recovery during gas injection in fractured reservoirs, especially when matrix permeability is low and fracture ...... into the oil when condensing drive mechanism takes place (Stalkup, 1983). ...... Michelsen, L.M.: The isothermal flash problem.
MASS TRANSFER DURING ENHANCED HYDROCARBON RECOVERY BY GAS INJECTION PROCESSES by Hasan Shojaei
A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirement for the Degree DOCTOR OF PHILOSOPHY (PETROLEUM ENGINEERING)
May 2014
Copyright 2014
Hasan Shojaei
UMI Number: 3643170
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“Yesterday I was clever, so I wanted to change the world. Today I am wise, so I am changing myself.” - Rumi
i
Dedication
To my parents For their unconditional love, support and encouragement
ii
Acknowledgements
I would like to express my sincere appreciation to anyone who has encouraged and supported me throughout my doctoral studies. In particular I would like to thank Professor Kristian Jessen, my PhD advisor who possesses every quality one could wish to find in an advisor. Besides his deep knowledge in various aspects of reservoir engineering, his curiosity and passion to solve challenging problems have constantly inspired me during the course of my PhD at the University of Southern California. He gives his students a lot of freedom in their research, which in turn nurtures self esteem and creativity in them. He also has great values and ethics. There are no words to describe my gratitude to him for being incredibly understanding when I was going through a difficult time. I have been privileged to work with two other world-renowned academics, Professor Iraj Ershaghi and Professor Roger Ghanem, as my dissertation committee members. Their valuable comments and recommendations have significantly improved the quality of my doctoral dissertation. Professor Jessen and Professor Ershaghi have been more than great mentors to me. Every time I was stuck in my research, or disappointed by hardships that an international student with Iranian passport and single-entry visa (!) may encounter in the U.S., I just needed to talk to either of them to regain my full enthusiasm and energy. Winning the first place in SPE student paper contest (PhD division) in Western North America, would have not been possible without their continuous encouragement and critical comments during rehearsal sessions. iii
I would like to extend my gratitude to the Graduate School at USC for supporting my doctoral studies through the Provost’s PhD Fellowship, one of the most generous scholarships one can find in a top U.S. university. I also owe my gratitude to the Graduate School, the Petroleum Engineering program and the Graduate Student Government at USC for providing me with travel grants to participate in professional conferences and exhibitions, and to Occidental Petroleum for the internship opportunity. I am grateful to my amazing friends in the greater Los Angeles area with whom I have shared wonderful memories. My special thanks go to Shahram Farhadi, Arman Khodabakhsh, Reza Rastegar, Cyrus Ashayeri, Siavash Hakim-Elahi, Mohammad Javaheri, Mohammad Evazi and Mehran Rahmani who have always been there to support and encourage me. I have also enjoyed the
company
of
Dalad Nattwongasem,
Hamidreza Mehran
Jahangiri, Hosseini,
Hamed
Mehrdad
Haddadzadegan, Mahdavi,
Bahar
Ehsan
Tajer,
Hosseini,
Sahra
Homayounian, Asal Rahimi, Sanaz Norouzi, Tayeb Ayatollahi, Mahshad Samnejad, Marjan Sherafati, Nadia Kadkhodayan, Gunel Murad and Lawrence Bustos. I am grateful to all of them, and to many more friends not mentioned here, for their constant support and encouragement. Having been born and raised in a far-off village in south of Iran, where even some fundamental educational services were not available at the time, I could have not managed to achieve the highest academic degree without the love, encouragement and support of my family. My deepest gratitude goes to my lovely parents, Hossein and Madineh, my brilliant brother, Mohammad Reza, and my beautiful sisters, Sakineh and Salimeh.
iv
Abstract
In order to estimate the potential incremental hydrocarbon recovery by gas injection, compositional reservoir simulators are commonly used in the industry. Successful design and implementation of gas injection processes (mainly CO2) rely in part on the accuracy by which the available simulation tools can represent the physics that govern the displacement behavior in a reservoir. In the first part of this thesis, we investigate the accuracy of some physical models that are frequently used to describe and interpret dispersive mixing and mass transfer in compositional reservoir simulation. The calculations from compositional simulation are compared with the experimental observations. A quaternary analog fluid system (alcohol-water-hydrocarbon) that mimics the phase behavior of CO2-hydrocarbon mixtures at high pressure and temperature has been designed in our research group. A porous medium was designed using Teflon materials to ensure that the analog oil acts as the wetting phase, and the properties of the porous medium were characterized in terms of porosity and permeability. Relative permeability and interfacial tension measurements were also performed to delineate interactions between the fluid system and the porous medium. Displacement experiments at First-contact miscible (FCM), near-miscible and multicontact miscible (MCM) conditions were consequently performed (Rastegar, 2010). The effluent concentrations from two-component FCM displacement experiments exhibit a tailing behavior that is attributed to imperfect sweep of the porous medium: A feature that is not v
captured by normal dispersion models. To represent this behavior in displacement calculations, we use dual-porosity (DP) models including mass transfer between flowing and stagnant porosities. The 4-component two-phase displacement experiments (near-miscible and MCM) are consequently simulated using the DP models constructed based on observations from FCM displacements. We demonstrate that the accuracy of our displacement calculations relative to the experimental observations is sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. We also demonstrate that numerical calculations substantially agree with the experimental observations for some physical models with limited need for model parameter adjustment. The second part of this thesis is devoted to diffusion and matrix-fracture interactions during gas injection in fractured reservoirs. Molecular diffusion can play a significant role in oil recovery during gas injection in fractured reservoirs, especially when matrix permeability is low and fracture intensity is high. Diffusion of gas components from a fracture into the matrix extracts oil components from matrix and delays, to some extent, the gas breakthrough. This in turn increases both sweep and displacement efficiencies. In current simulation models, molecular diffusion is commonly modeled using a classical Fick’s law approach with constant diffusion coefficients. In the classical Fick’s law approach, the dragging effects (off-diagonal diffusion coefficients) are neglected. In addition, the gas-oil diffusion at the fracture-matrix interface is normally modeled by assuming an average composition at the interface which does not have a sound physical basis. In this work, we present a dual-porosity model in which the generalized Fick’s law is used for molecular diffusion to account for the dragging effects; and gas-oil diffusion at the fracture-
vi
matrix interface is modeled based on film theory in which the gas in fracture and oil in the matrix are assumed to be at equilibrium. A novel matrix-fracture transfer function is introduced for gasoil diffusion based on film theory. Diffusion coefficients are calculated using the MaxwellStefan model and are pressure, temperature and composition dependent. A time-dependent transfer function is used for matrix-fracture exchange in which the shape factor is adjusted using a boost factor to differentiate between the transfer rate at early and late times. Field-scale examples are used to show that our approach, which is based on a sophisticated physical model for gas-oil diffusion (film theory), gives significantly different results from the conventional approach. It is also demonstrated that the dragging effects (off-diagonal diffusion coefficients) and time-dependency of matrix-fracture transfer function can moderately impact the oil recovery during gas injection in fractured reservoirs. We also show that miscibility is not developed in the matrix blocks even at pressures above minimum miscibility pressure (MMP) when molecular diffusion is the main recovery mechanism during gas injection in fractured reservoirs. In recent years, coalbed methane has become an important source of energy in the United States. Since primary production techniques typically recover less than half of the methane in a coalbed, enhanced coalbed methane (ECBM) recovery processes are needed in which CO2 and/or N2 are injected into the coalbed to recover more CH4. One of the main mechanisms that govern the dynamics of ECBM recovery processes is the sorption of gases onto the coal surfaces. Despite the well-documented complexity of multicomponent sorption phenomena, adsorption and desorption of CH4/CO2/N2 mixtures in the porous coal is commonly modeled with the extended Langmuir model. The extended Langmuir
vii
model has been proven unable to accurately describe the multicomponent sorption that is central to ECBM recovery processes and, therefore, more sophisticated sorption models are needed. In the third part of this thesis we apply potential theory to describe the multicomponent sorption of relevance to ECBM processes. In this approach for modeling multicomponent sorption, each component is assumed to be affected by a characteristic potential field emitted by the coal surface. We discuss the implementation of potential theory with emphasis on the simulation of ECBM processes where computational efficiency and accuracy must be balanced. The model must be solved by an iterative scheme, and is hence more computationally expensive than the extended Langmuir approach. The results and analysis presented in this work demonstrate that the application of potential theory of sorption to modeling of ECBM recovery processes can improve the accuracy of calculations. However, the additional complexity of the model and the associated increase in the computational efforts may not balance the gain in accuracy sufficiently to warrant application in general purpose reservoir simulation.
viii
Table of Contents Abstract ........................................................................................................................................... v List of Tables ............................................................................................................................... xiii List of Figures .............................................................................................................................. xiv Chapter 1 Introduction .................................................................................................................... 1 1.1 Background ........................................................................................................................... 1 1.2 Motivation ............................................................................................................................. 4 1.3 Objectives ............................................................................................................................. 5 1.4 Manuscript Organization ...................................................................................................... 7 Chapter 2 Literature Review ........................................................................................................... 9 2.1 Miscibility ............................................................................................................................. 9 2.2 Mixing Mechanisms............................................................................................................ 10 2.3 Effect of Mixing on Miscibility .......................................................................................... 12 2.4 Mass Transfer in Dual-Porosity (DP) Systems ................................................................... 13 2.5 Sorption in Coalbeds ........................................................................................................... 15 2.6 Mathematical Modeling ...................................................................................................... 16 Chapter 3 Experimental and Modeling Study of Multicontact Miscible Displacements ............. 19 3.1 Introduction ......................................................................................................................... 19 3.2 Fluid System ....................................................................................................................... 20 3.2.1 Phase Behavior............................................................................................................. 22 ix
3.2.2 Mixture Densities ......................................................................................................... 22 3.2.3 Mixture Viscosities ...................................................................................................... 24 3.2.4 Interfacial Tension ....................................................................................................... 25 3.3 Packed Column ................................................................................................................... 27 3.4 Solid-Fluid Interactions ...................................................................................................... 27 3.4.1 Wettability.................................................................................................................... 27 3.4.2 Relative Permeability ................................................................................................... 28 3.5 Displacement Experiments ................................................................................................. 29 3.5.1 FCM Displacements..................................................................................................... 30 3.5.2 Near-miscible and MCM Displacements ..................................................................... 30 3.6
Mathematical Modeling ................................................................................................. 36
3.7 Displacement Calculations.................................................................................................. 38 3.7.1 FCM Displacements..................................................................................................... 39 3.7.2 Near-miscible and MCM Displacements ..................................................................... 40 3.8 Effect of Mixing on Near-miscible and MCM Displacements ........................................... 41 3.9 Importance of Selected Physical Models ............................................................................ 45 3.10 Discussion ......................................................................................................................... 49 3.11 Conclusions ....................................................................................................................... 54 Chapter 4 Diffusion and Matrix-Fracture Interactions during Gas Injection in Fractured Reservoirs ..................................................................................................................................... 56 x
4.1 Introduction ......................................................................................................................... 56 4.2 Mathematical Model ........................................................................................................... 60 4.3 Molecular Diffusion ............................................................................................................ 63 4.3.1 Intra-phase Diffusion ................................................................................................... 64 4.3.2 Cross-phase Diffusion .................................................................................................. 65 4.4 Transfer Function ................................................................................................................ 68 4.5 Results ................................................................................................................................. 72 4.5.1 Example 1 .................................................................................................................... 74 4.5.2 Example 2 .................................................................................................................... 79 4.6 Discussion ........................................................................................................................... 82 4.7 Conclusions ......................................................................................................................... 87 Chapter 5 Application of Potential Theory of Adsorption to Modeling of ECBM Recovery ...... 89 5.1 Introduction ......................................................................................................................... 89 5.2 Potential Theory .................................................................................................................. 91 5.3 Numerical Approach ........................................................................................................... 93 5.4 Application of MPTA to Coal............................................................................................. 95 5.5 Discussion and Conclusions ............................................................................................. 103 Chapter 6 Summary and Future Research Directions ................................................................. 105 Nomenclature .............................................................................................................................. 108 References ................................................................................................................................... 111 xi
Appendix A:
UNIQUAC Model for Phase Equilibria ............................................................ 124
Appendix B:
UNIQUAC Viscosity Model ............................................................................. 126
Appendix C:
Numerical Dispersion ........................................................................................ 127
Appendix D:
Diffusion Coefficients ....................................................................................... 128
D.1 Example (Binary Mixtures) ............................................................................................. 130 D.2 Example (Ternary Mixture) ............................................................................................. 131 Appendix E:
Shape Factors for Gas-Oil Diffusion ................................................................. 133
Appendix F:
Transfer Rate due to Gravity Drainage ............................................................. 137
xii
List of Tables Table 3.1 Pure component property data for the analog fluid system .......................................... 22 Table 3.2 Parameters for UNIQUAC viscosity model (interactions and structural parameters).. 24 Table 3.3 Summary of displacement experiments ........................................................................ 31 Table 3.4 Model parameters obtained from vertical dispersivity experiment .............................. 39 Table 4.1 Initial and injection compositions and the MMP of the gas/oil pair used in example 1. ....................................................................................................................................................... 74 Table 4.2 Initial and injection compositions and the MMP of the gas/oil pair used in example 2. ....................................................................................................................................................... 79 Table 5.1 MPTA parameters from pure component isotherms..................................................... 97 Table 5.2 MPTA parameters from pure component isotherms using composite potential function. ..................................................................................................................................................... 100 Table 5.3 Model parameters for Extended Langmuir Model (ELM) ......................................... 102 Table 5.4 Average absolute relative error (%) in prediction of binary and ternary excess sorption ..................................................................................................................................................... 104 Table A.1 Parameters for UNIQUAC model (interactions and structural parameters) .............. 125 Table D.1 Calculated diffusion coefficient compared with experimental data from Kett and Anderson (1969) for a ternary liquid mixture at 298.15 K and 0.1 MPa. ................................... 132
xiii
List of Figures Figure 1.1 Evolution of CO2 injection projects and oil prices in the U.S. Data from Oil and Gas Journal EOR surveys 1980-2010 and U.S. EIA 2010 (Alvarado and Manrique, 2010). ................ 2 Figure 1.2 Percentage of current EOR projects in the U.S. by EOR methods. Data from Oil and Gas Journal (Koottungal, 2008). ..................................................................................................... 3 Figure 3.1 Quaternary phase diagrams (mass fractions). Top: CO2-CH4-nC4-C12 at 2280 psi and 212°F as calculated from the PR EOS. Bottom: Water-MeOH-IPA-iC8 at 68°F and 14.7 psi as calculated from the UNIQUAC model. .................................................................................... 21 Figure 3.2 Comparison of mixture densities (25°C and 1 atm) from Otero et al. (2000) with ideal mixing calculations as a function of Water mass fraction along the Water-IPA-iC8 binodal curve. ....................................................................................................................................................... 23 Figure 3.3 Experimental and calculated viscosity for the binary mixtures (25°C and 1 atm). Data from Tanaka et al., 1987 (MeOH-Water and Water-IPA), Ku, 2008 (IPA-iC8) and Soliman and Marschall, 1990 (MeOH-IPA). ..................................................................................................... 25 Figure 3.4 Simplistic IFT model based on tie-line lengths (mass fractions), IFT in mN/m. Data from Morrow et al., 1988 (Water-IPA-iC8) and Garcia-Flores et al., 2007 (Water-MeOH-iC8). 26 Figure 3.5 Interface between iC8 and MeOH in a PTFE capillary tube. While both pure substances spread on PTFE, iC8 preferentially wets the surface in this binary setting. ............... 28 Figure 3.6 iC8 (wetting phase) - Water drainage relative permeability: Steady state observations and Corey-type model. .................................................................................................................. 29 Figure 3.7 Effluent concentrations (in mass fractions) from FCM displacement experiments with IPA and iC8: Experimental data (circles), single-porosity model (solid line), DP I (broken line), and DP II (dotted line). Top: Vertical displacement. Bottom: Horizontal displacement. ............ 31
xiv
Figure 3.8 Natural logarithm of equilibrium K-values and saturation of analog gas along displacement length (zD=z/L) for experiment III at the dispersion-free limit after 0.7 PVI. ........ 32 Figure 3.9 Effluent concentrations (in mass fractions) for displacement experiment III from dualporosity model I (solid line), dual-porosity model II (dotted line), and experimental observations (circles). ........................................................................................................................................ 33 Figure 3.10 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment IV at the dispersion-free limit after 0.7 PVI. ................................................................................................................................. 34 Figure 3.11 Effluent concentrations (in mass fractions) for displacement experiment IV from dual-porosity model I (solid line), dual-porosity model II (dotted line), and experimental observations (circles). ................................................................................................................... 35 Figure 3.12 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment III with dispersion and mass transfer after 0.7 PVI..................................................................................................................... 42 Figure 3.13 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment IV with dispersion and mass transfer after 0.7 PVI..................................................................................................................... 43 Figure 3.14 Composition path (in mass fractions) for Exp. III from dispersion-free calculations (broken line) and calculations with dispersive mixing and mass transfer (dotted line)................ 44 Figure 3.15 Composition path (in mass fractions) for Exp. IV from dispersion-free calculations (broken line) and calculations with dispersive mixing and mass transfer (dotted line)................ 44
xv
Figure 3.16 Component recovery for Exp. III from dispersion-free calculations (IPA: solid line, iC8: broken line) and calculations with dispersive mixing and mass transfer (IPA: broken dotted line, iC8: dotted line). ................................................................................................................... 45 Figure 3.17 Component recovery for Exp. IV from dispersion-free calculations (IPA: solid line, iC8: broken line) and calculations with dispersive mixing and mass transfer (IPA: broken dotted line, iC8: dotted line). ................................................................................................................... 46 Figure 3.18 Effluent concentrations (in mass fractions) for displacement experiment III from DP I with IFT scaling of relative permeabilities (solid line), DP I without IFT scaling of relative permeabilities (broken dotted line), DP I with linear relative permeabilities (dotted line) and experimental observations (circles). ............................................................................................. 48 Figure 3.19 Effluent concentrations (in mass fractions) for displacement experiment IV from DP I with IFT scaling of relative permeabilities (solid line), DP I without IFT scaling of relative permeabilities (broken dotted line), DP I with linear relative permeabilities (dotted line) and experimental observations (circles). ............................................................................................. 49 Figure 3.20 Relative impacts of longitudinal dispersion and transverse mass transfer for FCM displacements. Top: Vertical displacement. Bottom: Horizontal displacement. .......................... 52 Figure 4.1 Schematic of two neighboring gridblocks containing oil and gas phases. .................. 63 Figure 4.2 Composition profiles near the interface during inter-phase mass transfer, after Taylor and Krishna (1993). ...................................................................................................................... 66 Figure 4.3 Schematic of the reservoir and wellbore locations. White lines represent the fracture system. .......................................................................................................................................... 73 Figure 4.4 Relative permeability curves for flowing (solid line) and stagnant (dashed line) domains. Gas and oil relative permeabilities are shown with red and green colors respectively. 73
xvi
Figure 4.5 Liquid recovery (left) and producing GOR (right) versus time for example 1 when diffusion is not considered (red), when diffusion is modeled using our approach (blue) and when diffusion is modeled using the conventional approach (black). ................................................... 76 Figure 4.6 Liquid recovery (left) and producing GOR (right) versus time for example 1 when diffusion is not considered (solid red), when our approach is used (solid blue), when our approach is used but PSS condition is assumed for M-F transfer function (dashed blue) and when our approach is used but PSS condition is assumed for M-F transfer function and off-diagonal diffusion coefficients are neglected (dotted blue). ........................................................................ 77 Figure 4.7 The CO2 composition (%) in the stagnant domain after 10 years (example 1) for the case without diffusion (left) and the case with diffusion based on calculations using our approach (right). ........................................................................................................................................... 79 Figure 4.8 Liquid recovery (left) and producing GOR (right) versus time for example 2 when diffusion is not considered (red), when diffusion is modeled using our approach (blue) and when diffusion is modeled using the conventional approach (black). ................................................... 80 Figure 4.9 Liquid recovery (left) and producing GOR (right) versus time for example 2 when diffusion is not considered (solid red), when our approach is used (solid blue), when our approach is used but PSS condition is assumed for M-F transfer function (dashed blue) and when our approach is used but PSS condition is assumed for M-F transfer function and off-diagonal diffusion coefficients are neglected (dotted blue). ........................................................................ 81 Figure 4.10 The CO2 composition (%) in the stagnant domain after 10 years (example 2) for the case without diffusion (left) and the case with diffusion based on calculations using our approach (right). ........................................................................................................................................... 82
xvii
Figure 4.11 The saturation profile of a matrix block (dotted line) near the injection well and its corresponding fracture (solid line) for example 1 as calculated using our approach. .................. 83 Figure 4.12 Composition profiles (mass fraction) of a matrix block (green) near the injection well and its corresponding fracture (red) for example 1 during the first 10 years as calculated using our approach. The blue curves show the two-phase boundaries in each ternary plane at 157 bar and 373.15 °K. The dashed brown line represents the dilution line. ...................................... 84 Figure 4.13 The saturation profile of a matrix block (dotted line) near the injection well and its corresponding fracture (solid line) for example 1 as calculated using the conventional approach. ....................................................................................................................................................... 85 Figure 4.14 Composition profiles (mass fraction) of a matrix block (green) near the injection well and its corresponding fracture (red) for example 1 during the first 10 years as calculated using the conventional approach. The blue curves show the two-phase boundaries in each ternary plane at 157 bar and 373.15 °K. The dashed brown line represents the dilution line................... 86 Figure 4.15 The CO2 composition (%) in the stagnant domain after 10 years (example 1) as calculated using the conventional approach (left) and calculations using our approach (right). .. 86 Figure 5.1 Pure component isotherms at 45 °C. Data from Ottiger et al. (2008b). Solid lines: MPTA after regression.................................................................................................................. 97 Figure 5.2 Experimental observations and MPTA model predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b). .................................................................................................................................... 98 Figure 5.3 Experimental observations and MPTA model predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b). .................................................................................................................................. 101
xviii
Figure 5.4 Experimental observations and ELM predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b). ..................................................................................................................................................... 102 Figure D.1 Calculated diffusion coefficient (solid lines) compared with experimental data (circles) from Sigmund (1976) for methane-propane mixtures at different values of pressure, temperature and composition. ..................................................................................................... 130 Figure D.2 Calculated diffusion coefficient (solid lines) compared with experimental data (circles) from Lo (1974) for binary liquid mixtures of alkanes at 298.15 K and 0.1 MPa with varying compositions. ................................................................................................................. 131 Figure E.1 Schematic of a matrix block surrounded by fractures............................................... 133
xix
Chapter 1 :
Introduction
1.1 Background With the increase in energy demands, there is a need for increased oil/gas production in the upcoming decades while alternative energy sources (e.g. solar, wind, geothermal, nuclear fusion, biofuels) are developed and implemented. Since the oil and gas reserves are limited, improved oil/gas recovery techniques are expected to play a crucial role in meeting the energy needs in the years to come by increasing recovery from existing reserves. Improved oil/gas recovery techniques include intelligent well monitoring and control, secondary recovery methods and tertiary recovery processes. In secondary recovery methods water (or gas) is injected into aquifer (or gas cap) to maintain the reservoir pressure, while tertiary recovery processes encompass thermal methods (steam flooding and in-situ combustion), gas injection (hydrocarbon, N2 and CO2) and chemical flooding (polymer, alkaline and surfactant). Injection of CO2 into oil/gas reservoirs, which is the main focus of this research project, has gained tremendous industrial application over the last three decades. Figure 1.1 depicts evolution of CO2 injection projects in the U.S. over the last three decades based on the data from Oil and Gas Journal EOR surveys 1980-2010 and U.S. EIA 2010 (Alvarado and Manrique, 2010). Even though the oil price dropped during the 1980s and 2008-2010, the number of CO2 injection projects increased during these time intervals. This is a very interesting observation and shows the economic feasibility of CO2 injection projects even at low oil prices. 1
One of the main reasons for the increase in CO2 injection projects in the U.S. is the cheap CO2 available from natural sources (Moritis, 2001). In addition, the miscible conditions for CO2 injection processes are achieved at significantly lower pressures compared to natural gas or Nitrogen. The solubility of CO2 in oil causes the oil to swell and flow at lower viscosities. The reduction in oil viscosity caused by CO2 offsets, to some extent, the unfavorable mobility ratios which result from low viscosity of CO2 at reservoir conditions (Stalkup, 1983). All of these factors have made CO2 injection the most favorable EOR process for medium and light oil reservoirs with API gravity greater than 25 (Taber, 1997; Alvarado and Manrique, 2010).
Figure 1.1 Evolution of CO2 injection projects and oil prices in the U.S. Data from Oil and Gas Journal EOR surveys 1980-2010 and U.S. EIA 2010 (Alvarado and Manrique, 2010).
In 2008, CO2 and other gas injection processes accounted for more than 60% of the EOR projects in the U.S. while approximately 30% of the EOR projects were based on thermal methods (Koottungal, 2008). Most of these gas injection projects are performed on non-fractured
2
reservoirs. But there are huge fractured reservoirs in different parts of the world which may have a potential for gas injection. Gas injection in fractured reservoirs is one of the focus areas of this research project.
Figure 1.2 Percentage of current EOR projects in the U.S. by EOR methods. Data from Oil and Gas Journal (Koottungal, 2008).
CO2 injection is not limited to oil reservoirs. In recent years, coalbed methane has become an important source of energy in the United States. Since primary production techniques typically recover less than half of the methane in a coalbed, enhanced coalbed methane (ECBM) recovery processes are used in which CO2 and/or N2 are injected into the coalbed to recover more CH4. Climate change due to greenhouse gas emissions has become an important concern over the last decade. With CO2 being one of the main greenhouse gases in the earth’s atmosphere, CO2 injection into depleted oil reservoirs and coalbeds provides additional benefit of sequestering carbon in the subsurface. 3
1.2 Motivation The initiation of an EOR project depends on the economic evaluation of the project which, in turn, depends on the potential incremental hydrocarbon recovery from the project. In order to estimate the potential incremental hydrocarbon recovery by gas injection, compositional reservoir simulators are commonly used in the industry. In addition, successful design and implementation of these processes rely, in part, on the accuracy by which the available simulation tools can represent the physics that govern the displacement behavior in a reservoir. Despite tremendous efforts and progress by different investigators over the past decades (see Chapter 2), the interaction of flow and phase behavior in porous media and its proper modeling in the context of CO2 injection (and more generally miscible displacements) is not completely understood due to the complex nature of these multiphase, multicomponent displacement processes. This has motivated us to investigate the accuracy of some physical models that are frequently used in compositional simulation of miscible displacements by comparing numerical calculations with observations from miscible and near-miscible displacement experiments (Chapter 3). In current simulation models, molecular diffusion which is an important recovery mechanism during gas injection in fractured reservoirs, is represented by simplified models. A classical Fick’s law approach (which neglects interactions among components) with constant diffusion coefficients is commonly used for molecular diffusion. In addition, the gas-oil diffusion at the fracture-matrix interface is normally modeled by assuming an average composition at the interface which does not have a sound physical basis. This has motivated us to develop and implement a dual-porosity model that uses sophisticated physical models for molecular diffusion and matrix-fracture interactions during gas injection in fractured reservoirs (Chapter 4).
4
The sorption of gases onto the coal surfaces is one of the main mechanisms that govern the dynamics of ECBM recovery processes. Despite the well-documented complexity of multicomponent sorption phenomena, adsorption and desorption of CH4/CO2/N2 mixtures in the porous coal is commonly modeled with the extended Langmuir model. The extended Langmuir model has been proven unable to accurately describe the multicomponent sorption that is central to ECBM recovery processes and, therefore, more sophisticated sorption models are needed. This has motivated us to apply the potential theory to describe the multicomponent sorption of relevance to ECBM processes (Chapter 5). 1.3 Objectives In the first part of this research, the objective is to investigate the accuracy of key physical models that are frequently used to describe and interpret dispersive mixing and mass transfer in compositional simulation of miscible displacements with emphasis on CO2 injection processes. A quaternary analog fluid system (alcohol-water-hydrocarbon) that mimics the phase behavior of CO2-hydrocarbon mixtures at high pressure and temperature has been designed in our
research
group
(Rastegar
2010).
A
porous
medium
was
designed
using
PolyTetraFlouroEthylene (PTFE) materials to ensure that the analog oil acts as the wetting phase, and the properties of the porous medium were characterized in terms of porosity and permeability. Relative permeability and interfacial tension (IFT) measurements were also performed to delineate interactions between the fluid system and the porous medium. Displacement experiments at First-contact miscible (FCM), near-miscible and multicontact miscible (MCM) conditions were consequently performed (Rastegar 2010). The effluent concentrations from two-component FCM displacement experiments exhibit a tailing behavior that is attributed to imperfect sweep of the porous medium: A feature that is not 5
captured by normal dispersion models. To represent this behavior in displacement calculations, we use dual-porosity (DP) models including mass transfer between flowing and stagnant porosities. The 4-component two-phase displacement experiments (near-miscible and MCM) are consequently simulated using the DP models constructed based on observations from FCM displacements. We demonstrate that the accuracy of our displacement calculations relative to the experimental observations is sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. We also demonstrate that numerical calculations substantially agree with the experimental observations for some physical models with limited need for model parameter adjustment. In the second part of this research, the objective is to formulate a dual-porosity model in which the generalized Fick’s law is used for molecular diffusion to account for the dragging effects (component interactions); and gas-oil diffusion at the fracture-matrix interface is modeled based on film theory in which the gas in fracture and oil in the matrix are assumed to be at equilibrium. A novel matrix-fracture transfer function is introduced for gas-oil diffusion based on film theory. Diffusion coefficients are calculated using the Maxwell-Stefan model and are pressure, temperature and composition dependent. A time-dependent transfer function is used for matrix-fracture exchange in which the shape factor is adjusted using a boost factor to differentiate between the transfer rate at early and late times. Field-scale calculation examples are used to demonstrate that our approach, which is based on a sophisticated physical model for gas-oil diffusion (film theory), gives significantly different results from the conventional approach. It is also demonstrated that the dragging effects (offdiagonal diffusion coefficients) and time-dependency of matrix-fracture transfer function can
6
moderately impact the oil recovery during gas injection in fractured reservoirs. We also show that miscibility is not developed in the matrix blocks even at pressures above minimum miscibility pressure (MMP) when molecular diffusion is the main recovery mechanism during gas injection in fractured reservoirs. The objective, in the third part of this research, is to apply potential theory to describe the multicomponent sorption of relevance to ECBM processes. In this approach for modeling multicomponent sorption, each component is assumed to be affected by a characteristic potential field emitted by the coal surface. We discuss the implementation of potential theory with emphasis on the simulation of ECBM processes where computational efficiency and accuracy must be balanced. The model must be solved by an iterative scheme, and is hence more computationally expensive than the extended Langmuir approach. The results and analysis presented in this work demonstrate that the application of potential theory of sorption to modeling of ECBM recovery processes can improve the accuracy of calculations. However, the additional complexity of the model and the associated increase in the computational efforts may not balance the gain in accuracy sufficiently to warrant application in general purpose reservoir simulation. 1.4 Manuscript Organization The remainder of this manuscript is organized as follows. In Chapter 2, a brief literature review on miscibility, mixing mechanisms, the effect of mixing on miscible displacements, mass transfer in dual-porosity systems, and the mathematical modeling of these processes is presented. Chapter 3 includes the experimental procedure and data together with mathematical modeling, displacement calculations and interpretations of the experiments performed in our research group to study mixing and mass transfer in multicontact miscible displacements. 7
In chapter 4, we present a dual-porosity model for gas injection in fractured reservoirs in which the generalized Fick’s law is used for molecular diffusion; and gas-oil diffusion at fracture-matrix interface is accounted for using the film theory. A time-dependent transfer function is presented that accounts for both early- and late-time behavior of matrix-fracture interactions. Using field-scale numerical examples, our approach is compared with the conventional approach in which the molecular diffusion is modeled using the classical Fick’s law with constant diffusion coefficients and the gas-oil diffusion is modeled by assuming average composition at the interface. In Chapter 5, we apply the potential theory to describe the multicomponent sorption of relevance to ECBM processes. We discuss the implementation of potential theory with emphasis on the simulation of ECBM processes where computational efficiency and accuracy must be balanced. The calculations from potential theory are compared with experimental sorption data of CH4, CO2, N2 and their binary and ternary mixtures on real coal samples. Finally, the future direction of this research project is presented in Chapter 6.
8
Chapter 2 :
Literature Review
2.1 Miscibility During gas/oil displacements in porous media, since the injected gas is not initially at equilibrium with the oil, the contact(s) between the two phases results in mass transfer; and therefore changes the properties of both oil and gas phases. If the pressure is high enough, the displacing gas and displaced oil become similar; and the displacement process becomes highly efficient. This condition is generally referred to as development of miscibility. First-contact miscibility is a condition where two fluids form a single phase at their first contact when they are mixed at any proportion at a given pressure and temperature. If two fluids form a single phase after a few contacts, they are called to be multicontact miscible at the given pressure and temperature. Multicontact miscibility can be achieved through three mechanisms: vaporizing, condensing and combined vaporizing/condensing drives. In the vaporizing drive mechanism, light and intermediate components transfer from the oil into gas phase (lean gas like CH4, N2 and CO2). In contrast, intermediate components transfer from the enriched gas phase into the oil when condensing drive mechanism takes place (Stalkup, 1983). Combined drives exhibit both features. Ternary diagrams have been used traditionally to explain development of miscibility. In ternary systems, a process is MCM by a vaporizing (or condensing) drive only if the initial (or injection) composition lies outside the region of tie-line extensions. However, it has been shown that a combination of vaporizing and condensing derives is responsible for development of multicontact miscibility for reservoir fluids where more than three components are present (Zick, 9
1986 and Stalkup, 1987). Johns et al. (1993) showed that at least four components are needed to describe combined condensing/vaporizing drive (quaternary diagrams). The minimum miscibility pressure (MMP) is the pressure above which the injected gas and initial oil become miscible by a multicontact mechanism. The MMP can be determined in the lab by slimtube experiments in which gas is injected into a long coiled slimtube that has been packed with sand and saturated with oil. The slimtube experiment is performed at increasing pressures; and the resulting oil recoveries at 1.2 pore volumes injected (PVI) are plotted versus displacement pressure. The break-point on the plot is considered as the MMP. Different approaches have been used to estimate the MMP including single mixing cell, multiple mixing cells, 1D slimtube simulation and the method of characteristics (MOC). A description of these methods and comparisons between calculations with different methods and experimental data can be found in Jessen (2000) and Abedini et al. (2014). 2.2 Mixing Mechanisms Diffusion and dispersion are often used to represent mixing mechanisms during displacement processes in porous media at scales where permeability heterogeneity is not represented explicitly. Perkins and Johnston (1963) presented an extensive review of studies of diffusion and dispersion in porous media. Diffusion takes place due to random motion of molecules when two miscible fluids are placed in contact. While diffusion is driven by gradients in concentrations (or more generally by gradients in the chemical potential), additional apparent mixing is caused by local variations in the fluid velocity in a porous medium. The variations in fluid velocity and the associated mixing are generally referred to as dispersion (Perkins and Johnston, 1963). Diffusion will oftentimes not play a significant role in regions of a porous medium where fluids velocities are high because the characteristic time for diffusion is relatively large. 10
However, in regions of the porous media where flow velocities are low (e.g. stagnant or bypassed pore space) diffusion can dominate mass transfer over viscous flow. For simplicity, the contribution of diffusion and dispersion are commonly lumped into an effective dispersion coefficient (or dispersivity) as discussed later. Several parameters are known to influence the effective mixing in a porous medium including: flow rate, heterogeneity, turbulence, viscosity and density differences, grain size distribution, and grain shapes (Perkins and Johnston, 1963). Coats and Smith (1964) observed asymmetric effluent concentrations from tracer experiments on consolidated cores in contrast to nearly symmetrical effluent concentrations for unconsolidated porous materials. They used a differential capacitance model to match their experimental observations. This model accounts for dead-end pore space (stagnant porosity) in a porous medium and includes mass transfer between flowing and stagnant pore space. Their approach was subsequently modified by Baker (1977) in a study of dispersion and mass transfer in FCM displacement experiments. Bretz and Orr (1987) used the Coats-Smith model and a porous-sphere model to interpret their experimental observations for FCM displacements in carbonate cores. The porous-sphere model assumes an assemblage of porous spheres with internal diffusive mass transfer limitations. They found that a porous-sphere model provided for a better match of their experimental observations and explained this by the actual structure of the cores being more similar to a porous-sphere model (from thin-section analysis). Arya et al. (1988) presented a numerical study of the effect of aspect ratio, heterogeneity and diffusion coefficients on the effective dispersive mixing at larger scales. They demonstrated that dispersive mixing in heterogeneous systems with large aspect ratio behaves like equivalent layered systems.
11
Peters et al. (1996) presented a method for measuring longitudinal dispersion coefficients in porous media through computed tomography imaging while Schulze-Makuch (2005) reviewed 109 dispersion related studies and compiled values of dispersivity for different formations/scales. Bijeljic and Blunt (2006) introduced a measure of transit times for particles between neighboring pores to overcome the limitations of conventional time independent dispersion coefficient to reproduce experimental observations at small scale. Garmeh et al. (2009) used pore-scale simulation to study dispersion in porous media. The results of their simulations align well with the classical work of Perkins and Johnston that dictate a relationship between the pore-scale Peclet number and the observed effective longitudinal dispersivity. Despite a broad industrial applicability, less effort has been devoted to the study of dispersion in multiphase systems. Sahimi et al. (1982) investigated dispersion in two-phase immiscible displacements using Monte Carlo simulation combined with percolation theory to determine phase distributions and mixing in artificial pore structures. They demonstrated that dispersive mixing depends strongly on phase saturation and saturation history. Delshad et al. (1985) performed multiphase experiments in sandpacks and Brea sandstone to measure relative permeabilities and dispersive mixing. They generalized single phase dispersion theory and applied it to interpret their experimental observations and arrived at dispersivities that differed from the equivalent single phase dispersivities. 2.3 Effect of Mixing on Miscibility It has long been known that numerical dispersion affects the efficiency in MCM displacements (Gardner, 1981 and Stalkup, 1988). Walsh and Orr (1990) studied the effect of dispersion (physical and numerical) on the composition path for one-dimensional flow of ternary, two12
phase mixtures. They demonstrated that the composition path passes through the two-phase region as a result of numerical/physical dispersion and that oil recovery is reduced as the Peclet number is decreased. They also demonstrated that the effect of dispersion on composition path, and therefore recovery, depends on the shape and size of two-phase region on ternary diagram. Johns et al. (1994) demonstrated that numerical dispersion causes some two-phase flow in a 4-component displacement that would be MCM in the absence of dispersion. They observed that as numerical dispersion is increased, the oil recovery is reduced, especially near the minimum miscibility enrichment (MME). Other investigators have also shown that the efficiency of enhanced oil recovery by multicontact miscible gas injection depends, in part, on the level of mixing that occurs between the injected gas and the oil in the reservoir (e.g. Solano et al., 2001; Johns et al., 2002; Jessen et al., 2004). 2.4 Mass Transfer in Dual-Porosity (DP) Systems In general, viscous, gravity, diffusion and capillary forces can contribute to mass transfer during flow in porous media. Different investigators have studied mass transfer between highpermeability fracture and low-permeability matrix in DP systems in the context of gas injection processes. da Silva and Belery (1989) showed by compositional simulations that Fick’s molecular diffusion was very important in gas injection processes in fractured reservoirs because of the large contact area for diffusion in these systems. They suggested that as the fracture spacing is reduced, the molecular diffusion becomes more significant and may dominate over viscous forces. Hoteit and Firoozabadi (2009) studied the role of diffusion during gas injection into fractured oil reservoirs and demonstrated that diffusion can have a significant effect on oil recovery away from the miscibility pressure. 13
In gas injection processes in fractured reservoirs low injection rates are desired to avoid early breakthrough and hence improve the sweep efficiency. This in turn reduces the contributions from viscous forces in such processes. Contributions from capillary forces will be marginal in miscible and near-miscible gas injection processes. Findings by Burger and Mohanty (1997) confirm that mass transfer from capillary forces is minimal in near-miscible displacements. Li et al. (2000) performed experiments in which CO2 was injected at miscible conditions after water flooding in artificially fractured cores containing dead oil. They found that gravity drainage can significantly improve oil recovery after water flooding, and that gravity drainage declines as initial water saturation increases and matrix permeability decreases. Asghari and Torabi (2008) observed from their experiments that CO2 gravity drainage improved the oil recovery from a DP system at miscible conditions. However, they could not match the experimental observations using a commercial compositional simulator. Darvish et al. (2006) performed a series of experiments to study the mass transfer between matrix and fracture during miscible CO2 injection. They observed that diffusion played a more significant role than gravity drainage in recovering oil from a long vertical tight matrix block when CO2 was injected at a low rate. Using a commercial compositional simulator, they failed to accurately reproduce the experimental observations mainly because the simulator did not take into account the gas-oil diffusion between gas in the fracture and oil in the matrix. To overcome this limitation of the commercial simulator, they placed a dummy two-phase layer between the matrix (initially containing oil) and fracture (initially containing gas) in their simulation model to initiate the diffusion between gas and oil. Then they manually changed the composition of the dummy two-phase layer until a reasonable match between the simulation results and experimental data was obtained.
14
However, one does not have to manually insert a dummy two-phase layer between gas and oil, and adjust its composition in order to model the gas-oil diffusion. A more sophisticated model, known as the film theory, has been used for gas-oil diffusion in the chemical engineering literature for several years. According to the film theory, when gas and oil phases are placed in contact, a two-phase interface forms between them in which the gas and oil are at equilibrium. Moreover, the molar flux of each component across the interface must be continuous (Taylor and Krishna, 1993). Moortgat and Firoozabadi (2013) used the film theory for gas-oil diffusion in their mixed finite element, discontinuous Galerkin approach and obtained reasonable agreement between their simulation results and experimental data from Darvish et al. (2006). Vega et al. (2010) performed miscible CO2 injection experiments on a vertical lowpermeability siliceous shale core with an artificial fracture. They found that both diffusion and convection were significant in their experiments, but failed to match the experimental observations using a commercial compositional simulator. 2.5 Sorption in Coalbeds In coalbeds, gas is present in two phases: a bulk gaseous phase that occupies the pore space and a liquid-like adsorbed phase on the coal surfaces/pores. In primary recovery, the coalbed is dewatered to reduce the overall reservoir pressure which causes CH4 to be desorbed from coal surfaces. Since primary production typically recovers less than half of the methane in a coalbed (Stevens et al., 1998), enhanced coalbed methane (ECBM) recovery processes are needed in which CO2 and/or N2 are injected into the coalbed to recover more CH4. Injection of CO2 into coalbeds also provides additional benefit of sequestering carbon in the subsurface.
15
Gas injection in ECBM recovery provides a method to maintain the overall coalbed pressure. In addition, injecting a second gas, or a mixture of gases, decreases the partial pressure of CH4 in the free gas. As a result, desorption of CH4 from coal surface is enhanced. The convective flow of injected gas sweeps desorbed CH4 through the coalbed. Therefore the sorption of gases onto the coal surfaces is one of the main mechanisms that govern the dynamics of ECBM recovery processes. Despite the well-documented complexity of multicomponent sorption phenomena (Stevenson et al., 1991; DeGance et al., 1993; Chaback et al., 1996), adsorption and desorption of CH4/CO2/N2 mixtures in ECBM recovery is usually modeled with the extended Langmuir model because of its simplicity and associated low computational cost (Guo, 2003; Zhu et al., 2003; Smith et al., 2005; Seto et al., 2009). The extended Langmuir has been proven unable to accurately describe the multicomponent sorption that is relevant to ECBM recovery processes (Clarkson, 2003; Wei et al., 2005). Jessen et al. (2008) demonstrated that extended Langmuir was able to model the sorption process in binary displacements; but failed to describe the behavior of ternary displacements. 2.6 Mathematical Modeling Description of flow in a porous medium is typically started by writing the mass conservation equation (also referred to as continuity equation). When convection, diffusion and dispersion are the main physical mechanisms that govern the flow dynamics, the multicomponent multiphase continuity equation for a single-porosity system can be written as (2.1)
16
where xij is the mole fraction of component i in phase j, phase j, ρj is the molar density of phase j,
is the porosity, Sj is the saturation of
is the velocity of phase j,
represents the effective
dispersion tensor (combined impact of dispersion and diffusion) for component i in phase j, t represents time, np represents the number of phases, and nc is the number of components. It should be noted that the effects of chemical reaction, adsorption and temperature variation have been neglected in derivation of Eq. (2.1) (Lake, 1989; Orr, 2007). The longitudinal (along the main direction of flow) and transverse (perpendicular to the main direction of flow) dispersion coefficients can be obtained from Eq. (2.2) and (2.3) respectively (Bear, 1972; Orr, 2007)
(2.2)
(2.3) where α denotes a material constant known as dispersivity, subscripts l and t refer to longitudinal and transverse directions, and Dij is the effective diffusion coefficient in the porous medium. We note that the molecular diffusion coefficient is multiplied by
to estimate the effective
diffusion coefficient in the porous medium, where FF is the formation factor (Perkins and Johnston, 1963). The phase velocities in Eq. (2.1) are calculated from the extended Darcy’s law for multiphase flow in porous media ,
17
(2.4)
where
is the relative permeability of phase j,
density of phase j,
is the viscosity of phase j,
is the mass
is the pressure of phase j, and k represents the absolute permeability. The
relationship between phase pressures are represented by capillary pressure functions (2.5) The phases are assumed to be in thermodynamic equilibrium which can be represented by equality of chemical potentials (2.6) It can be shown that the equality of chemical potentials leads to the equality of fugacities, and the fugacities are calculated using the equations of state (EOS) for reservoir fluids (Danesh, 1998; Orr, 2007). The phase saturations and mole fractions in each phase must sum to one
(2.7)
(2.8) The equations presented in this section provide enough information to obtain the solution to a flow and transport problem that models the effects of convection, diffusion, dispersion and phase equilibrium (Orr, 2007). Note that the governing equations presented in this section are valid for single-porosity systems. The governing equations for dual-porosity systems will be presented in the subsequent chapters.
18
Chapter 3 :
Experimental and Modeling Study of Multicontact Miscible Displacements1
3.1 Introduction In this chapter, we investigate the accuracy of some physical models that are commonly used to describe dispersive mixing and mass transfer in compositional reservoir simulation. We combine experimental and numerical efforts to investigate the effect of mixing and mass transfer in nearmiscible and MCM displacements. We use an analog fluid system that exhibit liquid-liquid phase behavior at ambient conditions that is comparable to the vapor-liquid equilibrium behavior of mixtures of CO2 and hydrocarbons at high pressure and temperature. Two-component FCM, and 4-component near-miscible and MCM displacement experiments were performed in a synthetic porous medium in our research group to form the basis of our modeling study with emphasis on physical models for representing mixing and mass-transfer in these relevant displacement processes (Rastegar, 2010). We start by presenting the analog fluid system and the design/characterization of our porous medium and solid-fluid interactions. We then present our observations from FCM, near-miscible and MCM displacement experiments and provide an interpretation of the observations based on numerical calculations with different physical models. A discussion and analysis of the presented results and observations concludes the chapter. 1
Most of the results presented in this chapter have been published in Transport in Porous Media: Shojaei, H., Rastegar, R., & Jessen, K.: Mixing and mass transfer in multicontact miscible displacements. Transport in Porous Media 94, 837-857 (2012)
19
3.2 Fluid System To select a suitable analog system for studying mixing and mass transfer in the context of CO2 injection processes, the quaternary system of CO2, Methane (CH4), Butane (nC4) and Dodecane (C12) at high pressure was initially examined. Figure 3.1 (top) shows the quaternary phase diagram for this system at 2280 psi and 212°F as calculated from the Peng-Robinson (PR) equation of state (EOS) (Peng and Robinson, 1976). CO2 represents the injected gas composition and a representative oil composition is shown with a circle in the base of the quaternary phase diagram. At the given conditions, the displacement of oil by CO2 will be MCM as compositions that are formed during the displacement process (assuming one-dimensional disperion-free flow) will pass through the critical locus close to the front ternary of the quaternary diagram (Orr, 2007). To facilitate experimental work at ambient conditions, analog solvents that mimic the phase behavior of high pressure CO2/hydrocarbon systems were used. The quaternary system of Water, Methanol (MeOH), Isopropanol (IPA), and Isooctane (iC8) was selected due to similarities with the high-pressure CO2/hydrocarbon system (Rastegar, 2010). Pure component properties of the analog fluids are summarized in Table 3.1. Other authors have used water-alcohol-hydrocarbon systems to study MCM displacements in porous media at ambient conditions (e.g. Batycky, 1994; Al-Wahaibi et al., 2007; and AlHamdan et al., 2011). However, all previous studies have utilized simplified representations of the associated phase behavior and transport properties that complicate the interpretation of the displacement experiments.
20
Figure 3.1 Quaternary phase diagrams (mass fractions). Top: CO2-CH4-nC4-C12 at 2280 psi and 212°F as calculated from the PR EOS. Bottom: Water-MeOH-IPA-iC8 at 68°F and 14.7 psi as calculated from the UNIQUAC model. 21
Table 3.1 Pure component property data for the analog fluid system Component
Mw (g/mol)
ρ (g/cm3)*
Viscosity (cP)*
Boiling Point (oC)**
Water
18.01
0.997
0.890
99.9
MeOH
32.04
0.787
0.544
64.7
IPA
60.10
0.781
1.960
82.3
iC8
114.23
0.687
0.473
99.3
*) density and viscosity at 25°C, **) Normal boiling point
Figure 3.1 (bottom) shows the phase diagram of the analog system at ambient conditions as calculated by the Universal Quasichemical Activity Coefficient (UNIQUAC) model (see Appendix A for more details). For the analog system, mixtures of Water and MeOH represent an injection gas composition (CO2 or CO2-rich gas) while compositions in the bottom ternary of the phase diagram represent an initial oil. 3.2.1 Phase Behavior To characterize and model the phase behavior of the analog fluid system, a range of ternary and quaternary equilibrium mixtures was created and analyzed by gas chromatography (GC). The UNIQUAC model, introduced by Abrams and Prausnitz (1975), was subsequently used to model the phase behavior of the analog fluid system and a set of binary interaction coefficients was extracted from the experimental observations. Additional details related to the phase behavior of the analog fluid system are presented in Rastegar and Jessen (2011). 3.2.2 Mixture Densities Mixture densities are commonly provided by the EOS model that is used in the equilibrium calculations for hydrocarbon systems. As the UNIQUAC model does not provide us with 22
mixture density information, a separate density model is required. We assume that the excess volume is negligible and calculate mixture densities based on the assumption of ideal mixing. This assumption is supported by the experimental density data reported by Otero et al. (2000) for mixtures of Water-IPA-iC8 at 25°C.
Figure 3.2 Comparison of mixture densities (25°C and 1 atm) from Otero et al. (2000) with ideal mixing calculations as a function of Water mass fraction along the Water-IPA-iC8 binodal curve.
Figure 3.2 compares the data from Otero et al. (2000) with ideal-mixing calculations along the binodal curve of the Water-IPA-iC8 ternary. The maximum error between calculated and observed mixture densities is less than 2% and is observed for water-rich equilibrium mixtures. Accordingly, we assume that the assumption of ideal mixing is sufficiently accurate for our modeling work.
23
3.2.3 Mixture Viscosities The viscosity of mixtures of the analog fluids is a highly nonlinear function of the composition due to the strong nonideal behavior of water-alcohol systems. Simple viscosity correlations are hence not suitable for this system. The UNIQUAC viscosity model, as proposed by Martin et al. (2001), was selected and used throughout our modeling work (see Appendix B for more details). The binary interaction coefficients of the UNIQUAC viscosity model were obtained from viscosity data for binary mixtures of MeOH-Water (Tanaka et al. 1987), Water-IPA (Tanaka et al. 1987), IPA-iC8 (Ku 2008), and MeOH-IPA (Soliman and Marschall 1990). Experimental observations are compared to model calculations for the relevant binary systems in Fig. 3.3 and the model parameters are summarized in Table 3.2.
Table 3.2 Parameters for UNIQUAC viscosity model (interactions and structural parameters) Comp
Water
MeOH
IPA
iC8
Water
0
-276.03
-536.44
0.00
0.9945
0.28486
MeOH
423.54
0
398.30
0.00
1.4320
1.4311
IPA
999.08
-210.98
0
-155.96
2.2571
3.3915
iC8
0.00
0.00
-8.32
0
5.0080
5.8463
The viscosity model was subsequently tested with viscosity data for 11 mixtures of MeOH-IPAWater (Soliman and Marschall, 1990) and a satisfactory agreement was observed with an average error of less than 3%. We note that the interaction coefficient between MeOH and iC8 is set to zero due to lack of experimental data for this binary. This results in a monotonic and almost linear compositional dependence of the viscosity for the associated binary system. We
24
believe that this is a reasonable representation for this polar-nonpolar binary as cross-association will not contribute to a nonmonotonic compositional dependence as seen for example for the IPA-Water binary.
Figure 3.3 Experimental and calculated viscosity for the binary mixtures (25°C and 1 atm). Data from Tanaka et al., 1987 (MeOH-Water and Water-IPA), Ku, 2008 (IPA-iC8) and Soliman and Marschall, 1990 (MeOH-IPA).
3.2.4 Interfacial Tension Pendant drop measurements were performed for equilibrated ternary mixtures of Water, IPA and iC8 to validate the experimental observations of Morrow et al. (1988) and our observations were in good agreement with reported values (Rastegar, 2010). Initial attempts to correlate the data from Morrow et al. (1998) including the Parachor method and the more complex model proposed by Bahramian and Danesh (2004) failed in predicting the IFT with reasonable accuracy. To 25
include IFT effects in our modeling efforts, we use a simple correlation derived from the data of Morrow et al. (1988). The proposed correlation has the form b,
(3.1)
where x denotes the Euclidian norm of a tie line based on mass fractions. A comparison of the observed and calculated IFT (along with the coefficients of Eq. 3.1) is provided in Fig. 3.4. The correlation was subsequently validated for a mixture of Water-MeOH-iC8 (circle in Fig. 3.4) from Garcia-Flores et al. (2007), and was found to be in reasonable agreement. We note that the correlation predicts a value of IFT equal to 4e-4 mN/m for critical mixtures which is sufficiently accurate for our modeling purposes. However, additional work is warranted to develop more rigorous IFT models for these complex mixtures.
Figure 3.4 Simplistic IFT model based on tie-line lengths (mass fractions), IFT in mN/m. Data from Morrow et al., 1988 (Water-IPA-iC8) and Garcia-Flores et al., 2007 (Water-MeOH-iC8).
26
3.3 Packed Column A porous material was selected to allow for the hydrocarbon-rich phase (oil compositions) to act as the wetting phase while water-rich mixtures (injection gas compositions) act as the nonwetting phase. To achieve this, PTFE materials were used to design the packed column. PTFE powder from Sigma-Aldrich with a 100 micron average particle diameter was used as the packing material. The column was designed with an inner diameter of 0.375 inches and length of 12 inches. This diameter is sufficiently large to minimize wall effects in the packed column. The porosity of the column was measured gravimetrically to 47% by saturating the column with iC8 under moderate vacuum. The permeability of the packed column was calculated to 0.39 Darcy from Darcy’s law using the stabilized pressure drop across the column when iC8 was injected at a constant rate (Rastegar, 2010). 3.4 Solid-Fluid Interactions 3.4.1 Wettability Additional considerations were made regarding the wettability of the individual components of the analog fluid system. From the pure components, Water is the only component that does not spread on a PTFE surface. Mixtures of Water and MeOH at sufficiently high MeOH concentrations (above 65% by mass) also spread on a PTFE surface and hence, the use of WaterMeOH mixtures to mimic injection gas compositions was questionable. With iC8 as the primary component in the analog oil phase, an additional experiment with a PTFE capillary tube was performed to identify the preferential wetting component of the MeOHiC8 pair. Figure 3.5 shows the interface between MeOH and iC8 at equilibrium in a PTFE capillary tube that confirms that iC8 will preferentially wet the surface and that MeOH will act as
27
a nonwetting phase in displacements where iC8 is present in the porous medium (Rastegar, 2010). 3.4.2 Relative Permeability Steady state relative permeability experiments were performed for the immiscible pair of iC8 and Water to establish the maximum residual oil saturation in the porous medium at the maximum attainable IFT of 39 mN/m (longest tie-line in Fig. 3.1). Figure 3.6 reports the relative permeabilities of the porous medium to Water and iC8 phases as a function of the iC8 saturation. From the experiments, a residual oil saturation of 29% is observed (Rastegar 2010). To model the observed relative permeability data, we use Corey-type relative permeability functions with saturation exponents for wetting and non-wetting phases of 1.5 and 3, respectively.
Figure 3.5 Interface between iC8 and MeOH in a PTFE capillary tube. While both pure substances spread on PTFE, iC8 preferentially wets the surface in this binary setting.
28
Figure 3.6 iC8 (wetting phase) - Water drainage relative permeability: Steady state observations and Corey-type model.
3.5 Displacement Experiments A series of displacement experiments was performed with the analog fluid system and the PTFE column. In all experiments, 3 droplets of effluent (less than 1% of the pore volume) were diverted to a clean sample vial, every 5-10 minutes, diluted with Ethanol and analyzed for composition by gas chromatography. This approach was used to minimize the additional apparent mixing during sampling. Displacement experiments were, unless otherwise noted, performed vertically to promote gravity stable fronts and to avoid phase segregation (override/underride) due to gravity: A moredense and more-viscous phase was injected to displace a less-dense and less-viscous phase from the bottom to the top of the column. According to Lake (1989), the upward vertical displacements performed in this work are unconditionally stable since the end-point mobility ratios are less than one and the injection phases are denser than the initial phases. 29
3.5.1 FCM Displacements Two FCM displacement tests (experiments I and II) were performed to study the effective mixing in the porous material. In the first displacement experiment, iC8 was displaced by IPA in the vertical direction (bottom to top) at 23°C at a volumetric flow rate of 0.05 ml/min. In a second FCM experiment, IPA was used to displace iC8 in the horizontal direction also at a rate of 0.05 ml/min (Rastegar, 2010). Based on the analysis of the scaling groups suggested by Zhou et al. (1997), the horizontal FCM test (exp. II) is not dominated by gravity forces. A 2D simulation with the conditions in experiment II verifies that gravity underride does not happen in this experiment. A flow rate of 0.05 ml/min corresponds to an average linear velocity of 3.3 ft/day. The effluent concentrations from these FCM displacements are reported in Fig. 3.7 (top and bottom figures are for vertical and horizontal displacements, respectively) as a function of pore volumes injected (PVI). We note that both FCM displacement experiments show a distinct tailing behavior and defer additional discussion/interpretation to the modeling section below. 3.5.2 Near-miscible and MCM Displacements Two 4-component near-miscible and MCM displacements were subsequently performed in the vertical direction (bottom to top). For these displacement experiments, the initial and injection compositions were selected to provide for a near-miscible and a MCM displacement in the absence of dispersion. The initial and injected compositions are reported in Table 3.3 (Rastegar, 2010).
30
Figure 3.7 Effluent concentrations (in mass fractions) from FCM displacement experiments with IPA and iC8: Experimental data (circles), single-porosity model (solid line), DP I (broken line), and DP II (dotted line). Top: Vertical displacement. Bottom: Horizontal displacement.
Table 3.3 Summary of displacement experiments Initial composition by mole Run
T (°C)
q (ml/min)
Water
I , II
23
0.05
0.00
III
19
0.05
IV
21
0.05
MeOH
Injection composition by mole
IPA
iC8
Water
MeOH
0.00
0.00
1.00
0.00
0.00
0.00
0.25
0.75
0.08
0.00
0.29
0.63
31
IPA
iC8
0.00
1.00
0.00
0.25
0.75
0.00
0.00
0.08
0.90
0.02
0.00
In the first 4-component displacement experiment (Exp. III), an initial oil composition of 75% iC8 and 25% IPA (by mole) was displaced by 75% MeOH and 25% water (by mole). This corresponds to a near-miscible displacement as demonstrated by a numerical calculation in 1D with 10,000 cells in the absence of physical dispersion/diffusion (hyperbolic form). Figure 3.8 reports the variation in equilibrium K-values (Ki = yi/xi) and analog gas saturation along the displacement length after 0.7 PVI. We observe that the trailing vaporization shock propagates at a specific velocity that is less than unity and that near-miscible conditions exist upstream of the leading edge of the displacement.
Figure 3.8 Natural logarithm of equilibrium K-values and saturation of analog gas along displacement length (zD=z/L) for experiment III at the dispersion-free limit after 0.7 PVI.
Figure 3.9 reports the effluent concentration of the four components in experiment III as a function of PVI. At the dispersion-free limit for MCM displacements, all compositions would propagate at a unit characteristic velocity through the porous medium. However, the impacts of mixing and mass transfer in the experiment at near-miscible conditions cause the characteristic 32
velocities of the various compositions to depart from unity. In addition, we observe that a bank of IPA is formed at the leading edge of the displacement process. This is, in part, due to the curvature of the two-phase boundary in the quaternary diagram that forces the compositional path of the displacement process to pass through a region of higher IPA concentrations relative to both initial and injection compositions (Orr, 2007).
PVI
Figure 3.9 Effluent concentrations (in mass fractions) for displacement experiment III from dual-porosity model I (solid line), dual-porosity model II (dotted line), and experimental observations (circles).
In a second 4-component displacement experiment (Exp. IV), a small amount of Water was added to the oil composition to place it on the Water–IPA–iC8 ternary while the injected
33
composition was enriched with IPA (see Table 3.3). The enrichment of the injected composition was performed to further promote the development of multicontact miscibility. To confirm the development of miscibility, a numerical calculation in 1D with 10,000 cells that represents the dispersion-free limit was performed. Figure 3.10 reports the variation in the equilibrium Kvalues and saturation along the displacement length after 0.7 PVI. The development of miscibility by a vaporizing/condensing mechanism is evident from the hour-glass shape observed in the top panel of Fig. 3.10.
Figure 3.10 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment IV at the dispersion-free limit after 0.7 PVI.
The observed effluent concentrations for the four components in the MCM displacement experiment (IV) are reported in Fig. 3.11. As for experiment III, we observe a departure from unit characteristic velocity for all components. Again, this is attributed to the mixing and mass transfer that influences the displacement characteristics relative to a hypothetical dispersion-free limit. In this experiment, we observe that banks of both Water and IPA are formed at the leading 34
edge of the displacement: Water and IPA are both more soluble in MeOH than in iC8 and are hence gradually stripped from the initial oil in the system to form these banks.
Figure 3.11 Effluent concentrations (in mass fractions) for displacement experiment IV from dual-porosity model I (solid line), dual-porosity model II (dotted line), and experimental observations (circles).
35
3.6
Mathematical Modeling
We observe a tailing effluent concentration in both FCM displacement experiments (see Fig. 3.7) that signifies an imperfect sweep of the column. This, in turn, suggests that a 1D single-porosity (SP) model will be insufficient to capture the dynamics of the displacement experiments. A 1D dual-porosity (DP) model with mass transfer between flowing and stagnant porosities (Coats and Smith, 1964), or a two (or three)-dimensional single-porosity model with explicit representation of heterogeneity can be used to capture the tailing behavior observed in dispersivity experiments. Since explicit information regarding the heterogeneity of the porous material is not available, we use the Coats-Smith model to interpret our displacement experiments. Advection, diffusion, dispersion, and gravity effects are the main physical mechanisms that control the displacement dynamics in the presented experiments. To represent the relevant physics, we write the 1D continuity equation for multicomponent two-phase flow in the packed column as
(3.2)
where xij is the mole fraction of component i in phase j, phase j, ρj is the molar density of phase j,
is the porosity, Sj is the saturation of
is the velocity of phase j,
represents the effective
dispersion (combined impact of dispersion and diffusion) for component i in phase j, f denotes the flowing fraction, t represents time,
denotes the direction along the packed column, np
represents the number of phases, and nc is the number of components. The mass transfer between the flowing and the stagnant porosities (qi) is represented by a simple linear mass-transfer model
(3.3)
36
where θi is the mass transfer coefficient of component i. Superscript * denotes the quantities of mixtures in the stagnant porosity. Phase velocities are calculated from Darcy’s law and we assume that capillary pressure is negligible for our displacement experiments. The effective dispersion coefficients in Eq. 3.2 are assumed to be identical for all components within a phase (by assuming that molecular diffusion is negligible compared to velocity dependent dispersion) and are calculated from the phase saturations and velocities as (3.4) where
is the effective longitudinal dispersivity of the packed column. This simplification is
introduced to reduce the number of model parameters. The phases are assumed to be in thermodynamic equilibrium and, as discussed in section 3.2.1, we use the UNIQUAC model to predict the phase behavior of the quaternary system used in our experiments. The UNIQUAC viscosity model (see section 3.2.3) is used for viscosity calculations while density calculations are based on ideal mixing (see section 3.2.2). To calculate IFT, we apply the simple correlation presented in section 3.2.4 while Corey-type relative permeability functions (see section 3.4.2) with IFT scaling are used in our displacement calculations to account for the effects of IFT on relative permeabilities (see Coats, 1980; Amaefule and Handy, 1982). A scaling factor, F, in the range of 0 to 1 that indicates how close a mixture is to a critical point (miscible condition) is calculated from (3.5) where σ is the mixture IFT, σ0 is a reference IFT, and β represent an adjustable scaling exponent. The scaling factor interpolates between straight lines and reference (σ0) relative permeability. According to Coats (1980), the scaling exponent β is generally in the range of 0.1 to 0.25. Since 37
our relative permeability experiments were performed at the maximum attainable IFT of 39 mN/m, we use this value as the reference IFT in Eq. 3.5. Residual oil saturations and critical gas saturations are then modified by the scaling factor F as (Coats, 1980) (3.6) Normalized oil and gas saturations are subsequently calculated from
(3.7)
and Oil and gas relative permeabilities are finally calculated from (3.8) where no and ng are Corey exponents (see section 3.4.2) for oil and gas respectively. Throughout our displacement calculations, we assume that Sgc is zero. An explicit finite volume formulation was used to simulate our displacement experiments. All simulations were performed in one dimension (as discussed above) with 1,000 grid blocks to minimize the effects of numerical diffusion. 3.7 Displacement Calculations Two representations of a dual-porosity porous medium were considered in this work. In the first model (DP I), the stagnant porosity is evenly distributed along displacement length, while in the second model (DP II), the stagnant (bypassed) porosity is located at the inlet and outlet of the packed column. This representation attempts to mimic imperfect distribution of the fluids at the inlet and outlet manifold.
38
3.7.1 FCM Displacements Three model parameters were obtained by matching the numerical calculations with DP I to the effluent concentrations from the vertical dispersivity experiment (Exp. I). The model parameters, reported in Table 3.4, include the flowing fraction (f = 0.965), the effective longitudinal dispersivity (α = 0.0015 m) and the mass transfer coefficient (θ = 0.0003 sec-1). The agreement between experimental observations and numerical calculations is reported in the top panel of Fig. 3.7. We note that a dispersivity of 0.0015 m corresponds to a Peclet number of 203 that indicates a moderate level of dispersive mixing in the porous medium.
Table 3.4 Model parameters obtained from vertical dispersivity experiment Flowing fraction: f
Dispersivity: α (m)
Mass transfer: θ (1/sec)
DP I
0.965
0.0015
0.0003
DP II
0.650
0.0015
0.0050
In DP II, we allocate 5% of the column length at the inlet and outlet of the system (10% of the total column length) to represent the stagnant porosity. This representation attempts to mimic imperfect distribution of the fluids at the inlet and outlet manifolds. The flowing fraction at the inlet and outlet of the packed column is calculated to 0.65 by assuming that the overall flowing porosity is the same for DP I and DP II. We apply the value of dispersivity obtained from DP I in our displacement calculations with DP II and match DP II model to the experimental data from vertical dispersivity test by adjusting the mass transfer coefficient (θ). Since the characteristic length for transverse mass transfer is larger for DP II (as compared to DP I), we expect a larger value of mass transfer coefficient (θ).
39
By matching DP II to the effluent concentrations from the vertical dispersivity experiment, a value of θ=0.005 sec-1 is obtained that is consistent with our expectations (see Table 3.4). Figure 3.7 (top) compares the calculated effluent concentrations from SP and DP models (I and II) with experimental effluent data from the vertical dispersivity experiment. From Fig. 3.7 (top) we observe that the tailing behavior is captured accurately by the DP models for vertical FCM experiment. In order to validate the matches obtained for DP models using the vertical dispersivity test, we predict the effluent concentrations for the horizontal dispersivity test and compare the predictions with the experimental data in Fig. 3.7 (bottom). From Fig. 3.7 (bottom) we observe that both DP models are able to accurately predict the effluent concentrations for the horizontal dispersivity test. We note also that there is no significant difference between the calculated effluent compositions from DP I and DP II. For comparison, Fig. 3.7 reports the calculations with a SP model and we observe, as expected, that the tailing behavior of the dispersivity experiments cannot be reproduced. We note that the value of dispersivity as obtained from DP I (α = 0.0015 m) was used with the SP model. 3.7.2 Near-miscible and MCM Displacements Next, we consider the near-miscible (Exp. III) and MCM (Exp. IV) displacement experiments. The model parameters from the dispersivity experiments (see Table 3.4) are used to simulate the four-component displacement experiments. The value of θ, as estimated from the dispersivity experiments, was used as the effective mass transfer coefficient for all components in the fourcomponent displacement calculations (see Eq. 3.3). From our modeling efforts, a value of β equal to 0.15 (see Eq. 3.5) was found to provide the best agreement between experimental and calculated effluent concentrations for experiments III 40
and IV. We note that the IFT scaling exponent (β=0.15) is the only parameter that was adjusted to match the four-component displacement experiments. The additional three model parameters obtained from the dispersivity experiments were used directly as input parameters to the displacement calculations without further adjustments. Figure 3.9 compares calculations with DP I and DP II to the experimental effluent concentrations from experiment III. Despite the very limited parameter adjustment, we find a reasonable agreement between calculated and measured effluent concentrations for all four components. A minor mismatch is observed for the Water effluent concentrations at later times and we attribute this to experimental errors introduced in the sampling and analysis of the effluent concentrations. We also observe that the numerical calculations with DP I and DP II are close to identical. Numerical calculations are compared to the experimental effluent concentrations for experiment IV in Fig. 3.11. The main features of the effluent concentrations are captured well by the calculations. The moderate mismatch observed for Water and IPA effluent concentrations is again attributed to experimental errors associated, in part, with sampling and subsequent compositional analysis. We observe, as for experiment III, that DP I and DP II provide close to identical effluent concentrations. 3.8 Effect of Mixing on Near-miscible and MCM Displacements For MCM displacements it has been shown that dispersive mixing (or alternatively mass transfer between flowing and stagnant regions) causes the composition paths to intersect the two-phase region (e.g. Walsh and Orr, 1990; Johns et al., 1994). Our simulations confirm that two-phase flow occur in both experiments III and IV. Figure 3.12 (3.13) reports the K-values and gas saturation along the displacement length (in flowing 41
porosity) when dispersive mixing and mass transfer are included in DP I calculations using 1,000 grid blocks for experiment III (IV). By comparing Fig. 3.12 (3.13) with Fig. 3.8 (3.10) we observe that dispersive mixing and mass transfer between flowing and stagnant porosities cause the displacement to depart from MCM conditions and promote two-phase flow inside the porous medium.
Figure 3.12 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment III with dispersion and mass transfer after 0.7 PVI.
42
Figure 3.13 Natural logarithm of equilibrium K-values for the 4 components and saturation of analog gas along the displacement length (zD=z/L) for experiment IV with dispersion and mass transfer after 0.7 PVI.
Figure 3.14 (3.15) compares the composition path for the dispersion-free calculations (SP simulation without physical dispersion using 10,000 cells) with the calculations with dispersive mixing and mass transfer (simulation with DP I using 1,000 cells) for experiment III (IV). As observed from Figures 3.14 and 3.15, dispersive mixing and mass transfer between flowing and stagnant porosities cause the composition path to move inside the two-phase region for both experiments III and IV. The compoent recoveries for the dispersion-free calculations and the calculations with dispersive mixing and mass transfer are compared for experiments III and IV in Figures 3.16 and 3.17 respectively. Since dispersion and mass transfer cause the displacements to depart from the MCM conditions, the analog oil recovery is reduced when these mechanisms are present.
43
Figure 3.14 Composition path (in mass fractions) for Exp. III from dispersion-free calculations (broken line) and calculations with dispersive mixing and mass transfer (dotted line).
Figure 3.15 Composition path (in mass fractions) for Exp. IV from dispersion-free calculations (broken line) and calculations with dispersive mixing and mass transfer (dotted line). 44
Figure 3.16 Component recovery for Exp. III from dispersion-free calculations (IPA: solid line, iC8: broken line) and calculations with dispersive mixing and mass transfer (IPA: broken dotted line, iC8: dotted line).
3.9 Importance of Selected Physical Models To investigate the impact of the selected dispersion model as well as the impact of IFT scaling of relative permeabilities on the calculated effluent concentrations, additional displacement calculations were performed, and the results are presented in this section. First we investigate the formulation of the dispersive mixing and replace the phase-velocity dependent model (see Eqs. 3.2 and 3.4) with a model that is based on the total fluid velocity. The effective dispersion coefficients for the latter model are calculated from the phase saturations and total velocity as 45
Figure 3.17 Component recovery for Exp. IV from dispersion-free calculations (IPA: solid line, iC8: broken line) and calculations with dispersive mixing and mass transfer (IPA: broken dotted line, iC8: dotted line).
(3.9) where
is the total linear velocity (summation of phase velocities). We note that when the
dispersion model is based on the total fluid velocity (Eq. 3.9), the continuity equation (Eq. 3.2) is rewritten as (3.10)
46
We found that the use of total velocity in the dispersion model does not have a significant effect on the simulation results or agreement with the experimental observations for both experiments III and IV. Experiment III is a near-miscible displacement and hence some two-phase flow is expected inside the column. In addition, as discussed in the previous section, dispersive mixing and mass transfer between flowing and stagnant porosities promote two-phase flow in both experiments III and IV. Hence the observation that the use of total or phase velocities in the dispersion model lead to substantially identical results is attributed to the moderate level of mixing observed in these experiments (α=0.0015 m). This argument was subsequently validated with additional simulations of experiments III and IV using larger values of dispersivity (smaller Pe numbers) exhibiting a more notable effect of the selected dispersion model. Next, we probe the sensitivity of our displacement calculations to the IFT scaling of relative permeability functions for experiment III. Figure 3.18 compares the experimental observations with 3 different numerical calculations (DP I) with values of F equal to 0.0, 1.0 and as calculated from Eq. 3.5 with β equal to 0.15. F = 0 corresponds to calculations with straight-line relative permeability functions (valid for fully miscible condition) while F = 1 defaults the relative permeability to the steady-state measurements reported in Fig. 3.6 (valid for immiscible condition). From Fig. 3.18, we observe that the calculated effluent is very sensitive to the scaling of the relative permeabilities and that the IFT scaling of relative permeabilities is required to match the experimental observations. The observed sensitivity can be explained by the variations in the interfacial tension due to composition changes that arise from the interaction of flow and phase behavior inside the column.
47
Similar observations regarding the sensitivity of the IFT scaling were obtained for displacement experiment IV as shown in Fig. 3.19. We note that additional steady-state relative permeability measurements can be performed with pre-equilibrated phase compositions at an IFT different from 39 mN/m to determine the scaling exponent (β) directly. This would allow for predictive calculations of the multicomponent displacements without any need for parameter adjustments.
Figure 3.18 Effluent concentrations (in mass fractions) for displacement experiment III from DP I with IFT scaling of relative permeabilities (solid line), DP I without IFT scaling of relative permeabilities (broken dotted line), DP I with linear relative permeabilities (dotted line) and experimental observations (circles). 48
Figure 3.19 Effluent concentrations (in mass fractions) for displacement experiment IV from DP I with IFT scaling of relative permeabilities (solid line), DP I without IFT scaling of relative permeabilities (broken dotted line), DP I with linear relative permeabilities (dotted line) and experimental observations (circles). 3.10 Discussion In the previous sections, we presented the experimental and modeling aspects of our work on mixing and mass transfer in near-miscible and MCM displacement processes. Both dispersivity (FCM) experiments display a tailing behavior in the effluent concentrations that is attributed to imperfect sweep in the column. A 1D SP model was not able to capture this behavior and,
49
therefore, a 1D DP model with mass transfer between flowing and stagnant porosities (CoatsSmith model) was applied to represent the tailing behavior observed in dispersivity experiments. From our interpretation of the dispersivity experiments, we find that dispersive mixing is moderate in the porous medium with a Peclet number of approximately 200. For comparison, the value of longitudinal dispersivity can be estimated using the correlation proposed by Perkins and Johnston (1963). They reported the following correlation to estimate the dispersivity in unconsolidated porous media (3.11) where
,
, FF,
, , and
denote longitudinal dispersion coefficient, molecular diffusion,
formation factor, porosity, interstitial velocity, and particle diameter, respectively. Using a typical value for molecular diffusion in liquids (Do=1e-5 cm2/sec), a formation factor for unconsolidated porous media (FF=2.5), and additional parameters as described in previous sections, we arrive at a value of α = 2.1e-4 m from the correlation. This is slightly lower than the value obtained from matching our experimental observations (α = 1.5e-3 m): According to the summary of experimental observations presented by Lake (1989), the value of dispersivity from laboratory-scale experiments are typically in the range of 1e-4 to 1e-2 m. Thus the agreement between the values of dispersivity obtained from our experiments and the correlation suggested by Perkins and Johnston (1963) is reasonable. We believe that the mass transfer between flowing and stagnant porosities in our experiments is diffusive. To verify this, we compare the characteristic time for diffusion in a porous medium to the characteristic time for mass transfer between flowing and stagnant porosities estimated from our experiments. The characteristic time for diffusion can be estimated as
50
(3.12) where L is the characteristic length and D is the diffusion coefficient in the porous medium. Assuming that the characteristic length for mass transfer is 10-20% of the column diameter combined with typical values of Do=1e-5 cm2/sec and FF=2.5, we estimate a characteristic time of 959-3837 sec for diffusion in the porous medium. The mass transfer coefficient obtained by matching DP I to the experimental data from dispersivity tests is θ = 3e-4 sec-1; thus the characteristic time for mass transfer between flowing and stagnant porosities is approximately which is in the same range as the estimated characteristic time for diffusion in the porous medium. It is therefore reasonable to assume that the mass transfer between flowing and stagnant porosities in DP I model is diffusive. To investigate the relative impacts of longitudinal dispersion (along streamlines) and transverse mass transfer (between stagnant and flowing regions), we compare simulations with the individual mechanisms turned on and compare the results with the experimental data for the FCM displacements in Fig. 3.20. From Fig. 3.20, we observe that the transverse mass transfer has a more significant impact on the agreement between calculated and experimental effluent concentrations for the vertical FCM displacement. However, for the horizontal FCM displacement, an explicit representation of both the longitudinal dispersion and the transverse mass transfer is important. The observation that longitudinal dispersion has a smaller impact on the effluent concentrations for the vertical experiment is attributed to the gravity stabilization that reduces the local variations in fluid velocity.
51
Figure 3.20 Relative impacts of longitudinal dispersion and transverse mass transfer for FCM displacements. Top: Vertical displacement. Bottom: Horizontal displacement.
To further establish that the level of numerical diffusion is small compared to the physical dispersion, we approximate numerical Peclet numbers for the relevant calculations (see Appendix C for details on estimating numerical diffusion in single- and two-phase flow). The numerical Peclet number is estimated in the range of 2200 to 3000 for experiments I through IV for simulations with 1,000 grid blocks. Accordingly, numerical Peclet numbers are one order of magnitude larger that the physical Peclet number of approximately 200 estimated from our experiments, and we can assume that the impact of numerical artifacts is marginal. From the displacement experiments, we observe that DP I and DP II provide for almost identical results. We note that in order to match the experimental data, separate parameter
52
adjustments were performed for DP I and DP II using the effluent concentrations from the vertical dispersivity experiment. The Coats-Smith model used in our interpretations is a 1D model that attempts to accounts for some unresolved 2D/3D effects by means of a mass transfer term. The two representations of the dual-porosity system (DP I and II) used in this work attempts to include additional spatial information regarding the location of stagnant porosity in our application of the Coats-Smith model. However, since both DP I and DP II are able to reproduce the experimental observations, we cannot argue which model (DP I or DP II) provides for a better representation of the unresolved heterogeneity. This, in turn, points out the need for experiments where the flowing and stagnant regions are well defined by e.g. CT imaging or by careful design of heterogeneity in bead packs. We observe that the calculated effluent concentrations are insensitive to the velocity dependence of the effective dispersion coefficient (phase velocities or total velocity). This is despite the fact that two phases are present inside the column due to near-miscible conditions (for experiment III) and dispersive mixing and mass transfer between flowing and stagnant porosities (for experiments III and IV). We attribute this insensitivity to the moderate level of mixing in our experiments. At larger scale, where the dispersivity is expected to increase, the impact of the dispersion model is likely to be more significant. IFT scaling of relative permeabilities was demonstrated to be of great importance in our modeling efforts. This importance is caused by the variations in the IFT that arise from interactions between flow and phase behavior during the displacements. This is in contrast to the theory of gas injection processes at the dispersion-free limit (Orr, 2007) where development of miscibility is independent of relative permeability. For MCM displacements in the absence of
53
dispersion, two-phase flow will not occur. However, the effect of dispersive mixing and mass transfer between flowing and stagnant porosities in our experiments results in some two-phase flow during both 4-component displacements. We show that if IFT scaling is not included, the numerical calculations fail in reproducing the experimental observations: Breakthrough time is underestimated and the cumulative effluent of Water and MeOH is overestimated (compared to the case where IFT scaling is performed) at the cost of iC8. In addition, the IPA is predicted to reside preferentially in the Water-rich phase rather than in the hydrocarbon-rich phase as observed in the experiments. Some degree of mismatch was observed between calculated and experimental effluent concentrations of Water and IPA. We attribute this to experimental errors introduced in effluent sampling and GC analysis. Additional error can be introduced via the phase behavior model used in the displacement calculations. However, we believe that the latter source of error is marginal based on the experiments and analysis presented by Rastegar and Jessen (2011). Finally, the use of a simple linear mass-transfer model appears to be sufficient to represent the component fluxes in and out of the stagnant porosities. While more sophisticated models exist (e.g. Maxwell-Stephan diffusion as discussed by Taylor and Krishna (1993)) and may be warranted in some displacement processes, the additional work required to determine component specific coefficients, does not appear to be crucial in the context of the displacement experiments reported in this work. 3.11 Conclusions Based on the results and analysis presented in this chapter, we conclude that:
The use of phase velocities or total velocity in dispersion modeling of near-miscible and MCM displacement processes with a moderate level of mixing does not have a significant 54
impact on the calculation results. This is despite the fact that two phases are flowing simultaneously inside the porous medium as a result of dispersive mixing and mass transfer between the stagnant and flowing porosities.
The two representations of a DP system used in our calculations provide for almost identical results. Therefore, it is not clear which model (DP I or DP II) provides for a better representation of the unresolved heterogeneity inside the column using the Coats-Smith model (A 1D model that accounts for some 2D/3D effects via a mass transfer term). This observation illustrates the need for 2D/3D displacement experiments where heterogeneity is carefully characterized to further test/validate the physical models that are used in compositional simulation.
IFT scaling of relative permeabilities is required in our calculations to match the experimental observations from displacement experiments where dispersive mixing and mass transfer between flowing and stagnant porosities promote some two-phase flow inside the porous medium. This observation contrasts the theory of gas injection processes that studies the dispersion-free limit of 1D flows where development of miscibility is independent of relative permeability.
55
Chapter 4 :
Diffusion and Matrix-Fracture Interactions during Gas Injection in Fractured Reservoirs1
4.1 Introduction Gravity drainage, molecular diffusion and viscous displacement are known as the main recovery mechanisms during gas injection in naturally fractured reservoirs. The relative significance of these mechanisms depends on several factors including matrix permeability, fracture intensity, fluids properties, injection rate and reservoir pressure and temperature. Viscous flow does not contribute directly to oil recovery because the injected gas channels through high-permeability fractures which comprise only a few percentage of the total pore volume. Gravity drainage, which is driven by density difference between oil and gas, plays a significant role when matrix permeability is high. Molecular diffusion may become the dominant recovery mechanism in cases with low matrix permeability and high fracture intensity. Contrary to conventional reservoirs where the impact of molecular diffusion is generally small, molecular diffusion can play an important role in fractured reservoirs because of the large surface area available for diffusion. Different investigators have demonstrated the efficiency of molecular diffusion in fractured reservoirs (e.g. da Silva and Belery, 1989; Hu and Whitson, 1991; Darvish et al., 2006; Hoteit and Firoozabadi, 2009; Vega et al., 2010; Jamili et al., 2011).
1
Most of the results in this chapter have been presented in the SPE Improved Oil Recovery Symposium: Shojaei, H., and Jessen, K.: Diffusion and matrix-fracture interactions during gas injection in fractured reservoirs. Paper SPE 169152 presented at the SPE Improved Oil Recovery Symposium, Tulsa, OK, 12-16 April (2014)
56
Diffusion of gas components from a fracture into the matrix extracts oil components from matrix and delays, to some extent, the gas breakthrough via high-permeability fractures. As a result, both displacement and sweep efficiencies are increased. Gravity drainage has a similar effect on displacement and sweep efficiencies. In this chapter we focus on molecular diffusion and matrix-fracture exchange while more details on other drive mechanisms in fractured reservoirs can be found elsewhere (e.g. Lemonnier and Bourbiaux, 2010; Chordia and Trivedi, 2010; Rezaveisi et al., 2012). Diffusive mass transfer in porous media can be attributed to three main mechanisms: Knudsen (free-molecule) diffusion, molecular (continuum) diffusion and surface diffusion. Knudsen diffusion occurs when the pore size is smaller than the mean free path of the molecules: Knudsen diffusion is dominant when the pore size is in the range of nanometers. Molecular diffusion takes place when the pore sizes are relatively large compared to the mean free path of the molecules. Surface diffusion describes the transport of matter in an adsorbed layer and involves interactions between surface and molecules (Mason and Malinauskas, 1983). In this work Knudsen and surface diffusions are neglected because it is assumed that the pores are in the range of micrometers and that there is no adsorbed layer on the walls of the pores. Molecular diffusion is driven by gradients in concentrations (or more generally by gradients in the chemical potential). Diffusion will oftentimes not play a significant role in regions of a porous media where fluids velocities are high (e.g. fractures) because the characteristic time for diffusion is relatively large. However, in regions of the porous media where flow velocities are low (e.g. matrix blocks) diffusion can dominate mass transfer over viscous flow (Perkins and Johnston, 1963; Shojaei et al., 2012).
57
Single-phase multicomponent molecular diffusion can be modeled using different approaches: The classical Fick’s law, the generalized Fick’s law and the Maxwell-Stefan (MS) model. In the classical Fick’s law approach, the diffusion flux of each component is only a function of its own concentration gradient. In other words, the interactions among different species (dragging effects) are neglected. The generalized Fick’s law approach, in contrary, takes into account the component interactions; i.e. the diffusion flux of each component depends on the concentration gradients of other components as well. In the MS model, diffusion is driven by a gradient in the chemical potential and component interactions are accounted for using component velocities (friction). It can be shown that under certain conditions the generalized Fick’s law and MS model are equivalent (Taylor and Krishna, 1993). In most reservoir simulation models (e.g. da Silva and Belery, 1989; Coats, 1989; Darvish et al., 2006; Vega et al., 2010; Jamili et al., 2011), multicomponent molecular diffusion is modeled using a classical Fick’s law approach with effective diffusion coefficients. This means the diffusion flux of each component will always be in the opposite direction of its concentration gradient. However, it has been shown that because of the dragging effects, diffusion may occur from a region of low to high concentration (reverse diffusion) (Duncan and Toor, 1962). Therefore ignoring the dragging effects (off-diagonal diffusion coefficients) may lead to considerable errors in simulation results as will be shown later. Effective diffusion coefficients for multicomponent mixtures (classical Fick’s law) are usually calculated using the approach of da Silva and Belery (1989) which is an extension of Sigmund’s correlation for binary mixtures (Sigmund, 1976). These coefficients correspond to diagonal elements of the diffusion coefficient matrix. The full-matrix diffusion coefficients
58
(generalized Fick’s law) can be calculated using the MS model (e.g. Taylor and Krishna, 1993; Ghorayeb and Firoozabadi, 2000; Leahy-Dios and Firoozabadi, 2007) as will be explained later. The gas-oil diffusion at the fracture-matrix interface (cross-phase diffusion) is usually modeled by assuming an average composition at the interface (equipartition hypothesis). da Silva and Belery (1989) argue that for practical purposes, this is a reasonable assumption for matrix blocks surrounded by interconnected fractures. However, the equipartition hypothesis and gas-oil diffusion coefficient calculations do not have a sound physical basis. In chemical engineering literature, the cross-phase diffusion has been modeled based on film theory in which oil and gas are assumed to be in equilibrium at the interface, and component fluxes to be continuous across the interface (e.g. Krishna and Standart, 1976; Taylor and Krishna, 1993). This approach has also been used in petroleum engineering to model gas-oil diffusion in lab experiments (e.g. Hu and Whitson, 1991; Irani et al., 2009; Guo et al., 2009; Haugen and Firoozabadi, 2009; Hoteit, 2013; Moortgat and Firoozabadi, 2013). In this chapter, we present a dual-porosity model for field-scale simulation of gas injection in fractured reservoirs in which the generalized Fick’s law is used to represent the molecular diffusion; and gas-oil diffusion at the fracture-matrix interface is modeled based on film theory. A novel matrix-fracture transfer function is introduced for gas-oil diffusion based on film theory. Diffusion coefficients are calculated using the MS model and are pressure, temperature and composition dependent. A time-dependent transfer function is used for the matrix-fracture interactions in which the shape factor is adjusted using a boost factor to differentiate between the transfer rate at early and late times. Field-scale examples are used to show that our approach, which is based on a sophisticated physical model for gas-oil diffusion (film theory), gives significantly different results from the
59
conventional approach. It is also demonstrated that the dragging effects (off-diagonal diffusion coefficients) and time-dependency of matrix-fracture transfer function can moderately impact the oil recovery during gas injection in fractured reservoirs. We also show that miscibility is not developed in the matrix blocks even at pressures above minimum miscibility pressure (MMP) when molecular diffusion is the main recovery mechanism during gas injection in fractured reservoirs. The remainder of this chapter is organized as follows: The governing equations for transport of fluid in a dual-porosity reservoir are first presented. Intra-phase and cross-phase diffusion are then discussed based on generalized Fick’s law and film theory. A generalized matrix-fracture transfer function is subsequently presented. Different physical models for molecular diffusion and matrix-fracture interactions are then compared using field-scale examples. A discussion and analysis of the presented results concludes the chapter. 4.2 Mathematical Model We use a dual-porosity approach to model gas injection in fractured reservoirs. Dual-porosity models assume there are two communicating domains in a fractured reservoir: a flowing domain (fracture) and a stagnant domain (matrix). Mass (or volume) balance equations are solved independently for these two domains. Transfer of fluids between these two overlapped domains is accounted for using a source/sink term in the mass (or volume) balance equations (Barenblatt et al., 1960; Warren and Root, 1963). We choose dual-porosity formulation because it provides a practical representation of fractured reservoirs with reasonable accuracy and computational efficiency. Other approaches such as fine-grid single-porosity and discrete fracture models are computationally expensive and need extensive data which are not generally available (e.g. fracture distribution in the reservoir). 60
In addition, a dual-porosity model can be constructed from detailed discrete fracture characterization using proper upscaling techniques (e.g. Karimi-Fard et al., 2006; Gong et al., 2008). The mass conservation equation for a component i in the flowing domain (fracture) containing np phases and nc components, where advection, diffusion and gravity are the main physical mechanisms, can be written as (Jessen et al., 2008) (4.1) while the mass conservation equation for a component i in the stagnant domain (matrix) is written as (4.2) where the overall molar density Ci is given by (4.3) the overall molar advective flux Fi is given by (4.4) and the overall molar diffusive flux Hi is given by
(4.5)
where Hi,j denotes the molar diffusive flux of component i within phase j (intra-phase diffusion) and Hi,jk represents the molar diffusive flux of component i at the interface between phases j and
61
k (cross-phase diffusion). More details on calculation of molar diffusive fluxes due to intra-phase and cross-phase diffusion are given in the subsequent section. In Equations (4.1) through (4.4), ϕ is the porosity; xij is the mole fraction of component i in phase j; ρj and Sj represent the molar density and saturation of phase j respectively; vj is the velocity vector of phase j; qi is the source term of component i; and Γi represents the transfer of component i from matrix to fracture. The phase velocities in Eq. (4.1) are evaluated from the pressure field which is obtained by solving the following volume-balance equation (4.6) where ct is the total fluid compressibility; p is the pressure; Vt is the total fluid volume; Vcell is the pore volume; and ni is the number of moles of component i. Capillarity is not included because we focus on gas injection processes in which IFT effects are less significant. We use a finite volume IMPES (implicit pressure explicit saturation/composition) formulation with a Cartesian grid in this work. This means at each new time level, Eq. (4.6) is solved with coefficients fixed at the old time level. The discrepancy between cell volume and fluid volume is minimized by carrying errors forward in time (Trangenstein and Bell, 1989; Jessen et al., 2008). Once the pressure field is obtained at the new time level using Eq. (4.6), phase velocities at the gridblock interfaces are obtained using Darcy’s law (4.7)
62
where K is the permeability tensor; λj and ρmj are the mobility and mass density of phase j; and g is the gravitational vector. The total number of moles of each component in a gridblock l is then updated using (4.8) where Alm denotes the area connecting gridblocks l and m; (F+H)i,lm is the total molar flux (advective+diffusive) of component i out of gridblock l at the interface lm, and qi,l represent the source term. The phase-equilibrium calculations are performed using Peng-Robinson (PR) equation of state (EOS). The advective flux is calculated using the standard single point upwind (SPU) scheme. 4.3 Molecular Diffusion Molecular diffusion may occur within a single phase (intra-phase) or between two different phases (cross-phase). For two neighboring gridblocks (l and m) that contain oil and gas phases (Fig. 4.1), the cross sectional areas that are available for gas-gas, oil-oil and gas-oil diffusion are Ag=Alm×min(Sg,l,Sg,m), Ao=Alm×min(So,l,So,m) and Ac=Alm-Ag-Ao respectively.
Figure 4.1 Schematic of two neighboring gridblocks containing oil and gas phases.
63
In the following subsections we explain how to calculate intra-phase and cross-phase molecular diffusion based on generalized Fick’s law and film theory. 4.3.1 Intra-phase Diffusion The molar diffusive flux of a component i in phase j (generalized Fick’s law) is given by (4.9) where Dik,j are the multicomponent diffusion coefficients in phase j. The sum of diffusive fluxes of all components in phase j must be zero (
); and the diffusive flux for the last
component (Hnc,j) is obtained accordingly (Taylor and Krishna, 1993). The diagonal elements in the diffusion coefficients matrix D are called the main diffusion coefficients; while the offdiagonal elements are called the cross-diffusion coefficients which are generally non-zero and non-symmetric (i.e. Dik ≠ Dki for i≠k). There are (nc-1)2 Fickian diffusion coefficients which are dependent on the numbering of components. Multicomponent diffusion coefficients can be calculated using the following equation which is obtained by comparing the generalized Fick’s law and MS model (4.10) where matrix B is a function of the inverse of the MS coefficients and γ represents the non-ideal behavior of the mixture. The elements of the matrices B and γ are given by the following equations
(4.11)
64
and (4.12) where Đik represent the MS coefficients; δik is the Kronecher delta function and φi is the fugacity coefficient of component i. The MS coefficients matrix is symmetric and its diagonal elements do not exist. Therefore there are nc(nc-1)/2 MS coefficients which can be obtained using (Wesselingh and Krishna, 1990; Kooijman and Taylor, 1991)
(4.13)
where Đoik is the diffusion coefficient of component i infinitely diluted in component k. In this work we use the correlation by Leahy-Dios and Firoozabadi (2007) to calculate the infinite dilution diffusion coefficients for both vapor and liquid phases. See Appendix D for relevant equations and also for comparisons between the calculated and experimental diffusion coefficients. 4.3.2 Cross-phase Diffusion Let us consider two different phases which have been placed in contact as shown in Fig. 4.2. According to the film theory, thermodynamic equilibrium will prevail at the interface between the two phases. In addition, the interface is assumed to offer no resistance to mass transfer. In other words, there will be no accumulation at the interface, provided there are no interface chemical reactions (continuity of molar fluxes). In Fig. 4.2, xi,I and yi,I are the interface compositions; and xi,b and yi,b denote the bulk phase compositions (Taylor and Krishna, 1993).
65
Figure 4.2 Composition profiles near the interface during inter-phase mass transfer, after Taylor and Krishna (1993).
Therefore the cross-phase mass transfer for a stationary interface is governed by two sets of equations: Continuity of molar fluxes at the interface (4.14) and thermodynamic equilibrium (4.15) where, assuming the positive direction is from left to right, component i from “x” phase into the interface; from interface into “y” phase; and
(
denotes the total molar flux of
denotes the total molar flux of component i
) is the chemical potential of component i in “x” (“y”)
phase at the interface. The molar flux consists of advective and diffusive fluxes. Since the advective flux is computed using an upwind scheme, the continuity of advective flux will always be honored. Therefore the continuity of the molar flux (Eq. 4.14) reduces to the continuity of diffusive flux which, based on the generalized Fick’s law, can be written in the following finite difference form
66
(4.16) where Lx (Ly) is the distance between the interface and the center of the gridblock containing the “x” (“y”) phase. To calculate the molar cross-phase diffusive fluxes, Equations (4.15) and (4.16) need to be solved simultaneously. The interface equilibrium calculations (Eq. (4.15)) can be done using an EOS at interpolated temperature and pressure. Multicomponent diffusion coefficients are obtained using Eq. (4.10). The solution method described in the above paragraph, is an iterative and hence timeconsuming approach. Hoteit (2013) proposed a non-iterative solution based on gradients of the chemical potential (instead of concentration gradients) which does not need explicit knowledge of interface compositions. In this approach, which is used in our work, the continuity of diffusive fluxes is given by (4.17) where
; B is the matrix given by Eq. (4.11); X is a diagonal matrix with diagonal
elements [1/xk]k=1,…,nc-1; ψk=ln(fk); and fk is the fugacity of component k. We note that because of the chemical equilibrium condition at the interface (Eq. (4.15)), we have
=
. Eq.
(4.17) can be rearranged to obtain the following expression in matrix form (4.18) where
and
. Once
is calculated, it can be substituted in either left-
hand side or right-hand side of Eq. (4.17) to calculate the molar diffusive flux of each component across the interface. The sum of diffusive fluxes of all components across the interface must be zero; and the diffusive flux of the last component is obtained accordingly. 67
4.4 Transfer Function Since the introduction of dual-porosity models in the 1960s, various transfer functions with different levels of sophistication have been suggested for matrix-fracture interactions. Warren and Root (1963) proposed a pseudo-steady state (PSS) transfer function for single-phase flow between matrix and fracture. In their model, the transfer rate per unit bulk volume is obtained by multiplying the average pressure difference between matrix and fracture by mobility and a parameter which is commonly referred to as the shape factor. The shape factor has a dimension of reciprocal area and is a characteristic of the fractured rock. The application of the shape factor in reservoir simulation was first introduced by Kazemi et al. (1976) for three-phase flow subject to a pseudo-steady state assumption. A gravity term was later included in the transfer function by Gilman and Kazemi (1983), Gilman (1986) and Sonier et al. (1988). However, it was added for flow between matrix and fracture in all three directions, which could lead to an overestimation of the transfer rate. Quandallet and Sabathier (1989) resolved this issue by treating flow in horizontal and vertical directions separately. Coats (1989) extended the dual-porosity model for compositional simulation and included molecular diffusion in the transfer function. He suggested the use of shape factors that are exactly twice those of Kazemi et al. (1976). By solving the single-phase pressure diffusion equation and approximating the exact solution with simplified but reasonably accurate functions, Zimmerman et al. (1993) and Lim and Aziz (1995) derived matrix-fracture transfer functions without assuming a pseudo-steady state condition. While pseudo-steady state transfer functions are only good for late times, the transfer functions developed by Zimmerman et al. (1993) and Lim and Aziz (1995) are able to provide reasonable accuracy for both early and late times.
68
Based on the work of Zimmerman et al. (1993), Lu et al. (2008) proposed a general transfer function that accounts for both early- and late-time behavior of multi-phase matrix-fracture interactions. Their approach, which is used in our work with some modifications, is explained in the following paragraphs. The transfer rate of a component i from matrix to fracture is given by (4.19) where Γi,e, Γi,d and Γi,gd are transfer rates due to expansion, diffusion and gravity drainage respectively. We note again that capillary effects are assumed to be negligible for gas injection processes. The expansion term is given by (4.20) where σ1 is the shape factor; Be is the boost or correction factor for the expansion term; Km is the (equivalent isotropic) matrix permeability; λjm is the mobility of phase j in the matrix; xij,m is the mole fraction of component i in phase j in the matrix; ρjm is the molar density of phase j in the matrix; and pm and pf are the pressure in matrix and fracture respectively. The shape factor proposed by Lim and Aziz (1995) for systems with anisotropic rectangular matrix blocks is used in this work (4.21) where Km,x, Km,y and Km,z are the matrix permeabilities in x-, y-, and z-direction respectively; and Lx, Ly, Lz are the dimensions of matrix block in x-, y-, and z-direction respectively. The equivalent isotropic matrix permeability is defined as (4.22)
69
The boost factor for expansion is given by the following equation (Zimmerman et al., 1993; Lu et al., 2008) (4.23) where pm,init stands for the initial pressure in the matrix. At late times, when matrix and fracture pressures are similar, the boost factor approaches unity and Eq. (4.20) reduces to Kazemi’s pseudo-steady state transfer function. However, at early times when the matrix pressure is close to its initial value, the correction factor is larger than one. To calculate the transfer rate of each component from matrix to fracture due to molecular diffusion, we sum the intra-phase and cross-phase diffusion rates of that component. For a twophase gas-oil system, the transfer rate of a component i is calculated from (4.24) where Γi,d,g, Γi,d,o and Γi,d,c are transfer rates due to gas-gas (intra-phase), oil-oil (intra-phase) and gas-oil (cross-phase) molecular diffusion respectively. The transfer rate of a component i due to gas-gas diffusion is given by (4.25) where Bd,g is the boost factor for gas-gas diffusion; ϕm is the matrix porosity and Sg=min(Sgm,Sgf) is the fraction of the total matrix-fracture surface area that is available for gas-gas diffusion. Since the molecular diffusion equation is analogous to the pressure diffusion equation, the following expression can be used to calculate the boost factor for gas-gas diffusion (4.26)
70
The transfer rate of a component i due to oil-oil diffusion is calculated similarly to gas-gas diffusion. The transfer rate of each component from matrix to fracture due to gas-oil diffusion is calculated based on the film theory as explained earlier. For the matrix-fracture system, the continuity of diffusive flux across the interface can be written as
(4.27)
where Sgo=1- min(Sgm,Sgf) - min(Som,Sof) is the fraction of the total matrix-fracture surface area that is available for gas-oil diffusion; σ1 is obtained from an isotropic version of Eq. (4.21); and σ2 is given by (4.28) where bx, by, and bz are the fracture openings in x-, y-, and z-direction respectively. The derivations of Eqs. (4.27) and (28) are given in Appendix E. By rearranging Eq. (4.27), we arrive at an equation similar to Eq. (4.18) in matrix form (4.29) where
and
. Once
is calculated, it can be substituted in
either left-hand side or right-hand side of Eq. (4.27) to calculate the cross-phase molar diffusive flux of each component from matrix to fracture. The boost factor for gas-oil diffusion can be obtained from (4.30)
71
The transfer rate of each component from matrix to fracture due to gravity drainage is calculated using the equations given in Appendix F. 4.5 Results Two numerical examples are presented in this section to demonstrate the significance of proper modeling of molecular diffusion and matrix-fracture interactions during gas injection in fractured reservoirs. In both cases, the size of reservoir in x-, y-, and z-direction is 250 m, 250 m, and 10 m respectively. 25×25×5 gridblocks are used for both flowing and stagnant domains. The flowing domain has a permeability of 500 md in all three directions; and a porosity of 1%. In the stagnant domain, the permeabilities in horizontal directions (Kx and Ky) are 1 md while the vertical permeability is 0.001 md. The stagnant domain has a porosity of 20%. The fracture spacing and opening in both x- and y-direction are 10 m and 1 mm respectively. In both examples, one injection and one production well are located at opposite corners of the reservoir. The injection well is completed at the top of formation while the production well is completed at the bottom of formation. The bottomhole pressure is kept constant for both injection and production wells, with values that will be specified for each example. A schematic of the reservoir and wellbore locations is shown in Fig. 4.3.
72
Figure 4.3 Schematic of the reservoir and wellbore locations. White lines represent the fracture system.
1
rel perm
0.8 0.6
kro_m krg_m
0.4
kro_f krg_f
0.2 0 0
0.2
0.4
0.6
0.8
1
Sg Figure 4.4 Relative permeability curves for flowing (solid line) and stagnant (dashed line) domains. Gas and oil relative permeabilities are shown with red and green colors respectively.
Linear relative permeabilities are used for the flowing domain (fractures); while Corey-type relative permeability curves are used for the stagnant domain (matrix). The saturation exponents 73
for gas and oil phases are 3.0 and 4.0 respectively; and the critical gas saturation and residual oil saturation are 0.0 and 0.25 respectively for the stagnant domain. Relative permeabilities for both flowing and stagnant domains are shown in Fig. 4.4. 4.5.1 Example 1 In this example, the initial reservoir pressure at the top of formation is 157 bar. The bottomhole pressure for the production well is kept constant at 157 bar and gas is injected at an average rate of 165 Rm3/day. The injected gas is composed of 90% CO2 and 10% CH4 by mole. The reservoir is initially saturated with oil of 30% CH4, 40% nC4H10 and 30% nC12H26 by mole. The reservoir temperature is 373.15 K at which the minimum miscibility pressure (MMP) of the oil/gas pair is 150.5 bar. The oil has a bubble point pressure of 91.6 bar at the reservoir temperature. The initial and injection compositions together with the MMP of the gas/oil pair used in example 1 are summarized in Table 4.1. Table 4.1 Initial and injection compositions and the MMP of the gas/oil pair used in example 1. Composition (mole %)
MMP (bar)
CO2
CH4
nC4H10
nC12H26
Initial
0
30
40
30
Injection
90
10
0
0
} 150.5
Fig. 4.5 compares the simulation results for example 1 for the cases when diffusion is not considered (red), when diffusion is modeled using our approach (blue), and when diffusion is modeled using the conventional approach (black). The differences between the conventional approach and our proposed approach are summarized below: In the conventional approach, 74
1- Molecular diffusion is modeled using the classical Fick’s law in which the off-diagonal diffusion coefficients are neglected. Moreover, constant diffusion coefficients are used by ignoring the dependency of diffusion coefficients on pressure, temperature and composition. The diagonal elements of the diffusion coefficient matrix obtained from Eq. (4.10) at initial and injection conditions are used for the entire simulation (constant diffusion coefficients) for oil and gas phases respectively. 2- The gas-oil diffusion is modeled by assuming an average composition at the interface. By assuming an average interface composition in Eq. (4.16), one can obtain expressions similar to Eqs. (4.27) and (4.29) for gas-oil diffusion. 3- Pseudo-steady state condition is assumed for the matrix-fracture transfer function. This is equivalent to setting the boost factor equal to one. Pseudo-steady state assumption is only valid for late times. In our proposed approach, 1- Molecular diffusion is modeled using the generalized Fick’s law. Diffusion coefficients (full matrix) are obtained based on the MS model (Eq. (4.10)) and are pressure-, temperature- and composition-dependent. 2- The gas-oil diffusion is modeled based on film theory. According to film theory, when gas and oil phases are placed in contact, a two-phase interface is formed between them in which gas and oil are at thermodynamic equilibrium. In addition, the molar flux of each component across the interface must be continuous. 3- A time-dependent transfer function is used for matrix-fracture interactions. The boost factor that is included in the transfer function differentiates between the early- and latetime behaviors of matrix-fracture interactions.
75
The left tab in Fig. 4.5 shows liquid recovery (%) while the right tab shows producing gas-oil ratio (GOR) versus time for ten years. In the absence of diffusion, the injected gas mainly sweeps the oil residing in the flowing domain (fracture) which in turn leads to very low liquid recovery and very high producing GOR. However, when molecular diffusion is present, diffusion of gas components from fracture into the matrix extracts oil components from matrix and leads to substantially higher liquid recoveries and lower producing GOR.
Figure 4.5 Liquid recovery (left) and producing GOR (right) versus time for example 1 when diffusion is not considered (red), when diffusion is modeled using our approach (blue) and when diffusion is modeled using the conventional approach (black).
Fig. 4.5 also shows that the simulation results obtained using the conventional approach are significantly different from the predictions based on our approach: Significantly higher liquid recoveries and lower gas-oil ratios are obtained using the conventional approach. We defer further analysis of the differences between predictions by these two different approaches until the discussion section.
76
The next step is to see how much the results would change if pseudo-steady state (PSS) condition was assumed for the matrix-fracture (M-F) interactions (i.e. setting the boost factor equal to one) and off-diagonal diffusion coefficients were neglected (i.e. the classical Fick’s law) in our approach. Fig. 4.6 compares the simulation results for example 1 when diffusion is not considered (solid red), when our approach is used (solid blue), when our approach is used but PSS condition is assumed for M-F transfer function (dashed blue) and when our approach is used but PSS condition is assumed for M-F transfer function and off-diagonal diffusion coefficients are neglected (dotted blue).
Figure 4.6 Liquid recovery (left) and producing GOR (right) versus time for example 1 when diffusion is not considered (solid red), when our approach is used (solid blue), when our approach is used but PSS condition is assumed for M-F transfer function (dashed blue) and when our approach is used but PSS condition is assumed for M-F transfer function and off-diagonal diffusion coefficients are neglected (dotted blue).
From Fig. 4.6 we observe that assuming PSS condition in M-F transfer function and neglecting the off-diagonal diffusion coefficients cause ~ %10 (%3 out of %28) difference in the predicted
77
liquid recoveries using different variants of our approach over the time period investigated. This is a notable difference, but much smaller than the ~ %60 difference (%17 out of %28) we observed between the cumulative liquid recoveries calculated using our approach (which uses film theory for gas-oil diffusion) and the conventional approach (which assumes an average composition at gas-oil interface) (see Fig. 4.5). Therefore the key factor in proper modeling of gas injection in fractured reservoirs when molecular diffusion is the main recovery mechanism, is using the appropriate physical model for gas-oil diffusion (i.e. film theory), not the timedependency of M-F transfer function or the off-diagonal diffusion coefficients. From Fig. 4.6 it is also observed that when off-diagonal diffusion coefficients are taken into account, higher liquid recoveries are obtained because of the larger contributions by the positive off-diagonal coefficients in this case. The effect of including boost factors in the M-F transfer function is more evident during the earlier times according to Fig. 4.6. The CO2 composition (mole %) in the stagnant domain (matrix) after ten years (example 1) is shown in Fig. 4.7 for the case with no diffusion (left) and the case with diffusion based on calculations using our approach (right). It is observed that, in the absence of molecular diffusion, very small amounts of CO2 have entered the stagnant domain due to viscous displacement and gravity drainage. However when molecular diffusion is present, considerable amounts of CO2 enter the stagnant domain and extract the oil components. It should be noted that the black lines in Fig. 4.7 represent the boundaries of simulation cells, not the actual natural fractures.
78
Injector
Injector
Producer
Producer
Figure 4.7 The CO2 composition (%) in the stagnant domain after 10 years (example 1) for the case without diffusion (left) and the case with diffusion based on calculations using our approach (right). 4.5.2 Example 2 In this example, the initial reservoir pressure is 150 bar, the bottomhole pressure of the production well is kept constant at 150 bar and gas is injected at an average rate of 180 Rm3/day. The injected gas is composed of 50% CO2 and 50% CH4 by mole. The reservoir is initially saturated with oil of the same composition as in example 1 (30% CH4, 40% nC4H10 and 30% nC12H26 by mole) at a temperature of 336.15 °K. The MMP of the oil/gas pair is 178.8 bar; and the oil has a bubble point pressure of 78.6 bar. The initial and injection compositions together with the MMP of the gas/oil pair used in example 2 are summarized in Table 4.2. Table 4.2 Initial and injection compositions and the MMP of the gas/oil pair used in example 2. Composition (mole %)
MMP (bar)
CO2
CH4
nC4H10
nC12H26
Initial
0
30
40
30
Injection
50
50
0
0
79
} 178.8
Fig. 4.8 compares the simulation results for example 2 for the cases when diffusion is not considered (red), when diffusion is modeled using our approach (blue), and when diffusion is modeled using the conventional approach (black). The left tab shows liquid recovery (%) while the right tab presents producing GOR versus time for ten years. Similar to the previous example, low liquid recovery and high producing GOR is observed in the absence of molecular diffusion. However, when molecular diffusion is present, substantially higher liquid recoveries and lower producing GOR are obtained. Significantly higher liquid recoveries and lower gas-oil ratios are calculated using the conventional approach as compared to our approach.
Figure 4.8 Liquid recovery (left) and producing GOR (right) versus time for example 2 when diffusion is not considered (red), when diffusion is modeled using our approach (blue) and when diffusion is modeled using the conventional approach (black).
The effect of PSS assumption in M-F transfer function and the off-diagonal diffusion coefficients on simulation results for example 2 is investigated in Fig. 4.9. The same color codes and line styles as in Fig. 4.6 are used to represent each physical model. 80
Similar to the previous example, assuming PSS condition in M-F transfer function and neglecting the off-diagonal diffusion coefficients cause notable differences among the results obtained using different variants of our approach, but much smaller than the difference we observed between the results of our approach and the conventional approach (see Fig. 4.8). Similar to the previous example, higher liquid recoveries are obtained when off-diagonal diffusion coefficients are taken into account, because of the larger contributions by the positive off-diagonal coefficients. According to Fig. 4.9, the effect of including boost factors in M-F transfer function is more evident during the earlier times.
Figure 4.9 Liquid recovery (left) and producing GOR (right) versus time for example 2 when diffusion is not considered (solid red), when our approach is used (solid blue), when our approach is used but PSS condition is assumed for M-F transfer function (dashed blue) and when our approach is used but PSS condition is assumed for M-F transfer function and off-diagonal diffusion coefficients are neglected (dotted blue).
Fig. 4.10 shows the CO2 composition (mole %) in the stagnant domain (matrix) after ten years (example 2) for the case without diffusion (left) and the case with diffusion as calculated using
81
our approach (right). Very small amounts of CO2 have entered the stagnant domain (due to viscous displacement and gravity drainage) when molecular diffusion is absent. However, considerable amounts of CO2 enter the stagnant domain and extract the oil components when molecular diffusion is present.
Injector
Injector
Producer
Producer
Figure 4.10 The CO2 composition (%) in the stagnant domain after 10 years (example 2) for the case without diffusion (left) and the case with diffusion based on calculations using our approach (right).
4.6 Discussion To understand how the oil is recovered by molecular diffusion during gas injection in fractured reservoirs, we analyze the saturation and composition profiles of a matrix block and its corresponding fracture. Fig. 4.11 shows the saturation profile observed during the first ten years of production in a matrix block (dotted line) near the injection well and its corresponding fracture (solid line) for example 1 as calculated using our approach. The fracture becomes fully saturated with gas in a short period of time; while the gas saturation in the matrix remains zero for almost one year, after which it gradually increases. 82
Figure 4.11 The saturation profile of a matrix block (dotted line) near the injection well and its corresponding fracture (solid line) for example 1 as calculated using our approach.
Fig. 4.12 shows the composition routes (mass fraction) of a matrix block (green) near the injection well and its corresponding fracture (red) for example 1 during the first ten years of production as calculated using our approach. The blue curves show the two-phase boundaries in each ternary plane at 157 bar and 373.15 °K; while the dashed brown line represents the dilution line. Although the pressures in example 1 are above the MMP, miscibility is not developed in the matrix block. As observed from Fig. 4.12, molecular diffusion generates a composition profile deep into the two-phase region. Initially, the CH4 composition increases with a slight increase in CO2 composition; while the nC4H10 (nC4) composition decreases and nC12H26 (C12) composition remains almost unchanged. Later, mainly CO2 composition increases and C12 composition decreases until the composition in the matrix approaches injection composition. It is also observed that the composition profile in the fracture departs from the multi-contact miscibility (MCM) condition due to exchange with the matrix (and the limited number of cells used in the representation of the domain).
83
Figure 4.12 Composition profiles (mass fraction) of a matrix block (green) near the injection well and its corresponding fracture (red) for example 1 during the first 10 years as calculated using our approach. The blue curves show the two-phase boundaries in each ternary plane at 157 bar and 373.15 °K. The dashed brown line represents the dilution line.
Fig. 4.13 shows the saturation profile observed during the first ten years of production at the same matrix block (dotted line) as in Fig. 4.11 and its corresponding fracture (solid line) as calculated using the conventional approach. As before, the fracture becomes fully saturated with gas in a short period of time; while the gas saturation in the matrix remains zero for almost seven years, after which it gradually increases. We note that a gaseous phase was formed in the matrix after ~ one year when our approach was used (see Fig. 4.11).
84
Figure 4.13 The saturation profile of a matrix block (dotted line) near the injection well and its corresponding fracture (solid line) for example 1 as calculated using the conventional approach.
Fig. 4.14 can explain why a gaseous phase does not form in the matrix until after ~ seven years as calculated using the conventional approach. During the first seven years, the CO2 composition largely increases with small changes in compositions of other components. This keeps the fluid inside the matrix in the single phase region for nearly seven years. The resulting swelling of oil cause large liquid recoveries. The composition profile in the matrix, as calculated by the conventional approach, differs significantly from what is calculated using our approach. This explains why the recovery calculations using these two different approaches differ significantly. The CO2 composition (mole %) in the stagnant domain (matrix) after ten years (example 1) as calculated by the conventional approach (left) is compared with calculations using our approach (right) in Fig. 4.15. It is observed that the two different approaches predict significantly different distributions of CO2 composition in the stagnant domain after 10 years.
85
Figure 4.14 Composition profiles (mass fraction) of a matrix block (green) near the injection well and its corresponding fracture (red) for example 1 during the first 10 years as calculated using the conventional approach. The blue curves show the two-phase boundaries in each ternary plane at 157 bar and 373.15 °K. The dashed brown line represents the dilution line.
Injector Injector
Producer Producer
Figure 4.15 The CO2 composition (%) in the stagnant domain after 10 years (example 1) as calculated using the conventional approach (left) and calculations using our approach (right).
86
Capillary forces have been neglected in our formulations by assuming that IFT effects are minimal for gas injection processes in oil reservoirs. When capillary forces are present, they act as a barrier to gas-oil gravity drainage which is driven by density difference between oil in the matrix and gas in fracture. Neglecting capillary effects will lead to overestimating oil recovery due to gravity drainage. To overcome this, we have used small matrix permeabilities in the vertical direction to minimize the role of gravity drainage in our example calculations. 4.7 Conclusions In this chapter, we presented a dual-porosity model in which the generalized Fick’s law is used to represent molecular diffusion; and the gas-oil diffusion at the fracture-matrix interface is modeled based on film theory. A novel matrix-fracture transfer function was introduced for gasoil diffusion based on film theory. Diffusion coefficients are calculated based on MS model and are pressure, temperature and composition dependent. In the matrix-fracture transfer function, a boost factor is used to adjust the shape factor in order to account for the differences between the transfer rate at early and late times. The following conclusions can be drawn based on the results presented in this chapter:
Molecular diffusion can substantially enhance the oil recovery by gas injection in fractured reservoirs. Diffusion of gas components from fracture into the matrix extracts oil components from matrix and leads to substantially higher liquid recoveries and lower producing GOR.
Our approach, which is based on a sophisticated physical model for gas-oil diffusion (film theory), gives significantly different results from the conventional approach which assumes an average composition at the gas-oil interface.
The dragging effects (off-diagonal diffusion coefficients) and time-dependency of matrixfracture transfer function can moderately impact the oil recovery during gas injection in 87
fractured reservoirs.
Miscibility is not developed in the matrix block even at pressures above MMP when molecular diffusion is the main recovery mechanism during gas injection in fractured reservoirs. In this case, molecular diffusion pushes the composition profile deep into the twophase region.
88
Chapter 5 :
Application of Potential Theory of Adsorption to Modeling of ECBM Recovery1
5.1 Introduction In coalbeds, gas is present in two phases: a bulk gaseous phase that occupies the pore space and a liquid-like adsorbed phase on the coal surfaces/pores. In primary recovery, the coalbed is dewatered to reduce the overall reservoir pressure which causes CH4 to be desorbed from coal surfaces. Since primary production typically recovers less than half of the methane in a coalbed (Stevens et al., 1998), enhanced coalbed methane (ECBM) recovery processes are needed in which CO2 and/or N2 are injected into the coalbed to recover more CH4. Injection of CO2 into coalbeds also provides additional benefit of sequestering carbon in the subsurface. Gas injection in ECBM recovery provides a method to maintain the overall coalbed pressure. In addition, injecting a second gas, or a mixture of gases, decreases the partial pressure of CH4 in the free gas. As a result, desorption of CH4 from coal surface is enhanced. The convective flow of injected gas sweeps desorbed CH4 through the coalbed. Therefore the sorption of gases onto the coal surfaces is one of the main mechanisms that govern the dynamics of ECBM recovery processes. Despite the well-documented complexity of multicomponent sorption phenomena (Stevenson et al., 1991; DeGance et al., 1993; Chaback et al., 1996), adsorption and desorption of CH4/CO2/N2 mixtures in ECBM recovery is usually modeled with the extended Langmuir model 1
Most of the results in this chapter have been presented in the SPE Western Regional Meeting: Shojaei, H. and Jessen, K.: Application of potential theory to modeling of ECBM recovery. Paper SPE 144612 presented at the SPE Western Regional Meeting, Anchorage, Alaska, 7-11 May (2011)
89
because of its simplicity and associated low computational cost (Guo, 2003; Zhu et al., 2003; Smith et al., 2005; Seto et al., 2009). The extended Langmuir has been proven unable to accurately describe the multicomponent sorption that is relevant to ECBM recovery processes (Clarkson, 2003; Wei et al., 2005). Jessen et al. (2008) demonstrated that extended Langmuir was able to model the sorption process in binary displacements; but failed to describe the behavior of ternary displacements. The ideal adsorbate solution (IAS) theory has been used to improve the agreement between calculated and measured values in ternary sorption processes. Although improvements have been attained over the extended Langmuir, still considerable mismatch between the calculated and measured values remained (Manik, 1999; Jessen et al., 2007). In addition, Manik et al. (2002) showed that at high pressures, predictions from IAS theory deviated from measured adsorption capacities due to non-ideality in the adsorbed phase. The real adsorbate solution (RAS) theory may be used to account for the non-ideality of adsorbate; but due to the complexity of required calculations and the large number of adjustable parameters involved in RAS modeling, it may not be suitable for ECBM recovery calculations. In recent years, a new model for the sorption behavior of gases and liquids in porous materials has been proposed (Shapiro and Stenby, 1998). The so called “multicomponent potential theory of adsorption” (MPTA) is based on the potential concept originally suggested by Polanyi (1932). The successful application of MPTA for sorption calculations in microporous materials (Shapiro and Stenby, 1998; Monsalvo and Shapiro, 2007; Monsalvo and Shapiro, 2009), and the lack of accuracy in previous ECBM calculations (Jessen et al., 2007; Jessen et al., 2008), motivated us to test this model in the context of ECBM recovery calculations. To the best of our knowledge, this is the first time that MPTA is used for sorption calculations in this context.
90
The remainder of this chapter is organized as follows: First, we describe the multicomponent potential theory of adsorption (MPTA) and its implementation with emphasis on ECBM recovery calculations. Subsequently, we apply the MPTA approach to multicomponent gas sorption on real coal and compare predictions to experimental observations. Several example calculations illustrate the improved predictive capability of the MPTA model. A discussion and conclusions section completes the manuscript. 5.2 Potential Theory Shapiro and Stenby (1998) developed the MPTA model based on the potential concept originally suggested by Polanyi (1932). In this theory, the mixture is considered to be a segregated phase in the potential field emitted by the adsorbent. Each component in the mixture is affected by the adsorption potential . The isothermal equilibrium state between the bulk and adsorbed phases in the external potential field is described by (5.1) where
is interpreted as porous volume,
is the pressure in the sorbed phase while
and x are
mole fractions of the bulk phase and adsorbed phase, respectively. The above equation can be written in a more convenient form using fugacities
that can be estimated by an appropriate
equation of state: (5.2) where R and T represent the gas constant and the temperature respectively. Given pressure and composition
in the bulk phase, the distribution of pressures
and mole fractions
are uniquely determined by Eq. (5.2). To solve the system of equations (5.2), we need a
91
representation of the adsorption potentials
. The generalized Dubinin dependence for
adsorption potentials in porous media has the following form: (5.3) where
is the total porous volume,
is the characteristic potential for component , and
the Dubinin exponent. A value of
is
corresponds to the standard Dubinin-Radushkevich
(DR) potential and is usually used for the activated carbon. ,
and
are commonly used as
adjustable parameters and are obtained by matching the MPTA model to relevant singlecomponent sorption isotherms. The surface excess is defined as the difference between the actual amount adsorbed on the rock surface and the amount present in the bulk phase of the pores pace if there was no adsorption potential. The value of surface excess Γ for each component is determined using: Γ where
(5.4)
is total pore volume, and
is the molar density (obtained from an equation of state).
Since the integrand in Eq. (5.4) cannot be written explicitly in terms of porous volume z, the above integral must be evaluated by numerical integration. After calculating the surface excess for each component, the total molar excess is calculated from Γ
Γ
,
(5.5)
and the average mole fraction of each component in the adsorbed phase is calculated from: Γ
Γ.
92
(5.6)
5.3 Numerical Approach Application of the MPTA model involves the solution of Eqs. (5.2) to (5.6) by a suitable numerical scheme. The evaluation of Eq. (5.6) requires numerical integration and hence repeated solution of the equilibrium problem stated by Eq. (5.2) at the nodes of any selected integration rule. The equilibrium problem stated by Eq. (5.2) is essentially similar to a dew point calculation where the solution is given by an incipient phase (here the adsorbed phase) and the associated equilibrium pressure (in the adsorbed phase). Given the similarity to dew point calculations, we adapt the approach by Michelsen (1985) for calculation of saturation points. At any given node (z) of the integration rule, we know the temperature (T), the bulk phase composition (y) and the pressure of the bulk phase (py). From these, we wish to calculate the composition (x) and pressure (p) of the adsorbate. We start by rewriting Eq. (5.2) in terms of the fugacity coefficients, φ .
(5.7)
Next, we introduce the equilibrium K-value γ
,
(5.8)
with γ
.
(5.9)
The requirement for equilibrium can now be stated as
(5.10)
93
and we can proceed by solving Eq. (5.10) for the equilibrium pressure (p) and the incipient phase composition (x) following the ideal solution based method proposed by Michelsen (1985). The iterative procedure is as follows: Given an initial estimate of p and x, we can calculate the Kvalues from Eq. (5.8) at iteration level k γ
.
(5.11)
Next, we evaluate the trial function (Eq. 5.10) and the derivative wrt p ,
(5.12)
.
(5.13)
The pressure is then updated to the next level by a Newton correction ,
(5.14)
and the mole fractions of the adsorbed phase are finally updated by direct substitution .
(5.15)
Equations (5.11) to (5.15) are then repeated until convergence. This approach for solving the vapor adsorbate equilibrium problem is simple and relatively inexpensive as the derivatives of the fugacity coefficients wrt composition are not needed. The solution strategy, does not guarantee that the equilibrium adsorbate phase is stable as pointed out by Shapiro and Stenby (1998). Stability testing (Michelsen, 1982) can be applied to the adsorbate phase but we have found this to be unnecessary for temperatures above the critical temperature of CO2 (most coalbeds). The overall algorithm for calculation of excess sorption is as follows: 94
1- Select a number of interior points, z, within the integration interval
as dictated by
the selected integration rule. In this work we use Simpson’s rule for simplicity. 2- Start at the upper integration limit and obtain initial estimates for adsorbate. For
and
in the
, py is a good initial estimate for p, and the extended Langmuir
approach provides initial estimates of x. 3- For subsequent (smaller) values of z, the converged values of p and x from previous integration node provide good initial estimates. The initial estimate of p can be refined by using the derivative of Eq. (5.10) wrt z from the previously converged level .
(5.16)
4- Evaluate the surface excess for each component using Eq. (5.4) and integration rule. 5- Evaluate the mole fraction of each component in the adsorbate phase using Eq. (5.6). Initial tests of our implementation were performed for experimental observations of pure component (CH4, CO2 and N2) and mixture sorption on well characterized activated carbon. We used the experimental data of Dreisbach et al. (1999) to compare our calculations with the work of Monsalvo and Shapiro (2007) and found a good agreement. In the next section, we apply the MPTA model to simulate the sorption behavior of CH4, CO2, N2 and their mixtures on real coal samples. 5.4 Application of MPTA to Coal Ottiger et al. (2008a, 2008b) presented a comprehensive study of the sorption characteristics of CH4, CO2, N2 and their binary and ternary mixtures on dried coal samples from the Sulcis Coal Province in Italy. They applied lattice density functional theory (DFT) and arrived at a good agreement between modeling and experimental observations. 95
Some of the key challenges in modeling sorption on real coal samples are related to the multiporosity and highly heterogeneous nature of coals. From careful characterization of the coal samples including low pressure CO2 sorption and Hg porosimetry, Ottiger et al. (2008b), report significant contributions from both micro- and meso- to macro-pores to the overall porosity of the coal with the micropores making up approximately 0.07 cc/gram of the 0.202 cc/gram of pore space. In their lattice DFT modeling efforts, they select three discrete pore sizes: Two in the microporous region (1.2 and 1.6 nm) and one to represent the entire meso- to macro-porous region (20nm). In addition, they include the assumption that CH4, CO2 and N2 have access to different portions of the pore volume in their modeling efforts. Both the multi-modal pore size distribution and the need for restricting species for some ranges of porosity suggests that MPTA may not be able to fully capture the complex sorption characteristics of coal given the inherent continuum scale equilibrium assumption. In addition, a simple potential function (e.g. Eq. (5.3)) may not be sufficient to represent the interactions between coal surfaces and gases over the entire range of the pore size distribution. With this in mind, we apply the MPTA model in its original form (Shapiro and Stenby, 1998) by fitting the model parameters (characteristic energies and pore volume) to the pure component isotherms. We use the Peng-Robinson equation of state with volume shift throughout this work. All binary interaction coefficients are set to zero to avoid additional parameter estimation. Figure 5.1 shows the pure component isotherms for the three gases at 45 °C in terms of the excess sorption at pressures of 5-188 bars. The associated model parameters are reported in Table 5.1.
96
Excess sorption (mmol/gram)
2.5 CH4 2
CO2 N2
1.5
1
0.5
0 0
50
100
150
200
Pressure (bar) Figure 5.1 Pure component isotherms at 45 °C. Data from Ottiger et al. (2008b). Solid lines: MPTA after regression.
Table 5.1 MPTA parameters from pure component isotherms.
ε0i/R (K)
CH4
CO2
N2
425.1
602.8
269.3
beta (-)
0.59
z0 (cc/gram)
0.122
We note that the pore volume calculated from simultaneous regression of the model parameters to the pure component isotherms is less than what is reported by Ottiger et al. (2008b). This can be interpreted in a multitude of ways including: a) the entire pore volume does not contribute to the excess sorption for this sample or b) the potential function that we use in this work does not adequately represent the interactions between solid and the gases and c) the equation of state does not capture the non-ideal behavior of the sorbed phase density accurately. While b) and c) are most likely both true, the fact that the pore volume needed in the potential function to match the experimental observations differs significantly from the measured pore 97
volume suggests that the MPTA model, in the original form, may not be suitable for modeling sorption on complex coal materials. In addition, we observe that the value of beta is somewhat low relative to what has been used for more homogeneous microporous materials such as activated carbon. We return to this after looking at the performance of the MPTA model for prediction of sorption for ternary mixtures of CH4, CO2 and N2. Based on the parameters listed in Table 5.1, we have applied the MPTA model, in predictive mode, to represent the sorption behavior of binary and ternary mixtures of CH4, CO2 and N2 at 45 °C. Figure 5.2 compares the experimental observations with the model predictions for a ternary mixture of equimolar bulk phase composition.
Calculated excess sorption (mmol/gram)
Excess Sorption (mmol/gram)
1.8 1.6 1.4 1.2 Total
1 0.8
CO2
0.6 CH4
0.4 0.2 0
1.6 1.4 1.2 1 0.8 Total
0.6
CH4
0.4
N2
0.2
CO2
0 0
50
100
150
200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Experimental excess sorption (mmol/gram)
Pressure (bar)
Figure 5.2 Experimental observations and MPTA model predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b).
We observe that the overall excess sorption is predicted accurately by the MPTA model but the excess sorption of CO2 is overestimated at the cost of both CH4 and N2. In this calculation
98
example, the excess sorption of N2 is predicted to be negative at moderate to high pressures causing a relatively large underestimation of the N2 concentration in the sorbed phase. Similar behavior is observed for binary mixtures (CO2-CH4 and CO2-N2) where the total excess sorption is reasonably accurate while the excess sorption of CO2 is overestimated. To improve the predictive performance of the MPTA model and to address the above concerns regarding the pore volume used in the potential function, we attempted to re-estimate the pure component parameters by fixing z0 to the reported value of 0.202cc/gram. However, it was not possible to arrive at a reasonably accurate description of the pure component isotherms by this approach. In the interest of integrating some of the pore volume information from the coal characterization into the MPTA model via the potential function, we propose to divide the potential function into a microporous section covering the pore volume from 0 to 0.07 cc/gram and a meso- to macro-porous section covering the pore volume from 0.07 to 0.202 cc/gram. In this initial attempt, we keep the functional form of the potential in both sections but allow the use of different characteristic parameters in each section. This can be done by introducing (5.17)
,
(5.18)
where b0 represents the pore volume of the microporous region. Eq. (5.17) ensures continuity in the chemical potential in the transition from the microporous to the mesoporous region. The new composite potential function is used with fixed values of b0 and z0 set to 0.07 and 0.202 cc/gram, respectively. In this form, a total of 8 parameters must be estimated simultaneously from the 3 pure-component isotherms relative to 5 parameters in the original version of the MPTA model. 99
Table 5.2 reports the estimated parameters for the composite potential function and Figure 5.3 compares model predictions with experimental observations for the ternary mixtures. The composite potential function improves the overall prediction for the ternary system slightly. Primarily, we see a better agreement between model and experiments for CO2 at lower pressures and for CH4 over the entire pressure range. However, the modified potential function does not improve the prediction of the excess sorption for N2 to any significant extent.
Table 5.2 MPTA parameters from pure component isotherms using composite potential function. CH4
CO2
N2
(ε0i I)/R (K)
600.3
722.4
485.6
(ε0i II)/R (K)
162.4
247.2
52.5
beta I (-)
0.688
beta II (-)
0.122
z0 (cc/gram)
0.202
b0 (cc/gram)
0.070
The modest improvement in the accuracy of the model predictions between the original MPTA model and composite version, suggests that additional segments may be needed. This would require additional information regarding the distribution of the pore volume in the meso- to macro-porous region. Such information is, however, not available and we choose instead to compare the MPTA predictions to those of the extended Langmuir model (ELM).
100
Calculated excess sorption (mmol/gram)
Excess sorption (mmol/gram)
1.8 1.6 1.4 1.2 1 Total
0.8 0.6
CO2
0.4
CH4
0.2
1.6 1.4 1.2 1 0.8 0.6
Total CH4 CO2 N2
0.4 0.2 0
0 0
50
100
150
200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Experimental excess sorption (mmol/gram)
Pressure (bar)
Figure 5.3 Experimental observations and MPTA model predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b).
An initial attempt to apply the ELM to predict the pure component isotherms suffered the same problem as our initial MPTA attempt: a significant adjustment of the pore volume was required to match the excess sorption of the individual gases. To overcome this problem, we apply the following composite version of ELM and force the pore volumes of the micro- and meso- to macro-porous regions to be honored. .
(5.19)
The parameters in Eq. (5.19) were obtained by regression of the pure component isotherms and are reported in Table 5.3.
101
Table 5.3 Model parameters for Extended Langmuir Model (ELM) CH4
CO2
N2
Vi,I (mmol/cc)
1.21E-02
1.89E-02
5.34E-03
bi,I (1/atm)
0.246389
0.443316
0.600095
Vi,II (mmol/cc)
4.57E-02
6.00E-02
3.88E-02
bi,II (1/atm)
2.69E-03
4.19E-03
2.94E-03
b0 (cc/gram)
0.070
z0 (cc/gram)
0.202
Figure 5.4 compares the experimental observations with ELM predictions for the ternary mixture
1.8
Calculated excess sorption (mmol/gram)
Excess sorption (mmol/gram)
of CH4/ CO2/ N2.
1.6 1.4 1.2 1 0.8
Total
0.6
CO2 CH4
0.4 0.2
1.6 1.4
Total
1.2
CH4
1
CO2
0.8
N2
0.6 0.4 0.2 0
0 0
50
100
150
200
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6
Experimental excess sorption (mmol/gram)
Pressure (bar)
Figure 5.4 Experimental observations and ELM predictions for a ternary mixture of CH4, CO2 and N2 at 45 °C and pressures from 5-188 bars. Excess sorption data from Ottiger et al. (2008b).
From Figure 5.4, we observe a significant difference in the predictive capabilities of ELM relative to MPTA. The prediction of the excess sorption of CH4 is improved at the cost of
102
underestimation of the total excess and the CO2 excess while the N2 excess is now slightly overestimated. The applied version of the ELM requires 12 parameters to be estimated from pure component data as compared to 5 parameters for the original MPTA and 8 parameters for the composite; suggesting that a 3 stage MPTA model (with 12 adjustable parameters) might be more valuable in terms of gains in accuracy. 5.5 Discussion and Conclusions In the previous sections, we have discussed the numerical and application aspects of the MPTA model and presented a comparison between calculations and experimental adsorption data from a real coal sample. In this work, we have used the MPTA in a purely predictive mode based on pure component isotherms alone. The model is able to accurately predict the total excess sorption over a range of binary and ternary mixtures but is found to overestimate the CO2 excess sorption and underestimate the CH4 and N2 excess sorptions. Table 5.4 summarizes the accuracy of the presented models for all experimental observations presented by Ottiger et al. (2008a, 2008b). The summary reports the errors in prediction of the component excess sorption as average absolute relative errors. The number of data points for each system and the number of adjustable parameters in the model are listed in parenthesis. We believe that the predictive capability of the MPTA model can be improved by improving the interactions between species in the adsorbed phase. Using more sophisticated model (relative to a standard cubic EOS) for density calculations in the high-pressure sorbed phase can also enhance the agreement between the model and experimental observations. The potential function(s) used in the MPTA model also deserve(s) additional attention as and more flexible functional form may need to be used. In addition, a careful consideration of the relevant pore size distribution seems essential in this context. From a comparison of the results 103
from the MPTA and the ELM models, we observe that the use of the MPTA model improves the agreement with experimental data and trends significantly. At the same time, however, the computational cost is substantially increased. Therefore the use of MPTA or ELM for sorption calculations is a matter of balancing the accuracy versus CPU time. We believe that at the current state, the MPTA is not suitable for large scale general purpose simulation of ECBM recovery processes; while it’s suggested for standalone sorption calculations and/or simulation of ECBM experiments at the laboratory scale.
Table 5.4 Average absolute relative error (%) in prediction of binary and ternary excess sorption Mixture CH4/N2 (30)
CH4/CO2 (41)
CO2/N2 (40)
CH4/CO2/N2 (10)
MPTA (5 par)
%
%
%
%
Ntot
4.1
3.9
4.3
2.4
nCH4
54.3
70.8
0.0
55.0
nCO2
0.0
20.9
28.6
23.1
nN2
70.4
0.0
94.4
97.5
Ntot
2.6
3.8
6.5
2.9
nCH4
42.9
58.6
0.0
34.4
nCO2
0.0
15.2
24.0
20.4
nN2
62.0
0.0
91.7
96.6
Ntot
8.4
9.0
17.8
12.7
nCH4
31.3
50.9
0.0
15.6
nCO2
0.0
15.8
38.9
30.5
nN2
31.1
0.0
81.2
78.2
MPTA (8 par)
ELM (12 par)
104
Chapter 6 :
Summary and Future Research Directions
This research project focuses on the role of mass transfer mechanisms and their impact on the displacement dynamics in enhanced oil/gas operations. In Chapter 3, we investigated the accuracy of some of the physical models that are frequently used to represent dispersive mixing and mass transfer in a 1D dual-porosity system by comparing the results from compositional simulations with experimental observations. We showed that the accuracy of our displacement calculations relative to the experimental observations was sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. The two representations of a dual-porosity model used in our calculations lead to almost identical results. Therefore, we were not able to conclude which model (DP I or DP II) provides for a better representation of the unresolved heterogeneity inside the column using the CoatsSmith model. Based on this observation, we recommend performing 2D/3D displacement experiments by carefully defining the heterogeneity in the porous medium. The porous medium should consist of two layers with high permeability contrast to represent a dual-porosity system. Such experiments will allow for further testing/validating the physical models that are used in compositional simulation of miscible displacements in dual-porosity systems, and coming up with the simplest accurate physical models to be used in this context. For the new set of experiments, it is suggested that steady-state relative permeability measurements with pre-equilibrated phase compositions at two different values of IFT be performed. This will allow one to determine the scaling exponent (β) and hence test/validate the
105
IFT scaling method suggested by Coats (1980) by comparing the experimental data and displacement calculations in a predictive mode (i.e. without adjusting β). As was discussed in section 2.4, several attempts by different investigators have failed in accurately reproducing the experimental observations for miscible gas injection into dualporosity systems using commercial compositional simulators. Therefore combined experimental and modeling study of multicontact miscible displacements in dual-porosity systems is needed for investigating the accuracy of the physical models that are commonly used in compositional simulations for dual-porosity systems. In Chapter 4, we presented a dual-porosity model in which the generalized Fick’s law is used for molecular diffusion; and gas-oil diffusion at the fracture-matrix interface is modeled based on film theory. A novel matrix-fracture transfer function was introduced for gas-oil diffusion based on film theory. A time-dependent transfer function is used for matrix-fracture exchange in which the shape factor is adjusted using a boost factor to differentiate between the transfer rate at early and late times. We used field-scale examples to show that our approach, which is based on a sophisticated physical model for gas-oil diffusion (film theory), gives significantly different results from the conventional approach in which gas-oil diffusion is modeled by simply assuming an average interface composition. It was also demonstrated that the off-diagonal diffusion coefficients and time-dependency of matrix-fracture transfer function can moderately impact the oil recovery during gas injection in fractured reservoirs. Capillary forces were neglected in our formulations by assuming that IFT effects are minimal for gas injection processes in oil reservoirs. When capillary forces are present, they act as a barrier to gas-oil gravity drainage which is driven by density difference between oil in the matrix
106
and gas in fracture. Neglecting capillary effects will lead to overestimating oil recovery due to gravity drainage. To overcome this, we have used small matrix permeabilities in the vertical direction to minimize the role of gravity drainage in our example calculations. Capillary effects can be included in our dual-porosity formulation in future research activities, where matrix-fracture transfer function needs to be re-defined carefully. The competition among molecular diffusion, capillary forces and gravity drainage in terms of oil recovery from matrix during gas injection in fractured reservoirs can be studied using such a dual-porosity model. In Chapter 5, we discussed the numerical and application aspects of the MPTA model and presented a comparison between calculations and experimental adsorption data from a real coal sample. We used the MPTA in a purely predictive mode based on pure component isotherms alone. The model was able to accurately predict the total excess sorption over a range of binary and ternary mixtures but was found to overestimate the CO2 excess sorption and underestimate the CH4 and N2 excess sorptions. The predictive capability of the MPTA model can be improved by improving the interactions between species in the adsorbed phase in future research activities. Using more sophisticated model (relative to a standard cubic EOS) for density calculations in the high-pressure sorbed phase can also enhance the agreement between the model and experimental observations. The potential function(s) used in the MPTA model also deserve(s) additional attention as and more flexible functional form may need to be used. In addition, a careful consideration of the relevant pore size distribution seems essential in this context.
107
Nomenclature A
Cross sectional area
b
Langmuir parameter
Bd
Boost factor for diffusion term
Be
Boost factor for expansion term
Bgd
Boost factor for gravity drainage term
ct
Total fluid compressibility
C
Overall molar density
dp
Particle diameter
D
Diffusion coefficient
Đ
MS coefficient
Đo
Infinite dilution diffusion coefficient
f
Flowing fraction, fugacity
F
IFT scaling factor, molar advective flux
FF
Formation factor
g
Gravitational vector
H
Molar diffusive flux
kr
Relative permeability
krg,e
End-point relative permeability of gas
kro,e
End-point relative permeability of oil
Ki
K-value of component i Effective dispersion coefficient of component i in phase j
Kl
Longitudinal dispersion coefficient 108
n
Number of moles
ng
Corey-exponent for gas
no
Corey-exponent for oil
N
Total molar flux
p
Pressure
q
Source term in the mass conservation equation
R
Universal gas constant
S
Saturation
Sgc
Critical gas saturation
Sor
Residual oil saturation
t
Time
T
Temperature
V
Langmuir parameter
Vt
Total fluid volume
x
Length of tie line (mass fraction)
xij
Mole fraction of component i in phase j
z
Direction along packed column, porous volume
α
Longitudinal dispersivity
β
IFT scaling exponent, Dubinin exponent
Γ
Matrix-fracture transfer rate, surface excess
Γd
Matrix-fracture transfer rate due to diffusion
Γe
Matrix-fracture transfer rate due to expansion
Γgd
Matrix-fracture transfer rate due to gravity drainage
109
δ
Kronecher delta function
ε
Sorption potential
θ
Mass transfer coefficient
λ
Mobility
μ
Viscosity, chemical potential
v
Velocity
ρ
Density
σ
Interfacial tension, shape factor
σ0
Reference interfacial tension
φ
Fugacity coefficient
ϕ
Porosity
ψ
Natural log of fugacity
110
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Appendix A:
UNIQUAC Model for Phase Equilibria
The Universal Quasichemical Activity Coefficient (UNIQUAC) model is an activity coefficient model used in description of phase equilibria. In this model the activity coefficient of component i in a multicomponent mixture is described by a combinatorial and a residual contribution (Abrams and Prausnitz, 1975; Poling et al., 2001) (A.1) where
is an entropic term which quantifies the deviation from ideal solubility due to
differences in the shape of molecule and
is an enthalpic correction resulting from the change
in interacting forces between different molecules upon mixing. The combinatorial term is calculated using the relative Van der Waals volumes ri and surface areas qi of the pure components
(A.2)
where
is the mole fraction, nc is the number of components, z is the coordination number,
the area fraction,
is the segment fraction, and
is
is defined as
(A.3)
A value of 10 is frequently used for the coordination number (number of close interacting molecules around a central molecule). The values of equations
124
and
are obtained from the following
(A.4)
(A.5)
The residual term is calculated using Eq. (A.6)
(A.6)
Where
is defined as (A.7)
The binary interaction parameters,
, are obtained using the experimental phase behavior data.
The binary interaction parameters together with ri and qi for the analog fluid system used in this work are given in Table A.1 (Rastegar and Jessen 2011). Table A.1 Parameters for UNIQUAC model (interactions and structural parameters) Comp
Water
MeOH
IPA
iC8
Water
0
-760.7
301.2
429.0
1.4000
0.9200
MeOH
2175.8
0.0
146.3
11.5
1.4320
1.4311
IPA
-241.1
-143.2
0.0
-83.2
3.1240
3.2491
iC8
2512.6
598.5
169.0
0.0
5.0080
5.8463
125
Appendix B:
UNIQUAC Viscosity Model
The UNIQUAC viscosity model has been developed based on the Eyring’s theory of viscous flow and the UNIQUAC phase behavior model (Martins et al., 2001). In this model, the mixture viscosity is calculated from
(B.1)
where
is the mixture viscosity,
component i,
is the viscosity of component i,
is the molar volume of component i,
the coordination number, qi is the surface area, (A.4)),
is the mole fraction of
is the molar volume of the mixture, z is
is the segment fraction (obtained from Eq.
is the area fraction (obtained from Eq. (A.5)), and
is the binary interaction
parameter between component k and i. The model parameters for the analog fluid system used in this work are given in Table 3.2.
126
Appendix C:
Numerical Dispersion
The truncation errors in finite-difference numerical calculations produce numerical diffusion. Lantz (1971) showed that the numerical Peclet number for single-phase displacements can be estimated from (C.1)
where
and
represent dimensionless distance and time, respectively. For simulations with
1000 grid locks, numerical Peclet numbers of 2636 and 2196 are estimated for experiments I and II respectively. The numerical Peclet number for two-phase displacements can be estimated from (Walsh and Orr, 1990; Orr, 2007) (C.2)
where df1/dS1 represents the slope of fractional flow curve. We use 3.33 and 3.13 for df1/dS1 (the maximum slope estimated from immiscible fractional flow curve) for experiments III and IV, respectively. For simulations with 1000 grid blocks, numerical Peclet numbers of 3057 and 2611 are estimated for experiments III and IV, respectively. The estimated values of numerical Peclet number for simulation of all experiments are one order of magnitude larger that the physical Peclet number of approximately 200 observed in our experiments. Hence, numerical artifacts should be marginal.
127
Appendix D:
Diffusion Coefficients
Fickian diffusion coefficients can be obtained from MS coefficients which are less sensitive to composition changes and hence easier to measure in the lab (Taylor and Krishna 1993). The relationship between Fickian diffusion coefficients and MS coefficients are given in section 4.3.1 of the manuscript (Eqs. 4.10 through 4.12). Multicomponent MS coefficients are commonly calculated using binary infinite dilution diffusion coefficients (see Eq. 3.13). In this work we use the correlation by Leahy-Dios and Firoozabadi (2007) for infinite dilution diffusion coefficients for both vapor and liquid phases as explained in the following paragraphs. By performing non-linear least-squares minimization on various relationships, Leahy-Dios and Firoozabadi (2007) found the following expression to best describe the experimental data for infinite dilution diffusion coefficients: (D.1)
where Đo21 is the diffusion coefficient of component 2 infinitely diluted in component 1; ρ is the molar density (mol/m3) of component 1; μ is the viscosity (Pa.s) of component 1; (ρĐ)0and μ0 are the dilute gas density-diffusion coefficient product (mol/m.s) and viscosity (Pa.s) respectively; Tr and pr are the reduced temperature and pressure respectively; and ω is the acentric factor. The parameters A0 to A1 are given by
(D.2)
128
where a1 = ─ 0.0472; a2 = 0.0103; a3 = ─ 0.0147; a4 = ─ 0.0053; a5 = ─ 0.337; a6 = ─ 0.1852; a7 = ─ 0.1914. The approach of Fuller et al. (1966, 1969) is used to calculate the dilute gas density-diffusion coefficient product (D.3)
where M is the molecular weight (g/mol); T is the absolute temperature (K); and Συ is the socalled “diffusion volume increments” which is calculated by summing the atomic diffusion volumes (see Poling et al., 2001). The low-pressure viscosity for each component is calculated using the correlation by Stiel and Thodos (1961) (D.4) where (D.5) The dilute gas viscosity of the mixture is obtained using a weighted average of dilute gas viscosity of the components (D.6) To validate our diffusion coefficient calculations (Eqs. 4.10 through 4.13, and D.3 through D.6), we compare our calculations with experimental data for binary and ternary mixtures. We note that SRK equation of state (Soave, 1972) with volume translation is used for density
129
calculations; and viscosities are obtained using LBC correlation (Lohrenz et al., 1964). We also note that similar comparisons can be found in Leahy-Dios and Firoozabadi (2007). D.1 Example (Binary Mixtures) Figure D.1 compares the calculated Fickian diffusion coefficients with experimental data from Sigmund (1976) for methane-propane binary gas mixtures at different temperatures and pressures. Very good agreements between the calculated and experimental values are observed.
Figure D.1 Calculated diffusion coefficient (solid lines) compared with experimental data (circles) from Sigmund (1976) for methane-propane mixtures at different values of pressure, temperature and composition.
130
Calculated Fickian diffusion coefficients are compared with experimental data from Lo (1974) for binary liquid mixtures of alkanes (n-C7, n-C10, n-C12, nC8 and nC14) at 298.15 K and 0.1 MPa in Fig. D.2. The agreements between calculated and experimental values are substantial.
Figure D.2 Calculated diffusion coefficient (solid lines) compared with experimental data (circles) from Lo (1974) for binary liquid mixtures of alkanes at 298.15 K and 0.1 MPa with varying compositions.
D.2 Example (Ternary Mixture) Table D.1 compares the calculated Fickian diffusion coefficients with experimental data from Kett and Anderson (1969) for a ternary liquid mixture of nC6 (33.3 mole %), nC12 (35 mole %)
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and nC16 (31.7 mole %) at 298.15 K and 0.1 MPa. Reasonable agreements between the calculated and experimental values are observed.
Table D.1 Calculated diffusion coefficient compared with experimental data from Kett and Anderson (1969) for a ternary liquid mixture at 298.15 K and 0.1 MPa. Dij (10-9 m2/sec) nC12(1), nC16(2), nC6 (3) Subscripts i j
Exp.
Model
∆ (%)
11
0.968
1.05
8
12
0.266
0.143
46
21
0.225
0.108
52
22
1.031
0.988
4
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Appendix E:
Shape Factors for Gas-Oil Diffusion
Consider a matrix block that is surrounded by fractures (Fig. E.1). Assume matrix and fractures are filled with different phases (e.g. oil in the matrix and gas in the fractures); and there is gas-oil equilibrium at the fracture-matrix interface at all six faces of the matrix block. Also assume the fluids in all fractures surrounding the matrix block have the same pressure and composition. The transfer rate of each component due to molecular diffusion from the center of matrix block to the matrix-fracture interface at each face is given by
Figure E.1 Schematic of a matrix block surrounded by fractures
(E.1)
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The total transfer rate of each component from the matrix block to the matrix-fracture interface is obtained by summing the transfer rates in Eq. (E.1)
(E.2)
Dividing both sides of Eq. (E.2) by
will give us the transfer rate per unit volume of
matrix block (E.3) where σ1 is given by (E.4) The transfer rate of each component due to molecular diffusion from the center of each fracture to the matrix-fracture interface is given by
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(E.5)
The total transfer rate of each component from the fracture to the matrix-fracture interface is obtained by summing the transfer rates in Eq. (E.5) (E.6) Dividing both sides of Eq. (E.6) by
will give us the transfer rate per unit volume of matrix
block (E.7) where σ2 is given by (E.8)
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Based on the film theory, the transfer rates given by Eqs. (E.3) and (E.7) must be equal. This leads to Eq. (27). Since the molecular diffusion equation is analogous to the pressure diffusion equation, one can replace 4 with π2 in Eqs. (E.4) and (E.8) and multiply the transfer function by a boost factor based on the derivations of Zimmerman et al. (1993) and Lim and Aziz (1995).
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Appendix F:
Transfer Rate due to Gravity Drainage
The transfer rate of a component i from matrix to fracture due to gravity drainage is given by
(F.1)
where Tjk represents the displacement of phase j by phase k due to gravity drainage which acts only in the vertical direction. Since we are looking at two-phase gas-oil systems, we need to calculate Tgo and Tog. Lu et al. (2008) proposed the following equations based on the observations of Di Donato et al. (2006) (F.2) where Bgd is the correction factor for the gravity drainage term; Sgm,init is the initial gas saturation in the matrix block; S*gm is the final gas saturation in the matrix at the end of displacement; b is the exponent of the oil relative permeability function; and βgo and F(Sgf) are parameters that are defined as follows (F.3) where λgm, λom and λtm are the gas, oil and total mobilities in the matrix respectively; ∆ρmgo is the difference between mass densities of gas and oil; g is the acceleration of gravity; and h denotes the hight of the matrix block. Since the gas is much more mobile than the oil, λgmλom/λtm can be approximated by
where
is the maximum oil relative permeability.
The parameter F(Sgf) is given by (F.4)
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where Sgf is the gas saturation in the fracture; and Kf is the fracture permeability. Since the gravity drainage acts in the vertical direction, vertical permeability is used in Eqs. (F.3) and (F.4). The correction factor for the gravity drainage term is given by (F.5) We note that the correction factor for gravity drainage has a value less than unity and will approach unity at late times. We also note that Tog = ─Tgo for incompressible flow settings (Lu et al., 2008).
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