4-8 : Summary of the sensitivity of TOP and DP data to the Corey relative permeability coefficients. ......................................................................................................................................................... 148 Table 4-9: Comparison of optimal CWI-Kr and MTC obtained by manual tuning and automatic tuning using optimiser; compositional simulation, CWI test. ..................................................................... 155 Table 4-10: Misfit valuse for manual tumimg and automtic histroy matching. ....................................... 157 Table 4-11: Optimal Corey coefficients of relative permeability obtained for WI and CWI coreflood tests, black-oil simulation. ........................................................................................................................ 158 Table 4-12: Optimal Corey coefficients of carbonated water-oil relative permeability obtained using two different methods, black-oil simulation, CWI test. .......................................................................... 160 Table 5-1: Core and fluid properties in SI unit. ........................................................................................ 186 Table 5-2: Capillary number using Equations 5-1 for both water-wet and mixed-wet cores. ................. 187 Table 5-3: Combination of core properties and operational conditions when the same result by the E300 and the developed simulators is obtained, water-wet coreflood experiment................................ 189 Table 5-4: Equilibrium Number for different combination of core properties and operational conditions when the same result by the E300 and the developed simulators is obtained, water-wet coreflood experiment. ..................................................................................................................................... 191 Table 5-5: Equilibrium Number for different combination of core properties and operational conditions when the same result by E300 and the developed simulators is obtained, mixed-wet coreflood experiment. ..................................................................................................................................... 192 Table 5-6: Details of the created ECLIPSE model. .................................................................................... 194

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Table 5-7: Grid size sensitivity analysis. ................................................................................................... 195 Table 5-8: Injection rate obtained based on the constraints imposed by Equilibrium and Capillary Numbers from the water-wet coreflood experiment. .................................................................... 197 Table 5-9:The screening criteria for the CO2 miscible injection technique(Taber et al., 1997a and 1997b). ......................................................................................................................................................... 198

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List of Figures Figure 1-1: Swelling rate during CWI compared to shrinkage rate during subsequent WI (Riazi et al., 2009; Riazi, 2011c)............................................................................................................................. 17 Figure 1-2: A typical simulation gridblock. ................................................................................................. 18 Figure 2-1: Density of decane-CO2 mixture as a function of CO2 mole fraction measured at P=138 atm, T=40 0C (Song et al. 2012). ................................................................................................................ 29 Figure 2-2: Density of water-CO2 mixture predicted by EOS as well as fitted linear equation as a function of CO2 mass fraction (P=136.1 atm, T=38 0C). ................................................................................... 33 Figure 2-3: Plot of effective diffusion coefficient of CO2 in decane from Table 2-5 as well as the fitted quadratic equation at T=38 0C. ......................................................................................................... 39 Figure 2-4: Longitudinal dispersion coefficient in a capillary network (Perkins and Johnston 1963). ....... 41 Figure 2-5: A chromosome. ........................................................................................................................ 44 Figure 2-6: Flowchart of the GA. ................................................................................................................ 47 Figure 3-1: A schematic of coreflood experiment. ..................................................................................... 50 Figure 3-2: Comparison of total oil production (TOP) data of the WI and CWI experiments. ................... 51 Figure 3-3: Comparison of recovery factor (RF) of the WI and CWI experiments. ..................................... 51 Figure 3-4: Comparison of differential pressure (DP) data of the WI and CWI experiments. ................... 52 Figure 3-5: Minimum and mean of Total misfit as a function of generation. ............................................ 54 Figure 3-6: Minimum and mean of TOP misfit as a function of generation. .............................................. 55 Figure 3-7: Minimum and mean of DP misfit as a function of generation. ................................................ 55 Figure 3-8: Total misfit as a function of nw and no. .................................................................................... 57 Figure 3-9:TOP misfit as a functin of nw and no. ......................................................................................... 58 Figure 3-10:DP misfit as a functin of nw and no. ......................................................................................... 58 Figure 3-11: Optimised Krw-o curve, WI test. .............................................................................................. 59 Figure 3-12: Experimental TOP versus predicted TOP data, predicted by the E100 black-oil simulator using the optimised Krw-o curve, WI test. ........................................................................................... 60 Figure 3-13: Experimental DP versus predicted DP data, predicted by the E100 black-oil simulator using the optimised Krw-o curve, WI test. .................................................................................................... 60 Figure 3-14: Predicted TOP data by the E100 and the developed simulator (model) under its black-oil mode versus injected PV using the optimised Krw-o curve, WI test. .................................................. 61 Figure 3-15: Predicted DP data by E100 and the developed simulator (model) under its black-oil mode using the optimised Krw-o curve, WI test. ........................................................................................... 62 Figure 3-16: Capillary pressure data for the WI test. ................................................................................. 63 Figure 3-17: Predicted total oil production (TOP) data by ECLIPSE100 (E100) with and without capillary pressure (Pc), WI test. ....................................................................................................................... 64 Figure 3-18: Predicted differential pressure (DP) data by ECLIPSE100 (E100) with and without capillary pressure (Pc), WI test. ....................................................................................................................... 64 Figure 3-19: Effect of the mass transfer coefficient (MTC) on prediction of TOP data by the developed simulator, CWI test. ........................................................................................................................... 66 Figure 3-20: Effect of the mass transfer coefficient (MTC) on prediction of DP data by the developed simulator, CWI test. ........................................................................................................................... 66 Figure 3-21: Minimum and mean misfit as a function of generation. ....................................................... 68 Figure 3-22: TOP data predicated by the developed simulator under its compositional mode compared to experimental TOP values, CWI test. .............................................................................................. 69 Figure 3-23: DP data predicated by the developed simulator under its compositional mode compared to experimental DP values, CWI test. .................................................................................................... 70 Figure 3-24: RF in the residual oil case. ...................................................................................................... 71 Figure 3-25: TOP data predicated by the developed simulator in its black-oil mode compared to the experimental TOP values, sor=0.27, CWI test. ................................................................................... 72

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Figure 3-26: DP data predicated by the developed simulator in its black-oil mode compared to experimental DP values, sor=0.27, CWI test....................................................................................... 73 Figure 3-27: Effect of the decane-CO2 density model on TOP data, predicted by the developed simulator in its compositional mode, CWI test. ................................................................................................. 74 Figure 3-28: Effect of decane-CO2 density model on DP data, predicted by the developed simulator in its compositional mode, CWI test. ......................................................................................................... 75 Figure 3-29: Oil density profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for three cases: PR-EOS Model, Ideal Solution Model and Constant Density, CWI test. ............................................................................................................... 76 Figure 3-30: CO2 mass fraction in oil phase through the core at end of the simulation as predicted by the developed simulator in its compositional mode, CWI test. .............................................................. 76 Figure 3-31: Effect of water-CO2 density model on TOP data as predicted by the developed simulator in its compositional mode, CWI test. .................................................................................................... 77 Figure 3-32: Effect of water-CO2 density model on DP data, predicted by the developed simulator in its compositional mode, CWI test. ......................................................................................................... 78 Figure 3-33: Water density profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for two cases: Model based on Measured Data and Constant Density, CWI test. ............................................................................................................... 79 Figure 3-34: CO2 mass fraction in water phase through the core at end of the simulation as predicted by the developed simulator in its compositional mode, CWI test. ........................................................ 79 Figure 3-35: Effect of decane-CO2 viscosity model on TOP data as predicted by the developed simulator in its compositional mode, CWI test. ................................................................................................. 80 Figure 3-36: Effect of decane-CO2 viscosity model on DP data as predicted by the developed simulator in its compositional mode, CWI test. .................................................................................................... 81 Figure 3-37: Oil viscosity profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for three cases: Constant Viscosity, Beggs and Robisnon Correlation and the Measured Data-Model, CWI test. ..................................................................... 82 Figure 3-38: Effect of water-CO2 viscosity model on TOP data as predicted by the developed simulator in its composition mode, CWI test. ....................................................................................................... 83 Figure 3-39: Effect of water-CO2 viscosity model on DP data as predicted by the developed simulator in its composition mode, CWI test. ....................................................................................................... 84 Figure 3-40: Water viscosity profile through the core at end of the simulation as predicted by the developed simulator in its composition mode for two cases: Constant Viscosity and Islam and Carlson Correlation, CWI test. ........................................................................................................... 85 Figure 3-41: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator, CWI test.................................................. 86 Figure 3-42: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the simulator, CWI test. ............................................................................................... 86 Figure 3-43: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator, MTC=5E-7, 15E-7 and 25E-7 1/sec, CWI test. .................................................................................................................................................... 88 Figure 3-44: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the simulator, MTC=5E-7 and 25E-7 1/sec, CWI test. .................................................. 89 Figure 3-45: RF predicted by the simulator in its compositional mode, MTC=5E-7, 15E-7 and 25E-7 1/sec, CWI test. ............................................................................................................................................ 90 Figure 3-46: Total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test. ...................................................... 91 Figure 3-47: Magnified plot of Figure 3-46, total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test. ........... 92

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Figure 3-48: Total CO2 production predicted by the simulator for the injection rates of 5, 10 and 20 cm3/hr without dispersion, CWI test. ................................................................................................ 93 Figure 3-49: Total CO2 production predicted by the simulator for dispersion coefficient (D) of zero and 171E-4 1/sec at injection rates of 5 and 20 cm3/hr , CWI test. ......................................................... 94 Figure 3-50: RF predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test. ........................................................................................................ 95 Figure 3-51: Profile of CO2 concentration in oil phase predicted by the simulator for D=0 and 171E-4 (cm2/sec) after 0.25 and 0.5 injected PV, CWI test. .......................................................................... 96 Figure 3-52: Profile of water saturation predicted by the simulator for D=0 and 171E-4 (cm2/sec) after 0.25 and 0.5 injected PV, CWI test. ................................................................................................... 96 Figure 3-53: RF predicted by the simulator for three different injection rates of 20, 10 and 5 cm 3/hr without dispersion, CWI test. ............................................................................................................ 98 Figure 3-54: Profile of CO2 concentration in oil phase predicted by the simulator at injection rates of 20 and 5 cm3/hr without dispersion after 0.5 injected PV, CWI test. .................................................... 99 Figure 3-55: RF predicted by the simulator for injection rates of 20, 10 and 5 cm 3/hr with and without dispersion, CWI test. .......................................................................................................................... 99 Figure 3-56: Profiles of RF for three different carbonation levels of 7, 5 and 3 wt % predicted by the developed simulator, CWI test. ....................................................................................................... 100 Figure 3-57: Profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S) versus injected PV of carbonated water, predicted by the simulator, CWI test. ............................ 102 Figure 3-58: The percentage of CO2 stored in water (CO2SW) and in oil phase(CO2SO). ........................ 103 Figure 3-59: (TCO2STCO2I ×100) versus injected PV of carbonated water predicted by the simulator, CWI test. .......................................................................................................................................... 103 Figure 3-60: RF predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, CWI test. ............................................................................................ 105 Figure 3-61: DP data predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, CWI test............................................................................. 105 Figure 3-62: TCO2P data predicted by the developed simulator compared to those predicted by E300, CWI test. .......................................................................................................................................... 106 Figure 3-63: CO2 concentration in oil phase through the core after 0.5 injected PV predicted by the developed simulator and E300. ....................................................................................................... 107 Figure 3-64: TOP data predicted by E300 simulator for injection rates of 20 and 5 cm 3/hr, CWI test. ... 108 Figure 3-65: TOP data predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, MTC=15E-7 1/sec, CWI test. ............................................. 109 Figure 3-66: Total CO2 production (TCO2P) data predicted by E300 simulator compared to those predicted by the developed simulator, MTC=15E-7 1/sec, CWI test. ............................................ 110 Figure 3-67: TOP data predicted by E300 simulator compared to those predicted by the developed simulator at injection rate of 6.67 cm3/hr, CWI test. ...................................................................... 111 Figure 3-68: Recovery factors predicted by the developed simulator at three diffrent MTC valuses of 15E-7, 25E-7 and 30E-7 1/sec. ......................................................................................................... 112 Figure 4-1: Schematic of coreflood experiment........................................................................................ 118 Figure 4-2: Comparison of TOP data of the WI and CWI experiments. .................................................... 119 Figure 4-3: Comparison of RF of the WI and CWI experiments. .............................................................. 120 Figure 4-4: Comparison of DP data of the WI and CWI experiments. ...................................................... 120 Figure 4-5: Optimised Krw-o curve, WI test. .............................................................................................. 123 Figure 4-6 : Experimental TOP versus predicted TOP data, predicted by the developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test. ............................................... 124 Figure 4-7: Experimental DP versus predicted DP data, predicted by the developed simulator (model) under the black-oil mode using optimised Krw-o curve, WI test....................................................... 124 Figure 4-8: TOP data predicted by ECLIPSE100 versus TOP data predicted by developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test. .................................. 125

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Figure 4-9: DP data predicted by ECLIPSE100 versus DP data predicted by the developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test. ................................. 126 Figure 4-10: Minimum and mean misfit as a function of generation. ..................................................... 128 Figure 4-11: Capillary pressure data for WI test. ..................................................................................... 129 Figure 4-12: TOP data predicted by ECLIPSE100 versus experimental values when capillary pressure is included, WI test. ............................................................................................................................ 130 Figure 4-13: DP data predicted by ECLIPSE100 versus experimental values when capillary pressure is included, WI test. ............................................................................................................................ 130 Figure 4-14: Effect of the mass transfer coefficient (MTC) on prediction of TOP data by the developed simulator, CWI test. ......................................................................................................................... 132 Figure 4-15: Effect of the mass transfer coefficient (MTC) on prediction of DP data by the developed simulator, CWI test. ......................................................................................................................... 133 Figure 4-16: Effect of nw Corey relative permeability exponent on prediction of TOP data. ................... 134 Figure 4-17: Effect of nw Corey relative permeability exponent on prediction of DP data. ..................... 135 Figure 4-18: Impact of 25% change in the base value of the nw Corey relative permeability exponent on Krw................................................................................................................................................... 136 Figure 4-19: Effect of no Corey relative permeability exponent on prediction of TOP data. ................... 137 Figure 4-20: Effect of no Corey relative permeability exponent on prediction of DP data. ..................... 137 Figure 4-21: Impact of 25% change in the base value of the no Corey relative permeability exponent on Kro. .................................................................................................................................................. 138 Figure 4-22: Effect of kwmax Corey relative permeability endpoint on prediction of TOP data. ............... 139 Figure 4-23: Effect of kwmax Corey relative permeability endpoint on prediction of DP data. ................. 139 Figure 4-24: Impact of 25% change in the base value of the kwmax Corey relative permeability endpoint on Krw................................................................................................................................................... 140 Figure 4-25: Effect of komax Corey relative permeability endpoint on prediction of TOP data. ................ 141 Figure 4-26: Effect of komax Corey relative permeability endpoint on prediction of DP data. .................. 141 Figure 4-27: Impact of change in the base value of the k omax Corey relative permeability endpoint on Kro. ......................................................................................................................................................... 142 Figure 4-28: Effect of sor Corey relative permeability saturation endpoint on prediction of TOP data. .. 143 Figure 4-29: Effect of sor Corey relative permeability saturation endpoint on prediction of DP data. .... 143 Figure 4-30: Impact of 25% change in the base value of the sor relative permeability saturation endpoint on Kro. ............................................................................................................................................. 144 Figure 4-31: Effect of swc Corey relative permeability saturation endpoint on prediction of TOP data. . 145 Figure 4-32: Effect of swc Corey relative permeability saturation endpoint on prediction of DP data..... 145 Figure 4-33: Impact of changes in the base value of the swc Corey relative permeability saturation endpoint on Krw. ............................................................................................................................. 146 Figure 4-34: Spider plot for TOP. .............................................................................................................. 147 Figure 4-35: Spider plot for DP. ................................................................................................................ 147 Figure 4-36: TOP data predicted by the developed simulator compared to experimental values, MTC=5E7 1/sec, CWI test. ............................................................................................................................ 150 Figure 4-37: DP data predicted by the developed simulator compared to experimental values, MTC=5E-7 1/sec, CWI test. ............................................................................................................................... 150 Figure 4-38: TOP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; WI-Kr (sor=0.36), CWI test............................................................................... 152 Figure 4-39: DP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; WI-Kr (sor=0.36), CWI test. ............................................................................... 152 Figure 4-40: TOP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; CWI-Kr (sor=0.36, no=1.7, kwmax=0.101), CWI test. ............................................ 154 Figure 4-41: DP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; CWI-Kr (sor=0.36, no=1.7, kwmax=0.101), CWI test. .......................................... 154

xiii

Figure 4-42: TOP data predicted by the developed simulator in its compositional mode compared to experimental values; CWI-Kr & MTC obtained by manual tuning and GA-based optimiser, CWI test. ......................................................................................................................................................... 156 Figure 4-43: DP data predicted by the developed simulator in its compositional mode compared to experimental values; CWI-Kr & MTC obtained by manual tuning and GA-based optimiser, CWI test. ......................................................................................................................................................... 156 Figure 4-44: Experimental TOP versus TOP data predicted by developed simulator (model) under the black-oil mode, using the CWI-relative permeability data (Table 4-11), CWI test. ......................... 159 Figure 4-45: Experimental DP versus DP data predicted by developed simulator (model) under the blackoil mode, using the CWI-relative permeability data (Table 4-11), CWI test. .................................. 159 Figure 4-46: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the developed simulator, CWI test. ............................. 161 Figure 4-47: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the developed simulator, CWI test............................................................................. 162 Figure 4-48: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator; MTC=2.2E-7, 6.5E-7 and 11E-7 1/sec, CWI test. .................................................................................................................................................. 164 Figure 4-49: Total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values; D=0, 950E-4 and 2000E-4 cm2/sec, CWI test. .................................................... 165 Figure 4-50: Total CO2 production predicted by the simulator for injection rates of 5, 10 and 20 cm3/hr, CWI test. .......................................................................................................................................... 166 Figure 4-51: Total oil production (TOP) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, MTC=2.2E-7 1/sec, CWI test. ..................... 167 Figure 4-52: Profile of CO2 concentration in oil phase predicted by the simulator for D=0 and 950E-4 cm2/sec after 0.3 and 0.5 injected PV, CWI test. ............................................................................. 169 Figure 4-53: RF predicted by the simulator for three different injection rates of 20, 10 and 5 cm 3/hr, CWI test. .................................................................................................................................................. 170 Figure 4-54: RF profile for three different carbonation levels of 7, 5 and 3 wt %, predicted by the developed simulator, MTC= 2.2E-7 1/sec, CWI test. ....................................................................... 171 Figure 4-55: Profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S) versus injected PV of carbonated water, predicted by the simulator, CWI test. ............................ 172 Figure 4-56: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator, CWI test. .......................................................................................................................................... 173 Figure 4-57: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator for MTC values of 2.2E-7, 6.5E-7 and 11E-7 1/sec, CWI test................................................................. 174 Figure 4-58: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator for injection rates of 20, 10 and 5 cm3/hr, CWI test. ............................................................................ 175 Figure 4-59: RF predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=2.2E-7 1/sec, CWI test. ............................................ 176 Figure 4-60: DP data predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=2.2E-7 1/sec, CWI test. ............................................ 176 Figure 4-61: TCO2P data predicted by the developed simulator compared to those predicted by the E300 simulator; MTC=2.2E-7 1/sec, CWI test. .......................................................................................... 177 Figure 4-62: TOP data predicted by the E300 simulator for injection rates of 20 and 5 cm3/hr, CWI test. ......................................................................................................................................................... 178 Figure 4-63: TCO2P data predicted by the E300 simulator for injection rates of 20 and 5 cm 3/hr, CWI test. .................................................................................................................................................. 178 Figure 4-64: TOP data predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=11E-7 1/sec, CWI test. ............................................. 180

xiv

Figure 4-65: Total CO2 production (TCO2P) data predicted by E300 simulator compared to those predicted by the developed simulator; MTC=11E-7 1/sec, CWI test. ............................................. 180 Figure 4-66: TOP data predicted by E300 simulator compared to those predicted by the developed simulator at injection rate of 4 cm3/hr, CWI test. ........................................................................... 181 Figure 4-67: Recovery factors prediced by the devlpoed simulator at three different MTC valuses of 7E-7, 11E-7 and 13E-7 1/sec. .................................................................................................................... 182 Figure 5-1: Schematic of the model. ........................................................................................................ 194 Figure 5-2: Grid size sensitivity analysis. .................................................................................................. 195 Figure 5-3: Shcmatic of the varaible area(blue rectangles) in front of fluid during traveling in the reservoir. ......................................................................................................................................... 197 Figure 5-4: Gas-Oil relative permeability curve........................................................................................ 199 Figure 5-5: RF of the secondary CWI compared to that of the secondary WI as predicted by the E300. 200 Figure 5-6: Gas saturation profile............................................................................................................. 201 Figure 5-7: CO2 concentration profile in water phase at 0.46 injected PV. ............................................. 201 Figure 5-8: CO2 concentration profile in oil phase at 0.46 injected PV. .................................................. 202 Figure 5-9: Water saturation profile at 0.46 injected PV. ........................................................................ 202 Figure 5-10: Total water production (TWP) and total CO2 production (TCO2P) profiles. ...................... 203 Figure 5-11: CO2 storage profile. .............................................................................................................. 204 Figure 5-12: Effect of KVKH ratio on oil recovery factor. ........................................................................ 205 Figure 5-13: Effect of KVKH ratio on CO2 production profile. ................................................................. 206 Figure 5-14: Recovery factor of the CWI at zero, 10% and 20% initial water saturations as predicted by the E300........................................................................................................................................... 207 Figure 5-15: Total water production (TWP) profile at zero, 10% and 20% initial water saturations as predicted by the E300. .................................................................................................................... 208 Figure 5-16: Total CO2 production (TCO2P) profile at zero,10% and 20% initial water saturations as predicted by the E300. .................................................................................................................... 209 Figure 5-17: RF profiles of secondary and tertiary CWI. .......................................................................... 210 Figure 5-18: Final water saturation distribution, secondary WI. ............................................................. 211 Figure 5-19: Final water saturation distribution, secondary CWI. ........................................................... 211 Figure 5-20: Final water saturation distribution, tertiary CWI. ................................................................ 212 Figure 5-21: The front view of the model. ............................................................................................... 213 Figure 5-22: Grid size sensitivity analysis using the developed simulator. .............................................. 213 Figure 5-23: RF of the WI predicted by E00 compared to that by the devlpoed simulator. .................... 214 Figure 5-24: RF of the CWI predicted by E00 compared to that by the devlpoed simulator. .................. 215

xv

Introduction It is expected that the global energy demand will increase by around 35% from 2011 to 2035. Fossil fuels will be the main source of energy and their share in world’s energy supply is anticipated to be around 80% in 2035. The contribution of crude oil to the world’s energy supply is estimated to be 26.3% in 2035 (OPEC publications, 2013). In an oil reservoir, first, oil is usually produced by natural depletion (primary production), while water is usually injected next (secondary production, 1st injection) and later enhanced oil recovery (EOR) techniques are applied (tertiary production, 2 nd injection). EOR techniques are increasingly being considered to satisfy the growing demand of oil in future. On the other hand, using fossil fuels has a negative impact on global warming by producing greenhouse gases (GHG), specifically CO2. Therefore, those techniques which help reducing CO2 concentration in atmosphere are desirable.

CO2-based injection EOR methods (CO2-EOR) are becoming more favourable due to simultaneous EOR and CO2 storage potentials. Continuous/cyclic CO2 injection and water alternating gas (WAG) are proficient CO2-EOR strategies which have been used and studied for some time by petroleum engineers. Carbonated water (CW) injection is a relatively new EOR method where CO2 is used efficiently. CW can be injected either in secondary mode (1st injection before water flooding) or can be injected in tertiary mode (after water flooding, 2nd injection). In the carbonated water injection (CWI) technique, carbon dioxide (CO2) is dissolved in the injected water. During CWI, and in comparison to the conventional water injection (WI), CO2 is transferred from carbonated water (CW) into the oil phase that results in increased oil mobility and higher oil recovery. Oil swelling, viscosity reduction and wettability alteration are the main mechanisms, which enhance oil recovery during CWI. Oil swelling refers to the ratio of the volume of oil with dissolved CO2 to the volume of oil without dissolved CO2.

During CWI, compared to typical CO2 injection, the problems of poor sweep efficiency and gas fingering are mitigated because CO2 exists as a dissolved phase. Typical CO2 injection projects need large quantities of CO2. Consequently, the availability of large sources of CO2 makes their application limited. CWI can be applied as an attractive alternative for such scenarios especially in offshore reservoirs, where the supply of CO2

1

Chapter 1: Introduction

is limited. Moreover, through CWI, some part of CO2 is stored in the reservoir in dissolved phase without the risk of CO2 leakage from caprock.

The CWI technique has not been studied extensively in the past. The first commercial application of CWI was in 1958 in United States called K&S project (Hickok et al., 1960). The K&S project was located 10-miles north of Bartlesville, Oklahoma. The project was first started by waterflood and then continued by carbonated waterflood. A substantial increase in water injectivity was observed during the CWI period. Additional oil recovery from watered-out area of the reservoir was also obtained. A simulation study was performed by Hasanvand et al. (2013) to evaluate CWI performance in an Iranian oil field. The simulation results showed that CWI could increase ultimate recovery by up to 20% compared to water injection.

CWI has been studied mainly experimentally with a focus on providing a better understanding of the involved mechanisms. Experimental study of CWI has mainly been focused on flooding tests including cores and sand packed set-ups. Direct visualisation of flow during CWI using a high-pressure transparent micro-model system (high pressure Hele-Shaw) has also been described in the literature.

1-1.

Experimental study of CWI

1-1.1.

Micro-Model Tests

Riazi et al. (2009 and 2011a) and Riazi (2011c) presented the results of some micro model tests that were carried out at 2000 Psia and 38 0C. The micro-model was constructed using two flat glass plates. To create an enclosed pore space, the surface of one glass plate was etched to create a two-dimensional pore structure and then covered by a second flat glass plate. The pore volume of the micro-model was 0.01 cc. The micro-model was saturated with n-decane (light oil sample with the viscosity of 0.83 cP at test conditions) when the established initial water saturation in the system was 33% (67% initial oil saturation).

The micro-model was determined to be water-wet when the direction and the shape of the water/oil interfaces were good indications of water-wet condition. Moreover, the smaller and dead-end pores were occupied by water. Initially water injection (WI) was performed 2


and then followed by CWI both at the same injection rate. The injection rate was 0.01 cm3/hr. At the end of WI period, the measured residual oil saturation was 49.1% which was equivalent to 27% oil recovery factor (RF) based on the original oil in place (OOIP). Significant expansion of oil (oil swelling) was observed. They reported that 64.3% oil saturation at the end of the CWI stage remained (swelled saturation), which was almost the same as the initial oil saturation (dead oil saturation). This indicated that swelling of the oil helps to mobilise the post water flood isolated ganglia when they are reconnected. CWI was later chased with another WI stage (second WI) to strip dissolved CO2 out of the swollen oil and obtain the volume of the remaining dead oil. The oil saturation at the end of the second WI stage was reduced to 33%. This means that the final oil recovery after 1st WI- CWI-2nd WI cycle was 51% (based on OOIP). It should be noted that in micro-model system the recovery factor was obtained by measuring the amount of dead oil saturation remained in the system using image analysis technique. However at end of the CW injection step, the remaining oil was not dead as oil had some dissolved CO2. Therefore the second WI was performed to obtain the dead residual saturation remained from CWI. That is, they expected that 2nd WI could not result in any more considerable oil production except striping of CO2 out of the oil. However to investigate this, they continued the test by 2nd CWI and 3rd WI and from their observation, they concluded that the 2nd WI had a contribution of around 2% in oil recovery. They attributed this oil production (occurred during 2nd WI) to local flow diversion and different shrinkage and swelling rates. They observed that the shrinkage rate during the second WI stage was meaningfully quicker than the swelling rate during CWI and it was not a reverse process of the previous swelling process. It was also observed that the oil swelling impacted the flow paths at the pore level. That is, a local flow diversion at the pore level was observed when oil swelling caused partial and or complete blockage of some pores. This in turn caused redistribution of fluids and diversion of the injected water towards the un-swept zones, which resulted in additional oil recovery. However, the mechanism for blockage is not explained. The test was repeated when CW was injected as a secondary mode (without preceding water injection). The recovery factor obtained at breakthrough time (BT) was almost the same as that of the first WI stage of the previous test. Based on that, they suggested that CO2 diffusion until BT of CW was not significant. They continued injection of CW and obtained more oil recovery after BT while for secondary WI of the previous test no more oil could be produced after BT. They observed swelling and reconnection of disconnect ganglia due to CO2 diffusion after BT which was observed in 3


the first test when CW was injected at the tertiary mode. They expressed that CWI resulted in more oil recovery at both secondary and tertiary injection modes and the main mechanism for the employed light oil sample was swelling and reconnection of oil blobs as well as flow diversion due to this swelling.

This work was continued by Sohrabi et al. (2009, 2011a and 2011b), who followed the same procedure and presented the results of CWI in the same micro-model system at the same test conditions but using a refined synthetic viscous oil sample (with the viscosity of 16.5 cP at test conditions). The goal was to compare the results with those of light oil sample (n-decane) mentioned above. For this oil sample the established initial oil saturation was 76.4% (23.6% initial water saturation) due to its higher viscosity compared to n-decane. First water was injected. For this oil sample after the WI stage, larger trapped oil ganglia were reported which resulted in 26 % RF. At this point CWI was started with the same injection rate. Oil swelling and reconnection of oil blobs was observed for this oil sample as well. However, less oil swelling had occurred, compared to that of the decane light oil due to the fact that the solubility of CO2 is higher for those oils with smaller viscosity values. The estimated swelling factors for the light oil (i.e. n-decane) and viscous oil in the experiments were reported to be about 105% and 23%, respectively. CWI was followed by the second WI test. 11.8% of residual oil that was remained after the first WI could be recovered at the end of the CWI and second WI (compared to 32.7% for light oil). They summarised that CWI for the lighter oil sample, with a higher CO2 solubility, resulted in larger swelling factor as the main reason for additional oil recovery while for the viscous oil sample, viscosity reduction and oil swelling played a major role.

Kechut et al. (2010) continued the experiments using the same micro-model system and conditions however oil was a crude viscous oil sample taken from a North Sea oil reservoir (with viscosity of 145 cP at 2000 Psia and 38 0C). Degassed distilled water was used to prepare carbonated water. After establishing the initial water saturation of 21% (with 79% initial oil saturation), WI was started at the rate of 0.008 cm3/hr. When no more oil was produced, WI was stopped and CWI was commenced with the same injection rate. Oil swelling and reconnection of disconnected oil ganglia in addition to flow diversion was observed. Moreover during CWI, the oil colour changed from dark brown to light brown. This change was attributed to the alteration of physical properties of the oil such as viscosity or density due to CO2 diffusion into the crude oil. At the end 4


of the CWI stage (after 140 hr) second WI was carried out to strip the CO2 out of the swelled remaining oil in the system. In order to estimate the oil saturation at different stages of the test, the area occupied by the oil was captured when an image analysis computer program was used. During tertiary CWI stage, 7% additional oil was recovered. Maximum swelling factor was around 15%.

1-1.2.

Coreflood Experiments

CW coreflood experiments have been reported in the literature. Sohrabi et al. (2011b and 2012a) carried out a series of CW coreflood experiments. Three types of core including a reservoir core (a relatively unconsolidated water-wet core taken from a North Sea sandstone oil reservoir) as well as two Clashach sandstone cores with different wettability (i.e. a mixed-wet and a water-wet sample) were used. Three oil samples including a light oil (n-decane), a refined viscous oil and a stock-tank crude oil were employed. Two different brine samples with different salinity values were used for preparation of carbonated water samples. Three sets of coreflood tests were performed. Two secondary CWI coreflood experiments (without preceding water injection) were carried out using Clashach water-wet and mixed-wet cores with no initial water saturation at 38 0C and 2000 Psig. Decane and refined viscous oil (with viscosity of 81 cP at test conditions) were used in these set of experiments. Reservoir core was used in the third set of tests. Here both secondary and tertiary CWI experiments were performed using crude oil sample at 38◦C and 2500 Psig (with viscosity of 158 cP at test conditions) as well as at 38 0C and 2000 Psig (with viscosity of 145 cP at test conditions). In this set of tests, the established initial water saturation was 10.6%. The performance of CWI was studied under various experimental conditions when the quantity of additional oil recovery and CO2 storage were measured. CWI, compared to WI, resulted in a higher oil recovery as a secondary injection recovery method. Additional oil recovery was also obtained in the tertiary recovery method after WI. Recovery improvement was observed for both secondary and tertiary CWI over the plain water injection. However, secondary injection of CW resulted in higher and earlier incremental oil recovery than the tertiary injection of CW. They explained that in addition to the oil swelling (based on their observation from micromodel tests), viscosity reduction and wettability of rock during CWI could also affect the oil recovery. They compared the contribution of CWI (comparing the recovery factors of CWI over WI) for decane and synthetic viscous oil and they realised that they are almost the same (approximately 3-4%). They concluded that as oil swelling is much lower for 5


viscous synthetic oil (based on their observation from micro-model tests), therefore viscosity reduction should also play a role for the viscous oil. However, it is not explained clearly how they concluded these mechanisms. Moreover, for mixed-wet (aged) core, they observed less pressure drop for CWI comparted to that for WI and also a better performance of CWI compared to the CWI performance in water-water-wet core. They interpreted these observations as being due to wettability effects.. Moreover, brine salinity influenced the CWI performance. Comparing the results of all experiments, the coreflood test using mixed-wet core and light oil sample with injection of low salinity carbonated brine resulted in higher oil recovery factor. Furthermore, a promising quantity of the injected CO2 could be stored in the form of dissolved CO2 in the remained fluids in the cores at the end of experiments.

Sohrabi et al. (2012b) presented the results of other series of coreflood tests using sandstone Clashach consolidated cores and a crude oil sample with a synthetic seawater. An injectivity improvement was reported when lower differential pressure (DP) across the core was observed in CWI compared to that in WI. Improvement in oil recovery was also achieved for secondary and tertiary injection of CW. Moreover, it took longer time for breakthrough of water in CWI than that in plain WI. Furthermore, significant storage of CO2 in the rock during CWI was also reported.

Tavakolian (2012) presented the results of some coreflood experiments investigating the potential of CO2 storage during CWI compared to pure CO2 injection. They compared the CO2 production against CO2 retention (storage) for pure CO2 injection with that for CWI observed in the coreflood experiments. They observed that during pure CO2 injection, higher retention of the gas was obtained as larger volumes of CO2 were injected. However, when CO2 production started at core outlet after breakthrough time, CO2 retention dropped quickly leading to a considerable production of CO2. During CWI, however, smaller amount of CO2 was retained in the pore space as less CO2 was injected in comparison with pure CO2 injection. On the other hand, during CWI, CO2 breakthrough occurred with a delay and the slope of CO2 production profile was not as sharp as the case of pure CO2 injection.

Alizadeh et al. (2014) presented the results of a series of tertiary (CWI) coreflood experiments followed by CO2 exsolution (liberation), as a consequence of pressure 6


depletion, leading to a significant recovery of trapped oil after WI stage. Coreflood experiments were carried out at both macro and micro scales and at low pressure (around 90 Psig) and ambient temperature using a mineral oil. At the macro-scale, a long consolidated Berea core (with length of 25.4 cm and diameter of 3.81 cm) was used in accompany with a x-ray CT scanner for in situ measurement of saturations. At the microscale a smaller Berea core sample (with length of 9.5 cm and diameter of 0.372 cm) along with a micro-CT scanner were utilized to capture high-resolution images of pore-scale fluid occupancies. When the injection of tertiary CW started, the pore pressure in the core was slowly reduced using a tightly-controlled back pressure pump letting gradual exsolution of the dissolved CO2. Observations at the pore-scale of three-phase flow including two immiscible disconnected phases (i.e., oil blobs and gas bubbles) and continuous wetting phase (i.e., water) were reported. In the macro-scale experiments, tertiary CWI process resulted in 34.6% additional recovery of the original oil in place and in the micro-scale experiments it resulted in 40.7% additional oil recovery. Pore scale visualisation using micro-CT scanner illustrated that the pressure drop applied after CWI stage led to exsolution of CO2, internal gas drive, mobilisation of oil ganglia, which consequently resulted in reduction of waterflood residual oil saturation. When oil ganglia contacted by liberated CO2, a thick spreading layer of oil blobs was formed between the brine and gas. As a result, moving gas clusters could displace this oil layer towards the outlet of the core. A substantial reconnection of disconnected oil ganglia were observed due to formation of oil layer right after CWI process. The oil layers formed were stable until very late stages of gas liberation process. Moreover, CO2 stayed entrapped in the rock at the end of the CWI process as the form of free gas as well as dissolved in brine suggesting the potential of CWI for geologic sequestration of CO2.

1-1.3.

Sand-packed Flooding Experiments

CW sand-packed flooding experiments have been reported in the literature as well. Dong et al. (2011) performed a series of experiments using sand packs with deionised water and dead oil from a deep-water field in Gulf of Mexico. The flooding experiments were performed at the pressure of 600 Psig and the temperature of 104 ºF. They observed recovery improvement during secondary CWI over conventional WI under the same conditions. Moreover they observed that tertiary CWI could result in some oil recovery as well. They explained that tertiary injection of CW could remobilise oil droplets trapped 7


during WI stage resulting in more oil recovery. In the secondary injection mode, CWI produced a higher volume of oil than WI when the same pore volume (PV) was injected. Various RF were presented based on different sand-pack systems and injection conditions. For instance at an injection rate of 4 PV/day, after injection of 8 PV of water (1st WI, secondary mode) a RF of 41% was obtained. The injection was continued using CW (tertiary mode) for another 8 PV (total 16 pore volume) and recovery factor was increased to 62%. The test was repeated however, CW was used (1st CWI, secondary mode) and after 16 PV injection of CW, a RF of 85% was obtained. This shows the better performance of CWI if it is injected before water flooding.

Mosavat and Torabi (2014a and 2014b) reported some CWI tests carried out in a sandpacked setup for a light oil sample at secondary and tertiary mode at different pressures and temperatures. They used a light stock tank oil sample which was a mixture of few samples taken from Bakken formation in Saskatchewan, Canada. Moreover, a synthetic brine of 20,000 ppm NaCl was used to prepare carbonated water. The length and internal diameter of the sand-pack holder were 30.48 and 2.54 cm, respectively. The main composition of sands used were silicon dioxide (SiO2) with a particle mesh size ranges from ASTM 40 to 270. They observed that the injection of CW at secondary mode gave better recovery compared to tertiary mode. Moreover they explained that, the CO2 content of injected water plays an important role in the CWI performance. they reported that CWI resulted in an average of 17% additional oil recovery at secondary mode of injection (compared to secondary WI) while 23-36% of total volume of injected CO2 was stored. However they noted that the performance of CWI was dependent on the pressure and temperature of the injected water. This was due to the fact that CO2 solubility in injected water is controlled by these parameters. They observed that, the injection of carbonated water with higher CO2 content resulted in higher oil recovery.

Generally, the solubility of CO2 in water plays an important role in performance of CWI process as it controls the amount of CO2 carried by carbonated water into the reservoir (Mosavat and Torabi, 2014a and 2014b). The solubility of CO2 in water is higher at higher pressures and lower temperatures and salinities (Stefan, 2011; Baviere, 1991). Moreover, CO2 solubility in oil increases with pressure and oil gravity (°API) and its value decreases with temperature (Emera and Sarma 2008).

8


A summary of the experiments presented above with more details are given in Table 1-1, Table 1-2 and Table 1-3. That is, these tables summarises the recovery factors presented above (after each injection step) for micro-model, coreflood and sand-packed experiments based on OOIP.

Table 1-1: Summary of micro-model experiments.

1st WI RF%

1 CWI RF%

2nd CWI (injection after 1st WI) RF%

27%

not applie d(NA)

not reported (NR)

51%

NA

NR

35%

NA

NR

41%

st

Experiment

Riazi et al. (2009 and 2011a)

Sohrabi et al. (2009, 2011a and 2011b)

Kechut et al. (2010)

P=2000 Psia, T=38 0C PV=0.01 cc, oil sample: n-decane 𝜇𝑜 =0.83 cP, 𝜌𝑜 =0.73 g/cm3 water: distilled water CO2 content: 5% wt 𝜇𝑐𝑤 =0.66 cP, 𝜌𝑐𝑤 =0.99 gr/cm3 Swi=33%, qinj= 0.01 cm3/hr P=2000 Psia, T=38 0C PV=0.01 cc, oil sample: synthetic oil 𝜇𝑜 =16.5 cP, water: distilled water CO2 content: 5% wt 𝜇𝑐𝑤 =0.66 cP, 𝜌𝑐𝑤 =0.99 gr/cm3 Swi=23.6%, qinj= 0.01 cm3/hr P=2000 Psia, T=38 0C PV=0.01 cc, oil sample: crude oil 𝜇𝑜 =145 cP, 𝜌𝑜 =0.93 gr/cm3 water: distilled water CO2 content: 5% wt 𝜇𝑐𝑤 =0.66 cP, 𝜌𝑐𝑤 =0.99 gr/cm3 Swi=21%, qinj= 0.008 cm3/hr

26%

32%

9

2nd WI (injection after 2nd CWI) RF%


Table 1-2: Summary of coreflood experiments.

Experiments

P=2000 Psia, T=38 0C, swi=0 water: synthetic brine (1% salinity) CO2 content: (5% wt) 𝜇𝑐𝑤 =0.7 cP, 𝜌𝑐𝑤 =1.01 gr/cm3, qinj= 20 cm3/hr Sohrabi et al. (2011b and 2012a ,2012b)

P=2500 Psia, T=38 0C, water: synthetic brine (3.5% salinity) CO2 content: (4.5% wt) 𝜇𝑐𝑤 =0.9 cP, 𝜌𝑐𝑤 =1.01 g/cm3

Alizadeh et al. (2014)

P=90 psig T: Ambient oil sample: mineral oil 𝜌𝑜 =0.8 g/cm3 water: synthetic bine CO2 content: 0.01 g/cm3 𝜌𝑐𝑤 =1.1 g/cm3

Clashach Sandstone, water -wet L=33.2 cm D=5cm k=1.3 D 𝜑=0.19 Aged Clashach Sandstone, mixed -wet L=61.3 cm D=5cm k=0.85 D 𝜑=0.17 reservoir core, water -wet L=8.1 cm D=3.7cm k=4.58 D 𝜑=0.35 swi=13%,

oil sample: decane 𝜇𝑜 =0.8 cP 𝜌𝑜 =0.73 g/cm3 oil sample: synthetic oil 𝜇𝑜 =81 cP 𝜌𝑜 =0.91 oil sample: decane 𝜇𝑜 =0.8 cP 𝜌𝑜 =0.73 g/cm3

oil sample: crude oil 𝜇𝑜 =154 cP 𝜌𝑜 =0.93 g/cm3

Clashach Sandstone, water -wet oil sample: L=33 cm crude oil D=5cm μo =8.5 cP k=1.123 D API=28.5 𝜑=0.24 swi=35%, qinj= 5 cm3/hr Berea, water -wet L=25.4 cm D=3.8 cm k=0.089 D 𝜑=0.2 swi=35%, qinj= 0.2 cm3/hr Berea, water -wet L=9.5 cm D=0.4 cm k=0.088 D 𝜑=0.19 qinj= 0.05 cm3/min

10

1st

1st

WI RF%

CWI RF%


73%

NA

50%

53%

NA

59%

68%

NA

59%

NA

68%

43%

55.5

53%

40%

NA

74%

35%

NA

75%

69%


Table 1-3: Summary of sand-packed experiments.

1st CWI RF%


41%

85%

62%

60%

71%

67%

1st Experiment

Dong et al. (2011)

Mosavat and Torabi (2014a and 2014b)

WI RF%

P=600 Psig, T=40 0C oil sample: dead oil 𝜇𝑜 =70.7 cP, 𝜌𝑜 =0.94 g/cm3 water: deionised water CO2 content: 3% wt 𝜇𝑐𝑤 =0.65 cP, 𝜌𝑐𝑤 =0.99 gr/cm3 Swi=0%, sand-packed: silicon dioxide (SiO2). mesh size: ASTM 50-200 k=3.5 D 𝜑=0.3 qinj= 4 PV/day P=4.1 MPa, T=25 0C oil sample: crude oil 𝜇𝑜 =2.7 cP, 𝜌𝑜 =0.799 g/cm3 water: brine salinity: 20000ppm CO2 content: 0.6 mol CO2/kg brine 𝜇𝑐𝑤 =0.9 cP, 𝜌𝑐𝑤 =1.01 gr/cm3 Swi=24%, sand-packed: silicon dioxide (SiO2). mesh size: ASTM 40-270 k=5.61 D 𝜑=.27 qinj= 1 cm3/min

Wettability alteration during CWI was also reported in the literature. Yang et al. (2008) experimentally measured the contact angle of a crude oil- reservoir rock-reservoir brine system with dissolution of CO2 at two temperatures ( 27 and 58 0C) and a range of pressures (0.4 t-34 MPa), using the axisymmetric drop shape analysis (ADSA) technique (sessile drop method) to determine the wettability. The reservoir brine and the crude oil samples were collected from the Weyburn Oilfield in Saskatchewan, Canada. The density of the reservoir brine was 1.045 g/cm3 at 15 °C. The density and viscosity of the Weyburn crude oil were 0.877 g/cm3 and 13.0 cP at the atmospheric pressure and 27 °C, respectively. The rock slides were taken from a core sample of a Vuggy formation in the Weyburn Oilfield. The Vuggy formation was characterized as limestone with small to 11


medium vugs to microcrystalline porosity. In addition, the Vuggy formation was determined to have an intermediate-wet using the USBM/Amott wettability analysis technique. A high-pressure cell having a sight glass window was used and prefilled with reservoir brine to submerge the rock slide. Then, at a constant temperature, CO2 was injected into the brine to pressurise the system to a planned pressure. After the CO2reservoir brine reached the equilibrium state, crude oil sample was introduced through a syringe needle to form an oil drop on the rock surface. A computer-aided image acquisition and processing techniques was used to measure the dynamic contact angles at different times by taking sequential images of the oil drop. The purpose was to measure the dynamic contact angles when CO2 was gradually dissolved into the oil drop and also the equilibrium contact angle when oil drop was fully saturated by CO2. They mentioned that the contact angle measurement was carried out for 6-24 hours; however the equilibrium contact angle was obtained after approximately after 7-8 minutes. It was observed that the dynamic contact angle between the crude oil and the reservoir rock in the presence of CO2-saturated brine was constant at a given pressure and temperature, although CO2 was gradually dissolved into the oil drop until the oil drop was completely saturated with the CO2. It was also observed that the equilibrium contact angle increased when the pressure was increased, whereas it decreased when the temperature was increased. This was attributed to the solubility of CO2 in brine which is higher at higher pressures and lower temperatures. Same measurements were repeated without any dissolution of CO2. In comparison with the equilibrium contact angle data without any dissolution of CO2, the equilibrium contact angles of the crude oil-reservoir brinereservoir rock system with dissolution of CO2 show a change (around 200) from oil-wet towards intermediate-wet.

Based on some observation during the micro-model tests, Sohrabi et al. (2012a) compared the shape of oil ganglia trapped in the micro-model by snapped off mechanism after secondary injection of water and carbonated water. They observed that the oil blobs had more rounded shape after CWI compared to those after the WI. Based on that, they concluded that micro-model surface became more water-wet after CWI. Kechut et al. (2010) observed more water droplets between oil and micro-model glass after tertiary CWI. They suggested that the tendency of water droplets to reach micro-model surface showed that the micro-model glass became more water-wet.

12


Seyyedi et al. (2015) performed a series of contact angle measurements to determine the wettability of three different minerals (substrates) of Quartz, Mica, and Calcite which were exposed to a crude oil and a brine as well as a carbonated brine at a wide range of pressures (from 100 to 3500 psi) and a constant temperature of 100 F. Quartz is a main mineral in sandstone rocks. Mica is a mineral which can be found in some shale and clay materials, and Calcite is the main mineral in carbonate rocks. The API of crude oil was 26.2. The brine was a relatively high salinity sea brine (54,597 ppm). In addition to clean substrates, the substrates were also aged in the same crude oil to measure the contact angle of aged minerals as well. A high-pressure, high-temperature Drop Shape Analyser rig was used to measure the contact angle by analysing the shape of oil drop introduced through a needle to the surface of the substance submerged in the brine or in the carbonated brine. Comparing the equilibrium contact angles with and without the present of CO2, the unaged quartz showed a contact angle change (around 60) from water wet towards less water-wet due to CO2 dissolution in brine while the clean (un-aged) Mica and Calcite showed a contact angle change (around 50) from water wet towards more water-wet due to CO2 dissolution in brine. Moreover, the aged Quartz was relatively neutral-wet while the un-aged Quartz was more water-wet. In comparison with the contact angle measured for the aged Quartz-crude oil-plain brine, an equilibrium contact angle towards water-wet (from 760 to 610) was observed due to CO2 dissolution (i.e. for aged Quartz-crude oilcarbonated brine). For the aged Mica the contact angle changed from 890 to 630. For the aged Calcite, a contact angle change from 1440 to 970 due to CO2 dissolution was observed.

All these studies show that the carbonated water can change the wettability of rock surfaces specifically the oil-wet surfaces to neutral-wet surfaces or neutral-wet surfaces to more water-wet surfaces. However it has minimal effect on water-wet or strong waterwet surfaces. To summarise the results from experiments, the CWI produces higher oil recovery over conventional WI at secondary injection mode and CWI has better performance at secondary (before water flooding) compared to tertiary injection mode (after water flooding). Moreover, oil swelling, viscosity reduction and wettability alteration may play a role during CWI.

13


1-2.

Simulation Study of CWI

In the literature, modelling and simulation of CWI process have not been given sufficient attentions and there is no comprehensive mathematical study capturing the actual physics of the CWI process. Modelling and simulation of CWI is complex due to multi-physics nature of the process. That is, during CWI process, fluid flow and mass transfer occur simultaneously. Moreover, flow of fluids in the reservoir during the course of CWI is accompanied by CO2 mass transfer between the phases leading to the variations of the fluid properties (i.e. viscosity and density changes). The literature data show that, both black-oil and compositional models have been used to simulate CWI process.

De Nevers (1964) presented a calculation method based on the Buckley–Leveret theory to predict the CWI performance. The developed method accounted for the effect of viscosity reduction and oil swelling during CWI process. It was assumed that in any point in the reservoir where CO2 is present, the CO2 concentration in oil and water phases is the equilibrium concentration. That is, CO2 partial pressure in both water and oil phases are same at each point (instantaneous equilibrium condition). Moreover molecular diffusion and dispersion were assumed to be zero and CO2 moved forward only by flow of water and oil phases. The mathematical model was developed to predict the performance of injection of a ‘slug’ of carbonated water into the reservoir. The objective was to see the effects of slug size as well as CO2: water ratio in that slug, on oil recovery. The viscosity and density of the CO2-oil mixture were considered to change as a function of CO2 concentration. In the presented calculation method, injected carbonated water can be deprived of all its CO2 and moves as plain water. In their model, a slug of carbonated water was injected (simultaneous injection of water and CO2) driven forward by plain water. That is, CO2 injection into the reservoir was stopped when a planned amount of carbonated water was injected however the plain water injection was continued. This determined the slug size of the injected CW which was moved through the reservoir by plain water.

Ramesh and Dixon (1972) presented a numerical black oil based model to predict the performance of carbon dioxide (CO2) flooding and CWI into heterogeneous oil reservoirs. The model was two-dimensional, developed based on transport equations describing simultaneous flow of three oil, water and CO2 phases. CO2 was allowed to dissolve in oil 14


and water phases and could also flow as a free phase. They assumed that the CO2 solubility in oil and water phases is only a function of pressure and the equilibrium between oil and water phases containing CO2 occurs instantaneously. They assumed that the presence of oil phase does not noticeably influence the CO2 solubility in water and vice versa. Therefore they use the solubility data from the separate CO2-oil and CO2water systems for the CO2-oil-water system.

Mansoori (1982) developed a compositional simulator to include the impact of CO2 solubility in water during CO2 injection. In the simulator, Soave–Redlich–Kwong (SRK) equation of state (EOS) was used to calculate phase equilibrium and density of CO2hydrocarbon mixture. CO2 solubility in water was taken into account using Henry's law. Component material balance and Darcy law were used to derive governing flow equations. Phase equilibrium calculation was based on the equality of components fugacities in all available phases (hydrocarbon and aqueous). In the simulator, three phases (oil, gas and water) could exist in each simulation grid block while they reach equilibrium state immediately. The finite difference technique was used to solve the equations. Final recovery reduced about 3-4% pore volume when CO2 solubility in water was included. However they mentioned that this reduction varies with the mode of operation.

Chang et al. (1998) presented a three-dimensional, three-phase compositional model to simulate CO2 flooding processes, when CO2 solubility in water is included. In this work, a cubic EOS was used to calculate oil and gas densities and fugacities and accordingly CO2 distribution between oil and gas phases. Correlations were used to calculate the oil and gas phase viscosities. No hydrocarbon components could dissolve in the water phase and only CO2 component could exist in three phases. Water phase properties were a function of the dissolved CO2 content. CO2 solubility in water and other properties of CO2-water system were computed using correlations. It was assumed that phases would reach thermodynamic equilibrium state instantaneously according to the equality of components fugacities in different phases. A method was presented to calculate the fugacity of CO2 component in aqueous phase as a function of pressure at reservoir temperature using available solubility data of a saturated binary CO2-water system. The model prediction showed that about 10% of the injected CO2 could dissolve in water and as a result there was not available to be mixed with oil. Consequently, including CO2 15


solubility in water reduced final oil recovery. A reduction of 5% of final recovery was reported when CO2 solubility in water was taken into account for a tertiary injection of CO2 (post-waterflood).

Kechut et al. (2010, 2011a and 2011b) used compositional simulators to simulate a set of carbonated water coreflood experiments. The coreflood experiments had been performed using three types of core (a reservoir water-wet core and two Clashach water-wet and mixed-wet sandstone cores) and three different oil samples: a light oil (n-decane), a refined viscous oil and a stock-tank crude oil. Secondary water injection and secondary and tertiary (post-waterflood) injection of carbonated water were simulated using the ECLIPSE commercial reservoir simulators. First water injection (WI) was simulated. The input water-oil relative permeability (Krw-o) curve into the simulator was obtained based on Corey correlations when production data was history matched. Next CWI was simulated. The ECLIPSE 300 compositional simulator was used with a tuned EOS, which could acceptably predict the PVT properties of oil-CO2 system. The Krw-o obtained for WI tests were input into the ECLIPSE 300 during CWI simulation. The simulation results by ECLIPSE 300 showed an over prediction of oil recovery in comparison with experimental data for all coreflood tests. They explained that compositional simulators were developed based on the inherent assumptions of instantaneous equilibrium and complex mixing while during carbonated water flooding, CO2 transferring process from water into the oil phase happens gradually. That is, the resistance against CO2 migration at the interface between water and oil phases plays an important role, which prevents immediate distribution of CO2 between the phases. Consequently, the assumption of instantaneous equilibrium may not be realistic during CWI process. To support their explanation, they mentioned the micro-model observation reported by Riazi et al. (2009), who used image processing technique during the conducted visualisation tests. They reported that in the micro-model system the swelling of oil blobs as a result of CO2 diffusion happens gradually and it needs time to stabilise and reach equilibrium state.

Figure 1-1 compares the changes in the shape of a specific oil blob in the micro-model during CWI process as a result of CO2 transfer from water into the oil phase(swelling process) versus shrinkage process during subsequent WI when CO2 comes out from the oil and moves into the water phase (Riazi et al., 2009). The swelling curve shows that the blob takes approximately 170 hour to reach its final shape corresponding to maximum 16


swelling value (equilibrium stage). This implies that CO2 diffusion from water into the oil is a slow process. The shrinkage curve, on the other hand, indicates faster mechanism of CO2 diffusion out of oil when the final shape and conditions reached quicker.

Figure 1-1: Swelling rate during CWI compared to shrinkage rate during subsequent WI (Riazi et al., 2009; Riazi, 2011c).

To explain the possible reason for this, Riazi (2011c) used a mathematical model and explained that the diffusion coefficient of CO2 in oil is variable and is a function of oil viscosity which in turn, is a function of CO2 concentration in oil. Therefore, at the early stage of shrinkage process, because the oil has higher CO2 concentration and in turn a lower viscosity, the diffusion coefficient of CO2 in oil is higher compared to the diffusion coefficient at early stage of swelling which oil has higher viscosity.

Based on these findings, Kechut et al. (2011b) concluded that the assumption of instantaneous equilibrium, which is used by available commercial compositional reservoir simulators, is not essentially true for all processes associated with the transfer of component between the phases. This was also mentioned before by Embid and Rivas (1994). They explained that this assumption can lead to a large error in the cases where mass transfer resistances are large. They mentioned that, this may happen when there are short contact times for mass transfer process (laboratory displacement in cores) or large diffusion patterns are available for components to diffuse through (field scale) and moreover if there is slow diffusion velocities due to large viscosity. All these can lead to not having an instantaneous thermodynamic equilibrium state. They developed a 17


compositional model for miscible gas injection when the mass transfer resistance at interface was included into the calculation.

In order to quantify the magnitude of mass transfer during a miscible displacement simulation and further evaluate the validity of instantaneous equilibrium assumption, a simple calculation was performed by Embid and Rivas (1994) using the correlation proposed by Collins (1976). The proposed formulation relates the saturation fraction sf , defined as the ratio of the volume of gas in the liquid, to the volume of gas in the liquid at equilibrium state, to other pertinent parameters based on:

sf =1-

8 -Dit( π )2 e zτ π2

(1 − 1)

where z is the oil pool depth (the thickness of oil which is in contact with gas) , τ is the tortuosity factor and Di is diffusion coefficient. They applied this equation for a typical simulation gridblock as shown in Figure 1-2. This typical simulation gridblock is one of several gridblocks in a simulation model.

Figure 1-2: A typical simulation gridblock.

X

They assumed that z is equal to 2 , i.e., oil has occupied the half of the gridblock volume as shown in the figure. In addition, they assumed a tortuosity equal to 2, x = 30 m, a time step (t) in the order of 30 days and a typical value of 2 × 10−9

m2 sec

for the CO2 diffusion

coefficient (Di ). Using these values sf = 0.19 was obtained. This sample calculation shows that, the volume of diffused gas in the oil in this gridblock during the assumed time step is 19 percent of the equilibrium value. This clearly demonstrates that the mass 18


transfer process is slow and a long time period is required to reach equilibrium which in the simulators is assumed to happen instantly.

Therefore, kinetics of mass transfer can be important at some conditions during gas injection into oil reservoirs. Kechut et al. (2011b) and Kechut (2011c) showed that mass transfer kinetics is important during CWI coreflood experiments. Thus, the current commercial compositional reservoir simulators cannot be used to simulate CWI coreflood experiments properly, as they are based on the inherent assumption of instantaneous equilibrium. That is, the current reservoir simulators apply the conventional concept of two phase fluid behaviour models with the constraint of thermodynamic equilibrium by equating the component fugacity in different phases in each gridblock (ECLIPSE Manual, 2014).This assumption results in a poor match between experimental and simulated recoveries with higher oil recovery from CWI predicted by the simulator. Thus a new simulation approach is required as suggested by Kechut et al. (2011b).

In summary, according to both experimental and mathematical data presented above, the role of rock wettability and the contribution of wettability alteration mechanism are uncertain during CWI process. Moreover with respect to simulation of CWI, the assumption that thermodynamic equilibrium is reached during CWI, as simulated, is questionable and there is not a kinetic model available to investigate this.

Therefore, the main objective of this study is to simulate the CWI process more realistically when the assumption of instantaneous equilibrium in the developed model (simulator) is relaxed. That is, the goal is to develop a mathematical model that captures the actual physics of CO2 mass transfer kinetics during CWI process. The simulator can be used to investigate the role of wettability during CWI process.

The mathematical model should describe two phase flow of oil and CW including dynamic process of CO2 transfer from water into oil phase and the resultant favourable changes in oil fluid properties. It should be noted that, to the best of author’s knowledge, there is not any compositional simulator available which captures the kinetics of mass transfer during multiphase flow in porous media. Accordingly, a one dimensional nonequilibrium based compositional simulator is developed. To control the amount and rate of CO2 transfer between water and oil, commonly used compositional component based 19


flow equations are modified by adding a source term to capture the kinetics of CO2 transfer between the two phases. The simulator is a two-phase model because during CWI, CO2 is only present in solution and not as a free gas.

The structure of the model will be discussed in detail in Chapter 2. Moreover in Chapter 2, the structure of an optimiser, which is developed based on the genetic algorithm theory for obtaining the unknown parameters of the model during simulation of CWI will be discussed. Chapter 3 will discuss the procedure followed to check the validity of the model and also the performance of the developed simulator for simulation of a carbonated water coreflood experiment carried out in a water-wet core. Chapter 4 will present the capability of the model when a different carbonated water coreflood experiment performed in a mixed-wet core, is simulated. Chapter 5 will present reservoir-scale simulation of CWI process. Chapter 6 will summarise the results and the outcomes of this thesis.

20

Model Structure This chapter presents the structure of the model developed in this study to simulate carbonated water injection (CWI) process. To develop a mathematical model for the simulation of CWI process, the governing partial differential equations representing the actual physics of the process were derived. Fully implicit numerical solution was applied to solve these equations. The model (simulator) aims to simulate CWI coreflood experiments and is developed for a homogeneous and isotropic porous medium with a length of L, a cross-sectional area of A, a constant porosity of φ and a constant permeability of k. After solving the equations, the phase pressure profile across the system as well as the distribution of saturation and composition of the components in the system will be obtained. To simulate the process and solve the governing equations, the following assumptions were made. 

The model is one-dimensional in cartesian coordinate.



No chemical reactions resulting from the formation of carbonic acid and its

subsequent dissociation are considered. 

Two phases, predominantly consisting of water and oil, are present, i.e. CO2 is in

solution and not as a free gas. 

The oil is a dead oil sample so there is no free hydrocarbon gas in the system.



The oil is single component and is not allowed to dissolve in water and vice versa.



Gravity effect is neglected.



Density and viscosity of oil and water change based on their CO2 content.



The water is fresh with no salinity.



Dispersion (or diffusion) coefficient is constant.



Temperature is constant.

It should be noted that based on the above assumptions, the model will have some limitations which may not be used directly for field-scale. For example, the oil is dead with one component, the model is one dimensional with no gravity effect and the injected water is fresh. Nevertheless, the application of model for simulation of real reservoirs is discussed in Chapter 5.

21

Chapter 2: Model Structure

2-1.

Governing Equations

Governing equations for carbonated water flooding system are derived based on the mass balance equations for each component existing in the system. Generally, the mass balance equation for component ‘I’ in phase ‘α’, which is displaced through a porous medium in ‘x’ direction by convection and dispersion (or diffusion) mechanisms, is shown by Equation 1:

φ

∂(ρα sα ωIα ) ∂t

=-

∂(ρα uα ωIα ) ∂x

∂ (ρα ωI ) ∂ α + (φ sα DI-α )+U ∂x ∂x

(2-1)

where, ωIα is the mass fraction of component I in the phase α. ρα and sα are the mass density and saturation of phase α, respectively. uα is the Darcy velocity of phase α and DI-α is the dispersion or diffusion coefficient of component I in phase α. The term on the left hand side (LHS) is accumulation term. The first term on the right hand side (RHS) is the convective term, which implies the displacement of component I through the porous medium by flow of the phase α. The second term on the RHS represents the displacement of component I within the phase α by dispersion or molecular diffusion mechanism. It should be noted that the formulations of dispersion and molecular diffusion terms in the mass balance equation are similar however the dispersion coefficient value is greater than the diffusion coefficient value. The last term on RHS,’U’, is the source or sink term, which describes the amount of component I entering (source) or exiting (sink) the phase α. During simulation of CWI, the U term can be representative of CO2 transfer between the phases.

Considering the simulations of CWI, two phases in the system are oil and water. Moreover, three components are oil, water and CO2 with the CO2 component existing in water or oil or both phases. The mass balance equation for each component in each phase results in four main expressions as follows:

φ

∂(ρo so ωoo ) ∂(ρo uo ωoo ) =∂t ∂x

(2-2a)

22

Chapter 2: Model Structure co

co

co

φ

∂(ρo so ωo 2 ) ∂(ρo uo ωo 2 ) ∂ ∂(ρo ωo 2 ) =+ (φ so Dco2 -o )+U ∂t ∂x ∂x ∂x

(2-2b)

φ

∂(ρw sw ωw ∂(ρw uw ωw w) w) =∂t ∂x

(2-2c)

co

co co ∂(ρw uw ωw 2 ) ∂ ∂(ρw sw ωw 2 ) ∂(ρw ωw 2 ) φ =+ (φ sw Dco2 -w )-U ∂t ∂x ∂x ∂x

(2-2d)

where ωIα is the mass fraction of component I (oil, water or CO2) in phase α (water or oil). It should be noted that the oil phase is a mixture of oil and CO2 and water phase is a mixture of water and CO2. ρα and sα are the mass density and saturation of phase α, respectively (α =oil or water). 𝑢α is the Darcy velocity of phase α defined as follows:

uα = -λα

∂pα ∂x

where

λα =

kk rα μα

α= o, w

(2-2e)

where k rα is the relative permeability of phase α (oil or water) and is a function of water saturation and pα is the pressure of phase α. The ‘p’, ‘ω’ and ‘s’ parameters are bounded by the following auxiliary equations: pc =po -pw =f(sw )

α=o, w

∑ sα =1

∑ ωIo =1

(2-2f)

I=o, co2

(2-2g)

and

∑ ωIw =1

I=w, co2

(2-2h)

where pc is the capillary pressure and is a function of water saturation. Equations (2-2a) and (2-2b) are the mass balance equations for oil and CO2 components in the predominantly oil phase. Equations (2-2c) and (2-2d) are the corresponding mass balance equations for the water and CO2 components in the predominantly water phase. The third term on the right hand side of Equations (2-2b) and (2-2d), ‘U’, expresses the absolute value of the CO2 mass being transferred from the water phase into the oil phase and has 23


a positive value as long as the mass transfer is in this direction. The parameters of these equations are explained in more details later in this chapter. It should be noted that similar equations can be found in literature for CWI however without dispersion term (Steffens, 2010) or for other processes (Valiollahi et al., 2012).

It is worth mentioning that, contrary to the model here, in conventional compositional approach, CO2 is partitioned between the oil and water phases based on equilibrium concentration which is calculated using the fugacity equilibration method. That is, in each time step of simulation, the fugacity of CO2 in oil phase is equated to the fugacity of CO2 oil water in water phase i.e. fCO =fCO (equilibrium criterion). Fugacities are calculated using 2 2

equations of states (EOS). In conventional compositional approach which is used for example by ECLIPSE 300, the flow equations similar to those mentioned above (Equations 2-2a to 2-2d) are used however without ‘U’ term (i.e. U=0) and are solved oil water together with equilibrium criterion (i.e. fCO =fCO ) to obtain the pressure and saturation 2 2

of each phase in addition to the CO2 concentration (ECLIPSE Manual, 2014).

2-2.

Initial and Boundary Conditions

As mentioned above, the flow is one-dimensional. Hence, for the four differential equations, which are second order with respect to space and first order to time, two boundary conditions and one initial condition for each are required. Consistent with core experimental conditions, the inlet of the core is set to have a constant flow rate (Neumann boundary condition) of water with constant CO2 content. The outlet of the core is set to have a constant pressure (Dirichlet boundary condition). The initial pressure and saturation of phases in the core are known. The initial mass fraction of CO2 in the core is co2

zero (ωo 2-3.

= 0 ).

Solution Technique

The four coupled governing equations together with the associated boundary and initial conditions were solved simultaneously for pressure, saturation and mass fractions as the main dependent variables, using the finite difference technique. The fully implicit solution technique was used to solve the governing equations for the variables of po , sw , co

ωoo and ωw 2 . 24


Substituting Equations (2-2e to 2-2h) in Equations (2-2a to 2-2d) and discretising the equations will give: φ∆t [ρo (1-sw )ωoo ]=∆x [ρo ωoo λo ∆x po ]

(2-3a)

φ∆t [ρo (1-sw )(1-ωoo )]=∆x [ρo (1-ωoo )λo ∆x po ]+φD co2 -o ∆x [(1-sw )∆x ρo (1-ωoo )] +U co

co

(2-3c)

φ∆t [ρw sw (1-ωw 2 )]=∆x [ρw (1-ωw 2 )λw ∆x (po -pc )] co

co

co

φ∆t [ρw sw ωw 2 ]=∆x [ρw ωw 2 λw ∆x (po -pc )]+φD co2 -w ∆x [sw ∆x (ρw ωw 2 )]-U where ∆x [ ]=

Δ[ ] Δx

and ∆t [ ]=

Δ[ ] Δt

(2-3b)

(2-3d)

. The equations can be written in the following residual

form: Roo n+1 =∆x [ρo ωoo λo ∆x po ]n+1 - φ ∆t [ρo (1-sw )ωoo ]

co2 n+1

Ro

(2-4a)

=∆x [ρo (1-ωoo )λo ∆x po ]n+1 +φ D co2 -o ∆x [(1-sw )∆x ρo (1-ωoo )]n+1

+U n+1 - φ ∆t [ρo (1-sw )(1-ωoo )] co

co

n+1 Rw =∆x [ρw (1-ωw 2 )λw ∆x (po -pc )] n+1 - φ ∆t [ρw sw (1-ωw 2 )] w

co n+1

Rw 2

co

=∆x [ρw ωw 2 λw ∆x (po -pc )]

n+1

co

(2-4c) n+1

+φ D co2-w ∆x [sw ∆x (ρw ωw 2 )]

co

-U n+1 -φ ∆t [ρw sw ωw 2 ]

(2-4b)

(2-4d)

where RIα is the residual value of each equation written for component I in phase α. ‘n+1’ shows that the calculation is performed at the new time step (i.e. implicit solution). The residual values of each equation should be zero that is achieved by employing the Newton-Raphson iterative method (Equation 2-5).

25


n+1 n+1 Xitr+1 =Xitr -

n+1 RIαn+1 (Xitr ) n+1 ∂RIαn+1 (Xitr ) n+1 ∂Xitr

n+1 n+1 n+1 n+1 or JR ( Xitr )( Xitr+1 -Xitr ) = - RIαn+1 (Xitr )

(2-5)

where ‘itr’ is the iteration number and ‘X’ is the vector of dependent variables (X = [p, co

n+1 sw, ωoo , ωw 2 ]). JR ( Xitr ) is the Jacobin matrix (or derivative matrix) of residual values

which its components are

n+1 ∂RIαn+1 (X itr ) n+1 ∂X itr

(Fanchi 2005, Aziz and Settari 1979). The details

of the solution technique are given in appendix A. The procedure followed to solve the equations is as follows:

1) Calculate the fluid properties (mixture density and viscosity of each phase) and saturation functions (relative permeability and capillary pressure) using Xnitr which ‘itr’ is reset to zero at the starting point of calculation for each time step. co

For the first time step (i.e. n=1), the initial condition data i.e. (p, sw, ωoo , ωw 2 ) is n=0 used as the initial state and initial guess i.e. Xitr=0 .

2) Calculate Jacobin matrix using Equations A-11 to A-58 given in Appendix A. 3) Solve Equation 2-5 to obtain Xnitr+1 . 4) If the difference between two Xnitr and Xnitr+1 is larger than a defined tolerance, replace Xnitr with Xnitr+1 and return to step 1. If not, the obtained Xnitr+1 is the solution at this time step (i.e. time step n). 5) Use the Xnitr+1 as the initial state and initial guess(i.e. the Xnitr )for the next time step (i.e. time step n+1) and repeat steps 1 to 4 to obtain Xn+1 itr+1 . Continue this algorithm to reach the planned simulation time.

In the above equations, the values of density and viscosity of phases are not constant during simulation. The density of mixture change according to either its CO2 content or pressure. However, the viscosity of mixture change only according to its CO2 content. It should be noted that during coreflood experiments, the temperature is constant and the pressure changes through the core is minimal. Therefore the viscosity and density of oilCO2 and water-CO2 mixtures change mainly due to CO2 concentration. To have a representative function, which is able to predict accurately the density and viscosity of systems of oil-CO2 mixture and or water-CO2 mixture, it is necessary to have some experimental PVT data. However, here there were no PVT data available for the system 26


under study. Consequently some correlations and approaches have been proposed to be used in the simulator. It is worth mentioning that the oil sample for this study is normal decane (n-C10H22) with well-defined fluid properties.

2-4.

Density Models

Densities of oil and water during CW flooding are allowed to change due to variation of their CO2 content. Having a reasonable relationship for calculation of oil density saturated with CO2 and water density saturated with CO2 is important. Density of the oil-CO2 mixture can be obtained using a tuned equation of state (EOS) or correlations. There are several different correlations in the literature for the calculation of the oil-CO2 mixture density, which are usually a function of temperature, pressure and composition (Marra et al. 1988, Emera and Sarma 2008). Generally, CO2 will increase or decrease the oil density depending on the temperature, pressure and type of the oil (Riazi 2011c). Some studies have shown that the density of pure oil (i.e. oil without any CO2 content) may be increased when CO2 is dissolved in it (DeRuiter et al. 1994) whilst there are other reports showing that it may be decreased (Miller and Jones 1981). In this study decane has been used as the oil sample and the density of decane and water slightly increases when CO2 is dissolved in them (Ashcroft and Ben Isa 1997, Farajzadeh et al. 2009). In this thesis, following models have been used for calculation of the mixture density.

2-4.1.

Oil-CO2 Density

Oil-CO2 Density Model 1 (Ideal Solution-Model): The solution of CO2 in oil is assumed to happen ideally. For such an ideal solution, the total solution volume is equal to the summation of the volume of pure components. That is, the mixture density is of the form below:

co

1 ωoo ωo 2 = + ρo ρoil ρco2

(2-6)

ωoo = Mass fraction of the oil component in the oil-CO2 mixture co

ωo 2 = Mass fraction of the CO2 component in the oil-CO2 mixture ρoil = Density of pure oil at test conditions (g/cm3) 27


ρco2 = Density of pure CO2 at test conditions (g/ cm3) ρo = Density of the oil-CO2 mixture at test conditions (g/cm3) This model predicts a reduction of dead oil density by dissolving CO2 if the CO2 density at test conditions is lower than that of the pure oil. However if the density of CO2 at test conditions is higher than oil, this equation gives higher oil-CO2 mixture density. Densities of pure CO2 and decane at test conditions are 0.775 and 0.727 g/cm3, respectively (NIST 2014). Therefore, using this model, higher oil-CO2 mixture density is obtained with an increase in CO2 content.

Oil-CO2 Density Model 2 (Measured Data-Model): This model calculates the mixture density based on some measured data for a decane-CO2 system using literature data (Song et al. 2012). Figure 2-1 shows the density of mixture for different mole fraction of CO2 content and at conditions of 138 atm and 40 0C, which is very close to the experimental conditions in this study (136.1 atm and 38 0C). Similar measurement has been reported by Cullick and Mathis (1984). Data points of Figure 2-1 are shown in Table 2-1. Equations 2-7a and 2-7b are used to convert mole fraction to mass fraction or vice versa in Table 2-1.

co ωo 2 =

co xo 2 =

co

co

xo 2 ×44

(2-7a)

co

(1-xo 2 )×142+xo 2 ×44 co

co

ωo 2 ×142

(2-7b)

co

(1-ωo 2 )×44+ωo 2 ×142 co

co

where ωo 2 and xo 2 are the mass and mole fraction of CO2 in the decane-CO2 mixture, respectively.

28


Figure 2-1: Density of decane-CO2 mixture as a function of CO2 mole fraction measured at P=138 atm, T=40 0C (Song et al. 2012).

Table 2-1: Density of decane-CO2 mixture measured at P=138 atm and T=40 0C (Song et al. 2012). CO2 mole fraction CO2 mass fraction decane-CO2 density (g/cm3) 0 0.00 0.729 0.2 0.07 0.737 0.5 0.24 0.75 0.72 0.44 0.77 0.8 0.55 0.79 0.87 0.67 0.795

It should be noted that the maximum solubility of CO2 in decane in this data is estimated around 67% (weight percent). However using the correlation expressed by Emera and Sarma (2008), the maximum solubility is around 58% (weight percent). Moreover, 65% (weight percent) of maximum solubility has been reported by Cullick and Mathis (1984). In this study maximum solubility of 65% at test conditions is considered. The data of Table 2-1 can be used as input to the simulator together with a linear interpolation technique to calculate the decane-CO2 mixture density.

Oil-CO2 Density Model 3 (PR EOS-Model): A tuned equation of state (EOS) can also be used for calculation of decane-CO2 mixture. However experimental data is needed to tune the EOS. Nourozieh et al. (2010) used the experimental data available in the literature for the decane-CO2 system to tune Peng-Robinson (PR) EOS as follows: 29


p=

RT a(T) − 2 v − b v +2vb − b 2

v 3 + (b −

RT

) v2+ ( p

a(T)= [Ωa

R2 Tc2 Pc

(2-8a)

a(T)−2RTb p

] × α(T)

− 3b2 ) v+ (b3 +

RTb2 −a(T)b p

) =0 T 0.5

α(T)= (1+k [1 − (T )

,

c

(2-8b) 2

])

k=0.379642+ 1.48503× ϖ- 0.164423× ϖ2 +0.016666×ϖ3 b= [Ωb

RTc ] Pc

where p is pressure, Pc is critical pressure, T is absolute temperature, Tc is critical temperature, v is molar volume of mixture and ϖ is acentric factor. R is the universal gas constant. Ωa and Ωb are constant and their values are given in Table 2-2. To obtain the EOS parameters for the mixtures, the van der Waals mixing rule is used:

am = ∑i ∑j xi xj √ai aj (1-δij )

i=co2 , decane j=co2 , decane

b m = ∑i x i b i

(2-9)

(2-10)

where δij is interaction coefficient between component i and j in the mixture (i.e. decane and CO2) which can be a tuning parameter. Furthermore, xi is the mole fraction of component ‘i’ in the mixture.

The PR EOS predicts the molar volume of mixture at given pressure and temperature. A volume translation (volume shift) presented by Peneloux et al. (1982) is applied to modify the molar volume of the system (v) predicted by the PR equation of state as follows (Danesh 1998):

v cor = v-c

where c= ∑ xi bi SE i

(2-11)

i

30


where v cor is the corrected molar volume and SE i is the dimensionless individual translation value for component i. SE can be obtained by correlation or can be a tuning parameter. Finally the density of decane-CO2 mixture is calculated as follows:

ρdecane-co2 =

∑i xi MWi

(2-12)

v cor

where, MW is molecular weight.

The input values for PR equation of state and tuned parameters are given in Table 2-2 for decane-CO2 system.

Table 2-2 : Thermodynamic properties of components (Yaws 2014, Nourozieh et al. 2010). decane CO2 142 44 MW(g/gmol) 618.45 304.19 𝐓𝐜 (K) 20.9 72.8 𝐩𝐜 (atm) 0.484 0.228 𝛚 0.45724 0.45724 𝛀𝐚 0.07780 0.07780 𝛀𝐛 0.0548 0.1971 𝐒𝐄 𝛅𝐢𝐣 0.1206 3 82.058 R (cm atm/ K gmol)

2-4.2.

Water-CO2 Density

Water-CO2 Density Model 1 (Measured Data-Model): This model predicts the waterCO2 mixture density as a function of CO2 mass fraction using a simple linear equation (Equation 2-13). The equation has been obtained when a commercial simulator with tuned Soave-Redlich–Kwong (SRK) equation of state was used to generate some data at test conditions. It should be noted that the tuned SRK-EOS in the commercial simulator is based on some measured data. Table 2-3 shows the water-CO2 mixture density data predicted by SRK EOS at test conditions.

31


Table 2-3: Density of water-CO2 mixture predicted by EOS as a function of CO2 calculated at P=136.1 atm, T=38 0C. CO2 mass fraction water-CO2 density (g/cm3) 0.0000 0.995 0.0100 0.997 0.0200 0.999 0.0300 1.002 0.0400 1.004 0.0500 1.006

Later a linear equation was fitted to these data (Equation 2-13). These data shows that the dissolved CO2 content of water increases its density minimally.

co

(2-13)

ρw =0.2264 ×ωw 2 +0.9948 ρw = Density of the water-CO2 mixture at test conditions (g/cm3) co

ωw 2 = Mass fraction of the CO2 component in the water-CO2 mixture Figure 2-2 shows the density data predicted by EOS as well as the fitted linear equation. It should be noted that the maximum solubility of CO2 in water predicted by EOS at test conditions is 5% (weight percent) and the measured data confirms this as well (Wiebe 1941).

32


1.010 1.008

Density(g/cm3)

1.006 1.004 1.002 1.000 0.998

EOS (SRK) Linear Equation

0.996 0.994 0.00

0.01

0.02 0.03 CO2 mass fraction

0.04

0.05

Figure 2-2: Density of water-CO2 mixture predicted by EOS as well as fitted linear equation as a function of CO2 mass fraction (P=136.1 atm, T=38 0C).

2-5.

Viscosity Models

There are different correlations in the literature for calculation of the oil-CO2 mixture viscosity, which are usually a function of temperature, pressure and composition (Emera and Sarma 2008). According to the literature, the viscosity of oil decreases with increasing amount of dissolved CO2 (Emera and Sarma 2008). Chang el. al (1998) reported that the effect of dissolved CO2 on water viscosity is minimal and hence the CO2-water viscosity mixture can be assumed constant. However, CO2 will increase slightly the viscosity of water (Islam and Carlson 2012). Following models can be used by the developed simulator for calculation of the mixture viscosity.

2-5.1.

Oil-CO2 Viscosity

Oil-CO2 Viscosity Model 1 (Linear-Model): In this model, viscosity of the oil-CO2 mixture is a linear function of the mass fraction of the two components in the mixture as follows: 33


co2

μo =μoil ×ωoo +μco2 ×ωo

(2-14)

μo = Viscosity of the oil-CO2 mixture at test conditions (cP) μoil = Viscosity of pure oil at test conditions (cP) μco2 = Viscosity of pure CO2 at test conditions (cP) co

ωo 2 = Mass fraction of the CO2 component in the oil-CO2 mixture ωoo = Mass fraction of the oil component in the oil-CO2 mixture Oil-CO2 Viscosity Model 2 (Beggs and Robinson–Model): This model is based on the Beggs and Robinson (1975) correlation that is a correlation for dissolved hydrocarbon gas and is used here as a hypothesis test for the oil-CO2 mixture. This model might be used in the absence of measured data. This correlation relates the oil viscosity to the corresponding dead oil value and the solution gas oil ratio as expressed below:

μo =A×μBoD

(2-15) Rs

A = 10.715 (5.614 + 150) B = 5.44 (

Rs

5.614

Rs =

+ 150)

−0.515

−0.338

co

ωo 2 ×ρoil ωo o ×ρco2

(Solution gas oil ratio based on the ideal solution theory (cm3/cm3))

where: μo = Viscosity of the oil-CO2 mixture at test conditions (cP) μoD = Viscosity of dead oil at test conditions (cP) ρoil = Density of pure oil at standard condition (g/cm3) ρco2 = Density of pure CO2 at standard condition (g/ cm3) co

ωo 2 = Mass fraction of the CO2 component in the oil-CO2 mixture ωoo = Mass fraction of the oil component in the oil-CO2 mixture

34


Oil-CO2 Viscosity Model 3 (Measured Data-Model): This model calculates the mixture viscosity based on some measured data for a decane-CO2 system presented in the literature (Cullick and Mathis 1984). Table 2-4 shows the decane-CO2 viscosity as a function of CO2 mass fraction.

Table 2-4: Viscosity of decane-CO2 mixture measured at test conditions (Cullick and Mathis 1984). CO2 mass fraction decane-CO2 viscosity (cP) 0 0.830 0.05 0.705 0.12 0.589 0.24 0.394 0.36 0.148 0.63 0.142

This data has been used and a cubic equation has been fitted as follows: co

co

co

μo =-1.75×(ωo 2 )3 +3.89×(ωo 2 )2 -2.9×ωo 2 +0.83 2-5.2.

(2-16)

Water-CO2 Viscosity

Water-CO2 Viscosity Model 1 (Islam and Carlson-Model): This model calculates the viscosity of water-CO2 mixture based on the Islam and Carlson (2012) correlation (Equation 2-17). The correlation was obtained using measured data and it shows that the viscosity of water increases slightly with its CO2 content. The correlation relates the water-CO2 mixture viscosity to the pure water viscosity, CO2 mass fraction in water and the system temperature. It should be noted that the temperature is constant in this study. (2-17)

μw = μr × μwater co

co

763.261×ωw 2 -9460.777× (ωw 2 ) μr =1+ -10471.874+36.833×T

2

μw = Viscosity of the water-CO2 mixture at test conditions (cP) μwater = Viscosity of pure water at test conditions (cP) 35


T = Temperature of the system (K) co

ωw 2 = Mass fraction of the CO2 component in the water-CO2 mixture

2-6.

Mass Transfer Term

The CO2 component is displaced through the system within the phases by fluid velocity (convection and or dispersion). CO2 can also be transferred across the phases if there is a CO2 concentration gradient between water and oil phases. During CWI process, when the carbonated water and oil are in contact, CO2 is transferred across the phases and this will continue until the CO2 concentration through the phases is at equilibrium concentration. Moreover, the equilibrium state can be reached immediately or it may take time. This depends on the existing resistances against CO2 transfer across the phases. During CWI, CO2 migrates out of the water phase and cross the interface before entering the oil phase. This may prevent reaching immediate equilibrium state in terms of CO2 distribution between the phases during CWI process. Therefore, to capture the actual physics of CWI process during simulation, it is necessary to develop the equations based on nonequilibrium conditions. The kinetics of CO2 distribution between the oil and water phases is included by adding a mass transfer term to the governing equations. That is, a source term, U, is added which is defined as follows: co

co

U=K×(ρw ×ωw 2 ×k eq -ρo ×ωo 2 ) if 𝑠𝑜 × 𝑠𝑤 ≠0, U=0

if

(2-18)

𝑠𝑜 × 𝑠𝑤 =0

where, K= (km×a) with ‘km’ is the overall mass transfer coefficient (cm/sec) and ‘a’ is the specific interfacial area (1/cm), which is the oil-water interfacial area per unit volume (Geller and Hunt 1993). so and sw are oil and water saturation, respectively. ρw and ρo co

co

are densities of water and oil phases, respectively. ωw 2 and ωo 2 are mass fractions of CO2 in the water and oil phases, respectively. keq is the distribution or partition coefficient that captures how CO2 is distributed between the phases at equilibrium state and it is a function of pressure and temperature of the system. This coefficient is a measure of the difference in solubility of the CO2 component in oil and water phases (Leo et al. 1971). keq is the ratio of concentration of CO2 component in oil to the water phase at equilibrium state defined as follows: 36

Chapter 2: Model Structure co

co

Co 2 ρo ωo 2 k eq = co2 = co Cw ρw ωw 2

(2-19)

co2 where Coco2 and Cw are CO2 concentration (g/cm3) in oil and water phases, respectively.

keq can be obtained experimentally using the solubility data or using EOS. At test conditions, CO2 mass fraction in the water phase is 5 % (Wiebe 1941) and water-CO2 mixture density is 1.01 (g/cm3) (Equation 2-13). Moreover CO2 mass fraction in decane is around 65% (Cullick and Mathis 1984) and decane-CO2 mixture density is around 0.75 (g/cm3) (Equation 2-6). Using these values, k eq value is approximately 9.6.

K is an unknown parameter here and is referred to as the mass transfer coefficient (MTC) and can be obtained by either history matching the coreflood production recovery results or be estimated using some mathematical theory with the help of experimental results as reported in the literature (Embid and Rivas 1994, Steffens, 2010; Valiollahi et al., 2012). Equation 2-18 implies that the driving force for the CO2 transfer between the water and oil phases is the concentration gradient. Moreover, the K parameter in the Equation 2-18 controls how fast the CO2 is transferred between the phases. The CO2 transfer happens faster at the beginning due to the higher concentration gradient and gradually decreases and finally stops, when the CO2 concentration through the phases is equal to the equilibrium concentration, which is controlled by the keq value. It should be noted that the developed simulator can work in black-oil mode when the K parameter is assigned to be zero. That is, no CO2 transfer between the phases happens.

2-7.

Molecular Diffusion

Molecular diffusion describes the migration of a substance from a high concentration region to a low concentration region until equilibrium is reached. The driving force for this process is an existing concentration gradient. The diffusional flux is expressed by Fick’s first law:

J=-D

∂C ∂x

(2-20)

where in SI unit: 37

Chapter 2: Model Structure gmol

J = Diffusional flux (sec×cm2) gmol

C = Concentration ( cm3 )

cm2

D = Diffusion coefficient ( sec ) In this study, the diffusion coefficient describes the molecular diffusivity of CO2 (solute) in the oil and water phases (solvent). Fick’s first law is based on the steady state conditions. However, usually, the diffusion rate decreases when the substance (solute) diffuses further into the solvent. Therefore Fick’s second law expresses the diffusion process under unsteady state conditions as shown in Equation 2-21 (Bird et al. 2002, Cussler 1997):. ∂C ∂2 C =D 2 ∂t ∂x

(2-21)

The importance of CO2 transport in porous media by molecular diffusion depends on contact time, length of diffusion and diffusion rate. The diffusion rate is proportional to the diffusion coefficient value. The diffusion length refers to the distance that the substance can propagate in a rock by diffusion during time t. The diffusion length is affected by rock heterogeneity and wettability, pore geometry and fluid properties (Grogan et al. 1988). Moreover, in porous media, the diffusion rate is slower compared to non-porous media as the diffusive molecules have to travel a longer path in a tortuous network. Therefore, the effective diffusion coefficient in porous media is smaller than the absolute diffusion coefficient in non-porous media (Perkins and Johnston 1963). For an unconsolidated sand pack, the absolute diffusion coefficient is approximately 1.4-1.7 times bigger than effective diffusion coefficient (Perkins and Johnston 1963). Renner (1988) measured the diffusion coefficient of CO2 in decane as a function of pressure and up to 850 Psia (57.7 atm) in a Berea core as shown in Table 2-5. Therefore these measurements are considered as effective diffusion coefficients in porous media.

38


Table 2-5: Diffusion coefficient of CO2 in decane as a function of pressure at T=38 0C (Renner 1988). 𝐜𝐦𝟐

Pressure(atm) Diffusion Coefficient ( ) 𝐬𝐞𝐜 15.2 28.8 44.76 57.6

1.97E-05 2.42E-05 3.77E-05 5.05E-05

Later these data have been used and a quadratic equation is fitted (Equation 2-22). D𝑐𝑜2 -decane =1×10-8 P 2 -1×10-7 P+2×10-5

(2-22) cm2

where D𝑐𝑜2 −decane is the CO2 effective diffusion coefficient in decane oil ( sec ) and P is the pressure of the system (atm).

Equation 2-22 is used to estimate the CO2 effective diffusion coefficient in decane at test pressure of 136.1 atm resulting in 1.9E-4 cm2/sec. Figure 2-3 shows the CO2 effective diffusion coefficient in decane of Table 2-5 as well as the fitted quadratic equation.

Diffusion Coefficient (cm2/sec)

6.00E-05

5.00E-05

4.00E-05

3.00E-05

2.00E-05 Diffusion Coefficient (Renner 1988)

1.00E-05

Quadratic Equation 0.00E+00 0

10

20

30 40 Pressure (atm)

50

60

Figure 2-3: Plot of effective diffusion coefficient of CO2 in decane from Table 2-5 as well as the fitted quadratic equation at T=38 0C. 39


Generally, the CO2 diffusion coefficient in oil should be measured experimentally. In the absence of such data, an empirical correlation developed by McManamey and Woollen (1973) can be used, which relates the CO2 diffusion coefficient to the viscosity of oil as follows: Dco2 -oil = 36.24×10-6 μ-0.47 oil

(2-23)

where μoil is the oil viscosity (cP) and D𝑐𝑜2 −oil is the diffusion coefficient of CO2 in oil cm2

( sec ). Thomas and Adams (1965) suggested the following correlation for calculation of the CO2 diffusion coefficient in water based on some measured data which relates diffusion coefficient to the temperature of the system, and viscosity of water as follows:

D𝑐𝑜2 -water = 5.72×10-8

T

(2-24)

μwater

where T is the temperature of the system (K), μwater is the water viscosity (cP) and D𝑐𝑜2 −water is the CO2 diffusion coefficient in water (

2-8.

cm2 sec

).

Dispersion

Dispersion refers to the physical mixing of fluid in porous media due to non-uniform fluid flow specifically in the heterogeneous rocks. Dispersion can be different in the longitudinal direction (same direction of bulk fluid movement) or in transverse direction to the direction of the bulk fluid movement. However longitudinal dispersion is higher than transverse dispersion. In this study only longitudinal dispersion is considered because during the coreflood tests, the radius of the core versus the length of the core is usually very small. Longitudinal dispersion in a capillary tube can be calculated by following equation (Perkins and Johnston 1963): U 2 a2 k L =D+ 48 D

(2-25)

40

Chapter 2: Model Structure cm2

cm2

where k L is longitudinal dispersion coefficient ( sec ), D is diffusion coefficient ( sec ), U is cm

average velocity (sec) and ‘a’ is the radius of capillary tube (cm). Moreover, in a capillary cm2

network, the longitudinal dispersion coefficient (k L ( sec )) is a function of average fluid cm2

cm

velocity through the network (U (sec)), diffusion coefficient (D ( sec )) and length of the capillary network (L (cm)) as shown in Figure 2-4 (Perkins and Johnston 1963). This plot k

UL

shows the relationship between ( DL ) and ( D ) and it can be seen that at higher values of UL

( D ), the longitudinal dispersion is more pronounced and it is increased by velocity. Moreover, at high velocities, the dispersion coefficient can be 100 times bigger than the diffusion coefficient.

Figure 2-4: Longitudinal dispersion coefficient in a capillary network (Perkins and Johnston 1963).

In this study UL/D value is around 96 for the mixed-wet core and is 48 for the water-wet core used during CWI coreflood experiments. U and L are given in next chapters and D has obtained from Equation 2-22. Therefore, the dispersion coefficient is calculated to be around 500 times bigger than the diffusion coefficient for the mixed-wet core and to be 90 times bigger than the diffusion coefficient for the water-wet core. Nevertheless, the dispersion coefficient can also be considered as a matching parameter to be obtained by history matching of the production data.

41


2-9.

Relative Permeability Relationship

For simulation of carbonated water injection, relative permeability curves (Kr curves) of oil and water phases are needed to be known. However, these are not available, i.e. they are unknown functions. To obtain the Kr curves, history matching approach can be used to find an optimised Kr curve, which can match the coreflood experimental data. Here to do that, Corey correlations (power law function) are used to define the Kr curves expressed as follows (Ahmed 2001):

Krw =k wmax s *

nw

,

Kro =k omax (1-s * )no ,

(s -swc ) wc -sor )

s* = (1-sw

(2-26)

where k wmax and k omax are the maximum relative permeabilities of water and oil phases respectively. Moreover, nw , and no are the exponents of water and oil relative permeability curves, respectively. The swc and sor are the saturations of connate water and residual oil, respectively. This model is attractive and widely used in the oil industry because of its simplicity and covering a reasonably wide range of Kr curves by changing only two parameters, i.e. the power law exponent (n) and maximum relative permeability endpoint (kmax). The unknown parameters of these correlations can be obtained by history matching of production data.

2-10.

Capillary Pressure

Capillary pressure curve (Pc curve) is not measured during the experiments. Therefore, it should be determined by matching the production data as part of history matching exercise. In this work the Brooks-Corey correlation is used to define the Pc curve expressed as follows:

sw -swc

Pc=pce (1-s

wc -sor

)

-

1 λ

(2-27)

where pce is the entry capillary pressure (atm), λ is the pore-size distribution index (Brooks and Corey 1964). The unknown parameters of this correlation (i.e. pce and λ) can be obtained by history matching of the production data.

42


2-11.

History Matching Using Genetic Algorithm Based Optimisation

During the CWI process, CO2 is transferred between oil and water phases and as a result, the composition of fluids changes dynamically. For simulation of CWI process, the input values including Kr and Pc functions as well as the mass transfer coefficient (MTC) are needed, however, they are unknown. A history matching technique can be used to determine these unknown values by matching simulator's prediction with the real core production data. To do that, a Genetic Algorithm (GA)-based optimiser was developed to automatically find the unknown parameters. That is, the optimiser can be used to determine the unknown Corey parameters of relative permeability curves, the unknown Brook-Corey parameters of capillary pressure curve and the unknown MTC. The GAbased optimiser finds the best values for these unknown parameters so that the mismatch between real and model predicted data can be minimised. The history matched data are differential pressure (DP) across the core and total (cumulative) oil production (TOP). It should be noted that the developed GA-based optimiser can be linked to the developed simulator as well as the ECLIPSE commercial reservoir simulators. The structure of GAbased optimiser is described in the following section. 2-11.1.

The Structure of Genetic Algorithm-Based Optimiser

The genetic algorithm (GA) is an optimisation technique based on the stochastic global search method that mimics the metaphor of natural biological evolution. GA operates on a population of potential solutions applying the principle of survival of the fittest to produce (hopefully) better and better approximations to a solution. GA allows a population composed of many individuals to evolve under specified selection rules to a state that maximises the “fitness” (i.e., minimises the cost function). At each generation, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals that they were created from, just as in natural adaptation. An overview of the structure of the designed optimiser and a glossary of the GA terms used in this study is presented below. More information on this optimisation tool can be found elsewhere (Haupt and Haupt 2004, MATLAB software manual, Shahverdi et al. 2011).

43


The GA-based optimisation program is used to optimise the Objective function/Cost function. In this study, the objective function (OF) that needs to be minimised, is defined as follows: DPreal -DPpredicted

OF=ωDP × |

DPreal

TOPreal -TOPpredicted

| +ωTOP × |

TOPreal

|

(2-28)

where DP is the differential pressure across the core and TOP is total oil production. Moreover, subscripts of ‘real’ and ‘predicted’ are presenting real and predicted values, respectively. ǀ ǀ is the absolute value function. ωDP and ωTOP show the weight of DP and TOP data during the matching process. That is, a higher weight value means finer match for that class of data (i.e. DP or TOP). Here in this work, equal values of one has been selected (i.e. ωDP = ωTOP =1) meaning both TOP and DP data have the same importance during matching process and the tuned simulator should predict both TOP and DP values accurately.

The goal of the optimisation task is to find the unknown parameters using GA-based optimiser. In GA, each unknown parameter is called a Gene. A Gene is a member of Chromosome (Individual). That is, chromosome is composed of genes and it is an array of parameters or genes that is passed to the cost function. It means that the size of each individual depends on the number of unknown parameters (genes) in each problem. In this study, each chromosome can have maximum of 8 genes (unknown parameters) including Corey coefficients of relative permeability curves (k wmax , nw , k omax , no and sor ), Brook-Corey coefficients of capillary pressure curve (pce and λ) and MTC (Figure 2-5). It is worth noting that the swc is known from the experiments and because of that is not included in the optimisation task.

Figure 2-5: A chromosome.

44


It should be noted that in this study, not all of these 8 parameters are always obtained by the optimisation program and the size of each chromosome (number of unknown parameters) is mentioned separately for each case in next chapters. During optimisation at any time step of the evolution, there is a group of individuals called Population. The Population Size in GA refers to the number of individuals in a population. The procedure used to design the optimiser is summarised below: 1. Generate a population composed of Npop number of chromosomes known as the initial population. The value of Npop is chosen by the user. Npop was selected to be 80 in this study.

2. To find the fitness of each chromosome, the objective function is calculated for the population, which gives corresponding Npop costs. Here, depending on the case under study, the optimiser is linked to the chosen simulator (i.e. ECLIPSE simulator or the developed simulator) and is run using each chromosome as an input to predict the DP and TOP output data. Later, the predicted data is used and the misfit value between experimental and predicted data of each chromosome is calculated which results in having Npop cost values.

3. Next, it must be decide which chromosome in the initial population set are fit enough to survive and possibly reproduce offspring in the next generation. Npop costs and associated chromosomes are ranked from the lowest to highest cost. Not all of the survivors are considered to be fit enough to mate. Considering all the sorted Npop chromosomes in a given generation, only the top Nkeep numbers which have lowest cost are kept for mating and the rest are discarded to make room for the new offspring. The value of Nkeep is chosen by the user. Here, Nkeep is assigned to 50%. A large Nkeep value (greater than 50%) can increase the time to reach the optimum value by GA as some nonqualified individuals will remain in the process. However a small Nkeep value (smaller than 50%) may causes some qualified individual to be discarded and in turn GA gives local and not global optimum.

4. Two parents (two chromosomes) for mating are selected using the weighted random pairing procedure. Each chromosome has a probability to be selected for mating. The 45


probability is inversely proportional to its cost. That is, a chromosome with the lower cost has higher probability of mating, while the chromosome with the higher cost has the lower probability of mating.

5. Crossover, which is one of the reproduction operations (mating), is performed at this stage. There are different combining methods, which two parents produce new offsprings. Two selected chromosomes exchange their genes to generate a new chromosome. The methodology used here is as follows:

A number as a crossover point is selected randomly (α). This number must be between 1 and the number of genes (variables) in chromosome (Nvar ) i.e. 1 ≤ α ≤ Nvar . For example two parents for mating after selection of a crossover point are: parent 1 =[pm1 ,pm2 ,…pmα …,pmNvar ] parent 2 =[pd1 ,pd2 ,…pdα …,pdNvar ] where the subscripts of ‘m’ and ‘d’ are presenting the mom and the dad. Then αth variables of each parent (i.e. pmα and pdα ) are combined to form new variables that appear in the children shown below: pnew1 = pmα − β(pmα − pdα )

(2-29a)

pnew2 = pdα + β(pmα − pdα )

(2-29b)

where β is also a random value between 0 and 1. Consequently new offsprings are born as follows: offspring 1 =[pm1 ,pm2 ,…pnew1 …,pmNvar ] offspring 2 =[pd1 ,pd2 ,…pnew2 …,pdNvar ] 6. After a crossover is performed, mutation takes place. This is to prevent all solutions of the population falling into a local optimum of the solved problem. Mutation randomly changes the new offspring. For example, a number as a mutation point is selected 46


randomly (α) and the parameter (gene or variable) in that point, is replaced with a new random value. For example: muted-offspring 1 =[pm1 ,pm2 muted ,…pnew1 …,pmNvar ] A mutation rate of 20% was selected. That is 20% of present individuals were selected randomly to be muted.

7. Now, a new generation is created. Again the cost function must be calculated and steps 2 to 6 be repeated until a convergence criterion is met. The convergence or stopping criteria can be achieved by setting a specific number of iterations (generations) to be repeated or by setting a specific tolerance to be crossed. In this study, stopping criterion was selected to repeat 100 iterations when it was set to 100 or to cross a misfit value of 0.6. That is, it was observed that after meeting these criteria, the optimiser would not considerably improve the results any more. The flowchart of the procedure of GA is shown in Figure 2-6.

Figure 2-6: Flowchart of the GA.

47

Simulation of CWI in a Water-Wet Core This chapter presents the numerical simulations of a water injection (WI) and a carbonated water injection (CWI) coreflood experiment from the literature (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c) performed in a water-wet core. The goal was to investigate the capability and performance of the developed simulator when a real CWI coreflood experiment was simulated. The simulation of the CWI was performed including the kinetics of CO2 transfer between the water and oil phases.

At the beginning, the experimental data of the CWI is investigated and compared with those of the WI. Next, the water injection (WI) experiment is simulated. Then, the data obtained from the simulation of the WI is used to simulate the CWI experiment. The procedure used to simulate CWI, is described in detail. Later, the effects of parameters which may influence the performance of the CWI process including dispersion coefficient, injection rate and carbonation level are studied. In addition, the CO2 storage capacity of the CWI process is investigated using the developed simulator. Finally the ECLIPSE300 compositional simulator is used to simulate CWI process and its performance is compared with the performance of the developed simulator.

3-1.

The WI and CWI Experiments

The WI and CWI coreflood experiments were selected from the literature (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c). A high-pressure high-temperature coreflood rig had been used in the displacement tests. The employed coreflood rig could operate at pressures up to 6000 psi and temperatures as high as 300 oF. Pressure and temperature at the inlet and outlet of the core holder had been continuously recorded. Temperature had been kept constant by putting the apparatus inside a temperature-controlled enclosure.

The porosity of the cores had been determined by a helium porosity test. The pore volume of the core had been calculated by the total volume of fluid injected to saturate the system minus the dead volumes of tubing connecting the core to the rest of the system.

48

Chapter 3: Simulation of CWI in a Water-Wet Core

The permeability of the core had been measured using brine at the test pressure and temperature. Moreover, during measurements no fines were observed in the effluent. Carbonated water had been prepared using CO2 with 99.9% purity before injection test. CO2 had been mixed and agitated with water in a pressure cell until the pressure had been stabilised (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c).

The production data of the secondary WI and the corresponding secondary CWI coreflood experiments performed in a water-wet sandstone core, under the same conditions, are examined in this section. It is worth investigating the results of the WI and CWI coreflood experiments first.

The core and fluid properties used during the experiments are given in Table 3-1 and Table 3-2 respectively. The core had been fully saturated by normal decane (n-C10H22) oil sample. During both WI and CWI tests, water or carbonated water (CW) was injected into the core at a constant rate and water and/or decane were collected at a constant pressure at the core outlet. The operational conditions of both WI and CWI experiments are the same and given in Table 3-3. It should be noted that the injection rate shown in Table 3-3 had been selected such that it was in the range of the flow rate in real oil reservoirs. That is, according to this flow rate, the velocity of the fluids in the core was in the range of the velocity in real oil reservoirs (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c).

Table 3-1: Core properties. Core

Length (cm)

Diameter (cm)

Porosity (fraction)

Pore Volume(cm3)

Permeability (mD)

water-wet Clashach sandstone

33.2

4.986

0.19

123.16

1300

Table 3-2: Fluid properties [Ref: NIST 2014]. Density (g/cm3) (Test conditions) (136.1 atm, 38 0C)

Density (g/cm3) Standard conditions (1 atm, 20 0C)

Fluid

Viscosity(cP) (Test conditions) (136.1 atm, 38 0C)

decane

0.83

0.730

0.727

water CO2

0.66 0.067

0.995 0.775

0.995 0.00184

49


It should be noted that the values given in Table 3-2 are from National Institute of Standard and Technology (NIST) based on measured data or using equation of states (NIST 2014).

Table 3-3: The operational conditions of the coreflood experiments. Injection rate (cm3/hr) Interstitial velocity(equivalent to the injection rate)(m/day) CO2 mass fraction of injected CW Outlet pressure (atm)

20 1.3 5% 136.1

Initial pressure (atm)

136.1

Initial water saturation

0

Figure 3-1 demonstrates a schematic of the coreflood experiment.

water + oil + CO2

carbonated water

Figure 3-1: A schematic of coreflood experiment.

It is worth comparing the production data of WI and CWI coreflood experiments first. Figure 3-2 and Figure 3-3 show TOP data and corresponding recovery factor (RF) of the WI experiment compared to those of CWI plotted versus injected pore volume (PV), respectively.

Figure 3-2 and Figure 3-3 show that during CWI, oil recovery has improved. That is, the final oil recovery factor (RF) obtained at the end of the CWI experiment is 73% (90.3 cm3) whereas it is 69% (85.4 cm3) at the end of the WI experiment. It can be seen that the production of secondary WI and its corresponding CWI is the same until the breakthrough point, i.e. at the breakthrough time, the CWI and WI have produced 78.2 cm3 oil (64% RF). During water injection, the recovery factor usually increases until the breakthrough point and usually after that point, the recovery factor stabilised promptly (i.e. no considerable oil is produced after the breakthrough time).

50


100 90 80

TOP(Scm3)

70 60 50 40 30 20

Exp-WI

10

Exp-CWI

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-2: Comparison of total oil production (TOP) data of the WI and CWI experiments.

80 70 60

RF(%)

50 40 30 20 Exp-WI 10

Exp-CWI

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-3: Comparison of recovery factor (RF) of the WI and CWI experiments.

51


Figure 3-3 shows that after the breakthrough time, recovery factor has increased by around 9% during CWI compared to that of 5% during WI. This additional 4% RF of CWI over WI is attributed to the contribution of CO2 available in the system during CWI process. It should be noted that, oil swelling, viscosity reduction, IFT reduction and wettability alteration are amongst the main mechanisms resulting in the additional oil recovery during the CWI. It will be discussed later that not all these mechanisms are important during this CWI experiment and the additional oil recovery obtained after the breakthrough time is attributed to the oil swelling. This conclusion is made with the support from the results of simulation.

Figure 3-4 compares DP values of WI and CWI experiments plotted versus injected PV. Figure 3-4 shows that the CWI and WI have approximately resulted in the same differential pressure (DP) values across the core.

0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 Exp-WI

0.1

Exp-CWI 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-4: Comparison of differential pressure (DP) data of the WI and CWI experiments.

After investigating the experimental data, CWI and WI coreflood experiments were simulated using the developed simulator (model). First, water injection (WI) coreflood experiment was simulated to obtain water-oil relative permeability (Krw-o) curve. Corey 52


correlation was used to define the relative permeability curves. To simulate the WI test, performing a history matching exercise, ECLIPSE100 (E100) commercial simulator was linked to the GA-based optimiser and an optimised Krw-o curve was obtained. Next, the black-oil mode of the developed simulator (with zero mass transfer) together with this optimised Krw-o curve was used to repeat the simulation. The goal was to check the reliability and validity of the simulator when the prediction of the simulator and that of the E100 were compared. Next, the simulator in its compositional mode (with mass transfer), was used to simulate the CWI experiment. Here, the Krw-o curve obtained from the WI test was used and the contribution of mass transfer was tuned to match the CWI production data. Details of these simulations are explained below.

3-2.

Simulation of the Water Injection Experiment

3-2.1.

Water-Oil Relative Permeability Curve

The WI experiment was simulated first. However, the required relative permeability and capillary pressure curves were unknown. In this step it was assumed that the capillary pressure is zero in the model and the unknown Krw-o curve was obtained by history matching the production data of WI experiment using the GA-based optimiser. Capillary pressure was assumed to be zero at this step because for optimisation purpose, it is easier and faster to start with less unknown parameters which here are only Corey parameters of Kr curve. Later, the optimised Kr can be used as an initial value and Kr and Pc can be obtained by optimisation simultaneously. This is performed and is described in the next section. Corey correlations (power law functions) were used to define the relative permeability curves. In order to obtain the Krw-o curve for the water injection test, the E100 black-oil simulator was linked to the developed optimiser. In this exercise, four parameters in the Corey correlation including kwmax, nw, no and sor were considered unknown. In this experiment, the core was fully saturated with decane with no initial water saturation; therefore swc was set to zero and komax to one. Table 3-4 shows the initial uncertainty range of each parameter used during optimisation task. These initial range were selected to be consistent with typical Corey parameters obtained for real oil reservoirs (Ahmed, 2001) and the same initial range were used for all history matching experiments in this thesis. Moreover, GA was run two times for each experiment to make sure that the reliable answer is obtained. 53


Table 3-4: Initial uncertainty range of parameters used in GA. Corey parameters

nw initial uncertainty range 1-5

no kwmax sor 1-5 0.01-0.8 0.05-0.6

Figure 3-5, Figure 3-6 and Figure 3-7 show the improvement of minimum and mean (average) Total misfit, TOP misfit and DP misfit respectively over the generations for this optimisation task. Figure 3-5 shows that the mean value of total misfits in each generation (iteration) is reducing and it has approximately stabilised after 85 generations. The best (minimum) total misfit obtained is 0.62.

14 Minimum_Totalmf 12

Mean_Totalmf

Total misfit

10 8 6 4 2 0 0

20

40 60 Generation

80

100

Figure 3-5: Minimum and mean of Total misfit as a function of generation.

54


Figure 3-6: Minimum and mean of TOP misfit as a function of generation.

12

Minimum_DPmf Mean_DPmf

10

DP misfit

8

6

4

2

0 0

20

40 60 Generation

80

100

Figure 3-7: Minimum and mean of DP misfit as a function of generation.

55


The optimal values of the Corey parameters are summarised in Table 3-5.

Table 3-5: Optimal Corey parameters of water-oil relative permeability, obtained using E100 linked to the GA-based optimizer as well as assigned komax and swc values, WI experiment. Corey parameters nw no kwmax komax sor swc 2.5 2.25 0.14 1 0.31 0 Krw-o curve

It is worth mentioning that, the results obtained by the history matching process using the optimisation technique may not be unique. Nevertheless, finding the optimised values by using other methods can help to assure that the optimiser has produced the right results. Therefore the residual oil saturation was also obtained using the material balance calculation. That is, at the end of WI experiment, 38 cm3 of oil is left in the core, which is equivalent to 31% residual oil saturation (sor). Moreover, the kwmax was also obtained by a manual calculation using the Darcy equation (Equation 3-1). The value of DP at the end of the WI experiment (endpoint value on the DP curve) as well as the core properties were employed during this calculation. qinj

(

)×μw ×L

A k wmax = K×∆P

(3-1)

endpoint

where qinj is the injection rate, A is the cross section area of the core, μw is the water viscosity, L is the core length, K is the absolute permeability and ∆Pendpoint is the endpoint value on the DP curve.

Furthermore, since the endpoint saturations and their relative permeability values were available or derivable for WI, the only remaining unknowns were Corey exponents, i.e., no and nw. Another practice was carried out to know where the unique solution lies. To do that, 17 values of each no and nw exponents over initial uncertain range i.e., between 1 and 5 as shown in Table 3-4, were picked and the misfits were calculated for various combinations of the two Corey exponents. The model was run with every combination of the exponents (289 runs in total), and TOP and DP misfits as well as the Total misfit (summation of TOP misfit and DP misfit) were calculated and shown on a distribution plot. Figure 3-8, Figure 3-9 and Figure 3-10 show Total misfit, TOP misfit and DP misfit 56


respectively as a function of nw and no values. The range of no and nw values at which the misfits have lower values can be observed on the figures. It should be noted that, due to the scale of the plots, it cannot clearly be observed a point as the optimal values of no and nw, however, an area can be recognised where the optimal values belong to that area. That is, a wide range of nw and no can result in similar and small but not exactly the same misfit values. As shown above, the optimal values of no and nw are 2.25 and 2.5 respectively which are consistent with these plots. Therefore, this practice can validate the accuracy of the values obtained for nw and no during optimisation task as presented in Table 3-5.

Figure 3-8: Total misfit as a function of nw and no.

57


Figure 3-9:TOP misfit as a functin of nw and no.

Figure 3-10:DP misfit as a functin of nw and no.

58


Figure 3-11 displays the relative permeability curves based on the obtained parameters.

1

1

0.9

0.9

0.7

Kro

0.8

Kro

0.7

Krw

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

Krw

0.8

0 0

0.15

0.3

0.45

0.6

Sw Figure 3-11: Optimised Krw-o curve, WI test.

It should be noted it was reported that this core had been water-wet (Kechut et al., 2011b; Kechut, 2011c). The Craig’s rule of thumb (Craig 1971) can be used to qualitatively estimate the wettability of the core based on relative permeability curves. Based on Craig’s rule of thumb for a water-wet system, end-point relative permeability to oil at swc (i.e. komax) should be higher than 95%, end-point relative permeability to water at sor (i.e. kwmax) should be less than 30%, connate water saturation (i.e. s wc) should be higher than 25% and the cross point at which Krw=Kro, should be greater than 50% water saturation. According to Figure 3-11, komax =100% which is greater than 95% as suggested by Craig, kwmax =14% which is less than 30% as suggested by Craig. However, the cross point is at 50% water saturation while according to Craig’s rule of thumb (Craig 1971), this should be higher than 50%. It is likely that due to the absence of connate water (i.e. s wc=0) the Craig’s rule of thumb cannot be correctly applied for this system.

Figure 3-12 and Figure 3-13 show experimental TOP and DP data respectively versus the E100’s prediction using the optimised Krw-o curve. 59


90 80 70

TOP(Scm3)

60 50 40 30 20 WI-Exp

10

WI-E100 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-12: Experimental TOP versus predicted TOP data, predicted by the E100 black-oil simulator using the optimised Krw-o curve, WI test.

0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 0.1

WI-Exp WI-E100

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-13: Experimental DP versus predicted DP data, predicted by the E100 black-oil simulator using the optimised Krw-o curve, WI test.

60


It can be observed that using the obtained Krw-o curve, E100 has predicted the production data of water injection experiment adequately.

Next, the simulation was repeated using the developed simulator in its black-oil mode (zero mass transfer) along with the obtained relative permeability curve (i.e. optimised Krw-o). The aim was to verify the performance of the developed simulator when the results of E100 and the developed simulator were compared. Figure 3-14 and Figure 3-15 show the outputs of ECLIPSE100 compared to those of the developed simulator. The close agreement between the data of these two figures clearly verifies the reliability of the developed simulator. It should be noted that the number of gridblocks used during simulation was optimised so that more refining of the grids did not change the results predicted by the simulator. 200 equal gridblocks were selected here during the simulations.

90 80 70

TOP(Scm3)

60 50 40 30 20 WI-Model WI-E100

10 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-14: Predicted TOP data by the E100 and the developed simulator (model) under its blackoil mode versus injected PV using the optimised Krw-o curve, WI test.

61


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 WI-Model

0.1

WI-E100 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-15: Predicted DP data by E100 and the developed simulator (model) under its black-oil mode using the optimised Krw-o curve, WI test.

3-2.2.

Capillary Pressure

The simulations presented so far have been performed assuming zero capillary pressure in the model (Pc=0). At this stage, the simulation was repeated when the capillary pressure was not zero. To do that, using the developed GA-based optimiser and E100, a history match task was performed and the capillary pressure and relative permeability curves were obtained simultaneously. Brook-Corey correlation described in Chapter 2, was used to define the Pc curve. Table 3-6 shows the optimised Krw-o and Pc curves. Figure 3-16 shows the capillary pressure curve based on the data shown in Table 3-6. Table 3-6: Optimal Corey and Brook-Corey coefficients of water-oil relative permeability and capillary pressure curves obtained by the optimiser linked to E100. no

kwmax komax sor

1 λ

nw

oil -water (Pc=0)

2.5 2.25

0.14

1

0.31

0

-

oil -water (Pc#0)

2.5 2.25

0.14

1

0.31

0

0.04 0.13

62

swc

pce

Relative Permeability

-

Pc (atm)


0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

0.2

0.4 sw

0.6

0.8

Figure 3-16: Capillary pressure data for the WI test.

It can be seen that including the capillary pressure has minimal effect on the results (i.e. original Kr curve). This was expected because the core used during the experiments was homogenous with high permeability. The results of the simulation including capillary pressure are shown in Figure 3-17 and Figure 3-18.

The data of Table 3-6 shows that the relative permeability curves obtained with and without capillary pressure are similar confirming the minimal impact of Pc. Considering this and also to exclude more uncertainty in the simulation process, it was concluded to ignore capillary pressure in all simulations to be presented from here onwards for both WI and CWI experiments under study in this chapter.

63


90 80 70

TOP(cm3)

60 50 40 30 20 WI-E100(Pc=0)

10

WI-E100(Pc#0)

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-17: Predicted total oil production (TOP) data by ECLIPSE100 (E100) with and without capillary pressure (Pc), WI test.

0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 WI-E100(Pc=0)

0.1

WI-E100(Pc#0) 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-18: Predicted differential pressure (DP) data by ECLIPSE100 (E100) with and without capillary pressure (Pc), WI test.

64


3-3.

Simulation of CWI

3-3.1.

Compositional Simulation

When injected CW contacts oil, dynamic transfer of CO2 from water into the oil occurs, resulting in the change of compositions of the phases during CWI process. Therefore, to model this process, the simulator was developed so that the kinetics of CO2 transfer between phases was captured and the viscosity and density of phases could change due to composition changes during the simulation. Moreover, the simulator was developed to include the dispersion mechanism as well. That is, when CO2 is transferred to the oil phase, it moves forward inside the oil phase by convection and/or dispersion mechanisms.

At this step, to simulate the CWI process, it was assumed that the dispersion is zero in the model. The oil mixture density and viscosity were calculated based on Ideal solution and measured data models. The water mixture density and viscosity were calculated based on measured data and Islam-Carlson models, respectively. Mass transfer coefficient (MTC) included in the mass transfer term as well as the relative permeability curves were unknown. Water-oil relative permeability curves obtained during the simulation of WI experiment (WI-Kr) was used. However, the unknown MTC was obtained by history matching of the production data as described below. 3-3.1.1.

Effect of Mass Transfer Coefficient (MTC)

The unknown MTC can be obtained during the simulation by history matching of the production data. To identify the MTC value so that the simulator can predict the production data of CWI experiment, a sensitivity analysis was performed to understand the impact of MTC on the TOP and DP data. Figure 3-19 and Figure 3-20 demonstrate how the value of MTC affects the prediction of TOP and DP data by the developed simulator in its compositional mode. It should be noted that higher value of MTC means faster and therefore more CO2 transfer from water to oil.

65


100 90 80

TOP(Scm3)

70 60 50 40 30 Model, MTC=1E-7 Model, MTC=5E-7 Model, MTC=10E-7

20 10 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-19: Effect of the mass transfer coefficient (MTC) on prediction of TOP data by the developed simulator, CWI test.

0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 Model, MTC=1E-7 Model, MTC=5E-7 Model, MTC=10E-7

0.1 0 0

1

2 Injected PV

3

4

Figure 3-20: Effect of the mass transfer coefficient (MTC) on prediction of DP data by the developed simulator, CWI test. 66


It can be observed that MTC affect mainly the post breakthrough part of TOP data and influence the post breakthrough part of DP data slightly (Figure 3-19 and Figure 3-20). In other words, increasing the MTC value increases mainly the TOP and slightly DP data after the breakthrough time.

3-3.1.2.

Tuning of MTC

Considering the role of MTC in prediction of production data by the developed simulator as demonstrated above, it was tried here to match the TOP and DP data of the CWI experiment by only tuning MTC when the water-oil relative permeability curve of WI (WI-Kr) was used. First, MTC was tuned manually. Using manual tuning, after a few trials the MTC value of 5E-7 1/sec was chosen as the optimal value. Misfit value obtained using WI-Kr and this optimal MTC was 0.52. Later, in another practice, the GA-based optimiser was linked to the developed simulator to obtain the relative permeability curve and MTC simultaneously by history matching the production data. The goal was to investigate the hypothesis proposed that the relative permeability curves for both WI and CWI are the same. Moreover, to check the accuracy and the uniqueness of the MTC value obtained manually. During this optimisation task, the initial uncertainty ranges selected for Kr curve’s parameters were the same as those presented in Table 3-4. The initial uncertainty range selected for MTC parameter was between 1E-8 to 1E-5. A minimum total misfit of 0.6 was obtained after 45 generations. It should be noted that the stopping criteria was selected to repeat 100 iterations or to cross a misfit value of 0.6. Figure 3-21 shows the improvement of minimum and mean (average) total misfits over the generations for this optimisation task.

67


Figure 3-21: Minimum and mean misfit as a function of generation.

Table 3-7 shows the optimal values obtained during this optimisation experiment compared to those obtained by manual tuning. It can be observed that GA resulted in similar MTC value as that obtained manually and also similar relative permeability as that obtained for WI experiment. This could help to assure about the uniqueness of the manually obtained MTC and also could verify the proposed hypothesis of having the same Kr for both WI and CWI for this coreflood experiment value when two different approaches resulted in similar results. It should be noted that the obtained misfit of the manual tuning step is slightly smaller than of automatic tuning step(using GA-optimiser) because during manual tuning step, one parameter was tuned only (i.e. the MTC) however, GA found optimal values for four parameters simultaneously.

Table 3-7: Optimal values of Corey parameters and MTC, obtained using devlpoed simulator linked to the GA-based optimiser compared to manual tuning,CWI experiment.

GA Manual tuning

nw no kwmax 2.35 2.25 0.14

sor 0.3

MTC 4.9 E-7

2.5

0.31

5E-7

2.25

68

0.14


Figure 3-22 and Figure 3-23 compare the TOP and DP data of the CWI test obtained from the simulator with those from the experiment. These results show the ability of the simulator in modelling the multi-physics process of CWI for this water-wet core by tuning MTC only.

100 90 80

TOP(Scm3)

70 60 50 40 30 20

CWI-Model(WI-Kr, MTC=5E-7)

10

CWI-Exp

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-22: TOP data predicated by the developed simulator under its compositional mode compared to experimental TOP values, CWI test.

69


0.75

DP(psi)

0.60

0.45

0.30

0.15 CWI-Model(WI-Kr, MTC=5E-7) CWI-Exp 0.00 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-23: DP data predicated by the developed simulator under its compositional mode compared to experimental DP values, CWI test.

The simulator also helps to better understand the mechanisms contributing to the additional oil recovery of CWI. As it was mentioned before the main mechanisms are possibly oil swelling, viscosity reduction, IFT reduction and wettability alteration. Here the oil viscosity reduction cannot be a noticeable mechanism because the viscosity value of decane (oil sample) is already small and the resultant change is not significant. This will also be shown later when the effect of different viscosity models for the mixture of CO2-decane will be presented. Moreover, if an effective wettability alteration occurs during this CWI, a different relative permeability curve compared to WI is expected to be used. However the simulator illustrated that for this specific water-wet coreflood test, the relative permeability curves were the same for both WI and CWI processes. That implies that the wettability alteration is not a dominant mechanism here as well. It should be noted that a significant change in IFT would also dictate a change in relative permeability but such a level of change is not expected here. The oil swelling mechanism is captured by the mass transfer term (‘U’) in the developed model. The prediction by the developed simulator showed that by only adjusting the mass transfer term in the model, the production data of the CWI experiment could be matched. Hence, it can be inferred that the swelling of oil as a result of CO2 transfer is the main mechanism here resulting in 70


higher recovery of CWI compared to WI. It should be noted that using experimental data and the material balance calculation, the residual oil saturation at the end of CWI experiment was estimated as 27%. This value was not adjusted in relative permeability curve during simulation of the CWI test. That is, the sor=31% from WI test was not changed and the production data of CWI could be matched. This is discussed more in the next section. Using a manual calculation, it can be estimated that the oil has around 15% swelling (i.e. 31/27=1.15) which is captured by MTC.

In another experiment, the developed simulator was used to estimate and quantify the amount of swelling in the residual oil case. To do that, the value of the initial oil saturation in the simulator was set to the residual oil saturation value according to the relative permeability curves, i.e., soi=sor=0.31. Figure 3-24 shows the RF obtained when carbonated water is injected. It can be observed that at the end of simulation a RF around 17% has obtained. This RF suggested a swelling factor of around 1.17 (equivalent to 17% swelling) which is consistent with the 15% swelling suggested above.

0.20 0.18 0.16 0.14

RF

0.12 0.10 0.08 0.06 0.04 0.02 0.00 0

0.5

1

1.5

2 2.5 Injected PV

Figure 3-24: RF in the residual oil case.

71

3

3.5

4


3-3.2.

Black-oil Simulation

In another exercise, the simulator in its black-oil mode was used to simulate CWI. The goal was to investigate the matching process of the production data of CWI experiment by reducing the residual oil saturation to 27% in relative permeability curve. That is, it was tried to realise if the compositional effect of oil swelling explained above, can be captured by only adjusting the residual oil saturation. Figure 3-25 and Figure 3-26 respectively show the TOP and DP data predicted by the black-oil mode of the developed simulator using the WI-Kr and the sor adjusted to 27%. It should be noted that, using the experimental data, 27% is the dead oil saturation left in the core at end of the experiment.

100 90 80

TOP(Scm3)

70 60 50 40 30 20 CWI-Model(Black-oil mode, Sor=0.27) 10

CWI-Exp

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-25: TOP data predicated by the developed simulator in its black-oil mode compared to the experimental TOP values, sor=0.27, CWI test.

72


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 0.1

CWI-Model(Black-oil mode, Sor=0.27) CWI-Exp

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-26: DP data predicated by the developed simulator in its black-oil mode compared to experimental DP values, sor=0.27, CWI test.

It can be seen that, TOP cannot be predicted by this procedure. The reason is that, reducing the sor value increases the TOP value at breakthrough time with no considerable contribution in oil production after the breakthrough time. That is, almost a flat line after the breakthrough point is observed, red curve, in Figure 3-25. However, it is clear that during the CWI experiment, and compared to the WI test, no change of TOP at breakthrough point has been seen and the increased production of CWI over WI happened after the breakthrough point which was captured by the compositional simulation and tuning the mass transfer coefficient as explained before. This practice showed that the role of mass transfer cannot be captured by the role of sor playing during the black-oil simulation approach. It should be noted that changing other parameters in Corey correlations in addition to sor, may help to history match the production data of CWI during the black-oil simulation approach, however this was not the goal here as the blackoil approach will not provide a generic model for the CWI process. That is because the black-oil model cannot capture the actual physics of the CWI process.

73


3-4.

Effect of Mixture Fluid Properties on the Simulation of CWI

3-4.1.

Effect of Mixture Density Model on the Simulation of CWI

When carbonated water is injected into the reservoir, due to dynamic transfer of CO2 from water into the oil, the density of oil will change. As explained in Chapter 2, the density of CO2-decane mixture increases depending on CO2 content. To evaluate the impact of accurate estimation of decane-CO2 mixture density on the CWI simulation, a sensitivity analysis was carried out. The developed simulator in its compositional mode was used when the density of decane-CO2 mixture was either assumed to be constant (same as pure decane density), predicted based on the ideal solution formulation (Equation 2-6) or predicted using the tuned PR EOS (Equation 2-8) as explained in Chapter 2. Figure 3-27 and Figure 3-28 show the results of this sensitivity analysis. It can be seen that the density model does not affect the prediction of the simulator.

100 90 80

TOP(Scm3)

70 60 50 40 30 CWI-Model(Density:Constant)

20

CWI-Model(Density Model:Ideal Soultion)

10

CWI-Model(Density Model:EOS)

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-27: Effect of the decane-CO2 density model on TOP data, predicted by the developed simulator in its compositional mode, CWI test.

74


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 CWI-Model(Density:Constant) CWI-Model(Density Mode:Ideal Soultion) CWI-Model(Density Model:EOS)

0.1 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-28: Effect of decane-CO2 density model on DP data, predicted by the developed simulator in its compositional mode, CWI test.

Figure 3-29 shows the density profile of oil phase (i.e. decane-CO2 mixture) at end of the simulation versus the distance from the inlet of the core as predicted by the simulator for two different models. It can be seen that at the beginning of the core, the calculated density of the mixture is higher. This is because the concentration of CO2 in decane is higher at the beginning of the core as shown in Figure 3-30 and CO2 increases the density of decane. It should be noted that the difference between the calculated values is negligible.

75


0.755 Density Profile, Density Model:EOS Density Profile, Density Model:Ideal Soultion Density Profile, Density:Constant

Oil Density(g/cm3)

0.75

0.745

0.74

0.735

0.73

0.725 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-29: Oil density profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for three cases: PR-EOS Model, Ideal Solution Model and Constant Density, CWI test. .

CO2 Mass Fraction in Oil Phase

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-30: CO2 mass fraction in oil phase through the core at end of the simulation as predicted by the developed simulator in its compositional mode, CWI test. 76


Moreover, water density is not constant as well because during CWI, CO2 leaves the aqueous phase and enters into the oil phase. To see the effect of water density on the simulation results, Figure 3-31 and

Figure 3-32 display the TOP and DP data, respectively, which are predicted by the simulator when the water density was constant in comparison with that when the water density calculated using a correlation developed based on some measured data presented in Chapter 2 (Equation 2-13). It can be seen that the model of water density used has no considerable effect on the prediction of data by the simulator.

100 90 80

TOP(Scm3)

70 60 50 40 30 20

CWI-Model(Density:Constant)

10

CWI-Model(Density Model:Measured Data)

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-31: Effect of water-CO2 density model on TOP data as predicted by the developed simulator in its compositional mode, CWI test.

77


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 CWI-Model(Density:Constant)

0.1

CWI-Model(Density Model:Measured Data) 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-32: Effect of water-CO2 density model on DP data, predicted by the developed simulator in its compositional mode, CWI test.

Figure 3-33 shows the density profile of water phase (i.e. water-CO2 mixture) at end of the simulation versus the distance from the inlet of the core as predicted by the simulator. It can be seen that at the beginning of the core, and when the density was allowed to change, the density of water-CO2 mixture is higher because the concentration of CO2 in water is higher at the beginning of the core as shown in Figure 3-34. It should be noted that CO2 increases the density of water slightly.

78


1.012

Water Density(g/cm3)

1.01 1.008 1.006 1.004 1.002

Density Profile, Denisty Model:Measured Data Density Profile, Density:Constant

1 0.998 0

10

20 Distance(cm)

30

Figure 3-33: Water density profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for two cases: Model based on Measured Data and Constant Density, CWI test.

CO2 Mass Fraction in Water Phase

0.06

0.05

0.04

0.03

0.02

0.01

0 0

10

20 Distance(cm)

30

40

Figure 3-34: CO2 mass fraction in water phase through the core at end of the simulation as predicted by the developed simulator in its compositional mode, CWI test. 79


3-4.2.

Effect of Mixture Viscosity Model on the Simulation of CWI

During simulation of CWI, due to dynamic transfer of CO2 from water into the oil, the viscosity of oil and water will also change based on their CO2 content. As explained in Chapter 2, the change of water viscosity with CO2 content is negligible and the viscosity of decane (oil sample used here) decreases by its CO2 content. To evaluate the impact of accurate estimation of the viscosity of decane-CO2 and water-CO2 mixtures on the CWI simulation results, a sensitivity analysis was performed. The developed simulator in its compositional mode was used when the viscosity of decane-CO2 mixture was assumed to be constant and the same as pure decane viscosity, and when predicted based on the Beggs and Robinson correlation (Equation 2-15) as well as using a correlation developed based on some measured data (Equation 2-16) as explained in Chapter 2. Figure 3-35 and Figure 3-36 show the results of this sensitivity analysis. It can be seen that the oil viscosity model slightly affects the DP data after breakthrough point predicted by the simulator.

100 90 80

TOP(Scm3)

70 60 50 40 30 CWI-Model(Viscosity:Constant)

20

CWI-Model(Viscosity Model:Beggs and Robinson)

10

CWI-Model(Viscosity Model:Measured Data)

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Injected PV Figure 3-35: Effect of decane-CO2 viscosity model on TOP data as predicted by the developed simulator in its compositional mode, CWI test.

80


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 CWI-Model(Viscosity:Constant) CWI-Model(Viscosity Mode:Beggs and Robinson) CWI-Model(Viscosity Model:Measured Data)

0.1 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-36: Effect of decane-CO2 viscosity model on DP data as predicted by the developed simulator in its compositional mode, CWI test.

It should be noted that the viscosity change of oil has minimal effect on oil recovery unless there is a big change in viscosity.

Figure 3-37 shows the viscosity of the oil-CO2 mixture at end of the simulation versus the distance from the inlet of the core as predicted by the simulator. It should be noted that Beggs and Robinson correlation (Beggs and Robinson, 1975) is a general correlation and is not derived specifically for CO2 gas, therefore, as can be seen on Figure 3-37, there is a relatively high difference between prediction of mixture viscosity based on Beggs and Robinson and Measured Data models. It can be concluded that, Beggs and Robinson model has poor prediction of CO2-decane mixture viscosity for this experiment at test conditions. This could be due to the supercritical nature of CO2 at reservoir conditions. Figure 3-37 shows that at the beginning of the core the viscosity of mixture is lower because the concentration of CO2 in oil is higher at the beginning of the core and CO2 decreases the viscosity of decane.

81


0.9

Oil Viscosity (cP)

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Viscosity:Constant Viscosity Model:Beggs and Robinson

0.1

Viscosity Model:Measured Data

0 0

10

20 Distance(cm)

30

40

Figure 3-37: Oil viscosity profile through the core at end of the simulation as predicted by the developed simulator in its compositional mode, for three cases: Constant Viscosity, Beggs and Robisnon Correlation and the Measured Data-Model, CWI test.

Figure 3-37 shows that the viscosity model based on measured data has predicted a reduction in viscosity value approximately half of the original viscosity of the n-decane when it has no CO2 content. However, on the other hand, Figure 3-36 shows that the DP across the core has not dropped by a factor of two when the Measured Data-Model has been used for prediction of the viscosity of the CO2-decane mixture in the developed simulator. It seems paradoxical because based on Darcy’s law when viscosity is reduced by a factor of 2, at a constant flow rate, DP should decrease by a factor of 2 as well. To realise this, using the created ECLIPSE 100 model for WI test (i.e. black-oil model), it was discovered that when the viscosity of water (i.e. displacing fluid) was reduced by a specific factor, DP was increased by the same factor, however this was not true for the viscosity of oil (i.e. displaced fluid). That is, changing the viscosity of oil, will not change the predicted DP value based on Darcy equation and when the decane viscosity was reduced by a relatively big factor compared to its original value, the DP slightly decreased and the TOP slightly increased. This was investigate and it was realised that overall total mobility reduction from the relative permeabilities has an order of magnitude greater control giving the impression of little effect of oil viscosity. It is worth mentioning that 82


the applied constraints in this ECLIPSE model were a constant water injection at the core inlet and a constant pressure at the core outlet. Therefore DP value is related to the viscosity of displacing fluid (here water) through Darcy equation. That is, DP values are controlled by properties of displacing fluid.

During CWI, and because of the change in the water phase composition, water viscosity also slightly changes. To see the effect of water viscosity on the simulation results, Figure 3-38 and Figure 3-39 display the TOP and DP data, respectively, which are predicted by the simulator when the water viscosity is constant in comparison with those when the water viscosity was calculated using Islam and Carlson correlation (Equation 2-17).

It can be seen that the viscosity model of water has no considerable effect on the prediction of data by the simulator because viscosity of water changes minimally due to CO2 dissolution.

100 90 80

TOP(Scm3)

70 60 50 40 30 20

CWI-Model(Viscosity:Constant)

10

CWI-Model(Viscosity Model:Islam and Carlson)

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-38: Effect of water-CO2 viscosity model on TOP data as predicted by the developed simulator in its composition mode, CWI test. 83


0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 CWI-Model(Viscosity:Constant)

0.1

CWI-Model(Viscosity Model:Islam and Carlson) 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-39: Effect of water-CO2 viscosity model on DP data as predicted by the developed simulator in its composition mode, CWI test.

Figure 3-40 shows the viscosity of the water-CO2 mixture at end of the simulation versus the distance from the inlet of the core as predicted by the simulator. It demonstrates the density of water phase predicted by the simulator using the Islam and Carlson correlation. It can be seen that at the beginning of the core the viscosity of mixture is higher because the concentration of CO2 in water is higher at the beginning of the core and CO2 increases the viscosity of water slightly.

84


0.67

Water Viscosity (cP)

0.67 0.67 0.67 Viscosity Model:Islam and Carlson

0.66

Viscosity:Constant 0.66 0.66 0.66 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-40: Water viscosity profile through the core at end of the simulation as predicted by the developed simulator in its composition mode for two cases: Constant Viscosity and Islam and Carlson Correlation, CWI test.

3-5.

CO2 Production Profile

The developed simulator can also predict the amount of produced CO2 during CWI process. CO2 production data is not available for the coreflood experiments under study in this thesis. Therefore the simulator was used to investigate the amount of CO2 produced during CWI test. Figure 3-41 shows the total CO2 production (TCO2P) and total water production (TWP) profile versus injected pore volume of carbonated water. It should be noted that Figure 3-41 is obtained when the simulator is used in its compositional mode with tuned MTC value of 5E-7 and WI-Kr curve as explained before. It should be noted that the mass fraction in injected water was 5 wt %.

Figure 3-41 illustrates that the breakthrough time of water and CO2 is the same. It can be inferred that water is not completely deprived of its CO2 content when it arrives at the production point (core outlet). To clarify this more, Figure 3-42 shows separately the total CO2 production from water (TCO2PW) and oil (TCO2PO) streams as predicted by the developed simulator. 85


450

9000 TWP-Model, MTC=5E-7 8000

TCO2P-Model, MTC=5E-7

350

7000

300

6000

250

5000

200

4000

150

3000

100

2000

50

1000

0

TCO2P(Scm3)

TWP(Scm3)

400

0 0.0

0.5

1.0

1.5

2.0 2.5 3.0 Injected PV

3.5

4.0

4.5

Figure 3-41: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator, CWI test.

9000

700 TCO2P-Model, MTC=5E-7 TCO2PW-Model, MTC=5E-7

600

TCO2PO-Model, MTC=5E-7

7000

500 6000 5000

400

4000

300

3000

TCO2PO(Scm3)

TCO2P(Scm3) , TCO2PW(Scm3)

8000

200 2000 100

1000 0

0 0.0

0.5

1.0

1.5


3.5

4.0

4.5

Figure 3-42: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the simulator, CWI test.

86


It can be seen that the main source of CO2 which is produced at the outlet is from the water stream. That is, CO2 arrives at outlet quickly via the water phase compared to that via the oil phase. It can also be inferred that the CO2 transfer from water into the oil phase is a slow process because the water does not lose all of its CO2 content and it arrives at the outlet when it has some amount of CO2. On the other hand, it seems that the CO2 displacement by convection through the oil phase has happened slowly because although oil has some CO2 content but it arrives late at the outlet. Figure 3-42 shows that CO2 breakthrough via the water stream happens after around 0.64 pore volume of injection while it occurs after around 0.8 pore volume of injection via the oil stream. To further support this hypothesis, i.e. slow CO2 transfer from water into the oil phase and slow CO2 displacement (if only have convection) through the oil phase, the simulator can be used to investigate the parameters which affect the CO2 production profile. It has been shown that MTC value controls the rate of CO2 transfer between the water and oil phases. Moreover, to accelerate the CO2 displacement through the oil phase, dispersion mechanism can play an important role, which affects the CO2 production profile. In addition, injection rate may also affect the CO2 production profile because its influence the present time of the fluids in the core. To study the effects of MTC, dispersion and injection rate on CO2 production profile, a sensitivity analysis was carried out as described below.

3-5.1.

Effect of MTC on the CO2 Production Profile

The optimised MTC value of 5E-7 was obtained when the production data (i.e. TOP and DP) of the CWI coreflood experiment was history matched. This results in having the same breakthrough time for water and CO2 production profiles. It is expected that if the MTC value is increased, the water may be deprived of its CO2 content and therefore the pure water will arrive at the outlet without having any CO2 content. This will lead to a delay in breakthrough time of the CO2 production compared to that of the water production. Figure 3-43 compares the water and CO2 production profiles at three different MTC values.

87


400 350

TWP(Scm3)

9000

TWP-Model, MTC=5E-7 TWP-Model, MTC=15E-7 TWP-Model, MTC=25E-7 TCO2P-Model, MTC=5E-7 TCO2P-Model, MTC=15E-7 TCO2P-Model, MTC=25E-7

8000 7000

300

6000

250

5000

200

4000

150

3000

100

2000

50

1000

0

TCO2P(Scm3)

450

0 0.0

0.5

1.0

1.5


3.5

4.0

4.5

Figure 3-43: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator, MTC=5E-7, 15E-7 and 25E-7 1/sec, CWI test.

Figure 3-43 shows that the water production profile is not changed by changing the mass transfer coefficient; however, the CO2 production profile is shifted to the right when the MTC is increased. That is, the water breakthrough time is not changed while CO2 breakthrough time is increased by increasing the MTC value. Moreover, for higher MTC values, the total produced CO2 is lower. This occurs because most of the CO2 is transferred to the oil phase while its displacement in the oil phase is slow which this in turn delays CO2 arrival at the outlet. On the other hand, lower MTC values cause the water to retain its CO2 content and as a result the water stream arrives at outlet with more CO2. This results in a steeper CO2 production profile at lower MTC values. To clarify this more, Figure 3-44 displays the proportion of CO2 produced from the water stream compared to that from the oil stream at two different MTC values.

88


7000

TCO2P(Scm3) , TCO2PW(Scm3)

4000

TCO2P-Model, MTC=5E-7 TCO2P-Model, MTC=25E-7 TCO2PW-Model, MTC=5E-7 TCO2PW-Model, MTC=25E-7 TCO2PO-Model, MTC=5E-7 TCO2PO-Model, MTC=25E-7

6000

3500 3000

5000

2500

4000

2000

3000

1500

2000

1000

1000

500

0

TCO2PO(Scm3)

8000

0 0.0

0.5

1.0

1.5


3.5

4.0

4.5

Figure 3-44: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the simulator, MTC=5E-7 and 25E-7 1/sec, CWI test.

It can be seen that regardless of the MTC value, the main part of the produced CO2 at core outlet comes from the water stream. However, when the MTC is increased, the oil phase has more contribution in the CO2 production (comparing dashed and solid dark blue lines). It can be concluded that at high MTC values, water can lose all its CO2 content at the front and pure water reaches the core outlet. This is expected if the rate of mass transfer is high and the system reaches the equilibrium state promptly. This will be later discussed when the results of ECLIPSE300 simulation which is worked based on the instantaneous equilibrium assumption are presented. Moreover, when the MTC is increased, more CO2 is transferred into the oil phase at a constant time scale and considering delayed breakthrough of CO2, it is expected to obtain higher oil recovery as the oil phase has higher CO2 content. This is clearly shown in Figure 3-45 when RF is compared at three different MTC values predicted by the simulator. It shows that, higher MTC values results in higher oil production. Figure 3-45 however indicates that RF is not a monotonic function of MTC and after a specific value; increasing MTC does not increase RF considerably. It seems at this value, the system may have reached the equilibrium state i.e. oil has received its equilibrium level of CO2 concentration. 89


90 80 70

RF(%)

60 50 40 30

Exp Model, MTC=5E-7

20

Model, MTC=15E-7

10

Model, MTC=25E-7

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-45: RF predicted by the simulator in its compositional mode, MTC=5E-7, 15E-7 and 25E-7 1/sec, CWI test.

It should be noted that, the CO2 is displaced by fluid velocity (i.e. convection mechanism) through the phases and water mobility controls the CO2 production profile. CO2 can also be displaced through the phases by dispersion mechanism and it is expected that if the rate of CO2 displacement through the oil phase is increased by the dispersion mechanism, the contribution of oil stream in CO2 production will increase. That is, the breakthrough time of CO2 may happen earlier. Impact of dispersion coefficient on the CO2 production profile is described below.

3-5.2.

Effect of Dispersion Coefficient on the CO2 Production Profile

All simulations presented so far, have been conducted assuming a zero dispersion coefficient (no dispersion term). However as explained above, dispersion can accelerate the CO2 displacement inside the oil phase and consequently affects the CO2 and oil production profile. To understand the effect of dispersion on the CO2 production data, a sensitivity analysis was carried out. Dispersion coefficient is not available for the 90


experiments under study here. Therefore, an approximated value as explained in Chapter 2 was used here.

Figure 3-46 shows the CO2 production profile at three different dispersion coefficient values (i.e. D=0, 171E-4 (from Chapter 2) and 1000E-4 cm2/sec). It can be seen that a bigger dispersion coefficient results in a steeper CO2 production profile because the dispersion helps the CO2 to move faster in oil phase. To clearly see the breakthrough time of CO2 production, a part of data on Figure 3-46 is re-plotted when its vertical axis is zoomed in (Figure 3-47).

9000 TCO2P-Model, D=0, MTC=5E-7 TCO2P-Model, D=171E-4, MTC=5E-7 TCO2P-Model, D=1000E-4, MTC=5E-7

8000 7000

TCO2P(Scm3)

6000 5000 4000 3000 2000 1000 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-46: Total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test.

91


1000 900 800

TCO2P(Scm3)

700 600 500 400 300 TCO2P-Model, D=0, MTC=5E-7 TCO2P-Model, D=171E-4, MTC=5E-7 TCO2P-Model, D=1000E-4, MTC=5E-7

200 100 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-47: Magnified plot of Figure 3-46, total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test.

Figure 3-47 clearly shows a reduction in breakthrough time when the dispersion coefficient is increased. It should be noted that, this result is obtained when the developed model is one dimensional and dispersion helps CO2 to move faster towards the core outlet. However, for a 2 or 3 dimensional model, dispersion may help CO2 to move in transverse direction to the flow resulting in a delay in CO2 production. Comparing Figure 3-43 and Figure 3-46, it can be seen that the CO2 production profile is less sensitive to the dispersion coefficient than the MTC. This may be related to the competition of the convection and dispersion mechanisms in the system during the CWI process. That is, when the CO2 is displaced inside the oil phase by both oil velocity and dispersion, at high velocities in homogenous porous media, convection is likely to be the dominant mechanism. In the other words, if the oil velocity is reduced, the effect of dispersion may be more pronounced. To support this hypothesis, a sensitivity analysis on the injection rate and dispersion was performed as described below.

92


3-5.3.

Effect of Injection Rate on the CO2 Production Profile

To see the effect of injection rate and fluid velocity on CO2 production profile, the injection rate was reduced from 20 to 10 and 5 cm3/hr. It should be noted that decreasing the injection rate will reduce the velocity of the flowing fluid in the system. The results are shown in Figure 3-48. Figure 3-48 shows the CO2 production profiles when the injection rates are set to 5, 10 and 20 cm3/hr without the dispersion effect. It can be seen that when the velocity is reduced, the CO2 breakthrough time is delayed as well as the total amount of CO2 at the same injected pore volume is lower (comparing red, green and dark blue lines). The reason is the availability of enough time for the CO2 to be transferred from the water into the oil at low injection rates. That is, at low injection rates, oil and water are in contact for a longer time, which, in turn, gives rise to more CO2 transfer into the oil phase. As mentioned before, at low injection rates, the CO2 displacement in the oil phase by the convection mechanism is slow. Hence, at low injection rates, CO2 arrives at the core outlet late compared to that at the high injection rates. It should be noted that, the length of the simulation run at the injection rate of 20 cm3/hr was 25 hours and in order to inject the same pore volume of the injected fluid, the length of simulation run for the injection rate of 10 and 5 cm3/hr was increased to 50 and 100 hours, respectively.

9000 8000

TCO2P-Model, D=0, Injection Rate=20 TCO2P-Model, D=0, Injection Rate=10 TCO2P-Model, D=0, Injection Rate=5

7000

TCO2P(Scm3)

6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-48: Total CO2 production predicted by the simulator for the injection rates of 5, 10 and 20 cm3/hr without dispersion, CWI test. 93


Now, to evaluate the performance of dispersion at low and high rates, the dispersion was activated and the CO2 production profile was studied at injection rates of 20 and 5 cm3/hr. Figure 3-49 compares CO2 production profiles with and without dispersion at injection rates of 20 and 5 cm3/hr.

Figure 3-49 indicates that the dispersion has no considerable effect at high flow rates which means at high flowrate convection is the dominant mechanism. However, when the fluid velocity is reduced, the dispersion effect can clearly be seen when it causes the CO2 breakthrough time to happen quicker.

9000 TCO2P-Model, D=0, Injection Rate=20 TCO2P-Model, D=0, Injection Rate=5 TCO2P-Model, D=171E-4, Injection Rate=20 TCO2P-Model, D=171E-4, Injection Rate=5

8000

TCO2P(Scm3)

7000 6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-49: Total CO2 production predicted by the simulator for dispersion coefficient (D) of zero and 171E-4 1/sec at injection rates of 5 and 20 cm3/hr , CWI test.

Therefore it can be concluded that both the dispersion and convection can be important mechanisms during the CWI process depending on the magnitude of the velocity and the dispersion coefficient. Generally, it seems that the CO2 is displaced through the oil phase by dispersion and convection at low velocity values while at high velocity values, the convection is the main mechanism and the effect of dispersion is minimal.

94


It is worth investigating the oil production profile as well. To do that, the effects of dispersion, injection rate and carbonation level on the oil production profile are studied below.

3-6.

Oil Production Profile

3-6.1.

Effect of Dispersion Coefficient on the Oil Production Profile

As explained before, dispersion accelerates the CO2 displacement through the oil phase. Therefore, it is expected that it also impacts the oil production profile. Figure 3-50 displays the effect of dispersion coefficient on the recovery factor (RF) profile.

80 70 60

RF(%)

50 40 30 Model, D=0, MTC=5E-7

20

Model, D=171E-4, MTC=5E-7 10

Model, D=1000E-4, MTC=5E-7

0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-50: RF predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, CWI test.

It can be seen that when the dispersion is activated, oil production is slightly reduced. This is due to the fact that when the dispersion is activated the CO2 content of the oil moves faster and therefore exits the system earlier. This in turn results in having more uniform CO2 distribution in oil phase when dispersion is activated. Figure 3-51 compares CO2 concentration in the oil phase through the core with and without dispersion effect. Figure 3-52 shows the corresponding water saturation profile through the core. 95


CO2 Concentration in Oil Phase(g/cm3)

0.045 0.25 Injected PV, D=0

0.04

0.25 Injected PV, D=171E-4 0.035

0.5 Injected PV, D=0 0.5 Injected PV, D=171E-4

0.03 0.025 0.02 0.015 0.01 0.005 0 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-51: Profile of CO2 concentration in oil phase predicted by the simulator for D=0 and 171E-4 (cm2/sec) after 0.25 and 0.5 injected PV, CWI test.

1 0.9

0.25 Injected PV, D=0 0.25 Injected PV, D=171E-4 0.5 Injected PV, D=0 0.5 Injected PV, D=171E-4

0.8 0.7

Sw

0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-52: Profile of water saturation predicted by the simulator for D=0 and 171E-4 (cm2/sec) after 0.25 and 0.5 injected PV, CWI test. 96


Figure 3-51 clearly displays that the CO2 concentration profile is more uniform when the CO2 is moved forward by dispersion mechanism as well. However, Figure 3-52 shows that water saturation profile is not changed by dispersion.

Figure 3-51 also shows that after 0.5 injected PV, the oil stream at the core outlet does not have any CO2 content if the dispersion is not activated (dashed dark blue line) however when the dispersion plays a role in the system, the CO2 concentration of the oil stream at the end of the core is above zero (solid dark blue line).

Moreover, comparing solid and dashed red lines in Figure 3-51, it can be observed that after 0.25 injected PV, the CO2 front in the oil phase is around 13 cm away from the core inlet without dispersion while when the dispersion is activated it is around 25 cm away from the core inlet. This shows faster movement of the CO2 in the presence of the dispersion. Moreover, Figure 3-51 also indicates that oil receives more CO2 with time regardless of the dispersion mechanism. This is seen by comparing red curves with corresponding dark blue curves plotted in Figure 3-51. Although the dispersion helps CO2 to contact more oil through the core, it also causes the CO2 to leave oil faster and as a result the CO2 content of oil is reduced. Moreover, considering that the CO2 does not change considerably the fluid properties of decane, the ultimate impact of the dispersion is a slight reduction of the total produced oil at high values of dispersion as shown in Figure 3-50. It should be noted that if the CO2 can diffuse in transverse direction to the flow by the dispersion mechanism and recover some oil which may be bypassed by the flowing carbonated water stream, the effect of dispersion would be a higher oil recovery factor. However this cannot be seen here as the model is one-dimensional. That is, it seems that a 2 or 3 dimensional model can better capture the contribution of dispersion in porous media particularly in heterogeneous systems if dispersion can help to increase the sweep efficiency.

3-6.2.

Effect of Injection Rate on the Oil Production Profile

As explained above, when the injection rate is reduced, the velocities of oil and water phases are reduced in the system. Therefore, oil and water phase can be in contact for a longer time before leaving the system. As a result, it is expected that more CO 2 can transfer from the water into the oil phase. To investigate this, a sensitivity analysis was 97


performed when the injection rate was reduced from 20 to 10 and 5 cm3/hr. Figure 3-53 presents the result of this exercise. It can be seen that when the injection rate is decreased, the oil production is increased. This can be explained that at low injection rates, more amount of CO2 is transferred into the oil phase and consequently oil swells more. This in

RF(%)

turn improves the mobility of the oil and results in a higher oil recovery.

85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Model, D=0, Injection Rate=20 Model, D=0, Injection Rate=10 Model, D=0, Injection Rate=5 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-53: RF predicted by the simulator for three different injection rates of 20, 10 and 5 cm3/hr without dispersion, CWI test.

Figure 3-54 shows the CO2 concentration of oil through the core after 0.5 PV of CW injected at injection rates of 20 and 5 cm3/hr. It can clearly be seen that at low injection rate, oil has higher CO2 concentration at the same distance away from the core inlet.

To see the impact of dispersion at different injection rates on oil production, Figure 3-55 shows the RF profiles at injection rates of 20 and 5 cm3/hr both with and without dispersion effect. It can be seen that at low injection rates, dispersion slightly decreases RF while at high flow rate the effect of dispersion is minimal. As explained above, this is due to the fact that, the dispersion is more pronounced at low injection rates and it helps the CO2 to leave the system quicker. This results in a reduction of oil production when the dispersion is activated at low injection rates. This trend can also be seen at high 98


injection rates; however the impact of dispersion on oil recovery factor at high injection rates is minimal.


0.14 0.12 0.5 Injected PV, Injection Rate=20

0.1

0.5 Injected PV, Injection Rate=5 0.08 0.06 0.04 0.02 0 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-54: Profile of CO2 concentration in oil phase predicted by the simulator at injection rates of 20 and 5 cm3/hr without dispersion after 0.5 injected PV, CWI test.

90 80 70

RF(%)

60 50 40 30 Model, D=0, Injection Rate=20 Model, D=0, Injection Rate=5 Model, D=171E-4, Injection Rate=20 Model, D=171E-4, Injection Rate=5

20 10 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-55: RF predicted by the simulator for injection rates of 20, 10 and 5 cm3/hr with and without dispersion, CWI test. 99


3-6.3.

Effect of Carbonation Level on the Oil Production Profile

Carbonation level (CL) refers to the amount of CO2, which is dissolved in water. During the CWI experiment, the amount of dissolved CO2 in the injected stream was 5 weight percent (i.e. 5 wt % carbonation level). It is obvious that if the solubility of CO2 in the injected water is decreased or increased, it will impact the performance of the CWI. The solubility of the CO2 in water depends on the pressure and temperature of the system. However, generally, the solubility of CO2 in water is low. The 5 wt % is the measured solubility of CO2 in water at test conditions. CO2 solubility in water is increased by increasing pressure and is decreased by increasing temperature (Wiebe 1941, Mosavat and Torabi 2014a). It is expected that increasing the carbonation level of the injected water during CWI, will increase the oil recovery and reducing it, will give rise to the reduction of oil recovery. Figure 3-56 shows the profiles of RF for three different

RF(%)

carbonation levels of 7, 5 and 3 wt % as predicted by the developed simulator.

85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Model, CL=3 wt %, MTC=5E-7 Model, CL=5 wt %, MTC=5E-7 Model, CL=7 wt %, MTC=5E-7 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-56: Profiles of RF for three different carbonation levels of 7, 5 and 3 wt % predicted by the developed simulator, CWI test.

100


It clearly indicates that a higher carbonation level of the injected water results in higher oil recovery. The reason is that, when the injected water has higher CO2 content, more CO2 is available to be transferred into the oil phase, which this in turn results in better oil mobility and consequently higher oil recovery. It should be noted that, a higher carbonation level in real oil field can be achieved if the injection pressure can be increased as at higher pressures more CO2 can be dissolved in water. However, due to economical or operational limitations, this may not always be possible.

3-7.

CO2 Storage

When carbonated water is injected into the oil reservoirs, CO2 starts its migration from water into oil. During CWI, no free CO2 is in the system, i.e. CO2 is dissolved in water and oil phases. This will mitigate the risk of CO2 leakage from a permeable cap rock as CO2 is more stable and less mobile when it is in the dissolved state. When injection is stopped, some amount of CO2 stays in the reservoir while the rest of it has been produced by the oil and water streams. The developed simulator can be used to estimate the amount of stored CO2 in the core at the end of experiment. Figure 3-57 shows the profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S) as predicted by the simulator. It can be seen that before breakthrough of CO2 at the core outlet, the profiles of injected and stored CO2 overlap with an increasing trend. It can be conclude that all the injected CO2 are stored inside the core until the breakthrough point of CO2. However, when the CO2 production starts, the stored CO2 curve also starts declining gradually. It is expected that the stored CO2 curve becomes flat after a period of time when the remained decane has received its equilibrium level of CO2 concentration at which, all the injected CO2 will be produced at the core outlet.

101


16000 TCO2P-Model

14000

TCO2I-Model TCO2S-Model

TCO2(Scm3)

12000 10000 8000 6000 4000 2000 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-57: Profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S) versus injected PV of carbonated water, predicted by the simulator, CWI test.

Figure 3-58 shows the total CO2 stored in water phase as a percentage of total stored CO2 TCO2SW

(

TCO2S

TCO2SO

×100) an also the total CO2 stored in oil phase ( TCO2S ×100). It can be observed

that, at the beginning, CO2 has been stored in water phase while later main retention of CO2 is by the oil phase. That is, Figure 3-58 displays that at the end of the experiment; approximately 70% of remained CO2 in the core is in the oil phase while 30% of it is in the water phase. TCO2S

Figure 3-59 shows the total stored CO2 as a percentage of injected CO2 ( TCO2I ×100). It can be observed that, at end of the experiment, approximately 44% of injected CO2 could be stored in the core.

102


Figure 3-58: The percentage of CO2 stored in water (CO2SW) and in oil phase(CO2SO).

110 100 90

TCO2S/TCO2I(%)

80 70 60 50 40 30 20 10 0 0

0.5

1

1.5

2 2.5 Injecetd PV

TCO2S

3

3.5

4

4.5

Figure 3-59: ( TCO2I ×100) versus injected PV of carbonated water predicted by the simulator, CWI test.

103


3-8.

Compositional Simulation of CWI Using ECLIPSE300 (E300)

At this stage, E300 compositional simulator was used to simulate CWI. The goal was to compare the results from the developed simulator which is based on non-equilibrium mass transfer assumption and those from E300 which works based on instantaneous equilibrium assumption. To use E300, an ECLIPSE model was created using CO2SOL KEYWORD. CO2 flooding in oil reservoirs for EOR purposes or in oil depleted reservoirs for the storage purpose can be simulated by the CO2SOL three-phase compositional option of E300. The CO2SOL option allows CO2 to be present in three phases of oil, gas and water while other components can be in the oil and gas phases. That is, the CO2SOL option allows CO2 to be dissolved in water phase. Water is allowed to be present as a separate aqueous phase only. The equality of CO2 fugacity in oil and gas phases is used to calculate the CO2 distribution between the gas and oil phases. Densities and fugacities of oil and gas phase are calculated using the Peng-Robinson cubic equation of state (EOS). Aqueous phase properties and the amount of CO2 dissolved in water are computed using solubility data available in the ECLIPSE software (ECLIPSE manuals 2014). Moreover, in E300, carbonated water cannot be explicitly defined in the injection stream. Therefore two individual injection wells were defined; one for injecting water and another for injecting CO2. These two wells commence injection at the same time and their flowrate are adjusted so as to represent 20 (cm3/hr) of carbonated water with 5 weight percent of carbon dioxide content. The input data in the E300 model was the same as those in the developed simulator including the same number of gridblocks and water-oil relative permeability curve. Figure 3-60, Figure 3-61 and Figure 3-62 show respectively the RF profile, differential pressure (DP) across the core and total CO2 production (TCO2P) predicted by E300 compared to those obtained from the developed simulator. Figure 3-60 shows that oil production predicted by E300 is higher with final recovery factor of 78% compared to that predicted by the developed simulator. Moreover, as shown in Figure 3-62, the developed simulator has predicted that the CO2 is produced quicker compared to that predicted by E300.

104

RF(%)


85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Exp E300 Model, MTC=5E-7 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-60: RF predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, CWI test.

0.7 0.6

DP(psi)

0.5 0.4 0.3 0.2 DP-Exp DP-E300 DP-Model, MTC=5E-7

0.1 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-61: DP data predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, CWI test. 105


9000 TCO2P-Model, MTC=5E-7

8000

TCO2P-E300

TCO2P(Scm3)

7000 6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-62: TCO2P data predicted by the developed simulator compared to those predicted by E300, CWI test.

It can be explained that due to the assumption of instantaneous equilibrium made by E300, CO2 is allowed to be quickly transferred into the oil phase resulting in higher CO 2 concentration in oil phase compared to that predicted by the developed simulator which assumes that the CO2 transfer into the oil phase is a slow process. Figure 3-63 shows the CO2 concentration in oil phase through the core after 0.5 injected pore volume predicted by the simulator compared to that by E300.

106



0.08 Model, 0.5 Injected PV, MTC=5E-7

0.07

E300, 0.5 Injected PV 0.06 0.05 0.04 0.03 0.02 0.01 0 0

5

10

15 20 Distance(cm)

25

30

35

Figure 3-63: CO2 concentration in oil phase through the core after 0.5 injected PV predicted by the developed simulator and E300.

Figure 3-63 displays that CO2 concentration in oil at the beginning of the core predicted by E300 is higher compared to that predicted by the developed simulator. However furtherer away, E300 predicts that the CO2 concentration profile declines quickly and it goes below that predicted by the developed simulator. It can be explained that based on the prediction by E300, because CO2 transfer at the beginning of the core is quick, water has lost most of its CO2 content at beginning of the core and hence it travels through the core while it does not have enough amount of CO2 to be transferred to the oil phase. Therefore, predicted CO2 concentration profile by E300 shows a rapid decline. Moreover, from Figure 3-63, it can be realised the reason that the predicted CO2 production by E300 has delayed compared to that predicted by the developed simulator.

To further support this hypothesis that higher oil production prediction by E300 is due to the assumption of instantaneous equilibrium, a sensitivity analysis was performed and the effect of injection rate was investigated using the E300 simulator. The aim was to understand if the injection rate was altered, will that change the TOP profile as it was observed during similar simulations conducted by the developed simulator. In other 107


words, contrary to those observations, it is expected that changing the injection rate should not affect the prediction of E300. That is, because the E300 simulations is based on the instantaneous equilibrium, the length of time which oil and water are in contact should not affect the amount of CO2 transfer between the phases. Figure 3-64 clearly shows that reduction of injection rate from 20 to 5 cm3/hr does not impact the TOP predicted by E300.

120

100

TOP(Scm3)

80

60

40 TOP-Exp TOP-E300, Injection Rate=20

20

TOP-E300, Injection Rate=5 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-64: TOP data predicted by E300 simulator for injection rates of 20 and 5 cm3/hr, CWI test.

It should also be noted that the developed simulator showed that when injection rate was decreased more oil was produced (i.e. TOP was increased) because more CO2 could be transferred to the oil phase at a higher contact time available at low injection rates.

The process of CWI by the developed simulator can be adjusted so that CO2 be transferred from water into the oil quicker. The goal is to see if the predictions of the developed simulator and E300 can get closer. Mass transfer coefficient (MTC) in the developed simulator controls the rate of CO2 transfer from water into the oil. It is expected that if the MTC is increased, the prediction of E300 and the developed simulator will be closer. 108


That is, higher MTC values accelerate the CO2 transfer between phases and reducing the time needed to reach the equilibrium state. Figure 3-65 and Figure 3-66 show the total oil and CO2 production profiles respectively as predicted by the developed simulator when MTC value was increased to 15E-7 1/sec, three times higher than the optimal value (i.e. 5E-7) mentioned before.

120

100

TOP(Scm3)

80

60

40 TOP-Exp 20

TOP-E300 TOP-Model, MTC=15E-7

0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-65: TOP data predicted by E300 simulator compared to the experimental values and those predicted by the developed simulator, MTC=15E-7 1/sec, CWI test.

109


9000 TCO2P-Model, MTC=15E-7 8000 TCO2P-E300 7000

TCO2P(Scm3)

6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5

2 2.5 Injected PV

3

3.5

4

4.5

Figure 3-66: Total CO2 production (TCO2P) data predicted by E300 simulator compared to those predicted by the developed simulator, MTC=15E-7 1/sec, CWI test.

It can be seen that TOP data predicted by the developed simulator and E300 overlap however TCO2P data are not completely matched. This can be due to the fact that the equations and procedures used for calculation of fluid properties and CO2 distribution between phases by E300 are not exactly same as those used by the developed simulator. It should be noted that it is not possible to control the amount of mass transfer in E300 to match those of the developed simulator’s prediction. Therefore the developed simulator was adjusted to match the prediction of E300.

Moreover as shown before, at low injection rates, the simulator predicts higher oil production while the TOP prediction by E300 is insensitive to the injection rate. Therefore it is expected that at low injection rates, the prediction of E300 and the developed simulator be very close. Figure 3-67 compares the TOP predicted by E300 and the developed simulator at injection rate of 6.67 cm3/hr (one third of 20 cm3/hr) showing that the predicted TOP data by both simulators are very close.

110


120 100

TOP(Scm3)

80 60 40 TOP-E300, Injection Rate=6.67

20

TOP-Model, Injection Rate=6.67, MTC=5E-7 0 0.0

0.5

1.0

1.5

2.0 2.5 Injected PV

3.0

3.5

4.0

4.5

Figure 3-67: TOP data predicted by E300 simulator compared to those predicted by the developed simulator at injection rate of 6.67 cm3/hr, CWI test.

This exercise clearly showed how the assumption of instantaneous equilibrium which is the basis of usual compositional simulation approach can affect the performance of carbonated coreflood experiments predicted by the available compositional simulators. In another experimnet, it was attempted to find a MTC value at which it can be expected that the developed simulator is working based on the instantaneous equilibrium assumption for this coreflood experiment. That is, how large the MTC value should be so that the CO2 can be immediately distributed between oil and water phases based on its equilibrium concentration (equilibrated MTC). Figure 3-68 shows recovery factors prediced by the developed simulator at three different MTC valuses of 15E-7, 25E-7 and 30E-7 1/sec. It can be observed that the RF at the MTC value of 25E-7 is approcimatly 2% higher than that at the MTC value of 25E-7. Morover, there is no change in the RF when MTC value has increased to 30E-7. Therefore it can be concluded that for this coreflood experiment, a MTC value

greather than 15E-7 can be assumed as the

equilibrated MTC with acceptable accuarcy. This was consistent with that MTC vlaue at which the results of the developed simulator and E300 were very similar. 111


Figure 3-68: Recovery factors predicted by the developed simulator at three diffrent MTC valuses of 15E-7, 25E-7 and 30E-7 1/sec.

3-9.

Summary and Conclusions

The WI and CWI coreflood experiments performed in a water-wet core, were studied in this chapter. First, the experimental data including total oil production (TOP) and differential pressure across the core (DP) of WI and those of CWI were compared. Experimental data showed that DP data of WI and CWI are same while TOP data of CWI is higher only after the breakthrough point. That is, the breakthrough point of WI and CWI happened at the same time while CWI produced more oil after the breakthrough time compared to that of WI. This increased oil recovery was attributed to the oil swelling due to CO2 transfer from water into the oil.

Next, simulation of WI and CWI experiments were studied using the developed simulator and ECLIPSE commercial simulators. The simulation of WI experiment was performed first using the E100 black-oil simulator. To simulate the WI test, an optimised water-oil relative permeability curve (WI-Kr) was obtained when E100 was linked to the developed GA-based optimiser. Then, the WI test was re-simulated using the developed simulator 112


in its black-oil mode (zero mass transfer). Good agreement between the results predicted by developed and E100 simulators could verify the performance of the developed simulator. The capillary pressure was also included in these simulations; however its effect was shown to be minimal. In addition to the verification of the developed simulator, WI experiment was simulated to use the simulation data of WI experiment as a base for the simulation of CWI experiment. That is, WI relative permeability curve was used for the CWI test.

Compositional simulation of CWI was performed using the developed simulator in its compositional mode. To simulate the CWI, mass transfer coefficient (MTC) was considered as the only unknown. A sensitivity analysis was carried out and it was shown that MTC mainly increases TOP data after breakthrough time with minimal effect on DP data. MTC was manually tuned and production data of CWI experiment was history matched adequately. MTC was also automatically obtained when the GA-based optimiser was linked to the simulator resulting in the same MTC value as that obtained by manual tuning. In addition, it was concluded that the additional oil recovery of CWI over WI is due to the oil swelling, which was captured by the mass transfer term. That is, although the residual oil saturation in WI-Kr curve was not changed during the simulation of CWI, TOP data of CWI experiment could be history matched by tuning the mass transfer term. An oil swelling of 15% was estimated to happen during this experiment. It was also tried to simulate CWI using a black-oil approach (i.e. without mass transfer term) by reducing the residual oil saturation of the WI-Kr curve, however a good match was not obtained.

Effects of different models used for calculation of mixture density and viscosity were also checked to see if the accuracy of fluid properties calculations of oil-CO2 and water-CO2 mixtures, will impact the outputs of the developed simulator. It has been shown in literature that, based on experimental observations, water density and viscosity are slightly increased by the CO2 dissolution and decane density is increased while its viscosity is decreased by increasing its CO2 content. The simulators results using these models were compared to the case when fluid properties were assumed to be constant. It was realised that the accuracy of the model used for the prediction of fluid properties has minimal effect on the results of this system.

113


The developed simulator was also used to obtain the CO2 production profile. The simulator predicted that the CO2 breakthrough time of water and CO2 were the same. It also was demonstrated that the main source of CO2 production at the core outlet was from the water and not the oil stream. It was suggested that if more CO2 was transferred into the oil phase, the breakthrough of CO2 would happen after the breakthrough of water. To demonstrate this, a sensitivity analysis was carried out and the effect of MTC on CO 2 production profile was evaluated. It was shown that higher MTC values resulted in delayed and lower CO2 production. Moreover, it was argued that CO2 displacement in the oil phase by convection, i.e. oil velocity, is slow. To accelerate the displacement of CO 2 within the oil phase, dispersion was also activated. CO2 concentration profile in oil was also obtained from the simulator showing that when the dispersion was activated, more uniform concentration profile was obtained. Higher dispersion coefficients also resulted in earlier CO2 production, however this was not considerable. It was therefore stated that, compared to the dispersion, convection was the dominant mechanism of CO2 transfer in oil for this system. To see the effect of dispersion clearly, it was suggested to decrease the injection rate of the carbonated water. It was observed that at low injection rates, dispersion was more pronounced. That is, by changing the dispersion coefficient at low injection rates, unlike in high injection rates, a considerable change in the CO2 production profile was observed. It was explained that dispersion helps CO2 to move faster within the oil and leaves the system quicker. This was consistent by noting that with inclusion of dispersion, contribution of oil stream in CO2 production was increased. Furthermore the effect of injection rate without dispersion on CO2 production profile was also studied by performing a sensitivity analysis and it was observed that at low injection rate, delayed and lower CO2 production was obtained compared to the corresponding high injection rates. This is due to the fact that at low injection rate, oil and water are in contact for longer time and therefore more CO2 can be transferred into the oil phase while contribution of oil stream in CO2 production is low. Moreover, comparing the corresponding profiles of CO2 concentration in oil phase at low and high injection rates, it was observed that CO2 concentration in oil phase at low injection rates is higher.

Next, the oil production profile was also studied when the effect of dispersion, injection rate and carbonation level on oil production was investigated. When dispersion was activated, the oil production slightly decreased. However the reduction of oil production due to dispersion was higher at low injection rates. Therefore it is more favourable if CO2 114


stays in oil phase for longer time and does not leave the oil phase fast by dispersion to help with more oil swelling. It was also observed that at low injection rates, more oil could be recovered by CWI compared to those at high injection rates. This exercise showed the importance of the kinetics of CO2 transfer in the model (simulator). At low injection rates, because oil and water are in contact for longer time, more CO2 could transfer into the oil phase and also CO2 could stay in oil phase for longer time. This resulted in higher oil recovery.

A sensitivity analysis on carbonation level (CO2 content) of the injected fluid was carried out demonstrating that at higher carbonation levels, more oil was produced as it was expected.

The simulator was also used to estimate the amount of CO2 stored during the CWI experiment. The profiles of cumulative injected, produced and stored CO2 were obtained from the simulator. Before the breakthrough time of CO2, all injected CO2 was stored in the system while after CO2 breakthrough time, the profile of stored CO2 declined gradually. It was also estimated that, at the end of the experiment, around 44 percentage of injected CO2 could be stored safely, as dissolved in water and oil phases, in the core.

Finally, the CWI experiment was also simulated using ECLIPSE300 (E300) compositional simulator which works based on the instantaneous equilibrium assumption. E300 predicted higher oil recovery and lower and delayed CO2 production compared to those predicted by the developed simulator. This was attributed to the assumption of instantaneous equilibrium used by E300 which resulted in quicker and higher CO2 transfer into the oil phase. This was confirmed by comparing the profiles of CO2 concentration in oil phase predicted by E300 and the developed simulator. Next, a sensitivity analysis was performed when the injection rate was reduced in E300 during simulations. E300 predicted the same CO2 and oil production profile as those at high injection rate. Moreover, it was shown that if the MTC value in the developed simulator could be increased by three times, the prediction of E300 and the developed simulator would be very close. This was attributed to the fact that, under the conditions with high MTC values, the procedure of mass transfer by the developed simulator is closer to the instantaneous equilibrium condition prevailing the CWI simulation by E300. Finally and in line with this, it was shown that at low injection rates the prediction of the developed 115


simulator and E300 were very close. It was shown that when the injection rate was decreased by three times, the predictions of the developed simulator and E300 became very close.

116

Simulation of CWI in a Mixed-Wet Core This chapter reports the numerical simulations of a different water injection (WI) and carbonated water injection (CWI) coreflood experiment from the literature (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c) performed in a mixed-wet core. The objective was to crosscheck and verify the performance of the developed simulator when a different coreflood experiment was simulated. This also could help to explore the generic capability of the developed simulator.

First, the performance of the CWI experiment is compared with that of the conventional water injection (WI) test by investigating the experimental data. Next, the WI experiment is simulated. Afterwards, the simulation of the CWI experiment is performed using compositional and black-oil approaches. A systematic procedure is described which was used to simulate CWI including history matching of the coreflood production data. In addition, the effects of parameters which may impact the performance of CWI are studied, including dispersion coefficient, injection rate and carbonation level. Additionally, the developed simulator is used to investigate the CO2 storage capacity of the CWI process. Finally the ECLIPSE300 compositional simulator is used to simulate the CWI process and its performance is compared with the performance of the developed simulator.

4-1.

The WI and CWI Experiments

In this section, the production data of the coreflood experiments performed in a mixedwet sandstone core presented in the literature (Sohrabi et al., 2012a; Kechut et al., 2011b; Kechut, 2011c), including both secondary WI and corresponding CWI under the same conditions, are examined. It is worth investigating the results of the WI and CWI coreflood experiments first.

117

Chapter 4: Simulation of CWI in a Mixed-Wet Core

As claimed by Sohrabi et al. (2012a); Kechut et al. (2011b) and Kechut (2011c), the core had been made mixed-wet at the beginning by an ageing process using a crude oil. However, it should be noted that, based on the results presented later in this chapter, for example relative permeability and capillary pressure curves, the core seems more to be intermediate-wet. Moreover, based the level of change to the contact angle after aging process reported in Chapter 1, the intermediate wettability might be a more correct term.

Decane oil sample had been used to saturate the core. Table 4-1 and Table 4-2 show the core and fluid properties, respectively, used in the experiments. During both WI and CWI tests, water or carbonated water (CW) was injected into the core at a constant rate and water and/or decane were collected at a constant pressure at the core outlet. Figure 4-1 shows a schematic of the coreflood experiment.

Table 4-1: Core properties. Core

Length (cm)

Diameter (cm)

Porosity (fraction)

Pore Volume(cm3)

Permeability (mD)

Mixed-wet Clashach sandstone

61.3

4.86

0.16

182

850

Table 4-2: Fluid properties [Ref: NIST 2014].

Fluid

Viscosity(cP) (Test conditions) (136.1 atm, 38 0C)

Density (g/cm3) (Test conditions) (136.1 atm, 38 0C)

Density (g/cm3) Standard conditions (1 atm, 20 0C)

Decane

0.83

0.730

0.727

Water CO2

0.66 0.067

0.995 0.775

0.995 0.00184

water + oil + CO2

carbonated water

Figure 4-1: Schematic of coreflood experiment

The operational conditions of both WI and CWI experiments were the same and are given in the Table 4-3.

118


Table 4-3: The operational conditions of the coreflood experiments. 20 Injection rate (cm3/hr) Interstitial velocity(equivalent to the injection rate)(m/day) 1.6 5% CO2 mass fraction of injected CW 136.1 Outlet pressure (atm) 136.1 Initial pressure (atm) 0 Initial water saturation

Figure 4-2 and Figure 4-3 show TOP data and corresponding recovery factor (RF) of the WI experiment compared to those of the CWI plotted versus injected pore volume (PV), respectively.

140 120

TOP(Scm3)

100 80 60 40 20

WI-Exp CWI-Exp

0 0

0.5

1

1.5 2 Injected PV

2.5

3

Figure 4-2: Comparison of TOP data of the WI and CWI experiments.

119

3.5

RF(%)


75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

WI-Exp CWI-Exp 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-3: Comparison of RF of the WI and CWI experiments.

Figure 4-4 compares DP values of the WI and the CWI experiments plotted versus injected PV.

3.5 3

DP(psi)

2.5 2 1.5 1 0.5

WI-Exp CWI-Exp

0 0

0.5

1

1.5 2 Injected PV

2.5

3

Figure 4-4: Comparison of DP data of the WI and CWI experiments. 120

3.5


Figure 4-2 and Figure 4-3 show that during CWI, oil recovery has improved. That is, it can be observed that, the final oil recovery factor (RF) obtained at end of the CWI experiment is 68% (123.4 cm3) whereas it is 59% (107.7 cm3) at the end of WI experiment. Moreover, during CWI, more oil has been produced at breakthrough time (red solid curve, 117.4 cm3, 64% RF) compared to the amount of produced oil during WI at breakthrough time (blue dashed curve, 103.5 cm3, 57% RF). These data suggest that 4% additional RF compared to that at breakthrough point, is obtained by CWI at end of the experiment.

Moreover, Figure 4-4 shows the CWI has resulted in less differential pressure (DP) across the core in comparison with the WI. That is, during the CWI for this specific mixed-wet core, an injectivity improvement is observed. It should be noted that based on a personal communication with the people involved (Mr Mojtaba Seyyedi, Mr Amir Farzaneh), this behavior has been observed for other aged core experiments and there was not any change in core properties (i.e. porosity and permeability) after CWI experiment.

The CWI and WI coreflood experiments were simulated using the developed simulator. First, the water injection (WI) coreflood experiment was simulated to obtain the wateroil relative permeability (Krw-o) curve. As mentioned in Chapter 2, the power law-based Corey correlation was used to define the relative permeability curves. To simulate the WI test, the developed simulator in its black-oil mode (with zero mass transfer) was linked to the GA-based optimiser and by performing a history matching exercise, an optimised Krw-o curve was obtained. To check the reliability and validity of the simulator, the optimised Krw-o curve was used and the WI experiment was re-simulated employing ECLIPSE100 reservoir simulator (E100) and then the predictions of the developed simulator and those of E100 were compared. Next, the simulator in its compositional mode was used to simulate the CWI experiment. First, it was attempted to use the Kr w-o curve obtained from the WI test for the simulation of CWI, but it was realised that the same water-oil relative permeability (i.e. Krw-o) curve could not be used for the CWI experiment. That is, using the Krw-o curve, it was tried to history match the production data of CWI test by tuning the mass transfer coefficient (MTC) in the model. However, the model could not predict the CWI’s production data based on that. Therefore, the water-oil relative permeability curve was also modified and a new relative permeability curve, i.e. the carbonated water (cw)-oil relative permeability (Krcw-o) curve, was 121


obtained. It was found that using the Krcw-o curve, along with a tuned MTC, the production data of the CWI could be predicted reasonably by the simulator. The relative permeability curve for CWI simulation (i.e. the Krcw-o curve) was obtained both manually and also automatically, when the GA based optimiser was linked to the simulator. To manually obtain the Krcw-o curve, a sensitivity analysis on the Corey parameters was performed and the influences of these parameters on TOP and DP data were determined. Details of these simulations are explained below.

4-2.

Simulation of the Water Injection Experiment

4-2.1.

Water-Oil Relative Permeability Curve

The WI experiment was simulated first. However, the required relative permeability and capillary pressure curves were unknown. Therefore, the Krw-o curve was obtained by performing an optimisation task. It should be noted that, in this step it was assumed that the capillary pressure is zero. In order to obtain the Corey-based krw-o curve for the water injection test (WI-Kr), the devolved simulator in its black-oil mode (zero mass transfer) was linked to the developed GA-based optimiser. In this exercise, four parameters of the Corey correlation (kwmax, nw, no, sor) were considered unknown. In this experiment, the core was fully saturated with decane and there was no initial water saturation, therefore swc was set to zero and komax was set to one. During this optimisation task the initial range of uncertainty for parameters were the same as those used for water-wet core system (Table 3-4). The GA setup was based on that explained in Chapter 2. The optimised values of the Corey parameters are summarised in Table 4-4. Figure 4-5 shows the relative permeability curves based on the obtained parameters. Table 4-4: Optimal Corey parameters of water-oil relative permeability, obtained using the developed simulator under the black-oil mode linked to the GA-based optimiser as well as assigned komax and swc values, WI experiment. Corey parameters WI-Kr

nw

no

kwmax

2.5 2.25 0.074

122

komax

sor

swc

1

0.41

0


1

1 Kro Krw

0.9 0.8

0.7

0.7

0.6

0.6

Kro

0.8

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

Krw

0.9

0 0

0.1

0.2

0.3 sw

0.4

0.5

0.6

Figure 4-5: Optimised Krw-o curve, WI test.

It should be noted that the residual oil saturation can also be obtained using the material balance calculations. At the end of the WI experiment, 74 cm3 of oil is left in the core, which is equivalent to 41% residual oil saturation (sor). Moreover, it should be noted that the kwmax was also obtained by a manual calculation using the Darcy equation (Equation 4-1). The endpoint value on the DP curve and the core properties were needed during this calculation. qinj

(

)×μw ×L

A k wmax = K×∆P

(4-1)

endpoint

where qinj is the injection rate, A is the cross section area of the core, μw is the water viscosity, L is the core length, K is the absolute permeability and ∆Pendpoint is the endpoint value on the DP curve (DP at end of the experiment). Figure 4-6 and Figure 4-7 show experimental TOP and DP data versus the simulator’s prediction using the optimised Krw-o curve, respectively.

123


120

100

TOP(Scm3)

80

60

40 WI-Exp WI-Model

20

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-6 : Experimental TOP versus predicted TOP data, predicted by the developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test.

3.5 3

DP(psi)

2.5 2 1.5 1

WI-Exp WI-Model

0.5 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-7: Experimental DP versus predicted DP data, predicted by the developed simulator (model) under the black-oil mode using optimised Krw-o curve, WI test.

124


It can be seen that the obtained Krw-o curve is acceptable however TOP data has been history matched better than DP data. That is, TOP misfit is 0.2 while DP misfit is 0.65 and total misfit is 0.85(i.e. TOP misfit+DP misfit).

To verify the performance of the developed simulator, the test was also simulated using ECLIPSE100 (E100) with the same relative permeability curve (i.e. the optimised Krw-o). Figure 4-8 and Figure 4-9 show the outputs of ECLIPSE100 compared to those of the developed simulator. The close agreement between the data of these two figures clearly verifies the reliability of the developed simulator.

120

100

TOP(Scm3)

80

60

40 WI-E100

20

WI-Model 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-8: TOP data predicted by ECLIPSE100 versus TOP data predicted by developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test.

125


3.5 3

DP(psi)

2.5 2 1.5 1

WI-E100 WI-Model

0.5 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-9: DP data predicted by ECLIPSE100 versus DP data predicted by the developed simulator (model) under the black-oil mode using the optimised Krw-o curve, WI test.

It is worth mentioning that the unstable flow is ruled out here and it is assumed that the flow is stable. That is, it is assumed that there is no instability in flow and the discrepancy in flow behavior of water-wet and mixed-wet cores reflected in production curves specifically at breakthrough point is not due to instability in flow i.e. is not due to for example viscous fingering. This is because, as mentioned above, from a personal communicating with involved people, similar behaviours have been observed for other coreflood experiments. Nevertheless, it was checked to see if the DP data of water-wet and mixed-wet cores are consistent with the core dimensions. That is, based on Darcy law presented above (Equation 4-1), the ratio of pressure drop of mixed-wet (mw) core to water-wet (ww) core at endpoint of DP curve was calculated and compared with the experimental value using the following relation derived from Darcy law. ∆Pendpoin-mw ∆Pendpoin-ww

K

k

A

L

= K ww kwmax-ww Aww Lmw

(4-2)

mw wmax-mw mw ww

The following data (Table 4-5) was used to calculate the ratio of pressure drops.

126


Table 4-5: The data needed for calculation of pressure drop ratio. Core

K Kwmax A L (mD) (Krw at sor) (cm2) (cm) 0.14 19.5 33.2 water-wet(ww) 1300 0.074 18.5 61.3 mixed-wet(mw) 850

DPendpoint (psi) 0.53 3

The pressure drop ratio calculated based on Equation (4-2), is 5.63. From Table 4-5, the ratio of pressure drop based on experimental measurement is 5.66 which is the same as that obtained by calculation. This experiment could check the consistency of DP data with the dimensions of the cores.

4-2.2.

Capillary Pressure

The simulations presented so far have been performed considering a zero capillary pressure (Pc = 0). At this stage, the simulation was repeated when the capillary pressure was not zero. That is, using the GA-based optimiser linked to the E100 simulator, a history match task was performed to obtain capillary pressure and relative permeability curves simultaneously. The Brooks-Corey correlation described in Chapter 2 is for a water-wet system however for a mixed-wet system the capillary pressure curve can have a negative part (Anderson, 1987; Bradford and Leij, 1995 and 1996; Helland and Skjaeveland, 2006) which cannot be captured by Brooks-Corey correlation. In this study to use a capillary pressure function which can be suitable for a mixed-wet system when it can predict both positive and negative Pc, Brooks-Corey correlation was modified as shown below:

sw -swc

Pc=pce (1-s

wc -sor

-

1 λ

(4-2)

) -Pcmax/β

In the above equation, a positive value which is a fraction of maximum Pc (i.e. Pc at lowest water saturation, for example, Pcmax=Pc at sw =0.000015) is subtracted to allow having a negative Pc as well. Therefore in the above equation, β determines the fraction and is also unknown in addition to pce and λ. Similar modification has been suggested in the literature (Bradford and Leij, 1995).

127


Table 4-6 shows the initial uncertainty range of each parameter used during optimization task. The GA setup was the same as described in Chapter 2.

Table 4-6: Initial uncertainty range of parameters used in GA. Kr and Pc parameters nw no kwmax sor pce λ β initial uncertainty range 1-5 1-5 0.001-0.8 0.05-0.6 0-15 0.2-10 1-25

Figure 4-10 shows the improvement of minimum and mean (average) misfits over the generations for this optimisation task. At end of the optimisation, a minimum misfit of 0.78 has been obtained.

Figure 4-10: Minimum and mean misfit as a function of generation.

Table 4-7 shows the optimised Krw-o and Pc curves. Figure 4-11 shows the capillary pressure curve based on the data shown in Table 4-7.

128


Table 4-7: Optimal Corey and modified Brook-Corey coefficients of water-oil relative permeability and capillary pressure curves obtained by the optimiser linked to the E100. Kr & Pc

nw

WI-Kr (Pc=0) WI-Kr (Pc#0)

no

kwmax

2.5 2.25 0.074 2.3 2.25 0.076

komax

sor

swc

1 1

0.41 0.40

0 0

pce

1 λ

β

0.02 0.22 17

0.25

Pc (atm)

0.2

0.15

0.1

0.05

0 0

0.2

0.4 sw

0.6

0.8

Figure 4-11: Capillary pressure data for WI test.

It can be seen that the capillary pressure is almost zero and positive (maximum value is 0.2 atm at sw=0.000015). This was expected because the core is homogeneous, with high permeability. It should be noted that based on the obtained capillary pressure curve which is positive, the intermediate wettability might be a more correct term for this core.

The results of simulation, including capillary pressure are shown in Figure 4-12 Figure 4-13.

129


120 100

TOP(Scm3)

80 60 40 Exp E100

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-12: TOP data predicted by ECLIPSE100 versus experimental values when capillary pressure is included, WI test.

3.5 3

DP(psi)

2.5 2 1.5 1 Exp 0.5

E100

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-13: DP data predicted by ECLIPSE100 versus experimental values when capillary pressure is included, WI test. 130


The data in Table 4-7 shows that the relative permeability curves obtained with and without capillary pressure are similar, confirming the minimal impact of Pc. Considering this and also to exclude more uncertainty in the simulation process, it was concluded to ignore capillary pressure in all simulations to be presented from here onwards, for both WI and CWI experiments under study in this chapter.

4-3.

Simulation of CWI

4-3.1.

Compositional Simulation

In this step, the CWI process is simulated used the developed simulator in its compositional model assuming zero dispersion in the system and only the convection term is in the model. The oil mixture density and viscosity were calculated based on Ideal solution and Measured data models. The water mixture density and viscosity were calculated based on Measured-data and Islam-Carlson models, respectively. The mass transfer coefficient (MTC) included in the mass transfer term, as well as the relative permeability curves were unknown for this exercise. Therefore it is worth investigating how TOP and DP data are sensitive to MTC value and relative permeability curve at beginning and also understanding the impact of MTC and Kr curve on the TOP and DP data during simulation of the CWI process.

4-3.1.1.

Effect of Mass Transfer Coefficient (MTC)

A sensitivity analysis was carried out to understand the role of MTC in the simulation of CWI. In this exercise, water-oil relative permeability curves from the simulation of the WI experiment (WI-Kr) were used. Figure 4-14 and Figure 4-15 demonstrate the effect of MTC on TOP and DP data respectively, predicted by the developed simulator. Similar behaviour as that observed in water-wet core, have been also observed here. That is, these figures show that increasing MTC values leads to an increase in the TOP data with minimal effect on DP data. Figure 4-14 shows that the TOP cannot be matched if mass transfer is used as the only unknown parameter of the history matching process to be tuned. Additionally, including mass transfer in the system, the DP of the system is increased slightly while, compared to the WI process, the DP values of the CWI have reduced during the test. Therefore, it was concluded that when using the water-oil relative 131


permeability obtained from the simulation of the WI test, it is not possible to match the CWI experimental data by only tuning the mass transfer coefficient. Hence, both the mass transfer and relative permeability need to be tuned so that the model can predict the experimental data of CWI appropriately.

140 120

TOP(Scm3)

100 80 60 CWI-Exp CWI-Model(WI-Kr, MTC=1E-7) CWI-Model(WI-Kr, MTC=5E-7) CWI-Model(WI-Kr, MTC=10E-7)

40 20 0 0

1

2 Injected PV

3

4

Figure 4-14: Effect of the mass transfer coefficient (MTC) on prediction of TOP data by the developed simulator, CWI test.

132


3.5 3

DP(psi)

2.5 2 1.5 CWI-Exp CWI-Model(WI-Kr, MTC=1E-7) CWI-Model(WI-Kr, MTC=5E-7) CWI-Model(WI-Kr, MTC=10E-7)

1 0.5 0 0

1

2 Injected PV

3

4

Figure 4-15: Effect of the mass transfer coefficient (MTC) on prediction of DP data by the developed simulator, CWI test.

To tune the relative permeability curve, it worth discovering the effect of Corey parameters values on the simulation results. Therefore, a sensitivity analysis on Corey type relative permeability curve was performed as described below.

4-3.1.2.

Relative Permeability Sensitivity Analysis

At this stage, and before continuing the simulation of CWI, a series of sensitivities were performed by individually altering the Corey coefficients of the relative permeability curves obtained for the WI test (WI-Kr), defined as the base case. It should be noted that during this sensitivity analysis task, the Corey parameters were individually changed by only around ±25. This value was chosen in order to have a different curve but not very far from the base curve, as it is not expected there would be a big relative permeability change for the same core in different experiments. It also helps to understand how the accuracy of the Corey parameters would affect the results of simulations.

It worth mentioning that E100 model developed for the simulation of WI test was used during this sensitivity analysis. The impact of each variation on TOP and DP data was 133


studied. This task was intended to help to obtain manually the unknown relative permeability curve during the simulation of the CWI process, if it is not the same as the relative permeability curve of the WI. That is, this exercise aimed to identify the important Corey parameters that need to be altered if the CWI production data are to be history matched by tuning the relative permeability curve.

nw Sensitivity

In this exercise, the base value of nw (the water power law exponent of the Corey relative permeability function) was changed by ±25%. Figure 4-16 and Figure 4-17 show the resultant impact on TOP and DP data.

120

100

TOP(Scm3)

80

60

40 Model-WI-Kr(nw=Base) Model-WI-Kr(nw=1.25 Base) Model-WI-Kr(nw=0.75 Base)

20

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-16: Effect of nw Corey relative permeability exponent on prediction of TOP data.

134


3.5 3

DP(psi)

2.5 2 1.5 Model-WI-Kr(nw=Base) 1

Model-WI-Kr(nw=1.25 Base) Model-WI-Kr(nw=0.75 Base)

0.5 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-17: Effect of nw Corey relative permeability exponent on prediction of DP data.

The data in these figures show that the predicted TOP and DP data are not sensitive to the value of nw, in this range of variation. It should be noted that increasing the value of nw results in lower water relative permeability, while decreasing it, gives rise to a higher water relative permeability. Figure 4-18 demonstrates the impact on Krw of a 25% change in the base value of the nw exponent.

135


1

0.1

0.9

0.09 Kro-Base Krw(nw=Base) Krw(nw=1.25 Base) Krw(nw=0.75 Base)

Kro

0.7

0.08 0.07

0.6

0.06

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.8

0 0

0.1

0.2

0.3 sw

0.4

0.5

0.6

Figure 4-18: Impact of 25% change in the base value of the nw Corey relative permeability exponent on Krw.

no Sensitivity

In this exercise, the base value of no (the oil power law exponent of the Corey relative permeability function) was changed by ±25%. Figure 4-19 and Figure 4-20 show the resultant impact on TOP and DP data. It is noted that changing the value of no alters both predicted TOP and DP data to some extent, albeit with a more pronounced effect on the DP data. Figure 4-20 shows that the effect of no on DP data is not linear: that is, increasing the base value of no by 25% causes a more pronounced DP change compared to decreasing it by 25 %.

136


120

100

TOP(Scm3)

80

60 Model-WI-Kr(no=Base) 40

Model-WI-Kr(no=1.25 Base) Model-WI-Kr(no=0.75 Base)

20

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-19: Effect of no Corey relative permeability exponent on prediction of TOP data.

4 3.5 3

DP(psi)

2.5 2 1.5 1

Model-WI-Kr(no=Base) Model-WI-Kr(no=1.25 Base)

0.5

Model-WI-Kr(no=0.75 Base) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-20: Effect of no Corey relative permeability exponent on prediction of DP data.

137


The reason for that can be explained by looking at Figure 4-20, which demonstrates the impact of 25% change in the base value of the no exponent on Kro. It can be seen that increasing the value of no results in lower oil relative permeability and decreasing it generates a higher oil relative permeability. However the magnitudes of the resultant relative permeability curves do not relate linearly to the magnitude of the base relative permeability curve. It should be noted that Krw was constant during this experiment. 1

0.1

0.9

0.09 Kro(no=Base) Kro(no=1.25 Base) Kro(no=0.75 Base) Krw-Base

Kro

0.7

0.08 0.07

0.6

0.06

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.8

0 0

0.2

0.4

0.6

sw

Figure 4-21: Impact of 25% change in the base value of the no Corey relative permeability exponent on Kro.

kwmax Sensitivity

In this exercise, the base value of kwmax (maximum water relative permeability at residual oil saturation) was changed by ±25%. Figure 4-22 and Figure 4-23 show the resultant impact on TOP and DP data.

138


120

100

TOP(Scm3)

80

60

40 Model-WI-Kr(kwmax=Base) 20

Model-WI-Kr(kwmax=1.25 Base) Model-WI-Kr(kwmax=0.75 Base)

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-22: Effect of kwmax Corey relative permeability endpoint on prediction of TOP data.

4.5 4 3.5

DP(psi)

3 2.5 2 1.5 Model-WI-Kr(kwmax=Base) 1

Model-WI-Kr(kwmax=1.25 Base)

0.5

Model-WI-Kr(kwmax=0.75 Base)

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-23: Effect of kwmax Corey relative permeability endpoint on prediction of DP data.

139


It is noted that kwmax affects the predicted DP data significantly, but has no noticeable impact on the predicted TOP data. It should also be noted that increasing kwmax results in higher water relative permeability and decreasing it produces a lower water relative permeability. Figure 4-24 demonstrates the impact on Krw of a 25% change in the base value of the kwmax Corey relative permeability endpoint. 1

0.1

0.9

0.09 Kro-Base Krw(kwmax=Base) Krw(kwmax=1.25 Base) Krw(kwmax=0.75 Base)

0.8 0.7

0.08 0.07

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.06

Kro

0.6

0 0

0.1

0.2

0.3 sw

0.4

0.5

0.6

Figure 4-24: Impact of 25% change in the base value of the kwmax Corey relative permeability endpoint on Krw.

komax Sensitivity

In this exercise, the base value of komax (maximum oil relative permeability at connate water saturation) was changed. Since the komax value in the base model was one, its value was decreased to 0.75 and 0.5. Figure 4-25 and Figure 4-26 show the resultant impact on TOP and DP data.

140


120

100

TOP(Scm3)

80

60

40 Model-WI-Kr(komax=Base) Model-WI-Kr(komax=0.75 Base)

20

Model-WI-Kr(komax=0.5 Base) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-25: Effect of komax Corey relative permeability endpoint on prediction of TOP data.

4 3.5 3

DP(psi)

2.5 2 1.5 1

Model-WI-Kr(komax=Base) Model-WI-Kr(komax=0.75 Base)

0.5

Model-WI-Kr(komax=0.5 Base) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-26: Effect of komax Corey relative permeability endpoint on prediction of DP data.

141


It is noted that komax variation has no effect on the TOP and DP data. It should be noted that increasing the komax results in higher oil relative permeability and decreasing it gives rise to a lower oil relative permeability. Figure 4-27 demonstrates the impact of the change in the base value of the komax Corey relative permeability endpoint on Kro. 1

0.1 Kro(komax=Base) Kro(komax=0.75 Base) Kro(komax=0.5 Base) Krw-Base

0.8

0.09 0.08 0.07

0.6

0.06

Kro

0.7

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.9

0 0

0.1

0.2

0.3 sw

0.4

0.5

0.6

Figure 4-27: Impact of change in the base value of the komax Corey relative permeability endpoint on Kro.

sor Sensitivity

In this exercise, the base value of sor (residual oil saturation) was changed by ±25%. Figure 4-28 and Figure 4-29 show the resultant impact on TOP and DP data. It is noted that changing sor affects the predicted TOP data significantly, whilst its impact on the predicted DP data is small.

142


140 120

TOP(cm3)

100 80 60 40

Model-WI-Kr(sor=Base) Model-WI-Kr(sor=1.25 Base)

20

Model-WI-Kr(sor=0.75 Base) 0 0

1

2 Injected PV

3

4

Figure 4-28: Effect of sor Corey relative permeability saturation endpoint on prediction of TOP data.

3.5 3

DP(psi)

2.5 2 1.5 1 Model-WI-Kr(sor=Base) Model-WI-Kr(sor=1.25 Base) Model-WI-Kr(sor=0.75 Base)

0.5 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-29: Effect of sor Corey relative permeability saturation endpoint on prediction of DP data.

143


It should be noted that increasing sor results in a higher amount of oil being trapped within the core, while decreasing it reduces the amount of trapped oil within the core. Figure 4-30 demonstrates the impact on Kro of a 25% change in the base value of sor. 1

0.1

0.9

0.09 Kro(sor=Base) Kro(sor=1.25 Base) Kro(sor=0.75 Base) Krw-Base

0.8 0.7

0.08 0.07

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.06

Kro

0.6

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sw

Figure 4-30: Impact of 25% change in the base value of the sor relative permeability saturation endpoint on Kro.

swc Sensitivity

In this exercise, the base value of swc (connate water saturation) was changed. Since the swc value in the base model was zero, in this study its value was increased to 0.25 and 0.4. Figure 4-31 and Figure 4-32 show the resultant impact on the TOP and DP data. It is observed that changing swc has minimal impact on the predicted TOP data and a small impact on the predicted DP data. It should be noted that swc reflects the minimum water saturation above which water is a mobile phase. Figure 4-33 demonstrates Krw at these different swc values.

144


120

100

TOP(Scm3)

80

60

40 Model-WI-Kr(swc=Base=0) Model-WI-Kr(swc=0.25)

20

Model-WI-Kr(swc=0.4) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-31: Effect of swc Corey relative permeability saturation endpoint on prediction of TOP data.

3.5 3

DP(psi)

2.5 2 1.5 1

Model-WI-Kr(swc=Base=0) Model-WI-Kr(swc=0.25)

0.5

Model-WI-Kr(swc=0.4) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-32: Effect of swc Corey relative permeability saturation endpoint on prediction of DP data.

145


1

0.1

0.9

0.09 Kro-Base Krw(swc=Base=0) Krw(swc=0.4) Krw(swc=0.25 )

0.8 0.7

0.08 0.07

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0

Krw

0.06

Kro

0.6

0 0

0.1

0.2

0.3 sw

0.4

0.5

0.6

Figure 4-33: Impact of changes in the base value of the swc Corey relative permeability saturation endpoint on Krw.

Summary of the Sensitivities

Figure 4-34 and Figure 4-35 are the spider plots, which show the impact of Corey relative permeability parameters on the predicted TOP and DP data, respectively. It is noted that the TOP data are very sensitive to the sor value, whilst DP data are very sensitive to the kwmax value. Moreover, the no value affects both TOP and DP data slightly. Table 4-8 summarises the sensitivity of all Corey relative permeability coefficients related to TOP and DP data, based on the observations presented in the previous sections and summarised in Figure 4-34 and Figure 4-35.

146


1.15

1.1

nw

1.05

no kwmax

1 0.75

0.85

0.95

1.05

1.15

1.25

komax sor

0.95

swc

0.9

0.85 Figure 4-34: Spider plot for TOP.

1.3

1.2 nw no 1.1

kwmax komax swc

1 0.75

0.85

0.95

1.05

1.15

0.9

0.8 Figure 4-35: Spider plot for DP.

147

1.25

sor


Table 4-8 : Summary of the sensitivity of TOP and DP data to the Corey relative permeability coefficients.

TOP DP

nw Not sensitive Not sensitive

no Slightly sensitive Slightly sensitive

kwmax Not sensitive Strongly sensitive

komax Not sensitive Not sensitive

sor Strongly sensitive Not sensitive

swc Not sensitive Not sensitive

Now, at this stage, the results of this sensitivity analysis can be used to tune the relative permeability curve together with the MTC value during the compositional simulation of the CWI as described below. It is worth mentioning that based on this sensitivity analysis, the RF and DP are insensitive to some Cory parameters, so it is possible to have different combinations of Kr curves which may match the production data due to these insensitive parameters.

4-3.1.3.

Tuning of MTC and Relative Permeability Curve

To history match the production data of the CWI coreflood experiment, as mentioned above, MTC and relative permeability values need to be tuned. That is, the results of sensitivity of production data to MTC values for this set of coreflood experiment were presented and it was seen that the production data of CWI cannot be predicted only by activating the mass transfer term in the equations and using WI-Kr obtained from the simulation of the WI experiment. In the other words, the role and contribution of the mass transfer term in the equations is such that it cannot capture the all mechanisms happing during the CWI process. The mass transfer term contributes mainly towards the oil swelling because it adds some mass to the oil phase, which, in turn, increases the oil volume, resulting in the swelling of the oil.

To find about the additional mechanisms contributing during CWI, it is worth comparing the DP data of CWI and WI tests (Figure 4-4). It is noted that, during CWI, the DP values reduce. This can be interpreted by suggesting that in mixed-wet (aged) cores, during CWI process, some part of the carbonated water possibly enters into the oil-wet pores which are occupied by the oil. The surfaces of these pores are wetted by the oil components and a layer of oil film adheres to the wall. Carbonated water can possibly extract and wash 148


away some part of this oil, i.e. the oil layer which is stuck to the surface of the pores (thus leading to a reduction of residual oil). This results in more oil recovery, as well as a reduction in the DP values. If this happens, the DP values will decrease as there is a larger area available for the water to pass through the pores (i.e. water mobility improvement). This mechanism, which is not seen in water-wet cores, can be related to the wettability modification (or alteration) of the rock surface, which allows the oil layer to be separated from the surface of the pores and be produced during the CWI process.

To incorporate this mechanism into the simulation, the relative permeability curve obtained from the WI simulation (WI-Kr) should be modified. That is, for this mixed-wet core, the relative permeability curve for the CWI test is not same as that for the WI experiment. Moreover, to correct and modify the WI-Kr relative permeability curve to capture the effect of wettability modification, it is important that the oil swelling effect is not captured through the relative permeability curve alone in order to have a generic model. That is, oil swelling should be captured by mass transfer term. Therefore, the main concern and aim at this stage is to quantify and discriminate the role and contribution of the oil swelling mechanism and wettability modification in CWI performance.

The oil swelling is captured by tuning the MTC, as has been shown for the water-wet cores. It should be noted that, it is difficult to estimate the oil swelling in this mixed-wet coreflood experiment explicitly, as wettability is also changed in this system. Therefore, to quantify the oil swelling here, it can be assumed that the same oil swelling as that of the water-wet core is also achieved for the mixed-wet core (i.e. 15%). That is, the MTC value obtained for the water-wet cores can also be used for this mixed-wet core (i.e. MTC=5E-7 1/sec).

Figure 4-36 and Figure 4-37 display the TOP and DP values predicted by the developed simulator compared to experimental values using the above-mentioned MTC and WI-Kr. It can be seen that only the endpoint of the TOP data is matched by this MTC value.

149


140 120

TOP(cm3)

100 80 60 40 CWI-Exp

20

CWI-Model(WI-Kr, MTC=5E-7) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-36: TOP data predicted by the developed simulator compared to experimental values, MTC=5E-7 1/sec, CWI test.

3.5 3

DP(psi)

2.5 2 1.5 1 CWI-Exp 0.5

CWI-Model(WI-Kr, MTC=5E-7)

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-37: DP data predicted by the developed simulator compared to experimental values, MTC=5E-7 1/sec, CWI test.

150


As explained before, oil swelling mainly contributes in oil production after the breakthrough point which is captured by the MTC value in the model. In water-wet coreflood experiments, as shown in Chapter 3, 9% additional oil recovery was obtained by CWI at end of the experiment over the oil recovery obtained at the breakthrough point. However, in the mixed-wet core, the additional oil recovery was 4%. Therefore, it seems that the importance of the oil swelling mechanism and the magnitude of the MTC in the mixed-wet core is not exactly the same as in water-wet core. The MTC value for the mixed-wet coreflood experiment can be possibly estimated to be 2.2E-7 1/sec, using 9% and 4% additional oil recovery obtained over the breakthrough point during the water4

wet and mixed-wet coreflood experiments respectively (i.e. 2.2E-7= ×5E-7). Moreover, 9

it should be considered that in the water-wet coreflood experiment, total pore volume of injected carbonated water was 4.1 while in the mixed-wet coreflood experiment it was 3.3. Therefore, the amount of mass transferred and resultant oil swelling in the mixedwet coreflood experiment should be lower than in the water-wet coreflood experiment (i.e. 15%). The oil swelling here can be possibly estimated to be 12%, using 4.1 and 3.3 total injected PV during water-wet and mixed-wet coreflood experiments respectively 3.3

(i.e. 12%= 4.1 ×15%). It should be noted that the suggested procedure for the estimation of MTC and swelling in this mixed-wet core, based on the data of the water-wet core and using the linear relations, is only a rough estimation and it could be checked if more experimental data were available. To match the TOP and DP data, first the residual oil saturation is adjusted to capture the swelling mechanism. Using the experimental data and based on the material balance, the calculated residual oil saturation (sor) for WI and CWI tests is 0.41 and 0.32, respectively. The 0.32 value is for dead oil saturation, with no CO2 content and hence, it is expected that the actual sor would be higher because of its CO2 content. The estimated oil swelling in this test is 12%. Therefore the swelled residual oil saturation is estimated to be 36% (i.e. 36% = 32% × 1.12). Figure 4-38 and Figure 4-39 show, respectively, the TOP and DP data when the residual oil saturation in WI-Kr is reduced from 41% to 36% and the MTC is set to 2.2E-7 1/sec. It can be seen that, at this stage, the TOP data is closer to the experimental values, while predicted DP data are still far away from the experimental data.

151


120

TOP(Scm3)

100 80 60 40 CWI-Exp CWI-Model(WI-Kr, MTC=2.2E-7)

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

. Figure 4-38: TOP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; WI-Kr (sor=0.36), CWI test.

3.5 3

DP(psi)

2.5 2

1.5 1 CWI-Exp

0.5

CWI-Model(WI-Kr, MTC=2.2E-7) 0 0

0.5

1

1.5

2

2.5

3

3.5

Injected PV Figure 4-39: DP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; WI-Kr (sor=0.36), CWI test.

152


It seems that the rest of data points on TOP and DP curves need to be matched by tuning the relative permeability curve to capture the wettability alteration effect. Before continuing, it is good to know the effect of wettability on Kr curves. As been mentioned in the literature, wettability influences the shape of relative permeabilities curves and it has been reported that when the wettability of rock is considered to be more water-wet, the intersection of the oil and water relative permeabilities shifts to the right, and also the maximum Krw (i.e. kwmax) decreases (Owens and Archer 1971, Morrow et al. 1973, McCaffery and Bennion 1974). However it has also been mentioned that in addition to wettability, the shape of relative permeabilities curves also depend on connate water saturations and pore-size distribution and interpretation of wettability from relative permeability curve is likely to have large error (Treiber and Owens 1972).

The Kr curve can be tuned manually as well as automatically using the GA-based optimiser. To manually tune the Kr curve, the results from the sensitivity analysis exercise were considered here. That is, it was observed that the TOP data are noticeably sensitive to the sor parameter and slightly to the no exponent, while DP data are mainly sensitive to the kwamx coefficient and slightly to the no exponent. Therefore, first the TOP data were matched. To do that, the rest of the TOP data points were history matched by tuning the no Corey component and after a few trials, the no Corey component from the water injection test was reduced by 25% (i.e. no=1.7). It should be noted that a lower no value means better oil mobility, as was shown during the sensitivity analysis on the Corey parameters. Later, the DP data were history matched manually and, after a few trials, kwmax from the water injection test was increased by 36% (i.e. kwmax=0.101). It should be noted that a higher value of kwmax means higher water mobility and it only affects the DP data, as shown during the sensitivity analysis on the Corey parameters. Higher kwmax was obtained because based on the hypothesis explained before; more area is available for water to move through due to some oil extraction from the rock surface by CO 2. Moreover, it should be emphasised that this kwmax value was also obtained by a manual calculation, in which the Darcy equation and the endpoint value on the DP curve, together with the core properties were employed. Figure 4-40 and Figure 4-41 respectively display the TOP and DP data predicted by the developed simulator after this systematic history matching step (tuning both MTC and Kr).

153


140 120

TOP(Scm3)

100 80 60 40 CWI-Exp

20

CWI-Model(CWI-Kr, MTC=2.2E-7) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-40: TOP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; CWI-Kr (sor=0.36, no=1.7, kwmax=0.101), CWI test.

2.5

2

DP(psi)

1.5

1

0.5 CWI-Exp CWI-Model(CWI-Kr, MTC=2.2E-7) 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-41: DP data predicted by the developed simulator compared to experimental values, MTC=2.2E-7 1/sec; CWI-Kr (sor=0.36, no=1.7, kwmax=0.101), CWI test.

154


It should be noted that the total misfit obtained based on this manually history matching step is 1.42.

In the second approach, the optimiser was used to estimate the optimised values of MTC, kwmax, no and sor automatically. The initial uncertainty range and GA setup were the same as those applied for water-wet core explained before. It should be noted that in this exercise, the rest of the Corey parameters were the same as those of the WI test. Table 4-9 compares CWI-Kr and MTC values obtained by the manual tuning and automatic tuning using the GA-based optimiser. It can be seen that the values obtained are almost the same.

Table 4-9: Comparison of optimal CWI-Kr and MTC obtained by manual tuning and automatic tuning using optimiser; compositional simulation, CWI test. Kr & MTC

Method

nw

no

kwmax

komax

sor

swc

MTC

CWI-Kr

Using optimiser

2.5

2

0.103

1

0.37

0.0

3E-7

CWI-Kr

Using WI-Kr as base and manual tuning

2.5

1.7

0.101

1

0.36

0.0

2.2E-7

Figure 4-42 and Figure 4-43 show TOP and DP data respectively predicted by the developed simulator using the relative permeability and mass transfer coefficient obtained by automatic history matching as well as manual tuning, compared to the experimental values.

Figure 4-42 shows that using CWI-Kr and MTC obtained by manual tuning, the developed simulator gives a closer match to the experimental data in comparison with using those values obtained by the optimiser.

155


140 120

TOP(Scm3)

100 80 60 40 CWI-Exp CWI-Model(CWI-Kr, MTC=2.2E-7, Manual Tuning) CWI-Model(CWI-Kr, MTC=3E-7, GA)

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-42: TOP data predicted by the developed simulator in its compositional mode compared to experimental values; CWI-Kr & MTC obtained by manual tuning and GA-based optimiser, CWI test.

2.5

2

DP(psi)

1.5

1

CWI-Exp CWI-Model(CWI-Kr, MTC=2.2E-7, Manual Tuning) CWI-Model(CWI-Kr, MTC=3E-7, GA)

0.5

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-43: DP data predicted by the developed simulator in its compositional mode compared to experimental values; CWI-Kr & MTC obtained by manual tuning and GA-based optimiser, CWI test. 156


Moreover, Figure 4-43 shows that the DP data predicted by the developed simulator is not completely matched to the experimental values, either when the input data (i.e. CWIKr and MTC) to the simulator are those obtained manually or automatically. However, the predictions of both cases are very close to the experimental values, with acceptable accuracy. The misfit values are shown in Table 4-10.

Table 4-10: Misfit valuse for manual tumimg and automtic histroy matching.

Manual tuning Automatic history matching (GA)

TOP misfit DP misfit Total misfit 0.4 1.02 1.42 0.6 1.05 1.65

It should be noted that during automatic history matching four parameters were optimised simultaneously while during manual tuning, parameters were tuned separately and step by step which can be the reason that the automatic history matching has a higher misfit compared to the misfit of manual tuning procedure. That is, the misfit values obtained by GA and manual tuning are not considerably different and data errors are large enough that both results are valid.

4-3.2.

Black-oil Simulation

At this stage, it is proposed to follow a black-oil approach to match the CWI production data with a relative permeability curve accounting for all changes as a result of CO2 mass transfer. However, it should be pointed out that this is not a general approach but is more specific for the case under study. In other words, the non-equilibrium mass transfer kinetic should also be considered as a part of this history matching exercise. It should be noted that the impact of variation of oil fluid properties (i.e. density and viscosity) as a result of CO2 transfer is minimal, because, at the test conditions, the decane properties are almost the same as CO2 properties. Hence, for the decane oil sample, a black-oil simulation approach is considered as well. To do that, a tuned relative permeability curve was obtained when the black-oil approach was used to simulate the CWI experiment.

To obtain a relative permeability curve that matches the CWI production data, it is proposed to alter the WI-relative permeability (krw-o) using the results of sensitivities 157


presented in the previous sections. For this purpose, it is worth looking at the experimental data (i.e. TOP and DP) again as shown in Figure 4-2 and Figure 4-4 before.

It is noted that the TOP of CWI is higher compared to that of WI, while its DP is lower. Based on the sensitivity results and noting that sor and kwmax have a noticeable effect on the simulation of the TOP or DP data respectively, it is proposed to concentrate on changing these two parameters only. Accordingly, and in a stepwise history matching process, it would be reasonable to change sor to match the TOP data, as it only affects these data but not the DP data. For matching DP, it is better to alter kwmax as it clearly affects the DP data, while it has no effect on the TOP data. Therefore, the base values of sor and kwmax were changed manually. It should be noted that for this manual exercise, the value of sor was estimated by material balance calculation using the CWI coreflood data. The kwmax value was obtained after a few manual iterations. All the Corey parameters of water-oil relative permeability are given in Table 4-11, for both WI and CWI coreflood tests.

Table 4-11: Optimal Corey coefficients of relative permeability obtained for WI and CWI coreflood tests, black-oil simulation. Experiment

Kr

Method

nw

CWI

CWI-Kr

Manual tuning using WI-Kr as base

WI

WI-Kr

Using GA linked to the developed simulator

no

kwmax

komax

sor

swc

2.5 2.25 0.107

1

0.32

0

2.5 2.25 0.074

1

0.41

0

Figure 4-44 and Figure 4-45 shows the corresponding match obtained by the developed simulator in its black-oil mode for the TOP and DP experimental data, using the corresponding manually-tuned CWI-relative permeability data given in Table 4-11.

158


140 120

TOP(Scm3)

100 80 60 40 CWI-Model CWI-Exp

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-44: Experimental TOP versus TOP data predicted by developed simulator (model) under the black-oil mode, using the CWI-relative permeability data (Table 4-11), CWI test.

2.5

2

DP(psi)

1.5

1 CWI-Model 0.5

CWI-Exp

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-45: Experimental DP versus DP data predicted by developed simulator (model) under the black-oil mode, using the CWI-relative permeability data (Table 4-11), CWI test.

159


Next, it was attempted to obtain black-oil based CWI-relative permeability curve automatically, using the GA-based optimiser, to see if its results are close to those obtained by the manual tuning. This can help with checking the uniqueness and accuracy of the optimised parameters if two different procedures can result in similar results. The developed simulator in its black-oil mode was linked to the optimiser. In this exercise, two sensitive unknown parameters (kwmax, sor) in the Corey correlation were optimised when the objective function was minimised. The initial ranges of uncertainty and the GA setup were the same as those used for obtaining WI-relative permeability curve (mentioned before). The obtained optimal values are given in Table 4-12 , compared to those obtained by the manual tuning. Table 4-12 shows that the optimal Corey relative permeability parameters that were automatically obtained by the optimiser are the same as those estimated by manual tuning.

Table 4-12: Optimal Corey coefficients of carbonated water-oil relative permeability obtained using two different methods, black-oil simulation, CWI test. Experiment

Kr

Method

nw

no

kwmax

komax

sor

swc

CWI

CWI-Kr

Using GA linked to the developed simulator

2.5 2.25 0.108

1

0.315 0.0

CWI

CWI-Kr

Manual tuning using water-oil relative permeability as base

2.5 2.25 0.107

1

0.32

0.0

In summary, it can be stated that, for this test, the CWI process could be simulated following a simple black-oil approach with only tuning of the relative permeability. However, as the actual physics, including the effect of existing CO2 in the system and mass transfer, has not been included, this is not a generic approach. For example, if the injection rate is changed, the contribution of the mass transfer will be changed (this will be shown later), and this cannot be captured using the same relative permeability curve. Therefore, a generic compositional approach when the effect of mass transfer is included must be used for the simulation of CWI, as described above.

4-4.

CO2 Production Profile

At this step, the developed simulator in its compositional mode is used to predict the total CO2 production (TCO2P) profile. The optimal MTC value of 2.2E-7 and the CWI-Kr 160


curve, as explained before, are used during this exercise. Figure 3-49 shows the total CO2 production (TCO2P) and total water production (TWP) profiles versus injected pore volume of carbonated water. It can be observed that the breakthrough points of water and CO2 have overlapped. That is, it can be deduced that water has some CO2 content when arrives at the core outlet and is not completely deprived of CO2.

To further clarify this, the simulator also predicts the total CO2 production from water (TCO2PW) and oil (TCO2PO) streams separately, as shown in Figure 3-50. It can be seen that the main amount of CO2 which is produced at the core outlet comes from the water stream. Moreover, the part of the CO2 in the water phase is produced quickly by the water stream, while the other part, in the oil phase, arrives at the core outlet later.

10000

600

9000 500

8000

400

6000 5000

300

4000

TWP(Scm3)

TCO2P(Scm3)

7000

200

3000 2000

100

TCO2P-Model TWP-Model

1000 0

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-46: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the developed simulator, CWI test.

161


10000

500 TCO2P-Model TCO2PW-Model TCO2PO-Model

8000

450 400

7000

350

6000

300

5000

250

4000

200

3000

150

2000

100

1000

50

0

TCO2PO(Scm3)

TCO2P(Scm3), TCO2PW(Scm3)

9000

0 0

1

2 Injected PV

3

4

Figure 4-47: Total CO2 production (TCO2P), total CO2 production from water stream (TCO2PW) and total CO2 production from oil stream (TCO2PO) versus injected pore volume of carbonated water predicted by the developed simulator, CWI test.

It can also be understood that the CO2 transfer from water into the oil phase does not happen quickly, because, when the water arrives at the core outlet, it has some CO2 content. On the other hand, the CO2 displacement through the oil phase seems to happen slowly, as the CO2 content of the oil arrives late at the core outlet. Figure 3-50 shows that CO2 breakthrough via the water stream happens after around 0.6 pore volume of injection, while it occurs after around 0.8 pore volume of injection via the oil stream.

To support these ideas, i.e. slow CO2 transfer from water into the oil phase and slow CO2 displacement through the oil phase, the simulator can be used to investigate the parameters which may affect the CO2 production profile.

The MTC, dispersion and injection rate values can influence the CO2 distribution and displacement in the system resulting in an impact on CO2 production profile as these parameters can affect the present time of CO2 in core. It has been shown that the MTC value controls the rate of CO2 transfer between the water and oil phases. In addition, the dispersion accelerates the CO2 displacement through the oil phase, which may affect the CO2 production profile. Furthermore, the injection rate affects the contact time between 162


oil and water phases in the system and accordingly may affect the CO2 production profile. To realise the roles of the mass transfer rate, dispersion and injection rate in the prediction of CO2 production profile by the developed simulator, a sensitivity analysis has been carried out, as described below.

4-4.1.

Effect of MTC on the CO2 Production Profile

It is expected that the MTC affects the CO2 production profile. This is due to the fact that higher MTC values gives rise to faster and more CO2 transfer to the oil phase and as it was observed before the CO2 displacement in the oil phase is slower than that in the water phase. The optimised MTC value of 2.2E-7 1/sec was obtained when the production data (i.e. the TOP and DP) of the coreflood CWI experiment was history matched. It was observed that using this MTC value (i.e. 2.2E-7 1/sec) results in having the same breakthrough time for water and CO2 production profiles. It is expected that if the MTC value is increased, the water gives a larger amount of its CO2 content to the oil and, as a result, it may be deprived of CO2 and therefore the pure water arrives at the outlet without having CO2 content. This will postpone the breakthrough time of CO2 compared to that of water. Figure 4-48 compares the water and CO2 production profiles for three different MTC values. Figure 4-48 shows that the MTC value does not affect the water production profile, as it is not changed by changing the mass transfer coefficient; however, MTC impacts the CO2 production profile as it is shifted to the right when MTC is increased. In other words, the water breakthrough time is not changed by the MTC, while CO2 breakthrough time is postponed by increasing the MTC. Moreover, a higher MTC value lowers the total produced CO2. This is due to the fact that a higher MTC value results in more CO2 being transferred into the oil phase and, as CO2 displacement in the oil phase is slower, CO2 arrives at the outlet with a delay. On the other hand, lower MTC values cause water to hold more of its CO2 content and, as a result, the water stream arrives at the outlet with a greater amount of CO2, leading the CO2 production profile to have a steeper slope for lower MTC values.

163


Figure 4-48: Total CO2 production (TCO2P) and total water production (TWP) versus injected pore volume of carbonated water predicted by the simulator; MTC=2.2E-7, 6.5E-7 and 11E-7 1/sec, CWI test.

4-4.2.

Effect of Dispersion Coefficient on the CO2 Production Profile

Movement and displacement of CO2 in the oil phase can be due to convection and/or dispersion. Displacement due to convection happens because the fluids carry their CO2 content within themselves when they travel through the core. Dispersion is a natural movement and mixing which happens in the porous media due to different flow paths being available for fluids flowing in the system. All the simulations presented so far, have been done without a dispersion term i.e. assuming zero dispersion coefficient. However, as explained above, dispersion can speed up the CO2 displacement through the oil phase which, in turn, affects the CO2 production profile. To reveal the influence of dispersion on CO2 production data, a sensitivity analysis was done. The dispersion coefficient is not available for the experiments under study here. However, an approximated value, as explained in Chapter 2, is used here. Figure 4-49 shows CO2 production profile for three different dispersion coefficient values (i.e. D=0, 950E-4 (from Chapter 2) and 2000E-4 cm2/sec). 164


10000 TCO2P-Model, D=0, MTC=2.2E-7 TCO2P-Model, D=950E-4, MTC=2.2E-7 TCO2P-Model, D=2000E-4, MTC=2.2E-7

9000 8000

TCO2P(Scm3)

7000 6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-49: Total CO2 production (TCO2P) predicted by the simulator for three different dispersion coefficient values; D=0, 950E-4 and 2000E-4 cm2/sec, CWI test.

It can be seen that increasing the dispersion coefficient results in a steeper CO2 production profile, as the oil stream will carry its CO2 content faster. However, the results seen here is similar to that for the water-wet core (Figure 3-46) and the effect of dispersion is not considerable. This is due to the fact that, at this specific injection rate, the convection effect is more pronounced in comparison with the dispersion and, as was shown in Chapter 3 for the water-wet core; at lower injection rates the dispersion has more pronounced effect. 4-4.3.

Effect of Injection Rate on the CO2 Production Profile

The CO2 production profile can also be affected by the injection rate because the contact time for water and oil in the core is different for different injection rates, which, in turn, affects the amount of CO2 transferred between the water and oil phases. To check the influence of the injection rate and fluid velocity on the CO2 production profile, a sensitivity analysis was performed. To do that, the injection rate was reduced from 20 to 10 and 5 cm3/hr. This is based on the fact that decreasing the injection rate reduces the 165


velocity of the fluids flowing in the system. Figure 4-50 compares the CO2 production profile at different injection rates.

11000 TCO2P-Model, Injection Rate=20 TCO2P-Model, Injection Rate=10 TCO2P-Model, Injection Rate=5

10000 9000

TCO2P(Scm3)

8000 7000 6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-50: Total CO2 production predicted by the simulator for injection rates of 5, 10 and 20 cm3/hr, CWI test.

It can be seen that when the velocity is reduced the CO2 breakthrough time is delayed, and also the total amount of CO2 at the same injected pore volume is lower (comparing the red, green and dark blue lines). This happens because, at low injection rates, the CO2 has more time to be transferred from the water into the oil phase. In the other words, at low injection rates, oil and water are in contact for longer time resulting in a larger amount of CO2 being transferred into the oil phase. Consequently, the oil phase receives more CO2 at low injection rates. On the other hand, as mentioned before, due to slow CO 2 displacement in the oil phase by the convection mechanism, at low injection rates, CO 2 arrives at the core outlet with a delay, compared to the time of arrival at high injection rates. It should be noted that the length of the simulation run at an injection rate of 20 cm3/hr was 30 hours, whereas in order to inject the same pore volume of injected fluid, the length of the simulation runs for the injection rates of 10 and 5 cm3/hr were increased to 60 and 120 hours, respectively. 166


It is worth investigating the oil production profile as well. To do this, the effect of dispersion, the injection rate and the carbonation level on the oil production profile are studied below.

4-5. 4-5.1.

Oil Production Profile Effect of Dispersion Coefficient on the Oil Production Profile

The effect of dispersion on oil production is investigated here. CO2 displacement in the oil phase can be accelerated by dispersion. Consequently, it is expected that the oil production profile can be influenced by dispersion as well. Figure 4-51 displays the effect of the dispersion coefficient on the oil production profile.

130 120 110 100

TOP(Scm3)

90 80 70 60 50 40 30

TOP-Model, D=0, MTC=2.2E-7 TOP-Model, D=950E-4, MTC=2.2E-7 TOP-Model, D=2000E-4, MTC=2.2E-7

20 10 0 0

1

2 Injected PV

3

4

Figure 4-51: Total oil production (TOP) predicted by the simulator for three different dispersion coefficient values, D=0, 171E-4 and 1000E-4 cm2/sec, MTC=2.2E-7 1/sec, CWI test.

Figure 4-51 shows that when the dispersion is activated in the model, the oil production is slightly reduced. This is due to the fact that when the dispersion is activated, the CO 2 content of the oil moves faster and therefore exits the system earlier. That is, dispersion helps the movement of CO2 within the oil phase leading to more CO2 effluent from the 167


oil stream at core outlet. On the other hand, dispersion makes CO2 distribution more uniform through the oil phase. Figure 4-52 compares CO2 concentration in the oil phase through the core, with and without the dispersion effect. It can clearly be seen that the CO2 concentration profile is not very steep when, in addition to convection; CO2 is moved forward by the dispersion mechanism as well. That is, CO2 is distributed through the oil phase more uniformly when the dispersion is also activated in the model. Figure 4-52 also shows that after 0.3 and 0.5 injected PV, the oil stream at the core outlet does not have any CO2 content if the dispersion is not activated (dashed red and dark blue lines); however, when dispersion plays a role in the system, the CO2 concentration of the oil stream at the end of the core is above zero (solid red and dark blue lines). Moreover, comparing the solid and dashed red lines, it can be observed that, after 0.3 injected PV, the CO2 front in the oil phase is in the middle of the core (around 32 cm away from the core inlet) without dispersion, while it has reached the core outlet when the dispersion is activated. This shows the faster movement of CO2 in presence of the dispersion. Moreover, Figure 4-52 also indicates that the oil receives more CO2 in the course of time regardless of dispersion mechanism. This is seen by comparing the red curves with the corresponding dark blue curves plotted in Figure 4-52. Although the dispersion helps the CO2 to contact a larger proportion of the oil as it passes through the core, it also causes the CO2 to leave oil faster and, as a result, the CO2 content of the oil is reduced. Moreover, considering that CO2 does not considerably change the fluid properties of decane, the ultimate impact of dispersion is a slight reduction of total produced oil at high values of dispersion, as shown in Figure 4-51. It should be noted that, as was explained for the water-wet coreflood experiment, CO2 can also diffuse in transverse direction to the flow by the dispersion mechanism and thus recovers some oil which would be bypassed by the flowing carbonated water stream. If this happens, the effect of dispersion might be a higher oil recovery factor. However this cannot be captured here, as the model is onedimensional.

168



0.035 0.3 Injected PV, D=950E-4 0.5 Injected PV, D=950E-4 0.3 Injected PV, D=0 0.5 Injected PV, D=0

0.03 0.025 0.02 0.015 0.01 0.005 0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

Distance(cm) Figure 4-52: Profile of CO2 concentration in oil phase predicted by the simulator for D=0 and 950E-4 cm2/sec after 0.3 and 0.5 injected PV, CWI test.

4-5.2.

Effect of Injection Rate on the Oil Production Profile

The injection rate affects the CO2 distribution between the oil and water phases. Hence, it is expected that the injection rate influences the oil production profile as well. As explained above, when the injection rate is reduced, the velocities of the oil and water phases are reduced in the system and fluids are flowing through the core more slowly. Therefore, the oil and water phases have more time to be in contact and exchange more CO2 before leaving the core. As a result, it is expected that more CO2 can be transferred from the water into the oil phase. A sensitivity analysis was carried out in which the injection rate was reduced from 20 to 10 and 5 cm3/hr. Figure 4-53 presents the results of this and it can be seen that when the injection rate is decreased, the recovery factor (RF) is increased. This can be explained by the fact that at low injection rates, more CO 2 can be transferred into the oil phase and consequently the contribution of the oil swelling in oil production can be boosted. This, in turn, improves the mobility of oil and results in higher oil recovery. 169

RF(%)


90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Model, Injection Rate=20 Model, Injection Rate=10 Model, Injection Rate=5 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-53: RF predicted by the simulator for three different injection rates of 20, 10 and 5 cm3/hr, CWI test.

4-5.3.

Effect of Carbonation Level on the Oil Production Profile

It is expected that the amount of dissolved CO2 in carbonated water affects the performance of CWI. The amount of existing CO2 plays an important role during CWI, as the presence of CO2 leads to the superiority of the CWI over the WI. The carbonation level (CL) refers to the amount of CO2 which is dissolved in water. During the CWI experiment, the amount of dissolved CO2 in the injected stream was 5 weight percent (i.e 5 wt % carbonation level). It is obvious that if the solubility of CO2 in the injected water is decreased or increased, it will influence the performance of CWI. It is expected that injection of carbonate water with higher carbonation levels will result in higher oil recovery. Figure 4-54 shows the profiles of RF for three different carbonation levels of 7, 5 and 3 wt %, predicted by the developed simulator.

170


80 70 60

RF(%)

50 40 30 20 Model,CL=3 wt % Model,CL=5 wt %

10

Model,CL=7 wt % 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-54: RF profile for three different carbonation levels of 7, 5 and 3 wt %, predicted by the developed simulator, MTC= 2.2E-7 1/sec, CWI test.

It demonstrates that a higher carbonation level of the injected water results in higher oil recovery. This is due to the fact that, when the injected water has higher CO2 content, more CO2 is available to be transferred into the oil phase, which, in turn, results in better oil mobility, thus resulting in higher oil recovery. It should be noted that having a higher amount of CO2 in the system might also result in a further wettability change in the system. However this cannot be shown by the simulator and some experimental data need to be available.

4-6.

CO2 Storage

The developed simulator can also be used to estimate the amount of stored CO2 in the core at the end of experiment. During carbonated water injection into the oil reservoir, when water is moving forward through the core, CO2 migrates from the water into the oil phase. As the CO2 content of the injected water is low, during CWI, no free CO2 phase will be in the system as all the existing CO2 is dissolved in the water and oil phases. This leads to less risk of CO2 leakage from a permeable cap rock, considering the dissolved 171


CO2 is less mobile and more stable. When injection is stopped, some proportion of the CO2 stays in the reservoir while the rest of it has been produced by the oil and water streams. Figure 4-55 shows the profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S), as predicted by the simulator. Figure 4-55 shows that the profiles of injected and stored CO2 overlap, with an increasing trend until the breakthrough point of CO2, revealing that all the injected CO2 is stored inside the core. However, when CO2 starts to be produced at the core outlet, the stored CO2 curve also declines gradually. It is expected that the stored CO2 curve becomes flat after a time period when the decane has received its equilibrium level of CO2 concentration and no more CO2 transfer between the phases happens. At that point, all the injected CO2 will be produced at the core outlet. TCO2S

Figure 4-56 shows the total stored CO2 as a percentage of injected CO2 ( TCO2I × 100). It can be seen that at the end of the experiment, approximately 44% of the injected CO2 is stored in the core.

16000

TCO2I-Model TCO2S-Model

14000

TCO2P-Model

TCO2(Scm3)

12000 10000 8000 6000 4000 2000 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-55: Profiles of total amount of CO2 produced (TCO2P), injected (TCO2I) and stored (TCO2S) versus injected PV of carbonated water, predicted by the simulator, CWI test.

172


100 90

TCO2S/TCO2I (%)

80 70 60 50 40 30 20 10 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-56: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator, CWI test.

4-6.1.

Effect of MTC on the CO2 Storage

It is expected that the MTC will affect the amount of CO2 stored in the core. This is based on the fact that a higher MTC causes more CO2 transfer into the oil and, considering that the CO2 displacement in oil is lower compared to that in water, CO2 can be stored better in the oil phase. Figure 4-57 clearly shows that the total stored CO2 as a percentage of TCO2S

injected CO2 ( TCO2I × 100) is higher for higher MTC values. Moreover, at the beginning all the injected CO2 is stored; however, after the CO2 breakthrough point, CO2 storage declines with a steeper slope for lower MTC values, as shown in Figure 4-57. It should be noted that this is true only if the system has not reached the equilibrium state. That is, if at a specific MTC value the system can reach the equilibrium state, a further increase in the MTC will not affect the storage process.

173


100 90

TCO2S/TCO2I (%)

80 70 60 50 40 30 MTC=2.2E-7 MTC=6.5E-7 MTC=11E-7

20 10 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-57: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator for MTC values of 2.2E-7, 6.5E-7 and 11E-7 1/sec, CWI test.

4-6.2.

Effect of Injection Rate on the CO2 Storage

It was explained that the injection rate affects the CO2 distribution between water and oil phases and that at low injection rates, more CO2 is transferred into the oil phase and thus, CO2 production is postponed, compared to that at high injection rates, resulting in higher oil production at low injection rates. Therefore, it is expected that the injection rate also affects the CO2 storage profile. A sensitivity analysis was performed and Figure 4-58 TCO2S

clearly shows that the total stored CO2 as a percentage of injected CO2 ( TCO2I × 100) is higher for lower injection rates.

174


100 90

TCO2S/TCO2I (%)

80 70 60 50 40 30 Injection Rate=20 20

Injection Rate=10

10

Injection Rate=5

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-58: (TCO2S/TCO2I ×100) versus injected PV of carbonated water predicted by the simulator for injection rates of 20, 10 and 5 cm3/hr, CWI test.

4-7.

Compositional Simulation of CWI Using ECLIPSE300 (E300)

At this stage, CWI was simulated using the E300 compositional simulator. The goal was to compare the results from the developed simulator which is based on the nonequilibrium assumption with those from the E300, which works based on the instantaneous equilibrium assumption. An ECLIPSE model similar to that created for the water-wet system was also created here for this mixed-wet system. The same input data as those for the developed simulator are used in the E300, including the same number of gridblocks and the carbonated water-oil relative permeability curve (CWI-Kr). Figure 4-59, Figure 4-60 and Figure 4-61 show the RF, the differential pressure (DP) across the core and the total CO2 production (TCO2P) respectively, predicted by E300, compared to the values obtained from the developed simulator. Figure 4-59 shows that the RF predicted by the E300 is higher compared to that predicted by the developed simulator.

175

RF(%)


80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Exp Model, MTC=2.2E-7 E300 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-59: RF predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=2.2E-7 1/sec, CWI test.

2.5

DP(psi)

2

1.5

1

DP-Exp DP-Model, MTC=2.2E-7 DP-E300

0.5

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-60: DP data predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=2.2E-7 1/sec, CWI test.

176


10000 TCO2P-Model, MTC=2.2E-7

9000

TCO2P-E300 8000

TCO2P(Scm3)

7000 6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-61: TCO2P data predicted by the developed simulator compared to those predicted by the E300 simulator; MTC=2.2E-7 1/sec, CWI test.

Moreover, as shown in Figure 4-61, the developed simulator predicts that the CO2 arrives and is produced at core outlet more quickly compared to that predicted by the E300. This could be due to the fact that the E300 is developed based on the instantaneous equilibrium assumption, which would cause CO2 to be quickly transferred into the oil phase, resulting in a higher CO2 concentration in oil compared to that predicted by the developed simulator, which assumes that the transfer of CO2 into the oil phase is a slow process.

To further support this idea, a sensitivity analysis was performed and the effect of injection rate was investigated using the E300 simulator. Figure 4-62 and Figure 4-63 clearly show these results.

177


160 140

TOP(Scm3)

120 100 80 60 40 TOP-E300, Injection Rate=20 20

TOP-E300, Injection Rate=5

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-62: TOP data predicted by the E300 simulator for injection rates of 20 and 5 cm3/hr, CWI test.

6000 TCO2P-E300, Injection Rate=20 TCO2P-E300, Injection Rate=5

TCO2P(Scm3)

5000

4000

3000

2000

1000

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-63: TCO2P data predicted by the E300 simulator for injection rates of 20 and 5 cm3/hr, CWI test.

178


The goal was to see if changing the injection rate would change the total oil and CO2 production profiles, as it was seen when using the developed simulator. That is, contrary to what was predicted by the developed simulator, it is not expected that changing the injection rate affects the prediction of the E300, as this is based on the assumption of instantaneous equilibrium and thus the length of time which the oil and water are in contact should not affect the amount of CO2 transfer between the phases. Figure 4-62 and Figure 4-63 clearly show that decreasing the injection rate from 20 to 5 cm3/hr has not impacted the TOP and TCO2P predicted by the E300 as it was expected.

It should be noted that the developed simulator conversely showed that when the injection rate was decreased more oil was produced (i.e. the TOP was increased) and the CO 2 breakthrough time on TCO2P profile was postponed and a smaller amount of CO2 was produced.

The developed simulator can be adjusted so that CO2 can be transferred quickly from the water into the oil phase. The goal is to see if the predictions of the developed simulator and the E300 can become closer. The mass transfer coefficient (MTC) in the model controls the rate of CO2 transfer from the water into the oil phase. It is expected that, if MTC is increased, the predictions of the E300 and the developed simulator will become closer. Figure 4-64 and Figure 4-65 indicate the total oil and CO2 production profiles, respectively, predicted by the developed simulator when the MTC value is increased to 11E-7 1/sec (five times higher than 2.2E-7 1/sec) in comparison with that of the E300. It can be seen that the TOP data predicted by the E300 and developed simulators almost overlap, whereas the TCO2P data are not matched. This could be due to the fact that the equations and procedure used by the E300 are not exactly same as those used by the developed simulator. It should be noted that the MTC value is increased by five times and the total oil production predicted by the E300 is approximately matched to the prediction by the developed simulator. However, for the water-wet core, the MTC was increased by three times and the TOP predicted by E300 and that by the developed simulator overlapped.

179


160 140

TOP(Scm3)

120 100 80 60 40

TOP-Exp TOP-Model, MTC=11E-7 TOP-E300

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-64: TOP data predicted by the E300 simulator compared to the experimental values and those predicted by the developed simulator; MTC=11E-7 1/sec, CWI test.

7000 TCO2P-Model, MTC=11E-7 6000

TCO2P-E300

TCO2P(Scm3)

5000 4000 3000 2000 1000 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-65: Total CO2 production (TCO2P) data predicted by E300 simulator compared to those predicted by the developed simulator; MTC=11E-7 1/sec, CWI test.

180


It should be noted that it is not possible to control the amount of mass transfer in the E300 to match the prediction of E300 to the developed simulator’s prediction. Therefore, the developed simulator was adjusted to match the prediction of the E300.

Moreover, as shown before, at low injection rates, the simulator predicts higher oil production while the TOP prediction by the E300 is insensitive to the injection rate. Considering that the TOP data predicted by the E300 are currently higher than those predicted by the developed simulator, it is expected that, at low injection rates, the prediction of the E300 and the developed simulator will be the same. Figure 4-66 compares the TOP predicted by the developed and E300 simulators at an injection rate of 4 cm3/hr (one fifth of 20 cm3/hr), showing that the TOP data predicted by both the simulators are almost the same.

160 140

TOP(Scm3)

120 100 80 60 40 TOP-Exp TOP-Model, Injection Rate=4, MTC=2.2E-7 TOP-E300, Injection Rate=4

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-66: TOP data predicted by E300 simulator compared to those predicted by the developed simulator at injection rate of 4 cm3/hr, CWI test.

This exercise clearly shows how the assumption of instantaneous equilibrium, which is the basis of the usual compositional simulation approach, affects the simulation of carbonated coreflood experiments using the available compositional simulators. 181


In another experiment, similar to that performed for water-wet core, here also attempted to find a large MTC value at which it can be expected that the developed simulator is working based on the instantaneous equilibrium assumption for this coreflood experiment. Figure 4-67 shows recovery factors prediced by the develpoed simulator at three different MTC valuses of 7E-7, 11E-7 and 13E-7 1/sec. It can be observed that RF has not changed when MTC is increased from 11E-7 to 13E-7. Therfeor it can be concluded that for this coreflood experimnt, a MTC value greather than 11E-7 can be assumed as the equilibrated MTC. This was consistent with that MTC vlaue at which the results of the developed simulator and E300 were very similar.

140 120

TOP(Scm3)

100 80 60 40 MTC=7E-7 MTC=11E-7 MTC=13E-7

20 0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 4-67: Recovery factors prediced by the devlpoed simulator at three different MTC valuses of 7E-7, 11E-7 and 13E-7 1/sec.

4-8.


To crosscheck and explore the generic capability of the developed simulator, simulations of a different WI and CWI coreflood experiment were carried out in this chapter. The coreflood experiments had been performed using a mixed-wet sandstone core. First, experimental data, including total oil production (TOP) and differential pressure across 182


the core (DP), of WI were compared with those of CWI. The experimental data showed that the DP of CWI is lower than that of WI for this mixed-wet coreflood experiment, which was attributed to the wettability effect. Moreover, the TOP of CWI was higher than that of WI, with a shift in breakthrough point which was also attributed to the wettability effect. Furthermore, some oil production after the breakthrough point was observed, which was explained as the effect of oil swelling.

Next, simulations of WI and CWI experiments were investigated using the developed simulator and ECLIPSE commercial simulators. Similar to the procedure followed for water-wet core, the simulation of the WI experiment was performed first when an optimised water-oil relative permeability curve (WI-Kr) was obtained. Later, the effect of capillary pressure was also discussed.

Next, the CWI was simulated using the developed simulator in its compositional mode when the effect of mass transfer was included. It was realised that for this mixed-wet core, in addition to MTC, WI-Kr curve needed to be tuned to predict the experimental data properly. Therefore, an optimal MTC and Kr curve were obtained manually and also automatically using the GA-based optimiser. In manual tuning step, the contribution of oil swelling and the wettability in the simulation of CWI process were discussed and quantified when the results of water-wet and mixed-wet coreflood experiments were investigated simultaneously. The results of automatic tuning were in good agreement with those obtained by manual tuning. Next, the CWI was simulated using the black-oil approach by obtaining an optimal Kr curve.

The developed simulator was also used to investigate the effects of MTC, dispersion and injection rate on CO2 and oil production production profiles. Similar results as those obtained for water-wet core, were also obtained here. In addition, a sensitivity analysis on the carbonation level (CO2 content) of the injected fluid was carried out and similar results as those for water-wet core, was also obtained here.

CO2 storage profile was also obtained by the simulator for this mixed-wet core. Similar profiles as those of water-wet core, were also predicted here. In addition, a sensitivity analysis was performed and the effects of both the MTC and the injection rate on the CO2 183


storage were investigated. It was observed that at low injection rates and high MTC values, more CO2 could be stored in the core.

Finally, the CWI experiment was also simulated using the ECLIPSE300 (E300). Similar to that predicted in water-wet core, in this mixed-wet core, E300 also predicted higher oil recovery and lower and delayed CO2 production in comparison with those predicted by the developed simulator. Later, a sensitivity analysis was carried out to realise the role of injection rate in simulation results predicted by E300. It was observed that, in contrast to the developed simulator, E300 predicted the same CO2 and oil production profiles as those at a high injection rate. In another practice, the tuned MTC value was increased and it was realised that if it can be increased by five times, the predictions of the E300 and the developed simulators became almost the same. Finally it was shown that, at low injection rates, the predictions of the E300 and the developed simulators are very close as oil and water are in contact for a longer time at low injection rates, therefore it is more possible to reach the equilibrium. It was observed that, if the carbonated water injection rate is reduced by five times during the simulation of CWI, the predictions of the E300 and the developed simulators will become very close.

184

Reservoir-scale Simulation of CWI This chapter presents the reservoir-scale simulation of the carbonated water injection (CWI). The objective was to highlight some difficulties of the reservoir-scale simulation of CWI using ECLIPSE 300 commercial reservoir simulator.

A study on dimensionless numbers is carried out and a new dimensionless number is introduced. This can help to use the results of the core-scale simulations and create the reservoir-scale model employing ECLIPSE300 (E300) compositional simulator. Initially, carbonated water injection in secondary mode is compared with the secondary water injection using the reservoir-scale simulation results. The effects of the gravity on the performance of CWI as well as on the CO2 storage during the CWI are studied. Moreover, tertiary injection of the carbonated water is examined and compared with the secondary CWI.

Core-scale simulation of CWI was comprehensively studied in the previous chapters. It was demonstrated that at some situations if the mass transfer coefficient can be increased or at low injection rates when there is enough time for oil and water phases to be in contact and exchange CO2, the E300 compositional simulator can be used to simulate CWI coreflood experiments.

The first difficulty of using ECLIPSE 300 for reservoir-scale simulation of the CWI, was the assumption of instantaneous equilibrium made by the ECLIPSE. To find a solution for this in order to be able to use E300 to study the reservoir-scale simulation of CWI, it was tried to design a model so that the instantaneous assumption is valid. As it was mentioned before, at low injection rates it can be presumed that the equilibrium state can be reached during CWI process at the time scale of the injection period. However to determine that how low the injection rate in ECLIPSE model should be in order to trust the predicted results by E300, a dimensionless number analysis was performed. Using the results of water-wet and mixed-wet coreflood experiments, the behavior of two dimensionless numbers of Capillary Number (Nc) and so-called Equilibrium Number (Ne), are investigated and their values are calculated as described below.

185

Chapter 5: Reservoir-scale Simulation of CWI

However to use a low injection rate in ECLIPSE model at which the simulation results of CWI by E300 can be trusted, a dimensionless number analysis was performed. Using the results of water-wet and mixed-wet coreflood experiments, the behavior of two dimensionless numbers of Capillary Number (Nc) and so called Equilibrium Number (Ne), are investigated and their values are calculated as described below.

5-1.

Dimensionless Number Analysis

5-1.1.

Capillary Number (Nc)

Capillary number is a dimensionless number used in the analysis of fluid flow and represents the ratio of viscous to capillary forces (Chandler et al. 1981, Dullien 1992). Capillary number is defined as follows: Nc=

uμ

(5-1)

σ

where u and μ is are the velocity and viscosity of displacing fluid, respectively. σ is the interfacial tension between displacing (carbonated water here) and displaced fluid(decane uμ

here). Sometimes the porosity is also included in the Capillary number Nc= φσ (Tiab and Donaldson, 1996; Forster 1973, Melrose and Brandner, 1974; Chatzis and Morrow, uμ

1984), and sometimes the contact angle Nc= cos(θ)σ (as a representative of wettability) (Moore and Slobod, 1955).

Table 5-1 shows the water-wet and mixed-wet cores properties presented in SI unit. Table 5-1: Core and fluid properties in SI unit. cross section Coreflood Diameter μw σ* area Experiment (D) (kg/m.sec) (N/m) (m2) (m) 19.5 E-04 0.66E-3 20E-3 Water-wet 4.986E-2 4.86E-2 18.5 E-04 0.66E-3 20E-3 Mixed -wet * : estimated from Georgiadis et al. (2011) for carbonated water-decane at reservoir conditions.

186


Table 5-2 shows the calculated capillary number using Equations 5-1 for both water-wet and mixed-wet cores presented in Chapters 3 and 4 using two different velocities. The first velocity (u1 ) is calculated based on the original injection rate (i.e. 20 cm3/hr) for both water-wet and mixed-wet core and second velocity (u2 ) is calculated based on the tuned injection rate at which the predictions of E300 and the developed simulator were almost the same (i.e. 6.67 cm3/hr for the water-wet core and 4 cm3/hr for the mixed-wet core). Table 5-2: Capillary number using Equations 5-1 for both water-wet and mixed-wet cores. Water-wet Coreflood Experiment

Nc

u1 =0.25 (m/day) u2 =0.08 (m/day)

9.39E-08

Mixed-wet Coreflood Experiment u1 =0.26 (m/day) u2 =0.05 (m/day)

3.13E-08 Nc

9.89E-08 1.98E-08

Generally for Nc values equal or less than 10E-5 based on the Equation 1, the flow regime is capillary dominant (Zendehboudi et al. 2011, Dullien 1992) and therefore from Table 5-2 it can be concluded that the flow is dominated by the capillary forces here.

It should be noted that, when flow regime is capillary dominant it does not mean that the capillary pressure is big and should be considered during simulation (Cense and Berg, 2009). This is because, first, the capillary number is the ratio of the viscous pressure drop to the capillary pressure at pore scale (Løvoll et al., 2005). That is, it is a microscopic concept. Second, as mentioned in the literature (Cense and Berg, 2009), although Darcy’s law seems to be a purely viscous law of two non-interacting fluid phases, viscous coupling (when one fluid exerts a drag force on another fluid at fluid-fluid interface) and capillary effects are captured via the relative permeabilities. That is, in the macroscopic view (i.e. based on Darcy’s law), capillary forces play a significant role through the relative permeabilities and in the macroscopic description there is no clear separation between viscous and capillary effects. 187


5-1.2.

Equilibrium Number (Ne)

It was explained in Chapters 3 and 4 that at some situations when using a specific values of injection rate and mass transfer coefficient (MTC), the results of E300 and the developed simulators will become almost the same. That is, it can be assumed that the system has reached the equilibrium state. Therefore it is helpful if an Equilibrium Number (Ne) can be introduced in order to relate the value of mass transfer coefficient to the core properties and the operational conditions of the flooding test. The goal is to find the parameters which affect the equilibrium conditions during CWI process in the core. It was shown that if tuned MTC value is increased by a factor of 3 and 5 for water-wet and mixed-wet cores respectively, the model shows that the system can reach the equilibrium state as the results are almost matched to those by E300. However, it seems that the MTC value is an intrinsic property of the system. Therefore, the MTC is a constant value and a basic characteristic of the fluid and core used in this study. Hence, for a different coreflood experiment when a different core and fluids system are used, a different MTC value can be obtained. For a specific fluid system, MTC values should be different in different cores with different pore geometries and heterogeneities. This is because at the pore-scale level (in microscopic view), pore geometry and heterogeneity affect the fluid distribution in core and in turn the interface area between the phases. That is, as explained in Section 2-6, mass transfer coefficient is a function of specific interfacial area which this should be different for different pore geometries. Nevertheless, based on Equation 218, MTC was defined as a lumped parameter. This has been widely used in the literature by researchers in hydrology (Powers et al., 1992 and 1994; Imhoff and Jaffe, 1994; Yoon et al., 2002). However defining MTC as a lumped parameter, is macroscopic view of MTC and based on this definition, Anwar (2008) realised that the mass transfer coefficient is dependent on the flow velocity, geometry of the porous medium and the properties of fluid. Anwar (2008) carried a series of non-aqueous phase liquids (NAPL) volatilization experiments. He used a one-dimensional sand-packed set-up. Three uniform size glass beads (d50=0.25mm, 0.50mm, 0.75mm) and a medium grain silica sand (d50=1.4mm) were selected as the porous media and Toluene was chosen as single component NAPL. Air was injected at different flow velocities. He presented that the dependency of mass transfer on pore gas velocity is linear. Yoon et al. (2002), also presented similar results. The results are consistent with the findings of this thesis that will be described later. 188


Therefore, to quantify the contribution of the MTC during simulation of CWI process, a sensitivity analysis was performed using the coreflood experiments data presented in Chapter 3 and 4. To do that, the core properties (i.e. core length and diameter, porosity and permeability) as well as the injection rate (operational conditions) were changed and based on these changes; the MTC was adjusted such that the developed simulator could produce the same results of those of a base-case. A base-case was defined when the injection rate and the core properties were same as the original values (tune-case) while the MTC was three times higher than the original(tuned) value(i.e. 15E-7=3×5E-7 1/sec). It was shown that (in Chapter 3) if the tuned MTC could be increased by three times the results of E300 and the developed simulator would become the same. It should be noted that this exercise was carried out for carbonated water-decane fluid system; therefore the effect of fluid properties was excluded. Table 5-3 shows the results of this sensitivity analysis. Table 5-3 demonstrates different combinations of core properties and operational conditions at which the output of E300 and the developed simulator are same for the water-wet coreflood experiment. That is, injection rate and core properties were changed from the corresponding base values and MTCs were adjusted such that the results of the developed simulator and E300 became the same. The oil recovery profile was used to compare the results.

Table 5-3: Combination of core properties and operational conditions when the same result by the E300 and the developed simulators is obtained, water-wet coreflood experiment.

1 2 3 4 5 6 7

Injection rate MTC Area L 𝐤 𝛗 (cm3/hr) (1/sec) (cm) (cm) (mD) (fraction) Tune-case 20 5E-7 4.986 33.2 1300 0.19 Base-case 20 15E-7 4.986 33.2 1300 0.19 Effect of injection rate 6.67 5E-7 4.986 33.2 1300 0.19 Effect of core length 20 30E-7 4.986 16.1 1300 0.19 Effect of core diameter 20 30E-7 3.526 33.2 1300 0.19 Effect of core permeability 20 15E-7 4.986 33.2 650 0.19 Effect of core porosity 20 15E-7 4.986 33.2 1300 0.095

Table 5-3 shows that when injection rate is decreased by a factor of three, the MTC needs to be also decreased by a factor of three from its base value in the developed simulator to produce the same result of that of E300. Moreover when the length of the core is reduced by half, the MTC needs to be increased by a factor of two. Furthermore, when the cross section area of the core is decreased by a factor of two (i.e. the core diameter is decreased by a factor of √2), the MTC needs to be increased by a factor of two. However, MTC is 189


not affected by the porosity and the permeability of the core. This exercise shows that the effect of MTC can be compensated by the parameters which control the length of the present time of the fluids inside the core. When the core length is increased or when the velocity of fluids in the core are decreased (by decreasing injection rate or increasing the core cross section area), the fluids have more time to exchange the CO 2 causing the system to reach the equilibrium state. Equilibrium Number (Ne) was introduced and defined using the results of this exercise (shown in Table 5-3) as follows:

Ne =

L ×MTC u

=

L ×MTC× A

(5-2)

qinj

where u is the superficial velocity obtained by injection rate (q inj ) divided by the core cross section area (A), L is the core length and MTC is the mass transfer coefficient. Table 5-4 shows the Equilibrium Number calculated for different combination of core properties and operational conditions presented in Table 5-3. It can be seen that in all cases, the Ne is equal to 0.175. However the Ne is 0.058 for tuned-case at which the equilibrium assumption is not valid and E300 over predicted TOP data compared to those predicted by the developed simulator. It should be noted that the pore level properties such as wettability or interfacial area is not captured by this equation as the lumpedparameter of MTC is defined in macroscopic scale.

Table 5-4 shows that for this specific coreflood system the equilibrium number is constant. Considering the base-case in Table 5-4, MTC value is 3 times higher than the tuned MTC value. However if all parameters are the same as that of the tuned case, E300 cannot be used to simulate the CWI process as it will over predict the production data. However if the equilibrium number (Ne) can be increased by 3 times (i.e. From 0.058 to 0.175), E300 can be used to simulate CWI as it will produce similar results as those by the developed simulator. That is, if the Ne value is 0.2 or higher it can be said that the amount of CO2 being transferred between the phases is such that the instantaneous equilibrium assumption used by the E300 is acceptable.

190


Table 5-4: Equilibrium Number for different combination of core properties and operational conditions when the same result by the E300 and the developed simulators is obtained, water-wet coreflood experiment. Injection rate (cm3/hr) 1 2

Tuned-case

20

Base-case

20

3

Effect of injection rate Effect of core length Effect of core cross section area

6.67 (20/3)

4 5

20 20

MTC (1/sec) 5E-7 15E-7 (5E-7×3) 5E-7 (15E-7/3) 30E-7 (15E-7×2) 30E-7 (15E-7×2)

cross section area (cm2) 19.5

L (cm)

Ne

33.2

0.058

19.5

33.2

0.175

19.5

33.2

0.175

19.5

16.1 (33.2/2)

0.175

9.75 (19.5/2)

33.2

0.175

To increase the Ne value, MTC cannot be changed as it is the inherent characteristic of the core and fluids used during the coreflood experiment. Therefore, for a specific MTC value, the injection rate can be reduced or core length can be increased to assure that the simulation performed based on the instantaneous equilibrium assumption, is acceptable. Hence, for this specific water-wet coreflood experiment, if a new CWI experiment is designed so that the original injection rate is decreased by three times or using a core plug when its length is three times higher than the original core length, it can be said that the E300 can be used to simulate it with acceptable accuracy.

Similar investigation was performed for the mixed-wet coreflood experiment. A basecase was defined when the injection rate and the core properties were the same as the original values (tuned-case) while the MTC was five times higher than the original(tuned) value (i.e. 11E-7=5×2.2E-7 1/sec). It was shown that (in Chapter 4) if the tuned MTC could be increased a factor of five, the results of E300 and the developed simulator would become similar. Injection rate and core properties were changed from the corresponding base values and it was attempted to adjust MTC such that the results of the developed simulator and E300 become the same. It was seen that the MTC value is insensitive to the permeability and the porosity of the core. It should be noted that, this was observed for the water-wet core as well. Table 5-5 demonstrates different combinations of core properties and operational conditions at which the output of the E300 and the developed

191


simulator are the same for the mixed-wet coreflood experiment as well as the corresponding Ne values. Table 5-5: Equilibrium Number for different combination of core properties and operational conditions when the same result by E300 and the developed simulators is obtained, mixed-wet coreflood experiment.

1 2 3 4 5

Injection rate (cm3/hr)

MTC (1/sec)

cross section area (cm2)

L (cm)

Tune-case

20

2.2E-7

18.5

61.2

Base-case

20

18.5

61.2

0.23

Effect of injection rate Effect of core length Effect of core cross section area

4 (20/5)

18.5

61.2

0.23

18.5

30.6 (61.2/2)

0.23

9.25 (18.5/2)

61.2

0.23

20 20

11E-7 (2.2E-7×5) 2.2E-7 (11E-7/5) 22E-7 (11E-7×2) 22E-7 (11E-7×2)

Ne 0.045

Table 5-5 shows the equilibrium number is constant for all combinations and is equal to the base-case value (i.e. Ne=0.23). Considering the base-case in Table 5-5, the MTC value is 5 times higher than the tuned MTC value. Table 5 shows that if the equilibrium number (Ne) can be increased by 5 times (i.e. from 0.045 to 0.23), E300 can be used to simulate CWI as it results in the similar data as those by the developed simulator. That is, if the Ne value is 0.23 or higher it can be mentioned that the amount of CO2 being transferred between the phases is such that the instantaneous equilibrium assumption used by E300 is acceptable.

Moreover, to increase the Ne value to 0.23 in this mixed-wet core, MTC cannot be changed as it cannot be controlled during the coreflood experiment, therefore, the injection rate can be reduced or the core length can be increased to assure that the simulation performed based on the instantaneous equilibrium assumption is acceptable. Hence, if a new experiment is designed so that the original injection rate is decreased by five times or using a core plug when its length is five times higher than the original core length, the E300 can be used to simulate that CWI coreflood experiment.

192


It should also be noted that this interpretation and introducing the Equilibrium Number is valid when the additional oil recovery of CWI is as a result of oil swelling which is captured by the mass transfer term in the developed simulator. Therefore, when the wettability alteration contributes to the oil recovery behaviour during CWI, the concept of introduced Equilibrium Number is not applicable. At this stage, the results of dimensionless numbers analysis is used to study the CWI process in the scale of the reservoir. That is, the goal is to make a reservoir-scale model using E300 simulator to study the performance of CWI. A synthetic model was made in E300 using the rock and fluid properties of the water-wet coreflood experiment. It should be noted that, the data of the mixed-wet coreflood experiment was not considered in this study because, as it was mentioned above, the concept of the Equilibrium Number may have some uncertainties for this mixed-wet core as both the oil swelling and wettability alteration had contribution during CWI process. Injection rate and the reservoir dimension were chosen such that the flow regime is capillary dominant and the order of Equilibrium Number value is in the range of the value explained before to assure the assumption of instantaneous equilibrium is valid and E300 can be used.

5-2.

Model Description

A synthetic three dimensional (3D) reservoir model was built to evaluate the performance of reservoir-scale CWI. The rock and fluid properties were the same as those of the waterwet coreflood experiment. That is, the same oil sample (i.e. decane) and the reservoir conditions of that of water-wet experiment were used to reduce the uncertainty due to the phase behaviour and fluid properties calculations. The reservoir was 1500 ft ×1500 ft ×120 ft. One producing vertical well and one injection vertical well were drilled into the reservoir. The locations of the vertical wells were chosen to represent a quarter 5-spot well pattern so that the production and injection wells were diagonally in opposite corners of the model (Figure 5-1). It should be noted that as CWI is a water based injection strategy, this well planning was selected which is typically used for water flooding projects (Ahmed, 2011). Table 5-6 shows the details of the created ECLIPSE model.

193


Figure 5-1: Schematic of the model.

Table 5-6: Details of the created ECLIPSE model. Initial reservoir pressure, psig

2500

No of gridblocks

15×15×6

Grid size, ft

100×100×20

Porosity, fraction

0.19

Permeability, mD (Kx=Ky=Kz)

1300

Reservoir depth

2500

Pore volume, MMbbl

9.137

injection rate, Rbbl/d (WI)

6000

CO2 volume fraction in injected carbonated water (%)

6.7

initial water saturation (swi), %

0

First, water injection was simulated using the water-oil relative permeability obtained from the water-wet coreflood experiment (Table 3-5). Later CWI was simulated. To define the model, selecting the number of gridblocks was challenging which was not a big issue during the core-scale simulations. In the reservoir simulations, it should be tried to select the size of the gridblocks so that the effect of numerical dispersion can be minimised and the results of the simulations become independent to the grid size. To achieve this goal, the number of gridblocks needs to be increased however this, in turn, will increase the time and the computer memory needed for the simulation. Therefore, an optimum number of gridblocks should be selected. A sensitivity analysis was performed. To to that, CWI process was considered because due to the presence of CO2, more refined 194


grids are expected for simulation of CWI compared to that for WI. The optimum size of the gridblocks can be determined when more refining of them will not affect the output of the simulation. The total numbers of gridblocks were increased from 675 to 9600 and the ultimate oil RF was obtained as shown in in Table 5-7. Figure 5-2 shows the predicted RF for different gridblock numbers in the model. It can be observed that by increasing the number of gridblocks from 675 to 1350, the breakthrough point has slightly shifted and further refining of gridblocks would not affect simulation results.

Table 5-7: Grid size sensitivity analysis.

1 2 3 4 5

No. of grids in ‘x’, ‘y’ and ‘z’ directions 15×15×3 15×15×6 15×15×12 20×20×12 40×40×6

Total numbers of gridblocks 675 1350 2700 4800 9600

Figure 5-2: Grid size sensitivity analysis.

195

Ultimate recovery factor 0.78 0.77 0.77 0.77 0.77


Finally, based on this sensitivity analysis, the model was discretised and divided into 15×15×6 gridblocks (i.e. 15 cells in x direction, 15 cells in y direction and 6 cells in z direction (6 layers)). The wells were penetrated in the vertical direction. The production and injection wells were penetrated through the entire reservoir interval. The initial reservoir pressure was 2500 Psia and the production well was constrained by the constant bottom-hole pressure of 1500 Psia. The injection well was operated under the constant injection rate. The operational conditions (i.e. the constraints) of the wells were chosen to mimic the operational conditions used during the coreflood experiments. The injection rate was set so that the calculated Capillary and Equilibrium Numbers were consistent with those values obtained for the core-scale simulations. To calculate Nc and Ne, the superficial velocity of the injected fluid was needed. In the coreflood experiment, the cross section area normal to the flowing flow was constant; therefore the velocity through the core was not changed due to the area effect. However, in the reservoir scale, based on the geometric shape of the model (Figure 5-1) and as the area normal to the flow is changing, the fluid velocity is variable through the reservoir and the injection fluid has maximum velocity near to the wellbore and the minimum velocity at the centre of the model. Therefore, the average velocity based on the estimated average area was used. The average area is the average between the cross section area at the wellbore and the cross sectional area at the centre of the model. Figure 5-3 shows schematically the variable area when fluids move towards the production well from the injection well. The minimum distance which the injected fluid should travel to reach the production well is through the diagonal line connecting the injection well to the producer. The minimum length was used to set the injection rate in order to assure that all over the reservoir, the calculated Capillary and Equilibrium Numbers were in the range of those values from the coreflood experiment. Table 5-8 shows the maximum injection rate which can be used in order to meet the criteria of the equilibrium conditions while having a capillary dominated flow regime.

196


Figure 5-3: Shcmatic of the varaible area(blue rectangles) in front of fluid during traveling in the reservoir.

Table 5-8: Injection rate obtained based on the constraints imposed by Equilibrium and Capillary Numbers from the water-wet coreflood experiment.

Ne

Nc

0.2 9.39E-08

MTC (1/sec)

average cross section area (ft2)

L (ft)

Maximum Injection rate based on Nc (bbl/d)

Maximum Injection rate based on Ne (bbl/d)

5E-7

120 × 750√2

1500√2

18285

1.04E7

Table 5-8 shows that the injection rate can be very large without concerning of the validity of the instantaneous equilibrium assumption. A typical injection rate of 6000 Rbbl/d was selected which meets both criteria imposed by the Capillary and Equilibrium Numbers. It should be noted that, for the injection well, it was not possible in E300 model with CO2SOL option to define carbonated water stream directly therefore at injection location, a pair of wells were defined so that one well could inject the CO2 and the second one could inject the water. Hence, in order to inject carbonated water at the rate of 6000 Rbbl/d, the water was injected at the rate of 5598 Rbbl/d and the CO2 was injected at the rate of 402 Rbbl/d through two injection wells drilled at the same location. This resulted in a ratio of 5 wt% (6.7 volume percent) CO2 solubility in the injected water stream at the reservoir conditions. Simulation was run for 13 years to inject around 3 PV fluids. 197


It should be noted that the ECLIPSE model was defined such that the screening criteria for a successful CO2-EOR project is met. That is, the reservoir/oil characteristics used in the ECLIPSE model is consistent with the optimum reservoir/oil characteristics (both technically and economically) suggested in the literature for a successful CO2 miscible injection project as given in Table 5-9 (Taber et al., 1997a and 1997b; Kang et al., 2014).

Table 5-9:The screening criteria for the CO2 miscible injection technique(Taber et al., 1997a and 1997b). Initial oil saturation (% PV) Oil gravity( 0API) Oil viscosity(cP) composition average permeability Type of formation Depth and temperature

>20 > 22 < 10 High percentage of intermediate hydrocarbons (especially C5 to C12) Not critical if sufficient injection rates can be maintained. Sandstone or carbonate and relatively thin unless dipping. For miscible displacement, depth must be great enough to allow injection pressures greater than the MMP, which increases with temperature. For example, for an oil with API gravity greater than 40, the depths should be greater than 2500 ft.

The capillary pressure was assumed to be zero and water-oil relative permeability from the core-scale simulation was used (Table 3-5). It should be noted that in addition to the water-oil relative permeability (Krw-o) curve, the gas-liquid relative permeability (Krg-l) was also needed when there was free CO2 in the system at some injection strategies. As it was mentioned, WI-Kr obtained from the water-wet coreflood experiment was used here. Moreover, for Krg-l, a typical gas-liquid relative permeability in sandstone reservoirs based on the Corey correlation was used (Ahmed 2001). Figure 5-4 shows the Krg-l curve which is plotted versus the oil phase saturation. It should be noted that during the coreflood experiments, the initial water saturation was zero. Therefore it was assumed that the initial water saturation is zero here as well.

198


1.00

1.00

0.90

0.90

0.80

0.80 Krg Kro

0.70

0.60

0.60

0.50

0.50

0.40

0.40

0.30

0.30

0.20

0.20

0.10

0.10

0.00 0.30

0.40

0.50

0.60

0.70

0.80

0.90

Kro

Krg

0.70

0.00 1.00

So Figure 5-4: Gas-Oil relative permeability curve

5-3.

Simulation Results

5-3.1.

Secondary WI and CWI

The objective of this section is to investigate and compare the performance of the secondary CWI and the secondary WI. It was shown that, during coreflood experiments, secondary CWI results in higher oil recovery compared to the secondary WI. This additional oil recovery was attributed to the existing CO2 in the system during CWI process which improves oil mobility. To investigate this in reservoir as well, using the results from the reservoir-scale simulation, Figure 5-5 shows the recovery factor (RF) of the secondary WI compared to that of the secondary CWI versus injected pore volume (PV) predicted by the E300.

199


0.9 0.8 0.7

RF

0.6 0.5 0.4 0.3 0.2 Secondary CWI 0.1

Secondary WI

0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-5: RF of the secondary CWI compared to that of the secondary WI as predicted by the E300.

Figure 5-5 shows that the final oil recovery factor obtained from the WI is 70% while it is 77% from CWI. That is, in this reservoir-scale model, 7% additional oil recovery has been obtained by the CWI over the WI. The 7% additional recovery factor of CWI over WI is attributed to the oil swelling due to existing CO2 in the system. It should be noted that the contribution of viscosity reduction is minimal as the decane viscosity (oil sample here) is very small. Moreover, IFT and wettability alteration do not have contribution in additional oil recovery during the simulation as these need to be captured by adjusting the relative permeability curve while the same relative permeability curve was used during the simulation of both WI and CWI. At breakthrough point, both WI and CWI have produced approximately 68% oil recovery. It can be calculated that the WI has given rise to 2% additional oil recovery after breakthrough point while this is 9% for the CWI. It can be concluded that the oil swelling contributes in additional oil recovery mainly after the breakthrough point similar to that seen during the coreflood experiments.

This practice demonstrated the capability of CWI technique in enhancing oil recovery if it is injected into a reservoir. It is worth investigating the gas saturating profile to check 200


if any gas has been evolved. Figure 5-6 shows the gas saturation distribution in the reservoir at end of the simulation predicted by the E300.

Figure 5-6: Gas saturation profile.

It can be observed than no free gas phase has been formed. That is, all present CO2 in the reservoir is dissolved in water and or oil. Figure 5-7 shows the CO2 concentration profile (based on mole fraction) in water phase at 0.46 injected PV.

Figure 5-7: CO2 concentration profile in water phase at 0.46 injected PV.

201


Figure 5-8 shows the CO2 concentration profile (based on mole fraction) in oil phase at 0.46 injected PV.

Figure 5-8: CO2 concentration profile in oil phase at 0.46 injected PV.

Figure 5-9 shows the water saturation profile at 0.46 injected PV.

Figure 5-9: Water saturation profile at 0.46 injected PV.

202


Comparing Figure 5-9 and Figure 5-7, it can be observed that at water front, water has been deprived of CO2 and is moving as plain water. As a result, it is expected that the water breakthrough happens faster than CO2 breakthrough. Figure 5-10 shows total water production (TWP) and total CO2 production (TCO2P) profiles. It can be observed that water breakthrough has happened after around 0.7 injected PV while CO2 breakthrough has happened after 1 injected PV.

25

1600 1400

20

1000

15

800 10

600

TCO2P(MMSCF)

TWP(MMSTB)

1200

400 5

TWP 200

TCO2P 0

0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-10: Total water production (TWP) and total CO2 production (TCO2P) profiles.

It is worth investigating the CO2 storage profile at reservoir-scale as well. Figure 5-11 displays CO2 storage profile. The ratio of the total CO2 stored (TCO2S) to the total CO2 injected (TCO2I) is plotted versus the pore volume injected of the carbonated water. It can be seen that at end of the simulation, around 65% of the injected CO2 has been stored in the reservoir. It can be observed that until the 1 injected PV point (breakthrough point), 100% of the injected CO2 has been stored and after that the storage curve has declined.

203


120

TCO2S/TCO2I(%)

100

80

60

40

20

0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-11: CO2 storage profile.

It is also important to look into the differences between the reservoir-scale and the corescale models as well. Here, in this reservoir-scale model, the flow is three-dimensional with while in the core-scale model; the flow was one-dimensional. Furthermore in real reservoirs, the permeability of the rock is usually different in different directions particularly in the vertical relative to horizontal direction. Therefore the ratio of vertical K

to horizontal permeabilities i.e., K V ratio, may affect the dynamic of flow in the reservoir H

and affects the simulation results. A sensitivity analysis was performed to investigate the K

effect of anisotropy ratio i.e., K V ratio, on oil recovery factor as described below. H

5-3.2.

Effect of

KV KH

ratio

During core-scale simulation of WI and CWI, the flow was one-dimensional and one permeability value was used. On the other hand, in reservoir-scale model, the fluid can flow both in the horizontal and vertical directions while the vertical permeability is usually smaller than the horizontal permeability (Ahmed 2001). To investigate the effect 204


of

KV KH

ratio on oil recovery, a sensitivity analysis was carried out and the ratio was

decreased from 1 to 0.3 and 0.03. Figure 5-12 and Figure 5-13 shows the profiles of oil RF and total CO2 production (TCO2P) respectively at these three different ratios. It can be observed that for this model, when the vertical permeability has been decreased, the results predicted by E300 are almost the same. It can be explained that, in this model the injection and production strategy is such that the oil is produced and water is injected through the all layers while the horizontal permeability is quite large. Therefore, it is possible that the fluids mainly flow in horizontal direction and therefore the RF and TCO2P are insensitive to the value of vertical permeability for this specific model setup.

0.9 0.8 0.7

RF

0.6 0.5 0.4 0.3 KV:KH=1

0.2

KV:KH=0.3 0.1

KV:KH=0.03

0 0

0.5

1

Figure 5-12: Effect of

1.5 Injected PV KV KH

2

2.5

ratio on oil recovery factor.

205

3


1600 KV:KH=1

1400

KV:KH=0.3 KV:KH=0.03

TCO2P(MMSCF)

1200 1000 800 600 400 200 0 0

0.5

Figure 5-13: Effect of

1

KV KH

1.5 Injected PV

2

2.5

3

ratio on CO2 production profile.

In next stage, the effect of initial water saturation (swi) was studied. The initial water saturation had been set to zero to be consisted with the conditions of the coreflood experiments presented in Chapter 3. However in real reservoir conditions, reservoirs usually have a non-zero initial water saturation which is called connate water saturation. To understand the effect of connate water saturation on CWI performance, a sensitivity analysis was performed as presented below.

5-3.3.

Initial Water Saturation Effect

The initial water saturation in the model was increased to 10% and 20% to investigate the effect of swi on simulation results. To be able to compare the results, the same relative permeability curve should be used however; changing the swi value in the model the relative permeability curve should be changed correspondingly. To do that, the relative permeability was scaled based on the swi value using ‘ENDSCALE’ Keyword in the ECLIPSE model (ECLIPSE Manual 2014). Figure 5-14 compares the oil recovery factor predicted by E300 for the CWI at zero, 10% and 20% initial water saturations. Figure 5-14

206


shows that when the initial water saturation in the reservoir model has been increased, lower oil recovery factor has been obtained.

0.9 0.8 0.7

RF

0.6 0.5 0.4 0.3 0.2

Swi=0% Swi=10% Swi=20%

0.1 0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-14: Recovery factor of the CWI at zero, 10% and 20% initial water saturations as predicted by the E300.

Moreover the water breakthrough time occurs earlier for carbonated water injection into the reservoir with more initial water saturation. This could be due to the fact that as swi is a saturation of non-flowing water, at higher swi values, there is less pore volume available to flow and so it fills up quicker.

It is also expected that the initial water saturation also impacts the water production profile. Figure 5-15 show the total water production (TWP) profile at three different values of initial water saturation.

207


25

TWP (MMSTB)

20

15

10


5

0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-15: Total water production (TWP) profile at zero, 10% and 20% initial water saturations as predicted by the E300.

It can be seen that higher initial water saturation in the reservoir causes higher and faster water production. As mentioned before, this is due to availability of continuous paths for the water to flow through at higher swi values which makes its displacement easier.

The initial water saturation may also affect the CO2 production profile. Figure 5-16 shows the CO2 production profile at swi values of zero, 10% and 20%. It can be seen that the CO2 production profile is not changed by changing the initial water saturation. This could be due to the fact that the CO2 solubility in water compared to that in oil is very small and therefore the amount of absorbed CO2 by the initial existing water in the reservoir has not considerable effect on the CO2 production profile.

208


1800 1600

TCO2P (MMSCF)

1400 1200 1000 800 600 400


200 0 0

0.5

1

1.5 Injected PV

2

2.5

3

Figure 5-16: Total CO2 production (TCO2P) profile at zero,10% and 20% initial water saturations as predicted by the E300.

5-4.

Tertiary CWI

The performance of the secondary carbonated water injection was investigated previously. It is worth studying the performance of the tertiary CWI as well when the carbonated water is injected following to a WI. It was shown that the RF of secondary CWI is higher than the secondary WI. Therefore, the residual oil left after the secondary WI is higher compared to the secondary CWI. The objective is to ascertain the success of the tertiary CWI in recovery of the residual oil left after the water flooding process. To investigate this, a new simulation was run and WI was stopped around the breakthrough point (after injecting 0.75 pore volume of the water) and immediately after that, 3 pore volume of the carbonated water was injected (the same pore volume of the secondary CWI). Therefore, in total, 3.75 PV of the fluids were injected. It should be noted at secondary WI and CWI presented previously, 3 PV of fluid was injected in total. WI did not produce any considerable oil after 0.75 pore volume injected as shown in Figure 5-5. Therefore it was attempted to start carbonated water injection in tertiary mode after the breakthrough point of the WI. Figure 5-17 compares the RF profiles of the secondary CWI and tertiary CWI. It can be observed that the performance of CWI at the secondary 209


mode is much better than that at the tertiary mode. The graph clearly displays that when 0.75 PV of water at secondary mode was injected and was followed with the injection of 3 PV of carbonated water at tertiary mode, the same recovery factor as that of 3 PV of injection of carbonated water at secondary mode has been obtained. Moreover the figure also shows that, for instance, after the injection of 1.5 pore volumes, the recovery factor of the secondary CWI is higher than that of the tertiary CWI.

It worth mentioning that the observation seen here is consistent with the data reported in the literature (Kechut 2010, Sohrabi 2012a). That is, experimental data show that carbonated water results in better oil recovery when it is injected at the secondary mode.

0.9 0.8 0.7

RF

0.6 0.5 0.4 0.3 Secondary CWI

0.2

Secondary WI 0.1

Tertiary CWI

0 0

0.5

1

1.5 2 Injected PV

2.5

3

3.5

Figure 5-17: RF profiles of secondary and tertiary CWI.

Moreover, the distribution of final water saturation in the reservoir for the secondary WI, CWI and tertiary CWI are shown in Figure 5-18, Figure 5-19 and Figure 5-20 respectively. Figure 5-18 shows that in this homogenies model, after 3 PV injected, the water has touched almost all the reservoir area and has swept oil throughout the reservoir to its residual saturation. Figure 5-19 and Figure 5-20 show that as a result of CO2 in the 210


system during CWI, the residual oil has probably expanded and occupied the pore space containing water and then has been transported to the producer. This results in less oil left in the reservoir and more oil recovery factor.

Figure 5-18: Final water saturation distribution, secondary WI.

Figure 5-19: Final water saturation distribution, secondary CWI. 211


Figure 5-20: Final water saturation distribution, tertiary CWI.

5-5.

Performance of the Developed Simulator for a Large-Scale Model

It is worth comparing the prediction of E300 and the develop simulator for a large scale model (i.e. for real reservoirs). This will help to discover if the results of the developed simulator and E300 are consistent. As explained in the beginning of this chapter, for a large scale model where oil and water have enough time to exchange the CO 2, the equilibrium assumption can be valid. To investigate this, a large synthetic onedimensional model is created. This is because the developed simulator is based on onedimensional flow. To do that, the previous model is used with the same set-up (Table 5-6), however, the dimensions in y and z directions are reduced to 10 ft and the injection rate is set to 6.77 Rbbl/d. That is, the dimensions of the model are 1500 ft ×10 ft ×10 ft and is only discretised in ‘x’ direction. As the dimension in the x direction is very large relative to the dimensions in the y and z directions, it can be assumed that the flow is onedimensional. The injection rate is selected such that the interstitial velocity is 2 ft/day as the typical velocity of fluids in reservoirs (Chen and Wood, 2001).

Figure 5-21 shows the front view of the model.

212


Figure 5-21: The front view of the model.

The model is used and both WI and CWI are simulated. However, first a sensitivity analysis is performed to select the optimum number of the gridblocks. CWI is simulated during this experiment because of existence of CO2, finer grid size is expected. The numbers of grids (Nx) are increased from 5 to 30. Figure 5-22 shows the results of this sensitivity. It can be observed that increasing the grid numbers above 10 has no considerable effect on RF predicted by the developed simulator. This sensitivity is repeated using E300. Similar results are observed. Based on this sensitivity analysis, the model is discretised and divided into 30 gridblocks in x direction and accordingly the length of the cells in x is 50 ft.

Figure 5-22: Grid size sensitivity analysis using the developed simulator.

213


Figure 5-23 compares the RF of the WI predicted by E300 and the developed simulator. It can be observed that the developed simulator has generated the same results as that by E300. This shows the good performance of the developed simulator for a large-scale model. Next the CWI is simulated. It was explained above that for a large scale model, the assumption of instantaneous equilibrium should be valid. This was discussed when the the equilibrium number was introduced and investigated. Therefore it is expected that the developed simulator and E300 result in the same RF if the original MTC value (i.e. 5E-7 1/sec) be used. Figure 5-24 compares the RF of the CWI predicted by E300 and the developed simulator. It can be observed that the results predicted by these two different simulators are almost the same. Presumably, it may be said that this experiment has been one example which has verified the hypothesis presented above that if the CO2 have enough time to diffuse into oil phase, it can reach its equilibrium concentration in time scale of the simulation. That is, during CWI, CO2 can be transferred and distributed between water and oil phases based on its equilibrium concentration if there is enough time for this transfer process. However, it is important to run more simulations and investigate different cases with more refined girds to support this results.

0.8 0.7 0.6

RF

0.5 0.4 0.3 0.2 WI-Model

0.1

WI-E300 0 0

0.5

1 Injected PV

1.5

2

Figure 5-23: RF of the WI predicted by E00 compared to that by the devlpoed simulator. 214


0.8 0.7 0.6

RF

0.5 0.4 0.3 0.2 CWI-Model

0.1

CWI-E300

0 0

0.5

1 Injected PV

1.5

2

Figure 5-24: RF of the CWI predicted by E00 compared to that by the devlpoed simulator.

It is worth mentioning that another model could be designed to better represent a one dimensional large scale model. For example If the reservoir was thin and was a rectangle with no flow boundaries due to faults and horizontal wells were completed at either end, the flow would behave as if it was one dimensional. This could be a better scenario to apply the one dimensional model to than the case with small ‘y’ dimension mentioned above. Nevertheless, this model was not considered here as mathematically, the developed simulator cannot handle a horizontal well.

5-6.


This chapter reported the simulations results of the carbonated water injection into the reservoir. The objective was to highlight the differences between the reservoir-scale and the core-scale simulations of CWI as well as to understand the difficulties of the reservoirscale simulation of CWI using ECLIPSE 300.

215


The results of the core-scale simulations were used to create the reservoir-scale model employing ECLIPSE300 (E300) compositional simulator. The first difficulty was the assumption of the instantaneous equilibrium made be ECLIPSE. In order to use E300 for simulation of CWI, a study on the behavior of two dimensionless numbers of capillary number and a new number introduced here, was performed. The new dimensionless number called equilibrium number (Ne) was introduced by performing a sensitivity analysis using the core-scale simulation results. It was shown that during the coreflood experiments, Ne is constant and at a specific range of Ne, it can be assumed that the equilibrium state will be reached during the CWI. A synthetic reservoir model was created to meet the criteria imposed by the capillary and equilibrium numbers. The data of the water-wet coreflood experiment including rock and fluid properties (i.e. decane-CO2water system) as well as the water-oil relative permeability were used in the reservoir model as well. Mixed-wet core data was not used to reduce the uncertainties due to wettability effect which was seen in in the CWI coreflood experiment in the mixed-wet core. Selecting the number of gridblocks was another issue in this reservoir-scale model. However, by performing a sensitivity analysis, the optimum numbers of the gridblock were selected. In E300, it was not possible to define directly the carbonated water stream for the injection well. Therefore two separate injection wells were defined.

Secondary CWI was compared with secondary WI using simulation results. It was realised that the CWI produces higher oil recovery compared to the WI in the reservoir model as well. The CO2 storage profile was also investigated. E300 showed that at end of the simulation, 65% of CO2 remained in the reservoir. Next, the role of

KV KH

ratio in the

simulation of the CWI was studied. The simulation results were insensitive to the three different

KV KH

ratios investigated here. In all simulations, the initial water saturation was

zero however a sensitivity analysis was carried out and the effect of initial water saturation was investigated. Higher oil recovery factors were obtained at lower initial water saturations.

Moreover, tertiary injection of the carbonated water was examined and compared with the secondary CWI. It was observed that the secondary CWI could produce better recovery factor which was consistent with the experimental data reported in the literature. Finally, a one-dimensional large-scale model was created and WI and CWI were 216


simulated using both the developed simulator and E300. The goal was to investigate the performance of the developed simulator for large scale models. The RF predicted by E300 were the same as those predicted by the developed simulator for both WI and CWI processes. This exercise could also verify the hypothesis suggested based on the results obtained by the introduced equilibrium number. That is, for large scale models, the assumption of equilibrium is valid. In other words, it can be concluded that when carbonated water is injected into real reservoirs, there is enough time for CO2 to be transferred and distributed between oil and water phases based on its equilibrium concentration.

217

Summary and Conclusions This chapter gives a summary of this study with presenting the main conclusions. In addition, some recommendations for future work are also made in this chapter.

This thesis presented a mathematical study on CWI process as a CO2-EOR technique. That is, mathematical modelling and numerical simulation of the carbonated water injection (CWI) process at both core-scale and reservoir-scale was the main objective of this study.

In this study, it was attempted to simulate the CWI coreflood experiments correctly by including the actual physics happening during the CWI process. Moreover, the results from core-scale simulation was used to study CWI process at reservoir-scale.

6-1.


When carbonated water (CW) contacts oil during carbonated water injection into an oil reservoir, CO2 migrates from water into oil. This happens because CO2 has higher solubility in hydrocarbons compared to water. CO2 improves the mobility of oil by increasing its volume (oil swelling) and reducing its viscosity. CO2 can also alter the wettability of the rock during CWI. To simulate this process, compositional simulation approach should be used in order to capture the physics of the CO2 exchange between the water and oil phases. According to both experimental and mathematical data presented in the literature, the effect of rock wettability and also the wettability alteration phenomenon, is not studied and understood very well during CWI process. That is, the role of rock wettability in CWI performance and the contribution of wettability alteration mechanism are still uncertain. Moreover the mechanism of CO2 transfer between oil and water phases occurring during CWI process is not studied comprehensively whilst it is very important for field-scale simulation of CWI process. That is, with respect to simulation of CWI, based on some observation during coreflood experiments presented in the literature, the rate of CO2 transfer is slow making the assumption of instantaneous equilibrium used by available compositional simulators invalid (Kechut et al., 2011b; Kechut, 2011c). Therefore, the assumption that thermodynamic equilibrium is reached

218

Chapter 6: Summary and Conclusions

during CWI, as simulated, is questionable and there is not a kinetic model available to investigate this. The main objective of this thesis was to explore the importance of kinetics of CO2 transfer during core-scale and reservoir scale simulation of CWI. Hence, a new compositional simulator was developed where the assumption of instantaneous equilibrium was relaxed. Moreover, the aim was to use the simulator (model) to discover the important mechanisms contributing towards incremental oil recovery during CWI. Particularly, the oil swelling and wettability alteration/modification mechanisms were studied. A GA-based optimiser was also employed to obtain the unknown parameters during CWI simulation by history matching the available experimental production data. Mass transfer coefficient (MTC) and Kr/Pc curves were obtained by the optimiser. Moreover, the role of oil swelling and wettability was understood by adjusting the MTC and relative permeability curves.

Two different sets of WI and CWI coreflood experiments were selected from the literature to be studied comprehensively using the developed simulator and E300. The first set of coreflood experiments had been performed in a water-wet sandstone core. The second set of coreflood experiments had been performed in an aged sandstone core. Different mechanisms were concluded for the water-wet and aged cores. Based on the results of this thesis, oil swelling, viscosity reduction and wettability alteration are amongst the important mechanisms resulting in incremental oil recovery during CWI. However, their roles are not the same. Reduction of oil viscosity provides a more favourable mobility ratio. Through the experiments studied here, standard decane was used as the oil sample. The viscosity of decane is very small and its reduction did not have any considerable effect on oil recovery. It seems that the original viscosity of dead oil (viscosity of oil without and dissolved CO2) and the level of its reduction due to CO2 dissolution will determine the importance of the viscosity reduction mechanism during CO2-EOR projects. For heavy and viscous oils, the viscosity reduction can be an important mechanism as the level of viscosity reduction should be high (Dong et al., 2011). Miller and Jones (1981) observed a viscosity reduction from 7000 cp to 100 cp for a 10º API oil at 140ºF when saturated with CO2. Oil swelling can be important during CO2-EOR projects and particularly during CWI. Oil swelling helps with more oil recovery in two ways. First, the amount of residual oil left in reservoir is inversely proportional to the swelling factor. Larger swelling factor (higher oil swelling) means less 219


oil in reservoir. Second, swollen oil droplets have higher volumes in pore space, i.e., larger saturation, resulting in higher oil relative permeability (Green and Willhite, 1998; Dong et al., 2011). In this study, during CWI, 15% swelling of decane was estimated due to CO2 dissolution resulted in an additional oil recovery factor of 4% over conventional water injection in water-wet core. In the aged core, in additional to oil swelling, it was concluded that wettability alteration will also play an impotent role. A change in breakthrough point and a shift in pressure drop curve observed during CWI in comparison with WI that were attributed to the change in the wettability of the rock due to CO2 effect. It is worth mentioning that, according to a personal communication with involved people (Mr Mojtaba Seyyedi, Mr Amir Farzaneh), this behavior has been observed for other aged core experiments. The aged core had been claimed to be mixed-wet by the original authors however it was suggested here that the intermediate-wet could be a more proper term. The importance of wettability alteration, is high when the original formation wettability is close to neutral, i.e. intermediate-wet. The wettability change of rock towards water wet enables later water breakthrough, higher recovery factor under the same injected pore volume (Ligtlelm, 2008; Masalmeh et al., 2014).

Comparing the experimental production data of CWI and WI in the water-wet core, a higher oil recovery after breakthrough point was observed which could be captured by adjusting MTC value in the simulator. However, in the aged core, a higher oil recovery due to a shift in breakthrough point together with a reduction in pressure drop across the core was observed. These could be captured by simulator through adjustment of MTC and Kr curves. However these should be verified with simulation of more coreflood experiments. The role of MTC in the simulator was explored and it was realised that MTC mainly increases the oil recovery factor after the breakthrough point with minimal effect on pressure drop data. MTC controls the rate of mass transfer between phases. That is, MTC controls how much CO2 is transferred between oil and water phases at a specific length of time during CWI. With respect to MTC value, it can be obtained for a specific CWI coreflood experiments by history matching of production data if all other necessary parameters including Kr curves are available. However, it should be mentioned that the accuracy of input parameters including Kr curves and fluid properties of CO2-oil and CO2-water mixtures can affect the accuracy of results predicted by the simulator. Nevertheless, this is the same for other available simulators such as ECLIPSE and it is 220


necessary to have accurate Kr values and measured fluid properties to reduce the level of inaccuracy in results predicted by the simulator.

It should be noted that in this thesis, MTC and Kr/Pc curves were obtained through a history matching experiment. That is, through an inversion study, these input parameter were calibrated such that the simulator could predict the same production data as those from the experiment. As a results, it is important to consider carefully the uncertainties of the inversion to understand if the answer is unique or not. More GA can be run with different initial samples and different combinations of Kr, Pc and MTC values to investigate the uniqueness and the uncertainty of parameters obtained. Moreover, it is useful to investigate how the uncertainty of input parameters could affect the output data predicted by the simulator. Furthermore, the measurement error and experimental uncertainty also affect inversion process as it dictates when the obtained answer is acceptable. If the measurement error is large, it is difficult to get a close match between experimental and predicted results. Therefore it is important to know the level of uncertainty in experimental data before history matching process.

Moreover, it could be useful to investigate the Kr curves fitted CWI experiment to see if they could also fit WI experiments. This could help to assure if the Kr curves of WI and CWI are really different in mixed-wet core and are the same in water-wet core as presented in previous chapters.

During CO2-EOR projects, a large volume of the injected CO2 can also be permanently stored underground. This technology has been used successfully in North America for a number of decades (Mezler, 2010; Uddin et al., 2013; Stewart and Haszeldine, 2014). The CO2 production and storage profiles were also studied in this study. The simulation results showed that CO2 displacement in oil phase during CWI was slower than that in water phase. This means that if the system is relatively large such that there is enough contact time for phases to exchange CO2 before leaving the system, more CO2 will be transferred to oil phase. As a results, less CO2 will be produced and more CO2 will stay in the system (more storage).

The effects of dispersion, injection rate and carbonation level on oil recovery during CWI were studied at core-scale. The simulation results showed that for the systems studied 221


here, the effect of dispersion was minimal. Perkins and Johnson (1963) studied the diffusion mechanism in porous media and they concluded that diffusion is important mechanism at low rates and convection controls the displacement at high rates. In addition, CO2 can also diffuse in transverse direction to the flow by the dispersion mechanism and thus recovers some oil which can be bypassed by the flowing carbonated water stream. That is, the effect of dispersion can be a higher oil recovery factor. However this could not be captured here, as the model was one-dimensional.

Furthermore, at low injection rates, higher oil recovery was obtained during core-scale simulation of CWI process, compared to high injection rates. Additionally, at higher carbonation levels, more oil recovery was obtained.

The core-scale simulation of CWI was investigated using ECLIPSE300 (E300) compositional simulator. E300 predicted higher oil recovery and lower and delayed CO2 production, compared to those predicted by the developed simulator for both the waterwet and aged core systems.

The results of the core-scale simulations were used to study the reservoir-scale simulation of CWI employing ECLIPSE300 (E300) compositional simulator. A new dimensionless number called Equilibrium number (Ne) was introduced by performing a sensitivity analysis using the core-scale simulation results. It was shown that during the coreflood experiments, Ne is constant and at a specific range of Ne, it can be assumed that the thermodynamic equilibrium state can be achieved during the CWI. The main conclusion was that E300 can be used to simulate CWI at reservoir-scale without any concern about equilibrium conditions. That is, the value of MTC is not important at reservoir-scale. However for core-scale simulation of CWI, first Ne number should be checked. In other words, if the dimensions of the core and the injection rate are such that the equilibrium conditions presented by Ne are met, E300 can be used.

It should be noted that, it was not possible in E300 to define directly the carbonated water stream for the injection well. Therefore two separate injection wells were defined.

Secondary CWI resulted in higher oil recovery compared to secondary WI in the reservoir model. In addition, the simulation results showed that higher initial water saturation in 222


the reservoir would cause a lower RF by CWI. Moreover, secondary CWI resulted in better performance compared to secondary CWI.

The main conclusions based on simulation results can be summaries as follows: 

CWI could produce higher oil recovery over conventional WI for both water-wet and aged cores at secondary injection mode.



Oil swelling was the main mechanism of oil recovery improvement during carbonated water injection into the water-wet core.



Oil swelling and wettability modification were the main mechanisms of oil recovery improvement during carbonated water injection into the aged core.



The developed simulator based on the non-equilibrium assumption could properly simulate the CWI coreflood experiments performed in the mixed-wet (aged) and water-wet cores.



The simulator was developed when convection and dispersion mechanisms were included. A mass transfer term was added to capture the kinetics of CO2 transfer between the phases.



A mass transfer coefficient (MTC) was introduced to control the rate of CO2 transfer from water to oil during CWI simulation.



Oil swelling mechanism could be captured by adjusting of MTC during simulation while wettability modification was captured by tuning of relative permeability curves.



Higher MTC values resulted in higher oil production with minimal effect on DP data.



With dispersion, the simulator predicted that the oil production decreased slightly.



At low injection rates, compared to high injection rates, a higher oil recovery was obtained.



At low injection rates, because oil and water were in contact for a longer time, more CO2 could transfer into the oil phase.



At higher carbonation levels, more oil was produced.



The developed simulator could also be used to predict the CO2 production profile.



The developed simulator showed that the CO2 breakthrough time of water and CO2 were the same. It also demonstrated that the main CO2 production at the core outlet is from the water and not from the oil stream. 223




Higher MTC values resulted in a lower and delayed CO2 production.



Higher dispersion coefficients resulted in earlier CO2 production; however, this was not considerable.



Convection was the dominant mechanism in these systems compared to the dispersion, causing the effect of dispersion to be insignificant.



Dispersion could be more important at low injection rates.



With dispersion, a more uniform CO2 concentration profile in oil phase was obtained.



At low injection rates, delayed and lower CO2 production was obtained compared to the corresponding high injection rates.



CO2 concentration in oil phase at low injection rates was higher.



The simulator could also predict the amount of CO2 stored during the CWI experiment.



Before the breakthrough time of CO2, all of the injected CO2 were stored in the system while after the CO2 breakthrough time, the profile of stored CO2 declined gradually.



It was also estimated that, at end of the experiment, around 44 % of the injected CO2 could be stored in both the water-wet and aged cores.



At low injection rates and high MTC values, more CO2 could be stored in the core, as more CO2 could transfer into the oil phase.



ECLIPSE300 (E300) compositional simulator was also used and it predicted a higher oil recovery and lower and delayed CO2 production, compared to those predicted by the developed simulator for both the water-wet and aged core systems.



The simulation results by E300 were insensitive to the injection rate.



At high MTC values, the results by E300 and the developed simulator were very close.



A new dimensionless number called Equilibrium number (Ne) was introduced.



At a specific range of Ne, it could be assumed that the equilibrium conditions could be achieved during the CWI. It was concluded that the equilibrium conditions would be met at reservoir-scale. Consequently, E300 can be used to simulate CWI at reservoir-scale.



Reservoir-scale simulation of CWI was also investigated.

224




The assumption of instantaneous equilibrium made by E300, defining the carbonated water stream explicitly for injection well, selecting the optimum number of gridblocks were the main difficulty during reservoir-scale simulation of CWI.



At reservoir-scale, secondary CWI could produce higher oil recovery compared to secondary WI.



Initial water saturation had a negative effect on CWI performance.



At reservoir-scale, secondary CWI could produce better oil recovery factor compared to tertiary CWI.

It should be noted that the results of this thesis may not be applied directly to a typical EOR project in real reservoirs. The main reasons are that the real reservoirs are usually heterogeneous while they are saturated with live oils and also may have aquifer support or gas cap. Furthermore, contrary to the models studied here, the real reservoirs do not have a specific geometric shape and may have some faults and fractures while these are not considered here.

The economics and operational conditions of CO2-EOR projects are also important. With respect to screening criteria, design and facility needed, it seems that CWI process is similar to CO2-WAG (water alternating gas) technique. However there are two different features differentiating CWI from WAG process. First, CWI needs less volume of CO2 during injection and the amount of CO2 used in CWI will not exceed the soluble level of CO2 in the injected brine/water under the reservoir pressure and temperature conditions. As a results, there should not be separate CO2 rich phase in reservoir during CWI. Second, mass transfer controls the process of CO2 distribution between two immiscible phases (water and oil). Therefore, the displacement efficiency of CWI, is not dictated by the minimum miscibility pressure, which is a function of several parameters, including CO2 purity, reservoir temperature and oil composition (Dong et al., 2011).CO2-WAG and its field application has been the subject of several studies reported in the literature (Kane, 1979; Fullbright et al., 1996; Hsie and Moore, 1988; Prieditis et al., 1991; Hancock, 1999; Elwy Amin et al., 2012; Duchenne et al., 2014).

With respect to the cost of CO2-EOR projects, the major factors affecting the profitability of CO2-EOR are the availability of CO2 at economic prices (generally within 2-3 225


$/MSCF) and the net utilisation ratio of CO2 per barrel of additional oil recovered (Elwy Amin et al., 2012). It is obvious that, the net utilization of CO2 for optimised EOR projects will vary from field to field. Nevertheless, in a US EOR overview study by Broome et al. (1984), it has been estimated at 5.5 MSCF CO2 per additional barrel. In a more recent study by Jeschke et al. (2000), it has been estimated to be between 4-6 MSCF/barrel. It should be noted that the design of a success CO2-EOR project depends on various parameters including, geology and geography of the area, availability of CO2, distance of CO2 source to oil field, oil price, the surface facility etc. Therefore, the economics and feasibility of each EOR project should be studied individually. For instance, CWI rather than pure CO2 injection is recommended for offshore projects with limited access to CO2 (Sohrabi et al., 2009, 2011a and 2011b).

Finally, corrosion and safety are also important issues during CO2-EOR projects needed to be taken into account during designing the projects. Under the conditions of injecting pure CO2, the corrosion rate of strings in CO2 injection wells is very low, whereas in CO2WAG and CWI processes, because CO2 is mixed with water, corrosion rate of downhole strings is high. This is extensively discussed in the literature (Jun et al., 2013; Meyer, 2007).

6-2.

Recommendations

In this study, a new compositional simulator (model) was developed to capture the kinetics of the CO2 transfer between the water and oil phases during the CWI process. To develop the model, some assumptions were made. For future work, the model can be improved by relaxing some of these assumptions as described below.

The model was developed to simulate the coreflood experiments when the core is positioned horizontally (no gravity effect). Therefore, the developed model was onedimensional as it was assumed that the flow in the core is one-dimensional and is in the direction of the core axis. The model can be extended to be used for the real reservoirs where the flow is three-dimensional. Therefore it is recommended to extend it to a threedimensional model when the effect of gravity is also included. Moreover, a threedimensional model can capture the effect of dispersion better as it was explained in the 226


previous chapters. Furthermore, if a three dimensional model can be developed, it is very useful and important to make very refined large scale models and check the equilibrium conditions by comparing the results of the three dimensional developed model and E300 similar to that experiment performed in last section of Chapter 5.

The model was developed when it was assumed that the oil is a dead oil sample without any dissolved gas. However, the real reservoir fluids are live samples with some dissolved gas. Therefore in real reservoirs, it is possible to have a free gas phase in the system during the CWI process. For instance, the methane content of the oil can be replaced by CO 2 during CWI and be liberated as a free gas. Hence, the model can be extended and a governing equation can be added to represent the flow dynamic of a gas phase. However, when the numbers of the components and phases are increased in the system, it is not easy to include the kinetic transfer of the components between the phases.

In this study, it was assumed that the mass transfer coefficient (MTC) is a tuning parameter. However the MTC can also be estimated using some theoretical methods. Therefore it is suggested to study the theoretical methods available in the literature and to estimate the MTC theoretically (Steffens, 2010; Valiollahi et al., 2012; Embid and Rivas, 1994). This can help to evaluate the MTC values obtained by history matching in this study. In addition, study of the theoretical methods for calculation of the MTC can help to find about the parameters which affect the MTC value in a specific system.

In this study, the developed model was employed and two different CWI coreflood experiments were simulated. The main recommendation is to use the developed model and simulate more coreflood experiments to further verify the performance of the simulator and particularly the procedures suggested for the history matching of the production data of CWI experiments in the water-wet and mixed-wet cores. Simulation of more CWI experiments in aged cores can help to know more about the wettability alteration mechanism during CWI.

227

Appendix A: Details of Equations and Solution Technique Equation of oil component in oil phase:

Roo n+1 =∆x [ρo ωoo λo ∆x po ]n+1 - φ ∆t [ρo (1-sw )ωoo ]

(A − 1)

Equation of CO2 component in oil phase: co2 n+1

Ro

=∆x [ρo (1-ωoo )λo ∆x po ]n+1 +φ D co2 -o ∆x [(1-sw )∆x ρo (1-ωoo )]n+1

+U n+1 - φ ∆t [ρo (1-sw )(1-ωoo )]

(A − 2)

Equation of water component in water phase: co

co

n+1 Rw =∆x [ρw (1-ωw 2 )λw ∆x (po -pc )] n+1 - φ ∆t [ρw sw (1-ωw 2 )] w

(A − 3)

Equation of CO2 component in water phase: co n+1

Rw 2

co

=∆x [ρw ωw 2 λw ∆x (po -pc )]

n+1

co

n+1

+φ D co2-w ∆x [sw ∆x (ρw ωw 2 )]

co

-U n+1 -φ ∆t [ρw sw ωw 2 ]

(A − 4)

Summation of Equations A-1 and A-2: R on+1 =∆x [ρo λo ∆x po ]n+1 +φ D co2 -o ∆x [(1-sw )∆x ρo (1-ωoo )]n+1 +U n+1 -φ∆t [ρo (1-sw )]

(A − 5)

Summation of Equations A-3 and A-4: co

n+1

R wn+1 =∆x [ρw λw ∆x (po -pc )]n+1 +φ D co2-w ∆x [sw ∆x (ρw ωw 2 )] -φ∆t [ρw sw ]

228

-U n+1

(A − 6)

Appendix A: Details of Equations and Solution Technique

It should be noted that these summations could help with the convergence problem during the solution of the equations.

Discretising Equation A-2:

co n+1 Ro i 2

n+1 n+1 (2 − ωoo n+1 − ωoo n+1 i i+1 ) ρo i λo i n+1 [pn+1 = o i+1 − po i ] 2Δx 2

n+1 n+1 (2 − ωoo n+1 − ωoo n+1 i i−1 )ρo i−1 λo i−1 n+1 [pn+1 + o i−1 − po i ] 2 2Δx

+

φDco2 −o (1−sw )n+1 i ∆x2

φDco2 −o (1−sw )n+1 i−1 ∆x2

−φ

o n+1 ] [(ρo (1 − ωoo ))n+1 + i+1 − (ρo (1 − ωo ))i

(A − 7)

o n+1 [(ρo (1 − ωoo ))n+1 ] + Uin+1 i−1 − (ρo (1 − ωo ))i

n+1 n o n o n+1 ((1−ωo −((1−ωo o )ρo )i o )ρo )i + (ρo sw (1−ωo ))i − (ρo sw (1−ωo ))i ∆t

Discretising Equation A-4: n+1

n+1 Rco2 = w

n+1 n+1 n+1 (ωco2 +ωco2 wi w i+1 ) ρw i λw i

2Δx2

n+1

+

n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 ) ρw i−1 λw i−1

2Δx2

φDco2 −w sw n+1 i ∆x2 φDco2 −w sw n+1 i−1 ∆x2

−φ

n+1 n+1 n+1 [pn+1 o i+1 − po i + pc i − pc i+1 ]

n+1 n+1 n+1 [pn+1 o i−1 − po i + pc i − pc i−1 ] +

co

co

2 n+1 [(ρw ωw 2 )n+1 ]+ i+1 −(ρw ωw )i

co

(A − 8)

co

2 n+1 [(ρw ωw 2 )n+1 ] − Uin+1 i−1 − (ρw ωw )i

n+1 n (ρw ωco2 − (ρw ωco2 w sw )i w sw )i

∆t


R on+1 i +

n+1 n+1 ρn+1 ρn+1 o i λo i o i−1 λo i−1 n+1 n+1 n+1 [po i+1 − po i ] + [pn+1 = o i−1 − po i ] Δx 2 Δx 2


φDco2 −o (1−sw )n+1 i−1 ∆x2

−φ

o n+1 ] [(ρo (1 − ωoo ))n+1 + i+1 − (ρo (1 − ωo ))i

o n+1 [(ρo (1 − ωoo ))n+1 ] + Uin+1 i−1 − (ρo (1 − ωo ))i

n n n+1 ρn+1 o i −ρo i + (ρo sw )i − (ρo sw )i

∆t

229

(A − 9)



R wn+1 i =

n+1 ρn+1 w i λw i

Δx2

n+1 ρn+1 w i−1 λw i−1 [pn+1 o i−1 Δx2

φDco2 −w sw n+1 i ∆x2 φDco2 −w sw n+1 i−1 ∆x2

φ

n+1 n+1 n+1 [pn+1 o i+1 − po i + pc i − pc i+1 ] + n+1 n+1 − pn+1 o i + pc i − pc i−1 ] + co

co

(A − 10)

2 n+1 [(ρw ωw 2 )n+1 ]+ i+1 −(ρw ωw )i

co

co

2 n+1 [(ρw ωw 2 )n+1 ] − Uin+1 − i−1 − (ρw ωw )i

(ρw sw )n+1 − (ρw sw )n i i ∆t

The components of Jacobin matrix are as follows: (A − 7):

n+1 ∂Rco2 oi

∂pn+1 o i+1

n+1

=

n+1 n+1 n+1 (2−ωo −ωo oi o i+1 ) ρo i λo i

+

2Δx2


(1 − (A − 11)

∂ρn+1 o i+1 ωoo )n+1 i+1 ∂pn+1 o i+1

n+1 ∂Rco2 oi

∂pn+1 oi

n+1

=−

n+1 n+1 n+1 (2−ωo −ωo oi o i+1 ) ρo i λo i

n+1

n+1 n+1 (2−ωo −ωo oi o i+1 )λo i

2Δx2

-[

φDco2 −o (1−sw )n+1 i

−φ

∆x2

+

2Δx2

∂ρn+1 oi

n+1 n+1 n+1 [po i+1 − po i ] −

∂po i

+

φDco2 −o (1−sw )n+1 i−1 ∆x2

](1 −

n+1

n+1 n+1 n+1 (ωo +ωo oi o i−1 )ρo i−1 λo i−1 2Δx2

∂ρn+1 ∂Un+1 oi i ωoo )n+1 + n+1 i ∂po i ∂pn+1 oi

(A − 12)

n+1 ∂ρn+1 (1−ωo (1− sw )n+1 o )i oi i

∂pn+1 oi

∆t

n+1

n+1 (2−ωo n+1 −ωo n+1 )ρn+1 λ ∂Rco2 oi o i−1 o i−1 o i−1 oi = n+1 ∂po i−1 2Δx2 n+1

n+1 n+1 n+1 (2−ωo −ωo oi o i−1 ) λo i−1 ∂ρo i−1 2Δx2 ∂pn+1 o i−1

ωoo )n+1 i−1

+ φDco2 −o (1−sw )n+1 i−1

n+1 [pn+1 o i−1 − po i ]+

∂ρn+1 o i−1

∂pn+1 o i−1

230

∆x2

(1 −

(A − 13)


n+1 ∂Rco2 oi

∂sn+1 w i+1

n+1 ∂Rco2 oi

∂sn+1 wi

φDco2 −o

=

n+1 ∂Rco2 oi

∂sn+1 w i−1

=

φDco2 −o

2Δx2

n+1 ∂Rco2 oi n+1 ∂ωo o i+1

ωoo ))n+1 i−1 n+1

=−

ρn+1 o I λo i 2Δx2

∂ρn+1 o i+1 ωoo )n+1 i+1 ∂ωo n+1 o i+1

n+1 ∂Rco2 oi n+1 ∂ωo oi

∂λo n+1 i ∂sn+1 wi

− (ρo (1 −

− ρn+1 o i+1

n+1

=(

2Δx2

ρn+1 o i−1 λo i−1 2Δx2

∂ρn+1 oi

n+1 ∂ωo oi

∆t

pn+1 oi ]+

n+1 ∂Rco2 oi

n+1 ∂ωco2 w i+1

n+1 [pn+1 o i−1 − po i ] +

φDco2 −o (1−sw )n+1 i−1 ∆x2

(A − 16)

∆x2

(A − 17)

n+1

−

ρn+1 o i λo i 2Δx2


+φ

∂ρn+1 oi n+1 ∂ωo oi

n+1 )[pn+1 o i+1 − po i ] +

+

] +

(A − 18)

∂Un+1 i n+1 ∂ωo oi

n+1 ρn+1 o i (1− sw )i

∆t

n+1

n+1

− pn+1 oi ]−


n+1 (2−ωo n+1 −ωo n+1 )λ n+1 ∂Rco2 o i−1 ∂ρo i−1 oi o i−1 n+1 oi = n+1 n+1 [po i−1 ∂ωo 2Δx2 ∂ωo o i−1 o i−1

ρn+1 o i−1 λo i−1 2Δx2

∆t

− pn+1 oi ]

o n+1 ) [ρn+1 o i − (1 − ωo )i

n+1 ∂ρn+1 (1−ωo (1− sw )n+1 o )i oi i

(A − 15)

ωoo ))n+1 ] i

n+1 ∂ωo oi

n+1 [pn+1 o i−1 − po i ] + (

φDco2 −o (1−sw )n+1 i−1 ∆x2

n+1 (ρo (1−ωo o ))i

]

n+1 n+1 n+1 n+1 (2−ωo −ωo n+1 oi o i+1 )ρo i ∂λo i n+1 [po i+1 2Δx2 ∂ωo oi

−φ

n+1 [pn+1 o i+1 − po i ] −

n+1 [pn+1 o i+1 − po i ] +

n+1 n+1 (2−ωo −ωo oi o i+1 )λo i

n+1

−

ρn+1 oi

n+1 n+1 ∂λo n+1 (2−ωo −ωo n+1 oi o i−1 ) n+1 i−1 ρ o i−1 ∂sn+1 [po i−1 2Δx2 w i−1

[(ρo (1 −

∆x2

+

n+1 n+1 (2−ωo −ωo oi o i+1 )

o n+1 ] [(ρo (1 − ωoo ))n+1 +φ i+1 − (ρo (1 − ωo ))i

∆x2

[(1 −

(A − 14)

=0

− pn+1 o i ]-

n+1 n+1 n+1 (2−ωo −ωo oi o i−1 )ρo i−1

2Δx2

[(1 − ωoo )n+1 i−1

∂ρn+1 o i−1

n+1 ∂ωo o i−1

=0

∂λo n+1 i−1

n+1 ∂ωo o i−1

[pn+1 o i−1 −

(A − 19)

− ρn+1 o i−1 ]

(A − 20) 231


n+1 ∂Rco2 oi

∂Un+1

(A − 21)

i = ∂ωco2 n+1 ∂ωco2 n+1 wi

wi

co2

∂Ron+1 i

n+1 ∂ωco2 w i−1

(A − 22)

=0

(A − 8): n+1

n+1 (ωco2 n+1 +ωco2 n+1 ) ρn+1 λ ∂Rco2 w i+1 w i wi w = wi ∂pn+1 2Δx2 o i+1

n+1 ∂Rco2 w

∂pn+1 oi

=−

− -(

∆x2

n+1 co n+1 ∂ρo i+1 ∂pn+1 o i+1

ωw 2 i+1

(A − 23)

2Δx2

n+1 n+1 (ωco2 +ωco2 wi w i+1 )λw i

∂ρn+1 wi

∂pn+1 oi

2Δx2


n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 )ρw i−1 λw i−1

(A − 24)

2Δx2

φDco2 −w sw n+1 i

φ


n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i+1 )ρw i λw i

n+1

+

+

∆x2 n+1 n+1 ωco2 sw i wi

∆t

+

φDco2 −w sw n+1 i−1 ∆x2

∂ρn+1

∂Un+1

n+1 w i i )ωco2 − ∂pn+1 − wi ∂pn+1 oi

oi

∂ρn+1 wi ∂pn+1 oi

n+1

n+1 (ωco2 n+1 +ωco2 n+1 ) ρn+1 λw ∂Rco2 w i−1 w i−1 w i−1 = wi ∂pn+1 2Δx2 o i−1 n+1

+

n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 )λw i−1 ∂ρo i−1 2 2Δx ∂pn+1 o i−1


n+1 ∂Rco2 w

∂sn+1 w i+1

n+1 ∂Rco2 w

∂sn+1 wi

n+1 n+1 n+1 [pn+1 o i−1 − po i + pc i − pc i−1 ] +

n+1 co n+1 ∂ρo i−1 ∂pn+1 o i−1

ωw 2 i−1

n+1

=−

=

(A − 25)


2Δx2

∂pn+1 c i+1

∂sn+1 w i+1

n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i+1 )ρw i ∂λw i [pn+1 o i+1 2Δx2 ∂sn+1 wi n+1


2Δx2

∂pn+1 ci ∂sn+1 wi

n+1 n+1 − pn+1 o i + pc i − pc i+1 ] +

(A − 27)

+

n+1

n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 ) ρw i−1 λw i−1 ∂pc i n+1 2 2Δx ∂sw i

−φ

(A − 26)

n+1 n+1 ωco2 ρw i wi

∆t

232


n+1 ∂Rco2 w

∂sn+1 w i−1

=

co

n+1 ∂ωo o i+1

n+1 ∂Rco2 w n+1 ∂ωo oi

co

(A − 29)

=0

∂Un+1

=− ∂ωoin+1

(A − 30)

=0

(A − 31)

oi

n+1 ∂Rco2 w n+1 ∂ωo o i−1

n+1 ∂Rco2 w

n+1 ∂ωco2 w i+1

n+1

=

ρn+1 w i λw i 2Δx2

φDco2 −w sw n+1 i ∆x2

= ∂ωco2 n+1

[ρn+1 w i+1 +

pn+1 c i+1 ] +

2Δx2

n+1

+

n+1

ρn+1 w i λw i−1 2Δx2

2Δx2

n+1 ∂ωco2 wi

n+1 n+1 [pn+1 o i+1 − po i + pc i −

∆x2


∂ρn+1

(A − 33)


∂Un+1

∆x2

co2 n+1 wi ) [ρn+1 ] − ∂ωco2i n+1 w i + ωw i ∂ωco2 n+1 wi

n+1 n+1 ωco2 sw i wi

∆t

n+1 − pn+1 o i + pc i −

n+1 n+1 n+1 [pn+1 o i−1 − po i + pc i − pc i−1 ] − (

φDco2 −w sw n+1 i−1

−φ

∂ρn+1 oi

n+1 n+1 n+1 ∂λ n+1 (ωco2 +ωco2 wi n+1 wi w i+1 )ρw i n+1 [po i+1 2Δx2 ∂ωco2 wi

ρn+1 w i λw i

(A − 32)

n+1 n+1 ∂ρo i+1 ωco2 co2 n+1 ] w i+1 ∂ω w i+1

n+1 n+1 (ωco2 +ωco2 wi w i+1 )λw i

wi

+


n+1

n+1 ∂Rco2 w

pn+1 c i+1 ]

(A − 28)

+

2 n+1 [(ρw ωw 2 )n+1 ] i−1 − (ρw ωw )i

∆x2

n+1 ∂Rco2 w

n+1 − pn+1 o i + pc i −

n+1

n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 ) ρw i−1 λw i−1 ∂pc i−1 2Δx2 ∂sn+1 w i−1

pn+1 c i−1 ] − φDco2 −w

n+1 n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 )ρw i−1 ∂λw i−1 [pn+1 o i−1 2Δx2 ∂sn+1 w i−1

∂ρn+1 wi

n+1 ∂ωco2 wi

−φ

wi

n+1 ρn+1 w i sw i

∆t

233

+


n+1

n+1 n+1 (ωco2 n+1 +ωco2 n+1 )λw ∂Rco2 wi w i−1 w n+1 i−1 ∂ρw i−1 co2 n+1 = co2 n+1 [po i−1 ∂ωw i−1 2Δx2 ∂ωw i−1 n+1

ρn+1 λw i−1 + w i−1 2Δx2

+

n+1 n+1 n+1 [pn+1 o i−1 − po i + pc i − pc i−1 ]

n+1 n+1 n+1 (ωco2 +ωco2 wi w i−1 )ρw i−1

2Δx2

φDco2 −w sw n+1 i−1

+

∆x2

n+1 n+1 − pn+1 o i + pc i − pc i−1 ]

∂λw n+1 i−1

[pn+1 o i−1 ∂ωco2 n+1 w i−1

−

(A − 34)

pn+1 oi

+

pn+1 ci

−

pn+1 c i−1 ]

∂ρn+1

co2 n+1 o i−1 [ρn+1 w i−1 + ωw i−1 ∂ωco2 n+1 ] w i−1

(A − 9): ∂Ron+1 i

∂pn+1 o i+1

n+1 ρn+1 o i λo i

=

Δx2

∂Ron+1 i

=− ∂pn+1

n+1 ρn+1 o i λo i

−( φ


((1 − ωoo )n+1 i+1

∂ρn+1 o i+1

∂pn+1 o i+1

(A − 35)

)

n+1 ρn+1 o i−1 λo i−1 Δx2

+

φDco2 −o (1−sw )n+1 i−1 ∆x2

∂ρn+1

∂Un+1

oi i )(1 − ωoo )n+1 + ∂pn+1 − i ∂pn+1 oi

(A − 36)

oi

∂ρn+1 (1− sw )n+1 oi i

∂pn+1 oi

∆t

∂Ron+1 i

∂pn+1 o i−1

+

∆x2

−

Δx2

oi


+

n+1 ρn+1 o i−1 λo i−1 Δx2

=

φDco2 −o (1−sw )n+1 i−1 ∆x2

∂Ron+1 i

∂sn+1 w i+1

∂Ron+1 i ∂sn+1 wi

+

n+1 λo n+1 i−1 ∂ρo i−1

((1 −

(A − 37)

n+1

∂ρo i−1 ωoo )n+1 i−1 ∂pn+1

o i−1

)

(A − 38)

n+1

ρn+1 o i ∂λo i Δx2

∂sn+1 wi

n+1 [pn+1 o i+1 − po i ] −

]+φ (ρo (1 − ωoo ))n+1 i

∂sn+1 w i−1

n+1 [pn+1 o i−1 − po i ]

=0

=

∂Ron+1 i

Δx2 ∂pn+1 o i−1

=

(ρo (1 −

φDco2 −o ∆x2

[(ρo (1 − ωoo ))n+1 i+1 −

ρn+1 oi

(A − 39)

∆t

n+1 ρn+1 o i−1 ∂λo i−1 [pn+1 o i−1 Δx2 ∂sn+1 w i−1

− pn+1 oi ]−

φDco2 −o ∆x2

] ωoo ))n+1 i

234

[(ρo (1 − ωoo ))n+1 i−1 −

(A − 40)



∂Ron+1 i

= ∂ωo n+1

∆x2

o i+1

−

∂ρn+1

o i+1 (1 − ωoo )n+1 i+1 ∂ωo n+1 o i+1

(A − 41)

φDco2 −o (1−sw )n+1 ρn+1 i o i+1 ∆x2

∂Ron+1 i

n+1 ∂ωo oi

−(

λo n+1 ∂ρn+1 n+1 i oi n+1 [po i+1 Δx2 ∂ωo oi

=


∂Un+1 i

−φ

n+1 ∂ωo oi

+

− pn+1 oi ]+

φDco2 −o (1−sw )n+1 i−1 ∆x2

φDco2 −o (1−sw )n+1 i−1 ∆x2

∂Ron+1 i

n+1 ∂ωco2 w i+1

(1 −

n+1

∂ρo i−1 ωoo )n+1 i−1 ∂ωo n+1 o i−1

ρn+1 o i−1 Δx2

−

∂λo n+1 i−1

n+1 ∂ωo o i−1

n+1 [pn+1 o i−1 − po i ] +

(A − 43)

n+1 φDco2 −o (1−sw )n+1 i−1 ρo i−1

∆x2

=0

∂Ron+1 i

(A − 44)

∂Un+1

(A − 45)

i = ∂ωco2 n+1 ∂ωco2 n+1 wi

wi

∂Ron+1 i


(A − 42)

n+1 ∂ωo oi

∆t

Δx2 ∂ωo i−1

o i−1

∂ρn+1

oi )[(1 − ωoo )n+1 − ρn+1 i oi ] + ∂ωo n+1

∂ρn+1 (1− sw )n+1 oi i

n+1 n+1 o n+1 [po i−1 − po i ] +

= ∂ωo n+1

− pn+1 oi ]

oi

n+1 λo n+1 i−1 ∂ρo i−1

∂Ron+1 i

n+1 ρn+1 o i ∂λo i [pn+1 o o i+1 Δx2 ∂ωo n+1 i

(A − 46)

=0

(A − 10): n+1 ∂Rw i

∂pn+1 o i+1

n+1 ∂Rwi

∂pn+1 oi

=

n+1 ρn+1 w i λw i

=−

Δx2

n+1 ρn+1 w i λw i

n+1 ρn+1 w i−1 λw i−1 Δx2

φ

+

Δx2

−(


+

λw n+1 ∂ρn+1 i wi Δx2

∂pn+1 oi


+

n+1 ∂ρn+1 w i+1

ωco2 w i+1

(A − 47)

∂pn+1 o i+1

n+1 n+1 n+1 [pn+1 o i+1 − po i + pc i − pc i+1 ] −


n+1 sn+1 w i ∂ρw i

∆t ∂pn+1 oi

235

∂ρn+1

∂Un+1

n+1 w i i )ωco2 − ∂pn+1 − wi ∂pn+1 oi

oi

(A − 48)


n+1

n+1 ρn+1 ∂Rw w i−1 λw i−1 i n+1 = ∂po i−1 Δx2


n+1 ∂Rw i

∂sn+1 w i+1

n+1 ∂Rw i

∂sn+1 wi

+

=

∂sn+1 wi

+

n+1 ∂ωo o i+1

n+1 ∂Rw i

n+1 ∂ωo oi

n+1 ∂Rw i

n+1 ∂ωo o i−1

n+1 n+1 − pn+1 o i + pc i − pc i+1 ]

n+1 n+1 ρn+1 w i−1 λw i−1 ∂pc i Δx2 ∂sn+1 wi

co

(A − 51)

+

co

n+1 ρn+1 w i−1 ∂λw i−1

∂sn+1 w i−1

Δx2

n+1 n+1 ρn+1 w i−1 λw i−1 ∂pc i−1 Δx2 ∂sn+1 w i−1

n+1 ∂Rw i

(A − 49)

(A − 50)

2 n+1 [(ρw ωw 2 )n+1 ]−φ i+1 −(ρw ωw )i

∆x2

−

n+1 ρn+1 ∂pn+1 w i λw i c i+1 Δx2 ∂sn+1 w i+1

Δx2

∂sn+1 w i−1


n+1

n+1 ρn+1 w i ∂λw i [pn+1 n+1 o i+1 Δx2 ∂sw i

φDco2 −w

∂pn+1 o i−1

Δx2

n+1 ∂ρw i−1 ωco2 w i−1 ∂pn+1

n+1 ρn+1 ∂pn+1 w i λw i ci

n+1 ∂Rw i

n+1 λw n+1 i−1 ∂ρo i−1

o i−1

=−

=

+

ρn+1 wi ∆t


+

φDco2 −w ∆x2

(A − 52) co co2 n+1 [(ρw ωw 2 )n+1 ] i−1 −(ρw ωw )i

(A − 53)

=0

∂Un+1

=− ∂ωoin+1

(A − 54)

=0

(A − 55)

oi

n+1 ∂Rw i

n+1 ∂ωco2 w i+1

n+1 ∂Rw i

n+1 ∂ωco2 wi


=

=

∆x2

λw n+1 i Δx2

∂ρn+1 oi

n+1 ∂ωco2 wi

∂λw n+1 ρn+1 i wi [pn+1 + Δx o i+1 2 ∂ωco2 n+1 wi

n+1 ∂ρn+1 w i+1 n+1 ∂ωco2 w i+1

[ ωco2 w i+1

−

+ ρn+1 w i+1 ]

(A − 56)


(A − 57) pn+1 oi

+

pn+1 ci

−

pn+1 c i+1 ]

236


−(


∂Un+1 i

−φ ∂ωco2 n+1

+

sn+1 wi

wi


w i−1

ρn+1 w i−1

∂λw n+1 i−1

Δx2



n+1 ∂ωco2 wi

+ ρn+1 wi ] −

n+1 ∆t ∂ωco2 wi

= ∂ωco2 n+1 +

∂ρn+1 wi

∂ρn+1 wi

n+1 λw n+1 n+1 i−1 ∂ρw i−1 n+1 [po i−1 Δx2 ∂ωco2 w i−1

n+1 ∂Rw i

n+1

)[ ωco2 w i

n+1 n+1 − pn+1 o i + pc i − pc i−1 ]

n+1 n+1 n+1 [pn+1 o i−1 − po i + pc i − pc i−1 ] n+1 ∂ρn+1 w i−1 n+1 ∂ωco2 w i−1

[ ωco2 w i−1

+ ρn+1 w i−1 ]

237

(A − 58)

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