CO2 SEQUESTRATION IN SALINE AQUIFER: GEOCHEMICAL MODELING,
REACTIVE TRANSPORT SIMULATION AND SINGLE-PHASE
FLOW EXPERIMENT
by
BINIAM ZERAI
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Dissertation Advisors: Dr. B. Saylor and Dr. J. Kadambi
Department of Geological Sciences
CASE WESTERN RESERVE UNIVERSITY
January, 2006
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______
candidate for the Ph.D. degree *.
(signed)______(chair of the committee)
______
______
______
______
______
(date) ______
*We also certify that written approval has been obtained for any proprietary material contained therein.
Dedicated to
My family
TABLE OF CONTENTS
TABLE OF CONTENTS………………………………………………...... i
LIST OF TABLES………………………………………………………...... v
LIST OF FIGURES...……………………………………………………………………vii
ACKNOWLEDGEMENTS……………………………………………...... xii
NOMENCLATURE……………………………………………………...... xiii
ABSTRACT……………………………………………………………...... xviii
INTRODUCTION……………………………………………………… ...... 1
PART I
GEOCHEMICAL MODELING AND REACTIVE TRANSPORT SIMULATION
CHAPTER 1: INTRODUCTION AND BACKGROUND
1.1. Introduction...... 6 1.2. Disposal in Geological Formation...... 10 1.3. Why Saline Aquifers...... 12 1.4. Purpose and Objective of Study ...... 13
CHAPTER 2: MECHANISM FOR AQUIFER STORAGE OF CO2 AND THE GEOLOGY OF ROSE RUN SANDSTONE
2.1. Mechanism for Aquifer Storage of CO2 ...... 16
2.1.1. Hydrodynamic Trapping...... 17 2.1.2. Solubility Trapping ...... 21 2.1.3. Mineral Trapping ...... 23
2.2. Geologic Reservoir - Rose Run Sandstone...... 27 2.3. Structure and Hydrologic Parameters...... 30 2.4. Rose Run Brine Chemistry ...... 31
i CHAPTER 3: CO2 SOLUBILITY MODELS
3.1. Introduction...... 33 3.2. CO2 Solubility Modeling ...... 34
3.2.1. PG-CSM...... 36 3.2.2. DS-CSM...... 37 3.2.3. SP-CSM ...... 41 3.2.4. Xu-CSM...... 42 3.2.5. MRK-CSM...... 44
3.3. Modeling of CO2 Solubility...... 46 3.4. Comparison with Experimental Data...... 48 3.5. Intermodel Comparison of CO2 Solubility Model...... 53 3.6. Discussion...... 57 3.7. Concluding Remark...... 61
CHAPTER 4: BATCH GEOCHEMICAL MODELING
4.1. Introduction...... 63 4.2. Geochemists’ Workbench...... 64 4.3. Rose Run Rock Assemblage...... 67 4.4. Brine Chemistry...... 68 4.5. Rate Constants ...... 69 4.6. Fugacity of CO2 ...... 70 4.7. Reactive Surface Area ...... 71 4.8. Activity Coefficients...... 73 4.9. Thermodynamic Database ...... 73
CHAPTER 5: RESULTS AND DISCUSSION ON GEOCHEMICAL MODELING
5.1. Computer Simulation – GWB ...... 74
5.1.1. Equilibrium Modeling...... 74 5.1.2. Path of Reaction...... 80 5.1.3. Kinetic Modeling ...... 83
5.2. Discussion...... 90 5.3. Recommendations and Concluding Remarks...... 96
CHAPTER 6: REACTIVE TRANSPORT CODE DEVELOPMENT
6.1. Introduction...... 99 6.2. Brief Assessment of Existing Codes...... 100 6.3. Reactive Transport Modeling ...... 103 6.4. Conceptual Model...... 104 6.5. Theoretical Background...... 105 6.6. Chemical, Hydrological and Physical Parameters...... 106
ii 6.6.1. Chemical Parameters and Thermodynamic Database...... 107 6.6.2. Hydrological and Physical Parameters ...... 120
6.7. CO2 Solubility Models...... 121 6.8. Mathematical Equations ...... 122
6.8.1. Flow and Transport...... 122 6.8.2. Chemical Reaction...... 127 6.8.3. Integral Finite Difference Discretization ...... 130
6.9. Solution Method ...... 135
CHAPTER 7: REACTIVE TRANSPORT SIMULATION RESULTS
7.1. Introduction...... 138 7.2. Rose Run Sandstone and CO2 Solubility Model ...... 139
7.2.1. CO2 Saturation, Pressure and pH...... 140 7.2.2. Fugacity, CO2 Solubility and Free CO2 ...... 144 7.2.3. Mineral Precipitation and Dissolution ...... 147 7.2.4. Porosity Change and CO2 uptake...... 157
7.3. Carbonate Aquifer ...... 160 7.4. Silicate Aquifer...... 161 7.5. Sensitivity Analysis ...... 166
7.5.1. Temperature and Pressure...... 168 7.5.2. Salinity and Reaction Rate...... 171
7.6. Concluding Remark...... 175
CHAPTER 8: DISCUSSIONS AND CONCLUSIONS
8.1. Discusion ...... 176 8.2. Conclusion ...... 185 8.3. Future Work...... 188
PART II
FLOW CHARACTERIZATION THROUGH A NETWORK CELL USING PARTICLE IMAGE VELOCIMETRY
CHAPTER 9: INTRODUCTION AND BACKGROUND
9.1. Introduction...... 191 9.2. Background...... 194
iii
CHAPTER 10: EXPERIMENTAL DESIGN AND SETUP
10.1. Experimental Flow Cell, Test Fluid and Seed Particles ...... 196 10.2. Image Acquisition and Processing...... 201
CHAPTER 11: EXPERIMENTAL RESULTS
11.1 Velocity Measurements ...... 207 12.1 Darcy’s Velocity ...... 215 13.1 Data Validation and Error Analysis...... 216 14.1 Numerical Simulation ...... 218
CHAPTER 12: DISCUSSIONS AND CONCLUSIONS
12.1 Discussion...... 222 13.1 Conclusions...... 226
APPENDIX A: UNCERTAINTY ANALYSIS…………………………...... 228
BIBLIOGRAPHY………………………………………………………...... 232
iv List of Tables
Table 1.1: World carbon emissions designated by region including both historical data and projections……………………………………..………...... 10
Table 3.1: CO2 solubility models and their range of temperature, pressure and salinity . 36
Table 3.2: Polynomial fitting coefficients……………………………… ...... 36
Table 3.3: Parameters for Eq. 3.4…………………………………………...... 39
Table 3.4: Interaction parameters for Eq. 3.7-3.9………………………...... 40
Table 3.5: Dimensionless coefficient parameters……………………… ...... 40
Table 3.6: The values of the coefficients b1, b2, b3, b4 and b5 are obtained from the logK values at 0, 25, 60, 100, 150, 200, 250, and 300 °C. The fitted constants of a, b, c, d, e and f are based on pressure 0-500 bar and temperature 50-350°C………………………………………………...... 43
Table 3.7: Second-order polynomial functions temperature dependent with empirically derived coefficients (dimensionless)….……………………...... 46
Table 3.8: Polynomial coefficients as a function of temperature for Eq. 3.27 ...... 48
Table 4.1: Values of model parameters used in the calculations……………...... 66
Table 4.2: The three main groups of mineral assemblages and each mineral mass recast based on a total of 10 kg rock mass……………………..……… ...... 68
Table 4.3: Composition of Rose Run, Clinton, Grand Rapids, and Mt Simon brines compiled from the literature……………………………………...... 69
Table 4.4: Rate constants for silicate and carbonate minerals compiled from literature and used in the computer simulations……………………..……...... 70
Table 5.1: Parameters deployed in each type of reaction modeling………...... 75
Table 5.2: Minerals precipitated or dissolved (in mole) at equilibrium for fugacity of 100 bar. Minus sign indicates dissolution of mineral……… ………………..77
Table 6.1: Some of the most commonly used reactive transport codes and their descriptions…………………………………..………………… ...... 102
Table 6.2: Primary and secondary minerals used in the modeling………… ...... 108
Table 6.3: Brine composition of Rose Run Sandstone, Clinton, Mt Simon, and Grand Rapids (Alberta)………………………………..……………...... 108
v
Table 6.4: Rate constants for silicate and carbonate minerals compiled from literature and used in the reactive transport simulations…………….……...... 109
Table 6.5: Thermodynamic parameters of aqueous and neutral species used in the computer modeling. Reference temperature is given in Kelvin ...... 111
Table 6.6: Thermodynamic parameters of minerals used in the computer modeling and a, b and c are heat capacity coefficients……………….……...... 112
Table 6.7: List of hydrological, physical and chemical parameters……...... 121
Table 6.8: List of all dependent variables………………………………...... 126
Table 7.1: List of hydrological, physical and chemical parameters………… ...... 139
Table 7.2: List of hydrological, physical and chemical parameters (carbonate aquifer)…..………………………………………...... 160
Table 7.3: List of hydrological, physical and chemical parameters (silicate aquifer) .... 164
Table 10.1: Fluid, seeding particle and PIV parameters…………………...... 198
vi List of Figures
Figure 1.1: Distribution of electric power plants and emissions of CO2 represented as metric tons of carbon emitted (Bureau of Economic Geology, 2004)...... 11
Figure 2.1: Phase diagram of CO2 under different pressure and temperature conditions (After Garcia, 2004) …………………………………………...... 19
Figure 2.2: Map showing depth to Knox Unconformity where it overlies the Rose Run Sandstone in eastern Ohio. Contour interval in meters. AEP = American Electric Power ………………………………………………...... 29
Figure 2.3: Cross-section for central Ohio. Cambrian sandstones that are candidates for CO2 storage are shaded (after Gupta & Bair 1997). PA = Pennsylvania. For location of cross-section, see Figure 2.2 ……………………………………30
Figure 2.4: Measured section of core through part of the Rose Run Sandstone. For location of core, see Figure 2.2. …………………………………………..... 31
Figure 3.1: Comparison of CSM with experimental data (Wiebe and Gaddy, 1939 and 1941, and King et al., 1992) as a function of temperature (P = 76 bars) in pure water……………………..…………………………………………………...50
Figure 3.2: Comparison of CSM with experimental data (Wiebe and Gaddy,...... 1939 and 1941, and King et al., 1992) as a function of temperature (P = 152 bars) in pure water……………………………………………………………...... 50
Figure 3.3: Comparison of CSM with experimental data (6 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of pressure ( at T = 313 K) CSM that do not include salting-out effect are shown by cross ...... 51
Figure 3.4: Comparison of CSM with experimental data (4 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of pressure (at T = 313 K) CSM that do not include salting-out effect are shown by cross ...... 52
Figure 3.5: Comparison of CSM with experimental data (6 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of temperature (at P = 70 bar). CSM that do not include salting-out effect are shown by cross..... 52
Figure 3.6: Comparison of CSM with experimental data (4 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of temperature (at P = 70 bar). CSM that do not include salting-out effect are shown by cross.... 53
Figure 3.7: Solubility of CO2 in a range of salinity as a function of temperature (P = 160 bar) based on Duan and Sun (2003) CO2 solubility model (DS-CSM) ...... 54
vii Figure 3.8: Solubility of CO2 as a function of temperature (at P = 150 bar and 6 molal of ionic strength) based on CO2 solubility model. CSM that do not include salting-out effect are shown by cross. The inserted box indicates the typical temperature range of saline aquifers appropriate for CO2 injection ...... 56
Figure 3.9: Solubility of CO2 as a function of pressure (at T = 300 K and 6 molal of ionic strength) based on CO2 solubility model. CSM that do not include salting-out effect are shown by cross. The inserted box indicates the typical pore pressure range of saline aquifers appropriate for CO2 injection …… ...... 57
Figure 4.1: Schematic diagram showing the decrease in CO2 pressure as a function of radial distance………………………………………………………………..72
Figure 5.1: Equilibrium modeling showing a) pH as a function of fCO2, and b) net CO2 consumption as a function of fCO2 …………………………………………...76
Figure 5.2: Equilibrium modeling at T = 54° C showing net precipitation/dissolution of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage …………………………………………………………………..78
Figure 5.3: Equilibrium modeling at T = 35° C showing net precipitation/dissolution of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage …………………………………………………………………..79
Figure 5.4: Equilibrium modeling at T = 75° C showing net precipitation/dissolution of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage …………………………………………………………………..80
Figure 5.5: Path of reaction showing the formation and dissolution of a) dawsonite, and b) siderite for the mixed assemblage (fCO2 ranges from 20 to 300 bar)...... 82
Figure 5.6: Fugacity of CO2 (a), dissolve CO2(aq) (b), and pH (c) as a function of time in kinetic model simulations ………………………………………… ...... 84
Figure 5.7: Computer model results of mineral precipitation/dissolution of a) carbonate, b) sandstone, and c) mixed assemblages as a function of time...... 85
Figure 5.8: Kinetic modeling for a range of brine to rock ratio a) pH and b) fCO2 ...... 87
Figure 5.9: Kinetic modeling for a range of brine to rock ratio that shows precipitation of a) dawsonite and b) siderite ………………………………………… ...... 88
Figure 5.10: Different brine composition for mixed assemblage a) uptake of CO2, b) precipitation of dawsonite, and c) precipitation of siderite…………………..89
Figure 6.1: Schematic representation of simplified aquifer ……………………………105
Figure 6.2: Space discretization and variable indexing in the integral finite difference method ………………………………………………………...... 130
viii
Figure 6.2: Flow chart of the 1-D reactive transport …..………………………………137
Figure 7.1: CO2 saturation as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ………………………………………………………………...141
Figure 7.2: The CO2 pressure distributions as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ………………………………………… ...... 142
Figure 7.3: The pH distribution as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ……………………………………………...... 143
Figure 7.4: Fugacity distribution as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ……………………………………………...... 145
Figure 7.5: Dissolved CO2 as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ………………………………………… ...... 146
Figure 7.6: Free CO2 as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ………………………………………… ...... 148
Figure 7.7: Precipitation/dissolution of Calcite as a function of time and space using CSM of a) DS, b) Xu, and c) MRK…………………………………………… 149
Figure 7.8: Precipitation/dissolution of Dolomite as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ……………………………………… 151
Figure 7.9: Precipitation/dissolution of Siderite as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ……………………………………… 152
Figure 7.10: Precipitation/dissolution of Dawsonite as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ……………………………………… 153
Figure 7.11: Precipitation/dissolution of Annite and Albite as a function of time and space using DS-CSM (a and d), Xu-CSM (b and e), and MRK-CSM (c and f) …………………………………………………………………….155
Figure 7.12: Precipitation/dissolution of Kaolinite and K-feldspar as a function of time and space using DS-CSM (a and d), Xu-CSM (b and e), and MRK-CSM (c and f) ……………………………………………………...... 156
Figure 7.13: Porosity change due to precipitation/dissolution as a function of time and space using CSM of (a) DS, (b) Xu, and (c) MRK ……………………...... 158 Figure 7.14: CO2 uptake by carbonate mineral precipitation as a function of time and space using CSM of a) DS, b) Xu, and c) MRK ………………………….. 159
Figure 7.15: Simulation result using DS-CSM in carbonate aquifer. a) CO2 fugacity b) Dissolved CO2 and c) CO2 uptake by precipitation/dissolution of carbonate minerals ……………………………………………………...... 162
ix
Figure 7.16: Simulation result using DS-CSM in silicate aquifer. a) Fugacity of CO2 b) Dissolved CO2 and c) CO2 uptake by precipitation/dissolution of carbonate minerals ……………………………………………………...... 165
Figure 7.17: Simulation result using DS-CSM in silicate (with anorthite) aquifer. a) Fugacity of CO2, b) Dissolved CO2 and c) CO2 uptake by precipitation/dissolution of carbonate minerals ……………………………167
Figure 7.18: Sensitivity analysis (Temperature) using DS-CSM in Rose Run aquifer at a) 35 °C, b) 45 °C, c) 55 °C, and d) 75 °C at pressure 200 bar ……………… 169
Figure 7.19: Sensitivity analysis (Pressure) using DS-CSM in Rose Run aquifer at a) 100 bar, b) 200 bar, and c) 300 bar at temperature 35°C…………………...... 170
Figure 7.20: Sensitivity analysis (Salinity) using DS-CSM in Rose Run aquifer at a) 3%, b) 7%, c) 11%, and d) 23% by wt, at temperature 35°C and pressure 200 bar ……………………………………………………… ...... 173
Figure 7.21: Sensitivity analysis (Reaction rate) using DS-CSM in Rose Run aquifer. a) two magnitude less than Table 6.4, and b) two magnitude higher than Table 6.4 …………………………………………………………...... 174
Figure 10.1: Diamond-lattice network flow cell showing locations where micro-PIV velocity data were collected………………………………………………...197
Figure 10.2: Diagram illustrating the experimental setup …………………………… . 199
Figure 10.3: Time response of the 2 m diameter silicon carbide seed particles. The y- axis shows the ratio of particle velocity to fluid velocity whereas x-axis represents time elapsed ……………………………………… ...... 201
Figure 10.4: A raw image of four throat channels (0.6 mm width) connected to a pore (2.5 mm diameter) obtained from PIVACQ ……………………………… 203
Figure 11.1: Velocity distribution mapped using micro-PIV (FOV 2.5 by 2.5 mm) in a cylindrical tube (2.5 mm diameter) at the outlet of the flow cell at location A………………………………………………………………..208
Figure 11.2: Comparison between experimental PIV data and analytical solution for flow at the cylindrical outlet tube of the flow cell at location A ……………….. 208
Figure 11.3: Velocity distribution mapped using micro-PIV and a 5 by 5 mm FOV at location B where the pore body is connected to four 1 mm wide pore throat ………………………………………………………...... 210
Figure 11.4: Velocity distribution mapped using micro-PIV and 5 by 5 mm FOV at location C where the pore body is connected to four 0.6 mm wide pore throats ………………………………………………………...... 211
x
Figure 11.5: Velocity distribution mapped at location C using PIV and a 2.5 by 2.5 mm FOV. This figure shows the same pore as Figure 8, but a smaller field of view ………………………………………………………...... 213
Figure 11.6: Velocity distribution mapped using micro-PIV and 5 by 5 mm FOV at location D where the pore body is connected to four 0.2 mm wide pore throats……………………………………………………………………….214
Figure 11.7: Uncertainty associate with the velocity measurements….. ………………217
Figure 11.8: Computed and measured flow conditions at the mid-section of a pore bodies B, C, and D …………………………………………………...... 220
xi ACKNOWLEDGEMENTS
I would like to thank my advisors, Prof. Beverly Saylor and Prof. J.R. Kadambi,
for their advice, guidance, and support. Without their constant advice and efforts, this
thesis would not have been completed. The time and efforts of my committee members
Dr. Matisoff and Dr. Van Orman are appreciated. I would like to acknowledge Dr Doug
Allen for providing me with the original code that calculates equilibrium constants and
Dr Brian McPherson for providing me with code of Equation of State for CO2 solubility.
I would like to recognize all the laser flow diagnostics lab members and Geological
Sciences colleagues for their technical expertise, moral support and friendship, and especially John Sankovic for getting me started on the PIV work. Last, but not least, I would like to thank my family, my fiancée, and all my friends for their tremendous love and support. This project is supported by Ohio Coal Research Consortium (OCRC) Grant
OCRC3-00-4.C4-1 and OCRC3-00-4.C3-15. The OCRC support is gratefully acknowledged.
xii NOMENCLATURE
K% : equilibrium constant [dimensionless]
ϒl : rate of reaction [mole/s]
ϑ : mole number component [mole]
Κ : absolute permeability [m2]
D : diffusion coefficient [m2/s] rr k : subscript to indicate kinetic rate of reaction
A : molal Helmholtz function for H2O [molal]
th a1…4j : coefficients unique to the j aqueous species
Al : mineral formed by combining components am : activity of mineral [dimensionless]
2 Amin : reactive surface area [m /g] aw : activity of water [dimensionless] aε,1…10 : dimensionless regression coefficients.
B and C : polynomial fit coefficients [dimensionless]
C : conversion factor [cal bar-1 cm-3]
C°PP,T : isobaric heat capacities [cal/mol K]
CO2 3 C : CO2 concentration [kg/m ]
th th 3 Ci,j : concentration of i and j species [kg/m ] dnm : nodal distance [m]
xiii Ea : activation energy [kJ/mol]
F : flux of advection [m/s] and diffusion [m2/s]
fg : gas fugacity [bar]
ĥ : conversion factor [bar cm3 cal-1]
H : Height/thickness [m] i : index for the primary variables.
Ir : reaction rate [mol/s]
KH : modified Henry’s coefficient [Pa]
kr : relative permeability [dimensionless]
2 Kr : rate constant [mol/s/m ]
kz± : a constant [Å]
l : equation index represents liquid and gases.
L : liquid phase index [dimensionless]
m : index for adjacent grid blocks connected to n
M : mass rate of components [kg/s]
Mwt : molecular weight [g/mol]
n : index of grid block [dimensionless]
: power term constant [dimensionless]
Nc : total number of chemical component
nm : mole number of minerals [mole]
nw : solvent mass [kg]
P : pressure [bar]
xiv Pr : reference pressure [bar]
Ql : activity product [dimensionless] r : radius [m]
R : index for sink/source reaction
: universal gas constant [bar m3/mol K].
: residual function [mol] rx,j : crystallographic radius [Å]
S : saturation [dimensionless]
S°P,T : standard molal entropy [cal/mol K]
SG : saturation of gas [dimensionless]
Sgr : irreducible gas saturation [dimensionless]
SI, Ωl , : saturation index [dimensionless]
SL : saturation of liquid [dimensionless]
Slr : irreducible water saturation [dimensionless]
Ss : solid saturation [dimensionless] t : number of the time step [dimensionless]
T : temperature [°C and K] t΄ : previous time step [s],
Td : dimensionless temperature ti : phase transition [dimensionless]
Tr : reference temperature [K] tr : triple-point (subscript)
xv U : Darcy’s velocity [m/s]
unm : Darcy’s velocity [m/s]
3 V°P,T : molar volume [cm /mol]
W : solvent Born function [dimensionless]
x : independent primary variables [dimensionless]
X : fraction of component [dimensionless]
XCO2free : mole fraction of free CO2 [dimensionless]
XG : fraction of component in gas [dimensionless]
XL : fraction of component in liquid [dimensionless]
z : ionic charge [dimensionless]
Z : solvent Born function [dimensionless]
∆G°f : standard molal Gibbs free energy [cal/mol]
∆G°P,T : Gibbs free energy [cal/mol]
∆H°f : standard molal enthalpy of formation [cal/mol]
∆H°P,T : enthalpies [cal/mol]
Φ : fugacity coefficient [dimensionless]
Λ : fractional length of pore bodies [dimensionless]
Θ : constant temperature for H2O [K]
Ξ and ξ : constants [dimensionless]
Ψ : constant pressure for H2O [bar]
α : isobaric expansivity
xvi β : Isothermal compressibility
χ : a constant [dimensionless]
δii΄ : Kronecker delta function
ε : dielectric constant
th th γi,j : activity coefficient of i and j secondary
η : a constant [Å cal mol-1]
ι : phase transition [dimensionless]
ϕ : porosity [dimensionless]
ϕr : fraction of original porosity [dimensionless]
κ : total number of equations
λ : a constant [dimensionless]
µ : dynamic viscosity [N s m-2]
νi, νj, νg and νH2O : stoichiometric reaction coefficient [dimensionless]
ρ : density [kg/m3]
τI : tortuosity [dimensionless]
ω : Born coefficient [cal mol-1]
ζ[0] : constant [dimensionless]
xvii CO2 Sequestration in Saline Aquifer: Geochemical Modeling, Reactive Transport
Simulation and Single-Phase Flow Experiment
Abstract
by
Biniam Zerai
Storage of CO2 in saline aquifers is one way to limit the buildup of greenhouse
gases in the atmosphere. Large-scale injection of CO2 into saline aquifers will induce
a variety of coupled physical and chemical processes including multiphase fluid flow,
solute transport, and chemical reactions between fluids and formation minerals.
These issues were addressed using CO2 solubility modeling, simulation using geochemical reaction, 1-D reactive transport and Particle Image Velocimetry (PIV).
Comparison of CO2 solubility model against experimental data suggest that
Duan and Sun (2003) CO2 solubility model (DS-CSM) accurately modeled solubility
of CO2 in brine for range of temperatures, pressures and salinities.
Modeling under equilibrium, path-of-reaction and kinetic rate using a reactor
type Geochemists Workbench demonstrate that dissolution of albite, K-feldspar, and glauconite, and the precipitation of dawsonite and siderite are very important for mineral trapping of CO2.
xviii A 1-D reactive transport was developed based on CO2 solubility model that take
in to account the high salinity of Rose Run brine and a module that calculates the
equilibrium constants based on temperature and pressure. The results indicate that the extent of sequestration through solubility and mineral trapping is sensitive to the choice of CO2 solubility model and the fugacity of CO2. Reactive transport modeling
underscores in the long-run siderite and dawsonite minerals are important sink in
trapping CO2 in the Rose Run Sandstone but over a short time-scale the
hydrodynamic trapping plays a crucial role. The calculated storage capacity using
DS-CSM suggest that for the first 100 years, 90 percent of the injected CO2 trapped as free CO2 whereas 6 percent are trapped in dissolve form and the rest
sequestered in minerals.
Micro-scale single-phase flow through a network model of porous rock was
investigated using experimental and numerical analysis. PIV with refractive index
matching was developed to map velocity of pore-scale fluid flow through acrylic two-
dimensional network without chemical reaction. Experimentally determined velocity
vectors for single-phase flow through pore bodies and adjoining throats as well as for the outlet of the flow cell were compared with numerical simulations of flow through the cell using FLUENT computer code.
xix
INTRODUCTION
Most energy used to meet human needs is derived from the combustion of
fossil fuels (natural gas, oil, and coal), which releases carbon to the atmosphere,
primarily as carbon dioxide (CO2). The atmospheric concentration of CO2, a greenhouse gas, is increasing, raising concerns, that solar heat will be trapped and the average surficial temperature of the Earth will rise in response. Global warming studies predict that climate changes resulting from increases in atmospheric CO2 will adversely affect life on Earth (IPCC, 2001).
There has been ongoing research to address the problem of rising
atmospheric CO2 concentrations through various stabilizing mechanisms (IPCC,
2001, 2002). Approaches to controlling the level of CO2 in the atmosphere
include a broad portfolio of strategies to reduce carbon emissions using methods
such as carbon capture and sequestration, enhanced efficiency of power generation and use, use of low carbon fuels, and the use of renewable energy sources. Carbon sequestration includes the removal of CO2 from the atmosphere by agricultural modifications and reforestation as well as the reduction of CO2 emissions by capture and storage. Storage of anthropogenic
CO2 within geologic reservoirs is one of the favorable methods of carbon
sequestration (Bachu, 2002). Geologic sequestration represents an immediately
available option for mitigating the global environmental impact of CO2 by
removing large amounts of the gas from the atmosphere. Implementation of this
1
approach requires knowledge of the fundamental science that governs the
physical and chemical behavior of CO2 during and after injection into the deep
subsurface.
The overall objective of this dissertation is to assess the factors that impact
injection and storage of CO2 in deep saline aquifers. The dissertation is divided
into two parts. The first consists of geochemical modeling studies of reactions
among CO2 and subsurface brines and minerals. The second concerns the
dynamics of flow through porous media and presents preliminary single-phase
flow experimental work in a two-dimensional lattice network.
Because of the large spatial extent and long time scale, it is difficult to study
the effect of CO2 sequestration in saline aquifers via laboratory or field studies
conducted over short periods of time. To address issues related to viability and
risk of injecting CO2 into the subsurface it is necessary to use geochemical
computer models and reactive transport simulations that take into account the
thermodynamics and kinetics of chemical reactions and mass transport. The first
part of the dissertation emphasizes CO2-brine and brine-mineral reactions, with
and without transport. Recently published equations of state for the solubility of
CO2 in brine are compared with each other and with experimental data. CO2- brine-mineral reactions are simulated for injection of CO2 into the Rose Run
Sandstone, a deep saline aquifer beneath eastern Ohio. The geochemical
simulations are based on a pre-existing batch geochemical model, the
Geochemist’s Workbench, and a newly developed one-dimensional reactive
transport code that includes three different equations of state for the solubility of
2
CO2 in brine. Sensitivity analysis using the two geochemical models investigates the effect of salinity, CO2 solubility model, pressure, temperature, brine
composition, brine to rock ratio, porosity, permeability, reactive surface area, rate
of reaction, and rock composition.
Even without geochemical reactions, the dynamics of the flow of CO2 through the complex geometries of an aquifer’s pore space may have tremendous impact on injectivity and storage capacity. The second part of the dissertation deals with the micro-scale dynamics of flow through porous media. It presents experimental and numerical analysis of single-phase flow through a network model of porous rock. Quantitative velocity data for single-phase flow through the individual pore bodies and pore throats of a transparent flow cell were obtained using a technique called micro-Particle Image Velocimetry (micro-PIV) based on refractive index matching. Particle Image Velocimetry (PIV) is a well-established technique for macroscopic flows that maps the instantaneous velocity field by periodically illuminating tracers, usually particles, suspended in a flow (Raffel et al., 1998). Advantages are that it is a non-intrusive, in-situ technique that can be used on two-dimensional or three-dimensional flow cells and is capable of characterizing multiphase flows. The intended purpose was to develop micro-PIV for a pore network model and to show the usefulness of micro-PIV in obtaining flow velocities at pore and throat scale. Ultimately, it should be possible to extend the technique to two-phase flow. In this work, preliminary results that focus on single-phase flow are discussed. Experimentally determined velocity vectors are compared with numerical simulations of flow through the flow cell using
3
FLUENTTM (version 6.02). FLUENTTM is a standard computational fluid dynamics code that allows computation of velocity distributions in complex geometries.
The two parts of this dissertation consist of a total 12 chapters. Part I covers chapters 1-8. Part II comprises chapter 9-12. Chapter 1 discusses the motivation selecting deep saline aquifers for geologic sequestration of CO2. Chapter 2 reviews mechanisms for aquifer storage of CO2 and provide geological and hydrological descriptions of the candidate aquifer, the Rose Run Sandstone.
Chapter 2 is based largely on a review paper by Saylor and Zerai (2004).
Chapter 3 presents a comparison study of equations of state for the solubility of
CO2 in brine. Chapter 4 introduces the batch geochemical model Geochemist’s
Workbench and the newly developed reactive transport code. Chapter 5 presents results of batch geochemical modeling of CO2-brine-mineral reactions in the
Rose Run Sandstone and is based on a paper submitted to Applied
Geochemistry and accepted pending revision. Chapter 6 describes the theoretical and mathematical equations of the newly developed one-dimensional reactive transport code and Chapter 7 details the results of the reactive transport modeling. Chapter 8 provides summary discussion and conclusions of all the geochemical modeling. Description of single-phase flow experimental work, experimental set up and techniques deployed are given in chapters 9 and 10.
Chapter 11 discusses the detail result of the experimental work. The discussion and conclusion part of part II is addressed in chapter 12. Chapters 9 through 12 are based on a paper by Zerai et al. (2005) in Transport in Porous Media, published by Kluwer.
4
PART I
Geochemical Modeling and Reactive Transport Simulation
of CO2 Trapped through Mineral Precipitation and Dissolution in the Rose Run Sandstone, Ohio
5
Chapter 1
INTRODUCTION AND BACKGROUND
1.1 Introduction
The burning of coal, oil, and natural gas, as well as deforestation and various agricultural and industrial practices, are altering the composition of the atmosphere and contributing to climate change (IPCC, 2001). These human activities have led to increased atmospheric concentrations of a number of greenhouse gases, including CO2, methane, nitrous oxide, chlorofluorocarbons, and ozone in the lower part of the atmosphere. CO2 has increased in the atmosphere by 25 percent over the last 100 years (Francey et al., 1999; IPCC,
2001, 2002) mostly due to fossil-fuel based energy industries and deforestation
(IPCC, 2001).
Four lines of evidence suggest that the recent increase in atmospheric concentrations of CO2 is largely from human activities. The nuclei of carbon atoms in CO2 emitted by burning coal, oil, and natural gas (fossil fuels) differ in
6 their characteristics from the nuclei of carbon atoms in CO2 emitted under natural conditions (Keeling, 1960; Stuiver et al., 1984). It takes millions of years for coal, oil, and natural gas to be formed deep inside the Earth. The fraction of their nuclei that were once radioactive has long ago changed to non-radioactive carbon. The CO2 emitted from natural sources on the Earth's surface retains a
measurable quantity of radioactive material. As CO2 has been emitted through fossil fuel combustion, the radioactive fraction of carbon in the atmosphere has decreased. Forty years ago scientists provided the first direct evidence that combustion of fossil fuels was causing a buildup of CO2 and thereby diluting
radioactive carbon in the atmosphere by measuring the decreasing fraction of
radioactive carbon-14 captured in tree rings, each year between 1800 and 1950
(Stuiver et al., 1984; Francey et al., 1999; IPCC, 2001, 2002).
Secondly, starting in the late 1950s, scientists have collected data through precise measurements of the total amount of CO2 in the atmosphere both at
Mauna Loa, Hawaii, and at the South Pole. Nowadays, many locations around the world are used to monitor CO2. The data so far collected show that the levels of CO2 have increased each year worldwide (Keeling, 1973, 1976, and 1978).
Furthermore, these increases are consistent with other estimates of the rise of
CO2 emissions due to human activity over this period.
Thirdly, ice buried below the surface of the Greenland and Antarctic ice caps
contains bubbles of air that were trapped when the ice formed. Measurements
from the youngest and most shallow segments of the ice cores, show increasing
CO2 concentrations. Layers with air from only a few decades ago produce CO2
7 concentrations nearly identical to those that were measured directly in the
atmosphere at the time the ice formed (Keeling, 1978; Francey et al., 1999;
IPCC, 2001, 2002). The CO2 content of the pre-industrial parts of the cores are
lower by almost 25% than what is measured today and are relatively constant
over a period of 10,000 years (Keeling, 1978; Francey et al., 1999; IPCC, 2001,
2002).
The fourth line of evidence comes from the geographic pattern of CO2 measured in air. Worldwide the CO2 distribution is not the same. According to
observation, the CO2 measurement on the northern hemisphere is slightly more
than in the southern hemisphere (Stuiver et al., 1984; IPCC, 2001, 2002). The
difference arises because most of the human activities that produce CO2 are in the north and it takes about a year for northern hemispheric emissions to circulate through the atmosphere and reach southern latitudes.
The human population’s dependency on fossil fuels as a source of energy is strong and the transition to alternate renewable energy sources is not in the current foreseeable future. The United States is by far the leading contributor of
CO2 emissions to the atmosphere followed by the Former Soviet Union and
China (Table 1.1). Fifteen-year projections show increases in carbon emissions by 100 to 200 percent in China, India, Mexico, Brazil, and South Korea.
Growing concern about human-induced climate change is driving technology development aimed at slowing the buildup of greenhouse gases in the atmosphere. The Kyoto Protocol, the first attempt to reach global consensus on how and how much to reduce greenhouse gas emissions, described in length
8 two key prospective solutions to the problem of balancing the carbon cycle:
reduce emissions and/or sequester more carbon. Improved efficiency and
increased use of alternative energy to reduce the use of fossil fuels almost certainly will be part of the solution. A second key prospect is to increase the storage of waste CO2 in reservoirs other than the atmosphere. Potential
reservoirs for sequestering CO2 include the terrestrial biosphere (Seneviratne
2003), the oceans (Drange et al. 2001), mineralized forms at the Earth’s surface
(Lackner 2002), and the deep underground formations (Holloway 2001).
Table 1.1 World carbon emissions designated by region including both historical data and projectionsWORLD CARBON EMISSIONS BY REGION, Reference Case (Millions of Metric Tons of Carbon) PERCENT HISTORY PROJECTIONS CHANGE REGION/COUNTRY 1990 2001 2010 2020 1990-2020 United States 1,352 1,559 1,800 2,082 54.00% Former Soviet Union 1,036 654 825 939 -9.40% China 617 832 1,109 1,574 155.00% Eastern Europe 301 202 213 248 -17.60%
Germany 271 233 232 241 -11.00% Japan 269 316 334 365 35.70% United Kingdom 164 153 163 176 7.30% India 153 250 321 435 184.00% Canada 129 155 186 196 52.00% France 102 108 108 122 19.60% Mexico 84 96 134 207 146.00% Brazil 68 95 127 180 165.00% South Korea 64 121 156 193 202.00%
TOTAL DEVELOPING 1,691 2,487 3,075 4,137 145.00% TOTAL WORLD 5,872 6,522 7,685 9,372 59.60%
Source: Based on U.S. Energy Information Administration, International Energy Outlook 2003, p. 191
9 Electric power plants and large-scale industrial process plants are ideal
candidates for CO2 sequestration because they are point-source emitters. Figure
1.1 shows the distribution of electric power plants throughout the United States
along with the carbon emission statistics for each plant. As shown in Figure 1.1,
the northeastern part of United States of America, especially power plants along
the Ohio River are the biggest point sources of anthropogenic CO2 in the United
States. Thus, it is logical to target this area in dealing with the reduction of CO2 emissions to the atmosphere.
Legend Metric Tons of Carbon Emitted 0 - 300000 300001 - 815000 815001 - 1650000 ¯ 1650001 - 3000000 3000001 - 5570000
0750,000 1,500,000 3,000,000 Kilometers
Figure 1.1 Distribution of electric power plants and emissions of CO2 represented as metric tons of carbon emitted (Bureau of Economic Geology, 2004).
1.2 Disposal in Geological Formation
Injection of power plant-generated CO2 into deep sedimentary formations is
one way to limit the buildup of greenhouse gases in the atmosphere. Potential
10 targets for CO2 injection include depleted oil and gas reservoirs, deep
unmineable coal seams, and deep saline aquifers. Technology for injecting CO2
has long been used in the oil and gas industry for enhanced oil recovery. In
addition, a pilot study of CO2 injection for enhanced methane recovery from deep
coal seams has been underway since 1996 (Gale & Freund 2001). These
technologies potentially can provide economic return while storing CO2 in the
subsurface for long times. Injection of CO2 into depleted oil and gas reservoirs, in particular, is a reasonable first approach to CO2 storage because the
infrastructure is largely in place. However, the storage capacity may not be
sufficient to meet long-term needs. The technology also exists for injecting CO2
into deep saline aquifers and pilot projects are underway in the Sleipner West oil
field of the North Sea (Gale et al. 2001), the West Pearl Micro-pilot in New
Mexico (Pawar et al., 2004), and the Frio Pilot Project in Texas (Doughty and
Hovorka, 2004).
The overall task of diverting power plant CO2 to deep geologic reservoirs
requires three steps: Capturing the flue gas, transport and injection of the CO2, and sequestration of CO2 in the reservoir. The present work focuses on the last
step, CO2 sequestration, in a saline aquifer. CO2 is less dense than formation
fluids. Once injected, some CO2 dissolves into the pore water. The remainder
rises buoyantly as a separate phase such that escape of CO2 to the surface is a
concern. Escape of CO2 would compromise the objectives of injection and also
could compromise safety, for example by contaminating shallow potable aquifers
(Saripalli & McGrail 2002; Klusman 2003). Safe, long- term storage requires that
11 the CO2 be immobilized or otherwise prevented from migrating upward. Closed structures immobilize buoyancy-driven flow in oil and gas reservoirs and demonstrably have retained oil, natural gas, and naturally occurring CO2 in the subsurface for millions of years (Pearce et al. 1996). As reported by Allis et al.
(2001), despite the abundance of CO2 reservoirs in the Colorado Plateau-
Southern Rocky Mountains region, no hazards from surface CO2 accumulations
are known. CO2 injected into deep coal seams is immobilized by absorption onto
the coal itself. Given a low permeability cap rock, deep saline aquifers
theoretically can trap CO2 hydrodynamically by slow moving, downward-directed
formation waters (Bachu et al. 1994), by solubility in the pore waters (Weir et al.
1995), and by reactions with minerals and pore waters that convert CO2 to
carbonate minerals (Gunter et al. 1993). These hydrodynamic and geochemical
trapping mechanisms may eliminate the need for geometric traps to retain CO2 in deep saline aquifers.
1.3 Why Saline Aquifers
Deep saline aquifers provide no economic return for CO2 injection, but they
are widespread, are geographically associated with fossil fuel sources, and,
because it is not necessary to identify and inject directly into closed structural
traps, are likely to have large storage volumes and suitable injection sites in
close proximity to power-plant sources of CO2 (Hitchon et al. 1999; Bachu, 2003).
Deep aquifers potentially have CO2-storage capacities sufficient to hold many
decades worth of CO2 emissions, but estimates of global capacity are poorly
12 constrained, varying from 300 to 10,000 Gigatons CO2 (Holloway 2001). In the
United States, deep saline aquifers have a larger potential storage capacity than
any other type of sedimentary formation, with estimates as high as 500 Gt of CO2 storage (Bergman and Winter, 1995). The variation in estimates of storage capacity reflects different assumptions about the effectiveness of trapping mechanisms. The low estimate counts only CO2 that could be stored as an
immiscible phase in closed structures within aquifers. The high number assumes
closed structures are not necessary and that CO2 can be stored through a
combination of hydrodynamic, solubility, and mineral trapping.
1.4 Purpose and Objective of Study
The overall objective of the study is to simulate changes in reservoir
properties and to evaluate storage capacity following the injection of CO2 into a deep aquifer. The Rose Run Sandstone, a deep aquifer and oil- and gas- containing formation in the Appalachian Basin area of eastern Ohio, USA, is used as a case study for much of the work. Geochemical reactions between CO2, brine, and formation minerals are thoroughly investigated because these reactions determine the ultimate fate of CO2.
Injection of foreign fluids into an aquifer with heterogeneous mineral
assemblages produces feedback loops between hydrological parameters,
geomechanics, energy transfer and chemical reactions that must be understood
in order to design safe and reliable injection strategies. In this work, the
chemical-thermal and the mechanical-chemical interactions are not considered.
13 The main emphasis in this study is the feedback between the chemical and
hydrologic regime caused by mineralization or dissolution within the host aquifer.
The introduction of fluids that are not necessarily in equilibrium with aquifer fluids
and minerals will most likely cause mineralization or dissolution within the rock
matrix to some degree and will affect the porosity and permeability and thus the
mass transfer of fluid within the reservoir.
Chemical reactions are affected by the system pressure, temperature,
salinity, rate of reaction, and solubility of CO2 solubility in the aquifer brine. These
factors ultimately affect the speed at which mineralization and dissolution reactions take place. Many of the hydrologic transport parameters and chemical reactions are well known to affect natural diagenesis to a great extent, but questions arise when the system involves a tremendous amount of mass transfer
from one domain (injected CO2) to another (subsurface reservoir). What is the extent of chemical alteration caused by the introduction of a foreign fluid into a reservoir? How will this in turn affect the total make up of the reservoir? All of these questions and many more have to be addressed prior to widespread implementation of geologic sequestration. Through sensitivity analysis, the hydrologic-chemical feedback loop is investigated throughout this study to provide a basis of the fundamental processes that underlie the geologic sequestration of carbon.
Hence, the goal of the present work is to assess the implications of geologic sequestration in the context of injection of CO2. It evaluates the capacities of
different geologic sequestration mechanisms applied to the Rose Run Sandstone
14 using geochemical modeling and reactive transport simulation. To attain these
goals, numerical modeling of CO2 sequestration in the Rose Run Sandstone
Reservoir was performed with special emphasis on the transport of dissolved
phase CO2, free CO2 and CO2 up-taken by the precipitation of carbonate
minerals over differing flow regimes as a function of time and space. The
simulations were performed using existing geochemical code (Geochemists’
Workbench) and a Reactive Transport computer code that has been developed
through this work. The reactive transport code utilizes different CO2 solubility
models developed by Duan and Sun (2003), Xu et al (2004) and McPherson and
Cole (2005).
The behavior of fluid flow in porous media is complex, even without chemical
reaction. Investigation of single-phase flow dynamics at the pore scale was
conducted using Particle Imaging Velocimetry (PIV). The strength of PIV is that it
produces velocity vector maps that quantify the variation in magnitude and
direction of flow velocity. CO2 sequestration at pore scale will be affected by the
dynamic flow behavior of two-phase flow. For example, PIV can document a
complex flow patterns such as deceleration at the pore mouth and the
development of interlocked regions with no or extremely low velocities, which
impact total fluid flux and may have great significance for solute transport or fluid- rock reactions during CO2 sequestration in deep saline aquifer. A preliminary
single-phase flow experimental and numerical simulation was investigated and
the detail work is given in part II of the dissertation.
15
Chapter 2
MECHANISM FOR AQUIFER STORAGE OF CO2 AND THE GEOLOGY OF THE ROSE RUN SANDSTONE
2.1 Mechanism for Aquifer Storage of CO2
Carbon dioxide is retained in geologic formations in three ways (Hitchon,
1996). First, CO2 can be trapped as a gas or supercritical fluid under a low-
permeability caprock. This process, commonly called hydrodynamic trapping, will
likely be, in the short term, the most important method of retention. Second, CO2
can dissolve into the groundwater, referred to as a solubility trapping. Third, CO2
can react directly or indirectly with minerals in the geologic formation leading to the precipitation of secondary carbonate minerals referred as mineral trapping.
These three different mechanisms of CO2 storage are described below.
16 2.1.1 Hydrodynamic Trapping
Hydrodynamic trapping refers to storage of free CO2 in the pore spaces of
sedimentary layers and the transport of that CO2 away from the surface by
regional groundwater flow (Bachu et al.1994). Free CO2 is the main form of
storage during injection, which can last 30-50 years. The injected CO2 is subject to injection-related hydrodynamic gradients and to buoyancy forces that cause it to form a plume that rises and spreads laterally until it meets a confining layer that impedes vertical ascent, causing the CO2 to accumulate as a cap. The
buoyant force of the CO2 cap will depend on the difference in density between
the CO2 and the brine and on the dip of the confining layer. Provided a near horizontal confining layer and relatively small density difference, the CO2 will
travel with the downward-directed regional groundwater flow (Bachu et al. 1994),
unless faults or other high permeability zones in the stratigraphic seal provide
escape routes to the surface. The over-pressuring required for reasonable rates
of CO2 injection and buoyancy forces exerted by the CO2 cap can widen small fractures, exacerbating the risk for CO2 escape (Saripalli and McGrail 2002;
Klusman 2003).
CO2 can exist in three different states under the pressure and temperature
conditions of deep saline aquifers: liquid, gas, and supercritical (Figure 2.1).
Supercritical CO2 behaves like a gas, filling all the volume available, but has a
density that varies with pressure and temperature from less than 200 kg/m3 to
more than 900 kg/m3 (Angus et al. 1976). To reduce costs associated with
injection and to limit the buoyancy forces and maximize the mass of free CO2 that
17 can fill a given pore volume, CO2 should be injected in a supercritical state
(Bachu, 2002). Early studies assumed an average surface temperature of 10˚C,
a geothermal gradient of 25˚C/km, and a hydrostatic pressure gradient of 10
MPa/km to determine the depth at which the critical point for CO2 (31.1 ˚C and
7.38 MPa) is reached (Holloway and Savage 1993). These studies set 800 m as a minimum depth for the confining layer. A depth corresponding to a pressure of
10-20 MPa, on the order of 1000 m, may be better because it avoids large CO2
density gradients in the vicinity of the critical point (Holloway 2001).
Figure 2.1 Phase diagram of CO2 under different pressure and temperature conditions (After Garcia, 2001)
18 In reality, the depths in the earth at which the pressure and temperature conditions for supercritical CO2 are reached vary depending on climate conditions and the geology of the basin. In warm, over-pressured basins, the critical point can lie as shallow as 400 meters and the pressure-temperature distribution can be such that CO2 passes with depth directly from a gas to a
supercritical state (Figure 2.1). In cold basins, the depth at which the pressure
and temperature conditions of the critical point are reached is likely to be deeper
than the gas-to-liquid phase change. All things being equal, old, cold, stable
basins are better for hydrodynamic trapping of CO2 because higher CO2 densities are reached at shallower depths and gradients in the density of CO2 are
more easily avoided (Bachu 2000). Old foreland and continental basins are best
suited for hydrodynamic trapping because they tend to be cold, stable, and close
to hydrostatic pressure, and to have erosion- or topography-driven, down-dip-
directed regional flow regimes (Bachu 2000).
3 At 10 MPa and 35˚C, CO2 has a density of approximately 700 kg/m . Under
these conditions, a cubic meter of sandstone with 10% porosity contains
approximately 70 kg of CO2 if the pore space is completely filled by CO2.
However, saturation of CO2 is not complete, some brine remains in the invaded
pore spaces (Pruess et al. 2003; Saripalli & McGrail 2002). In addition, non-
uniform flow of CO2 bypasses parts of the aquifer entirely. Darcy-flow based
analytical and numerical solutions are used to evaluate some of these effects by
simulating the advance of the CO2 front over timescales of decades to hundreds
of years and over distances of tens to hundreds of kilometers. To account for the
19 extreme changes in density and viscosity of CO2 with pressure and temperature,
these models must incorporate experimentally constrained equations of state
(Adams and Bachu 2002).
Initially the free CO2 is distributed in radially decreasing concentrations in
zones around the injection site (van der Meer 1996). Nearest the injection site
lies a zone of near completely saturated pores, containing isolated beads of
trapped brine, some of which evaporate into the CO2 (Pruess et al. 2003). The
middle zone contains mixed brine and CO2 (Pruess et al. 2003; Saripalli &
McGrail. 2002). In the outer zone, CO2 is present only as aqueous species.
Following injection, CO2 saturations around the injection site are predicted to
decrease over tens of years as the free CO2 rises buoyantly, spreads laterally,
and dissolves into the brine (Weir et al. 1995). Over timescales of hundreds of years dispersion, diffusion, and dissolution can reduce the concentration of both
free and aqueous CO2 to near zero (McPherson and Cole 2000).
Pruess et al. (2003) used the Darcy-flow based Buckley-Leverette two-phase
displacement theory to solve analytically the average saturation of injected CO2 under a range of conditions. Under Buckley-Leverette conditions, the radius of the region swept by the CO2 front increases with duration of injection and the
CO2 saturation decreases along that radius, but the average CO2 saturation in
the region is time independent. Assuming a homogeneous aquifer and uniformly
swept region, Pruess et al. (2003) found that average saturation is most sensitive
to permeability and for a range of rock types calculated a saturation range of 20-
40%, with higher average saturations corresponding to rocks with higher
20 permeabilities. Under these conditions, a rock with 10% porosity has a storage
capacity of 14-28 kg of CO2 per cubic meter. These maximum values ignore flow
patterns that may cause the CO2 to bypass large parts of the aquifer. For
example, taking into account buoyancy effects, Van der Meer (1995) estimated
only 6% average saturation by CO2. In addition, over certain ranges of viscosity
ratio and injection velocity, the displacement front between CO2 and brine can
develop fractal fingers, rather than advancing as straight front as modeled by
Darcy-flow. Fractal fingers may reduce the CO2 volume of the aquifer accessed
by CO2 to as low as 1%. Formation heterogeneity, including low permeability layers and lateral discontinuities in permeable rock types can compartmentalize the aquifer vertically and laterally, reducing the amount of the aquifer accessible to the flow of CO2. However, aquifer compartmentalization can also work in the
other direction, limiting buoyancy and fingering effects and increasing the access
of CO2 (Johnson et al. 2001). Any aquifer bypassing that does occur, due either to buoyancy effects, fingering, or formation heterogeneity, may be partially compensated by higher saturations in the layers swept by CO2 (Pruess et al.
2003).
2.1.2 Solubility Trapping
Solubility trapping refers to the CO2 that dissolves into the brine. The CO2- brine solution has a density greater than brine alone preventing buoyant flow of the CO2 toward the surface, even along high permeability vertical pathways such as faults.
21 Most models of solubility trapping assume instantaneous equilibrium between the
brine and free CO2. The solubility of CO2 varies as a function of pressure, temperature, and salinity. Numerous models for CO2 solubility in aqueous
solutions have been published to describe this relationship, though few deal with high ionic strength, multicomponent brines.
Under typical cold deep-aquifer conditions of 10 MPa, 35˚C, and 10% mass
3 fraction salt, the solubility of CO2 in brine is approximately 40 kg/m of brine. A
3 rock layer with 10% porosity has a storage capacity of as much as 4 kg CO2/m .
Under the conditions of interest for aquifer storage, CO2 solubility increases
approximately linearly by about 10 kg/m3 for each 10 MPa increase in pressure
over the range of 10-30 MPa (Pruess et al. 2003). It decreases by an average of
approximately 10 kg/m3 for each 25˚ C rise of temperature over the range of 35-
100˚C (Pruess et al. 2003). Thus under hydrostatic pressure gradients 10
MPa/km and average geothermal gradients of 25 ˚C/km the effects of increasing pressure and temperature with depth on the solubility of CO2 essentially cancel
each other out. However, around the injection site CO2 pressure increases to as
much as twice hydrostatic pressure and CO2 solubility increases with it. CO2 solubility decreases by about 30% as salinity increases from zero to saturated
NaCl (Moller et al. 1998, Pruess et al. 2003). CO2 solubility is sensitive not only
to the mass of salt dissolved (Bachu & Adams 2003), but also to the particular
kind of salt. However, there are few experimental constraints on this effect.
The extent to which CO2 dissolves into the brine is influenced by the migration
of the CO2 front and by the rate of dispersion and diffusion of CO2. Viscous
22 fingering and buoyancy flow, which tend to limit the storage of free CO2, may increase solubility trapping by increasing the surface area of the brine-CO2
contact, allowing more rapid solution. In addition, diffusion of CO2 into the brine
can set up reverse density gradients that lead to convective mixing and
increased rate of dissolution of free CO2 (Lindeberg & Wessel-Berg 1997).
Through these processes, dissolved CO2 becomes the dominant form of CO2 storage in aquifers over periods of tens to hundreds of years following injection
(Weir 1995). Over these timescales the CO2 disperses (Law & Bachu 1996) by
dispersion and diffusion, and dissolution into the brine (Lindeberg 1997).
Continued migration and dispersion drive both free and dissolved CO2 toward zero (McPherson & Cole 2000).
2.1.3 Mineral Trapping
Mineral trapping is the fixing of CO2 in carbonate minerals due to geochemical
reactions among aquifer brines, formation minerals, and aqueous species of
3 CO2. The density of CO2 in calcite is 1250 kg/m . In a rock with 10% porosity and
the pores completely filled with calcite the storage capacity would be 220 kg
3 CO2/m . However, the mass of CO2 sequestered as carbonate minerals is
sensitive to formation mineral and aquifer brine composition, pressure,
temperature, and brine-rock ratio. Time is also important because mineral
trapping reactions take hundreds to thousands of years and more to complete
(Gunter et al. 1997).
23 Four kinds of models are available to simulate mineral-brine-CO2 reactions: equilibrium, path of reaction, kinetic, and reactive transport models. Mineral equilibrium models and path of reaction models are used to calculate equilibrium solid phases and solution compositions for a given set of reactants based on a data set of equilibrium constants and activity coefficients. Equilibrium models calculate only the final state. Path of reactions models also calculate transitional phases along the way. These models do not provide information on the amount of time it takes to reach equilibrium or transition states. Kinetic models consider the rates of reactions. Widely available geochemical modeling codes such as
PATHARC (Hitchon 1996), SOLMINEQ (Kharaka et al. 1988), and Geochemists
Workbench (Bethke 1996) have been used for equilibrium, path of reaction, and kinetic simulations of CO2 storage in aquifers. Because these models have no transport components, these studies simulate closed-system batch conditions and do not take into account migration of CO2 through the aquifer (Gunter et al.
1993, 1996, 1997). Studies using full-scale reactive transport codes to simulate the flow, dissolution, and reaction of CO2 are just becoming available (e.g.
Johnson et al. 2001; Xu et al. 2003). In addition, experimental studies are investigating the kinetics of mineral-brine-CO2 reactions in mineral separate and rocks to refine and test model reliability (Kaszuba et al. 2003; Liu et al. 2003).
Early studies by Gunter et al. (1993) recognized three general cases for mineral-brine-CO2 reactions: 1) reactions with mafic minerals, including anorthite feldspar (CaAlSi2O8); 2) reactions with alkali feldspar (albite - NaAlSi3O8 and orthoclase – KAlSi3O8); and 3) reactions with carbonate minerals. They
24 concluded that the most promising reactions for mineral trapping involve mafic
minerals, which provide divalent cations (Fe2+, Mg2+, Ca2+) for precipitation of
carbonate. Mafic minerals such as olivine, pyroxene, and amphibole have the
highest CO2-fixing potential (Pruess et al. 2003), but are rare in sedimentary
basins. The most common sedimentary-mineral sources of divalent cations are
anorthite, mica-group minerals, especially glauconite, and clays, such that a
general equation for mineral trapping in aquifers is
Anorthite + Micas + Clays + CO2 + H2O ↔
Quartz + Kaolinite + Calcite + Dolomite + Siderite. (2.1)
The mineral trapping takes place in three steps as demonstrated by the example of anorthite dissolution:
+ - CO2(aq) + H2O ↔ H + HCO3 (2.2)
+ 2+ CaAl2Si2O8 + 2H + H2O ↔ Ca + Al2Si2O5(OH)4 (2.3)
2+ - + Ca + HCO3 ↔ CaCO3 + H (2.4)
which leads to the net reaction:
CaAl2Si2O8 + CO2(aq) + 2H2O ↔ CaCO3 + Al2Si2O5(OH)4 (2.5) Anorthite Calcite Kaolinite
25 First the aqueous CO2 dissociates in water, producing carbonic acid (Eq. 2.2);
second the acid attacks the anorthite, leaching Ca2+ and neutralizing the acid
(Eq. 2.3); and third, calcium carbonate precipitates (Eq. 2.4).
Reactions with alkali feldspars do not provide divalent cations for the precipitation of carbonate minerals and initially were thought to be of little significance for mineral trapping (Gunter et al. 1997). However, more recent work indicates that dissolution of alkali feldspars contributes to the fixing of CO2 as the sodium aluminocarbonate mineral dawsonite, NaAlCO3(OH)2 (Johnson et al.
2001, Saylor and Zerai, 2004, Zerai et al., 2005). In this case, the Na necessary
for dawsonite precipitation is available in abundance in the brine, but dissolution
of alkali feldspar provides a source of aluminum and neutralizes the acidic CO2 according to (Johnson et al. 2001):
+ + KAlSi3O8 + Na + CO2(aq) + 2H2O ↔ NaAlCO3(OH)2 + 3SiO2 + K (2.6)
The feasibility of mineral trapping of CO2 in dawsonite is demonstrated by the
Bowen-Gunnedah-Sydney Basin in Australia, which has abundant diagenetic dawsonite that formed in response to magmatic CO2 (Baker et al. 1995). In
addition, abundant dawsonite in the Green River Formation of Colorado and in
Pleistocene ash beds at Olduvai Gorge Tanzania formed by reactions of
aqueous carbonate species with nepheline in the sediment (Smith and Milton
1966).
26 Dissolution of carbonate minerals does not lead to mineral trapping of CO2
(Gunter 1993). However, carbonate dissolution, and other mineral precipitation-
dissolution reactions can affect sequestration capacity by altering the
permeability of the aquifer near the injection site.
2.2 Geologic Reservoir - Rose Run Sandstone
The potential for deep saline aquifer sequestration of CO2 in eastern
Appalachian basin, in general, and in Ohio in particular is high. One of the
candidate saline aquifers for CO2 sequestration in Ohio is the Rose Run
Sandstone.
Ohio is a populous and heavily industrialized state with high CO2 emissions
from its coal-fired electric plants. The largest power plants are concentrated in
the eastern part of the state, where coal is mined from the Appalachian Basin
(Figures 1.3). The Rose Run Sandstone is considered a potential candidate for
CO2 sequestration because it is located close to many of Ohio’s large coal-fired
power plants (Figures 1.3 and 2.2). It is also deep enough for CO2 to be
supercritical and is the shallowest of the sandstone formation, which makes
cheaper to inject. Oil and gas fields within the formation may also provide localized CO2 storage with a cost-incentive provided by enhanced oil recovery by
CO2 injection (Bachu, 2002). The Rose Run Sandstone also is the only one of the deep Cambrian sandstones that is known to retain its sandstone composition in the eastern part of the state rather than passing laterally into carbonate (Gupta et
al., 2004).
27 The Upper Cambrian Rose Run sandstone is a sandy layer in the middle of the Knox Dolomite, which across much of the eastern part of the state lies at depths suitable for injection of supercritical CO2 (Figures 2.3 and 2.4). The Rose
Run Sandstone was deposited during a passive margin phase of the
Appalachian Basin and consists of interbedded layers of carbonate, primarily
Location of cross section in Fig. 2.3
Location of core in Fig. 2.4
Figure 2.2 Map showing depth to Knox Unconformity where it overlies the Rose Run Sandstone in eastern Ohio. Contour interval in meters. AEP = American Electric Power.
dolostone, and sandstone. Like many deep aquifers of the eastern United States and Canada, the sandstone is compositionally mature, consisting of 83% by
28 volume quartz. Subordinate reactive minerals are the alkali feldspars and locally abundant glauconite. Dolomite and quartz are the dominant cements (Janssens
1973; Riley et al. 1993).
The Knox Unconformity is overlain by Mohawk limestone, followed by
Cincinnatian shale, which together form a low-permeablility confining unit that totals more than 500 m in thickness (Figure 2.3).
Figure 2.3 Cross-section for central Ohio. Cambrian sandstones that are candidates for CO2 storage are shaded (after Gupta & Bair 1997). PA = Pennsylvania. For location of cross-section, see Figure 2.2.
29
Rose Run Sandstone Core 3382 Columbiana County, OH
Figure 2.4 Measured section of core through part of the Rose Run Sandstone. For location of core, see Figure 2.2.
2.3 Structure and Hydrologic Parameters
Erosional truncation by the Knox unconformity restricts the Rose Run
sandstone to the eastern third of Ohio. It forms a wedge that thickens eastward
from 60 m across much of eastern Ohio to more than 200 m in central
30 Pennsylvania where it is brought to the surface by thrust sheets of the
Appalachian Mountains (Janssens 1973, Riley et al., 1993). Erosional remnants
and truncation along the Knox angular unconformity form stratigraphic traps, and
the subcrop belt, where the Rose Run Sandstone intersects the Knox
unconformity is the principal locale for oil and gas production.
The Rose Run Sandstone meets the general criteria for aquifer disposal of CO2
as outlined by Bergman and Winter (1995), Bergman et al (1997) and Bachu (2002).
Across much of eastern Ohio, it lies at depths greater than 800 m, the minimum
depth necessary for injection of CO2 as a dense, supercritical phase (Figure 2.3). It is sealed by impermeable cap rock consisting of Ordovician age carbonate and shale (Janssens 1973, Riley et al., 1993). The regional stratigraphic dip is gentle, approximately 5 degrees, down to the east and southeast. The regional flow is down-dip to the east and southeast (Gupta and Bair, 1997), as required for hydrodynamic trapping by downward-directed flow.
The typical ranges of porosity and permeability of the sandstone layers of the
Rose Run Sandstone are 7-15% and 1-15 md, respectively (Riley et al., 1993).
2.4 Rose Run Brine Chemistry
The brine of the Rose Run Sandstone is composed mainly of Na+, Cl-, with
2+ 2+ - + 2+ 2- 2+ + + 3+ subordinate Ca , Mg , Br , K , Sr , SO4 , Fe , HCO3 , H , SiO2(aq), and Al
(Breen et al., 1985). The median concentration of total dissolved solids is 278, 000 mg/kg (Breen et al., 1985). Recent work done by Gupta et al. (2004) indicates that the total dissolve solid concentration can reach more than 320,000 mg/kg.
31 The CO2 sequestration in Rose Run Sandstone was modeled using different brine chemistry including, Rose Run brine, Alberta brine, Clinton brine and Mt Simon brine. The detail methods and approach are discussed in chapters 4-7.
32
Chapter 3
CO2 SOLUBILITY MODELS
3.1 Introduction
To assess the long-term fate of injected CO2 and the effect of CO2-brine-
mineral reactions on the host environment, it is necessary to simulate the dissolution of CO2 in brine. For such purpose, CO2 solubility models (CSM) that
properly addresses the solubility of CO2 in brine solution over the range of
salinity, temperature and pressure conditions of deep saline aquifers need to be integrated into numerical codes. In this chapter recent CSM developed based on
the last few decades of theoretical and experimental work are reviewed and
compared, including CSM by Pruess and Garcia (2002), Duan and Sun (2003),
Spycher et al. (2003), Xu et al. (2004), and McPherson and Cole (2005). The
ranges of applicability of these CSM and their detailed descriptions are presented
below. CSM performance and comparison with experimental data are discussed
in this chapter. CSM developed by Duan and Sun (2003), Xu et al. (2004) and
McPherson and Cole (2005) are coded using Matlab v. 7.01 (2004) and TK
33 Solver 5.0 (2003) and incorporated into the one-dimensional reactive transport
code described in Chapter 6. The impact of some CSM on CO2 sequestration
using reactive transport simulation is discussed in Chapter 7.
The CO2 solubility is controlled by temperature, pressure and salinity. In
general, CO2 tends to dissolve more in cold than warm solutions, whereas more
CO2 can dissolve in solution at higher pressure. The solubility of CO2 decreases
at higher ionic strength due to a phenomenon called the salting-out effect. The
salting-out effect is a phenomenon where the increase in ionic strength forces the
activity coefficient of CO2 to decrease and hence the amount of CO2 dissolved in
a solution decreases.
3.2 CO2 Solubility Modeling
In the last few decades, much experimental work on the solubility of CO2 in pure H2O and in aqueous solutions over a range of temperature and pressure
conditions has been published (e.g. Wiebe and Gaddy, 1939, 1940 and 1941;
Tödheide and Frank, 1963; Takenouchi and Kennedy, 1964 and 1965; Coan and
King, 1971; Drummond, 1981; Helgeson et al., 1981; Gillepsie and Wilson, 1982;
Briones et al., 1987; Song and Kobayashi, 1987; D'Souza et al., 1988; Müller et
al., 1988; Mäder, 1991; King et al., 1992; Dohrn et al., 1993; Rumpf et al., 1994;
Jackson et al., 1995; Teng et al., 1997; Bamberger, et al., 2000; Anderson,
2002). Models of the solubility of CO2 in pure water and in aqueous solution and
methods to express the pressure, temperature, salinity properties of two-phase
CO2-H2O mixtures have been tabulated and formulated based on a variety of
34 approaches to correlating and curve fitting experimental data. Published solubility models include those by Redlich and Kwong (1949), Coan and King (1971),
Drummond (1981), Müller et al. (1988), Spycher and Reed (1988), Harvey and
Prausnitz (1989), Enick and Klara (1990), Duan et al. (1992), Crovetto (1991),
Zuo and Guo (1991), Carroll and Mather (1992), Bamberger, et al. (2000),
Pruess and Garcia (2002), Duan and Sun (2003), Spycher et al. (2003), Xu et al.
(2004) and McPherson and Cole (2005).
CSM describe the variation in solubility of CO2 as a function of pressure,
temperature, and salinity by applying correction factors to a modified Henry’s law
given by (Pruess and Garcia 2002):
ΦP = KHXCO2l (3.1)
where P (bar) is the partial pressure of CO2. XCO2l is the mole fraction solubility of
CO2 in the brine. The dimensionless fugacity coefficient, Φ, accounts for the
nonlinear increase in the solubility of CO2 with increasing P and T. The modified
Henry’s coefficient, KH (Pa), corrects for non-ideal solution behavior with
increasing temperature and mole fraction dissolved salt. Some CSM use an
equilibrium constant (directly related to the standard Gibbs free energy of
reaction through ∆G° = -RT ln K) rather than the Henry’s constant (KH). The CSM developed by Pruess and Garcia (2002), Duan and Sun (2003), Spycher et al.
(2003), Xu et al. (2004) and McPherson and Cole (2005) are discussed below.
These CSM are designed to cover a range of temperature, pressure and salinity as shown in Table 3.1.
35 Table 3.1: CO2 solubility models and their range of temperature, pressure and salinity. DS1 MRK2 PG3 SP4 Xu5
Temperature (K) 298 – 533 288 – 523 298 – 623 285 – 373 308 – 623
Pressure (bar) 0 – 2000 0 – 1000 0 – 500 0 – 600 0 – 500
Salinity (molal) ~ 6 0 ~ 5 0 ~ 6
Salt composition Na,Cl,K,Ca,Mg,SO4 Pure water Na and Cl Pure water Na and Cl 1 Duan and Sun (2003), 2 McPherson and Cole (2005), 3 Pruess and Garcia (2002), 4 Spycher et al. (2003) and 5 Xu et al. (2004)
3.2.1 PG-CSM
The Pruess and Garcia CSM (PG-CSM) for solubility of CO2 in brine (Pruess
and Garcia, 2002) uses an extended version of Henry’s law to account for the salting-out effect. For the fugacity coefficient, the authors use a correlation developed by Spycher and Reed (1988). The temperature and salinity dependency of Henry’s coefficient is represented using the correlation developed by Battistelli et al. (1997). The equation is given by:
234 2345mC()++ CT12 CT + CT 3 + CT 4 KBBTBTBTBTBTH =+++++()1012 3 4 5 × ab P2 de (3.2) Pc()()+++ ++ f TT22TT2 XeCO2 = ()/ PK H
where KH (Pa) is Henry’s coefficient, m (molal) is salt, XCO2 is mole fraction of
CO2, and B1..5 and C1…4 and a…f are polynomial fit coefficients (Table 3.2).
Table 3.2: Polynomial fitting coefficients.
B B1 B2 B3 B4 B5 7.8366x107 1.96025x106 8.20574x104 -7.40674x102 2.18380 -2.20999x107 C C1 C2 C3 C4 1.19784x10- -7.17823x10-4 4.93854x10-6 -1.03826x10-8 1.08233x10-11
36 3.2.2 DS-CSM
The Duan and Sun CSM (DS-CSM) for solubility of CO2 in brine (Duan and
Sun, 2003) takes into account not only NaCl rich solutions but also divalent
cations such as MgCl2 and CaCl2. The DS-CSM is based on potential chemical
energy; CO2 solubility in aqueous solutions is determined from the balance between its chemical potential in the liquid phase and that in the gas phase using
the specific interaction model of Pitzer (1973). DS-CSM adopted a virial expansion of excess Gibbs energy from Pitzer (1973) to calculate the activity coefficient of CO2. The authors calculate the fugacity coefficient of CO2 using
equation of state developed by Duan et al. (1992a and 1992b). The DS-CSM is
given by:
a aaaaa aaa++2 35689 ++ ++ 147TT23 TT 23 TT 23 lnΦ=−−++++ (T, P) Z 1 ln Z rr rr rr V2V4V24 rrr (3.3) aa11 12 a10 ++23 TTrr aaa13⎛⎞ 15 15 53++−++−⎜⎟a1(a114 14 22 exp() 5Vrr15rr 2T a⎝⎠ V V
where,
a 2 aaaaa35689 aaa147++23 ++ 23 ++ 23 PVrr TT rr TT rr TT rr Z1==+ +24 + + TVrr V r V r (3.4) aa11 12 a10 ++23 TTrraaa13⎛⎞ 15 15 53222++−⎜⎟aexp()14 VTVVVrrrrr⎝⎠
where, Pr = Ptot/Pc, Tr = T/Tc and Vr = VPc/RTc
37 l(0) ⎛⎞PPtot−µ H 2O CO2 ln mCO2=Φ−− ln⎜⎟ ( ) CO2 P tot ⎝⎠PRT
2λ+++−CO2− Na (m Na m K 2m Ca 2m Mg ) (3.5)
ζ++++CO2−− Na Clm(m Cl Na m K m Ca m Mg ) 0.07m SO4 where,
⎡⎤PT⋅ 1.9 P1b1(b2b3b5=⋅+−Α+⋅Α+Α+Αwc ⎡⎤⎡ 34⎤⎡⎤ H2O ⎢⎥⎣⎦() ⎣ ⎦⎣⎦ (3.6) ⎣⎦Twc
µl(0) c3 c5 CO2 =+⋅++⋅c1 c2 T c4 T2 + +⋅ c6 P RT T 630− T tot c8⋅⋅ P c9 P c10⋅ P 2 (3.7) +⋅c7 P ⋅ ln(T) +tot + tot +tot + c11 ⋅ Tln(P ) tot T 630−− T (630 T)2 tot
c3 c5 λ=+⋅++⋅++⋅c1 c2 T c4 T2 c6 P CO2− NaT 630− T tot c8⋅⋅ P c9 P c10⋅ P 2 (3.8) +⋅c7 P ⋅ ln(T) +tot + tot +tot + c11 ⋅ Tln(P ) tot T 630−− T (630 T)2 tot
c3 c5 ζ=+⋅++⋅++⋅c1 c2 T c4 T2 c6 P CO2−− Na ClT630T− tot c8⋅⋅ P c9 P c10⋅ P 2 (3.9) +⋅c7 P ⋅ ln(T) +tot + tot +tot + c11 ⋅ Tln(P ) tot T 630−− T (630 T)2 tot
where, Α = (T-Twc)/Twc, a1-15 are dimensionless parametric coefficients (Table
3.3), Tr is dimensionless reduced temperature, Vr is dimensionless reduced volume, mCO2 (molal) is concentration of CO2, Ptot (bar) is pressure (Ptot =
PCO2+PH2O), PH2O (bar) is pure water pressure, φCO2 is dimensionless fugacity coefficient, µCO2 is the standard chemical potential of CO2 in liquid phase, R (bar
38 m3 K-1 mol-1) is universal gas constant, T (K) is temperature, λ and ζ are
dimensionless second-order and third-order interaction parameters (Table 3.4),
respectively, mn (n = Na, K, Cl, Mg, Ca, SO4) are aqueous species present in the solution in molal, and c1-c11 are dimensionless interaction parameters (Table
3.4). Z is a compressibility factor, Pc is the critical pressure (73 bar), and Tc is
critical temperature (304 K). Pwc (=220.85 bar) is the critical pressure of water,
Twc (=647.29 K) is the critical temperature of water and b1-b5 are dimensionless coefficient parameters (Table 3.5).
Table 3.3: Parameters for Eq. 3.4
-2 a1 8.99288497x10 -1 a2 -4.94783127x10 -2 a3 4.77922245x10 -2 a4 1.03808883x10 -2 a5 -2.82516861x10 -2 a6 9.49887563x10 -4 a7 5.20600880x10 -4 a8 -2.93540971x10 -3 a9 -1.77265112x10 -5 a10 -2.51101973x10 -5 a11 8.93353441x10 -5 a12 7.88998563x10 -2 a13 -1.66727022x10
a14 1.398 -2 a15 2.96x10
39 Table 3.4: Interaction parameters for Eq. 3.7-3.9.
T-P µCO2/RT λCO2-Na ζCO2-Na-Cl coefficient c1 28.9447706 - 0.411370585 3.36389723x10-4 c2 -0.0354581768 6.07632013x10-4 -1.98298980x10-5 c3 - 4770.67077 97.5347708 0 c4 1.02782768x10-5 0 0 c5 33.8126098 0 0 c6 9.04037140x10-3 0 0 c7 -0.14934031x10-3 0 0 c8 - 0.307405726 - 0.0237622469 2.12220830x10-3 c9 - 0.0907301486 0.0170656236 - 5.24873303x10-3 c10 9.32713393x10-4 0 0 c11 0 1.41335834x10-5 0
Table 3.5: Dimensionless coefficient parameters.
b1 -38.640844 b2 5.894820 b3 59.876576 b4 26.654627 b5 10.637097
The solubility of CO2 using DS-CSM is calculated as follows. First, for any given pressure and temperature Pr and Tr are calculated. Then Vr is computed iteratively using Eq. 3.4 and using Vr, Z is calculated using Eq. 3.4. The values obtained for both Z and Vr are substituted into Eq. 3.3. The fugacity coefficient will be obtained from Eq. 3.3. The water pressure (PH2O) and interaction
40 parameters (λCO2-Na and ζCO2-Na-Cl) are computed from Eqs. 3.6-3.9. Finally, the
aqueous CO2 is calculated using Eq. 3.5.
3.2.3 SP-CSM
The SP-CSM (Spycher et al., 2003) only applies to CO2 solubility in pure
water. It does not take into account the effect of salt. The SP-CSM is based on
mutual solubility of CO2 and H2O in two coexisting phases. The composition of the compressed CO2 gas and liquid H2O phases, at equilibrium, are calculated
based on equating chemical potentials and using the Redlich-Kwong equation of
state to express departure from ideal behavior. SP-CSM is an extension of that
used by King et al. (1992) covering a much broader range of temperatures and
experimental data. A simplify version of SP-CSM is given by:
logKTT=+×−× 1.189 1.304 10−−25 5.446 10 (3.10)
11⎡⎤RT⎡⎤⎡⎤ RTb a 1 ab Vb=− −+−2 ⎢⎥ 23⎢⎥⎢⎥ (3.11) VP⎣⎦ V⎣⎦⎣⎦ PPT V PT
⎡⎤⎡⎤Vb2 aVbaVbbPV ⎡⎤++⎡⎤ ⎡⎤⎡⎤⎡⎤ lnφ =+− ln⎢⎥⎢⎥ ln ⎢⎥ +⎢⎥ ln ⎢⎥⎢⎥⎢⎥ −− ln (3.12) ⎣⎦⎣⎦Vb−− VbRPTb33 ⎣⎦ V RPTb⎣⎦ ⎣⎦⎣⎦⎣⎦ V Vb + RT
41 ⎛⎞−−()P PV0 ⎜⎟ ⎛⎞ΦP ⎝⎠RT XeCO2 = ⎜⎟ (3.13) ⎝⎠55.51K
where K is the equilibrium constant as a polynomial function of temperature at 1
0 bar (and H2O saturation pressure above 100° C), P is a reference pressure
(taken as 1 bar and H2O saturation pressure above 100° C) which is given by
0847352−−− − 3 PT=−×−×−×+10 3 10 T 6 10 T 1.1 10 T 1.0368 , (3.14)
V is average partial molar volume (m3/mol), a ( aT=×−×7.54 1074 4.13 10 ) and b (=
27.8) are parameters that represent the measure of intermolecular attraction and repulsion, respectively (dimensionless). This modified version of SP-CSM was intended for efficient calculation of mutual solubilities in numerical models used to study the feasibility of geologic sequestration of CO2 (Spycher et al., 2003).
3.2.4 Xu-CSM
Xu et al. (2004) adopted the following relationship for the Xu-CSM:
KΦPC= γ (3.15)
where K is the equilibrium constant, Φ is the gaseous CO2 fugacity coefficient
(dimensionless), P is the partial pressure (bar), γ is the aqueous CO2 activity
coefficient (dimensionless), and C (molal) is the aqueous concentration.
Equilibrium constants (K) at different temperatures were derived from the log K
values at eight different temperatures. Based on log K values at these temperatures, the coefficients of Eq. 3.15 were obtained without considering their dependency on pressure.
42 The authors correct fugacity coefficients according to temperatures and pressures using the Spycher and Reed (1988) approach and the equilibrium constant according Xu et al. (2004) model equation:
⎛⎞⎛⎞ab de P2 lnΦ=⎜⎟⎜⎟22 + + c P + + + f and, (3.16) ⎝⎠⎝⎠TT TT 2
b b logK=++++ b lnT b b T 4 5 (3.17) 123TT2 where T is absolute temperature (K), and a, b, b1-5, c, d, e, and f are dimensionless constants fitted from experimental data (Table 3.6). For high ionic strength solution, CO2 (aq) activity coefficient γ is corrected for salting out effect.
For such purpose, the authors adopted Drummond’s (1981) expression for activity coefficient for the neutral CO2 (aq) species using:
GI⎛⎞ lnγ= (C + FT + )I − (E + HT)⎜⎟ (3.18) TI1⎝⎠+ where I is ionic strength (molal), C, F G, E, and H are dimensionless constants
(Table 3.6).
Table 3.6: The values of the coefficients b1, b2, b3, b4 and b5 are obtained from the logK values at 0, 25, 60, 100, 150, 200, 250, and 300 °C. The fitted constants of a, b, c, d, e and f are based on pressure 0-500 bar and temperature 50-350°C.
3 b1 65.48 a -1.43087x10 C -1.0312 2 -3 b2 -4.255x10 b 3.598 F 1.2806x10 -2 -3 2 b3 -5.301x10 c -2.27376x10 G 2.559x10 4 b4 2.4010x10 d 3.47644 E 0.4445 6 -2 -3 b5 -1.22x10 e -1.04247x10 H -1.606x10 f 8.46271x10-6
43 3.2.5 MRK-CSM
The modified Redlich-Kwong-CSM (MRK-CSM) developed by McPherson
and Cole (2005) is applicable for the solubility of CO2 in pure water. It does not
include salting out effect. The MRK-CSM draws on the formulations of Anderson
et al. (1992), Carnahan and Starling (1972), Fenghour et al. (1998), Jacobs and
Kerrick (1981), Kerrick and Jacobs (1981), Patel and Eubank (1988), Reid et al.
(1987), Redlich and Kwong (1949), Vesovic et al. (1990), and Weir et al. (1996).
This CSM was developed to incorporate in to the TOUGH2 simulator (Pruess,
1999). The approach used in the TOUGH2 simulator is to use pressure,
temperature, and the mass fraction of the component dissolved in water as the
primary variables for those volume elements containing only one phase. If two-
phase conditions develop, the primary variables are switched to pressure,
temperature, and saturation of the gaseous phase. The MRK-CSM avoids the
primary variable switching method. Instead, it accommodates a set of four
persistent primary variables; pressure, dissolved mass fraction of CO2, saturation
of gaseous/supercritical CO2, and temperature. At equilibrium, the CO2 partitions between fluid phases were calculated according to Henry’s law.
In developing the MRK-CSM, the authors modified the attractive terms based on Kerrick and Jacobs (1981) and Weir et al. (1996) to the range of temperatures and pressures of typical sedimentary basins. The MRK-CSM is given as follows:
RT PZ= (3.19) ν
44 23 bb⎛⎞⎛⎞ b (1++⎜⎟⎜⎟ − ) 23 44νν 4 ν c(T)+ d(T)/ν+ e(T)/ ν + f(T)/ ν Z =−⎝⎠⎝⎠ (3.20) b 3/2 (1− ) 3 RT (ν+ b) 4ν
where,
2 c(T) = c1+c2T+c3T (3.21)
2 d(T) = d1+d2T+d3T (3.22)
2 e(T) = e1+e2T+e3T (3.23)
2 f(T) = f1+f2T+f3T (3.24)
23 bb⎛⎞ ⎛⎞ b (8−+ 9⎜⎟ 3 ⎜⎟ ) 44νν 4 ν c(T)⎛⎞ c(T) ν lnΦ=⎝⎠ ⎝⎠ − ln Z − + ln − b 3/2⎜⎟ 3/2 (1− ) 3 RT (ν+ b)⎝⎠ bRT ν+ b 4ν d(T) d(T)⎛⎞ d(T)ν+ b 3/2−+ 3/2⎜⎟ 2 3/2 ln − νν+νRT ( b) b RT⎝⎠ b RT ν e(T) e(T) e(T)⎛ e(T)ν+ b ⎞ (3.25) 23/2−+− 23/22 3/233/2⎜⎟ln − νν+νRT ( b) 2b RT b ν RT⎝ b RT ν⎠ f(T) f(T) f(T) f(T)⎛⎞ f(T)ν+ b 3 3/2−+ 3 3/2 2 2 3/2 −+ 3 3/2⎜⎟ 4 3/2 ln νν+νRT ( b) 3b RT 2b ν RT b ν RT⎝⎠ b RT ν
⎛⎞∞ Pvpl V XCO=Φ+−−− exp⎜⎟ ln ln P ln H P P vpl (3.26) 2 ⎜⎟() ⎝⎠RT where P is pressure (bar), T is temperature (K), R is universal gas constant, Z is compressibility factor, ν is molar volume (m3/mol), b (= 5.8x10-5 m3/mol) is co- volume, c, d, e and f are dimensionless second-order polynomial functions of temperature with empirically-derived coefficients (Table 3.7), Φ is fugacity
Pvpl coefficient (dimensionless), XCO2 is mole fraction, H is Henry’s coefficient (Pa) for the gaseous solute in the volatile liquid solvent (Reid et al., 1987), V∞ is the
45 average partial molar volume of the gaseous solute at infinite dilution in the
3 volatile liquid solvent (m /mol) and Pvp1 is vapor pressure of pure water (Pa).
Table 3.7: Second-order polynomial functions temperature dependent with empirically derived coefficients (dimensionless). i 1 2 3 c 2.39534x106 -4.55309x10-2 3.65168x10-5 d -4.09844x10-3 1.23158x10-5 -8.99791x10-9 e 2.89224 x10-7 -8.02594 x10- 7.30975 x10-13 f -6.43556x10-12 2.01284x10-14 -2.17304x10-17
3.3 Modeling of CO2 solubility
According to the U.S. Brine Wells Database (2003) compiled by DOE/NETL,
67 percent of sedimentary formations suitable for CO2 sequestration exceed 1
molal of ionic strength. Nearly 35 percent of formations have chloride
concentrations that exceed 2 molal. Most formations in the high CO2-emitting
states of the Appalachian Basin area, including the Rose Run Sandstone of
Ohio, have chloride concentrations that exceed 5 molal ionic strength. Therefore,
models involving interaction between aqueous fluid and rock matrix have to
adopt a versatile CSM to address adequately the issue of CO2 sequestration in
saline formations via solubility and mineral trapping mechanisms. To use a CSM
that does not include salting out effect would overestimate the uptake of CO2 through solubility and eventually mineral trapping.
Based on mean surface temperature of 288 to 298° K , a geothermal gradient of 288 to 303° K/km, and a hydrostatic pressure gradient of 100 to 160 bar/km, the window of temperature and pressure in typical sedimentary formations that
46 are suitable for CO2 sequestration is 303-348° K and 70-200 bar, respectively.
Experimental data were compiled within these ranges of temperatures and pressures in order to evaluate the performance of the different CSM. In addition, the salinity of the experimental data ranges from 0 up to 6 molal of NaCl solution,
which is typical, at least in terms of ionic strength, of most brine formations in US.
Most of the experimental data on CO2 solubility in aqueous solution were conducted in the presence of NaCl. Experimental data and CSM that depend on aqueous species other than NaCl are rare. Because of limited data availability, a model directly fit to experimental measurements is possible only for CO2-NaCl-
H2O system (Duan and Sun, 2003). In this section, the five CSM described above
are compared with each other and also with the Geochemists Workbench. The
Geochemist's Workbench® (GWB) (Bethke, 1998) is a chemical-reactor type,
module-based software program that simulates chemical reactions under both
equilibrium and kinetic reactions, calculates stability diagrams and equilibrium
states of natural waters, and traces reaction processes. GWB does not include
CSM that specifically addresses the solubility of CO2 as a function of
temperature, pressure and salinity. However, the effect of salinity on the solubility
of CO2 can be simulated by re-assigning the aqueous species CO2 (aq) an ionic
size of -0.5 in the thermodynamic database. The fugacity of CO2 needs to be
defined explicitly by the user. The effect of salting-out was modeled by adjusting
the activity coefficient of CO2 using a polynomial data fit developed by Helgeson
(1969) which are functions of temperature and ionic strength of the solution. The
47 activity coefficients of neutral species (like CO2(aq)) is calculated from ionic strength using an empirical relationship (Helgeson,1969; Bethke, 1996):
23 logγ=0 aI + bI + cI (3.27) where, a, b and c are polynomial coefficients that vary with temperatures as shown in Table 3.8 and I is ionic strength in molal. These values are coded in the
GWB to calculate the salt-out effect at higher ionic strength (Bethke, 1996).
In the following sections, the comparison with experimental data is first discussed followed by an intermodel comparison.
Table 3.8: Polynomial coefficients as a function of temperature for Eq. 3.27.
Temperature in °C a b c 25 0.1127 -0.01049 1.545x10-3 100 0.08018 -0.001503 0.5009x10-3 200 0.09892 -0.01040 1.386 x10-3 300 0.1967 -0.01809 -2.497x10-3
3.4 Comparison with Experimental Data
Models for the solubility of CO2 in pure water are compared with experimental data from Wiebe and Gaddy (1939 and 1941) and King et al. (1992) as a function of temperature at 76 bar (Fig. 3.1) and at 152 bar (Fig. 3.2). The
MRK-CSM and SP-CSM accurately predict the experimental data within temperature range of 300-325 K and DS-CSM accurately predicts solubility across the entire temperature range (Fig. 3.1). Across this same temperature range, PG-CSM and Xu-CSM tend to overestimate and underestimate the
48 solubility of CO2, respectively. At temperatures above 310° K, MRK-CSM underestimates the solubility of CO2 in pure water. At pressure of 152 bar, DS
and SP-CSM are well correlated with the experimental data (Fig. 3.2). The
solubility of CO2 is overestimated by as much as 32 percent using PG-CSM, 15
percent using Xu-CSM and underestimated by as much as 15 percent using
MRK-CSM.
The comparison of CSM with experimental data from Rumpf et al. (1994) and
Drummond (1981) in 6 and 4 molal of NaCl solution (Fig. 3.3) and (Fig. 3.4),
respectively, as a function of pressure at 313° K show the predictability power of
the CSM on CO2 solubility. DS-CSM accurately predicts the solubility of CO2 on
the range of pressure. PG-CSM consistently underestimates the solubility of CO2
by up to 10 percent and Xu-CSM overestimates it by up to 30 percent (Fig. 3.3).
SP-CSM and MRK-CSM overestimate the solubility of CO2 by as much as 65
percent at the lower end of the temperature. At lower ionic strength (4 molal),
DS-CSM predicts the experimental data very well as compare to the rest of CO2 solubility model (Fig. 3.4). PG-CSM underestimates the solubility of CO2 by up to
8 percent and Xu-CSM overestimate it by up to 45 percent. Both MRK and SP-
CSM overestimate the solubility of CO2 by more than 60 percent.
49 1.6
Experiment 1.4 Xu-EOS SP-EOS 1.2 DS_EOS MRK-EOS (molal) 2 PG- EOS 1
0.8 Dissolved CO 0.6
0.4 300 310 320 330 340 350 360 370 380
Temperature (K) Figure 3.1. Comparison of CSM with experimental data (Wiebe and Gaddy, 1939 and 1941, and King et al., 1992) as a function of temperature (P = 76 bars) in pure water.
2.4 Experiment 2.2 Xu-EOS 2 SP-EOS 1.8 DS_EOS
(molal) MRK-EOS 2 1.6 PG- EOS 1.4 1.2 1 Dissolved CO Dissolved 0.8 0.6 300 310 320 330 340 350 360 370 380 Temperature (K)
Figure 3.2. Comparison of CSM with experimental data (Wiebe and Gaddy, 1939 and 1941, and King et al., 1992) as a function of temperature (P = 152 bars) in pure water.
50
1.4 Xu-EOS Exp-Rumpf et al. (1994) 1.2 DS-EOS MRK-EOS 1 SP-EOS PG- EOS
(molal) Exp-Drummond (1981)
2 0.8
0.6
0.4 Dissolved CO
0.2
0 10 20 30 40 50 60 70 80 Pressure (bar)
Figure 3.3. Comparison of CSM with experimental data (6 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of pressure (at T = 313 K). CSM that do not include salting-out effect are shown by cross.
Experimental data compiled from Rumpf et al. (1994) and Drummond (1981) for CO2 solubility in 6 and 4 molal of aqueous solution were compared with results predicted by the different solubility models as a function of temperature at
70 bar (Fig. 3.5 and 3.6). The DS-CSM, PG-CSM and Xu-CSM correct for the effect of salinity on CO2 solubility whereas the SP-CSM and MRK-CSM do not. In
6 molal of NaCl solution, DS-CSM accurately predicts the experimental data with marginal error whereas PG-CSM consistently underestimates the solubility of
51
1.2 Xu-EOS Exp-Rumpf et al. (1994) DS-EOS 1 MRK-EOS SP-EOS PG- EOS 0.8 Exp-Drummond (1981)
(molal) 2 0.6
0.4 Dissolved CO Dissolved
0.2
0 10 20 30 40 50 60 70 80
Pressure (bar) Figure 3.4. Comparison of CSM with experimental data (4 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of pressure (at T = 313 K). CSM that do not include salting-out effect are shown by cross.
1.6 Xu-EOS 1.4 Exp-Rumpf et al. (1994) DS-EOS 1.2 MRK-EOS SP-EOS 1 PG- EOS (molal)
2 Exp-Drummond (1981) 0.8
0.6
Dissolved CO Dissolved 0.4
0.2
0 300 320 340 360 380 400 420 440
Temperature (K)
Figure 3.5. Comparison of CSM with experimental data (6 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of temperature (at P = 70 bar). CSM that do not include salting-out effect are shown by cross.
52 1.6 Xu-EOS 1.4 Exp-Rumpf et al. (1994) DS-EOS 1.2 MRK-EOS 1 SP-EOS (molal)
2 PG- EOS 0.8 Exp-Drummond (1981)
0.6 Dissolved CO Dissolved 0.4
0.2
0 300 320 340 360 380 400 420 440
Temperature (K)
Figure 3.6. Comparison of CSM with experimental data (4 molal of NaCl solution) from Rumpf et al. (1994) and Drummond (1981) as a function of temperature (at P = 70 bar). CSM that do not include salting-out effect are shown by cross.
CO2 by up to 10 percent and Xu-CSM overestimate it by up to 30 percent (Fig.
3.5). SP-CSM and MRK-CSM overestimate the solubility of CO2 by as much as
65 percent at the lower end of the temperature. At 4 molal of salt, DS-CSM predicts the experimental data very well as compared to the rest of CSM except the last two data points at higher temperature (Fig. 3.6). PG-CSM underestimates the solubility of CO2 by up to 8 percent and Xu-CSM overestimate it by up to 40 percent (Fig. 3.6).
3.5 Intermodel Comparison of CO2 Solubility Model
Figure 3.7 shows CO2 solubilities in aqueous solutions ranging from pure water to 7 molal ionic strength based on the DS-CSM at 160 bar. DS-CSM is strongly dependent on the type of salt dominant. Figure 3.7 was modeled based
53 on NaCl as the dominant salt and solved using Eq. 3.5. The effect of salinity on
CO2 solubility can be substantial, especially at the high ionic strength typical of the Rose Run Sandstone and other deep saline aquifers in Appalachian basin.
As Figure 3.7 shows, at 6 molal, the solubility of CO2 is overestimated by 60-70 percent if the salting out effect is ignored.
2 H2O 1.8 Sal 1 m 3 m 1.6 5 m 1.4 7 m
(molal) 2 1.2 1
0.8 Dissolved CO Dissolved 0.6
0.4
0.2 280 300 320 340 360 380 400 420 440 460
Temperature (K)
Figure 3.7. Solubility of CO2 in a range of salinity as a function of temperature (P = 160 bar) based on Duan and Sun (2003) CO2 solubility model (DS-CSM).
Figure 3.8 compares the solubility of CO2 at 6 molal of salt and 150 bar pressure as a function of temperature for the five CSM and GWB. The DS-CSM,
PG-CSM and Xu-CSM correct for the effect of salinity on CO2 solubility whereas the SP-CSM and MRK-CSM do not. As discussed above the ionic size was changed in the GWB modeling to correct for the salting out effect. The large
54 differences in CO2 solubility predicted by these CSM reflect, to some extent,
exclusion of the salting out effect. Specifically, the difference between DS-CSM, which includes the salting out effect, and MRK-CSM, which does not, is more than 45 percent. The difference between DS-CSM and SP-CSM, which also does not include the salting out effect, is more than 55 percent over the entire temperature range (Fig. 3.8). The smallest CO2 solubilities are given by DS-CSM
and PG-CSM. The largest difference in CO2 solubility is between SP-CSM and
GWB (more than 70 percent). Two CSM (MRK-CSM, SP-CSM) and GWB show
substantial changes in CO2 solubility with temperature in the temperature range
of saline aquifers. All CSM show little variability with temperature of CO2 solubility above 320° K.
Figure 3.9 compares the solubility of CO2 at 6 molal of salt and 300° K as a
function of pressure for the five CSM and GWB. All CSM and GWB reveal two
distinct slopes - a large positive slope, in general, at pressures below 60 bar and
a nearly flat trend at pressures above 60 bar (Fig. 3.9). All models indicate same
pattern of CO2 solubility as a function of pressure but portrays a large magnitude
difference between models that utilized salting out effect (DS, PG, Xu-CSM, and
GWB) and those modeled in pure water (MRK and SP-CSM). The Sp-CSM and
MRK-CSM overestimate the solubility of CO2 above 60 bar by more than 70 percent as compare to DS and PG-CSM (Fig. 3.9). The MRK and SP-CSM are closely correlated for the entire range of pressure. At high pressure, DS and PG-
CSM tend to close the gap whereas Xu-CSM tends to overestimate the solubility of CO2 by almost a factor of 2.5 as compare to DS and PG-CSM. The
55 comparisons of intra CSM as a function of pressure at the given conditions
suggest that pressure above 60 bar does not play a significant role in the
solubility of CO2 except PG-CSM, which has a steeper positive slope. GWB for range of pressure is also included in Figure 3.9 and has a similar pattern and magnitude as that of Xu-CSM.
MRK-EOS 1.6 DS-EOS Xu-EOS 1.4 SP-EOS PG- EOS 1.2 GWB 1.0 (molal) 2 0.8
0.6
CO Dissolved 0.4
0.2
0.0 300 320 340 360 380 400
Temperature (K)
Figure 3.8. Solubility of CO2 as a function of temperature (at P = 150 bar and 6 molal of ionic strength) based on CO2 solubility models. CSM that do not include salting-out effect are shown by cross. The inserted box indicates the typical temperature range of saline aquifers appropriate for CO2 injection.
56
MRK-EOS 1.8 DS-EOS Xu-EOS SP-EOS 1.6 PG- EOS GWB 1.4 1.2 (molal)
2 1
0.8
0.6 Dissolved CO 0.4
0.2
0 20 40 60 80 100 120 140 160 180 200 220 Pressure (bar)
Figure 3.9. Solubility of CO2 as a function of pressure (at T = 300 K and 6 molal of ionic strength) based on CO2 solubility models. CSM that do not include salting-out effect are shown by cross. The inserted box indicates the typical pore pressure range of saline aquifers appropriate for CO2 injection.
3.6 Discussion
In the United States, brackish or saline water are found in deep aquifers.
The US Environmental Protection Agency (EPA) excludes an aquifer with salinity
that exceeds 0.4 molal in total dissolved solids (TDS) as underground sources of
drinking water. Hence, they are logical targets for the eventual disposal of CO2.
The successful prediction of the short and long-term behavior of CO2
57 sequestration in a geological formation and estimation of its storage capacity depends on the utilization of the appropriate model for CO2 solubility that takes
into account the effect of higher salinity. An CSM that was developed based on
experimental work of CO2 solubility in brine solution has the advantage of
accurately predicting the dissolution of CO2. Salting out effect is determinant
factor in the assessment of CO2 sequestration in saline aquifer and ignoring its
impact on the overall trapping mechanism may lead to incorrect storage capacity
estimation.
The CO2 solubility model developed by Duan and Sun (2003), McPherson
and Cole (2005), Pruess and Garcia (2002), Spycher et al. (2003), and Xu et al.
(2004) have both advantages and disadvantages in evaluating the feasibility of
CO2 sequestration in geological formation. SP-CSM accurately predicts the
solubility of CO2 in pure water but overestimates the solubility of CO2 in aqueous
solution. Spycher et al. (2003) suggest that the CSM can be extended to
aqueous solution by including a correction for salting out effect. The prediction of
CO2 solubility in pure water demonstrates the capability of SP-CSM in effectively
correcting the departure from ideal solution within the range of injection pressure
and temperature. Adding a correction factor that accounts for the salting out effect would produce an CSM that can accurately predict the solubility of CO2 in brine.
MRK-CSM (McPherson and Cole, 2005) was also developed to predict solubility of CO2 only in pure water. However, most reservoirs below depths of a
few hundred meters contain saline brines. Increasing salinity of water decreases
58 the solubility of CO2. For example, for typical saline formation in US, the solubility
of CO2 in water containing 23 percent by weight at supercritical temperature and
pressure is approximately 55 percent of the solubility in pure water (Figure 3.9).
Applying MRK-CSM, without correcting the salting out effect would overestimate
the storage capacity of saline aquifer through both solubility and mineral trapping
mechanism.
The CSM developed by Pruess and Garcia (2002) overestimate the solubility
of CO2 by up to 25 percent in pure water, and in brine solution it is underestimate by up to 12 percent. Reacting CO2 and brine based on PG-CSM with formation rock over a range of fluid and rock types of variable chemical and mineralogical composition will affect the partitioning of CO2 trapped among the three sequestrating mechanism. If PG-CSM is included in any reactive transport code that uses to simulate CO2 sequestration potential in saline aquifer, then the
dissolved CO2 will be under represented by at least 12 percent.
The Xu-CSM developed by Xu et al. (2004) is valid for a typical CO2 injection
temperature and pressure. The solubility of CO2 in pure water at lower pressure
(76 bar) and higher pressure (152 bar) seems to depart from the experimental
data. The solubility of CO2 is underestimated at lower pressure and
overestimated at higher pressure. Xu-CSM has the capability to model solubility
of CO2 in brine solution. However, it consistently overestimates the solubility of
CO2 on in the entire range of temperature, pressure and salinity. Care should be taken when interpreting the results obtained using Xu-CSM in reactive transport
modeling. Applying the CO2 solubility models to case specific parameters and
59 conditions and a critical evaluation of modeling results may provide useful insight into sequestration mechanism and controlling geochemical conditions and
parameters.
The DS-CSM that has been developed based on the equation of state of
Duan et al. (1992b) and the theory of Pitzer (1973) by Duan and Sun (2003) is by
far the most accurate CO2 solubility model. It accurately modeled CO2 solubility
in pure water and aqueous solution for a range of temperature, pressure and
salinity (Figures 3.5-3.9). This model is extended to predict CO2 solubility in more
complex brines such as seawater with remarkable accuracy.
Equations of state, CO2 solubility models and mixing rules with various
degrees of complexity and accuracy have been presented in the literature to
calculate properties of CO2-H2O-Salt mixtures. Even though DS-CSM seems to
be a good choice in solubility modeling, its complex form may add a significant
burden for implementation into already computationally intensive fluid flow,
transport and chemical reaction model, as it requires an iterative solution. In
general, all CO2 solubility models developed by different scholars may suggest that large errors in CO2 sequestration capacity estimation are likely when
modeling the outcome of CO2 injection into deep saline aquifer. Because of the
chemical complexity of injection induced water-CO2-rock interaction processes,
an extensive theoretical framework is required to support modeling of such
system. Several features of the modeling results against the experimental data
serve to highlight the need for a more critical evaluation both of the fugacity coefficient correction and salting out effect correction in order to utilize the CO2
60 solubility model with confidence in predicting the solubility of CO2 in saline
formation. A unifying single model that incorporates and captures the complexity
of CO2 solubility in saline aquifer is very crucial in predicting with confidence the
effect of injecting CO2 into saline aquifer.
3.7 Concluding Remark
Effective sequestration of CO2 and assurance of public safety from escaping
CO2 require accurate prediction of long-term behavior of CO2 in the subsurface.
The CSM model developed by various authors are a step forward in developing a complete and interconnected models that has the predictability power of CO2
solubility in saline formation. They are applicable for a range of temperatures,
pressures and salinities. The models incorporate various fugacity coefficients, interacting parameters, correction for non-ideality behavior of the mixtures (H2O-
CO2-Salt) and salting out effect. MRK and SP-CSM are developed for CO2 solubility in pure water. Modeling CO2 sequestration in saline aquifer and
estimating the storage capacity using these CO2 solubility models would gravely
overestimate the short and long term partitioning of CO2 uptake among free CO2, dissolved CO2, and trapped carbon dioxide in carbonate mineral sequestering
mechanisms. The DS, Xu, and PG-CSM are applicable in both pure water and
brine solutions. Xu-CSM tends to overestimate the solubility of CO2 by a large
margin whereas PG-CSM underestimates the CO2 solubility in brine solution by a margin not more than 12%. Both CO2 solubility models overestimate the solubility
of CO2 in pure water at higher pressures. In comparison with all other CO2
61 solubility model, DS-CSM is accurately modeled CO2 solubility in pure water and
aqueous solution for range of temperatures, pressures and salinities. To predict
the short and long-term effect of CO2 sequestration in saline aquifer using reactive transport modeling, it is very crucial to use an appropriate CO2 solubility model that has the potential to represent the solubility of CO2 in brine at the injection pressure and temperature.
In the next chapter, an introduction to the geochemical reaction both under non-flow and flow conditions will be given. In the batch reaction modeling, the
DS-CSM is used to correct the fugacity coefficient for the pertinent pressure and temperature. The result of this approach is covered in chapters five and six. The
DS, Xu and MRK-CSM discussed in this chapter are used in the newly developed 1-D reactive transport. This is given in chapters seven, eight and nine.
62
Chapter 4
BATCH GEOCHEMICAL MODELING
4.1 Introduction
Fundamental issues of long-term geological carbon sequestration are very
important before embarking on a full scale pumping of supercritical carbon dioxide gas into geological formation. Experimental analyses of the long-term behaviors of CO2 injected into saline aquifer are not possible with relatively short-
term laboratory experiments. Comprehensive numerical models that incorporate
the underlying physics of CO 2 flow through porous media and the chemistry of
reactions are necessary to study effectively the fate of injected CO2 on longer
time scales. Robust numerical simulators of multiphase flow and multi- component transport coupled with geochemistry are needed to predict CO2 migration and fate, and formation responses to injection. These models must accurately represent processes over a wide range of spatial and temporal scales, and successfully integrate short-term injection with longer-term transport and reaction.
63 Several models are available to calculate batch geochemical reactions
among CO2, brine, and minerals at elevated temperatures and pressures. These
models do not consider the dynamics of flow and reactive transport. In this study,
a batch reaction geochemical model (based on Zerai et al. 2005 submitted to
Applied Geochemistry accepted bending revision) and a newly developed one-
dimensional reactive transport model were used to assess the fate of carbon
dioxide injected into geologic formations. This chapter introduces the batch
geochemical code as well as the compositional and geochemical parameters used in modeling. Results of the batch geochemical modeling are presented in chapter 5.
Batch geochemical modeling was conducted using the commercially available, flexible, and multipurpose geochemical software; Geochemist’s
Workbench (GWB) (Bethke, 1998). Simulations of water-rock-gas interactions under no flow conditions are important for identifying the reactions that are most important for trapping CO2 in the Rose Run Sandstone and for identifying the
parameters that have the greatest influence on the quantity and form of
sequestration. Detailed descriptions of the modeling are given below.
4.2 Geochemists’ Workbench
Geochemist’s Workbench (GWB™) version 3.2.2 was used for equilibrium, path
of reaction, and kinetic modeling of CO2-brine-mineral reactions. It is a chemical-
reactor type model that does not take into account the flow of brine through the
aquifer. Modeling the chemical reactions under no-flow conditions provides insight
into the important variables that control reaction progress and reaction products
64 such as the effect of fugacity, brine-to-rock ratio, type of mineral assemblage, and
rate constants on CO2 immobilization through carbonate precipitation can be
assessed.
Brine mass was set at 0.4 kg H2O plus total dissolved solids determined by
the brine chemistry. In order to attain a porosity of 10-15% of the Rose Run
Sandstone, the brine-to-rock ratio was set at 1:25 except in the case where the
Rose Run was compared with the other three brines (Alberta, Mount Simon, and
Clinton) for which the brine-to-rock ratio was 1:10. The temperature of the system
was assumed isothermal and set at 54° C for kinetic and path of reaction
simulations. In case of the equilibrium reaction, the temperature was set at 35° C,
54° C and 75° C in order to investigate the effect of temperature on the
precipitation/dissolution of carbonate rocks. The model parameters and
simulation conditions are given in Table 4.1.
The CO2-brine-rock reaction was modeled using three different simulation
methods. The first is equilibrium modeling. Equilibrium modeling of CO2-brine-
mineral reactions for typical mineral assemblages was applied to investigate the
impact of temperature, pressure, mineralogy, brine composition, and CO2 fugacity on mineral dissolution and precipitation, the amount of CO2 sequestered,
and the form of sequestration. The equilibrium reaction model is based on the
thermodynamic data for the minerals involved in the reaction. The initial
equilibrium state were constraints in many ways, depending on the nature of the
problem, but the number of pieces of information required is firmly set by the
65
Table 4.1 Values of model parameters used in the calculations.
Parameters Description Temperature 35° C, 54°C and 75° C
CO2 Dissolved in the brine Activity Coefficient Az2 I log γ =−i +BI& (B-Dot Equation) i 1 + aB&i I Reactive surface area 10 cm2/g Rate constants Table 4.4 Porosity 12% Brine – Rock ratio 1:25 and 1:10 Total Rock Mass 10 kg Fugacity Coefficient Based on Duan and Sun CSM
CO2 fugacity Atmospheric pressure to 100 bar
laws of thermodynamics. The main constraints are the mass of solvent water, the amounts of minerals in the system, the fugacities of any gases at known partial
pressure, the amount of any component dissolved in the fluid, and the activities
of species such as H+ as would be determined by pH measurement.
Second, path of reaction modeling was utilized to evaluate intermediate
products. Path of reaction modeling traces the progress of reactions by
progressively more minerals reacted with the CO2 enriched brine, and it is
important for investigating the precipitation and dissolution of phases as the
reaction progresses and it has important implications on secondary porosity.
Finally, kinetic modeling considers the rates of reactions based on
appropriate rate constants. It permits computation of the time the system needs
66 to start consuming and trapping CO2 precipitated mineral phases, as well as the
time it takes the system to approach ‘steady state’ or dynamic equilibrium.
Geochemist’s Workbench has an internal thermodynamic database and requires
user input of kinetic rate data. The following rate equation was adopted in the
modeling (Lasaga 1995):
dni −Ea Q ϒ=ir =KAmin exp( )[ − 1] (4.1) dt RT K%
th 2 where ϒ rate of i species, Kr is the rate constant (mol/m s), Amin is the reactive
2 surface area (m /g), Ea is the activation energy (J/mol), R is the gas constant
(J/K·mol), T is absolute temperature in (K), Q is the activity product, and K% is the
equilibrium constant. The model parameters that are used in GWB are given below.
4.3 Rose Run Rock Assemblage
Reactions with three types of mineral assemblages were investigated: a
carbonate assemblage composed of calcite, dolomite and siderite, which represents
the carbonate layers in the aquifer; a sandstone assemblage composed of quartz, K-
feldspar, kaolinite, albite, annite, and siderite, which represents the sandstone
layers; and a mixed assemblage, which represents the aquifer as a whole. Annite
was used as a proxy for glauconite due to the lack of thermodynamic and kinetic data for glauconite. The abundance of each mineral phase was compiled from
Janssens (1973) and Riley et al. (1993) and recast to 10 kg of rock (Table 4.2). The
67 simulation has been carried out based on the assumption of lateral homogeneous
rock assemblages.
Table 4.2 The three main groups of mineral assemblages and each mineral mass recast based on a total of 10 kg rock mass.
Carbonate wt. % Sandstone wt. % Mixed wt. % Dolomite 60 Quartz 83 Quartz 70 Calcite 39 K-feldspar 10 Dolomite 13.8 Siderite 1 Kaolinite 3 Calcite 8 Albite 2 K-feldspar 5 Annite 1 Annite 1 Siderite 1 Albite 1 Kaolinite 1 Siderite 0.2
4.4 Brine Chemistry
Four brine compositions were considered. The brine composition of the Rose
Run Sandstone is based on a sample from Coshocton County, Ohio (Breen et al.,
1985). In addition, brine compositions from the Clinton Formation in Ohio, the Grand
Rapids Formation in the Alberta Basin, and Mt. Simon Sandstone were used for
comparison with the Rose Run brine (Table 4.3).
Simulation begins by computing the initial equilibrium between brine and a given
fCO2. The initial brine composition of the Rose Run without added CO2 is reported in
Table 4.3. Carbon dioxide of a given fugacity is dissolved into and equilibrated with
the brine before adding mineral reactants. Once equilibrium is achieved between the
68 CO2 and the brine, reactions with mineral assemblages were simulated using equilibrium, path-of-reaction, and kinetic modeling.
Table 4.3 Composition of Rose Run, Clinton, Grand Rapids, and Mt Simon brines compiled from the literature.
Brine Rose Run(a) Clinton(b) Mt. Simon(c) Grand Rapids(d) Species (mg/kg) (mg/kg) (mg/kg) (mg/kg) Na+ 60122 67000 32000 26539 K+ 3354 850 1060 636 Ca2+ 37600 23200 12400 2737 Mg2+ 5880.6 1840 2190 533 HCO3- 122 200 71 182 Cl- 191203 160400 78700 47549 SO42- 326.4 523 1180 337 SiO2(aq) 3 1 5.00E-07 0.00046 Al3+ 2.16 1 5.00E-07 0.461 Fe2+ 140 5 1.54 4.6E-05 Sr2+ 455.52 753 236 - Br- 3760 - 362 - Rb+ - 51 - - pH 6.4 6.5 6.7 7.2 TDS 277,571 250,000 150,000 90,000 (a) Breen et al., 1985; (b) Lowry et al., 1988; (c) Ohio Geological Survey, 1990; (d) Gunter et al., 2000
4.5 Rate Constants
Rate constants for the kinetic reactions were compiled from published
literatures based on laboratory experiments; they can be several orders of
magnitude greater than rates of weathering measured in the field (Lasaga, 1995).
Much of the difficulty arises in kinetic rate of reaction steamed from the measured
69 rate constant that reflects the dominant reaction mechanism in the experiment from which the constant was derived. The precipitated minerals rate constants are assumed equal to the dissolution rate constants. The dissolution rate constants used in this work are given in Table 4.4.
Table 4.4 Rate constants for silicate and carbonate minerals compiled from literature and used in the computer simulations.
Rate constants 2 Mineral log Kr (mol/m s) References Albite -11 Sverdrup, 1990 Annite -10.5 Acker and Bricker, 1992 Calcite -5.8 Plummer et al., 1978 Dolomite -6.7 Busenberg and Plummer, 1982 Kaolinite -11.4 Sverdrup, 1990 K-feldspar -10.9 Helgeson et al., 1984 Siderite -6.7 Assumed Dolomite rate Quartz -12 Rimstidt and Barnes, 1980
4.6 Fugacity of CO2
The initial fugacity of CO2 (fCO2) was set at 100 bar for the kinetic modeling; it
was varied in 20 bar increments from 100 to 20 bar for the equilibrium modeling,
with a minimum fugacity set at 8x10-2 bar. The maximum fugacity was calculated
based on the maximum injection pressure that avoids rock fracturing, which is
85% of lithostatic pressure (Gunter et al., 2000). This is the maximum safe
injection pressure, above which pressure-induced rock fracturing poses a risk.
The lithostatic pressure in the Rose Run Sandstone ranges from 200 to 260 bar;
70 an injection pressure of 222 bar was assumed for an injection depth of 1000 m.
GWB requires that carbon dioxide pressure be directly input as fugacity. The
fugacity (fCO2) was calculated using CO2 solubility model developed by Duan and
Sun (2003). The minimum fugacity of 0.08 bar is the fugacity of dissolved bicarbonate reported in the brine, without injection of CO2. Initial fCO2 values between 100 and 8x10-2 bar are intended as simple representations of the
decrease in PCO2 with distance from the injection site (Figure 4.1). Low fCO2
values represent locations far from the injection site or long after injection has
ceased where the CO2 is present primarily as aqueous phases. Thus, the
investigation of how geochemical reactions vary as a function of initial fCO2 can approximate the sequence of reactions and the reaction progress as fCO2 decreases with increasing distance from the injection site and over time.
Because the program considers only the CO2 that is dissolved in the brine for a
given fugacity this approach does not take into account the decrease in the
saturation of the pores by free CO2 with increasing distance from the injection
site.
4.7 Reactive Surface Area
Spherical geometry of the minerals grains was assumed. The grain size of the
Rose Run Sandstone is in the range of fine to medium sand (Riley et al., 1993).
An average grain diameter of 200 micron was assumed. The estimated reactive
surface area was 1000 cm2/g. This was calculated based on surface roughness,
which is defined as the ratio of the true surface area to the equivalent geometric
71 surface area. A surface roughness of 10 was used (White and Peterson, 1990).
Interaction with the minerals is generally expected to occur only at selective sites
Injected CO2 (g)
Carbonate Rock
Sandstone Rock
Mixed Rock Decrease in CO2
Figure 4.1 Schematic diagram showing the decrease in CO2 pressure as a function of radial distance.
of the mineral surface and the actual reactive surface area can be between 1 to 3
orders of magnitude less than the surface roughness-based surface area
(Lasaga, 1995). The difference is attributed to the fact that only part of the
mineral surface is involved in the reaction due to coating or armoring, small
exposure area that is in contact with the brine, and channeling of the reactive
fluid flow. A conservative reactive surface area was set at 10 cm2/g for all mineral
assemblages.
72 4.8 Activity Coefficients
Activity coefficients were calculated using the B-dot equation (an extension of the
Debye Huckle equation) (Pitzer and Brewer, 1961; Table 4.1). The virial method
(Pitzer equations) is better suited to high ionic strength solutions such as the brine under consideration (Pitzer and Brewer, 1961), but GWB’s application of the Pitzer equations does not take into account the distribution of species in solution, only recognizes free ions as if each salt has fully dissociated in solution, and doesn’t take
3+ into consideration SiO2 and Al species. Those assumptions preclude use with minerals like albite, quartz, and feldspar. By using B-dot instead of Pitzer equations, the activity coefficients for the aqueous species are overestimated, which results in overestimates of the mass of minerals precipitates and dissolved.
4.9 Thermodynamic Database
The Geochemist’s Workbench has rich database that is capable to simulate a given chemical reaction. The database is compiled from a large database of the work done by Lawrence Livermore National Laboratory. In the modeling, the equilibrium constants are tabulated based on eight principal temperatures for any given reaction. For temperatures outside of the principal values, a polynomial fit was used to calculate the equilibrium constants. GWB provides default datasets where the equilibrium constants are only valid at 1 bar from 0-100°C and steam saturation pressures thereafter. These geochemical models do not have a built in method to adjust the values of the equilibrium constants both as a function of temperature and pressure explicitly.
73
Chapter 5
RESULTS AND DISCUSSIONS ON GEOCHEMICAL MODELING
5.1 Computer Simulation – GWB
The result of the complex water-brine-gas reaction is subdivided into
equilibrium reaction, path of reaction and kinetic reaction. As discussed in
chapter 3, the thermodynamic database of the GWB that contains the aqueous
species CO2 (aq) was re-assigned an ionic size of -0.5 to correct the effect of the salting-out in the solubility of CO2 (g). The results of geochemical reaction and
mineral trapping estimates in saline aqueous fluid at various temperatures and
pressures are given below. The parameters used in the modeling in each type of
reactions are given in Table 5.1.
5.1.1 Equilibrium Modeling
Figure 5.1 shows the pH and the change in dissolved concentration of CO2 (aq) as a consequence of equilibrating with the minerals plotted as a function of initial fCO2. The equilibrium pH for the carbonate mineral assemblage is low (< 5) for fCO2 of
74 Table 5.1: Parameters deployed in each type of reaction modeling.
Equilibrium Parameters Path of reaction Kinetic reaction reaction Temperature 35, 54 and 75 54 54 (°C) Fugacity (bar) 0.08, 20, 40, 20, 60, 100, 140 100 60, 80 and 100 and 300 Brine type Rose Run Rose Run Rose Run, Mt Simon, Clinton and Grand Rapids Rock type Carbonate, Mixed Carbonate, Silicate Silicate and and Mixed Mixed Simulation time 7000 (year) Brine-to-rock 1:25 1:25 1:50, 1:25, 1:17, ratio 1:12 and 1:10
20 or more, whereas the pH in the sandstone and mixed assemblages is buffered by
dissolution of silicate minerals at about 6 for all initial fCO2 (Figure 5.1a). Initially, the
brine at equilibrium with the CO2 contained 0.14, 0.28, 0.4, 0.58, and 0.7 molal of
CO2 for fugacities of 20, 40, 60, 80, and 100 bar, respectively. The change in CO2
(aq) is zero over the range of fugacities studied for equilibration with the carbonate
assemblage, whereas for equilibration with the sandstone and mixed assemblages,
for every initial fCO2 all the CO2 dissolved in the brine is consumed and trapped in the
precipitated carbonate minerals (Figure 5.1b).
75
6.5
6
5.5
pH 5
4.5
4 0 20 40 60 80 100
f CO2(g) (bar) a
0
-0.2
-0.4 (aq) (molal) 2 -0.6
-0.8 CO Delta -1 0 20406080100
f CO2(g) (bar) b Carbonate Silicate Silicate-Carbonate
Figure 5.1. Equilibrium modeling showing a) pH as a function of fCO2, and b) net CO2 consumption as a function of fCO2.
The minerals dissolved and precipitated at equilibrium include both carbonate
and silicate phases. There is a net precipitation of the carbonate minerals dolomite, siderite, and strontianite, whereas calcite is the only carbonate mineral subject to net dissolution (Table 5.2). In total, however, for the carbonate assemblage there is a net dissolution of 0.12 moles of carbonate minerals regardless of the CO2 fugacity
(Figure 5.2a). In the case of the sandstone and mixed mineral assemblages, the moles of carbonate minerals precipitated at equilibrium increases from 0.01 to 0.2 as
initial fCO2 increases from 20 to 100 bar (Figure 5.2a). In addition, for reactions with
76 the sandstone and mixed assemblages large quantities of quartz, muscovite and microcline minerals precipitate (Table 5.2). Figure 4.2b shows the breakdown of the individual carbonate minerals precipitated or dissolved for the mixed assemblage.
The mass of strontianite and dolomite that form is not affected by fCO2 but the mass of siderite that forms is proportional to fCO2.
Table 5.2: Minerals precipitated or dissolved (in mole) at equilibrium for fugacity of 100 bar. Minus sign indicates dissolution of mineral. Rock assemblage Dolomite Calcite Siderite Dawsonite Strontianite Quartz Muscovite
Carbonate 0.11 -0.31 0.09 0.000004 0.002 - -
Sandstone 0.064 0 0.15 - 0.001 1.86 0.25
Mixed 0.12 -0.13 0.2 - 0.002 1.9 1.9
77
0.25 0.2 Carbonate Silicate 0.15 Silicate-Carbonate 0.1 0.05 0
Total Minerals (mole) -0.05
-0.1
-0.15 0 20 40 60 80 100
a f CO2(g) (bar)
0.25 Siderite 0.2 Dolomite Calcite 0.15 Strontianite 0.1 0.05 0
Minerals (mole) Minerals -0.05
-0.1 -0.15 -0.2 0 20406080100 b f CO2(g) (bar)
Figure 5.2. Equilibrium modeling at T = 54° C showing net precipitation/dissolution in mole per 10 kg of rocks of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage.
The effect of temperature on the dissolution and precipitation of carbonate
minerals was studied for T = 35, 54, and 75 ˚ C. Temperature has little effect on reactions with the carbonate assemblage, but for the sandstone and mixed
assemblages, the total mass of carbonate minerals precipitated increases with
decreasing temperature (compare Figures 5.2a, 5.3a and 5.4a). Figures 5.2b,
78 5.3b, and 5.4b show the breakdown of carbonate mineral precipitation and dissolution for the mixed assemblage. The biggest difference with temperature is that more siderite forms at lower fCO2 for the lower temperature simulations.
Dawsonite is not a stable mineral in the equilibrium assemblages for any initial
fCO2 at T = 35˚, 54˚ or 75˚ C (Figure 5.2b, 5.3b and 5.4b).
0.4 Carbonate Sandstone 0.3 Mix ed
0.2
0.1
0.0
-0.1
mineral (mole) Carbonate Total -0.2 0 20406080100
a fCO2 (bar)
Siderite 0.50 Dolomite Calcite 0.40 Strontianite 0.30
0.20
0.10
Mineral (mole) Mineral 0.00
-0.10
-0.20 0 20406080100
fCO2 (bar) b
Figure 5.3. Equilibrium modeling at T = 35° C showing net precipitation/dissolution in mole per 10 kg of rocks of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage.
79
0.10 Carbonate Sandstone
0.05 Mix ed
0.00
-0.05
-0.10
-0.15 TotalCarbonate mineral (mole) -0.20 0 20406080100 a fCO2 (bar)
Siderite Dolomite Calcite 0.15 Strontianite 0.10
0.05
0.00
-0.05 Mineral (mole) Mineral
-0.10 -0.15 020406080100
fCO2 (bar) b
Figure 5.4. Equilibrium modeling at T = 75° C showing net precipitation/dissolution in mole per 10 kg of rocks of carbonate minerals for a) all three rock assemblages, and b) mixed assemblage.
5.1.2 Path of Reaction
Path of reaction modeling was conducted in order to understand the evolution of the system as it approached equilibrium. The path-of-reaction modeling follows the
80 incremental dissolution and precipitation of mineral phases as the system
equilibrates to small changes in composition due to the addition or removal of
mineral reactants along the approach to final equilibrium. As the model reactions progresses small aliquots of mineral reactant are added to the system and allowed to come to equilibrium; these steps are repeated until the total amount of reactant has been added. The measure of the reaction progress is a dimensionless number, which varies from 0 (no reactive minerals added to the system - initial state) to 1 (all reactive minerals added to the system - final equilibrium state). Figure 5.5 shows the masses of dawsonite and siderite precipitated over the path of the reaction for CO2-
brine- rock reactions with the mixed assemblage and initial fCO2 ranging from 20 to
300 bar. For the initial fCO2 of 20, 60,100 and 140, dawsonite precipitated until 2.5%,
6%, 10% and 14% of mineral reactants were added to the system, respectively.
When the total reactants added to the system exceeds the above percentages for
the respective fCO2, dawsonite starts to dissolve such that the system at final equilibrium (100% of the reactants reacted) contains no dawsonite (Figure 5.5a). For fCO2 of 300 bar, maximum quantities of dawsonite precipitated correspond to 25%
minerals reacted, respectively (Figure 5.5a). At final equilibrium, more than three
quarter of the precipitated dawsonite has dissolved.
Siderite formation shows a similar pattern over the course of reaction. The
maximum amount of siderite is precipitated as the total reactants added to the
system reaches 8%, 25%, 38% and 56% at initial fCO2 of 20, 60, 100 and 140 bar,
respectively (Figure 5.5b). As the total reactants added to the system exceeds
the above percentages, siderite starts to dissolve and such that at final
81 equilibrium, the total siderite left is 110% (beside all the newly precipitated, 10% of the initial rock was also dissolved), 40%, 35% and 30% less than the maximum amount formed during the path of the reaction (Figure 5.5b). For fCO2 of
300 bar, siderite is precipitated throughout the course of reaction (Figure 5.5b).
120
100
80
60
Dawsonite (grams) 40
20
0 0 0.2 0.4 0.6 0.8 1 a Path of Reaction (dimensionless)
80
60
40
20 Siderite (grams)
0
-20 0 0.2 0.4 0.6 0.8 1 Path of Reaction (dimensionless) b 20 bar 60 100 140 300 Figure 5.5. Path of reaction showing the formation and dissolution of a) dawsonite, and b) siderite for the mixed assemblage (fCO2 ranges from 20 to 300 bar) in grams per 10 kg of rocks.
82 5.1.3 Kinetic Modeling
Kinetic modeling was performed to simulate reactions over a 7,000 years period.
Initial fCO2 was set at 100 bar and the dispersal of CO2 over time due to flow and
diffusion was not considered. Figure 5.6 shows the evolution of fCO2 (and the
corresponding aqueous CO2 concentration) and pH through time for reactions with the carbonate, sandstone and mixed assemblages. The fCO2 and the pH for the
carbonate assemblage remain the same with time because there is little
sequestration of CO2 by mineral trapping and all the CO2 in the system remained in
dissolved form (Figure 5.6a, b, c). For both the sandstone and mixed assemblages
there are rapid initial declines in fCO2 as a result of CO2-brine-mineral reactions that
trap the CO2 as carbonate minerals, such that by 100 years fCO2 drops by 15% and
by 1000 years it drops by 99%. After 1000 years, the rate of consumption of CO2
slows down and the pH levels off (Figure 5.6a, b, c)
Figure 5.7 shows the masses of individual minerals precipitated and dissolved
over time for kinetic modeling of reactions with the carbonate, sandstone, and mixed
assemblages. For the carbonate rock assemblage a small quantity of calcite
dissolves rapidly over the first few tens of years and equilibrates with the brine
according to the following reaction (Figure 5.7a):
2+ - CaCO3 + H2O + CO2 Ca + 2HCO3 (5.1)
Much of the bicarbonate formed by the dissolution of calcite is removed from the
system and trapped in the precipitated dolomite phase. In addition, a small amount
83 120
100 80
60
fCO2 (bar) 40 20
0 0 1000 2000 3000 4000 5000 6000 7000 a Time (yeras)
0.7 0.6 0.5 0.4 0.3 (aq) (molal) 2 0.2
CO 0.1 0 0 1000 2000 3000 4000 5000 6000 7000 b Time (yeras)
6.2 6.0 5.8 5.6 5.4 5.2 5.0
pH 4.8 4.6 4.4 4.2 4.0 3.8 0 1000 2000 3000 4000 5000 6000 7000 Time (year)
c Carbonate Sandstone Mix ed
Figure 5.6. Fugacity of CO2 (a), CO2 (aq) (b) and pH (c) as a function of time in kinetic model simulations.
of siderite, strontianite, and dawsonite quickly precipitates. However, as shown by
Figure 5.7a, there was no net mineral trapping of CO2 because the precipitation of
dolomite, strontianite, and dawsonite is offset by the dissolution of calcite.
84
0.2
0.1
0
-0.1 Siderite Daw s onite Calcite -0.2 Minerals (mole) Minerals Strontianite Dolomite -0.3
-0.4 0 1000 2000 3000 4000 5000 6000 7000
a (Carbonate) Time (years)
0.4
0.3
0.2
0.1
0
-0.1 (mole) Minerals -0.2
-0.3
-0.4 0 1000 2000 3000 4000 5000 6000 7000 b (Sandstone) Time (years)
0.4
0.3
0.2
0.1
(mole) Minerals 0
-0.1
-0.2 0 1000 2000 3000 4000 5000 6000 7000
Time (years) c (Mixed) Siderite Daw s onite K-feldspar Annite Albite Kaolinite Strontianite
Figure 5.7. Computer model results of mineral precipitation/dissolution in mole per 10 kg of rocks of a) carbonate, b) sandstone, and c) mixed assemblages as a function of time.
85 For both the sandstone and the mixed assemblages, there is precipitation of
siderite, dawsonite, and strontianite, and dissolution of K-feldspar, annite, albite, and
kaolinite (Figures 5.7b, c). As much as 0.15 moles of dawsonite, 0.35 moles of
siderite, and 0.01 moles of strontianite precipitate per 10 kg of rock reacted for both
the sandstone and the mixed assemblages. The formation of dawsonite and siderite can be represented by Equations 5.2 and 5.3.
NaAlSi3O8 + CO2 + H2O ↔ NaAlCO3(OH)2 + 3SiO2(aq) (5.2)
KFe3AlSi3O10(OH)2 + 7CO2 + H2O ↔
+ - 3+ 3FeCO3 + 3SiO2 + K + 4HCO3 + Al (5.3)
In addition, dissolution of K-feldspar contributes Al, which reacts with Na in the initial
brine and dissolved CO2 to form dawsonite (Johnson et al., 2001). Dissolution of
albite, K-feldspar, annite and kaolinite is driven by the decrease in pH due to
dissolved CO2. The dissolution of aluminosilicate minerals buffers the pH and
provides Na, Al, and Fe needed for precipitation of carbonate minerals. Thus
precipitated mass of both siderite and dawsonite were determined by the rate of
dissolution of annite, albite, K-feldspar and kaolinite and the presence of
aluminosilicate phases in the sandstone and mixed assemblages significantly
increases the mineral trapping potential relative to the carbonate assemblage,
provided there is sufficient time for the reactions to occur.
The amount of CO2 trapped in carbonate minerals over time is also controlled by
the brine to rock ratio of the system (Figures 5.8 and 5.9). The pH decreases and
the fCO2 increases as the brine to rock ratio increases (Figure 5.8). The pH for brine
86
6.5
6
5.5
5
pH
4.5
4
3.5 0 1000 2000 3000 4000 5000 6000 7000
a Time (year)
100
80
60 (bar)
CO2 40 f
20
0 0 1000 2000 3000 4000 5000 6000 7000 b Time (year) Brine_Rock 1:50 Brine_Rock 1:25 Brine_Rock 1:17 Brine_Rock 1:12 Brine_Rock 1:10
Figure 5.8. Kinetic modeling for a range of brine to rock ratio a) pH and b) fCO2.
to rock ratio of 1:50 rises from 4 to 6 and then stabilizes by about 500 years, whereas the pH for the brine to rock ratio 1:10 rises from 4 to 4.7 and then continues to rise slowly throughout the 7000 year duration of the simulation (Figure 5.8a). The higher brine-rock ratios sequester more CO2 in the carbonate minerals dawsonite
87 and siderite (Figure 5.9a and 5.9b); the difference is particularly striking for
dawsonite. The modeling suggests that dawsonite is stable at lower pH values than
siderite. Once aluminosilicate dissolution reactions drive the pH sufficiently high the
dawsonite begins to dissolve. The impact of reactions on pH is more pronounced
and more rapid for lower brine-to-rock ratios than for higher brine-to-rock ratios
because there is less total CO2 available.
0.4
0.35
0.3
0.25
0.2
0.15
Dawsonite (mole) Dawsonite 0.1
0.05
0 0 1000 2000 3000 4000 5000 6000 7000
a Time (year)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 Siderite (mole) 0.2 0.1 0 0 1000 2000 3000 4000 5000 6000 7000 Time (year) b
Brine_Rock 1:50 Brine_Rock 1:25 Brine_Rock 1:17 Brine_Rock 1:12 Brine_Rock 1:10
Figure 5.9. Kinetic modeling for a range of brine to rock ratio that shows precipitation of a) dawsonite and b) siderite.
88
100 90 80 70 60
(bar) 50 CO2 f 40
30
20
10
0 0 1000 2000 3000 4000 5000 6000 7000 a Time (years)
0.7
0.6
0.5
0.4
0.3
(mole) Dawsonite 0.2
0.1
0 0 1000 2000 3000 4000 5000 6000 7000 b Time (years)
1
0.8
0.6
0.4 Siderite (mole) Siderite
0.2
0 0 1000 2000 3000 4000 5000 6000 7000
Time (years)
c Alberta Clinton Mt Simon Rose Run
Figure 5.10. Different brine composition for mixed assemblage a) uptake of CO2, b) precipitation of dawsonite, and c) precipitation of siderite.
89 Model results for three different brine compositions and a brine-rock ratio of
1:10 are shown in Figure 5.10 for the mixed rock assemblage. The fugacity of
CO2 decreases rapidly for the first 500 years for the Rose Run, 1800 years for
the Clinton and 2200 years for both the Mt Simon and Grand Rapids. More CO2
gas was consumed in both the Grand Rapids and Mt Simon brines compared to
the Rose Run and Clinton brines (Figure 5.10a) because more dawsonite was formed from the former brines than from latter brines (Figures 5.10b). The
important factor that plays a role is the degree of salinity of the three different brine compositions. Both Grand Rapids and Mt Simon are less saline than Rose
Run and Clinton (Table 4.3), which suggest that more CO2 would be dissolved in
the former ones. The higher concentration of aqueous Fe in the Clinton brine
(Table 4.3) consumes more CO2 gas when precipitating siderite (Figure 5.10c).
But the higher aqueous Na and Fe in the Rose Run brine do not seem to affect
the uptake of CO2 because of its higher salinity. During the 7000 years modeling,
0.2-0.66 moles of siderite were precipitated regardless of the brine composition
(Figures 5.10c). The largest dawsonite precipitate for Rose Run, Clinton, Mt
Simon and Grand Rapids was 0.12 (500 years), 0.14 (1800 years), 0.5 (2200
years) and 0.56 (3000 years) moles per 10 kg rocks, respectively (Figure 5.10b).
5.2 Discussion
Equilibrium, path of reaction, and kinetic modeling simulates the fate of CO2 and
geochemical reactions thousands of years after injection. The pH at equilibrium for
reactions with the carbonate assemblage decreases with increasing initial fCO2
90 because more CO2 is dissolved in the brine and the pH is not completely neutralized
by reactions with carbonate minerals. This relationship suggests that adding CO2
into a carbonate formation will increase acidity and dissolve, not precipitate,
carbonate minerals. The CO2 remains as dissolved aqueous species such that the
carbonate layers of the Rose Run are unlikely to contribute to mineral trapping of
CO2. The dominant CO2 sequestration mechanisms in carbonate host rocks are
solubility and hydrodynamic trapping.
In contrast, for both the sandstone and mixed assemblages the amount of CO2 trapped in carbonate minerals at equilibrium increases as the fCO2 increases (Figures
5.2a, 5.3a, and 5.4a). The most important minerals for trapping CO2 in the Rose Run
Sandstone are dawsonite (NaAlCO3(OH)2) and siderite (FeCO3). Precipitation of
these carbonate minerals requires that sufficient amounts of Na, Fe and Al be
available for reaction. These ions are added to brine by dissolution of aluminosilicate
minerals in response to the increase in acidity with the addition of CO2. In addition,
the Na needed for dawsonite precipitation is available in great quantities in the initial
brine (Johnson et al., 2001). Albite dissolution contributes additional Na and Al, and
K-feldspar dissolution contributes Al to the brine. Dissolution of the mineral
glauconite, which is inhomogeneously distributed, but locally very abundant in the
Rose Run Sandstone, can contribute Fe. That contribution was simulated here by dissolution of the mineral annite. In addition to providing sources of Na, Fe and Al, dissolution of aluminosilicate minerals neutralizes the acid formed by dissolution of
CO2 into the brine. Dissolution of CO2 and carbonate precipitation was confirmed
experimentally at 200° C and 20 MPa by Kaszuba et al. (2003).
91 Dawsonite and siderite are likely to be very important for mineral trapping of CO2 in many deep saline aquifers, like the Rose Run Sandstone, because many deep sandstone formations that are otherwise suitable for geologic sequestration lack abundant sources of Ca, Mg, and Sr that are necessary to induce precipitation of the carbonate minerals calcite, dolomite, and strontianite. Consequently, understanding the controls on dawsonite and siderite formation is very important. The equilibrium, path-of-reaction, and kinetic modeling demonstrate that the mass of these minerals precipitated depends strongly on initial fCO2, temperature, the brine to rock ratio, and
the kinetic rate of silicate mineral dissolution and time for reaction.
Specifically, equilibrium modeling shows that dawsonite does not precipitate
in the final equilibrium assemblage for simulations conducted at the range of
initial fCO2 and range of model temperatures. The path of reaction modeling,
however, demonstrates that for all initial fCO2 studied, dawsonite precipitates
during the initial stages of the reactions, when little mineral has reacted with the
system, but dissolves as the reaction proceeds to completion (Figure 5.5). This
pattern of initial precipitation of dawsonite followed by dissolution is repeated by
the kinetic modeling. For an initial fCO2 of 100 bar, dawsonite precipitates rapidly
for the first ca. 300 years for the sandstone assemblage and more slowly for the
first ca. 600 years for the mixed assemblage, but then begins to dissolve (Figure
5.7b, c). The time period over which precipitation takes place and the total
amount of dawsonite precipitated increase as brine-to-tock ratio increases
(Figure 5.9a).
92 Initial fCO2 controls how much CO2 is available in the model system for reaction
and consequently the evolution of the pH over the course of the reaction. The
transition from dawsonite precipitation to dawsonite dissolution with progress along the path of reaction reflects the consumption of the acidic CO2 by CO2-brine-mineral
reactions, and the resulting elevation of pH. The higher the initial fCO2 the more reaction is necessary to consume the CO2 dissolved in the brine to the point where
dawsonite is unstable. Increasing brine-to-rock ratio has the same effect as
increasing fCO2 by increasing the amount of CO2 available relative to the amount of minerals to react with it. Thus, the equilibrium, path-of-reaction, and kinetic modeling show that dawsonite is thermodynamically less stable at lower fCO2 and corresponding higher pH values. Patterns of siderite formation and dissolution are similar to those for dawsonite, but siderite is more stable than dawsonite as fCO2
decreases and pH rises, and at higher temperatures.
An important aspect of GWB modeling is the effect of salinity. As shown in Figure
5.10, modifying the database to correct for salting-out effect greatly affects the
amount of dawsonite and siderite formed through time. For the same pressure, if salinity of the formation were high, only a small part of the CO2 fugacity would
participate in the reaction. For example, Rose Run has higher TDS than Grand
Rapids; with the first 500 years 0.12 mole of dawsonite precipitated, but at the end of
1000 years all of the dawsonite dissolved (Figure 5.10b). Whereas in case of Grand
Rapids, around 0.6 mole (five times larger than the Rose Run) was formed in the
first 3000 years and it remains constant for the rest of modeling duration. The same
pattern is also exhibited by siderite (Figure 5.10c). This suggest that salinity plays
93 such an important role that ignoring its effect on CO2 solubility, particularly in brines
with high TDS would gravely overestimate the long term trapping of CO2 through the
precipitation of carbonate minerals.
High fCO2 is expected in close proximity to the injection site and immediately
following injection. The fugacity of CO2 decreases with distance from the injection
site and over hundreds of years after injection ceased due to dispersion and
migration. Given dawsonite’s increasing stability at higher fCO2 and the kinetic
modeling results that indicate relatively rapid precipitation of dawsonite during the
first thousand years, dawsonite is predicted to be important in converting mobile,
dissolved CO2 into immobile minerals, along with siderite in the Rose Run
Sandstone. Over time, as fCO2 decreases, dawsonite may begin to dissolve as predicted by thermodynamic simulations. However, kinetic inhibitions may slow or
prevent the dissolution. For example, dawsonite remains present in great abundance
in the Sydney Basin of Australia today, millions of years after the magmatic CO2 that induced its precipitation dissipated (Baker et al. 1995). Even as the dawsonite dissolves, siderite remains stable and it continues to precipitate and can take up some of the re-released CO2.
The modeling conducted here does not take into account free CO2, which will displace brine as it is injected. This displacement decreases the amount of brine in the pore space and therefore the brine-to-rock ratio with approach to the injection
site. Although the modeling indicates a decrease in the mass of carbonate minerals precipitated with decreasing brine-to-rock ratio, we expect this effect to have little
impact once free CO2 is included in the system. Free CO2 in the pore space provides
94 a large source of additional CO2 that was not considered in the model approach. The
equilibrium modeling indicates that for fCO2 of 100 bar and T=54˚ C, all the CO2 dissolved in the brine is consumed by reactions with formation minerals (Figure
5.2b). With more free CO2 available to dissolve into the brine more aluminosilicate
minerals would react with the aqueous CO2 and carbonate mineral precipitation
would be increased. In terms of the path of reaction and kinetic modeling, the impact of including free CO2 would be to increase the percentage of the reaction or the time period over which dawsonite precipitation is favored.
Overall, the equilibrium and kinetic modeling predicts a net rock mass gain of
10-15 grams at fCO2 of 100 bar, for the sandstone assemblage. The gain in mass
accompanies by an uptake of 1.5-2 grams of CO2 per kg of reacted rock. The
majority of the gain in rock mass is due not to the precipitation of carbonate
minerals, but instead to the precipitation of quartz, muscovite, and microcline in
response to release of silica from dissolving K-feldspar, kaolinite and albite. The
increase in rock mass is accompanied by a decrease in porosity of 0.1% to 0.2%,
which has the potential both to cause clogging, slowing injection, or, on the
positive side, can improve the cap rock sealing. Thus, constraints on the rates
and products of the CO2-brine-mineral reactions are very important for
understanding the long time effect on the safety and integrity of the seal of the formation.
95 5.3 Recommendations and Concluding Remarks
Geochemical modeling revealed the importance of the minerals dawsonite
and siderite for mineral trapping of CO2 in the Rose Run sandstone. The mineral
composition of the Rose Run sandstone is similar to other deep formations that
are being considered for sequestration of CO2, such as the Mount Simon
Sandstone beneath Ohio in the USA (Sass et al., 1998) and the glauconitic
sandstone in the Alberta Basin of Canada (Gunter, 1993). Consequently,
dawsonite and siderite are likely to be an important contributor to mineral trapping of CO2 in these formations as well. Like the Rose Run, these sandstone
layers were deposited in old, cold, stable basins, and consequently are well suited for hydrodynamic and solubility trapping, but have a paucity of the mafic minerals that are best suited for mineral trapping. The potential for dawsonite and siderite formation make these widespread basins better candidates for CO2 storage because mineral trapping can take place through reaction with alkali feldspars and glauconite. However, the thermodynamic conditions under which dawsonite forms remain poorly constrained.
Results from equilibrium modeling indicate that the extent of mineral trapping depends strongly on the fugacity of CO2. Consequently, the extent of mineral
trapping is sensitive to the rate of mineral-brine-CO2 reactions relative to the rate
of flow and dispersion of CO2, away from the site of injection. Reactions must be
fast enough to reach carbonate phase saturation before the CO2 is overly diluted
by outward radial flow, dispersion, and diffusion. Reactive transport modeling that
include a CO2 solubility model that properly address the solubility of CO2 in brine
96 is necessary in order to develop a more realistic form of CO2 storage in deep
saline formations. This is addressed in chapters 6-8.
Evaluating the long-term geochemical behavior of the injected CO2 and its
possible impact on the integrity of the formation is a crucial aspect of geological CO2
sequestration safety assessment. Computer simulations using the computer
program ‘Geochemist’s Workbench’ were used to model the equilibrium and kinetic
assemblages produced by reacting typical Rose Run carbonate, sandstone, and mixed mineral assemblages with measured brine compositions. Although the
computer code has no transport component to it, the effects of radially decreasing
CO2 pressure from an injection site were investigated in a simplistic way by varying
CO2 fugacity. This modeling of the chemical reactions under no-flow conditions
provides insight into what parameters control the reactions and their final products.
The results of equilibrium model demonstrate that dissolution of albite, K-feldspar,
and glauconite and the precipitation of siderite are potentially very important for
mineral trapping of CO2. The mass of dawsonite and siderite precipitated depends
on mineral assemblage composition, brine composition, temperature, and initial CO2
fugacity; brine composition and initial CO2 fugacity have by far the greatest impact.
The kinetic modeling indicates that over hundreds to thousands years it should be
possible to sequester large quantities of CO2 by mineral trapping in the sandstone
and mixed units of the Rose Run Sandstone, but over tens to hundreds years
solubility trapping plays the most crucial role.
This work demonstrates that the Rose Run Sandstone has a suitable mineral
composition for significant mineral trapping of CO2. However, it also indicates that
97 the extent of mineral trapping will be sensitive to the initial injected fCO2, the brine to
rock ratio, the initial brine composition, and the rate of reaction. The brine to rock
ratio, the kinetic rate of reaction, and the fCO2 has a strong impact on the pH and
saturation index of the precipitated/dissolved minerals. Quartz, muscovite and microcline minerals are by far the dominant phases precipitated as a result of the reactions and, if the reactions are fast enough, these minerals have the potential to cause clogging, slowing injection, or can locally improve the cap rock sealing capacity.
98
Chapter 6
REACTIVE TRANSPORT CODE DEVELOPMENT
6.1 Introduction
The focus of chapter 3 was on solubility of CO2 in brine and the focus of
chapters 4 and 5 was on reactions among dissolved CO2, brine, and minerals.
Flow of free CO2 and transport of dissolved CO2 and other solutes need to be
included in order to describe the full potential and long-term behavior of CO2 injection in geological formations. Reactive transport modeling integrates geochemical, hydrological, and mechanical processes that characterize dynamic geologic systems. Chemical reactions, fluid flow, heat-transfer, and mechanical stress-strain are interdependent and must be modeled simultaneously to simulate the true behavior of geologic systems. The work here consists of a simplified model, to identify the main constraints that govern chemical reaction and transport. In this work, the effect of heat, gravity, and mechanical processes are not included. The details of the conceptual and mathematical design of the
99 newly developed one-dimensional reactive transport are discussed in this
chapter.
6.2 Brief Assessment of Existing Codes
Multi-species reactive-transport modeling is an emerging research field,
which designs to give a comprehensive, quantitative and, ultimately, predictive
treatment of transport of dissolved chemical species and free CO2 in groundwater
that chemically interact with one another as well as with the solid matrix.
Reactive transport models have been used to simulate, among others, reactive
transport in geological porous media (e.g. McPherson et al., 2000; Johnson et
al., 2001; Pruess et al., 2003), microbial remediation of contaminants in porous
media (e.g. Murphy and Ginne, 2000), the fate and transport of contaminants in aquifers (e.g. Prommer et al., 2002 ), metasomatism and ore forming processes
(e.g. Xu et al., 2001), and enhanced oil recovery (e.g. Bijeljic et al., 2003).
Several multi-species reactive transport codes have been developed over the past 30 years. These numerous codes vary in the chemical processes that can be included, the hydrologic processes that can be considered, the equations of state adopted, the source of thermodynamic data, the numerical approach used to solve the coupled transport equations, and the extent to which the code is documented and generally available to the scientific community. RT3D
(Clement, 1997), NUFT (Nitao, 1998), FLOTRAN (Lichtner, 2001), CRUNCH
(Steefel, 2001), PHAST (Parkhurst and Kipp, 2002) and TOUGHREACT (Xu et al., 2004) are some of the most commonly used reactive transport codes that
100 have been developed in the past few years.
RT3D is a Fortran 90-based software package for simulating three-
dimensional, multi-species, reactive transport in groundwater. The code is based on the 1997 version of MT3D (Multiphase Transport in 3-D), but has several
extended reaction capabilities. NUFT is an integrated software package
containing five application-specific program modules (include discretization and
solving reaction and transport, thermodynamic database, relational kinetic
database, and graphic representation of data) that facilitates numerical
simulation of multiphase and multi-component flow and reactive transport within
a wide range of subsurface environments. This software has been used extensively in combination with other supporting modeling tools to simulate the behavior of injected CO2 into saline aquifer in the Sleipner gas field (Johnson,
2001). FLOTRAN is another popular code used to solve fluid flow and conjugate
heat transfer problems through finite element analysis. It is two-phase non-
isothermal coupled thermal-hydrologic-chemical (THC) reactive flow & transport
code. CRUNCH is one of the most recent computer programs used to simulate
multi-component multi-dimensional reactive transport in porous media. The code
uses an integrated finite volume approach currently restricted to orthogonal grids.
Using an automatic read of a thermodynamic and kinetic database, the code can
be used for reactive transport problems of arbitrary complexity and size. Another
code that was developed by Parkhurst and Kipp (2002) is the reactive transport
simulator PHAST; a combination of two U.S.G.S. groundwater simulators,
PHREEQC and HST3D (a groundwater flow and solute transport model). The
101 computer program PHAST simulates multi-component, reactive solute transport in three-dimensional saturated ground-water flow systems. The geochemical reactions are simulated with the geochemical model PHREEQC, which is embedded in PHAST. Xu et al. (2004) developed a non-isothermal reactive geochemical transport model, TOUGHREACT, by introducing reactive geochemistry into the framework of the existing multi-phase fluid and heat flow code TOUGH2 (Pruess, 1999). Most of the recent papers by Pruess et al. (2001),
Pruess and Garcia (2002), and Xu et al. (2004), among others, use
TOUGHREACT to model reaction and transport between various components during CO2 sequestration in geological formations.
Table 6.1: Some of the most commonly used reactive transport codes and their descriptions. RT3D NUFT FLOTRAN CRUNCH PHAST TOUGH REACT Parkhurst Clement Nitao Lichtner Steefel Xu et al. Reference & Kipp (1997) (1998a) (2001) (2001) (2004) (2002) 3-D yes yes yes yes yes yes Multiphase yes yes Two-phase yes yes yes Transport yes yes yes yes yes yes Reaction yes yes yes yes yes yes Public Research Research Research Public Public Availability domain code code code domain domain Coded Fortran Fortran Fortran Fortran Fortran Fortran using
102 All of the above codes are versatile ground-water flow and solute-transport
simulators with capabilities to model a wide range of equilibrium and kinetic
geochemical reactions. They can accommodate a variety of equilibrium chemical
reactions, such as aqueous complexation, linear adsorption and decay, gas
dissolution/exsolution, and cation exchange. Mineral dissolution/precipitation can
proceed subject to either local equilibrium or kinetic conditions. However, none of the codes includes the Duan and Sun CO2 Solubility Model (DS-CSM) (Duan and
Sun, 2003) as discussed in chapter 3, for simulating the behavior of injected CO2
into high ionic strength saline aquifers. In addition, the equilibrium constants in
the thermodynamic database were tabulated for atmospheric pressure. The
effect of high pressure on the equilibrium constant is ignored.
6.3 Reactive Transport Modeling
In this work, a reactive transport code is developed that includes the impact
of high pressure on the equilibrium constants. In addition, three CO2 solubility
models namely, DS-CSM (Duan and Sun, 2003), Xu-CSM (Xu et al., 2004) and
modified Redlich-Kwong MRK-CSM (McPherson and Cole, 2005) are included to
evaluate the effect of choice of solubility model on the overall CO2 uptake
through the partitioning of CO2 into free, dissolved and solid mineral forms. The
detailed descriptions of the conceptual model, the chemical, hydrological and
physical parameters deployed, the solubility models, the governing equations
and methods of solutions for the coupling flow, transport and chemical reactions
are discussed below.
103 6.4 Conceptual Model
Mathematical models of multiphase flow and reactive transport phenomena
provide a powerful means for understanding and predicting the flow and chemical
evolution of the subsurface environment in multiphase, multi-component, open-
system settings. However, the simulation of reactive transport is inherently
complex because a large number of transport and chemical reaction equations
must be solved simultaneously. Because of the natural complexity of aquifers,
the development of a conceptual model invariably requires a simplification of the natural system. In this work, fluid flow, solute transport and chemical reaction of injected CO2, brine, and minerals are modeled as a simplified 1-D flow regime in
2-D porous media. Vertical permeability is assumed zero and gravity effect is
ignored, but the modeled aquifer has a height (H) that is constant as a function of
distance(r). The simulation assumes radial flow from the center of injection as a
function of radius (r) (Figure 6.1). In the 1-D flow modeling, only a slice of Figure
6.1 is used. The model domain is 16000 m (rf - radius of formation) by 20 m (H –
Height), consisting of 2000 grid-blocks, each 8 m by 20m. The CO2 is injected
through 0.25 m radius injection well the evolution of the whole system is
simulated for 10,000 years. The aquifer is assumed homogenous and isotropic,
and gravity effect is neglected.
104
QCO2
H
rw rf
Figure 6.1 Schematic representation of simplified aquifer.
6.5 Theoretical Background
Major processes considered for isothermal fluid flow are: (1) fluid flow in both
liquid and gas phases occur under pressure and viscous forces; (2) interactions between flowing phases are represented by characteristic curves (relative permeability). The local chemical interactions in the transport equations are represented by reaction source/sink terms. Aqueous (dissolved) species are subject to transport in the liquid phase as well as to local chemical interactions with the solid and gas phases. Advection and diffusion processes are considered for both the liquid and gas phases. The diffusion coefficients for aqueous solute are assumed the same for all species. Transport and diffusion equations are
105 written in terms of total dissolved concentrations of chemical components. The
equations of chemical reactions are presented in section 6.8.2. For kinetic- controlled mineral dissolution and precipitation, a general rate law (Lasaga, 1984;
and Steefel and Lasaga, 1994) is used.
Temporal changes in porosity and permeability due to mineral dissolution
and precipitation processes can modify fluid flow. This feedback between
transport and chemistry is considered in this model. Changes in porosity during
the simulation are calculated from changes in mineral volume fractions. The
porosity permeability correlation in geologic media is very complex, depending on
an interplay of many factors, such as pore size distribution, pore shapes, and
connectivity (Verma and Pruess, 1988). The functional dependency of porosity-
permeability based on Verma and Pruess (1988) is given in section 6.8.1.
6.6 Chemical, Hydrological and Physical Parameters
One of the major challenges in the field of reactive transport modeling is the
realistic representation of the highly complex reaction networks including reaction
pathways, basin geometry, flow and transport of solute that characterize the
chemical dynamics of natural environments. Implementation of reactive transport
code requires the proper definition of model parameters such as the chemical reaction, thermodynamic database and hydrological inputs. Those parameters
and their associated assumption are given below.
106
6.6.1 Chemical Parameters and Thermodynamic Database
The reactive transport code developed in this study considers (1) heterogeneous reaction, (2) solubility of CO2 based on DS-CSM, Xu-CSM and
MRK-CSM, (3) kinetic reactions, and (4) diffusion transport and partitioning of
CO2 in liquid and gas phases. In heterogeneous reaction a variety of minerals are considered instead of just single mineral modeling. This is designed to evaluate the effect of actual rock composition on the CO2 sequestration through mineral trapping mechanism. To evaluate the effect of CSM on CO2 trapping mechanism, three of the latest CSM are selected as indicate above. Kinetic rate of reaction is very important as most of the silicate reactions are slow and it is considered in this work based on rate constants reported in the published literature. The transport and diffusion of CO2 were evaluated in both liquid phase and gas phase to track mass change of CO2 through time and space. The equilibrium constants are computed based on nine primary minerals (Table 6.2), three secondary minerals (Table 6.2), 13 aqueous species (Table 6.3), and CO2 gas. The minerals and aqueous species included are based on the dominant mineralogical and brine composition of the Rose Run Sandstone. The mineral anorthite, which is not present in the Rose Run Sandstone, was also added because it is important in other aquifers under consideration for CO2 sequestration Surface complexation, ion exchange, linear adsorption and decay, and solid solutions are not included. The initial mineral abundances, potential secondary minerals, reactive surface areas, and kinetic data were taken from the modeling work of chapter 5.
107 Table 6.2: Primary and secondary minerals used in the modeling Carbonate wt. % Sandstone wt. % Mixed wt. % Dolomite 60 Quartz 83 Quartz 70 Calcite 39 K-feldspar 10 Dolomite 13.8 Siderite 1 Kaolinite 3 Calcite 8 Albite 2 K-feldspar 5 Annite 1 Annite 1 Siderite 1 Albite 1 Kaolinite 1 Siderite 0.2 Secondary minerals: dawsonite, muscovite and strontianite
Table 6.3: Brine composition of Rose Run Sandstone, Clinton, Mt Simon, and Grand Rapids (Alberta).
Brine Rose Run(a) Clinton(b) Mt. Simon(c) Grand Rapids(d) Species (mg/kg) (mg/kg) (mg/kg) (mg/kg) Na+ 60122 67000 32000 26539 K+ 3354 850 1060 636 Ca2+ 37600 23200 12400 2737 Mg2+ 5880.6 1840 2190 533 HCO3- 122 200 71 182 Cl- 191203 160400 78700 47549 SO42- 326.4 523 1180 337 SiO2(aq) 3 1 5.00E-07 0.00046 Al3+ 2.16 1 5.00E-07 0.461 Fe2+ 140 5 1.54 4.6E-05 Sr2+ 455.52 753 236 - pH 6.4 6.5 6.7 7.2 TDS 277,571 250,000 150,000 90,000 (a) Breen et al., 1985; (b) Lowry et al., 1988; (c) Ohio Geological Survey, 1990; (d) Gunter et al., 2000
108 Calcite and dolomite dissolution and precipitation were assumed to take
place under local equilibrium in the silicate and mixed rock assemblage but
modeled under kinetic rate for the carbonate assemblage. The dissolution and
precipitation of the other minerals were treated under kinetic constraints (Table
6.4). The CO2 gas partial pressures used for the gas transport and the solubility
of CO2 in formation solution are computed using DS, Xu, and MRK-CSM. These
three solubility models are selected to evaluate the impact of the choice of CSM on the overall storage capacity. Activity coefficients for charged aqueous solutes are calculated using an extended form of the Debye-Huckel equation as discussed in chapter 4.
Table 6.4: Rate constants for silicate and carbonate minerals compiled from literature and used in the reactive transport simulations.
Rate constants Ea (kJ) 2 Mineral log Kr (mol/m s) (Activation References at 25° C energy) Albite -11 68 Sverdrup, 1990 Annite -10.5 45 Acker and Bricker, 1992 Calcite -5.8 63 Plummer et al., 1978 Dawsonite -8.4 64 Assumed Dolomite -6.7 55 Busenberg and Plummer, 1982 Kaolinite -11.4 64 Sverdrup, 1990 K-feldspar -10.9 58 Helgeson et al., 1984 Muscovite -11.7 64 Nagy 1995 Siderite -6.7 62.8 Assumed Dolomite rate Strontianite -7.35 41.9 Sonderegger et al., 1976 Quartz -12 87.5 Rimstidt and Barnes, 1980
109 The thermodynamic database used in this study incorporates the effect of
high pressure in the equilibrium constant. The original code (coded in TK Solver)
used to calculate equilibrium constant was provided by Doug Allen (personal communication). The code was revised and recoded into Matlab program. The most notable revisions were debugging the entire code (no manual or description
was available as how it was coded and what parameters use for what purpose)
and new data were added from various published literatures as shown in Table
6.5 and 6.6. All calculations were made based on a paper by Johnson et al.
(1992). The code has the potential to compute the equilibrium reaction constant
as a function of temperature and pressure for any reaction as long as all the
necessary parameters of that particular reaction are included in the database.
These parameters include Gibbs free energy, enthalpy, entropy, heat capacity,
molar volume of each specified reactions, coefficients pertinent to specific
aqueous solute species, and H2O dielectric constants at standard temperature
and pressure. The standard parameters were compiled from the published
literature (Table 6.5 and 6.6). Most of the database in GWB was generated using
SUPCRIT92 computer code (Johnson et al., 1992). The free Gibbs energy,
enthalpy, entropy and coefficients for the heat capacity in GWB were compiled
from the same source as used for this work. The references and all parameters
used to generate the equilibrium constant as a function of temperature and
pressure are given in Table 6.5 and 6.6.
110 Table 6.5: Thermodynamic parameters of aqueous and neutral species used in the computer modeling. Reference temperature is given in Kelvin.
IONT H+ Mg2+ Ca2+ OH- Fe2+ CO32- Cl- Shock and Shock and Shock and Shock and Shock and Shock and SUPCRIT Helgeson Helgeson Helgeson Helgeson Helgeson Helgeson Reference 92 (1988) (1988) (1988) (1988) (1988) (1988) Ionic charge 1 2 2 -1 2 -2 -1 Temperature 298.15 298.15 298.15 298.15 298.15 298.15 298.15 Gibbs free 0 -108505 -132120 -37595 -21870 -126191 -31379 energy (∆Gf) Entropy (S) 0 -33 -13.5 -2.56 -25.3 -11.95 13.56 Born Coeff. 0 153720 123660 172460 143820 339140 145600 Ion radius 2.14 .66 .99 1.4 .74 2.87 1.81 a1 0 -.08217 -.01947 .12527 -.07867 .28524 .4032 a2 0 -859.9 -725.2 7.38 -969.69 -398.44 480.1 a3 0 8.39 5.2966 1.8423 9.5479 6.4142 5.563 a4 0 -23900 -24792 -27821 -23780 -26143 -28470 c1 0 20.8 9 4.15 14.786 -3.3206 -4.4 c2 0 -58920 -25220 -103460 -46437 -171917 -57140 a1-a4 and c1-c2 are heat capacity coefficients.
IONT K+ Na+ Sr2+ Mn2+ HCO3- SiO2aq Al3+ Shock and Shock and Shock and Shock and Shock and SUPCRIT Helgeson Helgeson Helgeson Helgeson Helgeson SUPCRIT92 Reference 92 (1988) (1988) (1988) (1988) (1988) Ionic charge 1 1 2 2 -1 0 3 Temperature 298.15 298.15 298.15 298.15 298.15 298.15 298.15 Gibbs free -67510 -62591 -134760 -54500 -140282 -199190 -116543 energy (∆Gf) Entropy (S) 24.15 13.96 -7.53 -17.6 23.53 18 -77.7 Born Coef 19270 33060 245561 140060 127330 12910 287111.956 Ion radius 1.33 .97 .85 .8 2.26 .51 a1 .3559 .1839 .07071 .01016 .75621 .19 -.33802 a2 -147.3 -228.5 -1015.08 -802.95 115.05 170 -1700.71 a3 5.435 3.256 7.0027 8.906 1.2346 20 14.5185 a4 -27120 -27260 -23594 -24471 -28266 -27000 -20758 c1 7.4 18.18 10.7452 16.6674 12.9395 29.1 10.7 c2 -17910 -29810 -50818 -38698 -47579 -512000 -80600
111 Table 6.6: Thermodynamic parameters of minerals used in the computer modeling and a, b and c are heat capacity coefficients. Calcite Anorthite Albite Dolomite Quartz Helgeson et Helgeson et al. Helgeson et Helgeson et Helgeson et Reference al. (1978) (1978) al. (1978) al. (1978) al. (1978) Temperature 298.15 298.15 298.15 298.15 298.15 Gibbs free -269880 -954078 -886308 -517980 -204646 energy (∆Gf) Entropy (S) 22.15 49.1 49.51 37.09 9.88 Molar 36.934 100.79 100.07 64.365 22.688 volume (V) a 24.98 63.311 61.7 41.557 11.22 b .00524 .014794 .0139 .023952 .0082 c 620000 1544000 1501000 988400 270000
Muscovite Dawsonite Annite Siderite Microcline Helgeson et Robie (1995) Robie (1995) Robie (1995) Robie (1995) Reference al. (1978) Temperature 298.15 298.15 298.15 298.15 298.15 Gibbs free -1335667 -426558.4 -1145999.5 -163076.188 -895462.15 energy (∆Gf) Entropy (S) 68.8 31.526 415 22.809 51.16 Molar 140.71 59.3 154.3 29.38 108.72 volume (V) a 97.56 34.41 636.6 257.4 795.5 b .02638 .00016735 .04104 -.0231 -.10855 c 2544000 746.4 -4860000 1523000 4746000 Strontianite Kaolinite K-feldspar Magnesite Helgeson et al. Helgeson et al. Robie (1995) Robie (1995) Reference (1978) (1978) Temperature 298.15 298.15 298.15 298.15 Gibbs free -271698.11 -905614 -895374 -245880.11 energy (∆Gf) Entropy (S) 23.19 48.53 51.13 15.55 Molar 39.01 99.52 108.87 28.02 volume (V) a -161.8 72.77 76.617 81.12 b .06395 .0292 .00431 .026125 c -9018000 2152000 2994500 -1832000
112 The standard molal Gibbs free energies and enthalpies of minerals, gases,
and aqueous species are represented as apparent standard molal Gibbs free
energies (∆G°P,T) and enthalpies (∆H°P,T) of formation from the elements at the
subscripted pressure (P) and temperature (T). They are given by:
∆=∆+−GG(GG)°°°° P,T f P,T Prr ,T (6.1)
∆=∆+−HH(HH)°°°° P,T f P,T Prr ,T (6.2)
where, ∆G°f and ∆H°f denote the standard molal Gibbs free energy and enthalpy
of formation of the species from its elements in their stable phase at the
reference pressure (Pr = 1 bar) and temperature (Tr = 298.15 K), and G°P,T -
G°Pr,Tr and H°P,T - H°Pr,Tr refer to differences in the standard molal Gibbs free
energy and enthalpy of the species that arise from change in pressure (P – Pr) and temperature (T – Tr). The standard molal entropies, isobaric heat capacities,
and volumes of minerals, gases, and aqueous species are designated by S°P,T,
C°PP,T V°P,T respectively.
For the stable phase of H2O at any temperature and pressure, the standard
molal Gibbs free energy (∆G°H2O,P,T), enthalpy (∆Η°H2O,P,T) and entropy (S°H2O,P,T)
need to be referenced at the triple point. This conversion of the triple-point
(represents by subscript, tr) reference frame is accomplished using (Helgeson
and Kirkham, 1974)
S(SS)S° =−+ HO,P,T222r2 HO,P,T HO,T HO,Tr (6.3)
113 ∆=H(HH)H° −+ HO,P,T222r2 HO,P,T HO,T HO,Tr (6.4)
∆=G(HH)TSTSG° −−++ HO,P,T222222 HO,P,T HO,tr HO,P,T tr HO,tr HO,tr (6.5) =−++(G G ) T S G HO,P,T22 HO,tr tr HO,tr 22 HO,tr
where SH2O,P,T designate the molal entropy, Ttr = 273.16 K, SH2O,tr = 15.132 cal
-1 -1 mol K , HH2O,tr = -68,767 cal/mol, and GH2O,tr = -56290 cal/mol. The
mathematical formulation of standard molal properties of H2O, aqueous species,
minerals and gases as well as the calculated equilibrium constant are given
below.
Thermodynamic and electrostatic properties of H2O are calculated using
equations and data given by Helgeson and Kirkham (1974), Uematsu and Frank
(1980), Pitzer (1983), Haar et al. (1984), Johnson and Norton (1991), and Robie
and Hemingway (1995). The apparent standards free energy of aqueous species
are calculated using the revised HKF equation of state (Tanger and Helgeson,
1988; Shock et al., 1992) and for minerals and gases are calculated using
equations and data given by Maier-Kelly (1932) and Helgeson et al. (1978). The
H2O critical region is bounded by very high temperature (above 350°C) and pressure (250-400 bar). Since this region is way above the intended purpose of the present study, all calculations and mathematical formulations considered
here are appropriate only inside the subcritical region. The equation of state for
H2O is given by:
−0.5 0.5 ⎛⎞∂A ρ=(P × C × Mwt) ⎜⎟ (6.6) ⎝⎠∂ρ T
114 where, A is the molal Helmholtz function for H2O, C represents a conversion
factor (0.02390054 cal bar-1 cm-3), and Mwt stands for the molecular weight of
water (18.015 g/mole). The molar volume, V° can be calculated from V° = Mwt/ρ.
Values for ∆G°P,T, ∆Η°P,T, S°P,T of water can be calculated using Equations 6.3-7
together with
⎛⎞∂A GGAP,T−=+ρ tr ⎜⎟ (6.7) ⎝⎠∂ρ T
⎛⎞∂∂AA⎛⎞ HHATP,T−=− tr ⎜⎟ +ρ⎜⎟ (6.8) ⎝⎠∂T ρ ⎝⎠∂ρ T
⎛⎞∂A SSP,T−=− tr ⎜⎟ (6.9) ⎝⎠∂T ρ
Isothermal compressibility (β) and isobaric thermal expansivity (α) should be
calculated in order to describe the standard molal thermodynamic properties
aqueous species. These are given by:
−1 ⎡ 2 ⎤ 23⎛⎞∂∂AA⎛⎞ β=CMwt2 ×⎢ ρ⎜⎟ +ρ ⎜⎟2 ⎥ (6.10) ∂ρ ∂ρ ⎣⎢ ⎝⎠T ⎝⎠T ⎦⎥
−1 ⎛⎞∂∂∂22AAA⎡ ⎛⎞ ⎛⎞⎤ α=⎜⎟⎢2⎜⎟ +ρ ⎜⎟2 ⎥ (6.11) ∂ρ∂T ∂ρ ∂ρ ⎝⎠T,ρ ⎣⎢ ⎝⎠T ⎝⎠ T ⎦⎥
1 Z =− ε (6.12)
115 ⎛⎞∂∂εZ1 ⎛⎞ W ==⎜⎟2 ⎜⎟ (6.13) ⎝⎠∂ε∂PPTT ⎝⎠
⎛⎞∂∂εZ1 ⎛⎞ Y ==⎜⎟2 ⎜⎟ (6.14) ⎝⎠∂ε∂TTPP ⎝⎠ where ε is the dielectric constant of water and can be obtained using the following equations:
4 i ε=∑ κidd()T ρ (6.15) i0=
4 ⎛⎞∂ε i ⎜⎟= βκ∑iTidd() ρ (6.16) ⎝⎠∂P T i0=
4 ⎛⎞∂ε i ⎡⎛⎞∂κid(T ) ⎤ ⎜⎟=ρ∑ did⎢⎜⎟ −ακi(T)⎥ (6.17) ⎝⎠∂∂TTP i0= ⎣⎝⎠P ⎦
3 where the dimensionless variables are defined by Td = T/Tr and ρd = ρ/(1 g/cm ) and the κi(Td) are given by
κ=od(T ) 1 (6.18)
−1 κ=1d(T ) aε ,1d T (6.19)
−1 κ=2d(T ) aεεε ,2d T ++ a ,3,4d a T (6.20)
−12 κ=3d(T ) aεεε ,5d T + a ,6d T + a ,7d T (6.21)
−−21 κ=4d(T ) aε ,8d T + aεε ,9d T + a ,10 (6.22)
116 where the aε,1…10 are dimensionless adjustable regression coefficients. The
standard molar volume of the aqueous species was calculated using HKF
equation
⎡⎤aa a ⎛⎞∂ω o ˆ 2,j 3,j 4,j j Vhaj,P,T=+++⎢⎥ 1,j −ω−+ j W(Z1)⎜⎟ (6.23) Ψ+PT −Θ ( Ψ+ P)(T) −Θ ∂ P ⎣⎦⎢⎥⎝⎠T
where, ĥ is a conversion factor (41.84 bar cm3 cal-1), Ψ (2600 bar) and Θ (228 K)
are constants characteristics of the solvent, H2O. Z and W refer to the solvent
Born function defined by Equations (6.12) and (6.13), the PT independent a1…4j denote equation of state coefficients unique to the jth aqueous species, and ω is a
Born coefficient defined by
⎛⎞zz2 ω=η⎜⎟jj − j ⎜⎟r+ z k 3.082 (6.24) ⎝⎠x,j z±
5 -1 where, η is a constant (1.66027x10 Å cal mol ), rx,j ions crystallographic radius
and kz± is a constant equal to zero for anions and 0.94 for cations and z is the
ionic charge of the jth species.
The apparent standard molal Gibbs free energy of the jth aqueous solute
species can be calculated as a function of temperature and pressure from
Equations 6.1 together with (Johnson et al., 1992)
117 ⎡⎤⎛⎞T GGoo−=−−− S(TT)cTlnTT o −− j,P,T j,Prr ,T j,P rr ,T r 1, j⎢⎥⎜⎟ r ⎣⎦⎝⎠Tr ⎛⎞Ψ+P +−+a(PP)a1, j r 2, j ln⎜⎟ ⎝⎠Ψ+Pr ⎧⎫⎡⎤ ⎪⎪⎛⎞11⎛⎞ ⎡⎤Θ− TT ⎡T(Tr −Θ ) ⎤ −−×−cln2,j ⎨⎬⎢⎥⎜⎟⎜⎟ 2 ⎢ ⎥ TT−Θ −Θ⎢⎥ Θ Θ T(T) −Θ ⎩⎭⎪⎪⎣⎦⎝⎠⎝⎠rr ⎣⎦ ⎣ ⎦ (6.25) ⎛⎞1 ⎡⎤⎛⎞Ψ+P +−+⎜⎟⎢⎥a(PP)a3, j r 4, j ln⎜⎟ ⎝⎠T −Θ ⎣⎦⎝⎠Ψ+Pr −ω(Z + 1) +ω (Z + 1) +ω Y (T − T ) j j,Prr ,T P rr ,T j,P rr ,T P rr ,T r
The coefficients a1j…c1j…stand for PT independent adjustable regression
parameters unique to the jth aqueous solute species. For neutral species,
5 ωj = ωj,Pr,Tr = -1514.4×S°j,Pr,Tr +0.34×10 .
The apparent standard molal Gibbs free energy of a mineral or gas can be
calculated as a function of temperature and pressure from Equations 6.1 with
(Johnson et al., 1992)
11+ι +ι TTTT GGoo−=−−+ S(TT) oi1++ CdTTCdlnT o − i1 o P,T P,Trr P,T rr r∑∑∫∫TT P,i r P,i r i1==ii i1 ιι PPPT∆Ho (6.26) +−VdPoo VdP −ti (TT) − ∫∫PPP,Trti∑∑ ti rti i1== i1 Tti
th The subscript ti and ιP,T represents the i phase transitions (t) that occur
along the P-T path from Pr, Tr to P, T. In this study, ι is assigned zero which eliminate the last two parts of Equation 6.27. Using Maier-Kelly power function
-2 (C°Pr = a+b×T+c×T , where a, b and c correspond to adjustable regression
118 coefficients for each mineral or gas), the final result of Equation 6.27 for mineral
or gas is given by (Johnson et al., 1992)
⎡ ⎛⎞T ⎤ Goo−=−−+−− G S o (TT)aTTTln P,T Prr ,T P rr ,T r⎢ r ⎜⎟⎥ ⎣ ⎝⎠Tr ⎦ 22 (6.27) (c−×× b T Trr ) × (T − T ) ++−2 V(Prr P) 2T× Tr
For any reaction among (i) minerals, (j) aqueous solute species, (g) gases
(excluding H2O), and H2O, the change in apparent standard molal Gibbs free
energy of formation is given by
i j oo o ∆=ν∆+ν∆GGP,T∑∑ i i,P,T j G j,P,T i1== j1 g +ν∆GGoo +ν∆ (6.28) ∑ g g,P,T HO22 HO,P,T g1=
where, νi, νj, νg and νH2O refer to the stoichiometric reaction coefficient of the
subscripted species. Values for ∆G°P,T of reaction are obtained from Equations
6.1 through 6.27. Based on this, the corresponding equilibrium constant (KP,T) is given by
∆Go log K =− P,T 10 P,T 2.303× RT (6.29)
where R is gas constant. The equilibrium constant of the reaction fully accounts
for pressure and temperature.
119 6.6.2 Hydrological and Physical Parameters
The two-phase groundwater flow is modeled based on pressure using
Darcy’s equation. The two-phase flows were addressed using relative
permeability that modified the Darcy’s equation as discussed in detail in section
6.8.1. The velocity flux obtained using the Darcy’s law is used to solve the
transport of solutes and free CO2. The injected CO2 will be treated using solubility
model discussed in chapter 3 and section 6.7. The model solves the chemical
reaction based on mechanism discussed in section 6.8.2
In order to solve the flow and transport, the hydrological and physical
parameters need to be defined explicitly. The main hydrological and physical
parameters include permeability, porosity, temperature and pressure, rate of CO2 injection, the viscosity and density of both CO2 and brine. A range of
permeability, porosity, temperatures and pressures are used in the modeling
(Table 6.7). CO2 is injected uniformly at constant rate of 100 kg/s. This injection
rate is approximately equivalent to that generated by a 300 MW coal-fired power
plant. Viscosity and density of CO2 were calculated using MRK equation of state
whereas the density of the solution was computed using GWB. The main
assumptions are; the reservoir temperature is constant, brine is incompressible,
rock assemblages are laterally homogenous, permeability is isotropic, chemical
reaction is negligible at the injection well.
120
Table 6.7: List of hydrological, physical and chemical parameters. Variables Description
Rocks Rose Run (Table 6.2)
Temperature (°C) 35, 45, 55, and 75
Pressure (bar) 100, 200, and 300
Salinity (by wt %) 3, 7 (Alberta), 11 (Mt Simon), and 23 (Rose Run Sandstone)
Permeability (md) 100
Porosity (%) 12
Reactive surface area (m2/kg) 1
Reaction rate (mol/m2 s) Table 6.4, two magnitudes higher and lower than Table 6.4
3 CO2 density (kg/m ) 100-800
-2 -5 -7 CO2 viscosity (N s m ) 10 – 10
Solution density (kg/m3) 1050 – 1200
Solution viscosity (N s m-2) 10-3 – 10-5
CO2 solubility model (CSM) DS, Xu, and MRK
6.7 CO2 Solubility Models
The CO2 solubility models (CSM) are extensively discussed in chapter 3.
Three CSM, namely DS-CSM, Xu-CSM and MRK-CSM are implemented in the reactive transport code. DS- CSM and Xu- CSM have the capability to deal with ionic strength from diluted to highly saline water (up to 6 molal - any salt for DS-
121 CSM and with salt NaCl as the dominant solution for Xu- CSM). On the other
hand, MRK- CSM does not include salinity effect.
6.8 Mathematical Equations
The mathematical treatment of nonlinear transport equations is rather complicated. Numerical solution of the one-dimensional reactive transport is obtained using an implicit first order-finite-difference method, which guaranteed unconditional stability. In both transport and chemistry, the boundary condition at initial time and each grid points are explicitly defined according to Tables 6.2, 6.3 and 6.7. CO2 is injected at constant rate and the initial pressure is allowed to vary
to accommodate the boundary conditions and momentum conservation. The flow
and transport, chemical reactions and finite difference discretization equations
are given below.
6.8.1 Flow and Transport
The two-phase CO2 and solute transport is modeled based on pressure
using Darcy’s equation modified based on relative permeability and mass
balance equations. The velocity flux obtained using the Darcy’s law is used to
solve the transport of solutes and free CO2. Any reaction that may take place
during the flow and transport will be solved based on the mass action discussed
in section 6.8.2.
122 The mass balance equation for a specified chemical species, referred to as the transport equation, or advection-dispersion equation is the focus of this
section. The general continuum balance equation can be written as:
d M l dV = − F l • dΓ + R l dV dt ∫ n ∫ ad ,dif n ∫ n (6.30) V n Γn V n
where, M is the mass of components of l (kg/m3), F is total flux of advection (kg/s)
and diffusion (m2/s), R is sink/source reaction, and l is governing equation index
3 represents liquid and gases. Vn (m ) is the volume subdomain of the area under
2 study, which is bounded by surface area of each subdomain, Γn (m ). Eq. 6.30 is
for both chemical species and phases. This equation can put in differential form
by first transforming the surface integral using Gauss’s divergence theorem to
yield,
F l • dΓ = − ∇ • F l dV ∫ ad ,dif n ∫ ad ,dif n (6.31) Γn Vn
where, the general form of advection and diffusion are given by
F llllll=−ϕτ∇uM D M iaddif, i i i i (6.32)
The first component of the right hand side of Eq. 6.32 is the advection term and the second component is the diffusion term.
The mass conservation in a differential form, which is given by;
∂ l l l M + ∇ • F = R (6.33) ∂ t ad , dif
123 where,
M l = φ S l ρ X l ∑ i i i i , (6.34)
l l k ri u = −Κ (∂P ∂ x) (6.35) µi
R R l = υ I ∑ r =1 ir r (6.36)
where, ϕ is porosity (dimensionless), S is saturation of the CO2 or brine
(dimensionless), X is the fraction of component I of ith species (dimensionless), u
3 2 is Darcy’s velocity (volume flux, m /s), Κ absolute permeability (m ), kr is relative
permeability (dimensionless), ρ is density (kg/m3), µ is dynamic viscosity
-2 2 (N s m ), P is pressure (bar), D is diffusivity (m /s), and Ir is reaction rate
(discussed in next section). The subscript i represents water, aqueous solutes,
and CO2. The solid particles cause the diffusion and advection paths of solutes to
deviate from straight lines. Hence they need to be corrected for the tortuosity, τi
1/3 7/3 defined using τi = ϕ Si (Millington and Quirk, 1961).
The dimensionless relative permeability, krl (for liquid) is calculated using van
Genuchten function (1980)
2 1 / λ λ * ⎧ ⎛ * ⎞ ⎫ k rl = S ⎨1 − ⎜1 − []S ⎟ ⎬ (6.37) ⎩ ⎝ ⎠ ⎭
* S = ()Sl − Slr (1− Slr ) (6.38)
Slr = 0.30 (dimensionless irreducible water saturation) and λ = 0.457
124 The dimensionless relative permeability, krg (for gas) is computed using Corey curve (1954)
2 ˆ ˆ 2 krl = (1− S) (1− S ) (6.39) ()S − S Sˆ = l lr (6.40) ()1− Slr − S gr
Sgr = 0.05 (dimensionless irreducible gas saturation)
The functional dependency of formation permeability on porosity is calculated using an empirical formula (Verma and Pruess, 1988) and is given by
⎛⎞ ⎜⎟ 1/−Λ+Λ ϖ2 KK=ζ2 ⎜⎟ (6.41) 0 ⎜⎟2 ⎜⎟⎛⎞ζ ⎜⎟1 −Λ+Λ⎜⎟ ⎝⎠⎝⎠ζ+ϖ−1
1S−−ϕ ζ= sr (6.42) 1−ϕr
1/Λ ϖ=1 + (6.43) 1/ϕr − 1
2 where K is the final and K0 is the initial permeability (m ), 1-Ss is the original pore space that remains available to fluids (Ss is the ‘solid saturation’, defined as the
fraction of pore volume occupied by solid salt, dimenionless) , ϕr denotes the
dimensionless fraction of original porosity at which permeability is reduced to
zero, and Λ is the dimensionless fractional length of the pore bodies. Both ϕr and
Λ are set to 0.8 (based on Verma and Pruess, 1998).
Closure of the system of balance equations requires a sufficient number of equations such that all unknowns can be determined. For isothermal system in which there are three component (water, aqueous solute, CO2), there are three
equations. There are more additional dependent variables as shown in Table 6.8.
125 To close the system an additional 15 equations are required. These additional independent relationships are:
Table 6.8: List of all dependent variables Variables Number
SL, SS, SG 3
1 2 3 3 XL , X L, XL , XG 4
krL, krG 2
P 1
ρL, ρS, ρG 3
µL, µG 2
ϕ 1
τ0,G, τ0.L 2
T 1 Total 18 1 2 3 L G S water, aqueous solute, CO2; liquid, gas and salt
1. Phase saturation (of a liquid, gas and salt) sum to unity [total = 1]
SL+ SG + SS = 1
2. Mass fractions in each phase sum to unity [2]
1 2 3 3 1 XL + X L + XL = XG + (XG = 0) = 1
3. Relative permeability [2]
krG = krG(SG,SL)
KrL= krL(SG,SL)
4. Relationship based on CO2 solubility model, GWB and GWB database [5]
ρG, ρL, ρS, µG, and µL
126 5. Solubility and phase equilibria models [2]
2 XL - based on thermodynamic model (P, T)
3 XL – based on CO2 solubility model (P, T, Salinity)
6. Porosity [1]
ϕ = ϕ(SS)
7. Tortuosity is function of porosity and phase saturation [2]
τG, = τG(ϕ,SG) and τL = τL(ϕ,SL)
The total dependent variables and equations that relate the different aspect of the variables are equal. Hence, the system is fully defined and can be solved using the appropriate numerical techniques.
6.8.2 Chemical Reaction
The set of governing equations of the chemical reaction need to be formulated based on equilibrium and kinetic reaction. The mass action equation is given by
img ννννwj ij mj gj a(C)(a)(f)wiim∏∏∏γ g K% j = (6.44) γ jjC where K% is equilibrium constant of the j secondary species; a is activity of water
(w) and mineral (m); ν is reaction coefficient of water (w), base species (i) and
th th minerals (m); γi,j is activity coefficient of i base species and j secondary
th th species, Ci,j is activity of i base species and j secondary species, fg is gas fugacities, and νg reaction coefficients of gases.
127 The mass balance equations express conservation of mass in terms of the component. The mass of each chemical component is distributed among the species and minerals that make up the system. The water, aqueous solute species, minerals and gases component mass balance is given by (Bethke,
1996)
ϑ=wwn (55.51 +∑ ν wjj C ) (6.45) j
ϑ=iwin(C +∑ ν ijj C) (6.46) j
ϑ=mmwnn +∑ ν mjj C (6.47) j
ϑ=gwnC∑ ν gjj (6.48) j
where nw is solvent mass and nm is number of moles of minerals. ϑ is number of
moles of component of water (w), base species (i), minerals (m) and gases (g)
and ν is the stoichiometric reaction. The final form of the governing equations is
given by substituting the mass action equation for each occurrence of Cj in the
mass balance equations.
The principle of electroneutrality requires that the ionic species in an
electrolyte solution remain charge balanced on a macroscopic scale. It is
expressed using ionic charges, z, by the following formula:
∑ ziCi + ∑ z jC j = 0 (6.49) i j
Once the distribution of species in the fluid are calculated, the degree to which it
is undersaturated or supersaturated with respect to various minerals can be
128 determined. For any mineral Al in the thermodynamic database, we can write a reaction
⎯⎯→ AAAAAlwlwilimlmglg←⎯⎯ ν+ν+ν+ν∑ ∑∑ (6.50) im g in terms of basis. Al is a mineral that can be formed by combining components in the basis. The activity product, Ql can be calculated from Equation 6.50 using
img νwl ννil ml νgl a(C)(a)(f)wiim∏∏∏γ g Ql = (6.51) a l
Fluid’s saturation with respect to a mineral, Al is commonly expressed in terms of saturation index, Ωl , the ratio of activity product to equilibrium constant
Ω=lllQ/K%% ⇔ SI = log Ω= llog(Q ll / K ) (6.52)
Based on Equation 6.54, an undersaturated mineral has a negative saturation index (SI), a supersaturated mineral has a positive index, and a mineral at the point of saturation has an index of zero.
The above equations were derived using equilibrium approach. The kinetic part of the chemical reaction is rate dependent and the main equation is developed based on Lasaga et al., 1994. It is given by
Ξ ξ ϒ=±lrK1Amin −Ω l (6.53)
where ϒl is the rate of precipitation (if negative) or dissolution (if positive) of mineral l; Kr is the rate constant; Amin reactive surface area; Ξ and ξ are parameters that are determined from experimental work. In this work, Ξ and ξ are assigned one. The temperature dependency of rate constant is described using
129 Arrhenius equation and its dependency on neutral (nu), base (OH-) and acidic
(H+) mechanism at 25°C is given by (Lasaga et al., 1994; Palandri and Kharaka,
2004)
⎡⎤⎡⎤EEnu H nu−−aa⎛⎞11 H ⎛⎞ 11 nH Kr25=−+− k exp⎢⎥⎢⎥⎜⎟ k 25 exp ⎜⎟ a H ⎣⎦⎣⎦R⎝⎠ T 298.15 R ⎝⎠ T 298.15 (6.54) ⎡⎤OH OH −Ea ⎛⎞11 nOH kexp25⎢⎥⎜⎟− a OH ⎣⎦R⎝⎠ T 298.15 where Ea is activation energy (kJ/mole), a is the activity of the species, and n is power term constant.
6.8.3 Integral Finite Difference Discretization
The domain of interest is discretized into small blocks so that volume- normalized extensive quantities and fluxes can be evaluated using the integral finite difference approach. Figure 6.2 shows a schematic space discretization.
r = r = L 0 n m
Flow direction
d r Vn Anm Vm Fnm nm Figure 6.2. Space discretization and variable indexing in the integral finite difference method.
The continuum equations (Equation 6.30) are discretized in space using the integral difference method. The equations for multi-component chemical transport in the liquid phase is given by (Xu et al., 2004)
130 (j),t++ 1 (j),t 1 ∆t ⎡⎤t1++ (j),t1 CCmn− ∑ AuCnm⎢⎥ nm nm+ D nm Vdnnmm ⎣⎦ (6.55) (j),t++ 1 (j),t 1 =∆MRtnn − ∆
where j = 1…Nc (total number of chemical component), n labels the grid block, m
label the adjacent grid blocks connected to n, j labels the chemical component, t
labels the number of the time step, unm is Darcy’s velocity, D nm is diffusion
coefficient, dnm is the nodal distance, and R are the overall chemical reaction
source/sink terms.
The mass accumulation terms can be evaluated as
( j),t++++ 1 t 1 t 1 ( j),t 1 t t ( j),t ∆=ϕ−ϕMSCSCnLnnL,nnn (6.56)
where subscript L labels liquid phase. The overall equation in terms of unknowns’
total dissolved component concentration is given by (Xu et al., 2004)
⎡⎤ t1++ t1∆t ⎛⎞ t1 +D nm (j),t1 + ⎢⎥SAuCLϕ+ n nm⎜⎟ − nm + n Vd∑ ⎣⎦nnmm ⎝⎠ (6.57) tt(j),t(j),t1+ =ϕSCL,n n n + R n ∆ t
Equation 6.57 is for transport of total dissolved component in liquid phase.
The transport in the gas phase takes the same principle as above with a small modification pertinent to the gas phase. The CO2(g) concentration in gas phase
can be related to partial pressure by
100 CPCO2= CO2 (6.58) RT
131 CO2 3 where C is CO2 concentration (kg/m ) and P in Pascal . By following the same
principle as used for transport in liquid phase and by considering Equation 6.60,
the numerical formulation of gaseous transport in the gas phase is expressed by
(Xu et al., 2004)
⎡⎤22CO2 10t1++ t1∆ t 10 ⎛⎞ CO2,t1 +Dnm CO2,t1 + ⎢⎥SAuPCO2ϕ+ n∑ nm⎜⎟ − nm + n ⎣⎦RT Vnnmm RT⎝⎠ d (6.59) 102 =ϕ+∆SPRtttCO2,tt1+ RT CO2,n n n n
κ,t+1 To solve the above equations, a residual function (Rn ) is introduced for
each volume element, Vn using
κ+,t 1 κ+ ,t 1 κ ,t∆t κ+ ,t 1 κ+ ,t 1 RMMnnn=−−∑ AFRt nmnmn +∆ (6.60) Vn m
which κ is the total number of equations (equal to the number of components
species). These equations are solved by Newton-Raphson method by
introducing an iteration step index (p) and using first order of Taylor series as
shown in Equation 6.61
κ+,t 1 ∂R n κ+,t 1 ∑ p (xi,p+ 1−= x) i,p R n (x) i,p (6.61) i ∂x i
where x are the independent primary variables, i is the index for the primary
variables. All terms ∂∂R/xni in the Jacobian matrix are evaluated by numerical differentiation. Equation 6.60 is solved by means of preconditioned conjugate
κ,t+1 κ,t+1 gradients. Iteration is continued until the residuals Rn / Mn at each iteration step are reduced below a convergence tolerance of 10-4. In general, for a system
of n equations and n unknowns, the Jacobian is an nxn matrix with n2 entries:
132 ⎛⎞∂∂RR11 ⎜⎟∂∂xxK ⎜⎟1n []J = ⎜⎟MOM (6.62) ⎜⎟ ⎜⎟∂∂RRnn ⎜⎟L ⎝⎠∂∂xx1n
By writing the residual functions and corrections as vectors, the general equation
for determining the correction in Newton-Raphson iteration is given by
[Jx][∆=] [ R] (6.63)
Given an estimate (first guess) of the root to a system of equations, the residual
for each equation is calculated. If the residual is larger than the prescribed tolerance value, a new estimate of the solution is performed using a weak line search method (the reduced-Newton method) using
[0] −χ ζ=2 χ = 0, 1, 2, 3, … (6.64)
We accept the first value of ζ[0] generated by this sequence that satisfies
f(x[0]+ζ [0] ∆ x [0] ) < f [0] (6.65) 22
where the function, f is computed using 2-norm at the new value. Then the new
estimate of the solution is given by
[1] [0] [0] [0] xx=+ζ∆ x (6.66)
Since ∆x[0] is chosen from the solution of J[0]∆x[0] = -f[0], it can be shown that
unless the Jacobian is singular, there must be some positive value of ζ[0] that satisfies this condition.
Equations 6.57-6.60 are dealing with solution for the flow and transport part of the model. The second part, chemical reaction, is discussed next.
133 The solution for the chemical reaction can be written in the form of
instantaneous rate of change in the system’s bulk composition. This is accomplished using (Bethke, 1996)
′ ′′(t− t ) ϑ=ϑ−ww(t) (t ) νϒ+ϒ wrr (t) (t ) ∑ k ( ) (6.67) 2 r rk
′ ′′(t− t ) ϑ=ϑ−ii(t) (t ) νϒ+ϒ irr (t) (t ) (6.68) ∑ k ( ) 2 r rk
′ ′′(t− t ) ϑ=ϑ−mm(t) (t ) νϒ+ϒ mrr (t) (t ) ∑ k ( ) (6.69) 2 r rk
rr where t΄ is the previous time step, t is the new point in time and k represents
change of components’ composition due to kinetic reaction. To solve for the
chemical system at t, Newton-Raphson iteration is used to minimize a set of
residual functions. The Jacobian matrix is given by differentiating the residual
functions with respect to the independent variables, nw (H2O mass) and mi
(molality of basis species). The Jacobian matrixes are (Bethke, 1996)
∂Rw J55.51mww==+ν∑ wj j (6.70) ∂nw j
′ ∂Rnww (t− t ) JmA/m== νν+ ννΩΚrr r r% r wi∑∑ wj ij j wrkk ir ir k min r k r k i (6.71) ∂mm 2 r iijrk
∂Ri Jmmiw==+ν i∑ ij j (6.72) ∂nw j
134 ′ ∂Ri ⎡⎤(t− t ) Jn′′′′′′′==δ+νν+ m/m ννΩΚrr r A/m rr% ii wii⎢⎥∑∑ ijijj i iririrminrrkk k kk i (6.73) ∂m2r i′ ⎣⎦jrk
where i stands for basis species, j for secondary species, w for water, and δii΄ is
the Kronecker delta function (δii΄ = 1 if i=i΄ otherwise 0, if i≠i΄)
The reactive surface area (Amin) of the kinetic minerals must be evaluated after each iteration to account for changing mineral masses.
6.9 Solution Method
The large system of equations used in reactive transport modeling are often
highly nonlinear, make it important to choose an appropriate approach for
solution. Instead of substituting the chemical equations into the transport
equations and solving them simultaneously, a Sequential Non-Iteration Approach
(SNIA) is adopted in this work (Yeh and Tripathi, 1989). Operator splitting of the
flow, transport, and geochemical equations is used to separate the processes. It
consists of separately solving the chemical and the transport equations without
iterating between transport and reaction but iterating within each transport and
reaction using Newton–Raphson iteration method. The main advantage of the
SNIA is the small set of equations that one has to solve simultaneously, leading
to low computational costs per iteration. Flow and transport calculations are
based on space discretization by means of integral finite differences
(Narasimhan and Witherspoon, 1976). An implicit time-weighting scheme is used
for individual components of the model: flow, transport, and kinetic geochemical
reaction. The chemical transport equations are solved independently for each
135 component, whereas the reaction equations are solved on a grid-block basis using Newton–Raphson iteration.
The velocity is calculated at points within a finite-difference cell based on linearly interpolated estimates of specific discharge at those points divided by the effective porosity of the cell. After solution of the flow equations, the fluid velocities and phase saturations are used for chemical transport simulation.
Transport of solute and gas is simulated using the specific discharges computed at the end of the flow time step. The resulting concentrations obtained from the transport are substituted into the chemical reaction model and calculated independently at each time step and update the aqueous solute species before the next time step. These two sets of equations are coupled by updating chemical source/sink terms. The chemical transport and reactions are iteratively solved until convergence. The set of coupled linear equations arising at each iteration step is solved iteratively by means of preconditioned conjugate gradient methods (Barrett et al., 1994). Figure 6.3 shows the flow chart for solving isothermal fluid flow, solute transport, and chemical reaction.
136
Read Input for flow, transport and chemistry
Apply CSM, initialize chemical constants and assign chemical state variables to each grid block
Time Step: ∆t
Solve fluid equations
Fluid velocities Coupled transport and reaction
Solve transport of solute and CO2
Solve chemistry grid-block-by-grid-block base
NO Convergence
YES Update chemical state variables for next time step
New time step (∆tn) YES NO Stop
Figure 6.3 Flow chart of the 1-D reactive transport
137
Chapter 7
REACTIVE TRANSPORT SIMULATION RESULTS
7.1 Introduction
Coupling flow, transport and chemistry, and implementing an appropriate model of CO2 solubility and equilibrium constants that include corrections for their dependency on temperature and pressure and rates of reactions will increase the confidence in evaluating the potential of CO2 sequestration in saline aquifers.
The simulation results for various scenarios including sensitivity analysis will be discussed in this chapter. First, the dependency of CO2 solubility and the subsequent trapping mechanisms based on CO2 solubility model for the Rose
Run Sandstone aquifer will be covered. Then modeling results of carbonate, silicate, and anorthite aquifer will follow. At last sensitivity analysis based on temperature, pressure, salinity, and reaction rate will be discussed.
138 7.2 Rose Run Sandstone and CO2 Solubility Model
Results of reactive transport numerical simulation that was conducted using the mineral and brine composition of the Rose Run Sandstone are given below.
These results are obtained using DS-CSM, Xu-CSM and MRK-CSM, and according to parameters given in Table 7.1. In this section, simulations of the
temporal and spatial evolution of the saturation of CO2, pressure and pH; the
fugacity and CO2 solubility; and mineral precipitation/dissolution and the ultimate
CO2 uptake by precipitated carbonate minerals are presented.
Table 7.1: List of hydrological, physical and chemical parameters. Variables Description
Rocks Rose Run Aquifer
Temperature (°C) 35
Pressure (bar) 200
Salinity (by wt %) 23 (Rose Run Sandstone)
Permeability (md) 100
Porosity (%) 12
Reactive surface area (m2/kg) 1
Reaction rate (mol/m2 s) Table 6.4
CO2 solubility model (CSM) DS, Xu, and MRK
139 7.2.1 CO2 Saturation, Pressure and pH
Figure 7.1 shows the saturation of CO2 as a function of distance at t = 10,
100, 1000 and 10000 years using DS-CSM, Xu-CSM and MRK-CSM. The region near the injection site is dominated by free gas for the first 1000 years. Since injection was continued for 100 years, the first 100 m outward from the injection point is dominated by more than 50 percent of CO2. This liquid-gas fluid coexists in the intermediate region, which extends between 0.1 km and 8 km. It is characterized by a larger interval of CO2 saturation (40 to 0.01%). The outer part, beyond 12 km, is devoid of free CO2 and characterized by single-phase fluid. All of the CSM show essentially the same pattern and magnitude of CO2 saturation as a function of time and distance.
The pressure distribution with time and distance is given in Figure 7.2.
Pressures are high near the injection site and progressively decrease outward.
The pressure beyond 12.5 km approaches zero and any reactions that may take place beyond this point is solely based on the CO2 originally dissolved in the initial brine composition. All of the CO2 solubility models yield similar patterns as indicated by Figure 7.2a-c. The pressure exerted by DS-CSM within the inner region is 10 percent higher than the pressure exerted by Xu and MRK-CSM during the first 10 years of modeling. On longer time scale, the CO2 pressure distribution between DS and Xu-CSM is similar and it is 17 percent higher than
MRK-CSM.
The pH distribution as a function of time and distance is similar for all of the
CO2 solubility models (Figure 7.3). The lowest pH is near the injection site, fixed
140 around 5.5 and the largest is far away from the center of injection, and it is about
7.2. The lowest pH occurs at the highest partial pressure of CO2.
141
142
143 7.2.2 Fugacity, CO2 Solubility and Free CO2
Fugacity depends on the partial pressure of CO2 (which is equal to total
pressure because CO2 is the only gas in the modeling) and the CO2 solubility
model considered. The fugacity distribution as a function of distance for any given time progressively decreases outward from the center of injection (Figure
7.4). On the longer time scale (above 10 years), the fugacity calculated using
MRK is 25 and 14 percent higher than DS-CSM (Figure 7.4a) and Xu-CSM
(Figure 7.4b), respectively. After the injection of CO2 ceased (100 years), the
fugacities for any given CSM tend to cluster together on the entire distance.
Based on the CSM, fugacity and salinity, the total amount of CO2
dissolved in the formation water is calculated as shown in Figure 7.5. The pattern
mimics the pressure and fugacity shown in the previous figures (Figure 7.2 and
7.4). For a longer time scale (above 10 years), results obtained based on MRK-
CSM overestimate the total CO2 dissolved by 30-40 and 10-15 percent as compared to dissolved CO2 calculated based on DS-CSM and Xu-CSM,
respectively. The dissolved CO2 distribution with the time interval of 10-10,000
years remains the same for the first 8 km before it gradually decreases to below
0.1 molal. The largest amounts of CO2 dissolved in the formation water for any
given time are 1.4 molal (MRK-CSM), 1.2 molal (Xu-CSM), and 0.65 molal (DS-
CSM).
144
145
146 The total free CO2 within the pore space of the formation rock was
calculated based on the total pressure of CO2 excluding CO2 dissolved in brine
and CO2 trapped by precipitated carbonate minerals. The free CO2 was calculated using Eq. 6.58 and normalized by the volume of each grid cell. The distribution of free CO2 as a function of time and distance is shown in Figure 7.6.
For any given time scale, DS-CSM tend to have the largest free CO2 as compare
to the other CSM (Figure 7.6a, b, c). Free CO2 modeled based on MRK-CSM
tend to have the lowest value for any given time period. According modeling
results shown in Figure 7.6 and chapter 3, if the potential of CO2 sequestration
by geological media to be evaluated based the appropriate CO2 solubility model
that takes into account the many factors that have been discussed so far, then
the implementation of DS-CSM in any reactive transport modeling would be the
correct approach, given other factors equal.
7.2.3 Mineral Precipitation and Dissolution
The distribution of change of mineral abundance is presented in Figures
7.7-7.12. Large quantities of calcite are precipitated using all CO2 solubility
models (Figure 7.7). The largest occur in MRK-CSM, which is 20 and 5 percent
higher than DS-CSM and Xu-CSM, respectively. The locations of the break in pH
and CO2 saturation (Figures 7.1 and 7.3) coincide with the pattern of calcite
precipitation and dissolution. Far away from the injection site, more calcite is
formed owing to the bicarbonate species present in the original brine.
147
148
149 Dolomite, on the other hand dissolves in large quantities and the
dissolution pattern mimics the precipitation pattern of calcite (Figure 7.8). Up to
155 mol/m3 of dolomite is dissolved according to MRK-CSM modeling. Using DS-
CSM results in less dolomite dissolution (Figure 7.8a) as compared to the other
two CO2 solubility models.
Siderite was present in small quantities in the Rose Run Sandstone (Table
6.2). Modeling using the three CO2 solubility models produced more siderite at
the end of each simulation time (Figure 7.9). For the first 4 km, DS-CSM does not
precipitate more than 0.2 mol/m3 of siderite (Figure 7.9a). However, on the longer
time scale (1000 years) and beyond 8 km from the injection site, up to 54 mol/m3 of siderite is formed. The formation of siderite under both Xu and MRK-CSM modeling tends to show the same pattern and magnitude (Figure 7.9b, c).
Dawsonite is also formed using all of the CO2 solubility models (Figure 7.10), with
smallest mass precipitated in simulation using the DS-CSM (less than 0.01
mol/m3) and the largest mass produced by MRK-CSM (76 mol/m3). At a distance
beyond 8 km, the precipitation of dawsonite is almost negligible. There is a good
correlation between Figure 7.5 and 7.10: where the mass of dissolved CO2 in the brine is the largest, the most dawsonite is precipitated.
150
151
152
153 The precipitation and dissolution of annite and albite is shown in Figure
7.11. There is a solid correlation between the formation of siderite and the dissolution of annite (Figures 7.9 and 7.11a, b, c). Annite dissolution is the exact mirror copy of siderite precipitation indicating that the source of iron for siderites is from the dissolution of annite. The lowest and highest annite dissolution for both Xu and MRK-CSM is in the range of 0.1 to 23 mol/m3, and the largest dissolution of annite in DS-CSM occurred beyond the 8 km mark. As stated earlier the presence of dissolved aqueous species in the original brine plays a big role in the dissolution of annite in places that has not been subjected to any CO2 fluxes. Similarly, there is a strong relationship between the formation of dawsonite, and the dissolution of albite and kaolinite (Figure 7.10, Figure 7.11d, e, f, and Figure 7.12a, b, c). In particular, the large surge in the precipitation of dawsonite using Xu-CSM and MRK-CSM is a direct consequence of the large amount of albite dissolution (up to 20 mol/m3) within the first 4 km (Figure 7.11e and f). Large quantities of K-feldspar are formed in Xu and MRK-CSM (Figure
7.12d, e and f).
154
155
156 7.2.4 Porosity Change and CO2 uptake
Calcite, dawsonite, siderite, K-feldspar, muscovite, and quartz are
precipitated, whereas dolomite, annite, and kaolinite are dissolved. Albite tends
to dissolve for the first 4 km, but beyond this marker, it starts to precipitate. The
precipitation and dissolution of all the minerals is captured by the change in
porosity (Figure 7.13). Regardless of the CO2 solubility model used, overall there
is a decrease in porosity. The use of DS-CSM, Xu-CSM and MRK-CSM in the
modeling, for the first 4 km, regardless of the time scale, suggests that the porosity of the formation decreases by about 0.4, 0.5, and 0.6 percent,
respectively relative to the initial 0.12 porosity. On a longer time scale (1000 to
10000 years), the largest decrease in porosity observed is 2 percent (Xu-CSM)
and 2.4 percent (MRK-CSM). In case of DS-CSM, the largest decrease in
porosity recorded is 1 percent (at 100 year).
The total amount of CO2 taken up by the precipitated carbonate minerals
3 is given in Figure 7.14. Between 1.2 and 2.7 kg/m of CO2 is trapped in carbonate
minerals using DS-CSM (Figure 7.14a). For the other two CO2 solubility models,
the quantity of CO2 trapped is much higher than the DS-CSM. For Xu-CSM, it is
3 in the range of 1.6 to 5.5 kg/m of CO2 (Figure 7.14b) whereas the range of CO2 trapped in minerals under MRK-CSM modeling is 1.5-6 kg/m3 (Figure 7.14c). On
a longer time-scale (10,000 years), more CO2 is sequestered if the simulation is
performed based on Xu and MRK-CSM.
157
158
159 7.3 Carbonate Aquifer
The hydrological, physical and chemical parameters given in Table 7.2 are
used to model the response and the long-term behavior of carbonate aquifer
subjected to CO2 injection. The rock composition given in Table 7.2 is modified to reflect the carbonate assemblage portion of the Rose Run Sandstone (Table
6.2). Only DS-CSM is used in the carbonate aquifer modeling.
Table 7.2: List of hydrological, physical and chemical parameters carbonate aquifer. Variables Description
Rocks Carbonate Aquifer (Dolomite, Calcite and Siderite)
Temperature (°C) 35
Pressure (bar) 200
Salinity (by wt %) 23 (Rose Run Sandstone)
Permeability (md) 100
Porosity (%) 12
Reactive surface area (m2/kg) 1
Reaction rate (mol/m2 s) Table 6.4
CO2 solubility model (CSM) DS
160 On time scale above 10 years, the fugacity of CO2 remains the same
(around 60 bar) for the first 8 km (Figure 7.15a). The same pattern is also
displayed by the total amount of CO2 dissolved in the system (Figure 7.15b). The
total CO2 dissolved in the brine for times greater than 10 years is about 0.5 molal
(Figure 7.15b). Large amount of dissolved CO2 and higher fugacity value are registered during the first 10 years of simulation (Figure 7.15a, b). The total CO2
trapped in the precipitated carbonate rocks paints a completely different picture
as compared to that of the Rose Run Sandstone modeling (compare Figures
7.15c and 7.14c). Regardless of the time of simulation, more CO2 gas is added
into the brine from the dissolution of carbonate rocks, particularly dolomite. The
release of more CO2 into the formation brine is also traced beyond the 2 km
marker point except for the simulation period that lasts 10,000 years (Figure
7.15c).
161
162 7.4 Silicate Aquifer
The approach adopted in the silicate aquifer modeling is the same as that
of carbonate aquifer except the rock assemblage considered here is the silicate
portion of the Rose Run Sandstone (Table 7.3). No carbonate other than a small
percentage of siderite is considered in the original rock assemblage as described
in Table 6.2. Modeling results of CO2 fugacity, dissolved CO2 and the total CO2 trapped in precipitated carbonate rocks are given in Figure 7.16. The overall pattern of CO2 fugacity and dissolved CO2 distribution in time and distance
remains the same as that of carbonate aquifer. However, the magnitude of the
fugacity of CO2 simulated in the silicate aquifer is less than that of carbonate
aquifer by 10 to 25 percent (Figure 7.16a and Figure 7.15a). The total dissolved
CO2 in the formation waters within the first 2 km from the center of injection is
between 0.45 and 0.6 molal as shown in Figure 7.16b. This range is 7 to 14
percent less than the dissolved CO2 calculated using the carbonate aquifer.
The total CO2 trapped in the precipitated carbonate rock is higher than the
CO2 trapped by the carbonate aquifer (Figure 7.16c). From 0.5 up to 14 mole of
CO2 per cubic meter of rock can be trapped in the silicate aquifer according to
the modeling shown in Figure 7.16c. Unlike the carbonate aquifer, there is no
addition of CO2 (dissolution) into the brine in the silicate aquifer. The absence of
CO2 trapped in the carbonate rock or dissolve in to the formation brine, at
distance beyond 12 km from the center of injection coincides with the near zero
fugacity and dissolve CO2 (Figure 7.16a-c).
163
Table 7.3: List of hydrological, physical and chemical parameters (silicate aquifer). Variables Description
Rocks Silicate Aquifer (Albite, Annite, Kaolinite, K- feldspar, Quartz and Siderite)
Temperature (°C) 35
Pressure (bar) 200
Salinity (by wt %) 23 (Rose Run Sandstone)
Permeability (md) 100
Porosity (%) 12
Reactive surface area (m2/kg) 1
Reaction rate (mol/m2 s) Table 6.4
CO2 solubility model (CSM) DS
164
165 The silicate aquifer (Table 7.3) was modified to include anorthite in the
original rock composition. The modeling result is shown in Figure 7.17. The
pattern of the fugacity, dissolved CO2 and the total uptake of CO2 by minerals formation remains the same as that of silicate aquifer on the entire time and distance scale. The magnitudes of the fugacities and dissolved CO2, however,
are 18 and 10 percent higher than the silicate aquifer, respectively, whereas the
total CO2 uptake by minerals in the silicate aquifer is 40 percent higher than the anorthite-bearing aquifer (compare Figure 7.17a, b and Figure 7.16a, b).
7.5 Sensitivity Analysis
Reactive transport models depend on large collections of input
parameters, including temperature, pressure, porosity, permeability, solution
chemistry, and thermodynamic and kinetic constraints. A minor change in any of
these quantities may produce a significant change in the outcome and quality of
the model output. Sensitivity analyses attempt to address this problem in a
systematic manner and draw a conclusion regarding the overall effect each
parameter has on CO2 sequestration in saline aquifer. The range of input values
for temperature, pressure, salinity, and reaction rate are given in Table 6.7. The
sensitivity analysis has been performed using DS-CSM for the entire Rose Run
rock composition. Results of total CO2 trapped through mineral trapping are given in sections 7.5.1 and 7.5.2.
166
167 7.5.1 Temperature and Pressure
The temperature, pressure and salinity varied according to input data
given in Table 6.7. The increase in temperature accelerates the chemical reaction, which creates a favorable condition for more CO2 to be trapped as
carbonate minerals (Figure 7.18). For example, with 2 km from the center of CO2 injection for the time scale of 10,000 years, the CO2 trapped at 35, 45, 55, and
75°C is 1.2, 1.7, 3.4, and 8.4 kg/m3, respectively. According to Figure 7.18, a
jump in temperature from 35 to 75° C, would enhance the CO2 uptake through mineral trapping mechanism by more than 86 percent. At higher temperatures
(Figure 7.18c, d), the longer the time scale is, the higher the amount of CO2
trapped by precipitated carbonate minerals. At distances far away from the
center of injection (beyond 12 km), the amount of CO2 trapped due to
temperature variation is more or less similar regardless of the time scale
(compare a, b, c, and d in Figure 7.18).
Modeling results for three pressures, 100, 200 and 300 bar, are shown in
Figure 7.19. For the first 2 km, regardless of the time scale of modeling, the
uptake of CO2 by minerals at 200 bar (Figure 7.19b) is 20 percent less than the
CO2 uptake at 100 bar and 300 bar (Figure 7.19a, c). At a distance beyond 8 km,
more CO2 trapped in the precipitated carbonate rocks at lower pressure than
higher pressure.
168
169
170 7.5.2 Salinity and Reaction Rate
It is important to explore the impact of salinity on the total uptake of CO2 through the different trapping mechanisms. Four different brine compositions, namely brine from Grand Rapid, Mount Simon, and Rose Run and a hypothetical
3 percent salinity were used to evaluate the response of the Rose Run aquifer rock composition to the injection of carbon dioxide (Tables 6.3, 6.7 and Figure
7.20). For the 3 and 7% (Grand Rapids) salinity, the pattern and magnitude of trapped CO2 within the first 8 km remains the same as shown in Figure 7.20a and b, regardless of the time scale. The CO2 distribution pattern in Mount Simon is
also the same as the Grand Rapid, but the uptake of CO2 by minerals in the
former exceeds that of the latter by as much 37 percent. For the first 100 years,
more CO2 is added into the brine (dissolution) rather than being sequestered in
precipitated minerals (Figure 7.20a and b). The largest CO2 trapped in the Grand
Rapid and Mount Simon brines are 85 and 95 percent more than the Rose Run
brine, respectively (Figure 7.20b and d). All but the Rose Run brines sequester
large carbon dioxide when simulation time progressively increases from 10 to
10,000 years for the first 10 km (Figure 7.20). The increment in the total CO2 trapped with time collapsed to almost a single value beyond the 8 km marker point. The overall trend and magnitude is completely different in case of the Rose
Run brine in comparison to the rest. The CO2 trapped at time scale above 1000
years remains the same at 1.25 kg/m3 for the first 8 km and it takes off to reach
2.6 kg/m3 beyond the 8 km marker (Figure 7.20d). Unlike the 3 percent and
171 Grand Rapid brine compositions, the Rose Run shows lager CO2 being trapped
in the first 100 years within the first 8 km from the center of the injection site.
The effect of the rate constant on the chemical reaction is evaluated by both
increasing and decreasing the rate constant given in Table 6.4 by a factor of 100.
The modeling result is reported in Figure 7.21. The lowering of the rate constant
by a factor 100 forced the amount of CO2 to be trapped in the minerals to fall by
a factor of 2 to 6 (Figure 7.21a, and b). On the other hand, increasing the rate
constant by a magnitude of two enhanced the CO2 trapping mechanism through
carbonate mineral precipitation. For the time scale of 10, 100, 1000, and 10,000 years, the largest increments are 85, 88, 90 and 91 percent, respectively (Figure
7.21b and c).
172
173
174 7.6 Concluding Remarks
The dissolution of primary and precipitation of secondary minerals changes the formation porosity and permeability, and this will change the fluid flow pattern. The carbon dioxide pressure, the amount of CO2 dissolved in the brine, the type of minerals precipitated and/or dissolved, the effect of varying the magnitude of the modeling parameters, and the quantity of CO2 sequestered through mineral trapping mechanism underscore the strong link between geochemical and physical processes. The feedback loops between chemistry, flow and solute transport and the potential of the Rose Run aquifer in sequestering CO2 through different mechanisms are discussed in chapter 8.
175
Chapter 8
DISCUSSIONS AND CONCLUSIONS
8.1 Discussion
Modeling of the CO2 sequestration in saline aquifer demonstrated that many
parameters play a crucial role in partitioning the injected CO2 into free CO2 filling the pore space, CO2 dissolved in brine and CO2 trapped in carbonate minerals. These parameters includes the change in pH and CO2 fugacity, the dissolution and precipitation of carbonate minerals, the dependency of CO2 dissolution on
appropriate CO2 solubility model, and the effect of changing the chemical and physical factors.
The fugacity of CO2 controls how much CO2 is available in the model system for
reaction and consequently the evolution of the pH over the course of the reaction.
The pH of the systems remains below 5 for the first 2 km, regardless of time, because free CO2 is dominated in the inner region. Because of the constant supply
of CO2 for the first 100 years, more CO2 dissolved in the brine and the pH is not
completely neutralized by reactions. However, in regions beyond 4 km, the pH jumps
176 to 7, because the CO2 that was injected through the first 100 years at constant rate
gradually decreases through space and time. This coincides with the observation of
the total pressure and CO2 saturation distribution both in time and space (compare
Figures 7.1, 7.2, and 7.3). This relationship suggests that adding more CO2 into
silicate aquifer and Rose Run formation will enhance the uptake of CO2 by the
precipitated carbonate minerals. In carbonate aquifer, it paints a different picture: It
will increase acidity and dissolve, not precipitate carbonate minerals (compare
Figures 7.7, 7.9, 7.10, 7.14, and 7.15). Depending on the CO2 solubility model
considered, the CO2 remains as dissolved aqueous species such that the carbonate
layers of the Rose Run are unlikely to contribute to mineral trapping of CO2. This is in conformity with others work, that suggest the dominant CO2 sequestration
mechanisms in carbonate host rocks are solubility and hydrodynamic trapping.
The dissolution of dolomite tends to counterbalance the uptake of CO2 by the calcite precipitation (Figure 7.7). The total uptake and production of CO2 by these
two carbonate rocks offset each other with a net out come of a small mass of CO2 being trapped in the precipitated carbonate rocks. This is because one mole of CO2
is consumed in producing one mole of calcite (removal of CO2 though mineral
trapping) whereas in dissolving one mole of dolomite, two moles of CO2 are
produced (enhancing solubility trapping mechanism) (compare Figures 7.7 and 7.8).
Hence, if the aquifer is silicate rock assemblage or mixed rock assemblage the
contribution of calcite and dolomite minerals in trapping CO2, both in the long and
short-term is minimal.
177 Precipitation of siderite is pivotal for large sink of CO2 by the mineral trapping
mechanism. More siderite forms as serve as a sink for CO2 for CSM that predict higher solubilities of CO2 in the brine. The dissolution of annite provides Fe species.
This dissolve aqueous Fe bonded with the dissolve CO2 to produce siderite. The
annite dissolution and siderite precipitation, mirror each other, are shown in Figures
7.9 and 7.11. The formation of dawsonite also follows the exact trend of siderite
(Figure 7.10). Large quantity of dawsonite is formed when the total CO2 dissolve in
the brine is sufficient to meet the dynamic stability of dawsonite. The dissolved albite
provides the right amount of Na solute, which is added into the brine that has initial
Na and brings about a shift in the equilibrium of the system that is favorable for the
precipitation of dawsonite. In addition, the less than 0.1 mol/m3 dissolution of albite
using DS-CSM (Figure 7.11d) and the small quantity of precipitated dawsonite
(Figure 7.10a) strongly suggest that the amount of aqueous solute present in the
initial brine is not sufficient enough to initiate a large amount of dawsonite
precipitation. The large amount of kaolinite dissolution follows the same trend as that
of albite and dawsonite (Figure 7.12a, b, c). The dissolution of kaolinite as well as
albite provides the much-needed Al aqueous solute for the formation of dawsonite.
Modeling using Xu and MRK-CSM favored the formation of siderite and dawsonite in
large quantities.
Siderite and dawsonite are likely to be very important for mineral trapping of CO2 in many deep saline aquifers, like the Rose Run Sandstone, because many deep sandstone formations that are otherwise suitable for geologic sequestration lack abundant sources of Ca, Mg, and Sr that are necessary to induce precipitation of the
178 carbonate minerals calcite, dolomite, and strontianite. Precipitation of these
carbonate minerals requires that sufficient amounts of Fe, Na and Al be available for
reaction. These ions are added to the brine by the dissolution of aluminosilicate minerals in response to the increase in acidity due to the addition of CO2. In addition, the Na needed for dawsonite precipitation is available in great quantities in the initial brine. Dissolution of aluminosilicate, besides providing Fe, Na and Al minerals it neutralizes the acid formed by dissolution of CO2 into the brine.
Implementing an appropriate CO2 solubility model that quantifies the
amount of CO2 dissolve in brine solution is every important. The CO2 solubility
model developed by Duan and Sun (2003), McPherson and Cole (2005), and Xu
et al. (2004) have both advantages and disadvantages in evaluating the quantity
of CO2 sequestered through solubility and mineral trapping. MRK-CSM
(McPherson and Cole, 2005) is more appropriate in pure water because it
neglect the effect of salinity. As shown by the amount of CO2 dissolve in the brine
and the subsequent uptake by precipitated minerals, MRK tends to overestimate
the real partitioning of injected CO2 (among free, dissolve and trapped in mineral)
in saline aquifer. If MRK-CSM is included in any reactive transport code that uses to simulate CO2 sequestration potential in saline aquifer, then the dissolved CO2 and mineral trapping will be over estimated by up to 40 and 60 percent, respectively. The present work suggest that using Xu-CSM developed by Xu et al. (2004) in any reactive transport, though it has the capability to model solubility of CO2 in brine solution, however, it consistently overestimates the solubility of
CO2 on in the entire range of temperature, pressure and salinity as compare to
179 DS-CSM. Hence, any simulation result obtained using Xu-CSM need to be
carefully evaluated based on the intended purpose and degree of acceptance
sought.
The DS-CSM of CO2 solubility model that has been developed based on the
equation of state of Duan et al. (1992b) and the theory of Pitzer (1973) by Duan
and Sun (2003) is by far the most accurate CO2 solubility model. As discussed in
chapter 3, it accurately modeled CO2 solubility in pure water and aqueous
solution for a range of temperature, pressure and salinity. This model is used in
the1-D reactive transport modeling to predict the CO2 short and long-term fate
through solubility and mineral trapping. As compare to the other two CO2 solubility models, DS-CSM suggests that in the short-term the hydrodynamic trapping mechanism plays the leading role in CO2 sequestration in geological formation. It is vital to include DS-CSM in any reactive transport code development as it accurately modeled experimental data in a range of temperature, pressure and salinity. However, its complex form may add a
significant burden for implementation into already computationally intensive fluid
flow, transport and chemical reaction computer models, such as TOUGHREACT,
FLOTRAN, and CRUNCH as it requires an iterative solution.
DS-CSM has a built in methods to account for corrections to the equilibrium constant as a function of both temperature and pressure. However, many geochemical modeling software packages provide default datasets where the equilibrium constants are only valid at 1 bar from 0-100°C and steam saturation pressures thereafter. In this work, the effect of temperature and pressure on
180 equilibrium constant for a simplified and small database based on the theories
explained in chapter 6 is included. With the increase in pressure, the thermodynamic
equilibrium constants increase. The increase of equilibrium constant with the
increase in pressure forced more minerals to be dissolved, hence the amount of CO2 uptake by minerals would expect to decrease. However, as shown in Figure 7.19, an increase in pressure from 200 to 300 bar and a decrease from 200 to 100 bar produce a 20 percent increase in the CO2 uptake. This suggests beside the
equilibrium constant, there are other factors that play a role and provide a complex
feedback within the system. For example, fugacity increases as pressure increases
and this will intensify the activity of CO2. In addition, the activities calculated based on Debye Huckel equation were not corrected for such high pressure, as there is luck of data at elevated pressure. Hence, the increase or decrease in CO2 uptake by
minerals may not reflect the ‘actual’ chemical reaction that prevails at such high
pressures.
The dependency of the overall reaction on some of the physical and chemical
parameters was investigated through sensitivity analysis. Temperature, salinity, and
rate of reaction are the dominant factors that produce a large change in the
magnitude of chemical reaction output. Temperature plays such a pivotal role that a
jump from 35° to 75° can increase the CO2 trapped through mineral precipitation by
a factor of five. This is because the chemical reaction enhanced at higher temperature as manifested by the dissolution equilibrium constant. The increase in pressure did not produce major change as that of temperature. On the other hand, salinity and temperature effect have a comparable impact on the overall
181 geochemical reaction. For example, the decrease in salinity from 23% to 11%
accompany by a large uptake of CO2 through mineral trapping (compare Figures
7.20c and 7.20d). This is because large amount of CO2 can be dissolved in less
saline brine. The availability of enough aqueous carbon dioxide would lead to the
formation of carbonate minerals provide that there is enough source of divalent
cations within the system. This also subject to the dynamic stability and favorable
thermodynamic constant of the minerals participate in the consumption of the CO2.
The increase in CO2 uptake when the salinity increases from 3 to 11% (as opposed
to the preceding) suggest that a decrease in salinity by itself is not sufficient enough
to produce the difference observed between Rose Run and Mount Simon. The brine
need to be rich enough with dissolve aqueous solutes so that the reaction can take
place at the same time it should be dilute enough to dissolve more CO2 in the brine.
Hence, in the long-term, the amount of CO2 sequester though mineral trapping
mechanism strongly depends on salinity and temperature of the reservoir.
Particularly, salinity along with the choice of CO2 solubility model needs to be
properly addressed. Salinity plays such an important role that ignoring its effect on
CO2 solubility, particularly in brines with high total dissolve solid (TDS) would gravely
overestimate the long-term trapping of CO2 through the precipitation of carbonate
minerals.
The increase in kinetic rate constant accelerates chemical reaction. Most of rate
constants published in the literature are generated in the laboratory and they are
subjected to the intended purpose of the experiments. The rate constant produced in
the lab under different pH condition may be by several magnitudes higher than the
182 actual field reaction rate. As shown in Figure 7.21, the modeling is very sensitive to
the rate constant and an order of magnitude difference can produce a large
difference. Hence, it is important to consider rate constants that are close to the intended purpose, if available.
The modeling predicts a net rock mass gain from 0.5 to 8 kg of rock per meter cubic of medium for the Rose Run Sandstone. The gain in mass accompanies by an uptake of 1-6 kg of CO2 per cubic meter of medium. The gain in rock mass is due to
the precipitation of carbonate minerals, quartz, muscovite, and K-feldspar. The
increase in rock mass is accompanied by a decrease in porosity of 0.1% to 2%,
which has the potential both to cause clogging, slowing injection, or, on the positive
side, can improve the cap rock sealing.
In all cases discussed so far, CO2 saturation plays an important role as it controls
the amount of gas and liquid within each grid points. At the center of the injection
site, the gas saturation is close to one and it gradually decreases outward radially.
That means any reaction that may take place within each grid point will subject to
the balance of the right saturation of liquid that can generate dissolution or
precipitation of minerals. For example at the end of 10 year, except for calcite and
dolomite, close to the injection site, no mineral precipitate or dissolve, whereas with
increasing the saturation of liquid, more reactions take place. While it reacts, CO2 continues to migrate and covered large area through time depending on the initial pressure of the CO2. In most cases, it barely reaches the 16 km marker point. This is
because it loses strength when it flows radially outward and, through time, it is subjected to chemical reaction. In addition, the injection was carried out at a
183 constant rate for 100 years and there was no any external source of CO2 for the rest of the simulation period. Increasing pressure would increase the amount of CO2
available, however, in doing so the allowable limit of injection pressure may be
exceeded and reach the injection fracturing pressure. Exceeding the fracturing
pressure has a severe consequence on the integrity of the formation, which can
compromise the safety and intended purpose of the injection.
3 The effective storage density (kg CO2/m rock) for the different mechanisms of CO2 trapping estimated for the Rose Run Sandstone indicates that it depends on the type of CO2 solubility model adopted for a given time and distance from
the center of injection. Based on DS-CSM, close to the injection site (within the
first 2 km), the total CO2 trapped at the end of 10 years by free CO2, dissolve
3 CO2 and CO2 trapped in minerals are 52, 2.5 and 1.25 kg/m , respectively. This
suggests that more than 90% of CO2 is sequestered through hydrodynamic
3 trapping mechanism. At the end of 1000 years, 10, 2.2 and 1.25 kg/m of CO2
sequester through hydrodynamic, solubility and mineral trapping, respectively.
The amount of CO2 trapped in the precipitate minerals remains the same.
However, due to fluid flow, much of the free CO2 (more than 80%) disperse
through time to a wider area. The storage capacity estimated using Xu and
MRK-CSM give different figures. The model calculation using Xu-CSM, at the
3 end of 10 year indicate 48.4, 4.6 and 1.6 kg of CO2/m of rock are potentially
sequestered through free, dissolve and precipitate carbonate minerals,
respectively. Modeling using MRK-CSM suggests that CO2 trapped in free,
dissolve and precipitate carbonate minerals are 39.6 kg/m3, 5.3 kg/m3 and
184 1.7kg/m3, respectively. At 1000 years: modeling under Xu-CSM suggest 8.4, 3.1
and 5 kg/m3 and calculation using MRK-CSM shows 6.6, 3.6 and 5.5 kg/m3 of
CO2 can be potentially sequestered through hydrodynamic, solubility and mineral
trapping, respectively.
All CO2 solubility model developed by different scholars may suggest that
large errors in CO2 sequestration capacity estimation are likely when modeling
the outcome of CO2 injection into deep saline aquifer. Because of the chemical
complexity of injection induced water-CO2-rock interaction processes, an
extensive theoretical framework that is suitable for quantification of the storage
capacity on actual partitioning of sequestration mechanism, as a function of
space and time is required.
8.2 Conclusions
The effect of fluid dynamics and associate reaction of CO2 in the Rose
Run sandstone saline aquifer is explored in this dissertation. Improved CO2 solubility models and thermodynamic database are used to analyze the overall flow and reaction behavior under simplified hydrological condition using existing geochemical software and using the newly developed 1-D reactive transport. In the first three chapters the general concept, techniques and approach in light of
CO2 sequestration in saline aquifer as well as applicability of existing geochemical modeling were discussed. The potential of Rose Run Sandstone in trapping anthropogenic carbon dioxide were evaluated based on batch of reaction, in the absence of flow. The performance of trapping mechanisms were
185 evaluated through the existing Geochemists’ Workbench (GWB) software and
standalone solubility CO2 solubility models coded in matlab. The detail work and associate conclusions were discussed in chapter four and five.
To predict the short and long-term effect of CO2 sequestration in saline
aquifer using reactive transport modeling, it is very crucial to use an appropriate CO2
solubility model that has the potential to represent the solubility of CO2 in brine at the
injection pressure and temperature. Results from the reactive transport indicate that
the extent of mineral trapping depends strongly on the fugacity of CO2 and the
choice of CO2 solubility model that properly address the solubility of CO2 under a variety of temperature, pressure and salinity conditions. Particularly, the extent of sequestration through solubility and mineral trapping is sensitive to the choice of
CO2 solubility model. This has been demonstrated through the deployment of Duan and Sun (DS-CSM), Xu-CSM and modified Redlich-Kwong (MRK-CSM) CO2 solubility models. Duan and Sun CO2 solubility model is correlated well with existing
experimental works and incorporating it in reactive transport codes produce realistic
results that are pertinent to high saline aquifers. However, its complex form may add
a significant burden for implementation into already computationally intensive fluid
flow, transport and chemical reaction computer codes.
Reactive transport modeling underscores in the long-run siderite and
dawsonite minerals are important sink in trapping CO2 in the Rose Run
sandstone but over a short time-scale the hydrodynamic trapping plays a crucial
role. The mineral composition of the Rose Run sandstone is similar to other deep formations that are being considered for sequestration of CO2. The
186 potential for siderite and dawsonite formation make these widespread basins
better candidates for CO2 storage because mineral trapping can take place
through reaction with aluminium-silicates and glauconite minerals.
The rate of mineral-brine-CO2 reactions relative to the rate of flow and dispersion of CO2 away from the site of injection were evaluated. The modeling
suggest that, through time, chemical reactions tend to reach carbonate phase
saturation before the CO2 is overly diluted by outward radial flow as shown by the
large precipitation of calcite, siderite and dawsonite minerals. The necessary
ingredient for the precipitation of the above minerals provided through the dissolution of CO2, albite, annite, and kaolinite. The solubility of CO2 in the brine
depends on the CO2 solubility model adopted; the extent of carbonate minerals
formation strongly reflects this as discussed in chapter seven.
Sensitivity analyses were conducted to evaluate the long-term geochemical behavior of the injected CO2 and its possible impact on the integrity of the formation.
Sensitivity analysis indicates that the precipitation of carbonate minerals, hence the
uptake of CO2 through mineral trapping, depends on the composition of mineral
assemblage, salinity, temperature, CO2 fugacity, and rate of reaction. Carbonate aquifers favour sequestering CO2 through hydrodynamic and solubility trapping,
whereas silicate aquifers, on the long-term, serve as a sink for CO2 through mineral
trapping. A change in salinity, temperature, pressure of CO2 and rate of reaction can force the CO2 uptake to be dominated by either hydrodynamic, solubility or mineral
or any combination of the above sequestration mechanisms. In general, the
modeling indicates that over hundreds to thousands years it should be possible to
187 sequester large quantities of CO2 by mineral trapping in the silicate and Rose Run
Sandstone, but over tens years solubility trapping plays the decisive role.
8.3 Future Work
Reactive transport modeling is a powerful tool that can help to evaluate
the behavior of long term reaction and solute transport that can not be readily
studied at laboratory or field scale in short period of time. Nevertheless, it is
always important to keep in mind that a reactive transport model, like any science
or engineering simulation, is only a cartoon of reality. It produces output that is
only accurate to the same degree as the input data, and thus ultimately reflects
our understanding of a given system (complex or simple). In this regard, the
greatest challenge in applying reactive transport models is, and always will be,
assessing how well the conceptual model of a system is defined, based on real
field observations, and how well that conceptual model can be manifested in the
model. Hence, future work needs to include experimental and field observation
pertinent to the intended purpose.
Future work also need to focus on way to improve the data quality of the
main chemical, physical and hydrological parameters. It is imperative to model any conceptualized system or process based on appropriate CO2 solubility
model, realistic yet simple geometrical configuration, three dimensional flow
condition, and thermodynamic and kinetic data set. Formation heterogeneity,
both in terms of porosity and permeability, structure and mineral content, as well
as viscous fingering and gravity segregation are not considered in this study.
188 This heterogeneity influences how much of the aquifer is accessed by free CO2,
how quickly that CO2 dissolves into the brine, and what proportion of the reactive
minerals react with the dissolved CO2. Some questions that need to be addressed include; what is the impact of low permeability and anisotropic distribution, but highly reactive carbonate layers on the flow and dissolution of
CO2 through the Rose Run Sandstone? What is the impact of the uneven
distribution of glauconite, which is concentrated in layers, on the effectiveness of
mineral trapping reactions? In addition, faults, fractures, and plugged well holes,
which all have the potential to form vertical escape routes through the cap rock to
the surface, were not considered. Large faults and plugged well holes can be
identified through mapping and high resolution seismic, and possibly can be
avoided. Small fractures induced by injection and buoyancy pressures are more
difficult to avoid. Experimental and modeling research is needed to evaluate the
risks of leakage through large and small vertical pathways.
189
PART II
FLOW CHARACTERIZATION THROUGH A NETWORK CELL USING PARTICLE IMAGE VELOCIMETRY
190
Chapter 9
INTRODUCTION AND BACKGROUND
9.1 Introduction
Models of flow through porous media commonly apply Darcy’s law to describe the flux of a fluid as proportional to hydraulic head. The proportionality factor depends on permeability, a macroscopic property of the medium.
Modification of Darcy’s law to consider saturation-dependent relative permeabilities for both the invading and displaced fluids extends the applicability from single-phase to two-phase flow (Collins, 1961; Scheidegger, 1974; Bear,
1979). These approaches consider only macroscopic properties of the flow, not the details of flow through individual pores and pore throats (Adrian and Yao,
1985; Adrian, 1991). In reality, however, the geometries of pore bodies and pore fluids can have a greater impact on permeability and macroscopic transport rate than total pore volume (Fredrich, 1999). Pressure distributions, velocity fields, and fluid streamlines within the pores impact hydrodynamic dispersion and mixing (Rashidi et al., 1996; Yu et al. 1999; Gramling et al., 2002), fluid-solid and
191 fluid-fluid chemical reactions (Chen and Liu, 2002; Gramling et al., 2002;
Freedman et al., 2003; Keum and Hahn, 2003), and the efficiency with which one fluid displaces another (Browning et al., 2003). Our interests focus particularly on fluid transport in geological porous media where pore-scale flow affects such important environmental and industrial problems as contaminant transport, the displacement of pore phases during enhanced oil recovery or CO2 injection for
storage, and fluid-fluid and fluid-mineral reactions during deep well disposal of
acid waste.
Analytical solutions exist for flow through simple pore geometries, such as
periodic arrays of cylinders or spheres (Happel, 1959; Hasimoto, 1959). More
complex geometries of interconnected pore bodies require numerical solutions,
including finite difference and finite element methods (Zhao et al., 1999; Bryant and Thompson, 2001; Yeh et al., 2001; Pruess and Garcia, 2002), smoothed particle hydrodynamics (Zhu et al. 1999), and cellular automata models (Olson and Rothman 1997), to solve the relevant transport equations within specified pore bodies and pore throats. Hydraulic network models of porous flow, on the other hand, do not directly solve for the physics of flow within a pore body, but instead incorporate fundamental parameters of pore-scale flow into rules that govern transport properties and phase distributions across idealized networks of pores and pore throats. For example, the rules designate the differential pressure required to advance a fluid phase across a pore throat, to displace another phase, or to change the arrangement of phases within a pore body (e.g. Chen et al. 1985; Ferer et al. 2001; Blunt et al., 2002; see Blunt, 2001 for a review of
192 recent network modeling). By tracking the evolution of properties such as
pressure, relative permeability, relative phase saturation, and interfacial area,
averaged over the scale of the model, network models provide a link between
pore-scale physics and macroscale phase behavior.
New experimental techniques using thin section analysis (Ehrlich et al.,
1984; McCreesh et al., 1991; Coskun and Wardlaw, 1993, 1995), magnetic
resonance imaging (Bobroff et al., 1995; Irwin et al., 1999), confocal microscopy
(Fredrich, 1999), and x-ray computed tomography (Wildenschild et al. 2002) have been adapted to image the complex three-dimensional geometries of the pores and the fluid phases within them. In addition, early studies using laser anemometry have been used to visualize single-phase flow in highly simplified, up-scaled models of porous media (Dybbs and Edwards, 1984). Still, difficulties obtaining quantitative velocity data within individual pores have limited experimental validation of numerical models.
We report here on a technique using micro-Particle Image Velocimetry
(micro-PIV) with refractive index matching to obtain quantitative velocity data for single-phase flow through the individual pore bodies and pore throats of a transparent flow cell. Particle Image Velocimetry (PIV) is a well-established technique for macroscopic flows that maps the instantaneous velocity field by periodically illuminating tracers, usually particles, suspended in a flow (Raffel et al. 1998). Advantages are that it is a non-intrusive, in-situ technique that can be used on two-dimensional or three-dimensional flow cells and is capable of characterizing multiphase flows. PIV requires an optically transparent flow cell
193 that allows the tracers to be illuminated, repeatedly, by a sheet of pulsed light.
Thousands of image pairs, taken at a constant, known time interval, are captured
by a camera to provide a displacement record within the measurement plane.
The displacement record is then scaled to velocity. Refractive index matching of
the test cell and test fluid minimizes light reflection from the flow cell-fluid
interface, thereby maximizing image resolution. The refractive index of the
tracers is different from that of the flow cell and fluid such that sufficient light is
scattered from the particles for detection by the camera.
9.2 Background
Micro-PIV is a modification of PIV designed to measure fluid velocities in
microfluidic devices with spatial resolutions on the order of tens of microns
(Santiago et al.,1998; Meinhart et al. 1999; Devasenathipathy et al., 2003).
Complications associated with adapting PIV to the microscopic level include selection of appropriate tracers that are large enough to be imaged, yet small enough not to clog the device, minimization of errors due to the Brownian motion of tracers, and achievement of sufficient spatial resolution of the imaging system to make useful measurements of particle displacement. These problems can be solved by a combination of experimental methods and interrogation algorithms and the feasibility of micro-PIV has been demonstrated successfully for flow through straight channels and cylinders as small as 20 microns across
(Devasenathipathy et al., 2003).
194 This paper describes the application of micro-PIV to study flow through the
more complex, interconnected pore geometries that characterize porous media.
The feasibility of the technique is demonstrated with measurements of single-
phase, creeping flow through three pore bodies and adjacent pore throats of a two-dimensional, diamond-lattice, network flow cell. The geometry of the test cell
is three-dimensional but data were taken for a plane at the mid-section of the throats and pores. This work is a first step toward validating and extending studies of a two-dimensional network model based on an identical diamond- lattice structure that was developed for simulating two-phase flow through porous media (Chen and Wilkinson, 1985; Lenormand et al., 1988; Ferer et al., 1996,
2003a, b). Results are validated through error analysis, by comparing micro-PIV
measurements of velocity distribution with physically measured flow rate at the
outlet, and by mass conservation. Experimental results are compared with
numerical solutions produced by finite difference analysis. The limitations of the
technique and the potential for extending it to study two-phase porous flow, more
complex and smaller pore geometries, and three-dimensional porous flow cells
are discussed.
195
Chapter 10
EXPERIMENTAL DESIGN AND SETUP
10.1 Experimental Flow Cell, Test Fluid, and Seed Particles
The flow cell was constructed by computer-assisted micromachining the top surface of a sheet of optically transparent acrylic and affixing that sheet with screws to a second acrylic sheet. The 15.2 cm × 6.6 cm flow cell consists of 20 x
20, equal-size cylindrical pore bodies, 2.5 mm in diameter and 1.0 mm in height, connected on a diamond lattice by 2.5 mm long, square cross-section throats of widths that varied randomly among 0.2, 0.6, and 1.0 mm (Figure 10.1). A square cross-section manifold, 6 mm across, connects the pore bodies to inlet and outlet cylinders, each with a diameter of 2.5 mm. The machine error for the pore-throat and outlet-channel diameter is 1%. The refractive-index-matched fluid consists of
74% by volume sodium iodide, 21% glycerin and 5% distilled water, seeded with
196
C B
Inlet x Outle
D A
z
Figure 10.1. Diamond-lattice network flow cell showing locations where micro- PIV velocity data were collected.
2 micron diameter silicon carbide particles that have a refractive index higher
than those of the flow cell and test fluid (Table 10.1). A liquid detergent soap was
added to the NaI solution as a surfactant to minimize particle aggregation.
As shown in the schematic diagram of the experimental set-up (Figure 10.2),
the flow cell was oriented vertically with flow from the side. The flow was gravity
fed from a fluid reservoir 18 cm above the flow cell. A near constant differential
pressure of 827 Pa was maintained across the flow cell by monitoring the fluid elevation and refilling the reservoir before the level dropped by 1 cm. This differential pressure maintained a fluid flux through the flow cell of 1.1x10-7 m3/s as measured at the outlet of the tube.
197 Table 10.1. Fluid, seeding particle and PIV parameters.
LASER and CAMERA FLUID SEED PARTICLE
Nd:YAG laser - 120 NaI - 74% by volume Silicon Carbide mJ/pulse Beam diameter - Glycerin - 21%
5mm H2O - 5%
Wavelength - 532 nm Density - 1.7 g/cm3 Density - 3.65 g/cm3
Maximum frequency - 15 Hz Refractive index - 1.485 Refractive index - 2.65
Camera - Kodak ES 1.0 CCD Dynamic viscosity - 5.4 cP Settling Velocity - 1µm/s
Pixel size - 9.0 µm × 9.0 µm Mean Size - 2 µm (St. Dev. 1.4)
CCD camera array size - 1008 × 1018 pixels
Subregion size - 64 × 64 pixels with 50% overlap for cross correlation
Micro-PIV measurements were conducted for the outlet of the flow cell and for three pore bodies and their adjoining throats (Figure 10.1). Velocity data for the outlet channel were collected at location A, 20mm downstream of manifold outlet.
The pore bodies selected for analysis were each connected to four throats of equal size: 1 mm across at pore B, 0.6 mm at pore C, and 0.2 mm at pore D.
Prior to the experiment, the flow cell was cleaned ultrasonically to remove any trapped particles. The fluid reservoir was stirred approximately 10-20 minutes prior to the start of the experiment in order to distribute the particles evenly through the fluid. After velocity measurements were completed for the outlet location, A, and for pores B, C, and D, the FOV of the camera was changed to
2.5 mm by 2.5 mm and velocity measurements were repeated for pore C.
Because changing the FOV required adjustment of the flow cell position, the
198 influx to the cell was temporarily disrupted and then restarted to accommodate the change.
Figure 10.2. Diagram illustrating the experimental setup.
Gravitational and inertial effects on particle displacements were evaluated to assure the silicon carbide particles reliably follow the flow. Gravitational effects arise because the silicon carbide particles are denser than the sodium iodide solution. Because of the small size of the particles, the Stokes equation is valid
199 for calculating settling velocity (Vs) such that,
2 d p (ρ p − ρ f )g V = , (10.1) s 18µ
where dp is the particle diameter, ρp is the particle density, ρf is the fluid density,
µ is the viscosity of the fluid, and g is the gravitational constant. Because the
calculated settling velocity of 10-6 m/s is small compared to the minimum
measured flow velocity of 10-3 m/s the gravitational effects on instantaneous
particle velocities are negligible. Inertial effects impact how quickly the particles
respond to changes in flow velocity. They depend on the relaxation time, τs, a measure of the lag time response of the particles (Raffel et al., 1998), which is defined by: ρ τ = d 2 p s p 18µ . (10.2)
For particles denser than the fluid, the particle velocity, Vp exponentially
approaches the fluid velocity, V, with time, t, as given by
⎡ ⎛ t ⎞⎤ ⎜ ⎟ V p (t) = V ⎢1− exp⎜− ⎟⎥ (10.3) ⎣ ⎝ τ s ⎠⎦
The relaxation time of the silicon carbide particles in sodium iodide solution is 1.5
-7 x 10 seconds. A plot of the non-dimensional particle velocity, Vp/Vf, versus time
demonstrates that the particle velocity approaches the fluid velocity in
approximately 7x10-7 seconds (Figure 10.3), which is negligible compared to the
time separation of 0.1 to 5 ms between pairs of images used to for cross correlation and computation of particle displacements.
200
1
0.8
0.6 f /V p V 0.4
0.2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (µ s)
Figure 10.3. Time response of the 2 µm diameter silicon carbide seed particles. The y-axis shows the ratio of particle velocity to fluid velocity whereas x-axis represents time elapsed.
10.2 Image Acquisition and Processing
A schematic of the image acquisition and processing system is shown in
Figure 10.2. Two optically connected 120 mJ/pulse, 532 nm wavelength Nd:YAG
lasers provide a 5 mm diameter, highly directional, collimated light beam, which
is shaped by a combination of a cylindrical lens (-6.35 mm focal length) and a
spherical lens (300 mm focal length) into a planar light sheet. A mirror bends the
light sheet 90º so that it parallels the vertical plane of the flow cell. Image capture
and image analysis hardware consist of a Kodak ES1.0/10 bit (1018 x1008 pixel)
CCD camera oriented perpendicular to the light sheet, an EPIX PIXCI-D
framegrabber, and a computer for data collection and processing. The microscopic lens assembly consists of a zoom 6000 standard adapter (2x), zoom
201 lens (6.5x), and zoom lens attachment (2x). PIV Acquisition (PIVAQC) and PIV
Processing (PIVPROC) software provide robust image acquisition and
processing algorithms developed by Wernet (1999a and 2000).
Two different sizes for the FOV of the camera were selected for this study: 5
mm by 5 mm and 2.5 mm by 2.5 mm. The smaller FOV spans an entire pore
body and the larger FOV spans a pore body and much of the four adjoining pore
throats. The depth of field of the camera was determined with a depth of field
target to be approximately 1.5 mm for the 5 mm by 5 mm FOV and 1 mm for the
2.5 mm by 2.5 mm FOV. This depth of field is greater than the 0.16 mm thickness of the light sheet as required for well-focused images.
The laser is synchronized with the digital CCD camera in order to capture a separate, single-exposure image frame with every pulse of the light sheet. Pairs
of single exposure image frames collected at a constant exposure interval enable
cross correlation of particle locations for processing of particle displacements.
Frame straddling synchronizes the laser pulses such that the first pulse falls at the end of one frame and the second pulse falls at the beginning of the following frame, minimizing the inter-frame exposure time (Adrian, 1991; Raffel, et al
1998).
A 5 mm by 5 mm single exposure image frame taken of pore body C and its
four adjoining pore throats is shown in Figure 10.4. The image was obtained
using an image gain of 4.0 and threshold level of 10. The image gain is a
multiplier intensity applied to all pixels in an image. A higher gain setting implies
a brighter image. The threshold is the gray scale level below which particles are
202 removed from the image map. Adjustment of the threshold level aids background noise elimination. All data were taken under the same image gain and threshold level. Figure 10.4 shows a relatively homogenous distribution and medium
density of seed particles as required to produce high quality PIV images for
standard statistical PIV evaluation (Raffel et al., 1998).
5
4
3 ) mm (
Z-axis 2
1
0 0 1 2 3 4 5
X-axis (mm)
Figure 10.4. A raw image of four throat channels (0.6 mm width) connected to a pore (2.5 mm diameter) obtained from PIVACQ.
203 For cross correlation of the image pairs, both the 5 mm by 5 mm FOV and the 2.5 mm by 2.5 mm FOV were subdivided into subregions, each 64 pixels by
64 pixels. Because the cross-correlation software (Wernet, 1999a and 2000) uses Fourier transforms, which assume a periodic function, the maximum usable region of the output correlation plane is limited. According to the Nyquist sampling criteria associated with discrete Fourier transforms, the spatial displacement in any direction is limited to half of the subregion size in that direction (Adrian 1991). For a given subregion dimension N, the maximum particle displacement is N/4 (Wernet, 2000), which corresponds to 16 pixels.
Subregions were overlapped by 50% to fulfill these requirements and prevent over sampling the data.
Figure 10.2 shows a schematic view of an image pair for a subregion and the instantaneous velocity vector produced by cross correlation. Cross correlation maximizes the range of velocities that can be measured for any given minimum resolvable velocity and for a fixed set of instrumental parameters
(Adrian, 1991). Particle fallout, particle aggregation, and light scattering from the flow cell contribute noise to the images such that the cross correlation produces a cross-correlation plane with multiple peaks, rather than a single peak corresponding to average particle displacement. Three-point interpolation about the five peaks with the highest amplitudes in each subregion determines the centroids of the particle displacements for all subregions, which are then converted into an instantaneous vector map for the entire FOV. Spurious vectors are identified and corrected by a fuzzy logic processor. Multiple instantaneous
204 vector maps are collected and averaged for each FOV to reduce the effects of
small variations in the flow. 220 image pairs were required to bring the
uncertainty below 10%.
In order to determine the optimum time between laser pulses the velocity
should be known. However, due to the complex nature of the test cell, flow
velocities at individual locations were unknown a priori and it was necessary to
determine optimum time separation by trial and error. PIV images were collected
at each location and for each FOV using a small initial ∆t, determined based on
the measured velocity at the outlet to the flow cell. The ∆t was incremented repeatedly until a maximum displacement of 16 pixels per ∆t was reached and that ∆t was selected for velocity measurement. The velocity was calculated via the following formula:
∆x V = C ∆t (10.4)
where ∆x is the displacement in units of pixels, ∆t is the time in seconds between
exposures, and C is the optical system calibration factor having units of microns/pixel. The optical system calibration factor was obtained by placing a
ruler in the FOV of the camera, at the plane of the light sheet. The physically
viewed length of the image of the ruler was measured and then divided by the number of pixels yielding an optical calibration factor of 5 µm/pixel for the 5 mm
by 5 mm FOV and 2.5µm/pixel for the 2.5 mm by 2.5 mm FOV, respectively.
Converting to millimeters from pixel sizes the maximum displacements for a 5
mm by 5 mm FOV and a 2.5 mm by 2.5 mm FOV are 0.08 mm and 0.04 mm,
205 respectively. In addition, the spatial resolution was determined from the image
pixel size and the number of pixel centers, which gives 160µm/vector for the
5mm by 5mm FOV and 80µm/vector for the 2.5mm by 2.5 mm FOV.
Diffraction causes the imaged size of tracer particles to be greater than their geometric size. In this study, the imaged particle size is greater than 1 pixel for both the 5 mm by 5 mm and 2.5 mm by 2.5 mm FOV, as required to minimize
problems in data processing resulting from locking of the correlation peaks onto
integer pixel values.
206
Chapter 11
EXPERIMENTAL RESULTS
11.1 Velocity Measurements
Figure 11.5 shows the measured vector field for fluid velocities in the 2.5 mm
diameter outlet channel (Location A in Figure 10.2). Data were collected 20mm
down stream from the intersection of the outlet tube and the manifold. The
calculated hydrodynamic entry length is less than 10mm, which implies the flow
was fully developed and entrance effects are negligible. The axial velocity
averaged along streamlines decreases parabolically from a maximum of 0.06 m/sec near the center to a minimum of 0.02 m/sec near the wall (Figure 11.6).
The parabolic shape of the velocity profile and the uniformity of velocity vectors
207
Figure 11.1. Velocity distribution mapped using micro-PIV (FOV 2.5 by 2.5 mm) in a cylindrical tube (2.5 mm diameter) at the outlet of the flow cell at location A.
0.06 PIV data 0.05 Analytical
0.04
0.03
0.02
(m/s) Velocity 0.01
0.00 -1.5 -1 -0.5 0 0.5 1 1.5 Position (mm)
Figure 11.2. Comparison between experimental PIV data and analytical solution for flow at the cylindrical outlet tube of the flow cell at location A.
208 along streamlines (Figure 11.5) are consistent with fully developed laminar channel flow. Regions of anomalously low velocity near the channel walls are due to wall boundary effects and spurious vectors caused by surface flare of light reflected from particles deposited on the channel walls. In addition, the particle concentration within near-wall stagnation zones is insufficient to capture the details of the flow along the channel wall. The measured velocity profile corresponds well with the velocity profile calculated for Poiseuille flow (Figure
11.6). Discrepancy between the PIV measurements and the analytically predicted velocities is less than 2% throughout the field except for the near the channel walls, where the error reaches as high as 8%.
Figure 11.7 shows the velocity vector field measured at pore B (Figure 10.1), which intersects four 1mm diameter pore throats. Flow enters the pore through the top left and bottom left throats and exits through the top right and bottom right throats. The greatest velocities, however, are from top left to bottom right.
Velocities within individual pore throats decrease from center outward, consistent with channel flow. The maximum velocities near the center axis of the top left pore throat are 0.06 m/s. Velocities decrease by half and streamlines diverge where the flow crosses the top left entrance to the pore in response to the abrupt increase in cross-section diameter. Flow accelerates across the bottom right exit in response to the abrupt decrease in cross section.
209
Figure 11.3. Velocity distribution mapped using micro-PIV and a 5 by 5 mm FOV at location B where the pore body is connected to four 1 mm wide pore throats.
Figure 11.8 shows the velocity field for pore C, which intersects four 0.6mm
diameter pore throats. Flow enters the pore body from the bottom left pore throat
and exits through all three remaining throats, but maximum fluid flux is diagonally
across the pore body from the bottom left to the top right throat. As for the throats
connected to pore B, measured velocities decrease outward from the center
210 toward the walls of the pore throats. The maximum flow velocity in the entrance
throat is 0.05 m/s. The flow decelerates and streamlines expand across the entrance to the pore body. The maximum fluid velocities in the top right exit throat are 0.04 m/s.
Figure 11.4. Velocity distribution mapped using micro-PIV and 5 by 5 mm FOV at location C where the pore body is connected to four 0.6 mm wide pore throats.
211 Figure 11.9 shows the velocity distribution for pore C with the FOV changed
from 5 mm by 5 mm to 2.5 mm by 2.5 mm. The data were collected after a
temporary disruption of flow necessary to adjust the flow cell position in order to
accommodate the change in FOV. The inflow and outflow throats of the 2.5 mm
by 2.5 mm FOV (Figure 11.9) are consistent with those of the 5 mm by 5 mm
FOV (Figure 11.8). That is, in both cases, flow enters by the bottom left throat
and exits by the other three throats. However, there are differences in the pattern
and magnitude of the velocity vectors within the pore and throats. The maximum
entrance velocity for the 2.5mm by 2.5 mm FOV is 0.022m/s, less than half that of the 5 mm by 5 mm FOV. In addition, the fastest outflow in the 5 mm by 5 mm
FOV (Figure 11.8) is through the top right throat whereas in the 2.5 mm by 2.5 mm FOV (Figure 11.9) flow is nearly stagnant through the top right throat and the highest outflow velocities are through the top left throat. During aquistition of velocity data for the 2.5 mm by 2.5 mm FOV accumulations of aggregated particles were observed around both the top right and top left exit throats.
Particle aggregation and deposition during the temporary flow stoppage, leading to partial or complete clogging of pore throats is considered primarily responsible for the change in measure flow velocities between the two different
FOV’s. Some of the clogging is within the pore of interest, but additional clogging, upstream and downstream from the pore may have altered the preferential flow paths.
212
Figure 11.5. Velocity distribution mapped at location C using PIV and a 2.5 by 2.5 mm FOV. This figure shows the same pore as Figure 8, but a smaller field of view.
Figure 11.10 shows the velocity distribution at pore D, which intersects four
0.2 mm diameter pore throats. Flow enters through the bottom left and top left pore throats and exits through the bottom right and top right throats. Flow velocities in the throats are as high as about 0.03 m/s, comparable to flow entering and exiting through pores B and C through 0.6 and 1.0 mm throats, but
213 the vectors in the 0.2 mm throats are sparse and poorly aligned. Velocity vectors are less than 0.01 m/s within the pore and show little evidence for alignment along streamlines. An air bubble at the center of the pore was present and stationary throughout the period of data acquisition, disrupting flow.
Figure 11.6. Velocity distribution mapped using micro-PIV and 5 by 5 mm FOV at location D where the pore body is connected to four 0.2 mm wide pore throats.
214 11.2 Darcy’s Velocity
Simplified models, such as Darcy's law, provide a reasonable description of
the flow in the porous media using empirical coefficients. The average velocity
obtained using PIV at the outlet of the test cell was compared with Darcy's law for
a Newtonian fluid phase. Assuming that the porous medium is homogeneous and
isotropic, the fluid's density is constant, and the solid matrix is rigid and
stationary, the following equation can be used to calculate Darcy’s velocity:
22 −3 ϕ DP⎛⎞∂ V5.510=− × 2 ⎜⎟ (11.1) µ−ϕ()1 ⎝⎠∂x
where V, P, ρ and, µ denote the (intrinsic phase) average velocity, the pressure,
the density, and the viscosity of the fluid, respectively, ϕ is the effective porosity,
and D is the diameter of perfectly spherical granules. The Darcy’s equation is
modified using the Kozeny-Carman (Kozeny, 1927; Carman, 1937) empirical relationship that calculates permeability based on porosity and grain size as shown in Eq. 11.1.
The Reynold number (Re) for each throat width of 1mm, 0.6mm, and 0.2 mm were calculated to identify the applicability of Darcy’s law for the flow velocity
obtained using PIV within the pore and throat. The Re is given by:
ρVd Re = µ (11.2)
215 where d is the characteristic length. The average velocities for the 1 mm, 0.6
mm, and 0.2 mm throat width, respectively, are 0.025 m/s (Re = 8), 0.22 m/s (Re
= 4), and 0.01 m/s (Re = 1). Since the Re < 1 to 10, Darcy's law is valid for this
case.
The discharge rate of the fluid that was collected at the end of the tube was
1.1×10-7 m3/s. Based on the cross sectional area of the test cell (6.1 cm × 0.1
cm), the discharge rate corresponds to a Darcy velocity of about 0.0018 m/s.
This velocity was compared with velocity calculated using Eq. 11.1. The
calculated Darcy velocity is 0.0015 m/s, which differs by 15% from the average
velocity obtained by measuring fluid flux at the outlet of the test cell .
11.3 Data Validation and Error Analysis
Four steps were taken to ensure the validity of the PIV data. The first step is a
fuzzy logic processor applied to the correlation planes for each subregion and
each image pair to identify the correct displacement peak for the subregion.
Normally in PIV, spurious velocity vectors are identified by comparison with
neighboring vectors and are replaced by weighted averages of the surrounding
vectors (Wernet 1999a). The PIVPROC software uses a fuzzy logic processor to
compare each of the five displacement peaks in a subregion with the velocity
vectors for the neighboring subregions in order to determine which displacement
peak is correct (Wernet 2000). The velocity vector for the subregion is then recomputed based on the correct displacement peak rather than the centroid of the peaks, thus minimizing the number of spurious velocity vectors while
216 maintaining the number of independent measurements (Wernet 1999a and
2000).
The second step in the data validation was to apply Chauvenet’s criterion to
the repeat velocity measurements to identify spurious velocity vectors that result
from small temporal variations in the flow (Appendix A). Chauvenet’s criterion
determines the acceptable range around the mean beyond which points can
safely be rejected. Assuming a Gaussian distribution, mean and standard
deviation velocities of the repeat velocity measurements were calculated for each
vector location and each measurement. For each data point, the distance from
the mean was determined. All points falling outside two standard deviations of
the mean were removed and the mean was recalculated. The mean value of
uncertainty in the resultant velocity was determined to be 8% for 220 repeat image pairs (Figure 11.11).
30 Error (%) = 262.68N-0.7915 R2 = 0.9996 25 20
15
(%) Error 10 5
0 0 20 40 60 80 100 120 140 160 180 200 220
Number of Images (N)
Figure 11.7 Uncertainty associate with the velocity measurements.
217 The third step in data validation and error estimation was comparison of PIV- based measurements of average velocity at the outlet with average velocity based on measured flow rate from the outlet. PIV-obtained velocities were integrated over the outlet cylinder’s cross-sectional area to produce a total flow rate. The total PIV-based estimate of flow rate at the outlet was divided by the total cross-sectional area of the outlet to yield an average outlet velocity of
2.6×10-2 m/s. Similarly, the measured flow rate at the outlet was divided by the cross-sectional area of the outlet cylinder to yield an average outlet velocity of
2.5×10-2 m/s. The difference between these two estimates of average outlet velocity is 3.8%.
Lastly, mass conservation was evaluated for pore B by comparing the inflow with the outflow measured in the adjoining pore throats. The error incurred using the mass conservation approach across the two inlet and two outlet throats is
3.9%.
11.4 Numerical Simulation
Experimentally determined velocity vectors for pore bodies B, C, and D were compared with numerical simulations of flow through the flow cell using
FLUENTTM (version 6.02). FLUENTTM is a standard computational fluid dynamics code that allows computation of velocity distributions in complex geometries. The pore body and the throats were modeled using an unstructured grid of about
400,000 cells (Mazaheri et al., 2004). No-slip boundary conditions were imposed
218 and the Navier-Stokes and continuity equations were solved using the finite
difference and finite volume method under steady conditions.
Figure 11.12 shows a comparison between the PIV data and the numerical simulation results. Both the pattern of the velocity vectors and the maximum velocity coincide well between experimental data and numerical simulation for all three pores. The modeled flow direction perfectly matches with the experimental data for pore B (Figure 1112a). The average fluid velocity of 0.040 m/s, the distribution and overall velocity pattern of the numerical modeling perfectly matches the experimental results. The velocity distribution pattern of the numerical simulation in location C also coincides with the experimental data
(Figure 11.12b). In both the experimental and modeling cases fluid is flowing in through one inlet only. Both the experimental and the numerical one show an average inflow velocity of about 0.035 m/s in the lower left inlet throat. This figure shows a negligibly small measured velocity in the lower right throat of pore body, which is in agreement with the numerical simulation result. The outflow velocity of about 0.010m/s measured in the experimentation in the upper left outlet throat is also consistent with the numerical simulation results.
Figure 11.12c depicts the velocity distribution at location D, which is at the mid-section of the pore body with four 0.2 mm equal size throat connection. The velocity inside the pore body D is very small and a bubble was present during the experimentation that distorted the velocity field inside the pore body.
Nevertheless, the general features of the measured flow pattern are comparable with the computed one. In particular, as in the computer simulation results, the
219 measured velocity field shows that the flow enters the pore body from the two left throats and leaves through the right two connecting throats.
0.040 m/s
A
0.035 m/s
B 0.020 m/s
C Model Experiment
Figure 11.8. Computed and measured flow conditions at the mid-section of a pore bodies B, C, and D.
220 The good agreement between experimental work and computer simulation suggest that more result can be obtained through computer modeling by varying the variable parameters. Since the target of this work was to simulate single- phase flow, the complete two-phase flow and sensitivity analysis using FLUENT is left for future work. The implication and concluding remark of this work is discussed in chapter 12.
221
Chapter 12
DISCUSSIONS AND CONCLUSIONS
12.1 Discussion
The strength of PIV is that it produces planar velocity vector maps that
quantify the variation in magnitude and direction of flow velocity. The overall
good result of the mass conservation of the experimental data demonstrates the
ability of the experimental approach to accurately characterize the flow path for
single- phase flow within the complex interconnected pores and pore throats of a transparent, diamond-lattice flow cell. In the future, measured velocity data will be averaged for each pore throat to constrain differential pressures for direct comparison with, and validation of, network-model results for single-phase and potentially multiphase flow. However, because rules-based network models do not simulate velocity distributions within individual throats or pores, there is perhaps greater potential for PIV as a means to investigate the physical processes that govern flow within a pore and to validate physics-based models
222 that solve the flow equations within pore bodies. For example, even within the
relatively simple cylindrical pore bodies of the flow cell, PIV documented complex flow patterns such as deceleration at the pore mouth and the development of interlocked regions with no or extremely low velocities, which impact total fluid flux and may have great significance for solute transport or fluid-rock reactions.
There is great potential for PIV to further investigations of single-phase flow through more complex and realistic three-dimensional pore geometries. The PIV technique also may be adapted to study various aspects of multiphase flow, such as the impact of corners, wettability, and interfacial angle on pore-scale transport and phase displacement. However, a number of difficulties must be overcome for successful extension of PIV to studies of smaller, more complex pore geometries, three-dimensional pore geometries, or multi-phase flow.
One difficulty encountered during this study, the first to use PIV to measure flow through complex, interconnected channels, was the tendency of the particles to aggregate and settle, especially near the exits and entrances to pores and along pore walls. However, the generally good agreement with model results demonstrates that difficulties with pore clogging were successfully minimized by stirring the particle-seeded fluid and by adding surfactant such that clogging and modification of flow paths were problematic only after a temporary disruption of flow.
Other problems encountered during this study include increased background noise from the tracer particles, wall, and the sodium iodide solution. Background noise was minimized by using a high dynamic range CCD camera, applying
223 robust image processing algorithms during interrogation, and assuring well-
distributed, medium density tracer particles within the FOV, making it possible to
achieve consistent and reliable data for most conditions studied. Still, despite
efforts to identify and remove or correct false data points, spurious velocity
vectors appear near the walls of many pore throats and dominate the vector
maps of the narrowest pore throats (0.2 mm; compare Figures 11.10 and
11.12c). These velocity vectors deviate from the flow direction of neighbor
vectors. Spurious vectors near walls are due in part to a spatial resolution that is
inadequate to capture large changes in velocity approaching no-slip wall
surfaces. The 0.2 mm pore throats are of the same width scale as the spatial
resolution of the velocity vectors such that each vector includes the parabolic distribution of velocity across the throat. Difficulties with spatial resolution are
magnified by extraneous light scattering from the walls and, in the case of the 0.2 mm pore, the floor of the throat, and from particles deposited along them.
Thus, the main technical problems that need to be addressed to extend micro-PIV to smaller and more complex configurations in two-dimensions are selection of the appropriate tracers, improvement of imaging for smaller FOVs, and generation of a sufficiently thin light sheet. In selecting tracers, a balance must be struck between following the flow faithfully, not clogging the small channels, and scattering sufficient light to be recorded. A possible solution is to use sub-micron particles or dye that fluoresces when excited by photons. Such tracers would eliminate problems with particle aggregation, settling, and throat clogging, and would enhance the spatial resolution, thus reducing noise and
224 making it possible to view small-scale flow structures, including near-wall variations in flow velocity.
Extension of the micro-PIV studies of porous flow from two-dimensions to
three-dimensions is achievable because refractive index matching and an
optically transparent flow cell enable the light sheet to be focused on an image
plane at any depth within the flow cell. Stereo-camera systems can be used to
obtain three-component velocity vectors in the image plane. Two imaging
systems are required and algorithms need to be developed to improve the quality
of captured images. Improved image quality is necessary to capture complexities of the flow structure, including small-scale vortices, large spatial and temporal velocity gradients, as well as localized stagnation along wall and pore, which are known to be present in three-dimensional porous flow.
Micro-PIV measurements of single-phase flow can be extended to multiphase liquid-liquid flows by seeding one of the liquids and characterizing the flow of each liquid independently. Alternatively, the combination of PIV with laser- induced fluorescence (LIF) has enabled the simultaneous study of both phases of water-bubble flows (Lindken and Merzkirch, 2001). The water phase of the flow field was seeded with fluorescent particles while digital masking and shadowgraphy tracked the bubbles. This technique has the advantage that the signals of the two phases do not disturb each other because two cameras capture two different wavelengths through their filters. The main technical problem with extending this technique to micro-PIV studies of two-phase porous
225 flow would be to get the appropriate bubble size in the gas phase and achieve
the desired spatial resolution.
12.2 Conclusions
Micro-PIV was successfully developed to measure vector fields for single
phase, creeping flow through the sub-millimeter throats and pores of a transparent flow cell. The fluid, consisting of NaI solution with the same refractive index as the flow cell, was uniformly seeded with fine particles that follow the flow and the particles were subsequently illuminated with pulses of light from a laser sheet. Two-micron diameter silicon carbide particles were chosen for the seed particles due to their high refractive index. Images of the particles were captured and digitized with a large-format high-dynamic-range CCD camera. The resulting image fields were statistically analyzed to determine the most probable particle velocity vectors. Particles motion or displacement was tracked using cross- correlation methods.
Results were validated through error analysis, by comparing micro-PIV measurements of velocity distribution with physically measured flow rate at the outlet, and by mass conservation. Error analysis yielded a maximum measurement error of 8%. The difference between the average velocity at the outlet determined by flow rate and by micro-PIV was 3.8%. In addition, comparison of mass conservation across the inlet and outlet throat channels of a pore based on micro-PIV measurements yielded a measurement error of 3.9%.
226 Flow velocities at individual pore-throat network are different because of the
geometric configuration and throat size inhomogeneities. Result shows that the
highest velocity distribution was found at the center of the throat channel and
flow in the tube exhibits a laminar flow. The parabolic shape of the velocity profile and the uniformity of velocity vectors along streamlines are consistent with fully
developed laminar channel flow, and correspond well with the velocity profile
calculated for Poiseuille flow. The results of numerical simulations using the finite
difference and finite volume based program; Fluent, was validated by the micro-
PIV measurements of velocity vectors in individual pore and adjoining pore
throats in both direction and magnitude. Future work may make it possible to
adapt micro-PIV to study smaller and more complex pore-throat geometries in
two-dimensional and three-dimensional and to multiphase porous flow.
227
Appendix A: UNCERTAINTY ANALYSIS
1) Chauvenet’s Criterion
By assuming a Gaussian parent population distribution, Chauvenet’s
criterion was used to remove the spurious velocity data. Removals of the bad
vectors were accomplished using the computer coded in MATLAB. The general
description of the criteria is given below.
Suppose we have n measurements x1…xn of the same quantity x. Consider every sample to be suspicious sample xsus, then calculate tsus for any xsus using the following equation.
x − x t = sus sus , where σx is the standard deviation of x. σ x
Next step will be to find the probability that a valid measurement will differ from the mean (x’) by tsus standard deviations, P (outside tsusσx). It is given by:
t 1 2 P(inside tσ ) = e − z / 2 dz x ∫ 2π −t
P(outside tsusσx) = 1- P(inside tσx)
Then P(outside tsusσx) will be multiplied by N to obtain the number of
measurements expected to be at least as bad as xsus, Pbad.
Pbad (worse than xsus) = N* P(outside tsusσx)
228 If the number of measurements expected to be at least as bad as the suspect
measurement, Pbad < 0.5, then xsus fails Chauvenet’s criterion and is rejected. But
if it is > 0.5, then xsus will be kept. The mean and standard deviation are
recomputed after the elimination of the bad data.
2) Uncertainty Analysis The total uncertainty in velocity measurements using micro-PIV consists of
(a) bias error from determining the exact centroid of the particle on the CCD
array and (b) Random error.
(a) Bias Uncertainty
Bias uncertainty results from the timing uncertainty (time between the two
images resulting from firing of laser beams) and the displacement uncertainty
(location of the centeroids of particles):
2 2 σσud⎛⎞σ X ⎛⎞ =+⎜⎟⎜⎟ UT⎝⎠⎝⎠ D
Here, U is the velocity vector, T the time between two images, D the distance
between the centeroids, and σX and σd represent uncertainties in the time and
displacement measurements. For Nd:YAG laser timing error is negligible. The
displacement measurement accuracy is a measure of how well the cross-
correlation peak position can be determined on the CCD array. This number for
our system is approximately 0.1 to 0.2 pixels. The total displacement of the
particle is N/64 or 16 for a sub region size of 64. N is the number of pixels in the
sub-region. Thus neglecting timing uncertainty the total full-scale displacement uncertainty is the total bias uncertainty for the system can be calculated as 0.6%.
229 (b) Random Uncertainty
Random Uncertainty is reduced by increasing the number of image pairs during the experiment. Each image pair is divided into sub-regions (64 x 64) for our experiment. Each sub region is cross-correlated to determine a total displacement (every vector contains bias). Every image pair that is processed has 900 vectors that make up its velocity field—one vector for every sub-region of the image pairs. There are now as many vectors for a particular sub region as there are image pairs that were taken. Chauvenet’s criteria (Coleman and Steele,
1989) was applied to eliminate spurious velocity vectors. The more image pairs the closer the sample mean velocity will be to the actual population mean velocity if we were to take an infinite number of images. Once the bad data is removed, mean velocity, standard deviation and the new numbers of vectors after the removal of bad vectors are recalculated. Then relative error is calculated by assuming Gaussian distribution for N>30 and using 95% confidence interval.
The mean at each grid point:
Ns ∑ µijk,, µ = k =1 ij, Ns i,j are the vector grid indices and k is the index for the number of images in the sequence Ns . The relative error at each grid point:
N s 2 ∑ ()uuijk,, − ij, k =1 s ij, = N s − 1
230 s σ = z u c N s
The experimental error was calculated for increasing the number of readings i.e. image pair Ns. The results show that for increasing Ns the random error
decreased and for decreasing N the random uncertainty increased exponentially.
Note that Ns >30 is required for Gaussian considerations to hold. The variation in
the random uncertainty as a function of number of image pairs (data) is shown in
Figure 11.11. The uncertainty reduces from nearly 25% for 25 pairs to 4 % for
200 pairs. For some of data collected from different location of (Figure 10.1), the
uncertainty in velocity measurements fluctuate between 4 to 10%. A decision
was made to take 220 pairs resulting in a random uncertainty below 8%. The
bias uncertainty of 0.6% is negligible as compare to the random uncertainty and
the total uncertainty of the velocity vectors determined in this PIV experiment is
8%.
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